,transcriber's ,note  ,minor            
typographical errors and irregularities 
have been corrected4  ,footnotes have   
been numbered sequentially within       
chapters and moved to the end of each   
chapter4     ,,the ,,logic ,,of ,,chance
  ,,an ,,essay  ,,on ,,the ,,foundations
 ,,and ,,province ,,of ,,the ,,theory   
,,of ,,probability1  ,,with ,,especial  
,,reference ,,to ,,its ,,logical        
,,bearings ,,and ,,its ,,application    
,,to  ,,moral ,,and ,,social ,,science  
,,and ,,to ,,statistics1   ,,by  ,,john 
,,venn1 ,sc4,d41 ,f4,r4,s41  ,,fellow   
,,and ,,lecturer ,,in ,,the ,,moral     
,,sciences1 ,,gonville ,,and ,,caius    
,,college1 ,,cambridge4  ,,late         
,,examiner ,,in ,,logic ,,and ,,moral   
,,philosophy ,,in ,,the ,,university    
,,of ,,london4   ,so careful of the type
 she seems ,so careless of the single   
life4   ,,third ,,edition1              
,,re-,,written ,,and ,,enlarged4        
,london3 ,,macmillan ,,and ,,co4 ,,and  
,,new ,,york #ahhh  ,all ,rights        

reserved4     ,first ,edition printed   
#ahff4 ,second ,edition #ahgf4 ,third   
,edition #ahhh4     ,,preface ,,to      
,,first ,,edition  ,any work on         
,probability by a ,cambridge man will be
 so likely to have its scope and its    
general treatment of the subject        
prejudged1 that it may be well to state 
at the outset that the following ,essay 
is in no sense mathematical4 ,not only1 
to quote a common but often delusive    
assurance1 will 'no knowledge of        
mathematics beyond the simple rules of  
,arithmetic' be required to understand  
these pages1 but it is not intended that
 any such knowledge should be acquired  
by the process of reading them4 ,of the 
two or three occasions on which         
algebraical formul occur they will not  
be found to form any essential part of  
the text4  ,the science of ,probability 
occupies at present a somewhat anomalous
 position4 ,it is impossible1 ,i think1 
not to observe in it some of the marks  
and consequent disadvantages of a       

sectional study4 ,by a small body of    
ardent students it has been cultivated  
with great assiduity1 and the results   
they have obtained will always be       
reckoned among the most extraordinary   
products of mathematical genius4 ,but by
 the general body of thinking men its   
principles seem to be regarded with     
indifference or suspicion4 ,such persons
 may admire the ingenuity displayed1 and
 be struck with the profundity of many  
of the calculations1 but there seems to 
them1 if ,i may so express it1 an       
unreality about the whole treatment of  
the subject4 ,to many persons the       
mention of ,probability suggests little 
else than the notion of a set of rules1 
very ingenious and profound rules no    
doubt1 with which mathematicians amuse  
themselves by setting and solving       
puzzles4  ,it must be admitted that some
 ground has been given for such an      
opinion4 ,the examples commonly selected
 by writers on the subject1 though very 
well adapted to illustrate its rules1   

are for the most part of a special and  
peculiar character1 such as those       
relating to dice and cards4 ,when they  
have searched for illustrations drawn   
from the practical business of life1    
they have very generally1 but           
unfortunately1 hit upon just the sort of
 instances which1 as ,i shall endeavour 
to show hereafter1 are among the very   
worst that could be chosen for the      
purpose4 ,it is scarcely possible for   
any unprejudiced person to read what has
 been written about the credibility of  
witnesses by eminent writers1 without   
his experiencing an invincible distrust 
of the principles which they adopt4 ,to 
say that the rules of evidence sometimes
 given by such writers are broken in    
practice1 would scarcely be correct2 for
 the rules are of such a kind as        
generally to defy any attempt to appeal 
to them in practice4  ,this supposed    
want of harmony between ,probability and
 other branches of ,philosophy is       
perfectly erroneous4 ,it arises from the

 belief that ,probability is a branch of
 mathematics trying to intrude itself on
 to ground which does not altogether    
belong to it4 ,i shall endeavour to show
 that this belief is unfounded4 ,to     
answer correctly the sort of questions  
to which the science introduces us does 
generally demand some knowledge of      
mathematics1 often a great knowledge1   
but the discussion of the fundamental   
principles on which the rules are based 
does not necessarily require any such   
qualification4 ,questions might arise in
 other sciences1 in ,geology1 for       
example1 which could only be answered by
 the aid of arithmetical calculations4  
,in such a case any one would admit that
 the arithmetic was extraneous and      
accidental4 ,however many questions of  
this kind there might be here1 those    
persons who do not care to work out     
special results for themselves might    
still have an accurate knowledge of the 
principles of the science1 and even     
considerable acquaintance with the      

details of it4 ,the same holds true in  
,probability2 its connection with       
mathematics1 though certainly far closer
 than that of most other sciences1 is   
still of much the same kind4 ,it is     
principally when we wish to work out    
results for ourselves that mathematical 
knowledge is required2 without such     
knowledge the student may still have a  
firm grasp of the principles and even   
see his way to many of the derivative   
results4  ,the opinion that             
,probability1 instead of being a branch 
of the general science of evidence which
 happens to make much use of            
mathematics1 is a portion of            
mathematics1 erroneous as it is1 has yet
 been very disadvantageous to the       
science in several ways4 ,students of   
,philosophy in general have thence      
conceived a prejudice against           
,probability1 which has for the most    
part deterred them from examining it4   
,as soon as a subject comes to be       
considered 'mathematical' its claims    

seem generally1 by the mass of readers1 
to be either on the one hand scouted or 
at least courteously rejected1 or on the
 other to be blindly accepted with all  
their assumed consequences4 ,of         
impartial and liberal criticism it      
obtains little or nothing4  ,the        
consequences of this state of things    
have been1 ,i think1 disastrous to the  
students themselves of ,probability4 ,no
 science can safely be abandoned        
entirely to its own devotees4 ,its      
details of course can only be studied by
 those who make it their special        
occupation1 but its general principles  
are sure to be cramped if it is not     
exposed occasionally to the free        
criticism of those whose main culture   
has been of a more general character4   
,probability has been very much         
abandoned to mathematicians1 who as     
mathematicians have generally been      
unwilling to treat it thoroughly4 ,they 
have worked out its results1 it is true1
 with wonderful acuteness1 and the      

greatest ingenuity has been shown in    
solving various problems that arose1 and
 deducing subordinate rules4 ,and this  
was all that they could in fairness be  
expected to do4 ,any subject which has  
been discussed by such men as ,laplace  
and ,poisson1 and on which they have    
exhausted all their powers of analysis1 
could not fail to be profoundly treated1
 so far as it fell within their         
province4 ,but from this province the   
real principles of the science have     
generally been excluded1 or so meagrely 
discussed that they had better have been
 omitted altogether4 ,treating the      
subject as mathematicians such writers  
have naturally taken it up at the point 
where their mathematics would best come 
into play1 and that of course has not   
been at the foundations4 ,in the works  
of most writers upon the subject we     
should search in vain for anything like 
a critical discussion of the fundamental
 principles upon which its rules rest1  
the class of enquiries to which it is   

most properly applicable1 or the        
relation it bears to ,logic and the     
general rules of inductive evidence4    
,this want of precision as to ultimate  
principles is perfectly compatible here1
 as it is in the departments of ,morals 
and ,politics1 with a general agreement 
on processes and results4 ,but it is1 to
 say the least1 unphilosophical1 and    
denotes a state of things in which      
positive error is always liable to arise
 whenever the process of controversy    
forces us to appeal to the foundations  
of the science4  ,with regard to the    
remarks in the last few paragraphs1     
prominent exceptions must be made in the
 case of two recent works at least4#a   
,the first of these is ,professor de    
,morgan's ,formal ,logic4 ,he has there 
given an investigation into the         
foundations of ,probability as conceived
 by him1 and nothing can be more        
complete and precise than his statement 
of principles1 and his deductions from  
them4 ,if ,i could at all agree with    

these principles there would have been  
no necessity for the following essay1 as
 ,i could not hope to add anything to   
their foundation1 and should be far     
indeed from rivalling his lucid         
statement of them4 ,but in his scheme   
,probability is regarded very much from 
the ,conceptualist point of view2 as    
stated in the preface1 he considers that
 ,probability is concerned with formal  
inferences in which the premises are    
entertained with a conviction short of  
absolute certainty4 ,with this view ,i  
cannot agree4 ,as ,i have entered into  
criticism of some points of his scheme  
in one of the following chapters1 and   
shall have occasion frequently to refer 
to his work1 ,i need say no more about  
it here4 ,the other work to which ,i    
refer is the profound ,laws of ,thought 
of the late ,professor ,boole1 to which 
somewhat similar remarks may in part be 
applied4 ,owing however to his peculiar 
treatment of the subject1 ,i have       
scarcely anywhere come into contact with

 any of his expressed opinions4  ,the   
view of the province of ,probability    
adopted in this ,essay differs so       
radically from that of most other       
writers on the subject1 and especially  
from that of those just referred to1    
that ,i have thought it better1 as      
regards details1 to avoid all criticism 
of the opinions of others1 except where 
conflict was unavoidable4 ,with regard  
to that radical difference itself       
,bacon's remark applies1 behind which ,i
 must shelter myself from any change of 
presumption4--,quod ad universalem istam
 reprehensionem attinet1 certissimum    
vere est rem reputanti1 eam et magis    
probabilem esse et magis modestam1 quam 
si facta fuisset ex parte4  ,almost the 
only writer who seems to me to have     
expressed a just view of the nature and 
foundation of the rules of ,probability 
is ,mr ,mill1 in his ,system of         
,logic4#b ,his treatment of the subject 
is however very brief1 and a            
considerable portion of the space which 

he has devoted to it is occupied by the 
discussion of one or two special        
examples4 ,there are moreover some      
errors1 as it seems to me1 in what he   
has written1 which will be referred to  
in some of the following chapters4  ,the
 reference to the work just mentioned   
will serve to convey a general idea of  
the view of ,probability adopted in this
 ,essay4 ,with what may be called the   
,material view of ,logic as opposed to  
the ,formal or ,conceptualist1--with    
that which regards it as taking         
cognisance of laws of things and not of 
the laws of our own minds in thinking   
about things1--,i am in entire          
accordance4 ,of the province of ,logic1 
regarded from this point of view1 and   
under its widest aspect1 ,probability   
may1 in my opinion1 be considered to be 
a portion4 ,the principal objects of    
this ,essay are to ascertain how great a
 portion it comprises1 where we are to  
draw the boundary between it and the    
contiguous branches of the general      

science of evidence1 what are the       
ultimate foundations upon which its     
rules rest1 what the nature of the      
evidence they are capable of affording1 
and to what class of subjects they may  
most fitly be applied4 ,that the science
 of ,probability1 on this view of it1   
contains something more important than  
the results of a system of mathematical 
assumptions1 is obvious4 ,i am convinced
 moreover that it can and ought to be   
rendered both interesting and           
intelligible to ordinary readers who    
have any taste for philosophy4 ,in other
 words1 if the large and growing body of
 readers who can find pleasure in the   
study of books like ,mill's ,logic and  
,whewell's ,inductive ,sciences1 turn   
with aversion from a work on            
,probability1 the cause in the latter   
case must lie either in the view of the 
subject or in the manner and style of   
the book4  ,i take this opportunity of  
thanking several friends1 amongst whom  
,i must especially mention ,mr          

,todhunter1 of ,st ,john's ,college1 and
 ,mr ,h4 ,sidgwick1 of ,trinity         
,college1 for the trouble they have     
kindly taken in looking over the        
proof-sheets1 whilst this work was      
passing through the ,press4 ,to the     
former in particular my thanks are due  
for thus adding to the obligations which
 ,i1 as an old pupil1 already owed him1 
by taking an amount of trouble1 in      
making suggestions and corrections for  
the benefit of another1 which few would 
care to take for anything but a work of 
their own4 ,his extensive knowledge of  
the subject1 and his extremely accurate 
judgment1 render the service he has thus
 afforded me of the greatest possible   
value4  ,,gonville ,,and ,,caius        
,,college ,september #ahff4   #a4 ,i am 
here speaking1 of course1 of   those    
only who have expressly treated   of the
 foundations of the science4 ,mr        
,todhunter's admirable work on the      
,history of the ,theory of ,probability 
being1 as the name denotes1 mainly      

historical1 such enquiries have not     
directly fallen within his province4    
#b4 ,this remark1 and that at the       
commencement of the last paragraph1     
having been misunderstood1 ,i ought   to
 say that the only sense in which       
originality is claimed for this ,essay  
is in the thorough working out of the   
,material view of ,logic as applied to  
,probability4 ,i have given a pretty    
full discussion of the general          
principles   of this view in the tenth  
chapter1 and have there pointed out     
some of the peculiarities to which it   
leads4     ,,preface ,,to ,,second      
,,edition  ,the principal reason for    
designating this volume a second edition
 consists in the fact that the greater  
portion of what may be termed the first 
edition is incorporated into it4        
,besides various omissions 7principally 
where the former treatment has since    
seemed to me needlessly prolix71 ,i have
 added new matter1 not much inferior in 
amount to the whole of the original     

work4 ,in addition1 moreover1 to these  
alterations in the matter1 the general  
arrangement of the subject as regards   
the successive chapters has been        
completely changed2 the former          
arrangement having been 7as it now seems
 to me7 justly objected to as deficient 
and awkward in method4  ,after saying   
this1 it ought to be explained whether  
any change of general view or results   
will be found in the present treatment4 
,the general view of ,probability       
adopted is quite unchanged1 further     
reading and reflection having only      
confirmed me in the conviction that this
 is the soundest and most fruitful way  
of regarding the subject4 ,it is the    
more necessary to say this1 as to a     
cursory reader it might seem otherwise2 
owing to my having endeavoured to avoid 
the needlessly polemical tone which1 as 
is often the case with those who are    
making their first essay in writing upon
 any subject1 was doubtless too         
prominent in the former edition4 ,i have

 not thought it necessary1 of course1   
except in one or two cases1 to indicate 
points of detail which it has seemed    
necessary to correct4  ,a number of new 
discussions have been introduced upon   
topics which were but little or not at  
all treated before4 ,the principal of   
these refer to the nature and physical  
origin of ,laws of ,error 7,ch4 ,,ii472 
the general view of ,logic1 and         
consequently of ,probability1 termed the
 ,material view1 adopted here 7,ch4     
,x472 a brief history and criticism of  
the various opinions held on the subject
 of ,modality 7,ch4 ,,xii472 the logical
 principles underlying the method of    
,least ,squares 7,ch4 ,,xiii472 and the 
practices of ,insurance and ,gambling1  
so far as the principles involved in    
them are concerned 7,ch4 ,,xv474 ,the   
,chapter on the ,credibility of         
,extraordinary ,stories is also mainly  
new2 this was the portion of the former 
work which has since seemed to me the   
least satisfactory1 but owing to the    

extreme intricacy of the subject ,i am  
far from feeling thoroughly satisfied   
with it even now4  ,i have again to     
thank several friends for the assistance
 they have so kindly afforded4 ,amongst 
these ,i must prominently mention ,mr   
,c4 ,j4 ,monro1 late fellow of ,trinity4
 ,it is only the truth to say that ,i   
have derived more assistance from his   
suggestions and criticisms than has been
 consciously obtained from all other    
external sources together4 ,much of this
 criticism has been given privately in  
letters1 and notes on the proof-sheets2 
but one of the most elaborate of his    
discussions of the subject was          
communicated to the ,cambridge          
,philosophical ,society some years ago2 
as it was not published1 however1 ,i am 
unfortunately unable to refer the reader
 to it4 ,i ought to add that he is not  
in any way committed to any of my       
opinions upon the subject1 from some of 
which in fact he more or less dissents4 
,i am also much indebted to ,mr ,j4 ,w4 

,l4 ,glaisher1 also of ,trinity         
,college1 for many hints and references 
to various publications upon the subject
 of ,least ,squares1 and for careful    
criticism 7given in the midst of much   
other labour7 of the chapter in which   
that subject is treated4  ,i need not   
add that1 like every one else who has   
had to discuss the subject of           
,probability during the last ten years1 
,i have made constant use of ,mr        
,todhunter's ,history4  ,i may take this
 opportunity of adding that a           
considerable portion of the tenth       
chapter has recently appeared in the    
,january number of ,mind1 and that the  
substance of several chapters1          
especially in the more logical parts1   
has formed part of my ordinary lectures 
in ,cambridge2 the foundation and       
logical treatment of ,probability being 
now expressly included in the ,schedule 
of ,subjects for the ,moral ,sciences   
,tripos4  ,march #ahgf4     ,,preface   
,,to ,,third ,,edition  ,the present    

edition has been revised throughout1 and
 in fact rewritten4 ,three chapters are 
new1 viz4 the fifth 7,on the conception 
of ,randomness7 and the eighteenth and  
nineteenth 7,on the nature1 and on the  
employment1 of ,averages74 ,the eighth1 
tenth1 eleventh1 and fifteenth chapters 
have been recast1 and much new matter   
added1 and numerous alterations made in 
the remaining portions4#a ,on the other 
hand three chapters of the last edition 
have been nearly or entirely omitted4   
,these alterations do not imply any     
appreciable change of view on my part as
 to the foundations and province of     
,probability4 ,some of them are of      
course due to the necessary changes     
involved in the attempt to write up to  
date upon a subject which has not been  
stationary during the last eleven years4
 ,for instance the greatly increased    
interest now taken in what may be called
 the ,theory of ,statistics has rendered
 it desirable to go much more fully into
 the ,nature and treatment of ,laws of  

,error4 ,the omissions are mainly due to
 a wish to avoid increasing the bulk of 
this volume more than is actually       
necessary1 and to a feeling that the    
portions treating specially of          
,inductive ,logic 7which occupied some  
space in the last edition7 would be more
 suitable to a regular work on that     
subject4 ,i am at present engaged on    
such a work4  ,the publications which ,i
 have had occasion to notice have mostly
 appeared in various scientific         
journals4 ,the principal authors of     
these have been ,mr ,f4 ,galton and ,mr 
,f4 ,y4 ,edgeworth3 to the latter of    
whom ,i am also personally much obliged 
for many discussions1 oral and written1 
and for his kindness in looking through 
the proof-sheets4 ,his published        
articles are too numerous for separate  
mention here1 but ,i may say generally1 
in addition to the obligations specially
 noticed1 that ,i have been considerably
 indebted to them in writing the last   
two chapters4 ,two authors of works of a

 somewhat more substantial character1   
viz4 ,prof4 ,lexis and ,von ,kries1 only
 came under my notice unfortunately     
after this work was already in the      
printer's hands4 ,with the latter of    
these authors ,i find myself in closer  
agreement than with most others1 in     
respect of his general conception and   
treatment of ,probability4  ,december   
#ahhg4   #a4 ,i have indicated the new  
chapters and sections by printing them  
in   italics in the ,table of ,contents4
     ,,table ,,of ,,contents4#a  #a4    
,chapters and sections which are nearly 
or entirely new are printed in italics4 
,,part ,i4  ,,physical ,,foundations    
,,of ,,the ,,science ,,of ,,probability4
 ,chh4 ,i-,v4   ,,chapter ,i4  ,,the    
,,series ,,of ,,probability4   #a1 #b4  
,distinction between the proportional   
propositions of ,probability1        and
 the propositions of ,logic4    #c1 #d4 
,the former are best regarded as        
presenting a series of individuals1     
#e4 ,which may occur in any order of    

time1    #f1 #g4 ,and which present     
themselves in groups4    #h4 ,comparison
 of the above with the ordinary         
phraseology4    #i1 #aj4 ,these series  
ultimately fluctuate1    #aa4           
,especially in the case of moral and    
social phenomena1    #ab4 ,though in the
 case of games of chance the fluctuation
 is practically        inappreciable4   
#ac1 #ad4 ,in this latter case only can 
rigorous inferences be drawn4    #ae1   
#af4 ,the ,petersburg ,problem4         
,,chapter ,,ii4  ,,arrangement ,,and    
,,formation ,,of ,,the ,,series4 ,,laws 
,,of ,,error4   #a1 #b4 ,indication of  
the nature of a ,law of ,error or       
,divergence4    #c4 ,is there           
necessarily but one such law1    #d4    
,applicable to widely distinct classes  
of things8    #e1 #f4 ,this cannot be   
proved directly by statistics2    #g1   
#h4 ,which in certain cases show actual 
asymmetry4    #i1 #aj4 ,nor deductively2
    #aa4 ,nor by the ,method of ,least  
,squares4    #ab4 ,distinction between  

,laws of ,error and the ,method of      
,least        ,squares4    #ac4         
,supposed existence of types4    #ad-af4
 ,homogeneous and heterogeneous classes4
    #ag1 #ah4 ,the type in the case of  
human stature1 c4    #ai1 #bj4 ,the type
 in mental characteristics4    #ba1 #bb4
 ,applications of the foregoing         
principles and results4   ,,chapter     
,,iii4  ,,origin ,,or ,,process ,,of    
,,causation ,,of ,,the ,,series4    #a4 
,the causes consist of 7#a7 'objects1'  
#b1 #c4 ,which may or may not be        
distinguishable into natural kinds1     
#d-f4 ,and 7#b7 'agencies4'    #g4      
,requisites demanded in the above3      
#h1 #i4 ,consequences of their absence4 
#aj4 ,where are the required causes     
found8    #aa1 #ab4 ,not in the direct  
results of human will4    #ac-ae4       
,examination of apparent exceptions4    
#af-ah4 ,further analysis of some       
natural causes4   ,,chapter ,,iv4  ,,how
 ,,to ,,discover ,,and ,,prove ,,the    
,,series4    #a4 ,the data of           

,probability are established by         
experience2    #b4 ,though in practice  
most problems are solved deductively4   
#c-g4 ,mechanical instance to show the  
inadequacy of any  priori proof4    #h4 
,the ,principle of ,sufficient ,reason  
inapplicable4    #i4 ,evidence of actual
 experience4    #aj1 #aa4 ,further      
examination of the causes4    #ab1 #ac4 
,distinction between the succession of  
physical events and the        ,doctrine
 of ,combinations4    #ad1 #ae4 ,remarks
 of ,laplace on this subject4    #af4   
,bernoulli's ,theorem2    #ag1 #ah4 ,its
 inapplicability to social phenomena4   
#ai4 ,summation of preceding results4   
,,chapter ,v4  ,,the ,,conception ,,of  
,,randomness4    #a4 ,general           
,indication4    #b-e4 ,the postulate of 
ultimate uniform distribution at one    
stage or        another4    #f4 ,this   
area of distribution must be finite3    
#g1 #h4 ,geometrical illustrations in   
support3    #i4 ,can we conceive any    
exception here8    #aj1 #aa4            

,experimental determination of the      
random character when the        events 
are many3    #ab4 ,corresponding        
determination when they are few4    #ac1
 #ad4 ,illustration from the constant 4 
#ae1 #af4 ,conception of a line drawn at
 random4    #ag4 ,graphical             
illustration4   ,,part ,,ii4  ,,logical 
,,superstructure ,,on ,,the ,,above     
,,physical ,,foundations4 ,chh4         
,,vi-,,xiv4   ,,chapter ,,vi4           
,,measurement ,,of ,,belief4   #a1 #b4  
,preliminary remarks4    #c1 #d4 ,are we
 accurately conscious of gradations of  
belief8    #e4 ,probability only        
concerned with part of this enquiry4    
#f4 ,difficulty of measuring our belief2
    #g4 ,owing to intrusion of emotions1
    #h4 ,and complexity of the evidence4
    #i4 ,and when measured1 is it always
 correct8    #aj1 #aa4 ,distinction     
between logical and psychological views4
    #ab-af4 ,analogy of ,formal ,logic  
fails to show that we can thus detach   
and measure our belief4    #ag4         

,apparent evidence of popular language  
to the contrary4    #ah4 ,how is full   
belief justified in inductive enquiry8  
#ai-bc4 ,attempt to show how partial    
belief may be similarly justified4      
#bd-bh4 ,extension of this explanation  
to cases which cannot be repeated       
in experience4    #bi4 ,can other       
emotions besides belief be thus         
measured8    #cj4 ,errors thus arising  
in connection with the ,petersburg      
,problem4    #ca1 #cb4 ,the emotion of  
surprise is a partial exception4    #cc1
 #cd4 ,objective and subjective         
phraseology4    #ce4 ,the definition of 
probability1    #cf4 ,introduces the    
notion of a 'limit'1    #cg4 ,and       
implies1 vaguely1 some degree of belief4
   ,,chapter ,,vii4  ,,the ,,rules ,,of 
,,inference ,,in ,,probability4    #a4  
,nature of these inferences4    #b4     
,inferences by addition and subtraction4
    #c4 ,inferences by multiplication   
and division4    #d-f4 ,rule for        
independent events4    #g4 ,other rules 

sometimes introduced4    #h4 ,all the   
above rules may be interpreted          
subjectively1 i4e4 in terms        of   
belief4    #i-aa4 ,rules of so-called   
,inverse ,probability4    #ab1 #ac4     
,nature of the assumption involved in   
them3    #ad-af4 ,arbitrary character of
 this assumption4    #ag1 #ah4 ,physical
 illustrations4   ,,chapter ,,viii4     
,,the ,,rule ,,of ,,succession4    #a4  
,reasons for desiring some such rule3   
#b4 ,though it could scarcely belong to 
,probability4    #c4 ,distinction       
between ,probability and ,induction4    
#d1 #e4 ,impossibility of reducing the  
various rules of the latter under       
one head4    #f4 ,statement of the ,rule
 of ,succession2    #g4 ,proof offered  
for it4    #h4 ,is it a strict rule of  
inference8    #i4 ,or is it a           
psychological principle8   ,,chapter    
,,ix4  ,,induction4   #a-e4 ,statement  
of the ,inductive problem1 and origin of
 the ,inductive inference4    #f4       
,relation of ,probability to ,induction4

    #g-i4 ,the two are sometimes merged 
into one4    #aj4 ,extent to which      
causation is needed in ,probability4    
#aa-ac4 ,difficulty of referring an     
individual to a class3    #ad4 ,this    
difficulty but slight in ,logic1    #ae1
 #af4 ,but leads to perplexity in       
,probability3    #ag-ba4 ,mild form of  
this perplexity2    #bb1 #bc4 ,serious  
form4    #bd-bg4 ,illustration from     
,life ,insurance4    #bh1 #bi4 ,meaning 
of 'the value of a life'4    #cj1 #ca4  
,successive specialization of the       
classes to which objects are            
referred4    #cb4 ,summary of results4  
,,chapter ,x4  ,,change1 ,,causation    
,,and ,,design4    #a4 ,old ,theological
 objection to ,chance4    #b-d4         
,scientific version of the same4    #e4 
,statistics in reference to ,free-will4 
#f-h4 ,inconclusiveness of the common   
arguments here4    #i1 #aj4 ,chance as  
opposed to ,physical ,causation4    #aa4
 ,chance as opposed to ,design in the   
case of numerical constants4    #ab-ad4 

,theoretic solution between ,chance and 
,design4    #ae4 ,illustration from the 
dimensions of the ,pyramid4    #af1 #ag4
 ,discussion of certain difficulties    
here4    #ah1 #ai4 ,illustration from   
,psychical ,phenomena4    #bj4          
,arbuthnott's ,problem of the proportion
 of the sexes4    #ba-bc4 ,random or    
designed distribution of the stars4     
7,note on the proportion of the sexes47 
,,chapter ,,xi4  ,,material ,,and       
,,formal ,,logic4   #a1 #b4 ,broad      
distinction between these views2    #b1 
#c4 ,difficulty of adhering consistently
 to the objective view2    #d4          
,especially in the case of ,hypotheses4 
#e4 ,the doubtful stage of our facts is 
only occasional in ,inductive           
,logic4    #f-i4 ,but normal and        
permanent in ,probability4    #aj1 #aa4 
,consequent difficulty of avoiding      
,conceptualist phraseology4   ,,chapter 
,,xii4  ,,consequences ,,of ,,the       
,,distinctions ,,of ,,the ,,previous    
,,chapter4   #a1 #b4 ,probability has no

 relation to time4    #c1 #d4 ,butler   
and ,mill on ,probability before and    
after the event4    #e4 ,other attempts 
at explaining the difficulty4    #f-h4  
,what is really meant by the            
distinction4    #i4 ,origin of the      
common mistake4    #aj-ab4 ,examples in 
illustration of this view1    #ac4 ,is  
,probability relative8    #ad4 ,what is 
really meant by this expression4    #ae4
 ,objections to terming ,probability    
relative4    #af1 #ag4 ,in suitable     
examples the difficulty scarcely        
presents itself4   ,,chapter ,,xiii4    
,,on ,,modality4    #a4 ,various senses 
of ,modality2    #b4 ,having mostly some
 relation to ,probability4    #c4       
,modality must be recognized4    #d4    
,sometimes relegated to the predicate1  
#e1 #f4 ,sometimes incorrectly rejected 
altogether4    #g1 #h4 ,common practical
 recognition of it4    #i-aa4 ,modal    
propositions in ,logic and in           
,probability4    #ab4 ,aristotelian view
 of the ,modals2    #ac1 #ad4 ,founded  

on extinct philosophical views2    #ae4 
,but long and widely maintained4    #af4
 ,kant's general view4    #ag-ai4 ,the  
number of modal divisions admitted by   
various logicians4    #bj4 ,influence of
 the theory of ,probability4    #ba1    
#bb4 ,modal syllogisms4    #bc4 ,popular
 modal phraseology4    #bd-bf4 ,probable
 and ,dialectic syllogisms4    #bg1 #bh4
 ,modal difficulties occur in           
,jurisprudence4    #bi1 #cj4 ,proposed  
standards of legal certainty4    #ca4   
,rejected formally in ,english ,law1 but
 possibly recognized practically4       
#cb4 ,how1 if so1 it might be           
determined4   ,,chapter ,,xiv4          
,,fallacies4   #a-c4 7,i47 ,errors in   
judging of events after they have       
happened4    #d-g4 ,very various        
judgments may be thus involved4    #h1  
#i4 7,,ii47 ,confusion between random   
and picked selections4    #aj1 #aa4     
7,,iii47 ,undue limitation of the notion
 of ,probability4    #ab-af4 7,,iv47    
,double or ,quits3 the ,martingale4     

#ag1 #ah4 ,physical illustration4       
#ai1 #bj4 7,v47 ,inadequate realization 
of large numbers4    #ba-bd4 ,production
 of works of art by chance4    #be4     
,illustration from doctrine of heredity4
    #bf-cj4 7,,vi47 ,confusion between  
,probability and ,induction4    #ca-cc4 
7,,vii47 ,undue neglect of small        
chances4    #cd1 #ce4 7,,viii47 ,judging
 by the event in ,probability and in    
,induction4   ,,part ,,iii4  ,,various  
,,applications ,,of ,,the ,,theory ,,of 
,,probability4 ,chh4 ,,xv-,,xix4        
,,chapter ,,xv4  ,,insurance ,,and      
,,gambling4   #a1 #b4 ,the certainties  
and uncertainties of life4    #c-e4     
,insurance a means of diminishing the   
uncertainties4    #f1 #g4 ,gambling a   
means of increasing them4    #h1 #i4    
,various forms of gambling4    #aj1 #aa4
 ,comparison between these practices4   
#ab-ad4 ,proofs of the disadvantage of  
gambling3--        7#a7 on arithmetical 
grounds3    #ae1 #af4 ,illustration from
 family names4    #ag4 7#b7 from the    

'moral expectation'4    #ah1 #ai4       
,inconclusiveness of these proofs4      
#bj-bb4 ,broader questions raised by    
these attempts4   ,,chapter ,,xvi4      
,,application ,,of ,,probability ,,to   
,,testimony4   #a1 #b4 ,doubtful        
applicability of ,probability to        
testimony4    #c4 ,conditions of such   
applicability4    #d4 ,reasons for the  
above conditions4    #e1 #f4 ,are these 
conditions fulfilled in the case of     
testimony8    #g4 ,the appeal here is   
not directly to statistics4    #h1 #i4  
,illustrations of the above4    #aj1    
#aa4 ,is any application of ,probability
 to testimony valid8   ,,chapter ,,xvii4
  ,,credibility ,,of ,,extraordinary    
,,stories4    #a4 ,improbability before 
and after the event4    #b1 #c4 ,does   
the rejection of this lead to the       
conclusion that the       credibility of
 a story is independent of its nature8  
#d4 ,general and special credibility of 
a witness4    #e-h4 ,distinction between
 alternative and open questions1 and the

        usual rules for application of  
testimony to each of these4    #i4      
,discussion of an objection4    #aj1    
#aa4 ,testimony of worthless witnesses4 
#ab-ad4 ,common practical ways of       
regarding such problems4    #ae4        
,extraordinary stories not necessarily  
less probable4    #af-ah4 ,meaning of   
the term extraordinary1 and its         
distinction from        miraculous4     
#ai1 #bj4 ,combination of testimony4    
#ba1 #bb4 ,scientific meaning of a      
miracle4    #bc1 #bd4 ,two distinct     
prepossessions in regard to miracles1   
and the        logical consequences of  
these4    #be4 ,difficulty of discussing
 by our rules cases in which arbitrary  
interference can be postulated4    #bf1 
#bg4 ,consequent inappropriateness of   
many arguments4   ,,chapter ,,xviii4    
,,on ,,the ,,nature ,,and ,,use ,,of    
,,an ,,average1 ,,and ,,on ,,the        
,,different ,,kinds ,,of ,,average4     
#a4 ,preliminary rude notion of an      
average1    #b4 ,more precise           

quantitative notion1 yielding       7#a7
 the ,arithmetical ,average1    #c4 7#b7
 the ,geometrical4    #d4 ,in           
asymmetrical curves of error the        
arithmetic average must be              
distinguished from1    #e4 7#c7 the     
,maximum ,ordinate average1    #f4 7#d7 
and the ,median4    #g4 ,diagram in     
illustration4    #h-aj4 ,average        
departure from the average1 considered  
under the above        heads1 and under 
that of    #aa4 7#e7 ,the 7average of7  
,mean ,square of ,error1    #ab-ad4 ,the
 objects of taking averages4    #ae4 ,mr
 ,galton's practical method of          
determining the average4    #af1 #ag4   
,no distinction between the average and 
the mean4    #ah-bj4 ,distinction       
between what is necessary and what is   
experimental        here4    #ba1 #bb4  
,theoretical defects in the             
determination of the 'errors'4    #bc4  
,practical escape from these4  7,note   
about the units in the exponential      
equation and integral47   ,,chapter     

,,xix4  ,,the ,,theory ,,of ,,the       
,,average ,,as ,a ,,means ,,of          
,,approximation ,,to ,,the ,,truth4     
#a-d4 ,general indication of the        
problem3 i4e4 an inverse one            
requiring the previous consideration of 
a direct one4      ,i4 ,the direct      
problem3--given the central value and   
law of     dispersion of the single     
errors1 to determine those of the       
averages4  #f-bj4     #f4 7i7 ,the law  
of dispersion may be determinable       
priori1    #g4 7ii7 or experimentally1  
by statistics4    #h1 #i4 ,thence to    
determine the modulus of the error      
curve4    #aj-ad4 ,numerical example to 
illustrate the nature and amount of     
the contraction of the modulus of the   
average-error curve4    #ae4 ,this curve
 is of the same general kind as that of 
the single errors2    #af4 ,equally     
symmetrical1    #ag1 #ah4 ,and more     
heaped up towards the centre4    #ai1   
#bj4 ,algebraic generalization of the   
foregoing results4      ,,ii4 ,the      

inverse problem3--given but a few of the
 errors to     determine their centre   
and law1 and thence to draw the above   
deductions4  #ba-be4     #bb1 #bc4 ,the 
actual calculations are the same as     
before1    #bd4 ,with the extra demand  
that we must determine how probable are 
the        results4    #be4 ,summary4   
,,iii4 ,consideration of the same       
questions as applied to certain         
peculiar laws of error4  #bf-cg4        
#bf4 7i7 ,all errors equally probable4  
#bg1 #bh4 7ii7 ,certain peculiar laws of
 error4    #bi1 #cj4 ,further analysis  
of the reasons for taking averages4     
#ca-ce4 ,illustrative examples4    #cf1 
#cg4 ,curves with double centre and     
absence of symmetry4    #ch1 #ci4       
,conclusion4     ,,the ,,logic ,,of     
,,chance4     ,,chapter ,i4   ,,on      
,,certain ,,kinds ,,of ,,groups ,,or    
,,series ,,as ,,the ,,foundation ,,of   
,,probability4  #a4 ,it is sometimes not
 easy to give a clear definition of a   
science at the outset1 so as to set its 

scope and province before the reader in 
a few words4 ,in the case of those      
sciences which are more immediately and 
directly concerned with what are termed 
objects1 rather than with what are      
termed processes1 this difficulty is not
 indeed so serious4 ,if the reader is   
already familiar with the objects1 a    
simple reference to them will give him a
 tolerably accurate idea of the         
direction and nature of his studies4    
,even if he be not familiar with them1  
they will still be often to some extent 
connected and associated in his mind by 
a name1 and the mere utterance of the   
name may thus convey a fair amount of   
preliminary information4 ,this is more  
or less the case with many of the       
natural sciences2 we can often tell the 
reader beforehand exactly what he is    
going to study4 ,but when a science is  
concerned1 not so much with objects     
directly1 as with processes and laws1 or
 when it takes for the subject of its   
enquiry some comparatively obscure      

feature drawn from phenomena which have 
little or nothing else in common1 the   
difficulty of giving preliminary        
information becomes greater4 ,recognized
 classes of objects have then to be     
disregarded and even broken up1 and an  
entirely novel arrangement of the       
objects to be made4 ,in such cases it is
 the study of the science that first    
gives the science its unity1 for till it
 is studied the objects with which it is
 concerned were probably never thought  
of together4 ,here a definition cannot  
be given at the outset1 and the process 
of obtaining it may become by comparison
 somewhat laborious4  ,the science of   
,probability1 at least on the view taken
 of it in the following pages1 is of    
this latter description4 ,the reader who
 is at present unacquainted with the    
science cannot be at once informed of   
its scope by a reference to objects with
 which he is already familiar4 ,he will 
have to be taken in hand1 as it were1   
and some little time and trouble will   

have to be expended in directing his    
attention to our subject-matter before  
he can be expected to know it4 ,to do   
this will be our first task4  #b4 ,in   
studying ,nature1 in any form1 we are   
continually coming into possession of   
information which we sum up in general  
propositions4 ,now in very many cases   
these general propositions are neither  
more nor less certain and accurate than 
the details which they embrace and of   
which they are composed4 ,we are        
assuming at present that the truth of   
these generalizations is not disputed2  
as a matter of fact they may rest on    
weak evidence1 or they may be uncertain 
from their being widely extended by     
induction2 what is meant is1 that when  
we resolve them into their component    
parts we have precisely the same        
assurance of the truth of the details as
 we have of that of the whole4 ,when ,i 
know1 for instance1 that all cows       
ruminate1 ,i feel just as certain that  
any particular cow or cows ruminate as  

that the whole class does4 ,i may be    
right or wrong in my original statement1
 and ,i may have obtained it by any     
conceivable mode in which truths can be 
obtained2 but whatever the value of the 
general proposition may be1 that of the 
particulars is neither greater nor less4
 ,the process of inferring the          
particular from the general is not      
accompanied by the slightest diminution 
of certainty4 ,if one of these          
'immediate inferences' is justified at  
all1 it will be equally right in every  
case4  ,but it is by no means necessary 
that this characteristic should exist in
 all cases4 ,there is a class of        
immediate inferences1 almost            
unrecognized indeed in logic1 but       
constantly drawn in practice1 of which  
the characteristic is1 that as they     
increase in particularity they diminish 
in certainty4 ,let me assume that ,i am 
told that some cows ruminate2 ,i cannot 
infer logically from this that any      
particular cow does so1 though ,i should

 feel some way removed from absolute    
disbelief1 or even indifference to      
assent1 upon the subject2 but if ,i saw 
a herd of cows ,i should feel more sure 
that some of them were ruminant than ,i 
did of the single cow1 and my assurance 
would increase with the numbers of the  
herd about which ,i had to form an      
opinion4 ,here then we have a class of  
things as to the individuals of which we
 feel quite in uncertainty1 whilst as we
 embrace larger numbers in our          
assertions we attach greater weight to  
our inferences4 ,it is with such classes
 of things and such inferences that the 
science of ,probability is concerned4   
#c4 ,in the foregoing remarks1 which are
 intended to be purely preliminary1 we  
have not been able altogether to avoid  
some reference to a subjective element1 
viz4 the degree of our certainty or     
belief about the things which we are    
supposed to contemplate4 ,the reader may
 be aware that by some writers this     
element is regarded as the              

subject-matter of the science4 ,hence it
 will have to be discussed in a future  
chapter4 ,as however ,i do not agree    
with the opinion of the writers just    
mentioned1 at least as regards treating 
this element as one of primary          
importance1 no further allusion will be 
made to it here1 but we will pass on at 
once to a more minute investigation of  
that distinctive characteristic of      
certain classes of things which was     
introduced to notice in the last        
section4  ,in these classes of things1  
which are those with which ,probability 
is concerned1 the fundamental conception
 which the reader has to fix in his mind
 as clearly as possible1 is1 ,i take it1
 that of a series4 ,but it is a series  
of a peculiar kind1 one of which no     
better compendious description can be   
given than that which is contained in   
the statement that it combines          
individual irregularity with aggregate  
regularity4 ,this is a statement which  
will probably need some explanation4    

,let us recur to an example of the kind 
already alluded to1 selecting one which 
shall be in accordance with experience4 
,some children will not live to thirty4 
,now if this proposition is to be       
regarded as a purely indefinite or1 as  
it would be termed in logic1            
'particular' proposition1 no doubt the  
notion of a series does not obviously   
present itself in connection with it4   
,it contains a statement about a certain
 unknown proportion of the whole1 and   
that is all4 ,but it is not with these  
purely indefinite propositions that we  
shall be concerned4 ,let us suppose the 
statement1 on the contrary1 to be of a  
numerical character1 and to refer to a  
given proportion of the whole1 and we   
shall then find it difficult to exclude 
the notion of a series4 ,we shall find  
it1 ,i think1 impossible to do so as    
soon as we set before us the aim of     
obtaining accurate1 or even moderately  
correct inferences4 ,what1 for instance1
 is the meaning of the statement that   

two new-born children in three fail to  
attain the age of sixty-three8 ,it      
certainly does not declare that in any  
given batch of1 say1 thirty1 we shall   
find just twenty that fail3 whatever    
might be the strict meaning of the      
words1 this is not the import of the    
statement4 ,it rather contemplates our  
examination of a large number1 of a long
 succession of instances1 and states    
that in such a succession we shall find 
a numerical proportion1 not indeed fixed
 and accurate at first1 but which tends 
in the long run to become so4 ,in every 
kind of example with which we shall be  
concerned we shall find this reference  
to a large number or succession of      
objects1 or1 as we shall term it1 series
 of them4  ,a few additional examples   
may serve to make this plain4  ,let us  
suppose that we toss up a penny a great 
many times2 the results of the          
successive throws may be conceived to   
form a series4 ,the separate throws of  
this series seem to occur in utter      

disorder2 it is this disorder which     
causes our uncertainty about them4      
,sometimes head comes1 sometimes tail   
comes2 sometimes there is a repetition  
of the same face1 sometimes not4 ,so    
long as we confine our observation to a 
few throws at a time1 the series seems  
to be simply chaotic4 ,but when we      
consider the result of a long succession
 we find a marked distinction2 a kind of
 order begins gradually to emerge1 and  
at last assumes a distinct and striking 
aspect4 ,we find in this case that the  
heads and tails occur in about equal    
numbers1 that similar repetitions of    
different faces do so also1 and so on4  
,in a word1 notwithstanding the         
individual disorder1 an aggregate order 
begins to prevail4 ,so again if we are  
examining the length of human life1 the 
different lives which fall under our    
notice compose a series presenting the  
same features4 ,the length of a single  
life is familiarly uncertain1 but the   
average duration of a batch of lives is 

becoming in an almost equal degree      
familiarly certain4 ,the larger the     
number we take out of any mixed crowd1  
the clearer become the symptoms of      
order1 the more nearly will the average 
length of each selected class be the    
same4 ,these few cases will serve as    
simple examples of a property of things 
which can be traced almost everywhere1  
to a greater or less extent1 throughout 
the whole field of our experience4      
,fires1 shipwrecks1 yields of harvest1  
births1 marriages1 suicides2 it scarcely
 seems to matter what feature we single 
out for observation4#a ,the irregularity
 of the single instances diminishes when
 we take a large number1 and at last    
seems for all practical purposes to     
disappear4  ,in speaking of the effect  
of the average in thus diminishing the  
irregularities which present themselves 
in the details1 the attention of the    
student must be prominently directed to 
the point1 that it is not the absolute  
but the relative irregularities which   

thus tend to diminish without limit4    
,this idea will be familiar enough to   
the mathematician1 but to others it may 
require some reflection in order to     
grasp it clearly4 ,the absolute         
divergences and irregularities1 so far  
from diminishing1 show a disposition to 
increase1 and this 7it may be7 without  
limit1 though their relative importance 
shows a corresponding disposition to    
diminish without limit4 ,thus in the    
case of tossing a penny1 if we take a   
few throws1 say ten1 it is decidedly    
unlikely that there should be a         
difference of six between the numbers of
 heads and tails2 that is1 that there   
should be as many as eight heads and    
therefore as few as two tails1 or vice  
vers4 ,but take a thousand throws1 and  
it becomes in turn exceedingly likely   
that there should be as much as1 or more
 than1 a difference of six between the  
respective numbers4 ,on the other hand  
the proportion of heads to tails in the 
case of the thousand throws will be very

 much nearer to unity1 in most cases1   
than when we only took ten4 ,in other   
words1 the longer a game of chance      
continues the larger are the spells and 
runs of luck in themselves1 but the less
 their relative proportions to the whole
 amounts involved4  #d4 ,in speaking as 
above of events or things as to the     
details of which we know little or      
nothing1 it is not of course implied    
that our ignorance about them is        
complete and universal1 or1 what comes  
to the same thing1 that irregularity may
 be observed in all their qualities4    
,all that is meant is that there are    
some qualities or marks in them1 the    
existence of which we are not able to   
predicate with certainty in the         
individuals4 ,with regard to all their  
other qualities there may be the utmost 
uniformity1 and consequently the most   
complete certainty4 ,the irregularity in
 the length of human life is notorious1 
but no one doubts the existence of such 
organs as a heart and brains in any     

person whom he happens to meet4 ,and    
even in the qualities in which the      
irregularity is observed1 there are     
often1 indeed generally1 positive limits
 within which it will be found to be    
confined4 ,no person1 for instance1 can 
calculate what may be the length of any 
particular life1 but we feel perfectly  
certain that it will not stretch out to 
#aej years4 ,the irregularity of the    
individual instances is only shown in   
certain respects1 as e4g4 the length of 
the life1 and even in these respects it 
has its limits4 ,the same remark will   
apply to most of the other examples with
 which we shall be concerned4 ,the      
disorder in fact is not universal and   
unlimited1 it only prevails in certain  
directions and up to certain points4    
#e4 ,in speaking as above of a series1  
it will hardly be necessary to point out
 that we do not imply that the objects  
themselves which compose the series must
 occur successively in time2 the series 
may be formed simply by their coming in 

succession under our notice1 which as a 
matter of fact they may do in any order 
whatever4 ,a register of mortality1 for 
instance1 may be made up of deaths which
 took place simultaneously or           
successively2 or1 we might if we pleased
 arrange the deaths in an order quite   
distinct from either of these4 ,this is 
entirely a matter of indifference2 in   
all these cases the series1 for any     
purposes which we need take into        
account1 may be regarded as being of    
precisely the same description4 ,the    
objects1 be it remembered1 are given to 
us in nature2 the order under which we  
view them is our own private            
arrangement4 ,this is mentioned here    
simply by way of caution1 the meaning of
 this assertion will become more plain  
in the sequel4  ,i am aware that the    
word 'series' in the application with   
which it is used here is liable to some 
misconstruction1 but ,i cannot find any 
better word1 or indeed any as suitable  
in all respects4 ,as remarked above1 the

 events need not necessarily have       
occurred in a regular sequence of time1 
though they often will have done so4 ,in
 many cases 7for instance1 the throws of
 a penny or a die7 they really do occur 
in succession2 in other cases 7for      
instance1 the heights of men1 or the    
duration of their lives71 whatever may  
have been the order of their actual     
occurrence1 they are commonly brought   
under our notice in succession by being 
arranged in statistical tables4 ,in all 
cases alike our processes of inference  
involve the necessity of examining one  
after another of the members which      
compose the group1 or at least of being 
prepared to do this1 if we are to be in 
a position to justify our inferences4   
,the force of these considerations will 
come out in the course of the           
investigation in ,chapter ,,vi4  ,the   
late ,leslie ,ellis#b has expressed what
 seems to me a substantially similar    
view in terms of genus and species1     
instead of speaking of a series4 ,he    

says1 ,when individual cases are        
considered1 we have no conviction that  
the ratios of frequency of occurrence   
depend on the circumstances common to   
all the trials4 ,on the contrary1 we    
recognize in the determining            
circumstances of their occurrence an    
extraneous element1 an element1 that is1
 extraneous to the idea of the genus and
 species4 ,contingency and limitation   
come in 7so to speak7 together2 and both
 alike disappear when we consider the   
genus in its entirety1 or 7which is the 
same thing7 in what may be called an    
ideal and practically impossible        
realization of all which it potentially 
contains4 ,if this be granted1 it seems 
to follow that the fundamental principle
 of the ,theory of ,probabilities may be
 regarded as included in the following  
statement1--,the conception of a genus  
implies that of numerical relations     
among the species subordinated to it4   
,as remarked above1 this appears a      
substantially similar doctrine to that  

explained in this chapter1 but ,i do not
 think that the terms genus and species 
are by any means so well fitted to bring
 out the conception of a tendency or    
limit as when we speak of a series1 and 
,i therefore much prefer the latter     
expression4  #f4 ,the reader will now   
have in his mind the conception of a    
series or group of things or events1    
about the individuals of which we know  
but little1 at least in certain         
respects1 whilst we find a continually  
increasing uniformity as we take larger 
numbers under our notice4 ,this is      
definite enough to point out tolerably  
clearly the kind of things with which we
 have to deal1 but it is not            
sufficiently definite for purposes of   
accurate thought4 ,we must therefore    
attempt a somewhat closer analysis4     
,there are certain phrases so commonly  
adopted as to have become part of the   
technical vocabulary of the subject1    
such as an 'event' and the 'way in which
 it can happen4' ,thus the act of       

throwing a penny would be called an     
event1 and the fact of its giving head  
or tail would be called the way in which
 the event happened4 ,if we were        
discussing tables of mortality1 the     
former term would denote the mere fact  
of death1 the latter the age at which it
 occurred1 or the way in which it was   
brought about1 or whatever else in it   
might be the particular circumstance    
under discussion4 ,this phraseology is  
very convenient1 and will often be made 
use of in this work1 but without        
explanation it may lead to confusion4   
,for in many cases the way in which the 
event happens is of such great relative 
importance1 that according as it happens
 in one way or another the event would  
have a different name2 in other words1  
it would not in the two cases be        
nominally the same event4 ,the phrase   
therefore will have to be considerably  
stretched before it will conveniently   
cover all the cases to which we may have
 to apply it4 ,if for instance we were  

contemplating a series of human beings1 
male and female1 it would sound odd to  
call their humanity an event1 and their 
sex the way in which the event happened4
  ,if we recur however to any of the    
classes of objects already referred to1 
we may see our path towards obtaining a 
more accurate conception of what we     
want4 ,it will easily be seen that in   
every one of them there is a mixture of 
similarity and dissimilarity2 there is a
 series of events which have a certain  
number of features or attributes in     
common1--without this they would not be 
classed together4 ,but there is also a  
distinction existing amongst them2 a    
certain number of other attributes are  
to be found in some and are not to be   
found in others4 ,in other words1 the   
individuals which form the series are   
compound1 each being made up of a       
collection of things or attributes2 some
 of these things exist in all the       
members of the series1 others are found 
in some only4 ,so far there is nothing  

peculiar to the science of ,probability2
 that in which the distinctive          
characteristic consists is this2--that  
the occasional attributes1 as           
distinguished from the permanent1 are   
found on an extended examination to tend
 to exist in a certain definite         
proportion of the whole number of cases4
 ,we cannot tell in any given instance  
whether they will be found or not1 but  
as we go on examining more cases we find
 a growing uniformity4 ,we find that the
 proportion of instances in which they  
are found to instances in which they are
 wanting1 is gradually subject to less  
and less comparative variation1 and     
approaches continually towards some     
apparently fixed value4  ,the above is  
the most comprehensive form of          
description2 as a matter of fact the    
groups will in many cases take a far    
simpler form2 they may appear1 e4g4     
simply as a succession of things of the 
same kind1 say human beings1 with or    
without an occasional attribute1 say    

that of being left-handed4 ,we are using
 the word attribute1 of course1 in its  
widest sense1 intending it to include   
every distinctive feature that can be   
observed in a thing1 from essential     
qualities down to the merest accidents  
of time and place4  #g4 ,on examining   
our series1 therefore1 we shall find    
that it may best be conceived1 not      
necessarily as a succession of events   
happening in different ways1 but as a   
succession of groups of things4 ,these  
groups1 on being analysed1 are found in 
every case to be resolvable into        
collections of substances and           
attributes4 ,that which gives its unity 
to the succession of groups is the fact 
of some of these substances or          
attributes being common to the whole    
succession2 that which gives their      
distinction to the groups in the        
succession is the fact of some of them  
containing only a portion of these      
substances and attributes1 the other    
portion or portions being occasionally  

absent4 ,so understood1 our phraseology 
may be made to embrace every class of   
things of which ,probability can take   
account4  #h4 ,it will be easily seen   
that the ordinary expression 7viz4 the  
'event1' and the 'way in which it       
happens'7 may be included in the above4 
,when the occasional attributes are     
unimportant the permanent ones are      
sufficient to fix and appropriate the   
name1 the presence or absence of the    
others being simply denoted by some     
modification of the name or the addition
 of some predicate4 ,we may therefore in
 all such cases speak of the collection 
of attributes as 'the event1'--the same 
event essentially1 that is--only saying 
that it 7so as to preserve its nominal  
identity7 happens in different ways in  
the different cases4 ,when the          
occasional attributes however are       
important1 or compose the majority1 this
 way of speaking becomes less           
appropriate2 language is somewhat       
strained by our implying that two       

extremely different assemblages are in  
reality the same event1 with a          
difference only in its mode of          
happening4 ,the phrase is however a very
 convenient one1 and with this caution  
against its being misunderstood1 it will
 frequently be made use of here4  #i4 ,a
 series of the above-mentioned kind is1 
,i apprehend1 the ultimate basis upon   
which all the rules of ,probability must
 be based4 ,it is essential to a clear  
comprehension of the subject to have    
carried our analysis up to this point1  
but any attempt at further analysis into
 the intimate nature of the events      
composing the series1 is not required4  
,it is altogether unnecessary1 for      
instance1 to form any opinion upon the  
questions discussed in metaphysics as to
 the independent existence of           
substances4 ,we have discovered1 on     
examination1 a series composed of groups
 of substances and attributes1 or of    
attributes alone4 ,at such a series we  
stop1 and thence investigate our rules  

of inference2 into what these substances
 or attributes would themselves be      
ultimately analysed1 if taken in hand by
 the psychologist or metaphysician1 it  
is no business of ours to enquire here4 
#aj4 ,the stage then which we have now  
reached is that of having discovered a  
quantity of things 7they prove on       
analysis to be groups of things7 which  
are capable of being classified         
together1 and are best regarded as      
constituting a series4 ,the distinctive 
peculiarity of this series is our       
finding in it an order1 gradually       
emerging out of disorder1 and showing in
 time a marked and unmistakeable        
uniformity4  ,the impression which may  
possibly be derived from the description
 of such a series1 and which the reader 
will probably already entertain if he   
have studied ,probability before1 is    
that the gradual evolution of this order
 is indefinite1 and its approach        
therefore to perfection unlimited4 ,and 
many of the examples commonly selected  

certainly tend to confirm such an       
impression4 ,but in reference to the    
theory of the subject it is1 ,i am      
convinced1 an error1 and one liable to  
lead to much confusion4  ,the lines     
which have been prefixed as a motto to  
this work1 ,so careful of the type she  
seems1 so careless of the single life1  
are soon after corrected by the         
assertion that the type itself1 if we   
regard it for a long time1 changes1 and 
then vanishes and is succeeded by       
others4 ,so in ,probability2 that       
uniformity which is found in the long   
run1 and which presents so great a      
contrast to the individual disorder1    
though durable is not everlasting4 ,keep
 on watching it long enough1 and it will
 be found almost invariably to          
fluctuate1 and in time may prove as     
utterly irreducible to rule1 and        
therefore as incapable of prediction1 as
 the individual cases themselves4 ,the  
full bearing of this fact upon the      
theory of the subject1 and upon certain 

common modes of calculation connected   
with it1 will appear more fully in some 
of the following chapters2 at present we
 will confine ourselves to very briefly 
establishing and illustrating it4  ,let 
us take1 for example1 the average       
duration of life4 ,this1 provided our   
data are sufficiently extensive1 is     
known to be tolerably regular and       
uniform4 ,this fact has been already    
indicated in the preceding sections1 and
 is a truth indeed of which the popular 
mind has a tolerably clear grasp at the 
present day4 ,but a very little         
consideration will show that there may  
be a superior as well as an inferior    
limit to the extent within which this   
uniformity can be observed2 in other    
words whilst we may fall into error by  
taking too few instances we may also    
fail in our aim1 though in a very       
different way and from quite different  
reasons1 by taking too many4 ,at the    
present time the average duration of    
life in ,england may be1 say1 forty     

years2 but a century ago it was         
decidedly less2 several centuries ago it
 was presumably very much less2 whilst  
if we possessed statistics referring to 
a still earlier population of the       
country we should probably find that    
there has been since that time a still  
more marked improvement4 ,what may be   
the future tendency no man can say for  
certain4 ,it may be1 and we hope that it
 will be the case1 that owing to        
sanitary and other improvements1 the    
duration of life will go on increasing  
steadily2 it is at least conceivable1   
though doubtless incredible1 that it    
should do so without limit4 ,on the     
other hand1 and with much more          
likelihood1 this duration might         
gradually tend towards some fixed       
length4 ,or1 again1 it is perfectly     
possible that future generations might  
prefer a short and a merry life1 and    
therefore reduce their average          
longevity4 ,the duration of life cannot 
but depend to some extent upon the      

general tastes1 habits and employments  
of the people1 that is upon the ideal   
which they consciously or unconsciously 
set before them1 and he would be a rash 
man who should undertake to predict what
 this ideal will be some centuries      
hence4 ,all that it is here necessary   
however to indicate is1 that this       
particular uniformity 7as we have       
hitherto called it1 in order to mark its
 relative character7 has varied1 and1   
under the influence of future eddies in 
opinion and practice1 may vary still2   
and this to any extent1 and with any    
degree of irregularity4 ,to borrow a    
term from ,astronomy1 we find our       
uniformity subject to what might be     
called an irregular secular variation4  
#aa4 ,the above is a fair typical       
instance4 ,if we had taken a less simple
 feature than the length of life1 or one
 less closely connected with what may be
 called by comparison the great         
permanent uniformities of nature1 we    
should have found the peculiarity under 

notice exhibited in a far more striking 
degree4 ,the deaths from small-pox1 for 
example1 or the instances of duelling or
 accusations of witchcraft1 if examined 
during a few successive decades1 might  
have shown a very tolerable degree of   
uniformity4 ,but these uniformities have
 risen possibly from zero2 after various
 and very great fluctuations seem       
tending towards zero again1 at least in 
this century2 and may1 for anything we  
know1 undergo still more rapid          
fluctuations in future4 ,now these      
examples must be regarded as being only 
extreme ones1 and not such very extreme 
ones1 of what is the almost universal   
rule in nature4 ,i shall endeavour to   
show that even the few apparent         
exceptions1 such as the proportions     
between male and female births1 c41 may 
not be1 and probably in reality are not1
 strictly speaking1 exceptions4 ,a type1
 that is1 which shall be in the fullest 
sense of the words1 persistent and      
invariable is scarcely to be found in   

nature4 ,the full import of this        
conclusion will be seen in future       
chapters4 ,attention is only directed   
here to the important inference that1   
although statistics are notoriously of  
no value unless they are in sufficient  
numbers1 yet it does not follow but that
 in certain cases we may have too many  
of them4 ,if they are made too          
extensive1 they may again fall short1 at
 least for any particular time or place1
 of their greatest attainable accuracy4 
#ab4 ,these natural uniformities then   
are found at length to be subject to    
fluctuation4 ,now contrast with them any
 of the uniformities afforded by games  
of chance2 these latter seem to show no 
trace of secular fluctuation1 however   
long we may continue our examination of 
them4 ,criticisms will be offered1 in   
the course of the following chapters1   
upon some of the common attempts to     
prove  priori that there must be this   
fixity in the uniformity in question1   
but of its existence there can scarcely 

be much doubt4 ,pence give heads and    
tails about equally often now1 as they  
did when they were first tossed1 and as 
we believe they will continue to do1 so 
long as the present order of things     
continues4 ,the fixity of these         
uniformities may not be as absolute as  
is commonly supposed1 but no amount of  
experience which we need take into      
account is likely in any appreciable    
degree to interfere with them4 ,hence   
the obvious contrast1 that1 whereas     
natural uniformities at length          
fluctuate1 those afforded by games of   
chance seem fixed for ever4  #ac4 ,here 
then are series apparently of two       
different kinds4 ,they are alike in     
their initial irregularity1 alike in    
their subsequent regularity2 it is in   
what we may term their ultimate form    
that they begin to diverge from each    
other4 ,the one tends without any       
irregular variation towards a fixed     
numerical proportion in its uniformity2 
in the other the uniformity is found at 

last to fluctuate1 and to fluctuate1 it 
may be1 in a manner utterly irreducible 
to rule4  ,as this chapter is intended  
to be little more than explanatory and  
illustrative of the foundations of the  
science1 the remark may be made here    
7for which subsequent justification will
 be offered7 that it is in the case of  
series of the former kind only that we  
are able to make anything which can be  
interpreted into strict scientific      
inferences4 ,we shall be able however in
 a general way to see the kind and      
extent of error that would be committed 
if1 in any example1 we were to          
substitute an imaginary series of the   
former kind for any actual series of the
 latter kind which experience may       
present to us4 ,the two series are of   
course to be as alike as possible in all
 respects1 except that the variable     
uniformity has been replaced by a fixed 
one4 ,the difference then between them  
would not appear in the initial stage1  
for in that stage the distinctive       

characteristics of the series of        
,probability are not apparent2 all is   
there irregularity1 and it would be as  
impossible to show that they were alike 
as that they were different2 we can only
 say generally that each shows the same 
kind of irregularity4 ,nor would it     
appear in the next subsequent stage1 for
 the real variability of the uniformity 
has not for some time scope to make     
itself perceived4 ,it would only be in  
what we have called the ultimate stage1 
when we suppose the series to extend for
 a very long time1 that the difference  
would begin to make itself felt4#c ,the 
proportion of persons1 for example1 who 
die each year at the age of six months  
is1 when the numbers examined are on a  
small scale1 utterly irregular2 it      
becomes however regular when the numbers
 examined are on a larger scale2 but if 
we continued our observation for a very 
great length of time1 or over a very    
great extent of country1 we should find 
this regularity itself changing in an   

irregular way4 ,the substitution just   
mentioned is really equivalent to       
saying1 ,let us assume that the         
regularity is fixed and permanent4 ,it  
is making a hypothesis which may not be 
altogether consistent with fact1 but    
which is forced upon us for the purpose 
of securing precision of statement and  
definition4  #ad4 ,the full meaning and 
bearing of such a substitution will only
 become apparent in some of the         
subsequent chapters1 but it may be      
pointed out at once that it is in this  
way only that we can with perfect       
strictness introduce the notion of a    
'limit' into our account of the matter1 
at any rate in reference to many of the 
applications of the subject to purely   
statistical enquiries4 ,we say that a   
certain proportion begins to prevail    
among the events in the long run2 but   
then on looking closer at the facts we  
find that we have to express ourselves  
hypothetically1 and to say that if      
present circumstances remain as they    

are1 the long run will show its         
characteristics without disturbance4    
,when1 as is often the case1 we know    
nothing accurately of the circumstances 
by which the succession of events is    
brought about1 but have strong reasons  
to suspect that these circumstances are 
likely to undergo some change1 there is 
really nothing else to be done4 ,we can 
only introduce the conception of a      
limit1 towards which the numbers are    
tending1 by assuming that these         
circumstances do not change2 in other   
words1 by substituting a series with a  
fixed uniformity for the actual one with
 the varying uniformity4#d  #ae4 ,if the
 reader will study the following        
example1 one well known to              
mathematicians under the name of the    
,petersburg#e problem1 he will find that
 it serves to illustrate several of the 
considerations mentioned in this        
chapter4 ,it serves especially to bring 
out the facts that the series with which
 we are concerned must be regarded as   

indefinitely extensive in point of      
number or duration2 and that when so    
regarded certain series1 but certain    
series only 7the one in question being a
 case in point71 take advantage of the  
indefinite range to keep on producing   
individuals in it whose deviation from  
the previous average has no finite limit
 whatever4 ,when rightly viewed it is a 
very simple problem1 but it has given   
rise1 at one time or another1 to a good 
deal of confusion and perplexity4  ,the 
problem may be stated thus3--a penny is 
tossed up2 if it gives head ,i receive  
one pound2 if heads twice running two   
pounds2 if heads three times running    
four pounds1 and so on2 the amount to be
 received doubling every time that a    
fresh head succeeds4 ,that is1 ,i am to 
go on as long as it continues to give a 
succession of heads1 to regard this     
succession as a 'turn' or set1 and then 
take another turn1 and so on2 and for   
each such turn ,i am to receive a       
payment2 the occurrence of tail being   

understood to yield nothing1 in fact    
being omitted from our consideration4   
,however many times head may be given in
 succession1 the number of pounds ,i may
 claim is found by raising two to a     
power one less than that number of      
times4 ,here then is a series formed by 
a succession of throws4 ,we will        
assume1--what many persons will consider
 to admit of demonstration1 and what    
certainly experience confirms within    
considerable limits1--that the rarity of
 these 'runs' of the same face is in    
direct proportion to the amount ,i      
receive for them when they do occur4 ,in
 other words1 if we regard only the     
occasions on which ,i receive payments1 
we shall find that every other time ,i  
get one pound1 once in four times ,i get
 two pounds1 once in eight times four   
pounds1 and so on without any end4 ,the 
question is then asked1 what ought ,i to
 pay for this privilege8 ,at the risk of
 a slight anticipation of the results of
 a subsequent chapter1 we may assume    

that this is equivalent to asking1 what 
amount paid each time would on the      
average leave me neither winner nor     
loser8 ,in other words1 what is the     
average amount ,i should receive on the 
above terms8 ,theory pronounces that ,i 
ought to give an infinite sum3 that is1 
no finite sum1 however great1 would be  
an adequate equivalent4 ,and this is    
really quite intelligible4 ,there is a  
series of indefinite length before me1  
and the longer ,i continue to work it   
the richer are my returns1 and this     
without any limit whatever4 ,it is true 
that the very rich hauls are extremely  
rare1 but still they do come1 and when  
they come they make it up by their      
greater richness4 ,on every occasion on 
which people have devoted themselves to 
the pursuit in question1 they made      
acquaintance1 of course1 with but a     
limited portion of this series2 but the 
series on which we base our calculation 
is unlimited2 and the inferences usually
 drawn as to the sum which ought in the 

long run to be paid for the privilege in
 question are in perfect accordance with
 this supposition4  ,the common form of 
objection is given in the reply1 that so
 far from paying an infinite sum1 no    
sensible man would give anything        
approaching to #ej for such a chance4   
,probably not1 because no man would see 
enough of the series to make it worth   
his while4 ,what most persons form their
 practical opinion upon1 is such small  
portions of the series as they have     
actually seen or can reasonably expect4 
,now in any such portion1 say one which 
embraces #ajj turns1 the longest        
succession of heads would not amount on 
the average to more than seven or eight4
 ,this is observed1 but it is forgotten 
that the formula which produced these1  
would1 if it had greater scope1 keep on 
producing better and better ones without
 any limit4 ,hence it arises that some  
persons are perplexed1 because the      
conduct they would adopt1 in reference  
to the curtailed portion of the series  

which they are practically likely to    
meet with1 does not find its            
justification in inferences which are   
necessarily based upon the series in the
 completeness of its infinitude4  #af4  
,this will be more clearly seen by      
considering the various possibilities1  
and the scope required in order to      
exhaust them1 when we confine ourselves 
to a limited number of throws4 ,begin   
with three4 ,this yields eight equally  
likely possibilities4 ,in four of these 
cases the thrower starts with tail and  
therefore loses3 in two he gains a      
single point 7i4e4 #a72 in one he gains 
two points1 and in one he gains four    
points4 ,hence his total gain being     
eight pounds achieved in four different 
contingencies1 his average gain would be
 two pounds4  ,now suppose he be allowed
 to go as far as n throws1 so that we   
have to contemplate #bn possibilities4  
,all of these have to be taken into     
account if we wish to consider what     
happens on the average4 ,it will readily

 be seen that1 when all the possible    
cases have been reckoned once1 his total
 gain will be 7reckoned in pounds71     
#bn-#b  #bn-#cb  #bn-#dbb  444  #bbn-#c 
#bn-#b  #bn-#a1  viz4  7n  #a7 #bn-#b4  
,this being spread over #bn-#a different
 occasions of gain his average gain will
 be #a_/b 7n  #a74  ,now when we are    
referring to averages it must be        
remembered that the minimum number of   
different occurrences necessary in order
 to justify the average is that which   
enables each of them to present itself  
once4 ,a man proposes to stop short at a
 succession of ten heads4 ,well and     
good4 ,we tell him that his average gain
 will be #e4 #ajs4 #j;d43 but we also   
impress upon him that in order to       
justify this statement he must commence 
to toss at least #ajbd times1 for in no 
less number can all the contingencies of
 gain and loss be exhibited and         
balanced4 ,if he proposes to reach an   
average gain of #bj1 he will require to 
be prepared to go up to #ci throws1 ,to 

justify this payment he must commence to
 throw #bci times1 i4e4 about a million 
million times4 ,not before he has       
accomplished this will he be in a       
position to prove to any sceptic that   
this is the true average value of a     
'turn' extending to #ci successive      
tosses4  ,of course if he elects to toss
 to all eternity we must adopt the line 
of explanation which alone is possible  
where questions of infinity in respect  
of number and magnitude are involved4   
,we cannot tell him to pay down 'an     
infinite sum1' for this has no strict   
meaning4 ,but we tell him that1 however 
much he may consent to pay each time    
runs of heads occur1 he will attain at  
last a stage in which he will have won  
back his total payments by his total    
receipts4 ,however large n may be1 if he
 perseveres in trying #bn times he may  
have a true average receipt of #a_/b 7n 
#a7 pounds1 and if he continues long    
enough onwards he will have it4  ,the   
problem will recur for consideration in 

a future chapter4   #a4 ,the following  
statistics will give a fair idea of the 
wide range of   experience over which   
such regularity is found to exist3 ,as  
illustrations of equal amounts of       
fluctuation from totally   dissimilar   
causes1 take the deaths in the ,west    
district of ,london in   seven years    
7fluctuation #ac4ff71 and offences      
against the person   7fluctuation       
#ac4fa72 or deaths from apoplexy        
7fluctuation #e4ed71 and   offences     
against property1 without violence      
7fluctuation #e4dh72 or   students      
registered at the ,college of ,surgeons 
7fluctuation #a4he71   and the number of
 pounds of manufactured tobacco taken   
for home   consumption 7fluctuation     
#a4hi72 or out-door paupers 7fluctuation
   #c4de7 and tonnage of ,british       
vessels entered in ballast 7fluctuation 
#c4dc71 c4  ,extracted from a paper in  
the ,journal of the   ,statistical      
,society1 by ,mr ,guy1 ,march1 #aheh2   
the 'fluctuation' here   given is a     

measure of the amount of irregularity1  
that is of   departure from the average1
 estimated in a way which will be       
described hereafter4  #b4 ,transactions 
of the ,cambridge ,philosophical        
,society1 ,vol4 ,,ix4 p4 #fje4          
,reprinted in the collected edition of  
his writings1 p4 #ej4  #c4 ,we might    
express it thus3--a few instances are   
not sufficient to   display a law at    
all2 a considerable number will suffice 
to display   it2 but it takes a very    
great number to establish that a change 
is   taking place in the law4  #d4 ,the 
mathematician may illustrate the nature 
of this substitution by   the analogies 
of the 'circle of curvature' in         
geometry1 and the   'instantaneous      
ellipse' in astronomy4 ,in the cases in 
which these   conceptions are made use  
of we have a phenomenon which is        
continuously varying and also changing  
its rate of variation4 ,we   take it at 
some given moment1 suppose its rate at  
that moment to be   fixed1 and then     

complete its career on that supposition4
  #e4 ,so called from its first         
mathematical treatment appearing in the 
,commentarii of the ,petersburg         
,academy2 a variety of notices upon   it
 will be found in ,mr ,todhunter's      
,history of the ,theory of              
,probability4     ,,chapter ,,ii4       
,,further ,,discussion ,,upon ,,the     
,,nature ,,of ,,the ,,series ,,mentioned
 ,,in ,,the ,,last ,,chapter4  #a4 ,in  
the course of the last chapter the      
nature of a particular kind of series1  
that namely1 which must be considered to
 constitute the basis of the science of 
,probability1 has received a            
sufficiently general explanation for the
 preliminary purpose of introduction4   
,one might indeed say more than this2   
for the characteristics which were there
 pointed out are really sufficient in   
themselves to give a fair general idea  
of the nature of ,probability1 and of   
the sort of problems with which it      
deals4 ,but in the concluding paragraphs

 an indication was given that the series
 of this kind1 as they actually occur in
 nature or as the results of more or    
less artificial production1 are seldom  
or never found to occur in such a simple
 form as might possibly be expected from
 what had previously been said2 but that
 they are almost always seen to be      
associated together in groups after a   
somewhat complicated fashion4 ,a fuller 
discussion of this topic must now be    
undertaken4  ,we will take for          
examination an instance of a kind with  
which the investigations of ,quetelet   
will have served to familiarize some    
readers4 ,suppose that we measure the   
heights of a great many adult men in any
 town or country4 ,these heights will of
 course lie between certain extremes in 
each direction1 and if we continue to   
accumulate our measures it will be found
 that they tend to lie continuously     
between these extremes2 that is to say1 
that under those circumstances no       
intermediate height will be found to be 

permanently unrepresented in such a     
collection of measurements4 ,now suppose
 these heights to be marshalled in the  
order of their magnitude4 ,what we      
always find is something of the         
following kind2--about the middle point 
between the extremes1 a large number of 
the results will be found crowded       
together3 a little on each side of this 
point there will still be an excess1 but
 not to so great an extent2 and so on1  
in some diminishing scale of proportion1
 until as we get towards the extreme    
results the numbers thin off and become 
relatively exceedingly small4  ,the     
point to which attention is here        
directed is not the mere fact that the  
numbers thus tend to diminish from the  
middle in each direction1 but1 as will  
be more fully explained directly1 the   
law according to which this progressive 
diminution takes place4 ,the word 'law' 
is here used in its mathematical sense1 
to express the formula connecting       
together the two elements in question1  

namely1 the height itself1 and the      
relative number that are found of that  
height4 ,we shall have to enquire       
whether one of these elements is a      
function of the other1 and1 if so1 what 
function4  #b4 ,after what was said in  
the last chapter1 it need hardly be     
insisted upon that the interest and     
significance of such investigations as  
these are almost entirely dependent upon
 the statistics being very extensive4   
,in one or other of ,quetelet's works on
 ,social ,physics#a will be found a     
selection of measurements of almost     
every element which the physical frame  
of man can furnish3--his height1 his    
weight1 the muscular power of various   
limbs1 the dimensions of almost every   
part and organ1 and so on4 ,some of the 
most extensive of these express the     
heights of #be1jjj ,federal soldiers    
from the ,army of the ,potomac1 and the 
circumferences of the chests of #egch   
,scotch militia men taken many years    
ago4 ,those who wish to consult a large 

repertory of such statistics cannot be  
referred to any better sources than to  
these and other works by the same       
author4#b  ,interesting and valuable1   
however1 as are ,quetelet's statistical 
investigations 7and much of the         
importance now deservedly attached to   
such enquiries is1 perhaps1 owing more  
to his efforts than to those of any     
other person71 ,i cannot but feel       
convinced that there is much in what he 
has written upon the subject which is   
erroneous and confusing as regards the  
foundations of the science of           
,probability1 and the philosophical     
questions which it involves4 ,these     
errors are not by any means confined to 
him1 but for various reasons they will  
be better discussed in the form of a    
criticism of his explicit or implicit   
expression of them1 than in any more    
independent way4  #c4 ,in the first     
place then1 he always1 or almost always1
 assumes that there can be but one and  
the same law of arrangement for the     

results of our observations1            
measurements1 and so on1 in these       
statistical enquiries4 ,that is1 he     
assumes that whenever we get a group of 
such magnitudes clustering about a mean1
 and growing less frequent as they      
depart from that mean1 we shall find    
that this diminution of frequency takes 
place according to one invariable law1  
whatever may be the nature of these     
magnitudes1 and whatever the process by 
which they may have been obtained4      
,that such a uniformity as this should  
prevail amongst many and various classes
 of phenomena would probably seem       
surprising in any case4 ,but the full   
significance of such a fact as this 7if 
indeed it were a fact7 only becomes     
apparent when attention is directed to  
the profound distinctions in the nature 
and origin of the phenomena which are   
thus supposed to be harmonized by being 
brought under one comprehensive         
principle4 ,this will be better         
appreciated if we take a brief glance at

 some of the principal classes into     
which the things with which ,probability
 is chiefly concerned may be divided4   
,these are of a three-fold kind4  #d4   
,in the first place there are the       
various combinations1 and runs of luck1 
afforded by games of chance4 ,suppose a 
handful1 consisting of ten coins1 were  
tossed up a great many times in         
succession1 and the results were        
tabulated4 ,what we should obtain would 
be something of the following kind4 ,in 
a certain proportion of cases1 and these
 the most numerous of all1 we should    
find that we got five heads and five    
tails2 in a somewhat less proportion of 
cases we should have1 as equally        
frequent results1 four heads six tails1 
and four tails six heads2 and so on in a
 continually diminishing proportion     
until at length we came down1 in a very 
small relative number of cases1 to nine 
heads one tail1 and nine tails one head2
 whilst the least frequent results      
possible would be those which gave all  

heads or all tails4#c ,here the         
statistical elements under consideration
 are1 as regards their origin at any    
rate1 optional or brought about by human
 choice4 ,they would1 therefore1 be     
commonly described as being mainly      
artificial1 but their results ultimately
 altogether a matter of chance4  ,again1
 in the second place1 we might take the 
accurate measurements--i4e4 the actual  
magnitudes themselves1--of a great many 
natural objects1 belonging to the same  
genus or class2 such as the cases1      
already referred to1 of the heights1 or 
other characteristics of the inhabitants
 of any district4 ,here human volition  
or intervention of any kind seem to have
 little or nothing to do with the       
matter4 ,it is optional with us to      
collect the measures1 but the things    
measured are quite outside our control4 
,they would therefore be commonly       
described as being altogether the       
production of nature1 and it would not  
be supposed that in strictness chance   

had anything whatever to do with the    
matter4  ,in the third place1 the result
 at which we are aiming may be some     
fixed magnitude1 one and the same in    
each of our successive attempts1 so that
 if our measurements were rigidly       
accurate we should merely obtain the    
same result repeated over and over      
again4 ,but since all our methods of    
attaining our aims are practically      
subject to innumerable imperfections1   
the results actually obtained will      
depart more or less1 in almost every    
case1 from the real and fixed value     
which we are trying to secure4 ,they    
will be sometimes more wide of the mark1
 sometimes less so1 the worse attempts  
being of course the less frequent4 ,if a
 man aims at a target he will seldom or 
never hit it precisely in the centre1   
but his good shots will be more#d       
numerous than his bad ones4 ,here again1
 then1 we have a series of magnitudes   
7i4e4 the deflections of the shots from 
the point aimed at7 clustering about a  

mean1 but produced in a very different  
way from those of the last two cases4   
,in this instance the elements would be 
commonly regarded as only partially the 
results of human volition1 and chance   
therefore as being only a co-agent in   
the effects produced4 ,with these must  
be classed what may be called estimates1
 as distinguished from measurements4 ,by
 the latter are generally understood the
 results of a certain amount of         
mechanism or manipulation2 by the former
 we may understand those cases in which 
the magnitude in question is determined 
by direct observation or introspection4 
,the interest and importance of this    
class1 so far as scientific principles  
are concerned1 dates mainly from the    
investigations of ,fechner4 ,its chief  
field is naturally to be found amongst  
psychological data4  ,other classes of  
things1 besides those alluded to above1 
might readily be given4 ,these however  
are the classes about which the most    
extensive statistics are obtainable1 or 

to which the most practical importance  
and interest are attached4 ,the profound
 distinctions which separate their      
origin and character are obvious4 ,if   
they all really did display precisely   
the same law of variation it would be a 
most remarkable fact1 pointing doubtless
 to some deep-seated identity underlying
 the various ways1 apparently so widely 
distinct1 in which they had been brought
 about4 ,the questions now to be        
discussed are3 ,is it the case1 with any
 considerable degree of rigour1 that    
only one law of distribution does really
 prevail8 and1 in so far as this is so1 
how does it come to pass8  #e4 ,in      
support of an affirmative answer to the 
former of these two questions1 several  
different kinds of proof are1 or might  
be1 offered4  7,i47 ,for one plan we may
 make a direct appeal to experience1 by 
collecting sets of statistics and       
observing what is their law of          
distribution4 ,as remarked above1 this  
has been done in a great variety of     

cases1 and in some instances to a very  
considerable extent1 by ,quetelet and   
others4 ,his researches have made it    
abundantly convincing that many classes 
of things and processes1 differing      
widely in their nature and origin1 do   
nevertheless appear to conform with a   
considerable degree of accuracy to one  
and the same#e law4 ,at least this is   
made plain for the more central values1 
for those that is which are situated    
most nearly about the mean4 ,with regard
 to the extreme values there is1 on the 
other hand1 some difficulty4 ,for       
instance in the arrangements of the     
heights of a number of men1 these       
extremes are rather a stumbling-block2  
indeed it has been proposed to reject   
them from both ends of the scale on the 
plea that they are monstrosities1 the   
fact being that their relative numbers  
do not seem to be by any means those    
which theory would assign4#f ,such a    
plan of rejection is however quite      
unauthorized1 for these dwarfs and      

giants are born into the world like     
their more normally sized brethren1 and 
have precisely as much right as any     
others to be included in the formul we  
draw up4  ,besides the instance of the  
heights of men1 other classes of        
observations of a somewhat similar      
character have been already referred to 
as collected and arranged by ,quetelet4 
,from the nature of the case1 however1  
there are not many appropriate ones at  
hand2 for when our object is1 not to    
illustrate a law which can be otherwise 
proved1 but to obtain actual direct     
proof of it1 the collection of          
observations and measurements ought to  
be made upon such a large scale as to   
deter any but the most persevering      
computers from undergoing the requisite 
labour4 ,some of the remarks made in the
 course of the note on the opposite page
 will serve to illustrate the           
difficulties which would lie in the way 
of such a mode of proof4  ,we are       
speaking here1 it must be understood1   

only of symmetrical curves3 if there is 
asymmetry1 i4e4 if the ,law of ,error is
 different on different sides of the    
mean1--a comparatively very small number
 of observations would suffice to detect
 the fact4 ,but1 granted symmetry and   
rapid decrease of frequency on each side
 of the mean1 we could generally select 
some one species of the exponential     
curve which should pretty closely       
represent our statistics in the         
neighbourhood of the mean4 ,that is1    
where the statistics are numerous we    
could secure agreement2 and where we    
could not secure agreement the          
statistics would be comparatively so    
scarce that we should have to continue  
the observations for a very long time in
 order to prove the disagreement4  #f4  
,allowing the various statistics such   
credit as they deserve1 for their       
extent1 appropriateness1 accuracy and so
 on1 the general conclusion which will  
on the whole be drawn by almost every   
one who takes the trouble to consult    

them1 is that they do1 in large part1   
conform approximately to one type or    
law1 at any rate for all except the     
extreme values4 ,so much as this must be
 fully admitted4 ,but that they do not1 
indeed we may say that they cannot1     
always do so in the case of the extreme 
values1 will become obvious on a little 
consideration4 ,in some of the classes  
of things to which the law is supposed  
to apply1 for example1 the successions  
of heads and tails in the throws of a   
penny1 there is no limit to the         
magnitude of the fluctuations which may 
and will occur4 ,postulate as long a    
succession of heads or of tails as we   
please1 and if we could only live and   
toss long enough for it we should       
succeed in getting it at length4 ,in    
other cases1 including many of the      
applications of ,probability to natural 
phenomena1 there can hardly fail to be  
such limits4 ,deviations exceeding a    
certain range may not be merely         
improbable1 that is of very rare        

occurrence1 but they may often from the 
nature of the case be actually          
impossible4 ,and even when they are not 
actually impossible it may frequently   
appear on examination that they are only
 rendered possible by the occasional    
introduction of agencies which are not  
supposed to be available in the         
production of the more ordinary or      
intermediate values4 ,when1 for         
instance1 we are making observations    
with any kind of instrument1 the nature 
of its construction may put an absolute 
limit upon the possible amount of error4
 ,and even if there be not an absolute  
limit under all kinds of usage it may   
nevertheless be the case that there is  
one under fair and proper usage2 it     
being the case that only when the       
instrument is designedly or carelessly  
tampered with will any new causes of    
divergence be introduced which were not 
confined within the old limits4         
,suppose1 for instance1 that a man is   
firing at a mark4 ,his worst shots must 

be supposed to be brought about by a    
combination of such causes as were      
acting1 or prepared to act1 in every    
other case2 the extreme instance of what
 we may thus term 'fair usage' being    
when a number of distinct causes have   
happened to conspire together so as to  
tend in the same direction1 instead of1 
as in the other cases1 more or less     
neutralizing one another's work4 ,but   
the aggregate effect of such causes may 
well be supposed to be limited4 ,the man
 will not discharge his shot nearly at  
right angles to the true line of fire   
unless some entirely new cause comes in1
 as by some unusual circumstance having 
distracted his attention1 or by his     
having had some spasmodic seizure4 ,but 
influences of this kind were not        
supposed to have been available before2 
and even if they were we are taking a   
bold step in assuming that these        
occasional great disturbances are       
subject to the same kind of laws as are 
the aggregates of innumerable little    

ones4  ,we cannot indeed lay much stress
 upon an example of this last kind1 as  
compared with those in which we can see 
for certain that there is a fixed limit 
to the range of error4 ,it is therefore 
offered rather for illustration than for
 proof4 ,the enormous1 in fact          
inconceivable magnitude of the numbers  
expressive of the chance of very rare   
combinations1 such as those in question1
 has such a bewildering effect upon the 
mind that one may be sometimes apt to   
confound the impossible with the higher 
degrees of the merely mathematically    
improbable4  #g4 ,at the time the first 
edition of this essay was composed      
writers on ,statistics were1 ,i think1  
still for the most part under the       
influence of ,quetelet1 and inclined to 
overvalue his authority on this         
particular subject3 of late however     
attention has been repeatedly drawn to  
the necessity of taking account of other
 laws of arrangement than the binomial  
or exponential4  ,mr ,galton1 for       

instance1--to whom every branch of the  
theory of statistics owes so much1--has 
insisted#g that the assumption which    
lies at the basis of the well-known law 
of ',frequency of ,error'444 is         
incorrect in many groups of vital and   
social phenomena4444 ,for example1      
suppose we endeavour to match a tint2   
,fechner's law1 in its approximative and
 simplest form of sensation  log        
stimulus1 tells us that a series of     
tints1 in which the quantities of white 
scattered on a black ground are as #a1  
#b1 #d1 #h1 #af1 #cb1 c41 will appear to
 the eye to be separated by equal       
intervals of tint4 ,therefore1 in       
matching a grey that contains #h        
portions of white1 we are just as likely
 to err by selecting one that has #af   
portions as one that has #d portions4   
,in the first case there would be an    
error in excess1 of #h2 in the second   
there would be an error1 in deficiency1 
of #d4 ,therefore1 an error of the same 
magnitude in excess or in deficiency is 

not equally probable4 ,the consequences 
of this assumption are worked out in a  
remarkable paper by ,dr ,d4 ,mc,alister1
 to which allusion will have to be made 
again hereafter4 ,all that concerns us  
here to point out is that when the      
results of statistics of this character 
are arranged graphically we do not get a
 curve which is symmetrical on both     
sides of a central axis4  #h4 ,more     
recently1 ,mr ,f4 ,y4 ,edgeworth 7in a  
report of a ,committee of the ,british  
,association appointed to enquire into  
the variation of the monetary standard7 
has urged the same considerations in    
respect of prices of commodities4 ,he   
gives a number of statistics drawn from 
the prices of twelve commodities during 
the two periods #aghb-ahbj1 #ahbj-ahfe4 
,the maximum and minimum entry for each 
series having been noted1 it is found   
that the number of entries above the    
'middle point1' half-way between the    
maximum and minimum1#h is in every      
instance less than half the total number

 of entries in the series4 ,in the      
twenty-four trials there is not a single
 exception to the rule1 and in very few 
cases even an approach to an exception4 
,we may presume then that the curves are
 of the lop-sided character indicated by
 the accompanying diagram4 ,the same    
facts are also ascertained in respect to
 place variations as distinguished from 
time variations4 ,to these may be added 
some statistics of my own1 referring to 
the heights of the barometer taken at   
the same hour on more than #djjj        
successive days 7v4 ,nature1 ,sept4 #b1 
#ahhg74 ,so far as these go they show a 
marked asymmetry of arrangement4  ,in   
fact it appears to me that this want of 
symmetry ought to be looked for in all  
cases in which the phenomena under      
measurement are of a 'one-sided'        
character2 in the sense that they are   
measured on one side only of a certain  
fixed point from which their possibility
 is supposed to start4 ,for not only is 
it impossible for them to fall below    

this point3 long before they reach it   
the influence of its proximity is felt  
in enhancing the difficulty and         
importance of the same amount of        
absolute difference4  ,look at a table  
of statures1 for instance1 with a mean  
value of #fi inches4 ,a diminution of   
three feet 7were this possible7 is much 
more influential1--counts for much more1
 in every sense of the term1--than an   
addition of the same amount2 for the    
former does not double the mean1 while  
the latter more than halves it4 ,revert 
to an illustration4 ,if a vast number of
 petty influencing circumstances of the 
kind already described were to act upon 
a swinging pendulum we should expect the
 deflections in each direction to       
display symmetry2 but if they were to   
act upon a spring we should not expect  
such a result4 ,any phenomena of which  
the latter is the more appropriate      
illustration can hardly be expected to  
range themselves with symmetry about a  
mean4#i  #i4 7,,ii47 ,the last remarks  

will suggest another kind of proof which
 might be offered to establish the      
invariable nature of the law of error4  
,it is of a direct deductive kind1 not  
appealing immediately to statistics1 but
 involving an enquiry into the actual or
 assumed nature of the causes by which  
the events are brought about4 ,imagine  
that the event under consideration is   
brought to pass1 in the first place1 by 
some fixed cause1 or group of fixed     
causes4 ,if this comprised all the      
influencing circumstances the event     
would invariably happen in precisely the
 same way3 there would be no errors or  
deflections whatever to be taken account
 of4 ,but now suppose that there were   
also an enormous number of very small   
causes which tended to produce          
deflections2 that these causes acted in 
entire independence of one another2 and 
that each of the lot told as often1 in  
the long run1 in one direction as in the
 opposite4 ,it is easy#aj to see1 in a  
general way1 what would follow from     

these assumptions4 ,in a very few cases 
nearly all the causes would tell in the 
same direction2 in other words1 in a    
very few cases the deflection would be  
extreme4 ,in a greater number of cases1 
however1 it would only be the most part 
of them that would tell in one          
direction1 whilst a few did what they   
could to counteract the rest2 the result
 being a comparatively larger number of 
somewhat smaller deflections4 ,so on1 in
 increasing numbers1 till we approach   
the middle point4 ,here we shall have a 
very large number of very small         
deflections3 the cases in which the     
opposed influences just succeed in      
balancing one another1 so that no error 
whatever is produced1 being1 though     
actually infrequent1 relatively the most
 frequent of all4  ,now if all          
deflections from a mean were brought    
about in the way just indicated 7an     
indication which must suffice for the   
present7 we should always have one and  
the same law of arrangement of frequency

 for these deflections or errors1 viz4  
the exponential#aa law mentioned in #e4 
#aj4 ,it may be readily admitted from   
what we know about the production of    
events that something resembling these  
assumptions1 and therefore something    
resembling the consequences which follow
 from them1 is really secured in a very 
great number of cases4 ,but although    
this may prevail approximately1 it is in
 the highest degree improbable that it  
could ever be secured1 even             
artificially1 with anything approaching 
to rigid accuracy4 ,for one thing1 the  
causes of deflection will seldom or     
never be really independent of one      
another4 ,some of them will generally be
 of a kind such that the supposition    
that several are swaying in one         
direction1 may affect the capacity of   
each to produce that full effect which  
it would have been capable of if it had 
been left to do its work alone4 ,in the 
common example1 for instance1 of firing 
at a mark1 so long as we consider the   

case of the tolerably good shots the    
effect of the wind 7one of the causes of
 error7 will be approximately the same  
whatever may be the precise direction of
 the bullet4 ,but when a shot is        
considerably wide of the mark the wind  
can no longer be regarded as acting at  
right angles to the line of flight1 and 
its effect in consequence will not be   
precisely the same as before4 ,in other 
words1 the causes here are not strictly 
independent1 as they were assumed to be2
 and consequently the results to be     
attributed to each are not absolutely   
uninfluenced by those of the others4    
,doubtless the effect is trifling here1 
but ,i apprehend that if we were        
carefully to scrutinize the modes in    
which the several elements of the total 
cause conspire together1 we should find 
that the assumption of absolute         
independence was hazardous1 not to say  
unwarrantable1 in a very great number of
 cases4 ,these brief remarks upon the   
process by which the deflections are    

brought about must suffice for the      
present purpose1 as the subject will    
receive a fuller investigation in the   
course of the next chapter4  ,according1
 therefore1 to the best consideration   
which can at the present stage be       
afforded to this subject1 we may draw a 
similar conclusion from this deductive  
line of argument as from the direct     
appeal to statistics4 ,the same general 
result seems to be established2 namely1 
that approximately1 with sufficient     
accuracy for all practical purposes1 we 
may say that an examination of the      
causes by which the deflections are     
generally brought about shows that they 
are mostly of such a character as would 
result in giving us the commonly        
accepted ',law of ,error1' as it is     
termed4#ab ,the two lines of enquiry1   
therefore1 within the limits assigned1  
afford each other a decided mutual      
confirmation4  #aa4 7,,iii47 ,there     
still remains a third1 indirect and     
mathematical line of proof1 which might 

be offered to establish the conclusion  
that the ,law of ,error is always one   
and the same4 ,it may be maintained that
 the recognized and universal employment
 of one and the same method1 that known 
to mathematicians and astronomers as the
 ,method of ,least ,squares1 in all     
manner of different cases with very     
satisfactory results1 is compatible only
 with the supposition that the errors to
 which that method is applied must be   
grouped according to one invariable law4
 ,if all 'laws of error' were not of one
 and the same type1 that is1 if the     
relative frequency of large and small   
divergences 7such as we have been       
speaking of7 were not arranged according
 to one pattern1 how could one method or
 rule equally suit them all8  ,in order 
to preserve a continuity of treatment1  
some notice must be taken of this       
enquiry here1 though1 as in the case of 
the last argument1 any thorough         
discussion of the subject is impossible 
at the present stage4 ,for one thing1 it

 would involve too much employment of   
mathematics1 or at any rate of          
mathematical conceptions1 to be suitable
 for the general plan of this treatise3 
,i have accordingly devoted a special   
chapter to the consideration of it4     
,the main reason1 however1 against      
discussing this argument here1 is1 that 
to do so would involve the anticipation 
of a totally different side of the      
science of ,probability from that       
hitherto treated of4 ,this must be      
especially insisted upon1 as the neglect
 of it involves much confusion and some 
error4 ,during these earlier chapters we
 have been entirely occupied with laying
 what may be called the physical        
foundations of ,probability4 ,we have   
done nothing else than establish1 in one
 way or another1 the existence of       
certain groups or arrangements of things
 which are found to present themselves  
in nature2 we have endeavoured to       
explain how they come to pass1 and we   
have illustrated their principal        

characteristics4 ,but these are merely  
the foundations of ,inference1 we have  
not yet said a word upon the logical    
processes which are to be erected upon  
these foundations4 ,we have not         
therefore entered yet upon the logic of 
chance4  #ab4 ,now the way in which the 
,method of ,least ,squares is sometimes 
spoken of tends to conceal the magnitude
 of this distinction4 ,writers have     
regarded it as synonymous with the ,law 
of ,error1 whereas the fact is that the 
two are not only totally distinct things
 but that they have scarcely even any   
necessary connection with each other4   
,the ,law of ,error is the statement of 
a physical fact2 it simply assigns1 with
 more or less of accuracy1 the relative 
frequency with which errors or          
deviations of any kind are found in     
practice to present themselves4 ,it     
belongs therefore to what may be termed 
the physical foundations of the science4
 ,the ,method of ,least ,squares1 on the
 other hand1 is not a law at all in the 

scientific sense of the term4 ,it is    
simply a rule or direction informing us 
how we may best proceed to treat any    
group of these errors which may be set  
before us1 so as to extract the true    
result at which they have been aiming4  
,clearly therefore it belongs to the    
inferential or logical part of the      
subject4  ,it cannot indeed be denied   
that the methods we employ must have    
some connection with the arrangement of 
the facts to which they are applied2 but
 the two things are none the less       
distinct in their nature1 and in this   
case the connection does not seem at all
 a necessary one1 but at most one of    
propriety and convenience4 ,the ,method 
of ,least ,squares is usually applied1  
no doubt1 to the most familiar and      
common form of the ,law of ,error1      
namely the exponential form with which  
we have been recently occupied4 ,but    
other forms of laws of error may exist1 
and1 if they did1 the method in question
 might equally well be applied to them4 

,i am not asserting that it would       
necessarily be the best method in every 
case1 but it would be a possible one2   
indeed we may go further and say1 as    
will be shown in a future chapter1 that 
it would be a good method in almost     
every case4 ,but its particular merits  
or demerits do not interfere with its   
possible employment in every case in    
which we may choose to resort to it4 ,it
 will be seen therefore1 even from the  
few remarks that can be made upon the   
subject here1 that the fact that one and
 the same method is very commonly       
employed with satisfactory results      
affords little or no proof that the     
errors to which it is applied must be   
arranged according to one fixed law4    
#ac4 ,so much then for the attempt to   
prove the prevalence1 in all cases1 of  
this particular law of divergence4 ,the 
next point in ,quetelet's treatment of  
the subject which deserves attention as 
erroneous or confusing1 is the doctrine 
maintained by him and others as to the  

existence of what he terms a type in the
 groups of things in question4 ,this is 
a not unnatural consequence from some of
 the data and conclusions of the last   
few paragraphs4 ,refer back to two of   
the three classes of things already     
mentioned in #d4 ,if it really were the 
case that in arranging in order a series
 of incorrect observations or attempts  
of our own1 and a collection of natural 
objects belonging to some one and the   
same species or class1 we found that the
 law of their divergence was in each    
case identical in the long run1 we      
should be naturally disposed to apply   
the same expression ',law of ,error' to 
both instances alike1 though in         
strictness it could only be appropriate 
to the former4 ,when we perform an      
operation ourselves with a clear        
consciousness of what we are aiming at1 
we may quite correctly speak of every   
deviation from this as being an error2  
but when ,nature presents us with a     
group of objects of any kind1 it is     

using a rather bold metaphor to speak in
 this case also of a law of error1 as if
 she had been aiming at something all   
the time1 and had like the rest of us   
missed her mark more or less in almost  
every instance4#ac  ,suppose we make a  
long succession of attempts to measure  
accurately the precise height of a man1 
we should from one cause or another     
seldom or never succeed in doing so with
 absolute accuracy4 ,but we have no     
right to assume that these imperfect    
measurements of ours would be found so  
to deviate according to one particular  
law of error as to present the precise  
counterpart of a series of actual       
heights of different men1 supposing that
 these latter were assigned with        
absolute precision4 ,what might be the  
actual law of error in a series of      
direct measurements of any given        
magnitude could hardly be asserted      
beforehand1 and probably the attempt to 
determine it by experience has not been 
made sufficiently often to enable us to 

ascertain it2 but upon general grounds  
it seems by no means certain that it    
would follow the so-called exponential  
law4 ,be this however as it may1 it is  
rather a licence of language to talk as 
if nature had been at work in the same  
way as one of us2 aiming 7ineffectually 
for the most part7 at a given result1   
that is at producing a man endowed with 
a certain stature1 proportions1 and so  
on1 who might therefore be regarded as  
the typical man4  #ad4 ,stated as above1
 namely1 that there is a fixed          
invariable human type to which all      
individual specimens of humanity may be 
regarded as having been meant to attain1
 but from which they have deviated in   
one direction or another2 according to a
 law of deviation capable of  priori    
determination1 the doctrine is little   
else than absurd4 ,but if we look       
somewhat closer at the facts of the     
case1 and the probable explanation of   
these facts1 we may see our way to an   
important truth4 ,the facts1 on the     

authority of ,quetelet's statistics 7the
 great interest and value of which must 
be frankly admitted71 are very briefly  
as follows3 if we take any element of   
our physical frame which admits of      
accurate measurement1 say the height1   
and determine this measure in a great   
number of different individuals         
belonging to any tolerably homogeneous  
class of people1 we shall find that     
these heights do admit of an orderly    
arrangement about a mean1 after the     
fashion which has been already          
repeatedly mentioned4 ,what is meant by 
a homogeneous class8 is a pertinent and 
significant enquiry1 but applying this  
condition to any simple cases its       
meaning is readily stated4 ,it implies  
that the mean in question will be       
different according to the nationality  
of the persons under measurement4       
,according to ,quetelet1#ad in the case 
of ,englishmen the mean is about #e ft4 
#i in42 for ,belgians about #e ft4 #g   
in42 for the ,french about #e ft4 #d in4

 ,it need hardly be added that these    
measures are those of adult males4  #ae4
 ,it may fairly be asked here what would
 have been the consequence1 had we1     
instead of keeping the ,english and the 
,french apart1 mixed the results of our 
measurements of them all together8 ,the 
question is an important one1 as it will
 oblige us to understand more clearly   
what we mean by homogeneous classes4    
,the answer that would usually be given 
to it1 though substantially correct1 is 
somewhat too decisive and summary4 ,it  
would be said that we are here mixing   
distinctly heterogeneous elements1 and  
that in consequence the resultant law of
 error will be by no means of the simple
 character previously exhibited4 ,so far
 as such an answer is to be admitted its
 grounds are easy to appreciate4 ,in    
accordance with the usual law of error  
the divergences from the mean grow      
continuously less numerous as they      
increase in amount4 ,now1 if we mix up  
the ,french and ,english heights1 what  

will follow8 ,beginning from the        
,english mean of #e feet #i inches1 the 
heights will at first follow almost     
entirely the law determined by these    
,english conditions1 for at this point  
the ,english data are very numerous1 and
 the ,french by comparison very few4    
,but1 as we begin to approach the       
,french mean1 the numbers will cease to 
show that continual diminution which    
they should show1 according to the      
,english scale of arrangement1 for here 
the ,french data are in turn very       
numerous1 and the ,english by comparison
 few4 ,the result of such a combination 
of heterogeneous elements is illustrated
 by the figure annexed1 of course in a  
very exaggerated form4  ,figure3        
,superposition of two ,gaussians4  #af4 
,in the above case the nature of the    
heterogeneity1 and the reasons why the  
statistics should be so collected and   
arranged as to avoid it1 seemed         
tolerably obvious4 ,it will be seen     
still more plainly if we take a parallel

 case drawn from artificial proceedings4
 ,suppose that after a man had fired a  
few thousand shots at a certain spot1   
say a wafer fixed somewhere on a wall1  
the position of the spot at which he    
aims were shifted1 and he fired a few   
thousand more shots at the wafer in its 
new position4 ,now let us collect and   
arrange all the shots of both series in 
the order of their departure from either
 of the centres1 say the new one4 ,here 
we should really be mingling together   
two discordant sets of elements1 either 
of which1 if kept apart from the other1 
would have been of a simple and         
homogeneous character4 ,we should find1 
in consequence1 that the resultant law  
of error betrayed its composite or      
heterogeneous origin by a glaring       
departure from the customary form1      
somewhat after the fashion indicated in 
the above diagram4  ,the instance of the
 ,english and ,french heights resembles 
the one just given1 but falls far short 
of it in the stringency with which the  

requisite conditions are secured4 ,the  
fact is we have not here got the most   
suitable requirements1 viz4 a group     
consisting of a few fixed causes        
supplemented by innumerable little      
disturbing influences4 ,what we call a  
nation is really a highly artificial    
body1 the members of which are subject  
to a considerable number of local or    
occasional disturbing causes4 ,amongst  
,frenchmen were included1 presumably1   
,bretons1 ,provenals1 ,alsatians1 and so
 on1 thus commingling distinctions      
which1 though less than those between   
,french and ,english1 regarded as       
wholes1 are very far from being         
insignificant4 ,and to these differences
 of race must be added other            
disturbances1 also highly important1    
dependent upon varying climate1 food and
 occupation4 ,it is plain1 therefore1   
that whatever objections exist against  
confusing together ,french and ,english 
statistics1 exist also1 though of course
 in a less degree1 against confusing    

together those of the various provincial
 and other components which make up the 
,french people4  #ag4 ,out of the great 
variety of important causes which       
influence the height of men1 it is      
probable that those which most nearly   
fulfil the main conditions required by  
the ',law of ,error' are those about    
which we know the least4 ,upon the      
effects of food and employment1         
observation has something to say1 but   
upon the purely physiological causes by 
which the height of the parents         
influences the height of the offspring1 
we have probably nothing which deserves 
to be called knowledge4 ,perhaps the    
best supposition we can make is one     
which1 in accordance with the saying    
that 'like breeds like'1 would assume   
that the purely physiological causes    
represent the constant element2 that is1
 given a homogeneous race of people to  
begin with1 who freely intermarry1 and  
are subject to like circumstances of    
climate1 food1 and occupation1 the      

standard would remain on the whole      
constant4#ae  ,in such a case the man   
who possessed the mean height1 mean     
weight1 mean strength1 and so on1 might 
then be called1 in a sort of way1 a     
'type'4 ,the deviations from this type  
would then be produced by innumerable   
small influences1 partly physiological1 
partly physical and social1 acting for  
the most part independently of one      
another1 and resulting in a ,law of     
,error of the usual description4 ,under 
such restrictions and explanations as   
these1 there seems to be no reasonable  
objection to speaking of a ,french or   
,english type or mean4 ,but it must     
always be remembered that under the     
present circumstances of every political
 nation1 these somewhat heterogeneous   
bodies might be subdivided into various 
smaller groups1 each of which would     
frequently exhibit the characteristics  
of such a type in an even more marked   
degree4  #ah4 ,on this point the reports
 of the ,anthropometrical ,committee1   

already referred to1 are most           
instructive4 ,they illustrate the extent
 to which this subdivision could be     
carried out1 and prove1--if any proof   
were necessary1--that the discovery of  
,quetelet's homme moyen would lead us a 
long chase4 ,so far as their results go 
the mean ',english' stature 7in inches7 
is #fg4ff4 ,but this is composed of     
,scotch1 ,irish1 ,english and ,welsh    
constituents1 the separate means of     
these being1 respectively2 #fh4ga1      
#fg4ij1 #fg4cf1 and #ff4ff4 ,but these  
again may be subdivided2 for careful    
observation shows that the mean ,english
 stature is distinctly greater in       
certain districts 7e4g4 the             
,north-,eastern counties7 than in       
others4 ,then again the mean of the     
professional classes is considerably    
greater than that of the labourers2 and 
that of the honest and intelligent is   
very much greater than that of the      
criminal and lunatic constituents of the
 population4 ,and1 so far as the        

observations are extensive enough for   
the purpose1 it appears that every      
characteristic in respect of the        
grouping about a mean which can be      
detected in the more extensive of these 
classes can be detected also in the     
narrower4 ,nor is there any reason to   
suppose that the same process of        
subdivision could not be carried out as 
much farther as we chose to prolong it4 
#ai4 ,it need hardly be added to the    
above remarks that no one who gives the 
slightest adhesion to the ,doctrine of  
,evolution could regard the type1 in the
 above qualified sense of the term1 as  
possessing any real permanence and      
fixity4 ,if the constant causes1        
whatever they may be1 remain unchanged1 
and if the variable ones continue in the
 long run to balance one another1 the   
results will continue to cluster about  
the same mean4 ,but if the constant ones
 undergo a gradual change1 or if the    
variable ones1 instead of balancing each
 other suffer one or more of their      

number to begin to acquire a            
preponderating influence1 so as to put a
 sort of bias upon their aggregate      
effect1 the mean will at once begin1 so 
to say1 to shift its ground4 ,and having
 once begun to shift1 it may continue to
 do so1 to whatever extent we recognize 
that ,species are variable and          
,development is a fact4 ,it is as if the
 point on the target at which we aim1   
instead of being fixed1 were slowly     
changing its position as we continue to 
fire at it2 changing almost certainly to
 some extent and temporarily1 and not   
improbably to a considerable extent and 
permanently4  #bj4 ,our examples        
throughout this chapter have been almost
 exclusively drawn from physical        
characteristics1 whether of man or of   
inanimate things2 but it need not be    
supposed that we are necessarily        
confined to such instances4 ,mr ,galton1
 for instance1 has proposed to extend   
the same principles of calculation to   
mental phenomena1 with a view to their  

more accurate determination4 ,the       
objects to be gained by so doing belong 
rather to the inferential part of our   
subject1 and will be better indicated   
further on2 but they do not involve any 
distinct principle4 ,like other attempts
 to apply the methods of science in the 
region of the mind1 this proposal has   
met with some opposition2 with very     
slight reason1 as it seems to me4 ,that 
our mental qualities1 if they could be  
submitted to accurate measurement1 would
 be found to follow the usual ,law of   
,error1 may be assumed without much     
hesitation4 ,the known extent of the    
correlation of mental and bodily        
characteristics gives high probability  
to the supposition that what is proved  
to prevail1 at any rate approximately1  
amongst most bodily elements which have 
been submitted to measurement1 will     
prevail also amongst the mental         
elements4  ,to what extent such         
measurements could be carried out       
practically1 is another matter4 ,it does

 not seem to me that it could be done   
with much success2 partly because our   
mental qualities are so closely         
connected with1 indeed so run into one  
another1 that it is impossible to       
isolate them for purposes of            
comparison4#af ,this is to some extent  
indeed a difficulty in bodily           
measurements1 but it is far more so in  
those of the mind1 where we can hardly  
get beyond what can be called a good    
guess4 ,the doctrine1 therefore1 that   
mental qualities follow the now familiar
 law of arrangement can scarcely be     
grounded upon anything more than a      
strong analogy4 ,still this analogy is  
quite strong enough to justify us in    
accepting the doctrine and all the      
conclusions which follow from it1 in so 
far as our estimates and measurements   
can be regarded as trustworthy4 ,there  
seems therefore nothing unreasonable in 
the attempt to establish a system of    
natural classification of mankind by    
arranging them into a certain number of 

groups above and below the average1 each
 group being intended to correspond to  
certain limits of excellency or         
deficiency4#ag ,all that is necessary   
for such a purpose is that the rate of  
departure from the mean should be       
tolerably constant under widely         
different circumstances3 in this case   
throughout all the races of man4 ,of    
course if the law of divergence is the  
same as that which prevails in inanimate
 nature we have a still wider and more  
natural system of classification at     
hand1 and one which ought to be         
familiar1 more or less1 to every one who
 has thus to estimate qualities4  #ba4  
,perhaps one of the best illustrations  
of the legitimate application of such   
principles is to be found in ,mr        
,galton's work on ,hereditary ,genius4  
,indeed the full force and purport of   
some of his reasonings there can hardly 
be appreciated except by those who are  
familiar with the conceptions which we  
have been discussing in this chapter4   

,we can only afford space to notice one 
or two points1 but the student will find
 in the perusal1 of at any rate the more
 argumentive parts1 of that volume#ah an
 interesting illustration of the        
doctrines now under discussion4 ,for one
 thing it may be safely asserted1 that  
no one unfamiliar with the ,law of      
,error would ever in the least          
appreciate the excessive rapidity with  
which the superior degrees of excellence
 tend to become scarce4 ,every one1 of  
course1 can see at once1 in a numerical 
way at least1 what is involved in being 
'one of a million'2 but they would not  
at all understand1 how very little extra
 superiority is to be looked for in the 
man who is 'one of two million'4 ,they  
would confound the mere numerical       
distinction1 which seems in some way to 
imply double excellence1 with the       
intrinsic superiority1 which would      
mostly be represented by a very small   
fractional advantage4 ,to be 'one of ten
 million' sounds very grand1 but if the 

qualities under consideration could be  
estimated in themselves without the     
knowledge of the vastly wider area from 
which the selection had been made1 and  
in freedom therefore from any consequent
 numerical bias1 people would be        
surprised to find what a very slight    
comparative superiority was1 as a rule1 
thus obtained4  #bb4 ,the point just    
mentioned is an important one in        
arguments from statistics4 ,if1 for     
instance1 we find a small group of      
persons1 connected together by          
blood-relationship1 and all possessing  
some mental characteristic in marked    
superiority1 much depends upon the      
comparative rarity of such excellence   
when we are endeavouring to decide      
whether or not the common possession of 
these qualities was accidental4 ,such a 
decision can never be more than a rough 
one1 but if it is to be made at all this
 consideration must enter as a factor4  
,again1 when we are comparing one nation
 with another1#ai say the ,athenian with

 any modern ,european people1 does the  
popular mind at all appreciate what sort
 of evidence of general superiority is  
implied by the production1 out of one   
nation1 of such a group as can be       
composed of ,socrates1 ,plato1 and a few
 of their contemporaries8 ,in this      
latter case we are also1 it should be   
remarked1 employing the ',law of ,error'
 in a second way2 for we are assuming   
that where the extremes are great so    
will also the means be1 in other words  
we are assuming that every amount of    
departure from the mean occurs with a   
7roughly7 calculable degree of relative 
frequency4 ,however generally this truth
 may be accepted in a vague way1 its    
evidence can only be appreciated by     
those who know the reasons which can be 
given in its favour4  ,but the same     
principles will also supply a caution in
 the case of the last example4 ,they    
remind us that1 for the mere purpose of 
comparison1 the average man of any group
 or class is a much better object for   

selection than the eminent one4 ,there  
may be greater difficulties in the way  
of detecting him1 but when we have done 
so we have got possession of a securer  
and more stable basis of comparison4 ,he
 is selected1 by the nature of the case1
 from the most numerous stratum of his  
society2 the eminent man from a thinly  
occupied stratum4 ,in accordance        
therefore with the now familiar laws of 
averages and of large numbers the       
fluctuations amongst the former will    
generally be very few and small in      
comparison with those amongst the       
latter4   #a4 ,essai de ,physique       
,sociale1 #ahfi4 ,anthropomtrie1 #ahgj4 
#b4 ,as regards later statistics on the 
same subject the reader can   refer to  
the ,reports of the ,anthropometrical   
,committee of the   ,british            
,association 7#ahgi1 #ahhj1 #ahha1      
#ahhc2--;especially this   last74 ,these
 reports seem to me to represent a great
 advance on the   results obtained by   
,quetelet1 and fully to justify the     

claim of the   ,secretary 7,mr ,c4      
,roberts7 that their statistics are     
unique in range   and numbers4 ,they    
embrace not merely military             
recruits--like most   of the previous   
tables--but almost every class and age1 
and both   sexes4 ,moreover they refer  
not only to stature but to a number of  
other physical characteristics4  #c4 ,as
 every mathematician knows1 the relative
 numbers of each of these   possible    
throws are given by the successive terms
 of the expansion   of 7#a  #a7aj1 viz4 
#a1 #aj1 #de1 #abj1 #baj1 #beb1 #baj1   
#abj1 #de1 #aj1 #a4  #d4 ,that is they  
will be more densely aggregated4 ,if a  
space the size   of the bull's-eye be   
examined in each successive circle1 the 
number   of shot marks which it contains
 will be successively less4 ,the        
actual number of shots which strike the 
bull's-eye will not be the   greatest1  
since it covers so much less surface    
than any of the other   circles4  #e4   
,commonly called the exponential law2   

its equation being of the form   y      
,ae-hx#b4 ,the curve corresponding to it
 cuts the axis of y   at right angles   
7expressing the fact that near the mean 
there are a   large number of values    
approximately equal72 after a time it   
begins   to slope away rapidly towards  
the axis of x 7expressing the fact      
that the results soon begin to grow less
 common as we recede from   the mean72  
and the axis of x is an asymptote in    
both directions   7expressing the fact  
that no magnitude1 however remote from  
the   mean1 is strictly impossible2 that
 is1 every deviation1 however           
excessive1 will have to be encountered  
at length within the range of   a       
sufficiently long experience74 ,the     
curve is obviously symmetrical1         
expressing the fact that equal          
deviations from the mean1 in excess     
and in defect1 tend to occur equally    
often in the long run4    ,figure3      
,gaussian normal distribution4    ,a    
rough graphic representation of the     

curve is given above4 ,for the   benefit
 of those unfamiliar with mathematics   
one or two brief   remarks may be here  
appended concerning some of its         
properties4   7#a7 ,it must not be      
supposed that all specimens of the curve
 are   similar to one another4 ,the     
dotted lines are equally specimens of   
it4 ,in fact1 by varying the essentially
 arbitrary units in which x   and y are 
respectively estimated1 we may make the 
portion towards   the vertex of the     
curve as obtuse or as acute as we       
please4 ,this   consideration is of     
importance2 for it reminds us that1 by  
varying   one of these arbitrary units1 
we could get an 'exponential curve'     
which should tolerably closely resemble 
any symmetrical curve of   error1       
provided that this latter recognized and
 was founded upon the   assumption that 
extreme divergences were excessively    
rare4 ,hence it   would be difficult1 by
 mere observation1 to prove that the law
 of   error in any given case was not   

exponential2 unless the statistics      
were very extensive1 or the actual      
results departed considerably   from the
 exponential form4 7#b7 ,it is quite    
impossible by any graphic               
representation to give an adequate idea 
of the excessive rapidity   with which  
the curve after a time approaches the   
axis of x4 ,at the   point ,r1 on our   
scale1 the curve would approach within  
the   fifteen-thousandth part of an inch
 from the axis of x1 a distance   which 
only a very good microscope could       
detect4 ,whereas in the   hyperbola1    
e4g4 the rate of approach of the curve  
to its asymptote   is continually       
decreasing1 it is here just the reverse2
 this rate is   continually increasing4 
,hence the two1 viz4 the curve and the  
axis   of x1 appear to the eye1 after a 
very short time1 to merge into one      
another4  #f4 ,as by ,quetelet3 noted1  
amongst others1 by ,herschel1 ,essays1  
page   #dji4  #g4 ,proc4 ,r4 ,soc4 ,oct4
 #ba1 #ahgi4  #h4 ,we are here          

considering1 remember1 the case of a    
finite amount of   statistics2 so that  
there are actual limits at each end4    
#i4 ,it must be admitted that experience
 has not yet 7,i believe7 shown   this  
asymmetry in respect of heights4  #aj4  
,the above reasoning will probably be   
accepted as valid at this   stage of    
enquiry4 ,but in strictness1 assumptions
 are made here1   which however         
justifiable they may be in themselves1  
involve   somewhat of an anticipation4  
,they demand1 and in a future chapter   
will receive1 closer scrutiny and       
criticism4  #aa4 ,a definite numerical  
example of this kind of concentration of
   frequency about the mean was given in
 the note to #d4 ,it was of a   binomial
 form1 consisting of the successive     
terms of the expansion   of 7#a  #a7m4  
,now it may be shown 7,quetelet1        
,letters1 p4 #bfc2   ,liagre1 ,calcul   
des ,probabilits1 #cd7 that the         
expansion of such   a binomial1 as m    
becomes indefinitely great1 approaches  

as its limit   the exponential form2    
that is1 if we take a number of         
equidistant   ordinates proportional    
respectively to #a1 m1 m7m - #a7_/7ab7  
c41 and   connect their vertices1 the   
figure we obtain approximately          
represents some form of the curve y     
,ae-hx#b1 and tends to   become         
identical with it1 as m is increased    
without limit4 ,in other   words1 if we 
suppose the errors to be produced by a  
limited number   of finite1 equal and   
independent causes1 we have an          
approximation to   the exponential ,law 
of ,error1 which merges into identity as
 the   causes are increased in number   
and diminished in magnitude without     
limit4 ,jevons has given 7,principles of
 ,science1 p4 #cha7 a diagram   drawn to
 scale1 to show how rapid this          
approximation is4 ,one point   must be  
carefully remembered here1 as it is     
frequently overlooked   7by ,quetelet1  
for instance74 ,the coefficients of a   
binomial of two   equal terms--as 7#a   

#a7m1 in the preceding paragraph--are   
symmetrical in their arrangement from   
the first1 and very speedily   become   
indistinguishable in 7graphical7 outline
 from the final   exponential form4 ,but
 if1 on the other hand1 we were to      
consider the   successive terms of such 
a binomial as 7#a  #d7m 7which are      
proportional to the relative chances of 
#j1 #a1 #b1 #c1 444 failures in   m     
ventures1 of an event which has one     
chance in its favour to four   against  
it7 we should have an unsymmetrical     
succession4 ,if however   we suppose m  
to increase without limit1 as in the    
former   supposition1 the unsymmetry    
gradually disappears and we tend towards
   precisely the same exponential form  
as if we had begun with two   equal     
terms4 ,the only difference is that the 
position of the vertex   of the curve is
 no longer in the centre3 in other      
words1 the   likeliest term or event is 
not an equal number of successes and    
failures but successes and failures in  

the ratio of #a to #d4  #ab4 ',law of   
,error' is the usual technical term for 
what has been   elsewhere spoken of     
above as a ,law of ,divergence from a   
mean4 ,it is   in strictness only       
appropriate in the case of one1 namely  
the third1   of the three classes of    
phenomena mentioned in #d1 but by a     
convenient generalization it is equally 
applied to the other two2 so   that we  
term the amount of the divergence from  
the mean an 'error'   in every case1    
however it may have been brought about4 
#ac4 ,this however seems to be the      
purport1 either by direct assertion   or
 by implication1 of two elaborate works 
by ,quetelet1 viz4 his   ,physique      
,sociale and his ,anthropomtrie4  #ad4  
,he scarcely1 however1 professes to give
 these as an accurate   measure of the  
mean height1 nor does he always give    
precisely the   same measure4           
,practically1 none but soldiers being   
measured in any   great numbers1 the    
,english stature did not afford accurate

 data on   any large scale4 ,the        
statistics given a few pages further on 
are   probably far more trustworthy4    
#ae4 ,this statement will receive some  
explanation and correction in the   next
 chapter4  #af4 ,i am not speaking here 
of the now familiar results of          
,psychophysics1 which are mainly        
occupied with the measurement of        
perceptions and other simple states of  
consciousness4  #ag4 ,perhaps the best  
brief account of ,mr ,galton's method is
 to be   found in a paper in ,mind      
7,july1 #ahhj7 on the statistics of     
,mental   ,imagery4 ,the subject under  
comparison here--viz4 the relative      
power1 possessed by different persons1  
of raising clear visual   images of     
objects no longer present to us--is one 
which it seems   impossible to          
'measure'1 in the ordinary sense of the 
term4 ,but by   arranging all the       
answers in the order in which the       
faculty in   question seems to be       
possessed we can1 with some approach to 

accuracy1 select the middlemost person  
in the row and use him as a   basis of  
comparison with the corresponding person
 in any other   batch4 ,and similarly   
with those who occupy other relative    
positions   than that of the middlemost4
  #ah4 ,i refer to the introductory and 
concluding chapters3 the bulk of   the  
book is1 from the nature of the case1   
mainly occupied with   statistical and  
biographical details4  #ai4 ,see        
,galton's ,hereditary ,genius1 pp4      
#ccf-cej1 ,on the comparative   worth of
 different races4     ,,chapter ,,iii4  
,,on ,,the ,,causal ,,process ,,by      
,,which ,,the ,,groups ,,or ,,series    
,,of ,,probability ,,are ,,brought      
,,about4  #a4 ,in discussing the        
question whether all the various groups 
and series with which ,probability is   
concerned are of precisely one and the  
same type1 we made some examination of  
the process by which they are naturally 
produced1 but we must now enter a little
 more into the details of this process4 

,all events are the results of numerous 
and complicated antecedents1 far too    
numerous and complicated in fact for it 
to be possible for us to determine or   
take them all into account4 ,now1 though
 it is strictly true that we can never  
determine them all1 there is a broad    
distinction between the case of         
,induction1 in which we can make out    
enough of them1 and with sufficient     
accuracy1 to satisfy a reasonable       
certainty1 and ,probability1 in which we
 cannot do so4 ,to ,induction we shall  
return in a future chapter1 and         
therefore no more need be said about it 
here4  ,we shall find it convenient to  
begin with a division which1 though not 
pretending to any philosophical         
accuracy1 will serve as a preliminary   
guide4 ,it is the simple division into  
objects1 and the agencies which affect  
them4 ,all the phenomena with which     
,probability is concerned 7as indeed    
most of those with which science of any 
kind is concerned7 are the product of   

certain objects natural and artificial1 
acting under the influence of certain   
agencies natural and artificial4 ,in the
 tossing of a penny1 for instance1 the  
objects would be the penny or pence     
which were successively thrown2 the     
agencies would be the act of throwing1  
and everything which combined directly  
or indirectly with this to make any     
particular face come uppermost4 ,this is
 a simple and intelligible division1 and
 can easily be so extended in meaning as
 to embrace every class of objects with 
which we are concerned4  ,now if1 in any
 two or more cases1 we had the same     
object1 or objects indistinguishably    
alike1 and if they were exposed to the  
influence of agencies in all respects   
precisely alike1 we should expect the   
results to be precisely similar4 ,by one
 of the applications of the familiar    
principle of the uniformity of nature we
 should be confident that exact likeness
 in the antecedents would be followed by
 exact likeness in the consequents4 ,if 

the same penny1 or similar pence1 were  
thrown in exactly the same way1 we      
should invariably find that the same    
face falls uppermost4  #b4 ,what we     
actually find is1 of course1 very far   
removed from this4 ,in the case of the  
objects1 when they are artificial       
constructions1 e4g4 dice1 pence1 cards1 
it is true that they are purposely made 
as nearly as possible indistinguishably 
alike4 ,we either use the same thing    
over and over again or different ones   
made according to precisely the same    
model4 ,but in natural objects nothing  
of the sort prevails4 ,in fact when we  
come to examine them1 we find reproduced
 in them precisely the same             
characteristics as those which present  
themselves in the final result which we 
were asked to explain1 so that unless we
 examine them a stage further back1 as  
we shall have to do to some extent at   
any rate1 we seem to be merely          
postulating again the very peculiarity  
of the phenomena which we were          

undertaking to explain4 ,they will be   
found1 for instance1 to consist of large
 classes of objects1 throughout all the 
individual members of which a general   
resemblance extends4 ,suppose that we   
were considering the length of life4    
,the objects here are the human beings1 
or that selected class of them1 whose   
lives we are considering4 ,the          
resemblance existing among them is to be
 found in the strength and soundness of 
their principal vital organs1 together  
with all the circumstances which        
collectively make up what we call the   
goodness of their constitutions4 ,it is 
true that most of these circumstances do
 not admit of any approach to actual    
measurement2 but1 as was pointed out in 
the last chapter1 very many of the      
circumstances which do admit of such    
measurement have been measured1 and     
found to display the characteristics in 
question4 ,hence1 from the known analogy
 and correlation between our various    
organs1 there can be no reasonable doubt

 that if we could arrange human         
constitutions in general1 or the various
 elements which compose them in         
particular1 in the order of their       
strength1 we should find just such an   
aggregate regularity and just such      
groupings about the mean1 as the final  
result 7viz4 in this case the length of 
their lives7 presents to our notice4    
#c4 ,it will be observed therefore that 
for this purpose the existence of       
natural kinds or groups is necessary4   
,in our games of chance of course the   
same die may be thrown1 or a card be    
drawn from the same pack1 as often as we
 please2 but many of the events which   
occur to human beings either cannot be  
repeated at all1 or not often enough to 
secure in the case of the single        
individual any sufficient statistical   
uniformity4 ,such regularity as we trace
 in nature is owing1 much more than is  
often suspected1 to the arrangement of  
things in natural kinds1 each of them   
containing a large number of            

individuals4 ,were each kind of animals 
or vegetables limited to a single pair1 
or even to but a few pairs1 there would 
not be much scope left for the          
collection of statistical tables amongst
 them4 ,or to take a less violent       
supposition1 if the numbers in each     
natural class of objects were much      
smaller than they are at present1 or the
 differences between their varieties and
 sub-species much more marked1 the      
consequent difficulty of extracting from
 them any sufficient length of          
statistical tables1 though not fatal1   
might be very serious4 ,a large number  
of objects in the class1 together with  
that general similarity which entitles  
the objects to be fairly comprised in   
one class1 seem to be important         
conditions for the applicability of the 
theory of ,probability to any           
phenomenon4 ,something analogous to this
 excessive paucity of objects in a class
 would be found in the attempt to apply 
special ,insurance offices to the case  

of those trades where the numbers are   
very limited1 and the employment so     
dangerous as to put them in a class by  
themselves4 ,if an insurance society    
were started for the workmen in         
gunpowder mills alone1 a premium would  
have to be charged to avoid possible    
ruin1 so high as to illustrate the      
extreme paucity of appropriate          
statistics4  #d4 ,so much 7at present7  
for the objects4 ,if we turn to what we 
have termed the agencies1 we find much  
the same thing again here4 ,by the      
adjustment of their relative intensity1 
and the respective frequency of their   
occurrence1 the total effects which they
 produce are found to be also tolerably 
uniform4 ,it is of course conceivable   
that this should have been otherwise4   
,it might have been found that the      
second group of conditions so exactly   
corrected the former as to convert the  
merely general uniformity into an       
absolute one2 or it might have been     
found1 on the other hand1 that the      

second group should aggravate or disturb
 the influence of the former to such an 
extent as to destroy all the uniformity 
of its effects4 ,practically neither is 
the case4 ,the second condition simply  
varies the details1 leaving the         
uniformity on the whole of precisely the
 same general description as it was     
before4 ,or if the objects were supposed
 to be absolutely alike1 as in the case 
of successive throws of a penny1 it may 
serve to bring about a uniformity4      
,analysis will show these agencies to be
 thus made up of an almost infinite     
number of different components1 but it  
will detect the same peculiarity that we
 have so often had occasion to refer to1
 pervading almost all these components4 
,the proportions in which they are      
combined will be found to be nearly1    
though not quite1 the same2 the         
intensity with which they act will be   
nearly though not quite equal4 ,and they
 will all unite and blend into a more   
and more perfect regularity as we       

proceed to take the average of a larger 
number of instances4  ,take1 for        
instance1 the length of life4 ,as we    
have seen1 the constitutions of a very  
large number of persons selected at     
random will be found to present much the
 same feature2 general uniformity       
accompanied by individual irregularity4 
,now when these persons go out into the 
world1 they are exposed to a variety of 
agencies1 the collective influence of   
which will assign to each the length of 
life allotted to him4 ,these agencies   
are of course innumerable1 and their    
mutual interaction complicated beyond   
all power of analysis to extricate4     
,each effect becomes in its turn a      
cause1 is interwoven inextricably with  
an indefinite number of other causes1   
and reacts upon the final result4       
,climate1 food1 clothing1 are some of   
these agencies1 or rather comprise      
aggregate groups of them4 ,the nature of
 a man's work is also important4 ,one   
man overworks himself1 another follows  

an unhealthy trade1 a third exposes     
himself to infection1 and so on4  ,the  
result of all this interaction between  
what we have thus called objects and    
agencies is that the final outcome      
presents the same general               
characteristics of uniformity as may be 
detected separately in the two          
constituent elements4 ,or rather1 as we 
shall proceed presently to show1 it does
 so in the great majority of cases4  #e4
 ,it may be objected that such an       
explanation as the above does not really
 amount to anything deserving of the    
name1 for that instead of explaining how
 a particular state of things is caused 
it merely points out that the same state
 exists elsewhere4 ,there is a          
uniformity discovered in the objects at 
the stage when they are commonly        
submitted to calculation2 we then grope 
about amongst the causes of them1 and   
after all only discover a precisely     
similar uniformity existing amongst     
these causes4 ,this is to some extent   

true1 for though part of the objection  
can be removed1 it must always remain   
the case that the foundations of an     
objective science will rest in the last 
resort upon the mere fact that things   
are found to be of such and such a      
character4  #f4 ,this division1 into    
objects and the agencies which affect   
them1 is merely intended for a rough    
practical arrangement1 sufficient to    
point out to the reader the immediate   
nature of the causes which bring about  
our familiar uniformities4 ,if we go    
back a step further1 it might fairly be 
maintained that they may be reduced to  
one1 namely1 to the agencies4 ,the      
objects1 as we have termed them1 are not
 an original creation in the state in   
which we now find them4 ,no one supposes
 that whole groups or classes were      
brought into existence simultaneously1  
with all their general resemblances and 
particular differences fully developed4 
,even if it were the case that the first
 parents of each natural kind had been  

specially created1 instead of being     
developed out of pre-existing forms1 it 
would still be true that amongst the    
numbers of each that now present        
themselves the characteristic           
differences and resemblances are the    
result of what we have termed agencies4 
,take1 for instance1 a single           
characteristic only1 say the height2    
what determines this as we find it in   
any given group of men8 ,partly1 no     
doubt1 the nature of their own food1    
clothing1 employment1 and so on1        
especially in the earliest years of     
their life2 partly also1 very likely1   
similar conditions and circumstances on 
the part of their parents at one time or
 another4 ,no one1 ,i presume1 in the   
present state of knowledge1 would       
attempt to enumerate the remaining      
causes1 or even to give any indication  
of their exact nature2 but at the same  
time few would entertain any doubt that 
agencies of this general description    
have been the determining causes at     

work4  ,if it be asked again1 ,into what
 may these agencies themselves be       
ultimately analysed8 the answer to this 
question1 in so far as it involves any  
detailed examination of them1 would be  
foreign to the plan of this essay4 ,in  
so far as any general remarks1          
applicable to nearly all classes alike  
of such agencies1 are called for1 we are
 led back to the point from which we    
started in the previous chapter1 when we
 were discussing whether there is       
necessarily one fixed law according to  
which all our series are formed4 ,we    
there saw that every event might be     
regarded as being brought about by a    
comparatively few important causes1 of  
the kind which comprises all of which   
ordinary observation takes any notice1  
and an indefinitely numerous group of   
small causes1 too numerous1 minute1 and 
uncertain in their action for us to be  
able to estimate them or indeed to take 
them individually into account at all4  
,the important ones1 it is true1 may    

also in turn be themselves conceived to 
be made up of aggregates of small       
components1 but they are still best     
regarded as being by comparison simple  
and distinct1 for their component parts 
act mostly in groups collectively1      
appearing and disappearing together1 so 
that they possess the essential         
characteristics of unity4  #g4 ,now1    
broadly speaking1 it appears to me that 
the most suitable conditions for        
,probability are these3 that the        
important causes should be by comparison
 fixed and permanent1 and that the      
remaining ones should on the average    
continue to act as often in one         
direction as in the other4 ,this they   
may do in two ways4 ,in the first place 
we may be able to predicate nothing more
 of them than the mere fact that they   
act#a as often in one direction as the  
other2 what we should then obtain would 
be merely the simple statistical        
uniformity that is described in the     
first chapter4 ,but it may be the case1 

and in practice generally is so more or 
less approximately1 that these minor    
causes act also in independence of one  
another4 ,what we then get is a group of
 uniformities such as was explained and 
illustrated in the second chapter4      
,every possible combination of these    
causes then occurring with a regular    
degree of frequency1 we find one        
peculiar kind of uniformity exhibited1  
not merely in the mere fact of excess   
and defect 7of whatever may be the      
variable quality in question71 but also 
in every particular amount of excess and
 defect4 ,hence1 in this case1 we get   
what some writers term a 'mean' or      
'type1' instead of a simple average4    
,for instance1 suppose a man throwing a 
quoit at a mark4 ,here our fixed causes 
are his strength1 the weight of the     
quoit1 and the intention of aiming at a 
given point4 ,these we must of course   
suppose to remain unchanged1 if we are  
to obtain any such uniformity as we are 
seeking4 ,the minor and variable causes 

are all those innumerable little        
disturbing influences referred to in the
 last chapter4 ,it might conceivably be 
the case that we were only able to      
ascertain that these acted as often in  
one direction as in the other2 what we  
should then find was that the quoit     
tended to fall short of the mark as     
often as beyond it4 ,but owing to these 
little causes being mostly independent  
of one another1 and more or less equal  
in their influence1 we find also that   
every amount of excess and defect       
presents the same general               
characteristics1 and that in a large    
number of throws the quantity of        
divergences from the mark1 of any given 
amount1 is a tolerably determinate      
function1 according to a regular law1 of
 that amount of divergence4#b  #h4 ,the 
necessity of the conditions just hinted 
at will best be seen by a reference to  
cases in which any of them happen to be 
missing4 ,thus we know that the length  
of life is on the whole tolerably       

regular1 and so are the numbers of those
 who die in successive years or         
centuries of most of the commoner       
diseases4 ,but it does not seem to be   
the case with all diseases4 ,what1 for  
instance1 of the ,sweating ,sickness1   
the ,black ,death1 the ,asiatic         
,cholera8 ,the two former either do not 
recur1 or1 if they do1 recur in such a  
mild form as not to deserve the same    
name4 ,what in fact of any of the       
diseases which are epidemic rather than 
endemic8 ,all these have their causes   
doubtless1 and would be produced again  
by the recurrence of the conditions     
which caused them before4 ,but some of  
them apparently do not recur at all4    
,they seem to have depended upon such   
rare conditions that their occurrence   
was almost unique4 ,and of those which  
do recur the course is frequently so    
eccentric and irregular1 often so much  
dependent upon human will or want of    
will1 as to entirely deprive their      
results 7that is1 the annual number of  

deaths which they cause7 of the         
statistical uniformity of which we are  
speaking4  ,the explanation probably is 
that one of the principal causes in such
 cases is what we commonly call         
contagion4 ,if so1 we have at once a    
cause which so far from being fixed is  
subject to the utmost variability4      
,stringent caution may destroy it1      
carelessness may aggravate it to any    
extent4 ,the will of man1 as finding its
 expression either on the part of       
government1 of doctors1 or of the       
public1 may make of it pretty nearly    
what is wished1 though against the      
possibility of its entrance into any    
community no precautions can absolutely 
insure us4  #i4 ,if it be replied that  
this want of statistical regularity only
 arises from the fact of our having     
confined ourselves to too limited a     
time1 and that we should find           
irregularity disappear here1 as         
elsewhere1 if we kept our tables open   
long enough1 we shall find that the     

answer will suggest another case in     
which the requisite conditions for      
,probability are wanting4 ,such a reply 
would only be conclusive upon the       
supposition that the ways and thoughts  
of men are in the long run invariable1  
or if variable1 subject to periodic     
changes only4 ,on the assumption of a   
steady progress in society1 either for  
the better or the worse1 the argument   
falls to the ground at once4 ,from what 
we know of the course of the world1     
these fearful pests of the past may be  
considered as solitary events in our    
history1 or at least events which will  
not be repeated4 ,no continued          
uniformity would therefore be found in  
the deaths which they occasion1 though  
the registrar's books were kept open for
 a thousand years4 ,the reason here is  
probably to be sought in the gradual    
alteration of those indefinitely        
numerous conditions which we term       
collectively progress or civilization4  
,every little circumstance of this kind 

has some bearing upon the liability of  
any one to catch a disease4 ,but when a 
kind of slow and steady tide sets in1 in
 consequence of which these influences  
no longer remain at about the same      
average strength1 warring on about equal
 terms with hostile influences1 but on  
the contrary show a steady tendency to  
increase their power1 the statistics    
will1 with consequent steadiness and    
permanence1 take the impress of such a  
change4  #aj4 ,briefly then1 if we were 
asked where the distinctive             
characteristics of ,probability are most
 prominently to be found1 and where they
 are most prominently absent1 we might  
say that 7#a7 they prevail principally  
in the properties of natural kinds1 both
 in the ultimate and in the derivative  
or accidental properties4 ,in all the   
characteristics of natural species1 in  
all they do and in all which happens to 
them1 so far as it depends upon their   
properties1 we seldom fail to detect    
this regularity4 ,thus in men2 their    

height1 strength1 weight1 the age to    
which they live1 the diseases of which  
they die2 all present a well-known      
uniformity4 ,life insurance tables offer
 the most familiar instance of the      
importance of these applications of     
,probability4  7#b7 ,the same           
peculiarity prevails again in the force 
and frequency of most natural agencies4 
,wind and weather are seen to lose their
 proverbial irregularity when examined  
on a large scale4 ,man's work therefore1
 when operated on by such agencies as   
these1 even though it had been made in  
different cases absolutely alike to     
begin with1 afterwards shows only a     
general regularity4 ,i may sow exactly  
the same amount of seed in my field     
every year4 ,the yield may one year be  
moderate1 the next year be abundant     
through favourable weather1 and then    
again in turn be destroyed by hail4 ,but
 in the long run these irregularities   
will be equalized in the result of my   
crops1 because they are equalized in the

 power and frequency of the productive  
agencies4 ,the business of underwriters1
 and offices which insure the crops     
against hail1 would fall under this     
class2 though1 as already remarked1     
there is no very profound distinction   
between them and the former class4  ,the
 reader must be reminded again that this
 fixity is only temporary1 that is1 that
 even here the series belong to the     
class of those which possess a          
fluctuating type4 ,those indeed who     
believe in the fixity of natural species
 will have the best chance of finding a 
series of the really permanent type     
amongst them1 though even they will     
admit that some change in the           
characteristic is attainable in length  
of time4 ,in the case of the principal  
natural agencies1 it is of course       
incontestable that the present average  
is referable to the present geological  
period only4 ,our average temperature   
and average rainfall have in former     
times been widely different from what   

they now are1 and doubtless will be so  
again4  ,any fuller investigation of the
 process by which1 on the ,theory of    
,evolution1 out of a primeval simplicity
 and uniformity the present variety was 
educed1 hardly belongs to the scope of  
the present work3 at most1 a few hints  
must suffice4  #aa4 ,the above1 then1   
are instances of natural objects and    
natural agencies4 ,there seems reason to
 believe that it is in such things only1
 as distinguished from things           
artificial1 that the property in        
question is to be found4 ,this is an    
assertion that will need some discussion
 and explanation4 ,two instances1 in    
apparent opposition1 will at once occur 
to the mind of some readers2 one of     
which1 from its great intrinsic         
importance1 and the other1 from the     
frequency of the problems which it      
furnishes1 will demand a few minutes'   
separate examination4  7#a7 ,the first  
of these is the already mentioned case  
of instrumental observations4 ,in the   

use of astronomical and other           
instruments the utmost possible degree  
of accuracy is often desired1 a degree  
which cannot be reasonably hoped for in 
any one single observation4 ,what we do 
therefore in these cases is to make a   
large number of successive observations 
which are naturally found to differ     
somewhat from each other in their       
results2 by means of these the true     
value 7as explained in a future chapter1
 on the ,method of ,least ,squares7 is  
to be determined as accurately as       
possible4 ,the subjects then of         
calculation here are a certain number of
 elements1 slightly incorrect elements1 
given by successive observations4 ,are  
not these observations artificial1 or   
the direct product of voluntary agency8 
,certainly not3 or rather1 the answer   
depends on what we understand by        
voluntary4 ,what is really intended and 
aimed at by the observer1 is of course1 
perfect accuracy1 that is1 the true     
observation1 or the voluntary steps and 

preliminaries on which this observation 
depends4 ,whether voluntary or not1 this
 result only can be called intentional4 
,but this result is not obtained4 ,what 
we actually get in its place is a series
 of deviations from it1 containing      
results more or less wide of the truth4 
,now by what are these deviations       
caused8 ,by just such agencies as we    
have been considering in some of the    
earlier sections in this chapter4 ,heat 
and its irregular warping influence1    
draughts of air producing their         
corresponding effects1 dust and         
consequent friction in one part or      
another1 the slight distortion of the   
instrument by strains or the slow uneven
 contraction which continues long after 
the metal was cast2 these and such as   
these are some of the causes which      
divert us from the truth4 ,besides this 
group1 there are others which certainly 
do depend upon human agency1 but which  
are not1 strictly speaking1 voluntary4  
,they are such as the irregular action  

of the muscles1 inability to make our   
various organs and members execute      
precisely the purposes we have in mind1 
perhaps different rates in the rapidity 
of the nervous currents1 or in the      
response to stimuli1 in the same or     
different observers4 ,the effect        
produced by some of these1 and the      
allowance that has in consequence to be 
made1 are becoming familiar even to the 
outside world under the name of the     
'personal equation' in astronomical1    
psychophysical1 and other observations4 
#ab4 7#b7 ,the other example1 alluded to
 above1 is the stock one of cards and   
dice4 ,here1 as in the last case1 the   
result is remotely voluntary1 in the    
sense that deliberate volition presents 
itself at one stage4 ,but subsequently  
to this stage1 the result is produced or
 affected by so many involuntary        
agencies that it owes its characteristic
 properties to these4 ,the turning up1  
for example1 of a particular face of a  
die is the result of voluntary agency1  

but it is not an immediate result4 ,that
 particular face was not chosen1 though 
the fact of its being chosen was the    
remote consequence of an act of choice4 
,there has been an intermediate chaos of
 conflicting agencies1 which no one can 
calculate before or distinguish         
afterwards4 ,these agencies seem to show
 a uniformity in the long run1 and      
thence to produce a similar uniformity  
in the result4 ,the drawing of a card   
from a pack is indeed more directly     
volitional1 as in cutting for partners  
in a game of whist4 ,but no one         
continues to do this long without having
 the pack well shuffled in the interval1
 whereby a host of involuntary          
influences are let in4  #ac4 ,the once  
startling but now familiar uniformities 
exhibited in the cases of suicides and  
misdirected letters1 do not belong to   
the same class4 ,the final resolution1  
or want of it1 which leads to these     
results1 is in each case indeed an      
important ingredient in the individual's

 action or omission2 but1 in so far as  
volition has anything to do with the    
results as a whole1 it instantly        
disturbs them4 ,if the voice of the     
,legislature speaks out1 or any great   
preacher or moralist succeeds in        
deterring1 or any impressive example in 
influencing1 our moral statistics are   
instantly tampered with4 ,some further  
discussion will be devoted to this      
subject in a future chapter2 it need    
only be remarked here that 7always      
excluding such common or general        
influence as those just mentioned7 the  
average volition1 potent as it is in    
each separate case1 is on the whole     
swayed by non-voluntary conditions1 such
 as those of health1 the casualties of  
employment1 c41 in fact the various     
circumstances which influence the length
 of a man's life4  #ad4 ,such           
distinctions as those just insisted on  
may seem to some persons to be needless1
 but serious errors have occasionally   
arisen from the neglect of them4 ,the   

immediate products of man's mind1 so far
 indeed as we can make an attempt to    
obtain them1 do not seem to possess this
 essential characteristic of            
,probability4 ,their characteristic     
seems rather to be1 either perfect      
mathematical accuracy or utter want of  
it1 either law unfailing or mere        
caprice4 ,if1 e4g41 we find the trees in
 a forest growing in straight lines1 we 
unhesitatingly conclude that they were  
planted by man as they stand4 ,it is    
true on the other hand1 that if we find 
them not regularly planted1 we cannot   
conclude that they were not planted by  
man2 partly because the planter may have
 worked without a plan1 partly because  
the subsequent irregularities brought on
 by nature may have obscured the plan4  
,practically the mind has to work by the
 aid of imperfect instruments1 and is   
subjected to many hindrances through    
various and conflicting agencies1 and by
 these means the work loses its original
 properties4 ,suppose1 for instance1    

that a man1 instead of producing        
numerical results by imperfect          
observations or by the cast of dice1    
were to select them at first hand for   
himself by simply thinking of them at   
once2 what sort of series would he      
obtain8 ,it would be about as difficult 
to obtain in this way any such series as
 those appropriate to ,probability as it
 would be to keep his heart or pulse    
working regularly by direct acts of     
volition1 supposing that he had the     
requisite control over these organs4    
,but the mere suggestion is absurd4 ,a  
man must have an object in thinking1 he 
must think according to a rule or       
formula2 but unless he takes some       
natural series as a copy1 he will never 
be able to construct one mentally which 
shall permanently imitate the originals4
 ,or take another product of human      
efforts1 in which the intention can be  
executed with tolerable success4 ,when  
any one builds a house1 there are many  
slight disturbing influences at work1   

such as shrinking of bricks and mortar1 
settling of foundations1 c4 ,but the    
effect which these disturbances are able
 to produce is so inappreciably small1  
that we may fairly consider that the    
result obtained is the direct product of
 the mind1 the accurate realization of  
its intention4 ,what is the consequence8
 ,every house in the row1 if designed by
 one man and at one time1 is of exactly 
the same height1 width1 c4 as its       
neighbours2 or if there are variations  
they are few1 definite1 and regular4    
,the result offers no resemblance       
whatever to the heights1 weights1 c4 of 
a number of men selected at random4 ,the
 builder probably had some regular      
design in contemplation1 and he has     
succeeded in executing it4  #ae4 ,it may
 be replied that if we extend our       
observations1 say to the houses of a    
large city1 we shall then detect the    
property under discussion4 ,the         
different heights of a great number1    
when grouped together1 might be found to

 resemble those of a great number of    
human beings under similar treatment4   
,something of this kind might not       
improbably be found to be the case1     
though the resemblance would be far from
 being a close one4 ,but to raise this  
question is to get on to different      
ground1 for we were speaking 7as        
remarked above7 not of the work of      
different minds with their different    
aims1 but of that of one mind4 ,in a    
multiplicity of designs1 there may be   
that variable uniformity1 for which we  
may look in vain in a single design4    
,the heights which the different        
builders contemplated might be found to 
group themselves into something of the  
same kind of uniformity as that which   
prevails in most other things which they
 should undertake to do independently4  
,we might then trace the action of the  
same two conditions1--a uniformity in   
the multitude of their different        
designs1 a uniformity also in the       
infinite variety of the influences which

 have modified those designs4 ,but this 
is a very different thing from saying   
that the work of one man will show such 
a result as this4 ,the difference is    
much like that between the tread of a   
thousand men who are stepping without   
thinking of each other1 and their tread 
when they are drilled into a regiment4  
,in the former case there is the        
working1 in one way or another1 of a    
thousand minds2 in the latter1 of one   
only4  ,the investigations of this and  
the former chapter constitute a         
sufficiently close examination into the 
detailed causes by which the peculiar   
form of statistical results with which  
we are concerned is actually produced1  
to serve the purpose of a work which is 
occupied mainly with the methods of the 
,science of ,probability4 ,the great    
importance1 however1 of certain         
statistical or sociological enquiries   
will demand a recurrence in a future    
chapter to one particular application of
 these statistics1 viz4 to those        

concerned with some classes of human    
actions4  #af4 ,the only important      
addition to1 or modification of1 the    
foregoing remarks which ,i have found   
occasion to make is due to ,mr ,galton4 
,he has recently pointed out1--and was  
,i believe the first to do so1--that in 
certain cases some analysis of the      
causal processes can be effected1 and is
 in fact absolutely necessary in order  
to account for the facts observed4      
,take1 for instance1 the heights of the 
population of any country4 ,if the      
distribution or dispersion of these     
about their mean value were left to the 
unimpeded action of those myriad        
productive agencies alluded to above1 we
 should certainly obtain such an        
arrangement in the posterity of any one 
generation as had already been exhibited
 in the parents4 ,that is1 we should    
find repeated in the previous stage the 
same kind of order as we were trying to 
account for in the following stage4     
,but then1 as ,mr ,galton insists1 if   

such agencies acted freely and          
independently1 though we should get the 
same kind of arrangement or             
distribution1 we should not get the same
 degree of it3 there would1 on the      
contrary1 be a tendency towards further 
dispersion4 ,the 'curve of facility' 7v4
 the diagram on p4 #bi7 would belong to 
the same class1 but would have a        
different modulus4 ,we shall see this at
 once if we take for comparison a case  
in which similar agencies work their way
 without any counteraction whatever4    
,suppose1 for instance1 that a large    
number of persons1 whose fortunes were  
equal to begin with1 were to commence   
gambling or betting continually for some
 small sum4 ,if we examine their        
circumstances after successive intervals
 of time1 we should expect to find their
 fortunes distributed according to the  
same general law1--i4e4 the now familiar
 law in question1--but we should also   
expect to find that the poorest ones    
were slightly poorer1 and the richest   

ones slightly richer1 on each successive
 occasion4 ,we shall see more about this
 in a future chapter 7on ,gambling71 but
 it may be taken for granted here that  
there is nothing in the laws of chance  
to resist this tendency towards         
intensifying the extremes4  ,now it is  
found1 on the contrary1 in the case of  
vital phenomena1--for instance in that  
of height1 and presumably of most of the
 other qualities which are in any way   
characteristic of natural kinds1--that  
there is1 through a number of successive
 generations1 a remarkable degree of    
fixity4 ,the tall men are not taller1   
and the short men are not shorter1 per  
cent4 of the population in successive   
generations3 always supposing of course 
that some general change of             
circumstances1 such as climate1 diet1 c4
 has not set in4 ,there must therefore  
here be some cause at work which tends1 
so to say1 to draw in the extremes and  
thus to check the otherwise continually 
increasing dispersion4  #ag4 ,the facts 

were first tested by careful experiment4
 ,at the date of ,mr ,galton's original 
paper on the subject1#c there were no   
available statistics of heights of human
 beings2 so a physical element admitting
 of careful experiment 7viz4 the size or
 weight of certain seeds7 was accurately
 estimated4 ,from these data the actual 
amount of reversion from the extremes1  
that is1 of the slight pressure         
continually put upon the extreme members
 with the result of crowding them back  
towards the mean1 was determined1 and   
this was compared with what theory would
 require in order to keep the           
characteristics of the species          
permanently fixed4 ,since then1         
statistics have been obtained to a large
 extent which deal directly with the    
heights of human beings4  ,the general  
conclusion at which we arrive is that   
there are several causes at work which  
are neither slight nor independent4     
,there is1 for instance1 the observed   
fact that the extremes are as a rule not

 equally fertile with the means1 nor    
equally capable of resisting death and  
disease4 ,hence as regards their mere   
numbers1 there is a tendency for them   
somewhat to thin out4 ,then again there 
is a distinct positive cause in respect 
of 'reversion4' ,not only are the       
offspring of the extremes less numerous1
 but these offspring also tend to       
cluster about a mean which is1 so to    
say1 shifted a little towards the true  
centre of the whole group2 i4e4 towards 
the mean offspring of the mean parents4 
#ah4 ,for a full discussion of these    
characteristics1 and for a variety of   
most ingenious illustrations of their   
mode of agency and of their comparative 
efficacy1 the reader may be referred to 
,mr ,galton's original articles4 ,for   
our present purpose it will suffice to  
say that these characteristics tend     
towards maintaining the fixity of       
species2 and that though they do not    
affect what may be called the general   
nature of the 'probability curve' or    

'law of facility'1 they do determine its
 precise value in the cases in question4
 ,if1 indeed1 it be asked why there is  
no need for any such corrective         
influence in the case of1 say1 firing at
 a mark3 the answer is that there is no 
opening for it except where a cumulative
 influence is introduced4 ,the reason   
why the fortunes of our betting party   
showed an ever increasing divergency1   
and why some special correction was     
needed in order to avert such a tendency
 in the case of vital phenomena1 was    
that the new starting-point at every    
step was slightly determined by the     
results of the previous step4 ,the man  
who has lost a shilling one time starts1
 next time1 worse off by just a         
shilling2 and1 but for the corrections  
we have been indicating1 the man who was
 born tall would1 so to say1 throw off  
his descendants from a vantage ground of
 superior height4 ,the true parallel in 
the case of the marksmen would be to    
suppose that their new points of aim    

were always shifted a little in the     
direction of the last divergence4 ,the  
spreading out of the shot-marks would   
then continue without limit1 just as    
would the divergence of fortunes of the 
supposed gamblers4   #a4 ,as stated     
above1 this is really little more than a
 re-statement1 a   stage further back1  
of the existence of the same kind of    
uniformity   as that which we are called
 upon to explain in the concrete details
   presented to us in experience4  #b4  
,it would seem in fact that in coarse   
and rude observations the   errors      
proceed from a very few principal       
causes1 and in   consequence our        
hypothesis as to the ,exponential ,law  
of ,error will   probably represent the 
facts only imperfectly1 and the         
frequency of   the errors will only     
approximate roughly and vaguely to the  
law   which follows from it4 ,but when  
astronomers1 not content with the       
degree of accuracy they had reached1    
prosecuted their researches   into the  

remaining sources of error1 they found  
that not three or   four1 but a great   
number of minor sources of error of     
nearly   co-ordinate importance began to
 reveal themselves1 having been till    
then masked and overshadowed by the     
graver errors which had been now        
approximately removed4444 ,there were   
errors of graduation1 and many   others 
in the contraction of instruments2 other
 errors of their   adjustments2 errors  
7technically so called7 of observation2 
errors   from the changes of            
temperature1 of weather1 from slight    
irregular   motions and vibrations2 in  
short1 the thousand minute disturbing   
influences with which modern astronomers
 are familiar4 7,extracted   from a     
paper by ,mr ,crofton in the ,vol4 of   
the ,philosophical   ,transactions for  
#ahgj1 p4 #agg47  #c4 ,typical ,laws of 
,heredity2 read before the ,royal       
,institution1   ,feb4 #i1 #ahgg4 ,see   
also ,journal of the ,anthrop4 ,inst4   
,nov4 #ahhe4     ,,chapter ,,iv4   ,,on 

,,the ,,modes ,,of ,,establishing ,,and 
,,determining ,,the ,,existence ,,and   
,,numerical ,,proportions ,,of ,,the    
,,characteristic ,,properties ,,of ,,our
 ,,series ,,or ,,groups4  #a4 ,at the   
point which we have now reached1 we are 
supposed to be in possession of series  
or groups of a certain kind1 lying at   
the bottom1 as one may say1 and forming 
the foundation on which the ,science of 
,probability is to be erected4 ,we have 
described with sufficient particularity 
the characteristics of such a series1   
and have indicated the process by which 
it is1 as a rule1 actually brought about
 in nature4 ,the next enquiries which   
have to be successively made are1 how in
 any particular case we are to establish
 their existence and determine their    
special character and properties8 and   
secondly1#a when we have obtained them1 
in what mode are they to be employed for
 logical purposes8  ,the answer to the  
former enquiry does not seem difficult4 
,experience is our sole guide4 ,if we   

want to discover what is in reality a   
series of things1 not a series of our   
own conceptions1 we must appeal to the  
things themselves to obtain it1 for we  
cannot find much help elsewhere4 ,we    
cannot tell how many persons will be    
born or die in a year1 or how many      
houses will be burnt or ships wrecked1  
without actually counting them4 ,when we
 thus speak of 'experience' we mean to  
employ the term in its widest           
signification2 we mean experience       
supplemented by all the aids which      
inductive or deductive logic can afford4
 ,when1 for instance1 we have found the 
series which comprises the numbers of   
persons of any assigned class who die in
 successive years1 we have no hesitation
 in extending it some way into the      
future as well as into the past4 ,the   
justification of such a procedure must  
be sought in the ordinary canons of     
,induction4 ,as a special discussion    
will be given upon the connection       
between ,probability and ,induction1 no 

more need be said upon this subject     
here2 but nothing will be found there at
 variance with the assertion just made1 
that the series we employ are ultimately
 obtained by experience only4  #b4 ,in  
many cases it is undoubtedly true that  
we do not resort to direct experience at
 all4 ,if ,i want to know what is my    
chance of holding ten trumps in a game  
of whist1 ,i do not enquire how often   
such a thing has occurred before4 ,if   
all the inhabitants of the globe were to
 divide themselves up into whist parties
 they would have to keep on at it for a 
great many years1 if they wanted to     
settle the question satisfactorily in   
that way4 ,what we do of course is to   
calculate algebraically the proportion  
of possible combinations in which ten   
trumps can occur1 and take this as the  
answer to our problem4 ,so again1 if ,i 
wanted to know the chance of throwing   
six with a die whose faces were unequal1
 it would be a question if my best way  
would not be to calculate geometrically 

the solid angle subtended at the centre 
of gravity by the opposite face1 and the
 ratio of this to the whole surface of a
 sphere would represent sufficiently    
closely the chance required4  ,it is    
quite true that in such examples as the 
above1 especially the former one1 nobody
 would ever think of appealing to       
statistics4 ,this would be a tedious    
process to adopt when1 as here1 the     
mechanical and other conditions upon    
which the production of the events      
depend are comparatively few1           
determinate1 and admit of isolated      
consideration1 whilst the enormous      
number of combinations which can be     
constructed out of them causes an       
enormous consequent multiplicity of ways
 in which the events can possibly       
happen4 ,hence1 in practice1  priori    
determination is often easy1 whilst     
posteriori appeal to experience would be
 not merely tedious but utterly         
impracticable4 ,this1 combined with the 
frequent simplicity and attractiveness  

of such examples when deductively       
treated1 has made them very popular1 and
 produced the impression in many        
quarters that they are the proper       
typical instances to illustrate the     
theory of chance4 ,whereas1 had the     
science been concerned with those kinds 
of events only which in practice are    
commonly made subjects of insurance1    
probably no other view would ever have  
been taken than that it was based upon  
direct appeal to experience4  #c4 ,when1
 however1 we look a little closer1 we   
find that there is no occasion for such 
a sharp distinction as that apparently  
implied between the two classes of      
examples just indicated4 ,in such cases 
as those of dice and cards1 even1 in    
which we appear to reason directly from 
the determining conditions1 or possible 
variety of the events1 rather than from 
actual observation of their occurrence1 
we shall find that this procedure is    
only valid by the help of a tacit       
assumption which can never be determined

 otherwise than by direct experience4   
,it is1 no doubt1 an exceedingly natural
 and obvious assumption1 and one which  
is continually deriving fresh weight    
from every-day observation1 but it is   
one which ought not to be admitted      
without consideration4 ,as this is a    
very important matter1 not so much in   
itself as in connection with the light  
which it throws upon the theory of the  
subject1 we will enter into a somewhat  
detailed examination of it4  ,let us    
take a very simple example1 that of     
tossing up a penny4 ,suppose that ,i am 
contemplating a succession of two       
throws2 ,i can see that the only        
possible events are#b ,,hh1 ,,ht1 ,,th1 
,,tt4 ,so much is certain4 ,we are      
moreover tolerably well convinced from  
experience that these events occur1 in  
the long run1 about equally often4 ,this
 is of course admitted on all hands4    
,but on the view commonly maintained1 it
 is contended that we might have known  
the fact beforehand on grounds which are

 applicable to an indefinite number of  
other and more complex cases4 ,the form 
in which this view would generally be   
advanced is1 that we are enabled to     
state beforehand that the four throws   
above mentioned are equally likely4 ,if 
in return we ask what is meant by the   
expression 'equally likely'1 it appears 
that there are two and only two possible
 forms of reply4 ,one of these seeks the
 explanation in the state of mind of the
 observer1 the other seeks it in some   
characteristic of the things observed4  
7#a7 ,it might1 for instance1 be said on
 the one hand1 that what is meant is    
that the four events contemplated are   
equally easy to imagine1 or1 more       
accurately1 that our expectation or     
belief in their occurrence is equal4 ,we
 could hardly be content with this      
reply1 for the further enquiry would    
immediately be urged1 ,on what ground is
 this to be believed8 ,what are the     
characteristics of events of which our  
expectation is equal8 ,if we consented  

to give an answer to this further       
enquiry1 we should be led to the second 
form of reply1 to be noticed directly2  
if we did not consent we should1 it     
seems1 be admitting that ,probability   
was only a portion of ,psychology1      
confined therefore to considering states
 of mind in themselves1 rather than in  
their reference to facts1 viz4 as being 
true or false4 ,we should1 that is1 be  
ceasing to make it a science of         
inference about things4 ,this point will
 have to be gone into more thoroughly in
 another chapter2 but it is impossible  
to direct attention too prominently to  
the fact that ,logic 7and therefore     
,probability as a branch of ,logic7 is  
not concerned with what men do believe1 
but with what they ought to believe1 if 
they are to believe correctly4  7#b7 ,in
 the other form of reply the explanation
 of the phrase in question would be     
sought1 not in a state of mind1 but in a
 quality of the things contemplated4 ,we
 might assign the following as the      

meaning1 viz4 that the events really    
would occur with equal frequency in the 
long run4 ,the ground of this assertion 
would probably be found in past         
experience1 and it would doubtless be   
impossible so to frame the answer as to 
exclude the notion of our belief        
altogether4 ,but still there is a broad 
distinction between seeking an equality 
in the amount of our belief1 as before1 
and in the frequency of occurrence of   
the events themselves1 as here4  #d4    
,when we have got as far as this it can 
readily be shown that an appeal to      
experience cannot be long evaded4 ,for  
can the assertion in question 7viz4 that
 the throws of the penny will occur     
equally often7 be safely made  priori8  
,those who consider that it can seem    
hardly to have fully faced the          
difficulties which meet them4 ,for when 
we begin to enquire seriously whether   
the penny will really do what is        
expected of it1 we find that            
restrictions have to be introduced4 ,in 

the first place it must be an ideal     
coin1 with its sides equal and fair4    
,this restriction is perfectly          
intelligible2 the study of solid        
geometry enables us to idealize a penny 
into a circular or cylindrical lamina4  
,but this condition by itself is not    
sufficient1 others are wanted as well4  
,the penny was supposed to be tossed up1
 as we say 'at random4' ,what is meant  
by this1 and how is this process to be  
idealized8 ,to ask this is to introduce 
no idle subtlety2 for it would scarcely 
be maintained that the heads and tails  
would get their fair chances if1        
immediately before the throwing1 we were
 so to place the coin in our hands as to
 start it always with the same side     
upwards4 ,the difference that would     
result in consequence1 slight as its    
cause is1 would tend in time to show    
itself in the results4 ,or1 if we       
persisted in starting with each of the  
two sides alternately upwards1 would the
 longer repetitions of the same side get

 their fair chance8  ,perhaps it will be
 replied that if we think nothing       
whatever about these matters all will   
come right of its own accord4 ,it may1  
and doubtless will be so1 but this is   
falling back upon experience4 ,it is    
here1 then1 that we find ourselves      
resting on the experimental assumption  
above mentioned1 and which indeed cannot
 be avoided4 ,for suppose1 lastly1 that 
the circumstances of nature1 or my      
bodily or mental constitution1 were such
 that the same side always is started   
upwards1 or indeed that they are started
 in any arbitrary order of our own8     
,well1 it will be replied1 it would not 
then be a fair trial4 ,if we press in   
this way for an answer to such          
enquiries1 we shall find that these     
tacit restrictions are really nothing   
else than a mode of securing an         
experimental result4 ,they are only     
another way of saying1 ,let a series of 
actions be performed in such a way as to
 secure a sequence of a particular kind1

 viz41 of the kind described in the     
previous chapters4  #e4 ,an intermediate
 way of evading the direct appeal to    
experience is sometimes found by        
defining the probability of an event as 
being measured by the ratio which the   
number of cases favourable to the event 
bears to the total number of cases which
 are possible4 ,this seems a somewhat   
loose and ambiguous way of speaking4 ,it
 is clearly not enough to count the     
number of cases merely1 they must also  
be valued1 since it is not certain that 
each is equally potent in producing the 
effect4 ,this1 of course1 would never be
 denied1 but sufficient importance does 
not seem to be attached to the fact that
 we have really no other way of valuing 
them except by estimating the effects   
which they actually do1 or would        
produce4 ,instead of thus appealing to  
the proportion of cases favourable to   
the event1 it is far better 7at least as
 regards the foundation of the science1 
for we are not at this moment discussing

 the practical method of facilitating   
our calculations7 to appeal at once to  
the proportion of cases in which the    
event actually occurs4  #f4 ,the remarks
 above made will apply1 of course1 to   
most of the other common examples of    
chance2 the throwing of dice1 drawing of
 cards1 of balls from bags1 c4 ,in the  
last case1 for instance1 one would      
naturally be inclined to suppose that a 
ball which had just been put back would 
thereby have a better chance of coming  
out again next time1 since it will be   
more in the way for that purpose4 ,how  
is this to be prevented8 ,if we         
designedly thrust it to the middle or   
bottom of the others1 we may overdo the 
precaution2 and are in any case         
introducing human design1 that element  
so essentially hostile to all that we   
understand by chance4 ,if we were to    
trust to a good shake setting matters   
right1 we may easily be deceived2 for   
shaking the bag can hardly do more than 
diminish the disposition of those balls 

which were already in each other's      
neighbourhood1 to remain so4 ,in the    
consequent interaction of each upon all1
 the arrangement in which they start    
cannot but leave its impress to some    
extent upon their final positions4 ,in  
all such cases1 therefore1 if we        
scrutinize our language1 we shall find  
that any supposed  priori mode of       
stating a problem is little else than a 
compendious way of saying1 ,let means be
 taken for obtaining a given result4    
,since it is upon this result that our  
inferences ultimately rest1 it seems    
simpler and more philosophical to appeal
 to it at once as the groundwork of our 
science4  #g4 ,let us again take the    
instance of the tossing of a penny1 and 
examine it somewhat more minutely1 to   
see what can be actually proved about   
the results we shall obtain4 ,we are    
willing to give the pence fair treatment
 by assuming that they are perfect1 that
 is1 that in the long run they show no  
preference for either head or tail2 the 

question then remains1 ,will the        
repetitions of the same face obtain the 
proportional shares to which they are   
entitled by the usual interpretations of
 the theory8 ,putting then1 as before1  
for the sake of brevity1 ,h for head1   
and ,,hh for heads twice running1 we are
 brought to this issue2--,given that the
 chance of ,h is #a_/b1 does it follow  
necessarily that the chance of ,,hh     
7with two pence7 is #a_/d8 ,to say      
nothing of ',h ten times' occurring once
 in #ajbd times 7with ten pence71 need  
it occur at all8 ,the mathematicians1   
for the most part1 seem to think that   
this conclusion follows necessarily from
 first principles2 to me it seems to    
rest upon no more certain evidence than 
a reasonable extension by ,induction4   
,taking then the possible results which 
can be obtained from a pair of pence1   
what do we find8 ,four different results
 may follow1 namely1 7#a7 ,,ht1 7#b7    
,,hh1 7#c7 ,,th1 7#d7 ,,tt4 ,if it can  
be proved that these four are equally   

probable1 that is1 occur equally often1 
the commonly accepted conclusions will  
follow1 for a precisely similar argument
 would apply to all the larger numbers4 
#h4 ,the proof usually advanced makes   
use of what is called the ,principle of 
,sufficient ,reason4 ,it takes this     
form2--,here are four kinds of throws   
which may happen2 once admit that the   
separate elements of them1 namely1 ,h   
and ,t1 happen equally often1 and it    
will follow that the above combinations 
will also happen equally often1 for no  
reason can be given in favour of one of 
them that would not equally hold in     
favour of the others4  ,to a certain    
extent we must admit the validity of the
 principle for the purpose4 ,in the case
 of the throws given above1 it would be 
valid to prove the equal frequency of   
7#a7 and 7#c7 and also of 7#b7 and 7#d72
 for there is no difference existing    
between these pairs except what is      
introduced by our own notation4#c ,,th  
is the same as ,,ht1 except in the order

 of the occurrence of the symbols ,h and
 ,t1 which we do not take into account4 
,but either of the pair 7#a7 and 7#c7 is
 different from either of the pair 7#b7 
and 7#d74 ,transpose the notation1 and  
there would still remain here a         
distinction which the mind can          
recognize4 ,a succession of the same    
thing twice running is distinguished    
from the conjunction of two different   
things1 by a distinction which does not 
depend upon our arbitrary notation only1
 and would remain entirely unaltered by 
a change in this notation4 ,the         
principle therefore of ,sufficient      
,reason1 if admitted1 would only prove  
that doublets of the two kinds1 for     
example 7#b7 and 7#d71 occur equally    
often1 but it would not prove that they 
must each occur once in four times4 ,it 
cannot be proved indeed in this way that
 they need ever occur at all4  #i4 ,the 
formula1 then1 not being demonstrable   
priori1 7as might have been concluded17 
can it be obtained by experience8 ,to a 

certain extent it can2 the present      
experience of mankind in pence and dice 
seems to show that the smaller          
successions of throws do really occur in
 about the proportions assigned by the  
theory4 ,but how nearly they do so no   
one can say1 for the amount of time and 
trouble to be expended before we could  
feel that we have verified the fact1    
even for small numbers1 is very great1  
whilst for large numbers it would be    
simply intolerable4 ,the experiment of  
throwing often enough to obtain 'heads  
ten times' has been actually performed  
by two or three persons1 and the results
 are given by ,de ,morgan1 and          
,jevons4#d ,this1 however1 being only   
sufficient on the average to give 'heads
 ten times' a single chance1 the        
evidence is very slight2 it would take a
 considerable number of such experiments
 to set the matter nearly at rest4  ,any
 such rule1 then1 as that which we have 
just been discussing1 which professes to
 describe what will take place in a long

 succession of throws1 is only          
conclusively proved by experience within
 very narrow limits1 that is1 for small 
repetitions of the same face2 within    
limits less narrow1 indeed1 we feel     
assured that the rule cannot be         
flagrantly in error1 otherwise the      
variation would be almost sure to be    
detected4 ,from this we feel strongly   
inclined to infer that the same law will
 hold throughout4 ,in other words1 we   
are inclined to extend the rule by      
,induction and ,analogy4 ,still there   
are so many instances in nature of      
proposed laws which hold within narrow  
limits but get egregiously astray when  
we attempt to push them to great        
lengths1 that we must give at best but a
 qualified assent to the truth of the   
formula4  #aj4 ,the object of the above 
reasoning is simply to show that we     
cannot be certain that the rule is true4
 ,let us now turn for a minute to       
consider the causes by which the        
succession of heads and tails is        

produced1 and we may perhaps see reasons
 to make us still more doubtful4  ,it   
has been already pointed out that in    
calculating probabilities  priori1 as it
 is called1 we are only able to do so by
 introducing restrictions and           
suppositions which are in reality       
equivalent to assuming the expected     
results4 ,we use words which in         
strictness mean1 ,let a given process be
 performed2 but an analysis of our      
language1 and an examination of various 
tacit suppositions which make themselves
 felt the moment they are not complied  
with1 soon show that our real meaning   
is1 ,let a series of a given kind be    
obtained2 it is to this series only1 and
 not to the conditions of its           
production1 that all our subsequent     
calculations properly apply4 ,the       
physical process being performed1 we    
want to know whether anything resembling
 the contemplated series really will be 
obtained4  ,now if the penny were       
invariably set the same side uppermost1 

and thrown with the same velocity of    
rotation and to the same height1 c4--in 
a word1 subjected to the same           
conditions1--it would always come down  
with the same side uppermost4           
,practically1 we know that nothing of   
this kind occurs1 for the individual    
variations in the results of the throws 
are endless4 ,still there will be an    
average of these conditions1 about which
 the throws will be found1 as it were1  
to cluster much more thickly than       
elsewhere4 ,we should be inclined       
therefore to infer that if the same side
 were always set uppermost there would  
really be a departure from the sort of  
series which we ordinarily expect4 ,in a
 very large number of throws we should  
probably begin to find1 under such      
circumstances1 that either head or tail 
was having a preference shown to it4 ,if
 so1 would not similar effects be found 
to be connected with the way in which we
 started each successive pair of throws8
 ,according as we chose to make a       

practice of putting ,,hh or ,,tt        
uppermost1 might there not be a         
disturbance in the proportion of        
successions of two heads or two tails8  
,following out this train of reasoning1 
it would seem to point with some        
likelihood to the conclusion that in    
order to obtain a series of the kind we 
expect1 we should have to dispose the   
antecedents in a similar series at the  
start4 ,the changes and chances produced
 by the act of throwing might introduce 
infinite individual variations1 and yet 
there might be found1 in the very long  
run1 to be a close similarity between   
these two series4  #aa4 ,this is1 to a  
certain extent1 only shifting the       
difficulty1 ,i admit2 for the claim     
formerly advanced about the possibility 
of proving the proportions of the throws
 in the former series1 will probably now
 be repeated in favour of those in the  
latter4 ,still the question is very much
 narrowed1 for we have reduced it to a  
series of voluntary acts4 ,a man may put

 whatever side he pleases uppermost4 ,he
 may act consciously1 as ,i have said1  
or he may think nothing whatever about  
the matter1 that is1 throw at random2 if
 so1 it will probably be asserted by    
many that he will involuntarily produce 
a series of the kind in question4 ,it   
may be so1 or it may not2 it does not   
seem that there are any easily          
accessible data by which to decide4 ,all
 that ,i am concerned with here is to   
show the likelihood that the commonly   
received result does in reality depend  
upon the fulfilment of a certain        
condition at the outset1 a condition    
which it is certainly optional with any 
one to fulfil or not as he pleases4 ,the
 short successions doubtless will take  
care of themselves1 owing to the        
infinite complications produced by the  
casual variations in throwing2 but the  
long ones may suffer1 unless their      
interest be consciously or unconsciously
 regarded at the outset4  #ab4 ,the     
advice1 ',only try long enough1 and you 

will sooner or later get any result that
 is possible1' is plausible1 but it     
rests only on ,induction and ,analogy2  
mathematics do not prove it4 ,as has    
been repeatedly stated1 there are two   
distinct views of the subject4 ,either  
we may1 on the one hand1 take a series  
of symbols1 call them heads and tails2  
,h1 ,t1 c42 and make the assumption that
 each of these1 and each pair of them1  
and so on1 will occur in the long run   
with a regulated degree of frequency4   
,we may then calculate their various    
combinations1 and the consequences that 
may be drawn from the data assumed4     
,this is a purely algebraical process2  
it is infallible2 and there is no limit 
whatever to the extent to which it may  
be carried4 ,this way of looking at the 
matter may be1 and undoubtedly should   
be1 nothing more than the counterpart of
 what ,i have called the substituted or 
idealized series which generally has to 
be introduced as the basis of our       
calculation4 ,the danger to be guarded  

against is that of regarding it too     
purely as an algebraical conception1 and
 thence of sinking into the very natural
 errors both of too readily evolving it 
out of our own consciousness1 and too   
freely pushing it to unwarranted        
lengths4  ,or on the other hand1 we may 
consider that we are treating of the    
behaviour of things2--balls1 dice1      
births1 deaths1 c42 and drawing         
inferences about them4 ,but1 then1 what 
were in the former instance allowable   
assumptions1 become here propositions to
 be tested by experience4 ,now the whole
 theory of ,probability as a practical  
science1 in fact as anything more than  
an algebraical truth1 depends of course 
upon there being a close correspondence 
between these two views of the subject1 
in other words1 upon our substituted    
series being kept in accordance with the
 actual series4 ,experience abundantly  
proves that1 between considerable       
limits1 in the example in question1     
there does exist such a correspondence4 

,but let no one attempt to enforce our  
assent to every remote deduction that   
mathematicians can draw from their      
formul4 ,when this is attempted the     
distinction just traced becomes         
prominent and important1 and we have to 
choose our side4 ,either we go over to  
the mathematics1 and so lose all right  
of discussion about the things2 or else 
we take part with the things1 and so    
defy the mathematics4 ,we do not        
question the formal accuracy of the     
latter within their own province1 but   
either we dismiss them as somewhat      
irrelevant1 as applying to data of whose
 correctness we cannot be certain1 or we
 take the liberty of remodelling them so
 as to bring them into accordance with  
facts4  #ac4 ,a critic of any doctrine  
can hardly be considered to have done   
much more than half his duty when he has
 explained and justified his grounds for
 objecting to it4 ,it still remains for 
him to indicate1 if only in a few words1
 what he considers its legitimate       

functions and position to be1 for it can
 seldom happen that he regards it as    
absolutely worthless or unmeaning4 ,i   
should say1 then1 that when ,probability
 is thus divorced from direct reference 
to objects1 as it substantially is by   
not being founded upon experience1 it   
simply resolves itself into the common  
algebraical or arithmetical doctrine of 
,permutations and ,combinations4#e ,the 
considerations upon which these depend  
are purely formal and necessary1 and can
 be fully reasoned out without any      
appeal to experience4 ,we there start   
from pure considerations of number or   
magnitude1 and we terminate with them1  
having only arithmetical calculations to
 connect them together4 ,i wish1 for    
instance1 to find the chance of throwing
 heads three times running with a penny4
 ,all ,i have to do is first to         
ascertain the possible number of throws4
 ,permutations tell me that with two    
things thus in question 7viz4 head and  
tail7 and three times to perform the    

process1 there are eight possible forms 
of the result4 ,of these eight one only 
being favourable1 the chance in question
 is pronounced to be one-eighth4  ,now  
though it is quite true that the actual 
calculation of every chance problem must
 be of the above character1 viz4 an     
algebraical or arithmetical process1 yet
 there is1 it seems to me1 a broad and  
important distinction between a material
 science which employs mathematics1 and 
a formal one which consists of nothing  
but mathematics4 ,when we cut ourselves 
off from the necessity of any appeal to 
experience1 we are retaining only the   
intermediate or calculating part of the 
investigation2 we may talk of dice1 or  
pence1 or cards1 but these are really   
only names we choose to give to our     
symbols4 ,the ,h's and ,t's with which  
we deal have no bearing on objective    
occurrences1 but are just like the x's  
and y's with which the rest of algebra  
deals4 ,probability in fact1 when so    
treated1 seems to be absolutely nothing 

else than a system of applied           
,permutations and ,combinations4  ,it   
will now readily be seen how narrow is  
the range of cases to which any purely  
deductive method of treatment can apply4
 ,it is almost entirely confined to such
 employments as games of chance1 and1 as
 already pointed out1 can only be       
regarded as really trustworthy even     
there1 by the help of various tacit     
restrictions4 ,this alone would be      
conclusive against the theory of the    
subject being rested upon such a basis4 
,the experimental method1 on the other  
hand1 is1 in the same theoretical sense1
 of universal application4 ,it would    
include the ordinary problems furnished 
by games of chance1 as well as those    
where the dice are loaded and the pence 
are not perfect1 and also the           
indefinitely numerous applications of   
statistics to the various kinds of      
social phenomena4  #ad4 ,the particular 
view of the deductive character of      
,probability above discussed1 could     

scarcely have intruded itself into any  
other examples than those of the nature 
of games of chance1 in which the        
conditions of occurrence are by         
comparison few and simple1 and are      
amenable to accurate numerical          
determination4 ,but a doctrine1 which is
 in reality little else than the same   
theory in a slightly disguised form1 is 
very prevalent1 and has been applied to 
truths of the most purely empirical     
character4 ,this doctrine will be best  
introduced by a quotation from ,laplace4
 ,after speaking of the irregularity and
 uncertainty of nature as it appears at 
first sight1 he goes on to remark that  
when we look closer we begin to detect a
 striking regularity which seems to     
suggest a design1 and which some have   
considered a proof of ,providence4 ,but1
 on reflection1 it is soon perceived    
that this regularity is nothing but the 
development of the respective           
probabilities of the simple events1     
which ought to occur more frequently    

according as they are more probable4#f  
,if this remark had been made about the 
succession of heads and tails in the    
throwing up of a penny1 it would have   
been intelligible4 ,it would simply mean
 this3 that the constitution of the body
 was such that we could anticipate with 
some confidence what the result would be
 when it was treated in a certain way1  
and that experience would justify our   
anticipation in the long run4 ,but      
applied as it is in a more general form 
to the facts of nature1 it seems really 
to have but little meaning in it4 ,let  
us test it by an instance4 ,amidst the  
irregularity of individual births1 we   
find that the male children are to the  
female1 in the long run1 in about the   
proportion of #ajf to #ajj4 ,now if we  
were told that there is nothing in this 
but the development of their respective 
probabilities1 would there be anything  
in such a statement but a somewhat      
pretentious re-statement of the fact    
already asserted8 ,the probability is   

nothing but that proportion1 and is     
unquestionably in this case derived from
 no other source but the statistics     
themselves2 in the above remark the     
attempt seems to be made to invert this 
process1 and to derive the sequence of  
events from the mere numerical statement
 of the proportions in which they occur4
  #ae4 ,it will very likely be replied  
that by the probability above mentioned 
is meant1 not the mere numerical        
proportion between1 the births1 but some
 fact in our constitution upon which    
this proportion depends2 that just as   
there was a relation of equality between
 the two sides of the penny1 which      
produced the ultimate equality in the   
number of heads and tails1 so there may 
be something in our constitution or     
circumstances in the proportion of #ajf 
to #ajj1 which produces the observed    
statistical result4 ,when this          
something1 whatever it might be1 was    
discovered1 the observed numbers might  
be supposed capable of being determined 

beforehand4 ,even if this were the case1
 however1 it must not be forgotten that 
there could hardly fail to be1 in       
combination with such causes1 other     
concurrent conditions in order to       
produce the ultimate result2 just as    
besides the shape of the penny1 we had  
also to take into account the nature of 
the 'randomness' with which it was      
tossed4 ,what these may be1 no one at   
present can undertake to say1 for the   
best physiologists seem indisposed to   
hazard even a guess upon the subject4#g 
,but without going into particulars1 one
 may assert with some confidence that   
these conditions cannot well be         
altogether independent of the health1   
circumstances1 manners and customs1 c4  
7to express oneself in the vaguest way7 
of the parents2 and if once these       
influencing elements are introduced1    
even as very minute factors1 the results
 cease to be dependent only on fixed and
 permanent conditions4 ,we are at once  
letting in other conditions1 which1 if  

they also possess the characteristics   
that distinguish ,probability 7an       
exceedingly questionable assumption71   
must have that fact specially proved    
about them4 ,that this should be the    
case indeed seems not merely            
questionable1 but almost certainly      
impossible2 for these conditions        
partaking of the nature of what we term 
generally1 ,progress and ,civilization1 
cannot be expected to show any permanent
 disposition to hover about an average4 
#af4 ,the reader who is familiar with   
,probability is of course acquainted    
with the celebrated theorem of ,james   
,bernoulli4 ,this theorem1 of which the 
examples just adduced are merely        
particular cases1 is generally expressed
 somewhat as follows3--in the long run  
all events will tend to occur with a    
relative frequency proportional to their
 objective probabilities4 ,with the     
mathematical proof of this theorem we   
need not trouble ourselves1 as it lies  
outside the province of this work2 but  

indeed if there is any value in the     
foregoing criticism1 the basis on which 
the mathematics rest is faulty1 owing to
 there being really nothing which we can
 with propriety call an objective       
probability4  ,if one might judge by the
 interpretation and uses to which this  
theorem is sometimes exposed1 we should 
regard it as one of the last remaining  
relics of ,realism1 which after being   
banished elsewhere still manages to     
linger in the remote province of        
,probability4 ,it would be an           
illustration of the inveterate tendency 
to objectify our conceptions1 even in   
cases where the conceptions had no right
 to exist at all4 ,a uniformity is      
observed2 sometimes1 as in games of     
chance1 it is found to be so connected  
with the physical constitution of the   
bodies employed as to be capable of     
being inferred beforehand2 though even  
here the connection is by no means so   
necessary as is commonly supposed1 owing
 to the fact that in addition to these  

bodies themselves we have also to take  
into account their relation to the      
agencies which influence them4 ,this    
constitution is then converted into an  
'objective probability'1 supposed to    
develop into the sequence which exhibits
 the uniformity4 ,finally1 this very    
questionable objective probability is   
assumed to exist1 with the same faculty 
of development1 in all the cases in     
which uniformity is observed1 however   
little resemblance there may be between 
these and games of chance4  #ag4 ,how   
utterly inappropriate any such          
conception is in most of the cases in   
which we find statistical uniformity1   
will be obvious on a moment's           
consideration4 ,the observed phenomena  
are generally the product1 in these     
cases1 of very numerous and complicated 
antecedents4 ,the number of crimes1 for 
instance1 annually committed in any     
society1 is a function amongst other    
things1 of the strictness of the law1   
the morality of the people1 their social

 condition1 and the vigilance of the    
police1 each of these elements being in 
itself almost infinitely complex4 ,now1 
as a result of all these agencies1 there
 is some degree of uniformity2 but what 
has been called above the change of     
type1 which it sooner or later tends to 
display1 is unmistakeable4 ,the average 
annual numbers do not show a steady     
gradual approach towards what might be  
considered in some sense a limiting     
value1 but1 on the contrary1 fluctuate  
in a way which1 however it may depend   
upon causes1 shows none of the permanent
 uniformity which is characteristic of  
games of chance4 ,this fact1 combined   
with the obvious arbitrariness of       
singling out1 from amongst the many and 
various antecedents which produced the  
observed regularity1 a few only1 which  
should constitute the objective         
probability 7if we took all1 the events 
being absolutely determined1 there would
 be no occasion for an appeal to        
probability in the case71 would have    

been sufficient to prevent any one from 
assuming the existence of any such      
thing1 unless the mistaken analogy of   
other cases had predisposed him to seek 
for it4  ,there is a familiar practical 
form of the same error1 the tendency to 
which may not improbably be derived from
 a similar theoretical source4 ,it is   
that of continuing to accumulate our    
statistical data to an excessive extent4
 ,if the type were absolutely fixed we  
could not possibly have too many        
statistics2 the longer we chose to take 
the trouble of collecting them the more 
accurate our results would be4 ,but if  
the type is changing1 in other words1 if
 some of the principal causes which aid 
in their production have1 in regard to  
their present degree of intensity1      
strict limits of time or space1 we shall
 do harm rather than good if we overstep
 these limits4 ,the danger of stopping  
too soon is easily seen1 but in avoiding
 it we must not fall into the opposite  
error of going on too long1 and so      

getting either gradually or suddenly    
under the influence of a changed set of 
circumstances4  #ah4 ,this chapter was  
intended to be devoted to a             
consideration1 not of the processes by  
which nature produces the series with   
which we are concerned1 but of the      
theoretic basis of the methods by which 
we can determine the existence of such  
series4 ,but it is not possible to keep 
the two enquiries apart1 for here1 at   
any rate1 the old maxim prevails that to
 know a thing we must know its causes4  
,recur for a minute to the              
considerations of the last chapter4 ,we 
there saw that there was a large class  
of events1 the conditions of production 
of which could be said to consist of    
7#a7 a comparatively few nearly         
unchangeable elements1 and 7#b7 a vast  
number of independent and very          
changeable elements4 ,at least if there 
were any other elements besides these1  
we are assumed either to make special   
allowance for them1 or to omit them from

 our enquiry4 ,now in certain cases1    
such as games of chance1 the            
unchangeable elements may without       
practical error be regarded as really   
unchangeable throughout any range of    
time and space4 ,hence1 as a result1 the
 deductive method of treatment becomes  
in their case at once the most simple1  
natural1 and conclusive2 but1 as a      
further consequence1 the statistics of  
the events1 if we choose to appeal to   
them1 may be collected ad libitum with  
better and better approximation to      
truth4 ,on the other hand1 in all social
 applications of ,probability1 the      
unchangeable causes can only be regarded
 as really unchangeable under many      
qualifications4 ,we know little or      
nothing of them directly2 they are often
 in reality numerous1 indeterminate1 and
 fluctuating2 and it is only under the  
guarantee of stringent restrictions of  
time and place1 that we can with any    
safety attribute to them sufficient     
fixity to justify our theory4 ,hence1 as

 a result1 the deductive method1 under  
whatever name it may go1 becomes totally
 inapplicable both in theory and        
practice2 and1 as a further consequence1
 the appeal to statistics has to be made
 with the caution in mind that we shall 
do mischief rather than good if we go on
 collecting too many of them4  #ai4 ,the
 results of the last two chapters may be
 summed up as follows3--,we have        
extended the conception of a series     
obtained in the first chapter2 for we   
have found that these series are mostly 
presented to us in groups4 ,these groups
 are found upon examination to be formed
 upon approximately the same type       
throughout a very wide and varied range 
of experience2 the causes of this       
agreement we discussed and explained in 
some detail4 ,when1 however1 we extend  
our examination by supposing the series 
to run to a very great length1 we find  
that they may be divided into two       
classes separated by important          
distinctions4 ,in one of these classes  

7that containing the results of games of
 chance7 the conditions of production1  
and consequently the laws of statistical
 occurrence1 may be practically regarded
 as absolutely fixed2 and the extent of 
the divergences from the mean seem to   
know no finite limit4 ,in the other     
class1 on the contrary 7containing the  
bulk of ordinary statistical enquiries71
 the conditions of production vary with 
more or less rapidity1 and so in        
consequence do the results4 ,moreover it
 is often impossible that variations    
from the mean should exceed a certain   
amount4 ,the former we may term ideal   
series4 ,it is they alone which show the
 requisite characteristics with any     
close approach to accuracy1 and to make 
the theory of the subject tenable1 we   
have really to substitute one of this   
kind for one of the less perfect ones of
 the other class1 when these latter are 
under treatment4 ,the former class have1
 however1 been too exclusively          
considered by writers on the subject2   

and conceptions appropriate only to     
them1 and not always even to them1 have 
been imported into the other class4 ,it 
is in this way that a general tendency  
to an excessive deductive or  priori    
treatment of the science has been       
encouraged4   #a4 ,this latter enquiry  
belongs to what may be termed the more  
purely   logical part of this volume1   
and is entered on in the course of      
,chapter ,,vi4  #b4 ,for the use of     
those not acquainted with the common    
notation   employed in this subject1 it 
may be remarked that ,,hh is simply an  
abbreviated way of saying that the two  
successive throws of the   penny give   
head2 ,,ht that the first of them gives 
head1 and the   second tail2 and so on  
with the remaining symbols4  #c4 ,i am  
endeavouring to treat this rule of      
,sufficient ,reason in a way   that     
shall be legitimate in the opinion of   
those who accept it1 but   there seem   
very great doubts whether a             
contradiction is not involved   when we 

attempt to extract results from it4 ,if 
the sides are   absolutely alike1 how   
can there he any difference between the 
terms   of the series8 ,the succession  
seems then reduced to a dull            
uniformity1 a mere iteration of the same
 thing many times2 the   series we      
contemplated has disappeared4 ,if the   
sides are not   absolutely alike1 what  
becomes of the applicability of the     
rule8  #d4 ,formal ,logic1 p4 #ahe4     
,principles of ,science1 p4 #bjh4  #e4  
,the close connection between these     
subjects is well indicated in   the     
title of ,mr ,whitworth's treatise1     
,choice and ,chance4  #f4 ,essai        
,philosophique4 ,ed4 #ahbe1 p4 #gd4  #g4
 ,an opinion prevailed rather at one    
time 7quoted and supported by           
,quetelet amongst others7 that the      
relative ages of the parents had        
something to do with the sex of the     
offspring4 ,if this were so1 it   would 
quite bear out the above remarks4 ,as a 
matter of fact1 it   should be observed1

 that the proportion of #ajf to #ajj    
does not seem   by any means universal  
in all countries or at all times4 ,for  
various   statistical tables on the     
subject see ,quetelet1 ,physique        
,sociale1   ,vol4 ,i4 #aff1 #agc1 #bch4 
,,chapter ,v4   ,,the ,,conception      
,,randomness ,,and ,,its ,,scientific   
,,treatment4  #a4 ,there is a term of   
frequent occurrence in treatises on     
,probability1 and which we have already 
had repeated occasion to employ1 viz4   
the designation random applied to an    
event1 as in the expression 'a random   
distribution'4 ,the scientific          
conception involved in the correct use  
of this term is1 ,i apprehend1 nothing  
more than that of aggregate order and   
individual irregularity 7or apparent    
irregularity71 which has been already   
described in the preceding chapters4 ,a 
brief discussion of the requisites in   
this scientific conception1 and in      
particular of the nature and some of the
 reasons for the departure from the     

popular conception1 may serve to clear  
up some of the principal remaining      
difficulties which attend this part of  
our subject4  ,the original1#a and still
 popular1 signification of the term is  
of course widely different from the     
scientific4 ,what it looks to is the    
origin1 not the results1 of the random  
performance1 and it has reference rather
 to the single action than to a group or
 series of actions4 ,thus1 when a man   
draws a bow 'at a venture'1 or 'at      
random'1 we mean only to point out the  
aimless character of the performance2 we
 are contrasting it with the definite   
intention to hit a certain mark4 ,but it
 is none the less true1 as already      
pointed out1 that we can only apply     
processes of inference to such          
performances as these when we regard    
them as being capable of frequent1 or   
rather of indefinitely extended         
repetition4  ,begin with an             
illustration4 ,perhaps the best typical 
example that we can give of the         

scientific meaning of random            
distribution is afforded by the         
arrangement of the drops of rain in a   
shower4 ,no one can give a guess        
whereabouts at any instant a drop will  
fall1 but we know that if we put out a  
sheet of paper it will gradually become 
uniformly spotted over2 and that if we  
were to mark out any two equal areas on 
the paper these would gradually tend to 
be struck equally often4  #b4 ,i4 ,any  
attempt to draw inferences from the     
assumption of random arrangement must   
postulate the occurrence of this        
particular state of things at some stage
 or other4 ,but there is often          
considerable difficulty1 leading        
occasionally to some arbitrariness1 in  
deciding the particular stage at which  
it ought to be introduced4  7#a7 ,thus1 
in many of the problems discussed by    
mathematicians1 we look as entirely to  
the results obtained1 and think as      
little of the actual process by which   
they are obtained1 as when we are       

regarding the arrangement of the drops  
of rain4 ,a simple example of this kind 
would be the following4 ,a pawn1        
diameter of base one inch1 is placed at 
random on a chess-board1 the diameter of
 the squares of which is one inch and a 
quarter3 find the chance that its base  
shall lie across one of the intersecting
 lines4 ,here we may imagine the pawns  
to be so to say rained down vertically  
upon the board1 and the question is to  
find the ultimate proportion of those   
which meet a boundary line to the total 
of those which fall4 ,the problem       
therefore becomes a merely geometrical  
one1 viz4 to determine the ratio of a   
certain area on the board to the whole  
area4 ,the determination of this ratio  
is all that the mathematician ever takes
 into account4  ,now take the following4
 ,a straight brittle rod is broken at   
random in two places3 find the chance   
that the pieces can make a triangle4#b  
,since the only condition for making a  
triangle with three straight lines is   

that each two shall be greater than the 
third1 the problem seems to involve the 
same general conception as in the former
 case4 ,we must conceive such rods      
breaking at one pair of spots after     
another1--no one can tell precisely     
where1--but showing the same ultimate   
tendency to distribute these spots      
throughout the whole length uniformly4  
,as in the last case1 the mathematician 
thinks of nothing but this final result1
 and pays no heed to the process by     
which it may be brought about4          
,accordingly the problem is again       
reduced to one of mensuration1 though of
 a somewhat more complicated character4 
#c4 7#b7 ,in another class of cases we  
have to contemplate an intermediate     
process rather than a final result2 but 
the same conception has to be introduced
 here1 though it is now applied to the  
former stage1 and in consequence will   
not in general apply to the latter4     
,for instance3 a shot is fired at random
 from a gun whose maximum range 7i4e4 at

 #de elevation7 is #cjjj yards3 what is 
the chance that the actual range shall  
exceed #bjjj yards8 ,the ultimately     
uniform 7or random7 distribution here is
 commonly assumed to apply to the       
various directions in which the gun can 
be pointed2 all possible directions     
above the horizontal being equally      
represented in the long run4 ,we have   
therefore to contemplate a surface of   
uniform distribution1 but it will be the
 surface1 not of the ground1 but of a   
hemisphere whose centre is occupied by  
the man who fires4 ,the ultimate        
distribution of the bullets on the spots
 where they strike the ground will not  
be uniform4 ,the problem is in fact to  
discover the law of variation of the    
density of distribution4  ,the above is1
 ,i presume1 the treatment generally    
adopted in solving such a problem4 ,but 
there seems no absolute necessity for   
any such particular choice4 ,it is      
surely open to any one to maintain#c    
that his conception of the randomness of

 the firing is assigned by saying that  
it is likely that a man should begin by 
facing towards any point of the compass 
indifferently1 and then proceed to raise
 his gun to any angle indifferently4    
,the stage of ultimately uniform        
distribution here has receded a step    
further back4 ,it is not assigned       
directly to the surface of an imaginary 
hemisphere1 but to the lines of altitude
 and azimuth drawn on that surface4     
,accordingly1 the distribution over the 
hemisphere itself will not now be       
uniform1--there will be a comparative   
crowding up towards the pole1--and the  
ultimate distribution over the ground   
will not be the same as before4  #d4    
,difficulties of this kind1 arising out 
of the uncertainty as to what stage     
should be selected for that of uniform  
distribution1 will occasionally present 
themselves4 ,for instance3 let a book be
 taken at random out of a bookcase2 what
 is the chance of hitting upon some     
assigned volume8 ,i hardly know how this

 question would commonly be treated4 ,if
 we were to set our man opposite the    
middle of the shelf and inquire what    
would generally happen in practice1     
supposing him blindfolded1 there cannot 
be much doubt that the volumes would not
 be selected equally often4 ,on the     
contrary1 it is likely that there would 
be a tendency to increased frequency    
about a centre indicated by the height  
of his shoulder1 and 7unless he be      
left-handed7 a trifle to the right of   
the point exactly opposite his starting 
point4  ,if the question were one which 
it were really worth while to work out  
on these lines we should be led a long  
way back4 ,just as we imagined our      
rifleman's position 7on the second      
supposition7 to be determined by two    
independent coordinates of assumed      
continuous and equal facility1 so we    
might conceive our making the attempt to
 analyse the man's movements into a     
certain number of independent           
constituents4 ,we might suppose all the 

various directions from his starting    
point1 along the ground1 to be equally  
likely2 and that when he reaches the    
shelves the random motion of his hand is
 to be regulated after the fashion of a 
shot discharged at random4  ,the above  
would be one way of setting about the   
statement of the problem4 ,but the      
reader will understand that all which ,i
 am here proposing to maintain is that  
in these1 as in every similar case1 we  
always encounter1 under this conception 
of 'randomness'1 at some stage or other1
 this postulate of ultimate uniformity  
of distribution over some assigned      
magnitude3 either time2 or space1       
linear1 superficial1 or solid4 ,but the 
selection of the stage at which this is 
to be applied may give rise to          
considerable difficulty1 and even       
arbitrariness of choice4  #e4 ,some     
years ago there was a very interesting  
discussion upon this subject carried on 
in the mathematical part of the         
,educational ,times 7see1 especially1   

,vol4 ,,vii474 ,as not unfrequently     
happens in mathematics there was an     
almost entire accord amongst the various
 writers as to the assumptions          
practically to be made in any particular
 case1 and therefore as to the          
conclusion to be drawn1 combined with a 
very considerable amount of difference  
as to the axioms and definitions to be  
employed4 ,thus ,mr ,m4 ,w4 ,crofton1   
with the substantial agreement of ,mr   
,woolhouse1 laid it down unhesitatingly 
that at random has a very clear and     
definite meaning2 one which cannot be   
better conveyed than by ,mr ,wilson's   
definition1 'according to no law'2 and  
in this sense alone ,i mean to use it4  
,according to any scientific            
interpretation of 'law' ,i should have  
said that where there was no law there  
could be no inference4 ,but ultimate    
tendency towards equality of            
distribution is as much taken for       
granted by ,mr ,crofton as by any one   
else3 in fact he makes this a deduction 

from his definition3--,as this infinite 
system of parallels are drawn according 
to no law1 they are as thickly disposed 
along any part of the common            
perpendicular as along any other 7,,vii4
 p4 #he74 ,mr ,crofton holds that any   
kind of unequal distribution would imply
 law1--,if the points on a plane tended 
to become denser in any part of the     
plane than in another1 there must be    
some law attracting them there 7ib4 p4  
#hd74 ,the same view is enforced in his 
paper on ,local ,probability 7in the    
,phil4 ,trans41 ,vol4 #aeh74 ,surely if 
they tend to become equally dense this  
is just as much a case of regularity or 
law4  ,it may be remarked that wherever 
any serious practical consequences turn 
upon duly securing the desired          
randomness1 it is always so contrived   
that no design or awkwardness or        
unconscious one-sidedness shall disturb 
the result4 ,the principal case in point
 here is of course afforded by games of 
chance4 ,what we want1 when we toss a   

die1 is to secure that all numbers from 
#a to #f shall be equally often         
represented in the long run1 but that no
 person shall be able to predict the    
individual occurrence4 ,we might1 in our
 statement of a problem1 as easily      
postulate 'a number thought of at       
random' as 'a shot fired at random'1 but
 no one would risk his chances of gain  
and loss on the supposition that this   
would be done with continued fairness4  
,accordingly1 we construct a die whose  
sides are accurately alike1 and it is   
found that we may do almost what we like
 with this1 at any previous stage to    
that of its issue from the dice box on  
to the table1 without interfering with  
the random nature of the result4  #f4   
,,ii4 ,another characteristic in which  
the scientific conception seems to me to
 depart from the popular or original    
signification is the following4 ,the    
area of distribution which we take into 
account must be a finite or limited one4
 ,the necessity for this restriction may

 not be obvious at first sight1 but the 
consideration of one or two examples    
will serve to indicate the point at     
which it makes itself felt4 ,suppose    
that one were asked to choose a number  
at random1 not from a finite range1 but 
from the inexhaustible possibilities of 
enumeration4 ,in the popular sense of   
the term1--i4e4 of uttering a number    
without pausing to choose1--there is no 
difficulty4 ,but a moment's             
consideration will show that no         
arrangement even tending towards        
ultimately uniform distribution can be  
secured in this way4 ,no average could  
be struck with ever increasing          
steadiness4 ,so with spatial infinity4  
,we can rationally speak of choosing a  
point at random in a given straight     
line1 area1 or volume4 ,but if we       
suppose the line to have no end1 or the 
selection to be made in infinite space1 
the basis of ultimate tendency towards  
what may be called the equally thick    
deposit of our random points fails us   

utterly4  ,similarly in any other       
example in which one of the magnitudes  
is unlimited4 ,suppose ,i fling a stick 
at random in a horizontal plane against 
a row of iron railings and inquire for  
the chance of its passing through       
without touching them4 ,the problem     
bears some analogy to that of the       
chessmen1 and so far as the motion of   
translation of the stick is concerned   
7if we begin with this7 it presents no  
difficulty4 ,but as regards the rotation
 it is otherwise4 ,for any assigned     
linear velocity there is a certain      
angular velocity below which the stick  
may pass through without contact1 but   
above which it cannot4 ,and inasmuch as 
the former range is limited and the     
latter is unlimited1 we encounter the   
same impossibility as before in         
endeavouring to conceive a uniform      
distribution4 ,of course we might evade 
this particular difficulty by beginning 
with an estimate of the angular         
velocity1 when we should have to repeat 

what has just been said1 mutatis        
mutandis1 in reference to the linear    
velocity4  #g4 ,i am of course aware    
that there are a variety of problems    
current which seem to conflict with what
 has just been said1 but they will all  
submit to explanation4 ,for instance2   
,what is the chance that three straight 
lines1 taken or drawn at random1 shall  
be of such lengths as will admit of     
their forming a triangle8 ,there are two
 ways in which we may regard the        
problem4 ,we may1 for one thing1 start  
with the assumption of three lines not  
greater than a certain length n1 and    
then determine towards what limit the   
chance tends as n increases unceasingly4
 ,or1 we may maintain that the question 
is merely one of relative proportion of 
the three lines4 ,we may then start with
 any magnitude we please to represent   
one of the lines 7for simplicity1 say1  
the longest of them71 and consider that 
all possible shapes of a triangle will  
be represented by varying the lengths of

 the other two4 ,in either case we get a
 definite result without need to make an
 attempt to conceive any random         
selection from the infinity of possible 
length4  ,so in what is called the      
three-point problem3--,three points in  
space are selected at random2 find the  
chance of their forming an acute-angled 
triangle4 ,what is done is to start with
 a closed volume1--say a sphere1 from   
its superior simplicity1--find the      
chance 7on the assumption of uniform    
distribution within this volume72 and   
then conceive the continual enlargement 
without limit of this sphere4 ,so       
regarded the problem is perfectly       
consistent and intelligible1 though ,i  
fail to see why it should be termed a   
random selection in space rather than in
 a sphere4 ,of course if we started with
 a different volume1 say a cube1 we     
should get a different result2 and it is
 therefore contended 7e4g4 by ,mr       
,crofton in the ,educational ,times1 as 
already referred to7 that infinite space

 is more naturally and appropriately    
regarded as tended towards by the       
enlargement of a sphere than by that of 
a cube or any other figure4  ,again3 ,a 
group of integers is taken at random2   
show that the number thus taken is more 
likely to be odd than even4 ,what we do 
in answering this is to start with any  
finite number n1 and show that of all   
the possible combinations which can be  
made within this range there are more   
odd than even4 ,since this is true      
irrespective of the magnitude of n1 we  
are apt to speak as if we could conceive
 the selection being made at random from
 the true infinity contemplated in      
numeration4  #h4 ,where these conditions
 cannot be secured then it seems to me  
that the attempt to assign any finite   
value to the probability fails4 ,for    
instance1 in the following problem1     
proposed by ,mr ,j4 ,m4 ,wilson1 ,three 
straight lines are drawn at random on an
 infinite plane1 and a fourth line is   
drawn at random to intersect them3 find 

the probability of its passing through  
the triangle formed by the other three  
7,ed4 ,times1 ,reprint1 ,vol4 ,v4 p4    
#hb71 he offers the following solution3 
,of the four lines1 two must and two    
must not pass within the triangle formed
 by the remaining three4 ,since all are 
drawn at random1 the chance that the    
last drawn should pass through the      
triangle formed by the other three is   
consequently #a_/b4  ,i quote this      
solution because it seems to me to      
illustrate the difficulty to which ,i   
want to call attention4 ,as the problem 
is worded1 a triangle is supposed to be 
assigned by three straight lines4       
,however large it may be1 its size bears
 no finite ratio whatever to the        
indefinitely larger area outside it2    
and1 so far as ,i can put any           
intelligible construction on the        
supposition1 the chance of drawing a    
fourth random line which should happen  
to intersect this finite area must be   
reckoned as zero4 ,the problem ,mr      

,wilson has solved seems to me to be a  
quite different one1 viz4 ,given four   
intersecting straight lines1 find the   
chance that we should1 at random1 select
 one that passes through the triangle   
formed by the other three4  ,the same   
difficulty seems to me to turn up in    
most other attempts to apply this       
conception of randomness to real        
infinity4 ,the following seems an exact 
analogue of the above problem3--,a      
number is selected at random1 find the  
chance that another number selected at  
random shall be greater than the        
former2--the answer surely must be that 
the chance is unity1 viz4 certainty1    
because the range above any assigned    
number is infinitely greater than that  
below it4 ,or1 expressed in the only    
language in which ,i can understand the 
term 'infinity'1 what ,i mean is this4  
,if the first number be m and ,i am     
restricted to selecting up to n 7n  m7  
then the chance of exceeding m is n - m 
3 n2 if ,i am restricted to #bn then it 

is #bn - m 3 #bn and so on4 ,that is1   
however large n and m may be the        
expression is always intelligible2 but1 
m being chosen first1 n may be made as  
much larger than m as we please3 i4e4   
the chance may be made to approach as   
near to unity as we please4  ,i cannot  
but think that there is a similar       
fallacy in ,de ,morgan's admirably      
suggestive paper on ,infinity 7,camb4   
,phil4 ,trans4 ,vol4 #aa47 when he is   
discussing the three-point              
problem3--i4e4 given three points taken 
at random find the chance that they     
shall form an acute-angled triangle4    
,all that he shows is1 that if we start 
with one side as given and consider the 
subsequent possible positions of the    
opposite vertex1 there are infinitely as
 many such positions which would form an
 acute-angled triangle as an obtuse3    
but1 as before1 this is solving a       
different problem4  #i4 ,the nearest    
approach ,i can make towards true       
indefinite randomness1 or random        

selection from true indefiniteness1 is  
as follows4 ,suppose a circle with a    
tangent line extended indefinitely in   
each direction4 ,now from the centre    
draw radii at random2 in other words1   
let the semicircumference which lies    
towards the tangent be ultimately       
uniformly intersected by the radii4 ,let
 these radii be then produced so as to  
intersect the tangent line1 and consider
 the distribution of these points of    
intersection4 ,we shall obtain in the   
result one characteristic of our random 
distribution2 i4e4 no portion of this   
tangent1 however small or however       
remote1 but will find itself in the     
position ultimately of any small portion
 of the pavement in our supposed        
continual rainfall4 ,that is1 any such  
elementary patch will become more and   
more closely dotted over with the points
 of intersection4 ,but the other        
essential characteristic1 viz4 that of  
ultimately uniform distribution1 will be
 missing4 ,there will be a special form 

of distribution1--what in fact will have
 to be discussed in a future chapter    
under the designation of a 'law of      
error'1--by virtue of which the         
concentration will tend to be greatest  
at a certain point 7that of contact with
 the circle71 and will thin out from    
here in each direction according to an  
easily calculated formula4 ,the         
existence of such a state of things as  
this is quite opposed to the conception 
of true randomness4  #aj4 ,,iii4 ,apart 
from definitions and what comes of them1
 perhaps the most important question    
connected with the conception of        
,randomness is this3 ,how in any given  
case are we to determine whether an     
observed arrangement is to be considered
 a random one or not8 ,this question    
will have to be more fully discussed in 
a future chapter1 but we are already in 
a position to see our way through some  
of the difficulties involved in it4     
7#a7 ,if the events or objects under    
consideration are supposed to be        

continued indefinitely1 or if we know   
enough about the mode in which they are 
brought about to detect their ultimate  
tendency1--or even1 short of this1 if   
they are numerous enough to be beyond   
practical counting1--there is no great  
difficulty4 ,we are simply confronted   
with a question of fact1 to be settled  
like other questions of fact4 ,in the   
case of the rain-drops1 watch two equal 
squares of pavement or other surfaces1  
and note whether they come to be more   
and more densely uniformly and evenly   
spotted over3 if they do1 then the      
arrangement is what we call a random    
one4 ,if ,i want to know whether a      
tobacco-pipe really breaks at random1   
and would therefore serve as an         
illustration of the problem proposed    
some pages back1 ,i have only to drop   
enough of them and see whether pieces of
 all possible lengths are equally       
represented in the long run4 ,or1 ,i may
 argue deductively1 from what ,i know   
about the strength of materials and the 

molecular constitution of such bodies1  
as to whether fractures of small and    
large pieces are all equally likely to  
occur4  #aa4 ,the reader's attention    
must be carefully directed to a source  
of confusion here1 arising out of a     
certain cross-division4 ,what we are now
 discussing is a question of fact1 viz4 
the nature of a certain ultimate        
arrangement2 we are not discussing the  
particular way in which it is brought   
about4 ,in other words1 the antithesis  
is between what is and what is not      
random3 it is not between what is random
 and what is designed4 ,as we shall see 
in a few moments it is quite possible   
that an arrangement which is the        
result1--if ever anything were so1--of  
'design'1 may nevertheless present the  
unmistakeable stamp of randomness of    
arrangement4  ,consider a case which has
 been a good deal discussed1 and to     
which we shall revert again3 the        
arrangement of the stars4 ,the question 
here is rather complicated by the fact  

that we know nothing about the actual   
mutual positions of the stars1 all that 
we can take cognizance of being their   
apparent or visible places as projected 
upon the surface of a supposed sphere4  
,appealing to what alone we can thus    
observe1 it is obvious that the         
arrangement1 as a whole1 is not of the  
random sort4 ,the ,milky ,way and the   
other resolvable nebul1 as they present 
themselves to us1 are as obvious an     
infraction of such an arrangement as    
would be the occurrence here and there  
of patches of ground in a rainfall which
 received a vast number more drops than 
the spaces surrounding them4 ,if we     
leave these exceptional areas out of the
 question and consider only the stars   
which are visible by the naked eye or by
 slight telescopic power1 it seems      
equally certain that the arrangement is1
 for the most part1 a fairly            
representative random one4 ,by this we  
mean nothing more than the fact that    
when we mark off any number of equal    

areas on the visible sphere these are   
found to contain approximately the same 
number of stars4  ,the actual           
arrangement of the stars in space may   
also be of the same character3 that is1 
the apparently denser aggregation may be
 apparent only1 arising from the fact   
that we are looking through regions     
which are not more thickly occupied but 
are merely more extensive4 ,the         
alternative before us1 in fact1 is this4
 ,if the whole volume1 so to say1 of the
 starry heavens is tolerably regular in 
shape1 then the arrangement of the stars
 is not of the random order2 if that    
volume is very irregular in shape1 it is
 possible that the arrangement within it
 may be throughout of that order4  #ab4 
7#b7 ,when the arrangement in question  
includes but a comparatively small      
number of events or objects1 it becomes 
much more difficult to determine whether
 or not it is to be designated a random 
one4 ,in fact we have to shift our      
ground1 and to decide not by what has   

been actually observed but by what we   
have reason to conclude would be        
observed if we could continue our       
observation much longer4 ,this          
introduces what is called ',inverse     
,probability'1 viz4 the determination of
 the nature of a cause from the nature  
of the observed effect2 a question which
 will be fully discussed in a future    
chapter4 ,but some introductory remarks 
may be conveniently made here4  ,every  
problem of ,probability1 as the subject 
is here understood1 introduces the      
conception of an ultimate limit1 and    
therefore presupposes an indefinite     
possibility of repetition4 ,when we have
 only a finite number of occurrences    
before us1 direct evidence of the       
character of their arrangement fails us1
 and we have to fall back upon the      
nature of the agency which produces     
them4 ,and as the number becomes smaller
 the confidence with which we can       
estimate the nature of the agency       
becomes gradually less4  ,begin with an 

intermediate case4 ,there is a small    
lawn1 sprinkled over with daisies3 is   
this a random arrangement8 ,we feel some
 confidence that it is so1 on mere      
inspection2 meaning by this that        
7negatively7 no trace of any regular    
pattern can be discerned and            
7affirmatively7 that if we take any     
moderately small area1 say a square     
yard1 we shall find much about the same 
number of the plants included in it4    
,but we can help ourselves by an appeal 
to the known agency of distribution     
here4 ,we know that the daisy spreads by
 seed1 and considering the effect of the
 wind and the continued sweeping and    
mowing of the lawn we can detect causes 
at work which are analogous to those by 
which the dealing of cards and the      
tossing of dice are regulated4  ,in the 
above case the appeal to the process of 
production was subsidiary1 but when we  
come to consider the nature of a very   
small succession or group this appeal   
becomes much more important4 ,let us be 

told of a certain succession of 'heads' 
and 'tails' to the number of ten4 ,the  
range here is far too small for         
decision1 and unless we are told whether
 the agent who obtained them was tossing
 or designing we are quite unable to say
 whether or not the designation of      
'random' ought to be applied to the     
result obtained4 ,the truth must never  
be forgotten that though 'design' is    
sure to break down in the long run if it
 make the attempt to produce directly   
the semblance of randomness1#d yet for a
 short spell it can simulate it         
perfectly4 ,any short succession1 say of
 heads and tails1 may have been equally 
well brought about by tossing or by     
deliberate choice4  #ac4 ,the reader    
will observe that this question of      
randomness is being here treated as     
simply one of ultimate statistical fact4
 ,i have fully admitted that this is not
 the primitive conception1 nor is it the
 popular interpretation1 but to adopt it
 seems the only course open to us if we 

are to draw inferences such as those    
contemplated in ,probability4 ,when we  
look to the producing agency of the     
ultimate arrangement we may find this   
very various4 ,it may prove itself to be
 7a few stages back7 one of conscious   
deliberate purpose1 as in drawing a card
 or tossing a die3 it may be the outcome
 of an extremely complicated interaction
 of many natural causes1 as in the      
arrangement of the flowers scattered    
over a lawn or meadow3 it may be of a   
kind of which we know literally nothing 
whatever1 as in the case of the actual  
arrangement of the stars relatively to  
each other4  ,this was the state of     
things had in view when it was said a   
few pages back that randomness and      
design would result in something of a   
cross-division4 ,plenty of arrangements 
in which design had a hand1 a stage or  
two back1 can be mentioned1 which would 
be quite indistinguishable in their     
results from those in which no design   
whatever could be traced4 ,perhaps the  

most striking case in point here is to  
be found in the arrangement of the      
digits in one of the natural            
arithmetical constants1 such as  or e1  
or in a table of logarithms4 ,if we look
 to the process of production of these  
digits1 no extremer instance can be     
found of what we mean by the antithesis 
of randomness3 every figure has its     
necessarily pre-ordained position1 and a
 moment's flagging of intention would   
defeat the whole purpose of the         
calculator4 ,and yet1 if we look to     
results only1 no better instance can be 
found than one of these rows of digits  
if it were intended to illustrate what  
we practically understand by a chance   
arrangement of a number of objects4     
,each digit occurs approximately equally
 often1 and this tendency develops as we
 advance further3 the mutual            
juxtaposition of the digits also shows  
the same tendency1 that is1 any digit   
7say #e7 is just as often followed by #f
 or #g as by any of the others4 ,in     

fact1 if we were to take the whole row  
of hitherto calculated figures1 cut off 
the first five as familiar to us all1   
and contemplate the rest1 no one would  
have the slightest reason to suppose    
that these had not come out as the      
results of a die with ten equal faces4  
#ad4 ,if it be asked why this is so1 a  
rather puzzling question is raised4     
,wherever physical causation is involved
 we are generally understood to have    
satisfied the demand implied in this    
question if we assign antecedents which 
will be followed regularly by the event 
before us2 but in geometry and          
arithmetic there is no opening for      
antecedents4 ,what we then commonly look
 for is a demonstration1 i4e4 the       
resolution of the observed fact into    
axioms if possible1 or at any rate into 
admitted truths of wider generality4 ,i 
do not know that a demonstration can be 
given as to the existence of this       
characteristic of statistical randomness
 in such successions of digits as those 

under consideration4 ,but the following 
remarks may serve to shift the onus of  
unlikelihood by suggesting that the     
preponderance of analogy is rather in   
favour of the existence4  ,take the     
well-known constant  for consideration4 
,this stands for a quantity which       
presents itself in a vast number of     
arithmetical and geometrical relations2 
let us take for examination the best    
known of these1 by regarding it as      
standing for the ratio of the           
circumference to the diameter of a      
circle4 ,so regarded1 it is nothing more
 than a simple case of the measurement  
of a magnitude by an arbitrarily        
selected unit4 ,conceive then that we   
had before us a rod or line and that we 
wished to measure it with absolute      
accuracy4 ,we must suppose--if we are to
 have a suitable analogue to the        
determination of  to several hundred    
figures1--that by the application of    
continued higher magnifying power we can
 detect ever finer subdivisions in the  

graduation4 ,we lay our rod against the 
scale and find it1 say1 fall between #ca
 and #cb inches2 we then look at the    
next division of the scale1 viz4 that   
into tenths of an inch4 ,can we see the 
slightest reason why the number of these
 tenths should be other than independent
 of the number of whole inches8 ,the    
piece over which we are measuring may in
 fact be regarded as an entirely new    
piece1 which had fallen into our hands  
after that of #ca inches had been       
measured and done with2 and similarly   
with every successive piece over1 as we 
proceed to the ever finer and finer     
divisions4  ,similar remarks may be made
 about most other incommensurable       
quantities1 such as irreducible roots4  
,conceive two straight lines at right   
angles1 and that we lay off a certain   
number of inches along each of these    
from the point of intersection2 say two 
and five inches1 and join the           
extremities of these so as to form the  
diagonal of a right-angled triangle4 ,if

 we proceed to measure this diagonal in 
terms of either of the other lines we   
are to all intents and purposes         
extracting a square root4 ,we should    
expect1 rather than otherwise1 to find  
here1 as in the case of 1 that          
incommensurability and resultant        
randomness of order in the digits was   
the rule1 and commensurability was the  
exception4 ,now and then1 as when the   
two sides were three and four1 we should
 find the diagonal commensurable with   
them2 but these would be the occasional 
exceptions1 or rather they would be the 
comparatively finite exceptions amidst  
the indefinitely numerous cases which   
furnished the rule4  #ae4 ,the best way 
perhaps of illustrating the truly random
 character of such a row of figures is  
by appealing to graphical aid4 ,it is   
not easy here1 any more than in ordinary
 statistics1 to grasp the import of mere
 figures2 whereas the arrangement of    
groups of points or lines is much more  
readily seized4 ,the eye is very quick  

in detecting any symptoms of regularity 
in the arrangement1 or any tendency to  
denser aggregation in one direction than
 in another4 ,how then are we to dispose
 our figures so as to force them to     
display their true character8 ,i should 
suggest that we set about drawing a line
 at random2 and1 since we cannot trust  
our own unaided efforts to do this1 that
 we rely upon the help of such a table  
of figures to do it for us1 and then    
examine with what sort of efficiency    
they can perform the task4 ,the problem 
of drawing straight lines at random1    
under various limitations of direction  
or intersection1 is familiar enough1 but
 ,i do not know that any one has        
suggested the drawing of a line whose   
shape as well as position shall be of a 
purely random character4 ,for simplicity
 we suppose the line to be confined to a
 plane4  ,the definition of such a line 
does not seem to involve any particular 
difficulty4 ,phrased in accordance with 
the ordinary language we should describe

 it as the path 7i4e4 any path7 traced  
out by a point which at every moment is 
as likely to move in any one direction  
as in any other4 ,that we could not     
ourselves draw such a line1 and that we 
could not get it traced by any physical 
agency1 is certain4 ,the mere inertia of
 any moving body will always give it a  
tendency1 however slight1 to go on in a 
straight line at each moment1 instead of
 being instantly responsive to          
instantaneously varying dictates as to  
its direction of motion4 ,nor can we    
conceive or picture such a line in its  
ultimate or ideal condition4 ,but it is 
easy to give a graphical approximation  
to it1 and it is easy also to show how  
this approximation may be carried on as 
far as we please towards the ideal in   
question4  ,we may proceed as follows4  
,take a sheet of the ordinary ruled     
paper prepared for the graphical        
exposition of curves4 ,select as our    
starting point the intersection of two  
of these lines1 and consider the eight  

'points of the compass' indicated by    
these lines and the bisections of the   
contained right angles4#e ,for          
suggesting the random selection amongst 
these directions let them be numbered   
from #j to #g1 and let us say that a    
line measured due 'north' shall be      
designated by the figure #j1            
'north-east' by #a1 and so on4 ,the     
selection amongst these numbers1 and    
therefore directions1 at every corner1  
might be handed over to a die with eight
 faces2 but for the purpose of the      
illustration in view we select the      
digits #j to #g as they present         
themselves in the calculated value of 4 
,the sort of path along which we should 
travel by a series of such steps thus   
taken at random may be readily          
conceived2 it is given at the end of    
this chapter4  ,for the purpose with    
which this illustration was proposed1   
viz4 the graphical display of the       
succession of digits in any one of the  
incommensurable constants of arithmetic 

or geometry1 the above may suffice4     
,after actually testing some of them in 
this way they seem to me1 so far as the 
eye1 or the theoretical principles to be
 presently mentioned1 are any guide1 to 
answer quite fairly to the description  
of randomness4  #af4 ,as we are on the  
subject1 however1 it seems worth going  
farther by enquiring how near we could  
get to the ideal of randomness of       
direction4 ,to carry this out completely
 two improvements must be made4 ,for one
 thing1 instead of confining ourselves  
to eight directions we must admit an    
infinite number4 ,this would offer no   
great difficulty2 for instead of        
employing a small number of digits we   
should merely have to use some kind of  
circular teetotum which would rest      
indifferently in any direction4 ,but in 
the next place instead of short finite  
steps we must suppose them indefinitely 
short4 ,it is here that the actual      
unattainability makes itself felt4 ,we  
are familiar enough with the device1    

employed by ,newton1 of passing from the
 discontinuous polygon to the continuous
 curve4 ,but we can resort to this      
device because the ideal1 viz4 the      
curve1 is as easily drawn 7and1 ,i      
should say1 as easily conceived or      
pictured7 as any of the steps which lead
 us towards it4 ,but in the case before 
us it is otherwise4 ,the line in        
question will remain discontinuous1 or  
rather angular1 to the last3 for its    
angles do not tend even to lose their   
sharpness1 though the fragments which   
compose them increase in number and     
diminish in magnitude without any limit4
 ,and such an ideal is not conceivable  
as an ideal4 ,it is as if we had a rough
 body under the microscope1 and found   
that as we subjected it to higher and   
higher powers there was no tendency for 
the angles to round themselves off4 ,our
 'random line' must remain as 'spiky' as
 ever1 though the size of its spikes of 
course diminishes without any limit4    
,the case therefore seems to be this4   

,it is easy1 in words1 to indicate the  
conception by speaking of a line which  
at every instant is as likely to take   
one direction as another4 ,it is easy   
moreover to draw such a line with any   
degree of minuteness which we choose to 
demand4 ,but it is not possible to      
conceive or picture the line in its     
ultimate form4#f ,there is in fact no   
'limit' here1 intelligible to the       
understanding or picturable by the      
imagination 7corresponding to the       
asymptote of a curve1 or the continuous 
curve to the incessantly developing     
polygon71 towards which we find         
ourselves continually approaching1 and  
which therefore we are apt to conceive  
ourselves as ultimately attaining4 ,the 
usual assumption therefore which        
underlies the ,newtonian infinitesimal  
geometry and the ,differential          
,calculus1 ceases to apply here4  #ag4  
,if we like to consider such a line in  
one of its approximate stages1 as above 
indicated1 it seems to me that some of  

the usual theorems of ,probability1     
where large numbers are concerned1 may  
safely be applied4 ,if it be asked1 for 
instance1 whether such a line will      
ultimately tend to stray indefinitely   
far from its starting point1            
,bernoulli's ',law of ,large ,numbers'  
may be appealed to1 in virtue of which  
we should say that it was excessively   
unlikely that its divergence should be  
relatively great4 ,recur to our         
graphical illustration1 and consider    
first the resultant deviation of the    
point 7after a great many steps7 right  
or left of the vertical line through the
 starting point4 ,of the eight          
admissible motions at each stage two    
will not affect this relative position1 
whilst the other six are equally likely 
to move us a step to the right or to the
 left4 ,our resultant 'drift' therefore 
to the right or left will be analogous  
to the resultant difference between the 
number of heads and tails after a great 
many tosses of a penny4 ,now the        

well-known outcome of such a number of  
tosses is that ultimately the           
proportional approximation to the       
priori probability1 i4e4 to equality of 
heads and tails1 is more and more nearly
 carried out1 but that the absolute     
deflection is more and more widely      
displayed4  ,applying this to the case  
in point1 and remembering that the      
results apply equally to the horizontal 
and vertical directions1 we should say  
that after any very great number of such
 'steps' as those contemplated1 the     
ratio of our distance from the starting 
point to the whole distance travelled   
will pretty certainly be small1 whereas 
the actual distance from it would be    
large4 ,we should also say that the     
longer we continued to produce such a   
line the more pronounced would these    
tendencies become4 ,so far as concerns  
this test1 and that afforded by the     
general appearance of the lines         
drawn1--this last1 as above remarked1   
being tolerably trustworthy1--,i feel no

 doubt as to the generally 'random'     
character of the rows of figures        
displayed by the incommensurable or     
irrational ratios in question4  ,as it  
may interest the reader to see an actual
 specimen of such a path ,i append one  
representing the arrangement of the     
eight digits from #j to #g in the value 
of 4 ,the data are taken from ,mr       
,shanks' astonishing performance in the 
calculation of this constant to #gjg    
places of figures 7,proc4 of ,r4 ,s41   
,,xxi4 p4 #cai74 ,of these1 after       
omitting #h and #i1 there remain #efh2  
the diagram represents the course traced
 out by following the direction of these
 as the clue to our path4 ,many of the  
steps have of course been taken in      
opposite directions twice or oftener4   
,the result seems to me to furnish a    
very fair graphical indication of       
randomness4 ,i have compared it with    
corresponding paths furnished by rows of
 figures taken from logarithmic tables1 
and in other ways1 and find the results 

to be much the same4  ,figure3 ,digits  
of  as a path4  #a4 ,according to ,prof4
 ,skeat 7,etymological ,dictionary7 the 
earliest   known meaning is that of     
furious action1 as in a charge of       
cavalry4 ,the etymology1 he considers1  
is connected with the   ,teutonic word  
rand 7brim71 and implies the furious and
 irregular   action of a river full to  
the brim4  #b4 ,see the problem paper of
 ,jan4 #ah1 #ahed1 in the ,cambridge    
,mathematical ,tripos4  #c4 ,as1        
according to ,mr ,h4 ,godfray1 the      
majority of the candidates   did assume 
when the problem was once proposed in an
   examination4 ,see the ,educational   
,times 7,reprint1 ,vol4 ,,vii4   p4     
#ii47  #d4 ,vide p4 #fh4  #e4 ,it would 
of course be more complete to take ten  
alternatives of   direction1 and thus to
 omit none of the digits2 but this is   
much   more troublesome in practice than
 to confine ourselves to eight4  #f4    
,any more than we picture the shape of  
an equiangular spiral at the   centre4  

,,chapter ,,vi4#a   ,,the ,,subjective  
,,side ,,of ,,probability4 ,,measurement
 ,,of ,,belief4  #a4 ,originally written
 in somewhat of a spirit of protest     
against what   seemed to me the         
prevalent disposition to follow ,de     
,morgan in taking   too subjective a    
view of the science4 ,in reading it     
through now ,i   cannot find any single 
sentence to which ,i could take distinct
   objection1 though ,i must admit that 
if ,i were writing it entirely   afresh 
,i should endeavour to express myself   
with less emphasis1 and   ,i have made  
alterations in that direction4 ,the     
reader who wishes to   see a view not   
substantially very different from mine1 
but expressed   with a somewhat opposite
 emphasis1 can refer to ,mr ,f4 ,y4     
,edgeworth's   article on ,the          
,philosophy of ,chance 7,mind1 ,vol4    
,,ix47   #a4 ,having now obtained a     
clear conception of a certain kind of   
series1 the next enquiry is1 ,what is to
 be done with this series8 ,how is it to

 be employed as a means of making       
inferences8 ,the general step that we   
are now about to take might be described
 as one from the objective to the       
subjective1 from the things themselves  
to the state of our minds in            
contemplating them4  ,the reader should 
observe that a substitution has1 in a   
great number of cases1 already been made
 as a first stage towards bringing the  
things into a shape fit for calculation4
 ,this substitution1 as described in    
former chapters1 is1 in a measure1 a    
process of idealization4 ,the series we 
actually meet with are apt to show a    
changeable type1 and the individuals of 
them will sometimes transgress their    
licensed irregularity4 ,hence they have 
to be pruned a little into shape1 as    
natural objects almost always have      
before they are capable of being        
accurately reasoned about4 ,the form in 
which the series emerges is that of a   
series with a fixed type4 ,this         
imaginary or ideal series is the basis  

of our calculation4  #b4 ,it must not be
 supposed that this is at all at        
variance with the assertion previously  
made1 that ,probability is a science of 
inference about real things2 it is only 
by a substitution of the above kind that
 we are enabled to reason about the     
things4 ,in nature nearly all phenomena 
present themselves in a form which      
departs from that rigorously accurate   
one which scientific purposes mostly    
demand1 so we have to introduce an      
imaginary series1 which shall be free   
from any such defects4 ,the only        
condition to be fulfilled is1 that the  
substitution is to be as little         
arbitrary1 that is1 to vary from the    
truth as slightly1 as possible4 ,this   
kind of substitution generally passes   
without notice when natural objects of  
any kind are made subjects of exact     
science4 ,i direct distinct attention to
 it here simply from the apprehension   
that want of familiarity with the       
subject-matter might lead some readers  

to suppose that it involves1 in this    
case1 an exceptional deflection from    
accuracy in the formal process of       
inference4  ,it may be remarked also    
that the adoption of this imaginary     
series offers no countenance whatever to
 the doctrine criticised in the last    
chapter1 in accordance with which it was
 supposed that our series possessed a   
fixed unchangeable type which was merely
 the development of the probabilities of
 things1 to use ,laplace's expression4  
,it differs from anything contemplated  
on that hypothesis by the fact that it  
is to be recognized as a necessary      
substitution of our own for the actual  
series1 and to be kept in as close      
conformity with facts as possible4 ,it  
is a mere fiction or artifice           
necessarily resorted to for the purpose 
of calculation1 and for this purpose    
only4  ,this caution is the more        
necessary1 because in the example that  
,i shall select1 and which belongs to   
the most favourite class of examples in 

this subject1 the substitution becomes  
accidentally unnecessary4 ,the things1  
as has been repeatedly pointed out1 may 
sometimes need no trimming1 because in  
the form in which they actually present 
themselves they are almost idealized4   
,in most cases a good deal of alteration
 is necessary to bring the series into  
shape1 but in some--prominently in the  
case of games of chance--we find the    
alterations1 for all practical purposes1
 needless4  #c4 ,we start then1 from    
such a series as this1 upon the enquiry1
 ,what kind of inference can be made    
about it8 ,it may assist the logical    
reader to inform him that our first step
 will be analogous to one class of what 
are commonly known as immediate         
inferences1--inferences1 that is1 of the
 type1--',all men are mortal1 therefore 
any particular man or men are mortal4'  
,this case1 simple and obvious as it is 
in ,logic1 requires very careful        
consideration in ,probability4  ,it is  
obvious that we must be prepared to form

 an opinion upon the propriety of taking
 the step involved in making such an    
inference4 ,hitherto we have had as     
little to do as possible with the       
irregular individuals2 we have regarded 
them simply as fragments of a regular   
series4 ,but we cannot long continue to 
neglect all consideration of them4 ,even
 if these events in the gross be        
tolerably certain1 it is not only in the
 gross that we have to deal with them2  
they constantly come before us a few at 
a time1 or even as individuals1 and we  
have to form some opinion about them in 
this state4 ,an insurance office1 for   
instance1 deals with numbers large      
enough to obviate most of the           
uncertainty1 but each of their          
transactions has another party          
interested in it--,what has the man who 
insures to say to their proceedings8 for
 to him this question becomes an        
individual one4 ,and even the office    
itself receives its cases singly1 and   
would therefore like to have as clear   

views as possible about these single    
cases4 ,now1 the remarks made in the    
preceding chapters about the subjects   
which ,probability discusses might seem 
to preclude all enquiries of this kind1 
for was not ignorance of the individual 
presupposed to such an extent that even 
7as will be seen hereafter7 causation   
might be denied1 within considerable    
limits1 without affecting our           
conclusions8 ,the answer to this enquiry
 will require us to turn now to the     
consideration of a totally distinct side
 of the question1 and one which has not 
yet come before us4 ,our best           
introduction to it will be by the       
discussion of a special example4  #d4   
,let a penny be tossed up a very great  
many times2 we may then be supposed to  
know for certain this fact 7amongst many
 others7 that in the long run head and  
tail will occur about equally often4    
,but suppose we consider only a moderate
 number of throws1 or fewer still1 and  
so continue limiting the number until we

 come down to three or two1 or even one8
 ,we have1 as the extreme cases1        
certainty or something undistinguishably
 near it1 and utter uncertainty4 ,have  
we not1 between these extremes1 all     
gradations of belief8 ,there is a large 
body of writers1 including some of the  
most eminent authorities upon this      
subject1 who state or imply that we are 
distinctly conscious of such a variation
 of the amount of our belief1 and that  
this state of our minds can be measured 
and determined with almost the same     
accuracy as the external events to which
 they refer4 ,the principal mathematical
 supporter of this view is ,de ,morgan1 
who has insisted strongly upon it in all
 his works on the subject4 ,the clearest
 exposition of his opinions will be     
found in his ,formal ,logic1 in which   
work he has made the view which we are  
now discussing the basis of his system4 
,he holds that we have a certain amount 
of belief of every proposition which may
 be set before us1 an amount which in   

its nature admits of determination1     
though we may practically find it       
difficult in any particular case to     
determine it4 ,he considers1 in fact1   
that ,probability is a sort of sister   
science to ,formal ,logic1#a speaking of
 it in the following words3 ,i cannot   
understand why the study of the effect1 
which partial belief of the premises    
produces with respect to the conclusion1
 should be separated from that of the   
consequences of supposing the former to 
be absolutely true4#b ,in other words1  
there is a science--,formal             
,logic--which investigates the rules    
according to which one proposition can  
be necessarily inferred from another2 in
 close correspondence with this there is
 a science which investigates the rules 
according to which the amount of our    
belief of one proposition varies with   
the amount of our belief of other       
propositions with which it is connected4
  ,the same view is also supported by   
another high authority1 the late ,prof4 

,donkin1 who says 7,phil4 ,mag4 ,may1   
#ahea71 ,it will1 ,i suppose1 be        
generally admitted1 and has often been  
more or less explicitly stated1 that the
 subject-matter of calculation in the   
mathematical theory of ,probabilities is
 quantity of belief4  #e4 ,before       
proceeding to criticise this opinion1   
one remark may be made upon it which has
 been too frequently overlooked4 ,it    
should be borne in mind that1 even were 
this view of the subject not actually   
incorrect1 it might be objected to as   
insufficient for the purpose of a       
definition1 on the ground that variation
 of belief is not confined to           
,probability4 ,it is a property with    
which that science is concerned1 no     
doubt1 but it is a property which meets 
us in other directions as well4 ,in     
every case in which we extend our       
inferences by ,induction or ,analogy1 or
 depend upon the witness of others1 or  
trust to our own memory of the past1 or 
come to a conclusion through conflicting

 arguments1 or even make a long and     
complicated deduction by mathematics or 
logic1 we have a result of which we can 
scarcely feel as certain as of the      
premises from which it was obtained4 ,in
 all these cases then we are conscious  
of varying quantities of belief1 but are
 the laws according to which the belief 
is produced and varied the same8 ,if    
they cannot be reduced to one harmonious
 scheme1 if in fact they can at best be 
brought to nothing but a number of      
different schemes1 each with its own    
body of laws and rules1 then it is vain 
to endeavour to force them into one     
science4  ,this opinion is strengthened 
by observing that most of the writers   
who adopt the definition in question do 
practically dismiss from consideration  
most of the above-mentioned examples of 
diminution of belief1 and confine their 
attention to classes of events which    
have the property discussed in ,chap    
,i41 viz4 'ignorance of the few1        
knowledge of the many4' ,it is quite    

true that considerable violence has to  
be done to some of these examples1 by   
introducing exceedingly arbitrary       
suppositions into them1 before they can 
be forced to assume a suitable form4    
,but still there is little doubt that1  
if we carefully examine the language    
employed1 we shall find that in almost  
every case assumptions are made which   
virtually imply that our knowledge of   
the individual is derived from          
propositions given in the typical form  
described in ,chap ,i4 ,this will be    
more fully proved when we come to       
consider some common misapplications of 
the science4  #f4 ,even then1 if the    
above-mentioned view of the subject were
 correct1 it would yet1 ,i consider1 be 
insufficient for the purpose of a       
definition2 but it is at least very     
doubtful whether it is correct4 ,before 
we could properly assign to the belief  
side of the question the prominence     
given to it by ,de ,morgan and others1  
certainly before the science could be   

defined from that side1 it would be     
necessary1 it appears1 to establish the 
two following positions1 against both of
 which strong objections can be brought4
  7#a7 ,that our belief of every        
proposition is a thing which we can1    
strictly speaking1 be said to measure2  
that there must be a certain amount of  
it in every case1 which we can realize  
somehow in consciousness and refer to   
some standard so as to pronounce upon   
its value4  7#b7 ,that the value thus   
apprehended is the correct one according
 to the theory1 viz4 that it is the     
exact fraction of full conviction that  
it should be4 ,this statement will      
perhaps seem somewhat obscure at first2 
it will be explained presently4  #g4    
7,i47 ,now1 in the first place1 as      
regards the difficulty of obtaining any 
measure of the amount of our belief4    
,one source of this difficulty is too   
obvious to have escaped notice2 this is 
the disturbing influence produced on the
 quantity of belief by any strong       

emotion or passion4 ,a deep interest in 
the matter at stake1 whether it excite  
hope or fear1 plays great havoc with the
 belief-meter1 so that we must assume   
the mind to be quite unimpassioned in   
weighing the evidence4 ,this is noticed 
and acknowledged by ,laplace and others2
 but these writers seem to me to assume 
it to be the only source of error1 and  
also to be of comparative unimportance4 
,even if it were the only source of     
error ,i cannot see that it would be    
unimportant4 ,we experience hope or fear
 in so very many instances1 that to omit
 such influences from consideration     
would be almost equivalent to saying    
that whilst we profess to consider the  
whole quantity of our belief we will in 
reality consider only a portion of it4  
,very strong feelings are1 of course1   
exceptional1 but we should nevertheless 
find that the emotional element1 in some
 form or other1 makes itself felt on    
almost every occasion4 ,it is very      
seldom that we cannot speak of our      

surprise or expectation in reference to 
any particular event4 ,both of these    
expressions1 but especially the former1 
seem to point to something more than    
mere belief4 ,it is true that the word  
'expectation' is generally defined in   
treatises on ,probability as equivalent 
to belief2 but it seems doubtful whether
 any one who attends to the popular use 
of the terms would admit that they were 
exactly synonymous4 ,be this however as 
it may1 the emotional element is present
 upon almost every occasion1 and its    
disturbing influence therefore is       
constantly at work4  #h4 ,another cause1
 which co-operates with the former1 is  
to be found in the extreme complexity   
and variety of the evidence on which our
 belief of any proposition depends4     
,hence it results that our actual belief
 at any given moment is one of the most 
fugitive and variable things possible1  
so that we can scarcely ever get        
sufficiently clear hold of it to measure
 it4 ,this is not confined to the times 

when our minds are in a turmoil of      
excitement through hope or fear4 ,in our
 calmest moments we shall find it no    
easy thing to give a precise answer to  
the question1 ,how firmly do ,i hold    
this or that belief8 ,there may be one  
or two prominent arguments in its       
favour1 and one or two corresponding    
objections against it1 but this is far  
from comprising all the causes by which 
our state of belief is produced4        
,because such reasons as these are all  
that can be practically introduced into 
oral or written controversies1 we must  
not conclude that it is by these only   
that our conviction is influenced4 ,on  
the contrary1 our conviction generally  
rests upon a sort of chaotic basis      
composed of an infinite number of       
inferences and analogies of every       
description1 and these moreover         
distorted by our state of feeling at the
 time1 dimmed by the degree of our      
recollection of them afterwards1 and    
probably received from time to time with

 varying force according to the way in  
which they happen to combine in our     
consciousness at the moment4 ,to borrow 
a striking illustration from ,abraham   
,tucker1 the substructure of our        
convictions is not so much to be        
compared to the solid foundations of an 
ordinary building1 as to the piles of   
the houses of ,rotterdam which rest     
somehow in a deep bed of soft mud4 ,they
 bear their weight securely enough1 but 
it would not be easy to point out       
accurately the dependence of the        
different parts upon one another4       
,directly we begin to think of the      
amount of our belief1 we have to think  
of the arguments by which it is         
produced--in fact1 these arguments will 
intrude themselves without our choice4  
,as each in turn flashes through the    
mind1 it modifies the strength of our   
conviction2 we are like a person        
listening to the confused hubbub of a   
crowd1 where there is always something  
arbitrary in the particular sound we    

choose to listen to4 ,there may be      
reasons enough to suffice abundantly for
 our ultimate choice1 but on examination
 we shall find that they are by no means
 apprehended with the same force at     
different times4 ,the belief produced by
 some strong argument may be very       
decisive at the moment1 but it will     
often begin to diminish when the        
argument is not actually before the     
mind4 ,it is like being dazzled by a    
strong light2 the impression still      
remains1 but begins almost immediately  
to fade away4 ,i think that this is the 
case1 however we try to limit the       
sources of our conviction4  #i4 7,,ii47 
,but supposing that it were possible to 
strike a sort of average of this        
fluctuating state1 should we find this  
average to be of the amount assigned by 
theory8 ,in other words1 is our natural 
belief in the happening of two different
 events in direct proportion to the     
frequency with which those events happen
 in the long run8 ,there is a lottery   

with #ajj tickets and ten prizes2 is a  
man's belief that he will get a prize   
fairly represented by one-tenth of      
certainty8 ,the mere reference to a     
lottery should be sufficient to disprove
 this4 ,lotteries have flourished at all
 times1 and have never failed to be     
abundantly supported1 in spite of the   
most perfect conviction1 on the part of 
many1 if not of most1 of those who put  
into them1 that in the long run all will
 lose4 ,deductions should undoubtedly be
 made for those who act from            
superstitious motives1 from belief in   
omens1 dreams1 and so on4 ,but apart    
from these1 and supposing any one to    
come fortified by all that mathematics  
can do for him1 it is difficult to      
believe that his natural impressions    
about single events would be always what
 they should be according to theory4    
,are there many who can honestly declare
 that they would have no desire to buy a
 single ticket8 ,they would probably say
 to themselves that the sum they paid   

away was nothing worth mentioning to    
lose1 and that there was a chance of    
gaining a great deal2 in other words1   
they are not apportioning their belief  
in the way that theory assigns4  ,what  
bears out this view is1 that the same   
persons who would act in this way in    
single instances would often not think  
of doing so in any but single instances4
 ,in other words1 the natural tendency  
here is to attribute too great an amount
 of belief where it is or should be     
small2 i4e4 to depreciate the risk in   
proportion to the contingent advantage4 
,they would very likely1 when argued    
with1 attach disparaging epithets to    
this state of feeling1 by calling it an 
unaccountable fascination1 or something 
of that kind1 but of its existence there
 can be little doubt4 ,we are speaking  
now of what is the natural tendency of  
our minds1 not of that into which they  
may at length be disciplined by         
education and thought4 ,if1 however1    
educated persons have succeeded for the 

most part in controlling this tendency  
in games of chance1 the spirit of       
reckless speculation has scarcely yet   
been banished from commerce4 ,on        
examination1 this tendency will be found
 so prevalent in all ages1 ranks1 and   
dispositions1 that it would be          
inadmissible to neglect it in order to  
bring our supposed instincts more       
closely into accordance with the        
commonly received theories of           
,probability4  #aj4 ,there is another   
aspect of this question which has been  
often overlooked1 but which seems to    
deserve some attention4 ,granted that we
 have an instinct of credence1 why      
should it be assumed that this must be  
just of that intensity which subsequent 
experience will justify8 ,our instincts 
are implanted in us for good purposes1  
and are intended to act immediately and 
unconsciously4 ,they are1 however1      
subject to control1 and have to be      
brought into accordance with what we    
believe to be true and right4 ,in other 

departments of psychology we do not     
assume that every spontaneous prompting 
of nature is to be left just as we find 
it1 or even that on the average1        
omitting individual variations1 it is   
set at that pitch that will be found in 
the end to be the best when we come to  
think about it and assign its rules4    
,take1 for example1 the case of         
resentment4 ,here we have an instinctive
 tendency1 and one that on the whole is 
good in its results4 ,but moralists are 
agreed that almost all our efforts at   
self-control are to be directed towards 
subduing it and keeping it in its right 
direction4 ,it is assumed to be given as
 a sort of rough protection1 and to be  
set1 if one might so express oneself1 at
 too high a pitch to be deliberately and
 consciously acted on in society4 ,may  
not something of this kind be the case  
also with our belief8 ,i only make a    
passing reference to this point here1 as
 on the theory of ,probability adopted  
in this work it does not appear to be at

 all material to the science4 ,but it   
seems a strong argument against the     
expediency of commencing the study of   
the science from the subjective side1 or
 even of assigning any great degree of  
prominence to this side4  ,that men do  
not believe in exact accordance with    
this theory must have struck almost     
every one1 but this has probably been   
considered as mere exception and        
irregularity2 the assumption being made 
that on the average1 and in far the     
majority of cases1 they do so believe4  
,as stated above1 it is very doubtful   
whether the tendency which has just been
 discussed is not so widely prevalent   
that it might with far more propriety be
 called the rule than the exception4    
,and it may be better that this should  
be so3 many good results may follow from
 that cheerful disposition which induces
 a man sometimes to go on trying after  
some great good1 the chance of which he 
overvalues4 ,he will keep on through    
trouble and disappointment1 without     

serious harm perhaps1 when the cool and 
calculating bystander sees plainly that 
his 'measure of belief' is much higher  
than it should be4 ,so1 too1 the        
tendency also so common1 of underrating 
the chance of a great evil may also work
 for good4 ,by many men death might be  
looked upon as an almost infinite evil1 
at least they would so regard it        
themselves2 suppose they kept this      
contingency constantly before them at   
its right value1 how would it be        
possible to get through the practical   
work of life8 ,men would be stopping    
indoors because if they went out they   
might be murdered or bitten by a mad    
dog4 ,to say this is not to advocate a  
return to our instincts2 indeed when we 
have once reached the critical and      
conscious state1 it is hardly possible  
to do so2 but it should be noticed that 
the advantage gained by correcting them 
is at best but a balanced one4#c ,what  
is most to our present purpose1 it      
suggests the inexpediency of attempting 

to found an exact theory on what may    
afterwards prove to be a mere instinct1 
unauthorized in its full extent by      
experience4  #aa4 ,it may be replied1   
that though people1 as a matter of fact1
 do not apportion belief in this exact  
way1 yet they ought to do so4 ,the      
purport of this remark will be examined 
presently2 it need only be said here    
that it grants all that is now contended
 for4 ,for it admits that the degree of 
our belief is capable of modification1  
and may need it4 ,but in accordance with
 what is the belief to be modified8     
obviously in accordance with experience2
 it cannot be trusted to by itself1 but 
the fraction at which it is to be rated 
must be determined by the comparative   
frequency of the events to which it     
refers4 ,experience then furnishing the 
standard1 it is surely most reasonable  
to start from this experience1 and to   
found the theory of our processes upon  
it4  ,if we do not do this1 it should be
 observed that we are detaching         

,probability altogether from the study  
of things external to us1 and making it 
nothing else in effect than a portion of
 ,psychology4 ,if we refuse to be       
controlled by experience1 but confine   
our attention to the laws according to  
which belief is naturally or            
instinctively compounded and distributed
 in our minds1 we have no right then to 
appeal to experience afterwards even for
 illustrations1 unless under the express
 understanding that we do not guarantee 
its accuracy4 ,our belief in some single
 events1 for example1 might be correct1 
and yet that in a compound of several   
7if derived merely from our instinctive 
laws of belief7 very possibly might not 
be correct1 but might lead us into      
practical mistakes if we determined to  
act upon it4 ,even if the two were in   
accordance1 this accordance would have  
to be proved1 which would lead us round1
 by what ,i cannot but think a          
circuitous process1 to the point which  
has been already chosen for commencing  

with4  #ab4 ,de ,morgan seems to imply  
that the doctrine criticised above finds
 a justification from the analogy of    
,formal ,logic4 ,if the laws of         
necessary inference can be studied apart
 from all reference to external facts   
7except by way of illustration71 why not
 those of probable inference8 ,there    
does not1 however1 seem to be much force
 in any such analogy4 ,formal ,logic1 at
 any rate under its modern or ,kantian  
mode of treatment1 is based upon the    
assumption that there are laws of       
thought as distinguished from laws of   
things1 and that these laws of thought  
can be ascertained and studied without  
taking into account their reference to  
any particular object4 ,now so long as  
we are confined to necessary or         
irreversible laws1 as is of course the  
case in ordinary ,formal ,logic1 this   
assumption leads to no special          
difficulties4 ,we mean by this1 that no 
conflict arises between these subjective
 and objective necessities4 ,the two    

exist in perfect harmony side by side1  
the one being the accurate counterpart  
of the other4 ,so precise is the        
correspondence between them1 that few   
persons would notice1 until study of    
metaphysics had called their attention  
to such points1 that there were these   
two sides to the question4 ,they would  
make their appeal to either with equal  
confidence1 saying indifferently1 'the  
thing must be so1' or1 'we cannot       
conceive its being otherwise4' ,in fact 
it is only since the time of ,kant that 
this mental analysis has been to any    
extent appreciated and accepted4 ,and   
even now the dominant experience school 
of philosophy would not admit that there
 are here two really distinct sides to  
the phenomenon2 they maintain either    
that the subjective necessity is nothing
 more than the consequence by inveterate
 association of the objective           
uniformity1 or else that this so-called 
necessity 7say in the ,law of           
,contradiction7 is after all merely     

verbal1 merely a different way of saying
 the same thing over again in other     
words4 ,whatever the explanation        
adopted1 the general result is that     
fallacies1 as real acts of thought1 are 
impossible within the domain of pure    
logic2 error within that province is    
only possibly by a momentary lapse of   
attention1 that is of consciousness4    
#ac4 ,but though this perfect harmony   
between subjective and objective        
uniformities or laws may exist within   
the domain of pure logic1 it is far from
 existing within that of probability4   
,the moment we make the quantity of our 
belief an integral part of the subject  
to be studied1 any such invariable      
correspondence ceases to exist4 ,in the 
former case1 we could not consciously   
think erroneously even though we might  
try to do so2 in the latter1 we not only
 can believe erroneously but constantly 
do so4 ,far from the quantity of our    
belief being so exactly adjusted in     
conformity with the facts to which it   

refers that we cannot even in           
imagination go astray1 we find that it  
frequently exists in excess or defect of
 that which subsequent judgment will    
approve4 ,our instincts of credence are 
unquestionably in frequent hostility    
with experience2 and what do we do then8
 ,we simply modify the instincts into   
accordance with the things4 ,we are     
constantly performing this practice1 and
 no cultivated mind would find it       
possible to do anything else4 ,no man   
would think of divorcing his belief from
 the things on which it was exercised1  
or would suppose that the former had    
anything else to do than to follow the  
lead of the latter4 ,hence it results   
that that separation of the subjective  
necessity from the objective1 and that  
determination to treat the former as a  
science apart by itself1 for which a    
plausible defence could be made in the  
case of pure logic1 is entirely         
inadmissible in the case of probability4
 ,however we might contrive to 'think'  

aright without appeal to facts1 we      
cannot believe aright without           
incessantly checking our proceedings by 
such appeals4 ,whatever then may be the 
claims of ,formal ,logic to rank as a   
separate science1 it does not appear    
that it can furnish any support to the  
theory of ,probability at present under 
examination4  #ad4 ,the point in        
question is sometimes urged as follows4 
,suppose a man with two1 and only two1  
alternatives before him1 one of which he
 knows must involve success and the     
other failure4 ,he knows nothing more   
about them than this1 and he is forced  
to act4 ,would he not regard them with  
absolutely similar and equal feelings of
 confidence1 without the necessity of   
referring them to any real or imaginary 
series8 ,if so1 is not this equivalent  
to saying that his belief of either1    
since one of them must come to pass1 is 
equal to that of the other1 and         
therefore that his belief of each is    
one-half of full confidence8 ,similarly 

if there are more than two alternatives3
 let it be supposed that there are any  
number of them1 amongst which no        
distinctions whatever can be discerned  
except in such particulars as we know   
for certain will not affect the result2 
should we not feel equally confident in 
respect of each of them8 and so here    
again should we riot have a fractional  
estimate of our absolute amount of      
belief8 ,it is thus attempted to lay the
 basis of a pure science of             
,probability1 determining the           
distribution and combination of our     
belief hypothetically2 viz4 if the      
contingencies are exactly alike1 then   
our belief is so apportioned1 the       
question whether the contingencies are  
equal being of course decided as the    
objective data of ,logic or ,mathematics
 are decided4  ,to discuss this question
 fully would require a statement at some
 length of the reasons in favour of the 
objective or material view of ,logic1 as
 opposed to the ,formal or              

,conceptualist4 ,i shall have to speak  
on this subject in another chapter1 and 
will not therefore enter upon it here4  
,but one conclusive objection which is  
applicable more peculiarly to           
,probability may be offered at once4 ,to
 pursue the line of enquiry just        
indicated1 is1 as already remarked1 to  
desert the strictly logical ground1 and 
to take up that appropriate to          
psychology2 the proper question1 in all 
these cases1 being not what do men      
believe1 but what ought they to believe8
 ,admitting1 as was done above1 that in 
the case of ,formal ,logic these two    
enquiries1 or rather those corresponding
 to them1 practically run into one1     
owing to the fact that men cannot       
consciously 'think' wrongly2 it cannot  
be too strongly insisted on that in     
,probability the two are perfectly      
separable and distinct4 ,it is of no use
 saying what men do or will believe1 we 
want to know what they will be right in 
believing2 and this can never be settled

 without an appeal to the phenomena     
themselves4  #ae4 ,but apart from the   
above considerations1 this way of       
putting the case does not seem to me at 
all conclusive4 ,take the following     
example4 ,a man#d finds himself on the  
sands of the ,wash or ,morecambe ,bay1  
in a dense mist1 when the spring-tide is
 coming in2 and knows therefore that to 
be once caught by the tide would be     
fatal4 ,he hears a church-bell at a     
distance1 but has no means of knowing   
whether it is on the same side of the   
water with himself or on the opposite   
side4 ,he cannot tell therefore whether 
by following its sound he will be led   
out into the mid-stream and be lost1 or 
led back to dry land and safety4 ,here  
there can be no repetition of the event1
 and the cases are indistinguishably    
alike1 to him1 in the only circumstances
 which can affect the issue3 is not then
 his prospect of death1 it will be said1
 necessarily equal to one-half8 ,a      
proper analysis of his state of mind    

would be a psychological rather than a  
logical enquiry1 and in any case1 as    
above remarked1 the decision of this    
question does not touch our logical     
position4 ,but according to the best    
introspection ,i can give ,i should say 
that what really passes through the mind
 in such a case is something of this    
kind3 ,in most doubtful positions and   
circumstances we are accustomed to      
decide our conduct by a consideration of
 the relative advantages and            
disadvantages of each side1 that is by  
the observed or inferred frequency with 
which one or the other alternative has  
succeeded4 ,in proportion as these      
become more nearly balanced1 we are more
 frequently mistaken in the individual  
cases2 that is1 it becomes more and more
 nearly what would be called 'a mere    
toss up' whether we are right or wrong4 
,the case in question seems merely the  
limiting case1 in which it has been     
contrived that there shall be no        
appreciable difference between the      

alternatives1 by which to decide in     
favour of one or other1 and we          
accordingly feel no confidence in the   
particular result4 ,having to decide1   
however1 we decide according to the     
precedent of similar cases which have   
occurred before4 ,to stand still and    
wait for better information is certain  
death1 and we therefore appeal to and   
employ the only rule we know of2 or     
rather we feel1 or endeavour to feel1 as
 we have felt before when acting in the 
presence of alternatives as nearly      
balanced as possible4 ,but ,i can       
neither perceive in my own case1 nor    
feel convinced in that of others1 that  
this appeal1 in a case which cannot be  
repeated1#e to a rule acted on and      
justified in cases which can be and are 
repeated1 at all forces us to admit that
 our state of mind is the same in each  
case4  #af4 ,this example serves to     
bring out very clearly a point which has
 been already mentioned1 and which will 
have to be insisted upon again1 viz4    

that all which ,probability discusses is
 the statistical frequency of events1   
or1 if we prefer so to put it1 the      
quantity of belief with which any one of
 these events should be individually    
regarded1 but leaves all the subsequent 
conduct dependent upon that frequency1  
or that belief1 to the choice of the    
agents4 ,suppose there are two          
travellers in the predicament in        
question3 shall they keep together1 or  
separate in opposite directions8 ,in    
either case alike the chance of safety  
to each is the same1 viz4 one-half1 but 
clearly their circumstances must decide 
which course it is preferable to adopt4 
,if they are husband and wife1 they will
 probably prefer to remain together2 if 
they are sole depositaries of an        
important state secret1 they may decide 
to part4 ,in other words1 we have to    
select here between the two alternatives
 of the certainty of a single loss1 and 
the even chance of a double loss2       
alternatives which the common           

mathematical statement of their chances 
has a decided tendency to make us regard
 as indistinguishable from one another4 
,but clearly the decision must be       
grounded on the desires1 feelings1 and  
conscience of the agents4 ,probability  
cannot say a word upon this question4   
,as ,i have pointed out elsewhere1 there
 has been much confusion on this matter 
in applications of the science to       
betting1 and in the discussion of the   
,petersburg problem4  ,we have thus     
examined the doctrine in question with a
 minuteness which may seem tedious1 but 
in consequence of the eminence of its   
supporters it would have been           
presumptuous to have rejected it without
 the strongest grounds4 ,the objections 
which have been urged might be          
summarised as follows3--the amount of   
our belief of any given proposition1    
supposing it to be in its nature capable
 of accurate determination 7which does  
not seem to be the case71 depends upon a
 great variety of causes1 of which      

statistical frequency--the subject of   
,probability--is but one4 ,that even if 
we confine our attention to this one    
cause1 the natural amount of our belief 
is not necessarily what theory would    
assign1 but has to be checked by appeal 
to experience4 ,the subjective side of  
,probability therefore1 though very     
interesting and well deserving of       
examination1 seems a mere appendage of  
the objective1 and affords in itself no 
safe ground for a science of inference4 
#ag4 ,the conception then of the science
 of ,probability as a science of the    
laws of belief seems to break down at   
every point4 ,we must not however rest  
content with such merely negative       
criticism4 ,the degree of belief we     
entertain of a proposition may be hard  
to get at accurately1 and when obtained 
may be often wrong1 and may need        
therefore to be checked by an appeal to 
the objects of belief4 ,still in popular
 estimation we do seem to be able with  
more or less accuracy to form a         

graduated scale of intensity of belief4 
,what we have to examine now is whether 
this be possible1 and1 if so1 what is   
the explanation of the fact8  ,that it  
is generally believed that we can form  
such a scale scarcely admits of doubt4  
,there is a whole vocabulary of common  
expressions such as1 ',i feel almost    
sure1' ',i do not feel quite certain1'  
',i am less confident of this than of   
that1' and so on4 ,when we make use of  
any one of these phrases we seldom doubt
 that we have a distinct meaning to     
convey by means of it4 ,nor do we feel  
much at a loss1 under any given         
circumstances1 as to which of these     
expressions we should employ in         
preference to the others4 ,if we were   
asked to arrange in order1 according to 
the intensity of the belief with which  
we respectively hold them1 things       
broadly marked off from one another1 we 
could do it from our consciousness of   
belief alone1 without a fresh appeal to 
the evidence upon which the belief      

depended4 ,passing over the looser      
propositions which are used in common   
conversation1 let us take but one simple
 example from amongst those which       
furnish numerical data4 ,do ,i not feel 
more certain that some one will die this
 week in the whole town1 than in the    
particular street in which ,i live8 and 
if the town is known to contain a       
population one hundred times greater    
than that in the street1 would not      
almost any one be prepared to assert on 
reflection that he felt a hundred times 
more sure of the first proposition than 
of the second8 ,or to take a            
non-numerical example1 are we not often 
able to say unhesitatingly which of two 
propositions we believe the most1 and to
 some rough degree how much more we     
believe one than the other1 at a time   
when all the evidence upon which each   
rests has faded from the mind1 so that  
each has to be judged1 as we may say1   
solely on its own merits8  ,here then a 
problem proposes itself4 ,if popular    

opinion1 as illustrated in common       
language1 be correct1--and very         
considerable weight must of course be   
attributed to it1--there does exist     
something which we call partial belief  
in reference to any proposition of the  
numerical kind described above4 ,now    
what we want to do is to find some test 
or justification of this belief1 to     
obtain in fact some intelligible answer 
to the question1 ,is it correct8 ,we    
shall find incidentally that the answer 
to this question will throw a good deal 
of light upon another question nearly as
 important and far more intricate1 viz4 
,what is the meaning of this partial    
belief8  #ah4 ,we shall find it         
advisable to commence by ascertaining   
how such enquiries as the above would be
 answered in the case of ordinary full  
belief4 ,such a step would not offer the
 slightest difficulty4 ,suppose1 to take
 a simple example1 that we have obtained
 the following proposition1--whether by 
induction1 or by the rules of ordinary  

deductive logic1 does not matter for our
 present purpose1--that a certain       
mixture of oxygen and hydrogen is       
explosive4 ,here we have an inference1  
and consequent belief of a proposition4 
,now suppose there were any enquiry as  
to whether our belief were correct1 what
 should we do8 ,the simplest way of     
settling the matter would be to find out
 by a distinct appeal to experience     
whether the proposition was true4 ,since
 we are reasoning about things1 the     
justification of the belief1 that is1   
the test of its correctness1 would be   
most readily found in the truth of the  
proposition4 ,if by any process of      
inference ,i have come to believe that a
 certain mixture will explode1 ,i       
consider my belief to be justified1 that
 is to be correct1 if under proper      
circumstances the explosion always does 
occur2 if it does not occur the belief  
was wrong4  ,such an answer1 no doubt1  
goes but a little way1 or rather no way 
at all1 towards explaining what is the  

nature of belief in itself2 but it is   
sufficient for our present purpose1     
which is merely that of determining what
 is meant by the correctness of our     
belief1 and by the test of its          
correctness4 ,in all inferences about   
things1 in which the amount of our      
belief is not taken into account1 such  
an explanation as the above is quite    
sufficient2 it would be the ordinary one
 in any question of science4 ,it is     
moreover perfectly intelligible1 whether
 the conclusion is particular or        
universal4 ,whether we believe that     
'some men die'1 or that 'all men die'1  
our belief may with equal ease be tested
 by the appropriate train of experience4
  #ai4 ,but when we attempt to apply the
 same test to partial belief1 we shall  
find ourselves reduced to an awkward    
perplexity4 ,a difficulty now emerges   
which has been singularly overlooked by 
those who have treated of the subject4  
,as a simple example will serve our     
purpose1 we will take the case of a     

penny4 ,i am about to toss one up1 and  
,i therefore half believe1 to adopt the 
current language1 that it will give     
head4 ,now it seems to be overlooked    
that if we appeal to the event1 as we   
did in the case last examined1 our      
belief must inevitably be wrong1 and    
therefore the test above mentioned will 
fail4 ,for the thing must either happen 
or not happen3 i4e4 in this case the    
penny must either give head1 or not give
 it2 there is no third alternative4 ,but
 whichever way it occurs1 our           
half-belief1 so far as such a state of  
mind admits of interpretation1 must be  
wrong4 ,if head does come1 ,i am wrong  
in not having expected it enough2 for ,i
 only half believed in its occurrence4  
,if it does not happen1 ,i am equally   
wrong in having expected it too much2   
for ,i half believed in its occurrence1 
when in fact it did not occur at all4   
,the same difficulty will occur in every
 case in which we attempt to justify our
 state of partial belief in a single    

contingent event4 ,let us take another  
example1 slightly differing from the    
last4 ,a man is to receive #a if a die  
gives six1 to pay #as4 if it gives any  
other number4 ,it will generally be     
admitted that he ought to give #bs4     
#f;d4 for the chance1 and that if he    
does so he will be paying a fair sum4   
,this example only differs from the last
 in the fact that instead of simple     
belief in a proposition1 we have taken  
what mathematicians call 'the value of  
the expectation'4 ,in other words1 we   
have brought into a greater prominence1 
not merely the belief1 but the conduct  
which is founded upon the belief4 ,but  
precisely the same difficulty recurs    
here4 ,for appealing to the event1--the 
single event1 that is1--we see that one 
or other party must lose his money      
without compensation4 ,in what sense    
then can such an expectation be said to 
be a fair one8  #bj4 ,a possible answer 
to this1 and so far as appears the only 
possible answer1 will be1 that what we  

really mean by saying that we half      
believe in the occurrence of head is to 
express our conviction that head will   
certainly happen on the average every   
other time4 ,and similarly1 in the      
second example1 by calling the sum a    
fair one it is meant that in the long   
run neither party will gain or lose4 ,as
 we shall recur presently to the point  
raised in this form of answer1 the only 
notice that need be taken of it at this 
point is to call attention to the fact  
that it entirely abandons the whole     
question in dispute1 for it admits that 
this partial belief does not in any     
strict sense apply to the individual    
event1 since it clearly cannot be       
justified there4 ,at such a result      
indeed we cannot be surprised2 at least 
we cannot on the theory adopted         
throughout this ,essay4 ,for bearing in 
mind that the employment of ,probability
 postulates ignorance of the single     
event1 it is not easy to see how we are 
to justify any other opinion or         

statement about the single event than a 
confession of such ignorance4  #ba4 ,so 
far then we do not seem to have made the
 slightest approximation to a solution  
of the particular question now under    
examination4 ,the more closely we have  
analysed special examples1 the more     
unmistakeably are we brought to the     
conclusion that in the individual       
instance no justification of anything   
like quantitative belief is to be found2
 at least none is to be found in the    
same sense in which we expect it in     
ordinary scientific conclusions1 whether
 ,inductive or ,deductive4 ,and yet we  
have to face and account for the fact   
that common impressions1 as attested by 
a whole vocabulary of common phrases1   
are in favour of the existence of this  
quantitative belief4 ,how are we to     
account for this8 ,if we appeal to an   
example again1 and analyse it somewhat  
more closely1 we may yet find our way to
 some satisfactory explanation4  ,in our
 previous analysis 7#ah7 we found it    

sufficient to stop at an early stage1   
and to give as the justification of our 
belief the fact of the proposition being
 true4 ,stopping however at that stage1 
we have found this explanation fail     
altogether to give a justification of   
partial belief2 fail1 that is1 when     
applied to the individual instance4 ,the
 two states of belief and disbelief     
correspond admirably to the two results 
of the event happening and not happening
 respectively1 and unless for           
psychological purposes we saw no reason 
to analyse further2 but to partial      
belief there is nothing corresponding in
 the result1 for the event cannot       
partially happen in such cases as we are
 concerned with4 ,suppose then we       
advance a step further in the analysis1 
and ask again what is meant by the      
proposition being true8 ,this introduces
 us1 of course1 to a very long and      
intricate path2 but in the short        
distance along it which we shall        
advance1 we shall not1 it is to be      

hoped1 find any very serious difficulty4
 ,as before1 we will illustrate the     
analysis by first applying it to the    
case of ordinary full belief4  #bb4     
,whatever opinion then may be held about
 the essential nature of belief1 it will
 probably be admitted that a readiness  
to act upon the proposition believed is 
an inseparable accompaniment of that    
state of mind4 ,there can be no         
alteration in our belief 7at any rate in
 the case of sane persons7 without a    
possible alteration in our conduct1 nor 
anything in our conduct which is not    
connected with something in our belief4 
,we will first take an example in       
connection with the penny1 in which     
there is full belief2 we will analyse it
 a step further than we did before1 and 
then attempt to apply the same analysis 
to an example of a similar kind1 but one
 in which the belief is partial instead 
of full4  ,suppose that ,i am about to  
throw a penny up1 and contemplate the   
prospect of its falling upon one of its 

sides and not upon its edge4 ,we feel   
perfectly confident that it will do so4 
,now whatever else may be implied in our
 belief1 we certainly mean this2 that we
 are ready to stake our conduct upon its
 falling thus4 ,all our betting1 and    
everything else that we do1 is carried  
on upon this supposition4 ,any risk     
whatever that might ensue upon its      
falling otherwise will be incurred      
without fear4 ,this1 it must be         
observed1 is equally the case whether we
 are speaking of a single throw or of a 
long succession of throws4  ,but now let
 us take the case of a penny falling1   
not upon one side or the other1 but upon
 a given side1 head4 ,to a certain      
extent this example resembles the last4 
,we are perfectly ready to stake our    
conduct upon what comes to pass in the  
long run4 ,when we are considering the  
result of a large number of throws1 we  
are ready to act upon the supposition   
that head comes every other time4 ,if   
e4g4 we are betting upon it1 we shall   

not object to paying #a every time that 
head comes1 on condition of receiving #a
 every time that head does not come4    
,this is nothing else than the          
translation1 as we may call it1 into    
practice1 of our belief that head and   
tail occur equally often4  ,now it will 
be obvious1 on a moment's consideration1
 that our conduct is capable of being   
slightly varied3 of being varied1 that  
is1 in form1 whilst it remains identical
 in respect of its results4 ,it is clear
 that to pay #a every time we lose1 and 
to get #a every time we gain1 comes to  
precisely the same thing1 in the case   
under consideration1 as to pay ten      
shillings every time without exception1 
and to receive #a every time that head  
occurs4 ,it is so1 because heads occur1 
on the average1 every other time4 ,in   
the long run the two results coincide2  
but there is a marked difference between
 the two cases1 considered individually4
 ,the difference is two-fold4 ,in the   
first place we depart from the notion of

 a payment every other time1 and come to
 that of one made every time4 ,in the   
second place1 what we pay every time is 
half of what we get in the cases in     
which we do get anything4 ,the          
difference may seem slight2 but mark the
 effect when our conduct is translated  
back again into the subjective condition
 upon which it depends1 viz4 into our   
belief4 ,it is in consequence of such a 
translation1 as it appears to me1 that  
the notion has been acquired that we    
have an accurately determinable amount  
of belief as to every such proposition4 
,to have losses and gains of equal      
amount1 and to incur them equally often1
 was the experience connected with our  
belief that the two events1 head and    
tail1 would occur equally often4 ,this  
was quite intelligible1 for it referred 
to the long run4 ,to find that this     
could be commuted for a payment made    
every time without exception1 a payment1
 observe1 of half the amount of what we 
occasionally receive1 has very naturally

 been interpreted to mean that there    
must be a state of half-belief which    
refers to each individual throw4  #bc4  
,one such example1 of course1 does not  
go far towards establishing a theory4   
,but the reader will bear in mind that  
almost all our conduct tends towards the
 same result2 that it is not in betting 
only1 but in every course of action in  
which we have to count the events1 that 
such a numerical apportionment of our   
conduct is possible4 ,hence1 by the     
ordinary principles of association1 it  
would appear exceedingly likely that1   
not exactly a numerical condition of    
mind1 but rather numerical associations1
 become inseparably connected with each 
particular event which we know to occur 
in a certain proportion of times4 ,once 
in six times a die gives ace2 a         
knowledge of this fact1 taken in        
combination with all the practical      
results to which it leads1 produces1 one
 cannot doubt1 an inseparable notion of 
one-sixth connected with each single    

throw4 ,but it surely cannot be called  
belief to the amount of one-sixth2 at   
least it admits neither of justification
 nor explanation in these single cases1 
to which alone the fractional belief1 if
 such existed1 ought to apply4  ,it is  
in consequence1 ,i apprehend1 of such   
association that we act in such an      
unhesitating manner in reference to any 
single contingent event1 even when we   
have no expectation of its being        
repeated4 ,a die is going to be thrown  
up once1 and once only4 ,i bet #e to #a 
against ace1 not1 as is commonly        
asserted1 because ,i feel one-sixth part
 of certainty in the occurrence of ace2 
but because ,i know that such conduct   
would be justified in the long run of   
such cases1 and ,i apply to the solitary
 individual the same rule that ,i should
 apply to it if ,i knew it were one of a
 long series4 ,this accounts for my     
conduct being the same in the two cases2
 by association1 moreover1 we probably  
experience very similar feelings in     

regard to them both4  #bd4 ,and here1 on
 the view of the subject adopted in this
 ,essay1 we might stop4 ,we are bound to
 explain the 'measure of our belief' in 
the occurrence of a single event when we
 judge solely from the statistical      
frequency with which such events occur1 
for such a series of events was our     
starting-point2 but we are not bound to 
inquire whether in every case in which  
persons have1 or claim to have1 a       
certain measure of belief there must be 
such a series to which to refer it1 and 
by which to justify it4 ,those who start
 from the subjective side1 and regard   
,probability as the science of          
quantitative belief1 are obliged to do  
this1 but we are free from the          
obligation4  ,still the question is one 
which is so naturally raised in         
connection with this subject1 that it   
cannot be altogether passed by4 ,i think
 that to a considerable extent such a   
justification as that mentioned above   
will be found applicable in other cases4

 ,the fact is that we are very seldom   
called upon to decide and act upon a    
single contingency which cannot be      
viewed as being one of a series4        
,experience introduces us1 it must be   
remembered1 not merely to a succession  
of events neatly arranged in a single   
series 7as we have hitherto assumed them
 to be for the purpose of illustration71
 but to an infinite number belonging to 
a vast variety of different series4 ,a  
man is obliged to be acting1 and        
therefore exercising his belief about   
one thing or another1 almost the whole  
of every day of his life4 ,any one      
person will have to decide in his time  
about a multitude of events1 each one of
 which may never recur again within his 
own experience4 ,but by the very fact of
 there being a multitude1 though they   
are all of different kinds1 we shall    
still find that order is maintained1 and
 so a course of conduct can be          
justified4 ,in a plantation of trees we 
should find that there is order of a    

certain kind if we measure them in any  
one direction1 the trees being on an    
average about the same distance from    
each other4 ,but a somewhat similar     
order would be found if we were to      
examine them in any other direction     
whatsoever4 ,so in nature generally2    
there is regularity in a succession of  
events of the same kind4 ,but there may 
also be regularity if we form a series  
by taking successively a number out of  
totally distinct kinds4  ,it is in this 
circumstance that we find an extension  
of the practical justification of the   
measure of our belief4 ,a man1 say1 buys
 a life annuity1 insures his life on a  
railway journey1 puts into a lottery1   
and so on4 ,now we may make a series out
 of these acts of his1 though each is in
 itself a single event which he may     
never intend to repeat4 ,his conduct1   
and therefore his belief1 measured by   
the result in each individual instance1 
will not be justified1 but the reverse1 
as shown in #ai4 ,could he indeed repeat

 each kind of action often enough it    
would be justified2 but from this1 by   
the conditions of life1 he is debarred4 
,now it is perfectly conceivable that in
 the new series1 formed by his          
successive acts of different kinds1     
there should be no regularity4 ,as a    
matter of fact1 however1 it is found    
that there is regularity4 ,in this way  
the equalization of his gains and       
losses1 for which he cannot hope in     
annuities1 insurances1 and lotteries    
taken separately1 may yet be secured to 
him out of these events taken           
collectively4 ,if in each case he values
 his chance at its right proportion 7and
 acts accordingly7 he will in the course
 of his life neither gain nor lose4 ,and
 in the same way if1 whenever he has the
 alternative of different courses of    
conduct1 he acts in accordance with the 
estimate of his belief described above1 
i4e4 chooses the event whose chance is  
the best1 he will in the end gain more  
in this way than by any other course4   

,by the existence1 therefore1 of these  
cross-series1 as we may term them1 there
 is an immense addition to the number of
 actions which may be fairly considered 
to belong to those courses of conduct   
which offer many successive             
opportunities of equalizing gains and   
losses4 ,all these cases then may be    
regarded as admitting of justification  
in the way now under discussion4  #be4  
,in the above remarks it will be        
observed that we have been giving what  
is to be regarded as a justification of 
his belief from the point of view of the
 individual agent himself4 ,if we       
suppose the existence of an enlarged    
fellow-feeling1 the applicability of    
such a justification becomes still more 
extensive4 ,we can assign a very        
intelligible sense to the assertion that
 it is #iii to #a that ,i shall not get 
a prize in a lottery1 even if this be   
stated in the form that my belief in my 
so doing is represented by the fraction 
#a_/ajjjth of certainty4 ,properly it   

means that in a very large number of    
throws ,i should gain once in #ajjj     
times4 ,if we include other             
contingencies of the same kind1 as      
described in the last section1 each     
individual may be supposed to reach to  
something like this experience within   
the limits of his own life4 ,he could   
not do it in this particular line of    
conduct alone1 but he could do it in    
this line combined with others4 ,now    
introduce the possibility of each man   
feeling that the gain of others offers  
some analogy to his own gains1 which we 
may conceive his doing except in the    
case of the gains of those against whom 
he is directly competing1 and the above 
justification becomes still more        
extensively applicable4  ,the following 
would be a fair illustration to test    
this view4 ,i know that ,i must die on  
some day of the week1 and there are but 
seven days4 ,my belief1 therefore1 that 
,i shall die on a ,sunday is            
one-seventh4 ,here the contingent event 

is clearly one that does not admit of   
repetition2 and yet would not the belief
 of every man have the value assigned it
 by the formula8 ,it would appear that  
the same principle will be found to be  
at work here as in the former examples4 
,it is quite true that ,i have only the 
opportunity of dying once myself1 but ,i
 am a member of a class in which deaths 
occur with frequency1 and ,i form my    
opinion upon evidence drawn from that   
class4 ,if1 for example1 ,i had insured 
my life for #ajjj1 ,i should feel a     
certain propriety in demanding #gjjj in 
case the office declared that it would  
only pay in the event of my dying on a  
,sunday4 ,i1 indeed1 for my own private 
part1 might not find the arrangement an 
equitable one2 but mankind at large1 in 
case they acted on such a principle1    
might fairly commute their aggregate    
gains in such a way1 whilst to the      
,insurance ,office it would not make any
 difference at all4  #bf4 ,the results  
of the last few sections might be       

summarised as follows3--the different   
amounts of belief which we entertain    
upon different events1 and which are    
recognized by various phrases in common 
use1 have undoubtedly some meaning4 ,but
 the greater part of their meaning1 and 
certainly their only justification1 are 
to be sought in the series of           
corresponding events to which they      
belong2 in regard to which it may be    
shown that far more events are capable  
of being referred to a series than might
 be supposed at first sight4 ,the test  
and justification of belief are to be   
found in conduct2 in this test applied  
to the series as a whole1 there is      
nothing peculiar1 it differs in no way  
from the similar test when we are acting
 on our belief about any single event4  
,but so applied1 from the nature of the 
case it is applied successively to each 
of the individuals of the series2 here  
our conduct generally admits of being   
separately considered in reference to   
each particular event2 and this has been

 understood to denote a certain amount  
of belief which should be a fraction of 
certainty4 ,probably on the principles  
of association1 a peculiar condition of 
mind is produced in reference to each   
single event4 ,and these associations   
are not unnaturally retained even when  
we contemplate any one of these single  
events isolated from any series to which
 it belongs4 ,when it is found alone we 
treat it1 and feel towards it1 as we do 
when it is in company with the rest of  
the series4  #bg4 ,we may now see1 more 
clearly than we could before1 why it is 
that we are free from any necessity of  
assuming the existence of causation1 in 
the sense of necessary invariable       
sequence1 in the case of the events     
which compose our series4 ,against such 
a view it might very plausibly be urged1
 that we constantly talk of the         
probability of a single event2 but how  
can this be done1 it may reasonably be  
said1 if we once admit the possibility  
of that event occurring fortuitously8   

,take an instance from human life2 the  
average duration of the lives of a batch
 of men aged thirty will be about       
thirty-four years4 ,we say therefore to 
any individual of them1 ,your           
expectation of life is thirty-four      
years4 ,but how can this be said if we  
admit that the train of events composing
 his life is liable to be destitute of  
all regular sequence of cause and       
effect8 ,to this it may be replied that 
the denial of causation enables us to   
say neither more nor less than its      
assertion1 in reference to the length of
 the individual life1 for of this we are
 ignorant in each case alike4 ,by       
assigning1 as above1 an expectation in  
reference to the individual1 we mean    
nothing more than to make a statement   
about the average of his class4 ,whether
 there be causation or not in these     
individual cases does not affect our    
knowledge of the average1 for this by   
supposition rests on independent        
experience4 ,the legitimate inferences  

are the same on either hypothesis1 and  
of equal value4 ,the only difference is 
that on the hypothesis of non-causation 
we have forced upon our attention the   
impropriety of talking of the 'proper'  
expectation of the individual1 owing to 
the fact that all knowledge of its      
amount is formally impossible2 on the   
other hypothesis the impropriety is     
overlooked from the fact of such        
knowledge being only practically        
unattainable4 ,as a matter of fact the  
amount of our knowledge is the same in  
each case2 it is a knowledge of the     
average1 and of that only4#f  #bh4 ,we  
may conclude1 then1 that the limits     
within which we are thus able to justify
 the amount of our belief are far more  
extensive than might appear at first    
sight4 ,whether every case in which     
persons feel an amount of belief short  
of perfect confidence could be forced   
into the province of ,probability is a  
wider question4 ,even1 however1 if the  
belief could be supposed capable of     

justification on its principles1 its    
rules could never in such cases be made 
use of4 ,suppose1 for example1 that a   
father were in doubt whether to give a  
certain medicine to his sick child4 ,on 
the one hand1 the doctor declared that  
the child would die unless the medicine 
were given2 on the other1 through a     
mistake1 the father cannot feel quite   
sure that the medicine he has is the    
right one4 ,it is conceivable that some 
mathematicians1 in their conviction that
 everything has its definite numerical  
probability1 would declare that the     
man's belief had some 'value' 7if they  
could only find out what it is71 say    
nine-tenths2 by which they would mean   
that in nine cases out of ten in which  
he entertained a belief of that         
particular value he proved to be right4 
,so with his belief and doubt on the    
other side of the question4 ,putting the
 two together1 there is but one course  
which1 as a prudent man and a good      
father1 he can possibly follow4 ,it may 

be so1 but when 7as here7 the           
identification of an event in a series  
depends on purely subjective conditions1
 as in this case upon the degree of     
vividness of his conviction1 of which no
 one else can judge1 no test is         
possible1 and therefore no proof can be 
found4  #bi4 ,so much then for the      
attempts1 so frequently made1 to found  
the science on a subjective basis2 they 
can lead1 as it has here been           
endeavoured to show1 to no satisfactory 
result4 ,still our belief is so         
inseparably connected with our action1  
that something of a defence can be made 
for the attempts described above2 but   
when it is attempted1 as is often the   
case1 to import other sentiments besides
 pure belief1 and to find a             
justification for them also in the      
results of our science1 the confusion   
becomes far worse4 ,the following       
extract from ,archbishop ,thomson's     
,laws of ,thought 7#abb1 ,ed4 ,,ii47    
will show what kind of applications of  

the science are contemplated here3 ,in  
applying the doctrine of chances to that
 subject in connexion with which it was 
invented--games of chance1--the         
principles of what has been happily     
termed 'moral arithmetic' must not be   
forgotten4 ,not only would it be        
difficult for a gamester to find an     
antagonist on terms1 as to fortune and  
needs1 precisely equal1 but also it is  
impossible that with such an equality   
the advantage of a considerable gain    
should balance the harm of a serious    
loss4 ',if two men1' says ,buffon1 'were
 to determine to play for their whole   
property1 what would be the effect of   
this agreement8 ,the one would only     
double his fortune1 and the other reduce
 his to naught4 ,what proportion is     
there between the loss and the gain8    
,the same that there is between all and 
nothing4 ,the gain of the one is but a  
moderate sum1--the loss of the other is 
numerically infinite1 and morally so    
great that the labour of his whole life 

may not perhaps suffice to restore his  
property4'  ,as moral advice this is all
 very true and good4 ,but if it be      
regarded as a contribution to the       
science of the subject it is quite      
inappropriate1 and seems calculated to  
cause confusion4 ,the doctrine of       
chances pronounces upon certain kinds of
 events in respect of number and        
magnitude2 it has absolutely nothing to 
do with any particular person's feelings
 about these relations4 ,we might as    
well append a corollary to the rules of 
arithmetic1 to point out that although  
it is very true that twice two are four 
it does not follow that four1 horses    
will give twice as much pleasure to the 
owner as two will4 ,if two men play on  
equal terms their chances are equal2 in 
other words1 if they were often to play 
in this manner each would lose as       
frequently as he would gain4 ,that is   
all that ,probability can say2 what     
under the circumstances may be the      
determination and opinions of the men in

 question1 it is for them and them alone
 to decide4 ,there are many persons who 
cannot bear mediocrity of any kind1 and 
to whom the prospect of doubling their  
fortune would outweigh a greater chance 
of losing it altogether4 ,they alone are
 the judges4  ,if we will introduce such
 a balance of pleasure and pain the     
individual must make the calculation for
 himself4 ,the supposition is that total
 ruin is very painful1 partial loss     
painful in a less proportion than that  
assigned by the ratio of the losses     
themselves2 the inference is therefore  
drawn that on the average more pain is  
caused by occasional great losses than  
by frequent small ones1 though the money
 value of the losses in the long run may
 be the same in each case4 ,but if we   
suppose a country where the desire of   
spending largely is very strong1 and    
where owing to abundant production loss 
is easily replaced1 the calculation     
might incline the other way4 ,under such
 circumstances it is quite possible that

 more happiness might result from       
playing for high than for low stakes4   
,the fact is that all emotional         
considerations of this kind are         
irrelevant2 they are1 at most1 mere     
applications of the theory1 and such as 
each individual is alone competent to   
make for himself4 ,some more remarks    
will be made upon this subject in the   
chapter upon ,insurance and ,gambling4  
#cj4 ,it is by the introduction of such 
considerations as these that the        
,petersburg ,problem has been so        
perplexed4 ,having already given some   
description of this problem we will     
refer to it very briefly here4 ,it      
presents us with a sequence of sets of  
throws for each of which sets ,i am to  
receive something1 say a shilling1 as   
the minimum receipt4 ,my receipts       
increase in proportion to the rarity of 
each particular kind of set1 and each   
kind is observed or inferred to grow    
more rare in a certain definite but     
unlimited order4 ,by the wording of the 

problem1 properly interpreted1 ,i am    
supposed never to stop4 ,clearly        
therefore1 however large a fee ,i pay   
for each of these sets1 ,i shall be sure
 to make it up in time4 ,the            
mathematical expression of this is1 that
 ,i ought always to pay an infinite sum4
 ,to this the objection is opposed1 that
 no sensible man would think of         
advancing even a large finite sum1 say  
#ej4 ,certainly he would not2 but why8  
,because neither he nor those who are to
 pay him would be likely to live long   
enough for him to obtain throws good    
enough to remunerate him for one-tenth  
of his outlay2 to say nothing of his    
trouble and loss of time4 ,we must not  
suppose that the problem1 as stated in  
the ideal form1 will coincide with the  
practical form in which it presents     
itself in life4 ,a carpenter might as   
well object to ,euclid's second         
postulate1 because his plane came to a  
stop in six feet on the plank on which  
he was at work4 ,many persons have      

failed to perceive this1 and have       
assumed that1 besides enabling us to    
draw numerical inferences about the     
members of a series1 the theory ought   
also to be called upon to justify all   
the opinions which average respectable  
men might be inclined to form about     
them1 as well as the conduct they might 
choose to pursue in consequence4 ,it is 
obvious that to enter upon such         
considerations as these is to diverge   
from our proper ground4 ,we are         
concerned1 in these cases1 with the     
actions of men only1 as given in        
statistics2 with the emotions they      
experience in the performance of these  
actions we have no direct concern       
whatever4 ,the error is the same as if  
any one were to confound1 in political  
economy1 value in use with value in     
exchange1 and object to measuring the   
value of a loaf by its cost of          
production1 because bread is worth more 
to a man when he is hungry than it is   
just after his dinner4  #ca4 ,one class 

of emotions indeed ought to be excepted1
 which1 from the apparent uniformity and
 consistency with which they show       
themselves in different persons and at  
different times1 do really present some 
better claim to consideration4 ,in      
connection with a science of inference  
they can never indeed be regarded as    
more than an accident of what is        
essential to the subject1 but compared  
with other emotions they seem to be     
inseparable accidents4  ,the reader will
 remember that attention was drawn in   
the earlier part of this chapter to the 
compound nature of the state of mind    
which we term belief4 ,it is partly     
intellectual1 partly also emotional2 it 
professes to rest upon experience1 but  
in reality the experience acts through  
the distorting media of hopes and fears 
and other disturbing agencies4 ,so long 
as we confine our attention to the state
 of mind of the person who believes1 it 
appears to me that these two parts of   
belief are quite inseparable4 ,indeed1  

to speak of them as two parts may convey
 a wrong impression2 for though they    
spring from different sources1 they so  
entirely merge in one result as to      
produce what might be called an         
indistinguishable compound4 ,every kind 
of inference1 whether in probability or 
not1 is liable to be disturbed in this  
way4 ,a timid man may honestly believe  
that he will be wounded in a coming     
battle1 when others1 with the same      
experience but calmer judgments1 see    
that the chance is too small to deserve 
consideration4 ,but such a man's belief1
 if we look only to that1 will not      
differ in its nature from sound belief4 
,his conduct also in consequence of his 
belief will by itself afford no ground  
of discrimination2 he will make his will
 as sincerely as a man who is           
unmistakeably on his death-bed4 ,the    
only resource is to check and correct   
his belief by appealing to past and     
current experience4#g ,this was advanced
 as an objection to the theory on which 

probability is regarded as concerned    
primarily with laws of belief4 ,but on  
the view taken in this ,essay in which  
we are supposed to be concerned with    
laws of inference about things1 error   
and difficulty from this source vanish4 
,let us bear clearly in mind that we are
 concerned with inferences about things1
 and whatever there may be in belief    
which does not depend on experience will
 disappear from notice4  #cb4 ,these    
emotions then can claim no notice as an 
integral portion of any science of      
inference1 and should in strictness be  
rigidly excluded from it4 ,but if any of
 them are uniform and regular in their  
production and magnitude1 they may be   
fairly admitted as accidental and       
extraneous accompaniments4 ,this is     
really the case to some extent with our 
surprise4 ,this emotion does show a     
considerable degree of uniformity4 ,the 
rarer any event is the more am ,i1 in   
common with most other men1 surprised at
 it when it does happen4 ,this surprise 

may range through all degrees1 from the 
most languid form of interest up to the 
condition which we term 'being          
startled'4 ,and since the surprise seems
 to be pretty much the same1 under      
similar circumstances1 at different     
times1 and in the case of different     
persons1 it is free from that extreme   
irregularity which is found in most of  
the other mental conditions which       
accompany the contemplation of          
unexpected events4 ,hence our surprise1 
though1 as stated above1 having no      
proper claim to admission into the      
science of ,probability1 is such a      
constant and regular accompaniment of   
that which ,probability is concerned    
with1 that notice must often be taken of
 it4 ,references will occasionally be   
found to this aspect of the question in 
the following chapters4  ,it may be     
remarked in passing1 for the sake of    
further illustration of the subject1    
that this emotional accompaniment of    
surprise1 to which we are thus able to  

assign something like a fractional      
value1 differs in two important respects
 from the commonly accepted fraction of 
belief4 ,in the first place1 it has what
 may be termed an independent existence2
 it is intelligible by itself4 ,the     
belief1 as we endeavoured to show1 needs
 explanation and finds it in our        
consequent conduct4 ,not so with the    
emotion2 this stands upon its own       
footing1 and may be examined in and by  
itself4 ,hence1 in the second place1 it 
is as applicable1 and as capable of any 
kind of justification1 in relation to   
the single event1 as to a series of     
events4 ,in this respect1 as will be    
remembered1 it offers a complete        
contrast to our state of belief about   
any one contingent event4 ,may not these
 considerations help to account for the 
general acceptance of the doctrine1 that
 we have a certain definite and         
measurable amount of belief about these 
events8 ,i cannot help thinking that    
what is so obviously true of the        

emotional portion of the belief1 has    
been unconsciously transferred to the   
other or intellectual portion of the    
compound condition1 to which it is not  
applicable1 and where it cannot find a  
justification4  #cc4 ,a further         
illustration may now be given of the    
subjective view of ,probability at      
present under discussion4  ,an appeal to
 common language is always of service1  
as the employment of any distinct word  
is generally a proof that mankind have  
observed some distinct properties in the
 things1 which have caused them to be   
singled out and have that name          
appropriated to them4 ,there is such a  
class of words assigned by popular usage
 to the kind of events of which         
,probability takes account4 ,if we      
examine them we shall find1 ,i think1   
that they direct us unmistakeably to the
 two-fold aspect of the question1--the  
objective and the subjective1 the       
quality in the events and the state of  
our minds in considering them1--that    

have occupied our attention during the  
former chapters4  ,the word             
'extraordinary'1 for instance1 seems to 
point to the observed fact1 that events 
are arranged in a sort of ordo or rank4 
,no one of them might be so exactly     
placed that we could have inferred its  
position1 but when we take a great many 
into account together1 running our eye1 
as it were1 along the line1 we begin to 
see that they really do for the most    
part stand in order4 ,those which stand 
away from the line have this divergence 
observed1 and are called extraordinary1 
the rest ordinary1 or in the line4 ,so  
too 'irregular' and 'abnormal' are      
doubtless used from the appearance of   
things1 when examined in large numbers1 
being that of an arrangement by rule or 
measure4 ,this only holds when there are
 a good many2 we could not speak of the 
single events being so arranged4 ,again 
the word 'law'1 in its philosophical    
sense1 has now become quite popularised4
 ,how the term became introduced is not 

certain1 but there can be little doubt  
that it was somewhat in this way3--,the 
effect of a law1 in its usual           
application to human conduct1 is to     
produce regularity where it did not     
previously exist2 when then a regularity
 began to be perceived in nature1 the   
same word was used1 whether the cause   
was supposed to be the same or not4 ,in 
each case there was the same generality 
of agreement1 subject to occasional     
deflection4#h  ,on the other hand1      
observe the words 'wonderful'1          
'unexpected'1 'incredible'4 ,their      
connotation describes states of mind    
simply2 they are of course not confined 
to ,probability1 in the sense of        
statistical frequency1 but imply simply 
that the events they denote are such as 
from some cause we did not expect would 
happen1 and at which therefore1 when    
they do happen1 we are surprised4  ,now 
when we bear in mind that these two     
classes of words are in their origin    
perfectly distinct2--the one denoting   

simply events of a certain character2   
the other1 though also denoting events1 
connoting simply states of mind2--and   
yet that they are universally applied to
 the same events1 so as to be used as   
perfectly synonymous1 we have in this a 
striking illustration of the two sides  
under which ,probability may be viewed1 
and of the universal recognition of a   
close connection between them4 ,the     
words are popularly used as synonymous1 
and we must not press their meaning too 
far2 but if it were to be observed1 as  
,i am rather inclined to think it could1
 that the application of the words which
 denote mental states is wider than that
 of the others1 we should have an       
illustration of what has been already   
observed1 viz4 that the province of     
,probability is not so extensive as that
 over which variation of belief might be
 observed4 ,probability only considers  
the case in which this variation is     
brought about in a certain definite     
statistical way4  #cd4 ,it will be found

 in the end both interesting and        
important to have devoted some attention
 to this subjective side of the         
question4 ,in the first place1 as a mere
 speculative inquiry the quantity of our
 belief of any proposition deserves     
notice4 ,to study it at all deeply would
 be to trespass into the province of    
,psychology1 but it is so intimately    
connected with our own subject that we  
cannot avoid all reference to it4 ,we   
therefore discuss the laws under which  
our expectation and surprise at isolated
 events increases or diminishes1 so as  
to account for these states of mind in  
any individual instance1 and1 if        
necessary1 to correct them when they    
vary from their proper amount4  ,but    
there is another more important reason  
than this4 ,it is quite true that when  
the subjects of our discussion in any   
particular instance lie entirely within 
the province of ,probability1 they may  
be treated without any reference to our 
belief4 ,we may or we may not employ    

this side of the question according to  
our pleasure4 ,if1 for example1 ,i am   
asked whether it is more likely that ,a4
 ,b4 will die this year1 than that it   
will rain to-morrow1 ,i may calculate   
the chance 7which really is at bottom   
the same thing as my belief7 of each1   
find them respectively1 one-sixth and   
one-seventh1 say1 and therefore decide  
that my 'expectation' of the former is  
the greater1 viz4 that this is the more 
likely event4 ,in this case the process 
is precisely the same whether we suppose
 our belief to be introduced or not2 our
 mental state is1 in fact1 quite        
immaterial to the question4 ,but1 in    
other cases1 it may be different4       
,suppose that we are comparing two      
things1 of which one is wholly alien to 
,probability1 in the sense that it is   
hopeless to attempt to assign any degree
 of numerical frequency to it1 the only 
ground they have in common may be the   
amount of belief to which they are      
respectively entitled4 ,we cannot       

compare the frequency of their          
occurrence1 for one may occur too seldom
 to judge by1 perhaps it may be unique4 
,it has been already said1 that our     
belief of many events rests upon a very 
complicated and extensive basis4 ,my    
belief may be the product of many       
conflicting arguments1 and many         
analogies more or less remote2 these    
proofs themselves may have mostly faded 
from my mind1 but they will leave their 
effect behind them in a weak or strong  
conviction4 ,at the time1 therefore1 ,i 
may still be able to say1 with some     
degree of accuracy1 though a very slight
 degree1 what amount of belief ,i       
entertain upon the subject4 ,now we     
cannot compare things that are          
heterogeneous3 if1 therefore1 we are to 
decide between this and an event        
determined naturally and properly by    
,probability1 it is impossible to appeal
 to chances or frequency of occurrence4 
,the measure of belief is the only      
common ground1 and we must therefore    

compare this quantity in each case4 ,the
 test afforded will be an exceedingly   
rough one1 for the reasons mentioned    
above1 but it will be better than none2 
in some cases it will be found to       
furnish all we want4  ,suppose1 for     
example1 that one letter in a million is
 lost in the ,post ,office1 and that in 
any given instance ,i wish to know which
 is more likely1 that a letter has been 
so lost1 or that my servant has stolen  
it8 ,if the latter alternative could1   
like the former1 be stated in a         
numerical form1 the comparison would be 
simple4 ,but it cannot be reduced to    
this form1 at least not consciously and 
directly4 ,still1 if we could feel that 
our belief in the man's dishonesty was  
greater than one-millionth1 we should   
then have homogeneous things before us1 
and therefore comparison would be       
possible4  #ce4 ,we are now in a        
position to give a tolerably accurate   
definition of a phrase which we have    
frequently been obliged to employ1 or   

incidentally to suggest1 and of which   
the reader may have looked for a        
definition already1 viz4 the probability
 of an event1 or what is equivalent to  
this1 the chance of any given event     
happening4 ,i consider that these terms 
presuppose a series2 within the         
indefinitely numerous class which       
composes this series a smaller class is 
distinguished by the presence or absence
 of some attribute or attributes1 as was
 fully illustrated and explained in a   
previous chapter4 ,these larger and     
smaller classes respectively are        
commonly spoken of as instances of the  
'event1' and of 'its happening in a     
given particular way4' ,adopting this   
phraseology1 which with proper          
explanations is suitable enough1 we may 
define the probability or chance 7the   
terms are here regarded as synonymous7  
of the event happening in that          
particular way as the numerical fraction
 which represents the proportion between
 the two different classes in the long  

run4 ,thus1 for example1 let the        
probability be that of a given infant   
living to be eighty years of age4 ,the  
larger series will comprise all infants1
 the smaller all who live to eighty4    
,let the proportion of the former to the
 latter be #i to #a2 in other words1    
suppose that one infant in ten lives to 
eighty4 ,then the chance or probability 
that any given infant will live to      
eighty is the numerical fraction #a_/aj4
 ,this assumes that the series are of   
indefinite extent1 and of the kind which
 we have described as possessing a fixed
 type4 ,if this be not the case1 but the
 series be supposed terminable1 or      
regularly or irregularly fluctuating1 as
 might be the case1 for instance1 in a  
society where owing to sanitary or other
 causes the average longevity was       
steadily undergoing a change1 then in so
 far as this is the case the series     
ceases to be a subject of science4 ,what
 we have to do under these              
circumstances1 is to substitute a series

 of the right kind for the inappropriate
 one presented by nature1 choosing it1  
of course1 with as little deflection as 
possible from the observed facts4 ,this 
is nothing more than has to be done1 and
 invariably is done1 whenever natural   
objects are made subjects of strict     
science4  #cf4 ,a word or two of        
explanation may be added about the      
expression employed above1 'the         
proportion in the long run4' ,the run   
must be supposed to be very long indeed1
 in fact never to stop4 ,as we keep on  
taking more terms of the series we shall
 find the proportion still fluctuating a
 little1 but its fluctuations will grow 
less4 ,the proportion1 in fact1 will    
gradually approach towards some fixed   
numerical value1 what mathematicians    
term its limit4 ,this fractional value  
is the one spoken of above4 ,in the     
cases in which deductive reasoning is   
possible1 this fraction may be obtained 
without direct appeal to statistics1    
from reasoning about the conditions     

under which the events occur1 as was    
explained in the fourth chapter4  ,here 
becomes apparent the full importance of 
the distinction so frequently insisted  
on1 between the actual irregular series 
before us and the substituted one of    
calculation1 and the meaning of the     
assertion 7,ch4 ,i4 #ac71 that it was in
 the case of the latter only that strict
 scientific inferences could be made4   
,for how can we have a 'limit' in the   
case of those series which ultimately   
exhibit irregular fluctuations8 ,when we
 say1 for instance1 that it is an even  
chance that a given person recovers from
 the cholera1 the meaning of this       
assertion is that in the long run one   
half of the persons attacked by that    
disease do recover4 ,but if we examined 
a sufficiently extensive range of       
statistics1 we might find that the      
manners and customs of society had      
produced such a change in the type of   
the disease or its treatment1 that we   
were no nearer approaching towards a    

fixed limit than we were at first4 ,the 
conception of an ultimate limit in the  
ratio between the numbers of the two    
classes in the series necessarily       
involves an absolute fixity of the type4
 ,when therefore nature does not present
 us with this absolute fixity1 as she   
seldom or never does except in games of 
chance 7and not demonstrably there71 our
 only resource is to introduce such a   
series1 in other words1 as has so often 
been said1 to substitute a series of the
 right kind4  #cg4 ,the above1 which may
 be considered tolerably complete as a  
definition1 might equally well have been
 given in the last chapter4 ,it has been
 deferred however to the present place1 
in order to connect with it at once a   
proposition involving the conceptions   
introduced in this chapter2 viz4 the    
state of our own minds1 in reference to 
the amount of belief we entertain in    
contemplating any one of the events     
whose probability has just been         
described4 ,reasons were given against  

the opinion that our belief admitted of 
any exact apportionment like the        
numerical one just mentioned4 ,still1 it
 was shown that a reasonable explanation
 could be given of such an expression   
as1 'my belief is #a_/ajth of           
certainty'1 though it was an explanation
 which pointed unmistakeably to a series
 of events1 and ceased to be            
intelligible1 or at any rate            
justifiable1 when it was not viewed in  
such a relation to a series4 ,in so far1
 then1 as this explanation is adopted1  
we may say that our belief is in        
proportion to the above fraction4 ,this 
referred to the purely intellectual part
 of belief which cannot be conceived to 
be separable1 even in thought1 from the 
things upon which it is exercised4 ,with
 this intellectual part there are       
commonly associated various emotions4   
,these we can to a certain extent       
separate1 and1 when separated1 can      
measure with that degree of accuracy    
which is possible in the case of other  

emotions4 ,they are moreover            
intelligible in reference to the        
individual events4 ,they will be found  
to increase and diminish in accordance1 
to some extent1 with the fraction which 
represents the scarcity of the event4   
,the emotion of surprise does so with   
some degree of accuracy4  ,the above    
investigation describes1 though in a    
very brief form1 the amount of truth    
which appears to me to be contained in  
the assertion frequently made1 that the 
fraction expressive of the probability  
represents also the fractional part of  
full certainty to which our belief of   
the individual event amounts4 ,any      
further analysis of the matter would    
seem to belong to ,psychology rather    
than to ,probability4   #a4 ,in the     
ordinary signification of this term4 ,as
 ,de ,morgan uses it he   makes ,formal 
,logic include ,probability1 as one of  
its branches1 as   indicated in his     
title ,formal ,logic1 or the ,calculus  
of ,inference1   necessary and probable4

  #b4 ,formal ,logic4 ,preface1 page v4 
#c4 ,an illustration of the points here 
insisted on has recently #ahgf   been   
given in a quarter where few would have 
expected it2 ,i allude1   as many       
readers will readily infer1 to ,j4 ,s4  
,mill's exceedingly   interesting       
,essays on ,theism4 ,it is not within   
our province here to   criticise any of 
their conclusions1 but they have        
expressed in a   very significant way   
the conviction entertained by him that  
beliefs   which are not justified by    
evidence1 and possibly may not be       
capable   of justification 7those for   
instance of immortality and the         
existence of the ,deity71 may           
nevertheless not only continue to exist 
in cultivated minds1 but may also be    
profitably encouraged there1 at   any   
rate in the shape of hopes1 for certain 
supposed advantages   attendant on their
 retention1 irrespective even of their  
truth4  #d4 ,it is necessary to take an 
example in which the man is forced to   

act1 or we should not be able to shew   
that he has any belief on the   subject 
at all4 ,he may declare that he neither 
knows nor cares   anything about the    
matter1 and that therefore there is     
nothing of   the nature of belief to be 
extracted out of his mental condition4  
,he   very likely would take this ground
 if we asked him1 as ,de ,morgan   does1
 with a slightly different reference    
7,formal ,logic1 p4 #ahc71   whether he 
considers that there are volcanoes on   
the unseen side of   the moon larger    
than those on the side turned towards   
us2 or1 with   ,jevons 7,principles of  
,science1 ,ed4 ,,ii4 p4 #bab7 whether he
   considers that a ,platythliptic      
,coefficient is positive4 ,these do not 
therefore seem good instances to        
illustrate the position that we   always
 entertain a certain degree of belief on
 every question which   can be stated1  
and that utter inability to give a      
reason in favour   of either alternative
 corresponds to half belief4  #e4       

,except indeed on the principles        
indicated further on in #bd1 #be4  #f4  
,for a fuller discussion of this1 see   
the ,chapter on ,causation4  #g4 ,the   
best example ,i can recall of the       
distinction between judging   from the  
subjective and the objective side1 in   
such cases as these1   occurred once in 
a railway train4 ,i met a timid old lady
 who was in   much fear of accidents4 ,i
 endeavoured to soothe her on the usual 
statistical ground of the extreme rarity
 of such events4 ,she   listened        
patiently1 and then replied1 ,yes1 ,sir1
 that is all very   well2 but ,i don't  
see how the real danger will be a bit   
the less   because ,i don't believe in  
it4  #h4 ,this would still hold of      
empirical laws which may be capable of  
being broken3 we now have very much     
shifted the word1 to denote an          
ultimate law which it is supposed cannot
 be broken4     ,,chapter ,,vii4   ,,the
 ,,rules ,,of ,,inference ,,in          
,,probability4  #a4 ,in the previous    

chapter1 an investigation was made into 
what may be called1 from the analogy of 
,logic1 ,immediate ,inferences4 ,given  
that nine men out of ten1 of any        
assigned age1 live to forty1 what could 
be inferred about the prospect of life  
of any particular man8 ,it was shown    
that1 although this step was very far   
from being so simple as it is frequently
 supposed to be1 and as the             
corresponding step really is in ,logic1 
there was nevertheless an intelligible  
sense in which we might speak of the    
amount of our belief in any one of these
 'proportional propositions1' as they   
may succinctly be termed1 and justify   
that amount4 ,we must now proceed to the
 consideration of inferences more       
properly so called1 ,i mean inferences  
of the kind analogous to those which    
form the staple of ordinary logical     
treatises4 ,in other words1 having      
ascertained in what manner particular   
propositions could be inferred from the 
general propositions which included     

them1 we must now examine in what cases 
one general proposition can be inferred 
from another4 ,by a general proposition 
here is meant1 of course1 a general     
proposition of the statistical kind     
contemplated in ,probability4 ,the rules
 of such inference being very few and   
simple1 their consideration will not    
detain us long4 ,from the data now in   
our possession we are able to deduce the
 rules of probability given in ordinary 
treatises upon the science4 ,it would be
 more correct to say that we are able to
 deduce some of these rules1 for1 as    
will appear on examination1 they are of 
two very different kinds1 resting on    
entirely distinct grounds4 ,they might  
be divided into those which are formal1 
and those which are more or less        
experimental4 ,this may be otherwise    
expressed by saying that1 from the kind 
of series described in the first        
chapters1 some rules will follow        
necessarily by the mere application of  
arithmetic2 whilst others either depend 

upon peculiar hypotheses1 or demand for 
their establishment continually renewed 
appeals to experience1 and extension by 
the aid of the various resources of     
,induction4 ,we shall confine our       
attention at present principally to the 
former class2 the latter can only be    
fully understood when we have considered
 the connection of our science with     
,induction4  #b4 ,the fundamental rules 
of ,probability strictly so called1 that
 is the formal rules1 may be divided    
into two classes1--those obtained by    
addition or subtraction on the one hand1
 corresponding to what are generally    
termed the connection of exclusive or   
incompatible events2#a and those        
obtained by multiplication or division1 
on the other hand1 corresponding to what
 are commonly termed dependent events4  
,we will examine these in order4  7#a7  
,we can make inferences by simple       
addition4 ,if1 for instance1 there are  
two distinct properties observable in   
various members of the series1 which    

properties do not occur in the same     
individual2 it is plain that in any     
batch the number that are of one kind or
 the other will be equal to the sum of  
those of the two kinds separately4 ,thus
 #cf4d infants in #ajj live to over     
sixty1 #ce4d in #ajj die before they are
 ten2#b take a large number1 say        
#aj1jjj1 then there will be about #cfdj 
who live to over sixty1 and about #cedj 
who do not reach ten2 hence the total   
number who do not die within the        
assigned limits will be about #bhbj     
altogether4 ,of course if these         
proportions were accurately assigned1   
the resultant sum would be equally      
accurate3 but1 as the reader knows1 in  
,probability this proportion is merely  
the limit towards which the numbers tend
 in the long run1 not the precise result
 assigned in any particular case4 ,hence
 we can only venture to say that this is
 the limit towards which we tend as the 
numbers become greater and greater4     
,this rule1 in its general algebraic    

form1 would be expressed in the language
 of ,probability as follows3--,if the   
chances of two exclusive or incompatible
 events be respectively #a_/m and #a_/n 
the chance of one or other of them      
happening will be #a_/m  #a_/n or 7m    
n7_/mn4 ,similarly if there were more   
than two events of the kind in question4
 ,on the principles adopted in this     
work1 the rule1 when thus algebraically 
expressed1 means precisely the same     
thing as when it is expressed in the    
statistical form4 ,it was shown at the  
conclusion of the last chapter that to  
say1 for example1 that the chance of a  
given event happening in a certain way  
is #a_/f1 is only another way of saying 
that in the long run it does tend to    
happen in that way once in six times4   
,it is plain that a sort of corollary to
 this rule might be obtained1 in        
precisely the same way1 by subtraction  
instead of addition4 ,stated generally  
it would be as follows3--,if the chance 
of one or other of two incompatible     

events be #a_/m and the chance of one   
alone be #a_/n1 the chance of the       
remaining one will be #a_/m - #a_/n or  
7n - m7_/nm4  ,for example1 if the      
chance of any one dying in a year is    
#a_/aj1 and his chance of dying of some 
particular disease is #a_/ajj1 his      
chance of dying of any other disease is 
#i_/ajj4  ,the reader will remark here  
that there are two apparently different 
modes of stating this rule1 according as
 we speak of 'one or other of two or    
more events happening1' or of 'the same 
event happening in one or other of two  
or more ways4' ,but no confusion need   
arise on this ground2 either way of     
speaking is legitimate1 the difference  
being merely verbal1 and depending 7as  
was shown in the first chapter1 #h7 upon
 whether the distinctions between the   
'ways' are or are not too deep and      
numerous to entitle the event to be     
conventionally regarded as the same4    
,we may also here point out the         
justification for the common doctrine   

that certainty is represented by unity1 
just as any given degree of probability 
is represented by its appropriate       
fraction4 ,if the statement that an     
event happens once in m times1 is       
equivalently expressed by saying that   
its chance is #a_/m1 it follows that to 
say that it happens m times in m times1 
or every time without exception1 is     
equivalent to saying that its chance is 
m_/m or #a4 ,now an event that happens  
every time is of course one of whose    
occurrence we are certain2 hence the    
fraction which represents the 'chance'  
of an event which is certain becomes    
unity4  ,it will be equally obvious that
 given that the chance that an event    
will happen is #a_/m1 the chance that it
 will not happen is #a - #a_/m or 7m -  
#a7_/m4  #c4 7#b7 ,we can also make     
inferences by multiplication or         
division4 ,suppose that two events      
instead of being incompatible1 are      
connected together in the sense that one
 is contingent upon the occurrence of   

the other4 ,let us be told that a given 
proportion of the members of the series 
possess a certain property1 and a given 
proportion again of these possess       
another property1 then the proportion of
 the whole which possess both properties
 will be found by multiplying together  
the two fractions which represent the   
above two proportions4 ,of the          
inhabitants of ,london1 twenty-five in a
 thousand1 say1 will die in the course  
of the year2 we suppose it to be known  
also that one death in five is due to   
fever2 we should then infer that one in 
#bjj of the inhabitants will die of     
fever in the course of the year4 ,it    
would of course be equally simple1 by   
division1 to make a sort of converse    
inference4 ,given the total mortality   
per cent4 of the population from fever1 
and the proportion of fever cases to the
 aggregate of other cases of mortality1 
we might have inferred1 by dividing one 
fraction by the other1 what was the     
total mortality per cent4 from all      

causes4  ,the rule as given above is    
variously expressed in the language of  
,probability4 ,perhaps the simplest and 
best statement is that it gives us the  
rule of dependent events4 ,that is2 if  
the chance of one event is #a_/m1 and   
the chance that if it happens another   
will also happen #a_/n1 then the chance 
of the latter is #a_/mn4 ,in this case  
it is assumed that the latter is so     
entirely dependent upon the former that 
though it does not always happen with   
it1 it certainly will not happen without
 it2 the necessity of this assumption   
however may be obviated by saying that  
what we are speaking of in the latter   
case is the joint event1 viz4 both      
together if they are simultaneous       
events1 or the latter in consequence of 
the former1 if they are successive4  #d4
 ,the above inferences are necessary1 in
 the sense in which arithmetical        
inferences are necessary1 and they do   
not demand for their establishment any  
arbitrary hypothesis4 ,we assume in them

 no more than is warranted1 and in fact 
necessitated by the data actually given 
to us1 and make our inferences from     
these data by the help of arithmetic4   
,in the simple examples given above     
nothing is required beyond arithmetic in
 its most familiar form1 but it need    
hardly be added that in practice        
examples may often present themselves   
which will require much profounder      
methods than these4 ,it may task all the
 resources of that higher and more      
abstract arithmetic known as algebra to 
extract a solution4 ,but as the         
necessity of appeal to such methods as  
these does not touch the principles of  
this part of the subject we need not    
enter upon them here4  #e4 ,the formula 
next to be discussed stands upon a      
somewhat different footing from the     
above in respect of its cogency and     
freedom from appeal to experience1 or to
 hypothesis4 ,in the two former         
instances we considered cases in which  
the data were supposed to be given under

 the conditions that the properties     
which distinguished the different kinds 
of events whose frequency was discussed1
 were respectively known to be          
disconnected and known to be connected4 
,let us now suppose that no such        
conditions are given to us4 ,one man in 
ten1 say1 has black hair1 and one in    
twelve is short-sighted2 what           
conclusions could we then draw as to the
 chance of any given man having one only
 of these two attributes1 or neither1 or
 both8 ,it is clearly possible that the 
properties in question might be         
inconsistent with one another1 so as    
never to be found combined in the same  
person2 or all the short-sighted might  
have black hair2 or the properties might
 be allotted#c in almost any other      
proportion whatever4 ,if we are         
perfectly ignorant upon these points1 it
 would seem that no inferences whatever 
could be drawn about the required       
chances4  ,inferences however are drawn1
 and practically1 in most cases1 quite  

justly drawn4 ,an escape from the       
apparent indeterminateness of the       
problem1 as above described1 is found by
 assuming that1 not merely will         
one-tenth of the whole number of men    
have black hair 7for this was given as  
one of the data71 but also that         
one-tenth alike of those who are and who
 are not short-sighted have black hair4 
,let us take a batch of #abjj1 as a     
sample of the whole4 ,now1 from the data
 which were originally given to us1 it  
will easily be seen that in every such  
batch there will be on the average #abj 
who have black hair1 and therefore #ajhj
 who have not4 ,and here in strict right
 we ought to stop1 at least until we    
have appealed again to experience2 but  
we do not stop here4 ,from data which we
 assume1 we go on to infer that of the  
#abj1 #aj 7i4e4 one-twelfth of #abj7    
will be short-sighted1 and #aaj 7the    
remainder7 will not4 ,similarly we infer
 that of the #ajhj1 #ij are             
short-sighted1 and #iij are not4 ,on the

 whole1 then1 the #abjj are thus        
divided3--black-haired short-sighted1   
#aj2 short-sighted without black hair1  
#ij2 black-haired men who are not       
short-sighted1 #aaj2 men who are neither
 short-sighted nor have black hair1     
#iij4  ,this rule1 expressed in its most
 general form1 in the language of       
,probability1 would be as follows3--,if 
the chances of a thing being p and q are
 respectively #a_/m and #a_/n1 then the 
chance of its being both p and q is     
#a_/mn1 p and not q is 7n - #a7_/mn1 q  
and not p is 7m - #a7_/mn1 not p and not
 q is 77m - #a77n - #a77_/mn1 where p   
and q are independent4 ,the sum of these
 chances is obviously unity2 as it ought
 to be1 since one or other of the four  
alternatives must necessarily exist4    
#f4 ,i have purposely emphasized the    
distinction between the inference in    
this case1 and that in the two          
preceding1 to an extent which to many   
readers may seem unwarranted4 ,but it   
appears to me that where a science makes

 use1 as ,probability does1 of two such 
very distinct sources of conviction as  
the necessary rules of arithmetic and   
the merely more or less cogent ones of  
,induction1 it is hardly possible to lay
 too much stress upon the distinction4  
,few will be prepared to deny that very 
arbitrary assumptions have been made by 
many writers on the subject1 and none   
will deny that in the case of what are  
called 'inverse probabilities'          
assumptions are sometimes made which are
 at least decidedly open to question4   
,the best course therefore is to make a 
pause and stringent enquiry at the point
 at which the possibility of such error 
and doubtfulness first exhibits itself4 
,these remarks apply to some of the best
 writers on the subject2 in the case of 
inferior writers1 or those who appeal to
 ,probability without having properly   
mastered its principles1 we may go      
further4 ,it would really not be        
asserting too much to say that they seem
 to think themselves justified in       

assuming that where we know nothing     
about the distribution of the properties
 alluded to we must assume them to be   
distributed as above described1 and     
therefore apportion our belief in the   
same ratio4 ,this is called 'assuming   
the events to be independent1' the      
supposition being made that the rule    
will certainly follow from this         
independence1 and that we have a right1 
if we know nothing to the contrary1 to  
assume that the events are independent4 
,the validity of this last claim has    
already been discussed in the first     
chapter2 it is only another of the      
attempts to construct  priori the series
 which experience will present to us1   
and one for which no such strong defence
 can be made as for the equality of     
heads and tails in the throws of a      
penny4 ,but the meaning to be assigned  
to the 'independence' of the events in  
question demands a moment's             
consideration4  ,the circumstances of   
the problem are these4 ,there are two   

different qualities1 by the presence and
 absence respectively of each of which1 
amongst the individuals of a series1 two
 distinct pairs of classes of these     
individuals are produced4 ,for the      
establishment of the rule under         
discussion it was found that one        
supposition was both necessary and      
sufficient1 namely1 that the division   
into classes caused by each of the above
 distinctions should subdivide each of  
the classes created by the other        
distinction in the same ratio in which  
it subdivides the whole4 ,if the        
independence be granted and so defined  
as to mean this1 the rule of course will
 stand1 but1 without especial attention 
being drawn to the point1 it does not   
seem that the word would naturally be so
 understood4  #g4 ,the above1 then1     
being the fundamental rules of inference
 in probability1 the question at once   
arises1 ,what is their relation to the  
great body of formul which are made use 
of in treatises upon the science1 and in

 practical applications of it8 ,the     
reply would be that these formul1 in so 
far as they properly belong to the      
science1 are nothing else in reality    
than applications of the above          
fundamental rules4 ,such applications   
may assume any degree of complexity1 for
 owing to the difficulty of particular  
examples1 in the form in which they     
actually present themselves1 recourse   
must sometimes be made to the           
profoundest theorems of mathematics4    
,still we ought not to regard these     
theorems as being anything else than    
convenient and necessary abbreviations  
of arithmetical processes1 which in     
practice have become too cumbersome to  
be otherwise performed4  ,this          
explanation will account for some of the
 rules as they are ordinarily given1 but
 by no means for all of them4 ,it will  
account for those which are demonstrable
 by the certain laws of arithmetic1 but 
not for those which in reality rest only
 upon inductive generalizations4 ,and it

 can hardly be doubted that many rules  
of the latter description have become   
associated with those of the former1 so 
that in popular estimation they have    
been blended into one system1 of which  
all the separate rules are supposed to  
possess a similar origin and equal      
certainty4 ,hints have already been     
frequently given of this tendency1 but  
the subject is one of such extreme      
importance that a separate chapter 7that
 on ,induction7 must be devoted to its  
consideration4  #h4 ,in establishing the
 validity of the above rules1 we have   
taken as the basis of our               
investigations1 in accordance with the  
general scheme of this work1 the        
statistical frequency of the events     
referred to2 but it was also shown that 
each formula1 when established1 might   
with equal propriety be expressed in the
 more familiar form of a fraction       
representing the 'chance' of the        
occurrence of the particular event4 ,the
 question may therefore now be raised1  

,can those writers who 7as described in 
the last chapter7 take as the primary   
subject of the science not the degree of
 statistical frequency1 but the quantity
 of belief1 with equal consistency make 
this the basis of their rules1 and so   
also regard the fraction expressive of  
the chance as a merely synonymous       
expression8 ,de ,morgan maintains that  
whereas in ordinary logic we suppose the
 premises to be absolutely true1 the    
province of ,probability is to study    
'the effect which partial belief of the 
premises produces with respect to the   
conclusion4' ,it would appear therefore 
as if in strictness we ought on this    
view to be able to determine this       
consequent diminution at first hand1    
from introspection of the mind1 that is 
of the conceptions and beliefs which it 
entertains2 instead of making any       
recourse to statistics to tell us how   
much we ought to believe the conclusion4
  ,any readers who have concurred with  
me in the general results of the last   

chapter1 will naturally agree in the    
conclusion that nothing deserving the   
name of logical science can be extracted
 from any results of appeal to our      
consciousness as to the quantity of     
belief we entertain of this or that     
proposition4 ,suppose1 for example1 that
 one person in #ajj dies on the sea     
passage out to ,india1 and that one in  
#i dies during a #e years residence     
there4 ,it would commonly be said that  
the chance that any one1 who is now     
going out1 has of living to start       
homewards #e years hence1 is #hh_/ajj2  
for his chance of getting there is      
#ii_/ajj2 and of his surviving1 if he   
gets there1 #h_/i2 hence the result or  
dependent event is got by multiplying   
these fractions together1 which gives   
#hh_/ajj4 ,here the real basis of the   
reasoning is statistical1 and the       
processes or results are merely         
translated afterwards into fractions4   
,but can we say the same when we look at
 the belief side of the question8 ,i    

quite admit the psychological fact that 
we have degrees of belief1 more or less 
corresponding to the frequency of the   
events to which they refer4 ,in the     
above example1 for instance1 we should  
undoubtedly admit on enquiry that our   
belief in the man's return was affected 
by each of the risks in question1 so    
that we had less expectation of it than 
if he were subject to either risk       
separately2 that is1 we should in some  
way compound the risks4 ,but what ,i    
cannot recognise is that we should be   
able to perform the process with any    
approach to accuracy without appeal to  
the statistics1 or that1 even supposing 
we could do so1 we should have any      
guarantee of the correctness of the     
result without similar appeal4 ,it      
appears to me in fact that but little   
meaning1 and certainly no security1 can 
be attained by so regarding the process 
of inference4 ,the probabilities        
expressed as degrees of belief1 just as 
those which are expressed as fractions1 

must1 when we are put upon our          
justification1 first be translated into 
their corresponding facts of statistical
 frequency of occurrence of the events1 
and then the inferences must be drawn   
and justified there4 ,this part of the  
operation1 as we have already shown1 is 
mostly carried on by the ordinary rules 
of arithmetic4 ,when we have obtained   
our conclusion we may1 if we please1    
translate it back again into the        
subjective form1 just as we can and do  
for convenience into the fractional1 but
 ,i do not see how the process of       
inference can be conceived as taking    
place in that form1 and still less how  
any proof of it can thus be given4 ,if  
therefore the process of inference be so
 expressed it must be regarded as a     
symbolical process1 symbolical of such  
an inference about things as has been   
described above1 and it therefore seems 
to me more advisable to state and       
expound it in this latter form4   ,on   
,inverse ,probability and the ,rules    

required for it4  #i4 ,it has been      
already stated that the only fundamental
 rules of inference in ,probability are 
the two described in #b1 #c1 but there  
are of course abundance of derivative   
rules1 the nature and use of which are  
best obtained from the study of any     
manual upon the subject4 ,one class of  
these derivative rules1 however1 is     
sufficiently distinct in respect of the 
questions to which it may give rise1 to 
deserve special examination4 ,it        
involves the distinction commonly       
recognised as that between ,direct and  
,inverse ,probability4 ,it is thus      
introduced by ,de ,morgan3--  ,in the   
preceding chapter we have calculated the
 chances of an event1 knowing the       
circumstances under which it is to      
happen or fail4 ,we are now to place    
ourselves in an inverted position3 we   
know the event1 and ask what is the     
probability which results from the event
 in favour of any set of circumstances  
under which the same might have         

happened4#d ,the distinction might      
therefore be summarily described as that
 between finding an effect when we are  
given the causes1 and finding a cause   
when we are given effects4  ,on the     
principles of the science involved in   
the definition which was discussed and  
adopted in the earlier chapters of this 
work1 the reader will easily infer that 
no such distinction as this can be      
regarded as fundamental4 ,one common    
feature was traced in all the objects   
which were to be referred to            
,probability1 and from this feature the 
possible rules of inference can be      
immediately derived4 ,all other         
distinctions are merely those of        
arrangement or management4  ,but        
although the distinction is not by any  
means fundamental1 it is nevertheless   
true that the practical treatment of    
such problems as those principally      
occurring in ,inverse ,probability1 does
 correspond to a very serious source of 
ambiguity and perplexity4 ,the arbitrary

 assumptions which appear in ,direct    
,probability are not by any means       
serious1 but those which invade us in a 
large proportion of the problems offered
 by ,inverse ,probability are both      
serious and inevitable4  #aj4 ,this will
 be best seen by the examination of     
special examples2 as any1 however       
simple1 will serve our purpose1 let us  
take the two following3--  7#a7 ,a ball 
is drawn from a bag containing nine     
black balls and one white3 what is the  
chance of its being the white ball8     
7#b7 ,a ball is drawn from a bag        
containing ten balls1 and is found to be
 white2 what is the chance of there     
having been but that one white ball in  
the bag8  ,the class of which the first 
example is a simple instance has been   
already abundantly discussed4 ,the      
interpretation of it is as follows3 ,if 
balls be continually drawn and replaced1
 the proportion of white ones to the    
whole number drawn will tend towards the
 fraction #a_/aj4 ,the contemplated     

action is a single one1 but we view it  
as one of the above series2 at least our
 opinion is formed upon that assumption4
 ,we conclude that we are going to take 
one of a series of events which may     
appear individually fortuitous1 but in  
which1 in the long run1 those of a given
 kind are one-tenth of the whole2 this  
kind 7white7 is then singled out by     
anticipation4 ,by stating that its      
chance is #a_/aj1 we merely mean to     
assert this physical fact1 together with
 such other mental facts1 emotions1     
inferences1 c41 as may be properly      
associated with it4  #aa4 ,have we to   
interpret the second example in a       
different way8 ,here also we have a     
single instance1 but the nature of the  
question would seem to decide that the  
only series to which it can properly be 
referred is the following3--,balls are  
continually drawn from different bags   
each containing ten1 and are always     
found to be white2 what is ultimately   
the proportion of cases in which they   

will be found to have been taken from   
bags with only one white ball in them8  
,now it may be readily shown#e that time
 has nothing to do with the question2   
omitting therefore the consideration of 
this element1 we have for the two series
 from which our opinions in these two   
examples respectively are to be         
formed3--7#a7 balls of different colours
 presented to us in a given ultimate    
ratio2 7#b7 bags with different contents
 similarly presented4 ,from these data  
respectively we have to assign their due
 weight to our anticipations of 7#a7 a  
white ball2 7#b7 a bag containing but   
one white ball4 ,so stated the problems 
would appear to be formally identical4  
,when1 however1 we begin the practical  
work of solving them we perceive a most 
important distinction4 ,in the first    
example there is not much that is       
arbitrary2 balls would under such       
circumstance really come out more or    
less accurately in the proportion       
expected4 ,moreover1 in case it should  

be objected that it is difficult to     
prove that they will do so1 it does not 
seem an unfair demand to say that the   
balls are to be 'well-mixed' or 'fairly 
distributed1' or to introduce any of the
 other conditions by which1 under the   
semblance of judging  priori1 we take   
care to secure our prospect of a series 
of the desired kind4 ,but we cannot say 
the same in the case of the second      
example4  #ab4 ,the line of proof by    
which it is generally attempted to solve
 the second example is of this          
kind2--,it is shown that there being one
 white ball for certain in the bag1 the 
only possible antecedents are of ten    
kinds1 viz4 bags1 each of which contains
 ten balls1 but in which the white balls
 range respectively from one to ten in  
number4 ,this of course imposes limits  
upon the kind of terms to be found in   
our series4 ,but we want more than such 
limitations1 we must know the           
proportions in which these terms are    
ultimately found to arrange themselves  

in the series4 ,now this requires an    
experience about bags which may not1 and
 indeed in a large proportion of similar
 cases1 cannot1 be given to us4 ,if     
therefore we are to solve the question  
at all we must make an assumption2 let  
us make the following2--that each of the
 bags described above occurs equally    
often1--and see what follows4 ,the bags 
being drawn from equally often1 it does 
not follow that they will each yield    
equal numbers of white balls4 ,on the   
contrary they will1 as in the last      
example1 yield them in direct proportion
 to the number of such balls which they 
contain4 ,the bag with one white and    
nine black will yield a white ball once 
in ten times2 that with two white1      
twice2 and so on4 ,the result of this1  
it will be easily seen1 is that in #ajj 
drawings there will be obtained on the  
average #ee white balls and #de black4  
,now with those drawings that do not    
yield white balls we have1 by the       
question1 nothing to do1 for that       

question postulated the drawing of a    
white ball as an accomplished fact4 ,the
 series we want is therefore composed of
 those which do yield white4 ,now what  
is the additional attribute which is    
found in some members1 and in some      
members only1 of this series1 and which 
we mentally anticipate8 ,clearly it is  
the attribute of having been drawn from 
a bag which only contained one of these 
white balls4 ,of these there is1 out of 
the #ee drawings1 but one4 ,accordingly 
the required chance is #a_/ee4 ,that is 
to say1 the white ball will have been   
drawn from the bag containing only that 
one white1 once in #ee times4  #ac4     
,now1 with the exception of the passage 
in italics1 the process here is         
precisely the same as in the other      
example2 it is somewhat longer only     
because we are not able to appeal       
immediately to experience1 but are      
forced to try to deduce what the result 
will be1 though the validity of this    
deduction itself rests1 of course1      

ultimately upon experience4 ,but the    
above passage is a very important one4  
,it is scarcely necessary to point out  
how arbitrary it is4  ,for is the       
supposition1 that the different         
specified kinds of bags are equally     
likely1 the most reasonable supposition 
under the circumstances in question8    
,one man may think it is1 another may   
take a contrary view4 ,in fact in an    
excellent manual#f upon the subject a   
totally different supposition is made1  
at any rate in one example2 it is taken 
for granted in that instance1 not that  
every possible number of black and white
 balls respectively is equally likely1  
but that every possible way of getting  
each number is equally likely1 whence it
 follows that bags with an intermediate 
number of black and white balls are far 
more likely than those with an extreme  
number of either4 ,on this supposition  
five black and five white being         
obtainable in #beb ways against the ten 
ways of obtaining one white and nine    

black1 it follows that the chance that  
we have drawn from a bag of the latter  
description is much less than on the    
hypothesis first made4 ,the chance1 in  
fact1 becomes now #a_/eab instead of    
#a_/ee4 ,in the one case each distinct  
result is considered equally likely1 in 
the other every distinct way of getting 
each result4  #ad4 ,uncertainties of    
this kind are peculiarly likely to arise
 in these inverse probabilities1 because
 when we are merely given an effect and 
told to look out for the chance of some 
assigned cause1 we are often given no   
clue as to the relative prevalence of   
these causes1 but are left to determine 
them on general principles4 ,give us    
either their actual prevalence in       
statistics1 or the conditions by which  
such prevalence is brought about1 and we
 know what to do2 but without the help  
of such data we are reduced to guessing4
 ,in the above example1 if we had been  
told how the bag had been originally    
filled1 that is by what process1 or     

under what circumstances1 we should have
 known what to do4 ,if it had been      
filled at random from a box containing  
equal numbers of black and white balls1 
the supposition in ,mr ,whitworth's     
example is the most reasonable2 but in  
the absence of any such information as  
this we are entirely in the dark1 and   
the supposition made in #ab is neither  
more nor less trustworthy and reasonable
 than many others1 though it doubtless  
possesses the merit of superior         
simplicity4#g ,if the reader will recur 
to ,ch4 ,v4 #d1 #e1 he will find this   
particular difficulty fully explained4  
,everybody practically admits that a    
certain characteristic arrangement or   
distribution has to be introduced at    
some prior stage2 and that1 as soon as  
this stage has been selected1 there are 
no further theoretic difficulties to be 
encountered4 ,but when we come to       
decide1 in examples of the class in     
question1 at what stage it is most      
reasonable to make our postulate1 we are

 often left without any very definite or
 rational guidance4  #ae4 ,when1        
however1 we take what may be called1 by 
comparison with the above purely        
artificial examples1 instances presented
 by nature1 much of this uncertainty    
will disappear1 and then all real       
distinction between direct and inverse  
probability will often vanish4 ,in such 
cases the causes are mostly determined  
by tolerably definite rules1 instead of 
being a mere cloud-land of capricious   
guesses4 ,we may either find their      
relative frequency of occurrence by     
reference to tables1 or may be able to  
infer it by examination of the          
circumstances under which they are      
brought about4 ,almost any simple       
example would then serve to illustrate  
the fact that under such circumstances  
the distinction between direct and      
inverse probability disappears          
altogether1 or merely resolves itself   
into one of time1 which1 as will be more
 fully shown in a future chapter1 is    

entirely foreign to our subject4  ,it is
 not of course intended to imply that   
difficulties similar to those mentioned 
above do not occasionally invade us here
 also4 ,as already mentioned1 they are1 
if not inherent in the subject1 at any  
rate almost unavoidable in comparison   
with the simpler and more direct        
procedure of determining what is likely 
to follow from assigned conditions4     
,what is meant is that so long as we    
confine ourselves within the            
comparatively regular and uniform field 
of natural sequences and co-existences1 
statistics of causes may be just as     
readily available as those of effects4  
,there will not be much more that is    
arbitrary in the one than in the other4 
,but of course this security is lost    
when1 as will be almost immediately     
noticed1 what may be called metaphysical
 rather than natural causes are         
introduced into the enquiry4  ,for      
instance1 it is known that in ,london   
about #bj people die per thousand each  

year4 ,suppose it also known that of    
every #ajj deaths there are about #d    
attributable to bronchitis4 ,the odds   
therefore against any unknown person    
dying of bronchitis in a given year are 
#abdi to #a4 ,exactly the same          
statistics are available to solve the   
inverse problem3--,a man is dead1 what  
is the chance that he died of           
bronchitis8 ,here1 since the man's death
 is taken for granted1 we do not require
 to know the general average mortality4 
,all that we want is the proportional   
mortality from the disease in question  
as given above4 ,if ,probability dealt  
only with inferences founded in this way
 upon actual statistics1 and these      
tolerably extensive1 it is scarcely     
likely that any distinction such as this
 between direct and inverse problems    
would ever have been drawn4  #af4       
,considered therefore as a contribution 
to the theory of the subject1 the       
distinction between ,direct and ,inverse
 ,probability must be abandoned4 ,when  

the appropriate statistics are at hand  
the two classes of problems become      
identical in method of treatment1 and   
when they are not we have no more right 
to extract a solution in one case than  
in the other4 ,the discussion however   
may serve to direct renewed attention to
 another and far more important         
distinction4 ,it will remind us that    
there is one class of examples to which 
the calculus of ,probability is         
rightfully applied1 because statistical 
data are all we have to judge by2       
whereas there are other examples in     
regard to which1 if we will insist upon 
making use of these rules1 we may either
 be deliberately abandoning the         
opportunity of getting far more         
trustworthy information by other means1 
or we may be obtaining solutions about  
matters on which the human intellect has
 no right to any definite quantitative  
opinion4  #ag4 ,the nearest approach to 
any practical justification of such     
judgments that ,i remember to have seen 

is afforded by cases of which the       
following example is a specimen3-- ,of  
#aj cases treated by ,lister's method1  
#g did well and #c suffered from        
blood-poisoning3 of #ad treated with    
ordinary dressings1 #i did well and #e  
had blood-poisoning2 what are the odds  
that the success of ,lister's method was
 due to chance84#h ,or1 to put it into  
other words1 a short experience has     
shown an actual superiority in one      
method over the other3 what are the     
chances that an indefinitely long       
experience1 under similar conditions1   
will confirm this superiority8  ,the    
proposer treated this as a 'bag and     
balls' problem1 analogous to the        
following3 #aj balls from one bag gave  
#g white and #c black1 #ad from another 
bag gave #i white and #e black3 what is 
the chance that the actual ratio of     
white to black balls was greater in the 
former than in the latter8--this actual 
ratio being of course considered a true 
indication of what would be the ultimate

 proportions of white and black         
drawings4 ,this seems to me to be the   
only reasonable way of treating the     
problem1 if it is to be considered      
capable of numerical solution at all4   
,of course the inevitable assumption has
 to be made here about the equal        
prevalence of the different possible    
kinds of bag1--or1 as the supporters of 
the justice of the calculation would put
 it1 of the obligation to assume the    
equal  priori likelihood of each        
kind1--but ,i think that in this        
particular example the arbitrariness of 
the assumption is less than usual4 ,this
 is because the problem discusses simply
 a balance between two extremely similar
 cases1 and there is a certain set-off  
against each other of the objectionable 
assumptions on each side4 ,had one set  
of experiments only been proposed1 and  
had we been asked to evaluate the       
probability of continued repetition of  
them confirming their verdict1 ,i should
 have felt all the scruples ,i have     

already mentioned4 ,but here we have got
 two sets of experiments carried on     
under almost exactly similar            
circumstances1 and there is therefore   
less arbitrariness in assuming that     
their unknown conditions are tolerably  
equally prevalent4  #ah4 ,examples of   
the description commonly introduced seem
 objectionable enough1 but if we wish to
 realize to its full extent the         
vagueness of some of the problems       
submitted to this ,inverse ,probability1
 we have not far to seek4 ,in natural as
 in artificial examples1 where          
statistics are unattainable the enquiry 
becomes utterly hopeless1 and all       
attempts at laying down rules for       
calculation must be abandoned4 ,take1   
for instance1 the question which has    
given rise to some discussion1#i whether
 such and such groups of stars are or   
are not to be regarded as the results of
 an accidental distribution2 or the     
still wider and vaguer question1 whether
 such and such things1 or say the world 

itself1 have been produced by chance8   
,in cases of this kind the insuperable  
difficulty is in determining what sense 
exactly is to be attached to the words  
'accidental' and 'random' which enter   
into the discussion4 ,some account was  
given1 in the fourth chapter1 of their  
scientific and conventional meaning in  
,probability4 ,there seem to be the same
 objections to generalizing them out of 
such relation1 as there is in           
metaphysics to talking of the ,infinite 
or the ,absolute4 ,infinite magnitude1  
or infinite power1 one can to some      
extent comprehend1 or at least one may  
understand what is being talked about1  
but 'the infinite' seems to me a term   
devoid of meaning4 ,so of anything      
supposed to have been produced at       
random3 tell us the nature of the       
agency1 the limits of its randomness and
 so on1 and we can venture upon the     
problem1 but without such data we know  
not what to do4 ,the further            
consideration of such a problem might1  

,i think1 without arrogance be relegated
 to the ,chapter on ,fallacies4         
,accordingly any further remarks which  
,i have to make upon the subject will be
 found there1 and at the conclusion of  
the chapter on ,causation and ,design4  
#a4 ,it might be more accurate to speak 
of 'incompatible hypotheses with        
respect to any individual case'1 or     
'mutually exclusive classes of          
events'4  #b4 ,the examples1 of this    
kind1 referring to human mortality are  
taken   from the ,carlisle tables4      
,these differ considerably1 as is well  
known1 from other tables1 but we have   
the high authority of ,de   ,morgan for 
regarding them as the best              
representative of the average           
mortality of the ,english middle classes
 at the present day4  #c4 ,i say1 almost
 any proportion1 because1 as may easily 
be seen1   arithmetic imposes certain   
restrictions upon the assumptions that  
can be made4 ,we could not1 for         
instance1 suppose that all the          

black-haired men are short-sighted1 for 
in any given batch of men   the former  
are more numerous4 ,but the range of    
these restrictions is   limited1 and    
their existence is not of importance in 
the above   discussion4  #d4 ,essay on  
,probabilities1 p4 #ec4 ,i have been    
reminded that in his   article on       
,probability in the ,encyclopdia        
,metropolitana he has   stated that such
 rules involve no new principle4  #e4   
,this point will be fully discussed in a
 future chapter1 after the   general    
stand-point of an objective system of   
logic has been   explained and          
illustrated4  #f4 ,whitworth's ,choice  
and ,chance1 ,ed4 ,,ii41 p4 #abc4 ,see  
also ,boole's   ,laws of ,thought1 p4   
#cgj4  #g4 ,opinions differ about the   
defence of such suppositions1 as they do
   about the nature of them4 ,some      
writers1 admitting the above            
assumption to be doubtful1 call it the  
most impartial   hypothesis4 ,others    
regard it as a sort of mean hypothesis4 

#h4 ,educational ,times2 ,reprint1 ,vol4
 xxxvii4 p4 #dj4 ,the question was      
proposed by ,dr4 ,macalister and gave   
rise to considerable controversy4   ,as 
usual with problems of this inverse kind
 hardly any two of the   writers were in
 agreement as to the assumptions to be  
made1 or   therefore as to the numerical
 estimate of the odds4  #i4 ,see        
,todhunter's ,history1 pp4 #ccc1 #d4    
,there is an interesting discussion upon
 this question by the late   ,j4 ,d4    
,forbes in a paper in the ,philosophical
 ,magazine for   ,dec4 #ahej4 ,it was   
replied to in a subsequent number by    
,prof4 ,donkin4     ,,chapter ,,viii4   
,,the ,,rule ,,of ,,succession4#a  #a4  
,a word of apology may be offered here  
for the introduction of a new   name4   
,the only other alternative would have  
been to entitle the rule   one of       
,induction4 ,but such a title ,i cannot 
admit1 for reasons   which will be      
almost immediately explained4   #a4 ,in 
the last chapter we discussed at some   

length the nature of the kinds of       
inference in ,probability which         
correspond to those termed1 in ,logic1  
immediate and mediate inferences4 ,we   
ascertained what was the meaning of     
saying1 for example1 that the chance of 
any given man ,a4 ,b4 dying in a year is
 #a_/c1 when concluded from the general 
proposition that one man out of three in
 his circumstances dies4 ,we also       
discussed the nature and evidence of    
rules of a more completely inferential  
character4 ,but to stop at this point   
would be to take a very imperfect view  
of the subject4 ,if ,probability is a   
science of real inference about things1 
it must surely lead up to something more
 than such merely formal conclusions2 we
 must be able1 if not by means of it1 at
 any rate by some means1 to step beyond 
the limits of what has been actually    
observed1 and to draw conclusions about 
what is as yet unobserved4 ,this leads  
at once to the question1 ,what is the   
connection of ,probability with         

,induction8 ,this is a question into    
which it will be necessary to enter now 
with some minuteness4  ,that there is a 
close connection between ,probability   
and ,induction1 must have been observed 
by almost every one who has treated of  
either subject2 ,i have not however seen
 any account of this connection that    
seemed to me to be satisfactory4 ,an    
explicit description of it should rather
 be sought in treatises upon the        
narrower subject1 ,probability2 but it  
is precisely here that the most         
confusion is to be found4 ,the province 
of ,probability being somewhat narrow1  
incursions have been constantly made    
from it into the adjacent territory of  
,induction4 ,in this way1 amongst the   
arithmetical rules discussed in the last
 chapter1 others have been frequently   
introduced which ought not in strictness
 to be classed with them1 as they rest  
on an entirely different basis4  #b4    
,the origin of such confusion is easy of
 explanation2 it arises1 doubtless1 from

 the habit of laying undue stress upon  
the subjective side of ,probability1    
upon that which treats of the quantity  
of our belief upon different subjects   
and the variations of which that        
quantity is susceptible4 ,it has been   
already urged that this variation of    
belief is at most but a constant        
accompaniment of what is really         
essential to ,probability1 and is       
moreover common to other subjects as    
well4 ,by defining the science therefore
 from this side these other subjects    
would claim admittance into it2 some of 
these1 as ,induction1 have been         
accepted1 but others have been somewhat 
arbitrarily rejected4 ,our belief in a  
wider proposition gained by ,induction  
is1 prior to verification1 not so strong
 as that of the narrower generalization 
from which it is inferred4 ,this being  
observed1 a so-called rule of           
probability has been given by which it  
is supposed that this diminution of     
assent could in many instances be       

calculated4  ,but time also works       
changes in our conviction2 our belief in
 the happening of almost every event1 if
 we recur to it long afterwards1 when   
the evidence has faded from the mind1 is
 less strong than it was at the time4   
,why are not rules of oblivion inserted 
in treatises upon ,probability8 ,if a   
man is told how firmly he ought to      
expect the tide to rise again1 because  
it has already risen ten times1 might he
 not also ask for a rule which should   
tell him how firm should be his belief  
of an event which rests upon a ten      
years' recollection8#a ,the infractions 
of a rule of this latter kind could     
scarcely be more numerous and extensive1
 as we shall see presently1 than those  
of the former confessedly are4 ,the fact
 is that the agencies1 by which the     
strength of our conviction is modified1 
are so indefinitely numerous that they  
cannot all be assembled into one        
science2 for purposes of definition     
therefore the quantity of belief had    

better be omitted from consideration1 or
 at any rate regarded as a mere         
appendage1 and the science1 defined from
 the other or statistical side of the   
subject1 in which1 as has been shown1 a 
tolerably clear boundary-line can be    
traced4  #c4 ,induction1 however1 from  
its importance does merit a separate    
discussion2 a single example will show  
its bearing upon this part of our       
subject4 ,we are considering the        
prospect of a given man1 ,a4 ,b4 living 
another year1 and we find that nine out 
of ten men of his age do survive4 ,in   
forming an opinion about his surviving1 
however1 we shall find that there are in
 reality two very distinct causes which 
aid in determining the strength of our  
conviction2 distinct1 but in practice so
 intimately connected that we are very  
apt to overlook one1 and attribute the  
effect entirely to the other4  7,i47    
,there is that which strictly belongs to
 ,probability2 that which 7as was       
explained in ,chap ,,vi47 measures our  

belief of the individual case as deduced
 from the general proposition4 ,granted 
that nine men out of ten of the kind to 
which ,a4 ,b4 belongs do live another   
year1 it obviously does not follow at   
all necessarily that he will4 ,we       
describe this state of things by saying1
 that our belief of his surviving is    
diminished from certainty in the ratio  
of #aj to #i1 or1 in other words1 is    
measured by the fraction #i_/aj4        
7,,ii47 ,but are we certain that nine   
men out of ten like him will live       
another year8 we know that they have so 
survived in time past1 but will they    
continue to do so8 ,since ,a4 ,b4 is    
still alive it is plain that this       
proposition is to a certain extent      
assumed1 or rather obtained by          
,induction4 ,we cannot however be as    
certain of the inductive inference as we
 are of the data from which it was      
inferred4 ,here1 therefore1 is a second 
cause which tends to diminish our       
belief2 in practice these two causes    

always accompany each other1 but in     
thought they can be separated4  ,the two
 distinct causes described above are    
very liable to be confused together1 and
 the class of cases from which examples 
are necessarily for the most part drawn 
increases this liability4 ,the step from
 the statement 'all men have died in a  
certain proportion' to the inference    
'they will continue to die in that      
proportion' is so slight a step that it 
is unnoticed1 and the diminution of     
conviction that should accompany it is  
unsuspected4 ,in what are called  priori
 examples the step is still slighter4   
,we feel so certain about the permanence
 of the laws of mechanics1 that few     
people would think of regarding it as an
 inference when they believe that a die 
will in the long run turn up all its    
faces equally often1 because other dice 
have done so in time past4  #d4 ,it has 
been already pointed out 7in ,chapter   
,,vi47 that1 so far as concerns that    
definition of ,probability which regards

 it as the science which discusses the  
degree and modifications of our belief1 
the question at issue seems to be simply
 this3--,are the causes alluded to above
 in 7,,ii47 capable of being reduced to 
one simple coherent scheme1 so that any 
universal rules for the modification of 
assent can be obtained from them8 ,if   
they are1 strong grounds will have been 
shown for classing them with 7,i471 in  
other words1 for considering them as    
rules of probability4 ,even then they   
would be rules practically of a very    
different kind1 contingent instead of   
necessary 7if one may use these terms   
without committing oneself to any       
philosophical system71 but this         
objection might perhaps be overruled by 
the greater simplicity secured by       
classing them together4 ,this view is1  
with various modifications1 generally   
adopted by writers on ,probability1 or  
at least1 as ,i understand the matter1  
implied by their methods of definition  
and treatment4 ,or1 on the other hand1  

must these causes be regarded as a vast 
system1 one might almost say a chaos1 of
 perfectly distinct agencies2 which may 
indeed be classified and arranged to    
some extent1 but from which we can never
 hope to obtain any rules of perfect    
generality which shall not be subject to
 constant exception8 ,if so1 but one    
course is left2 to exclude them all     
alike from ,probability4 ,in other      
words1 we must assume the general       
proposition1 viz4 that which has been   
described throughout as our             
starting-point1 to be given to us2 it   
may be obtained by any of the numerous  
rules furnished by ,induction1 or it may
 be inferred deductively1 or given by   
our own observation2 its value may be   
diminished by its depending upon the    
testimony of witnesses1 or its being    
recalled by our own memory4 ,its real   
value may be influenced by these causes 
or any combinations of them2 but all    
these are preliminary questions with    
which we have nothing directly to do4   

,we assume our statistical proposition  
to be true1 neglecting the diminution of
 its value by the process of attainment2
 we take it up first at this point and  
then apply our rules to it4 ,we receive 
it in fact1 if one may use the          
expression1 ready-made1 and ask no      
questions about the process or          
completeness of its manufacture4  #e4   
,it is not to be supposed1 of course1   
that any writers have seriously         
attempted to reduce to one system of    
calculation all the causes mentioned    
above1 and to embrace in one formula the
 diminution of certainty to which the   
inclusion of them subjects us4 ,but on  
the other hand1 they have been unwilling
 to restrain themselves from all appeal 
to them4 ,from an early period in the   
study of the science attempts have been 
made to proceed1 by the ,calculus of    
,probability1 from the observed cases to
 adjacent and similar cases4 ,in        
practice1 as has been already said1 it  
is not possible to avoid some extension 

of this kind4 ,but it should be         
observed1 that in these instances the   
divergence from the strict ground of    
experience is not in reality recognized1
 at least not as a part of our logical  
procedure4 ,we have1 it is true1        
wandered somewhat beyond it1 and so     
obtained a wider proposition than our   
data strictly necessitated1 and         
therefore one of less certainty4 ,still 
we assume the conclusion given by       
induction to be equally certain with the
 data1 or rather omit all notice of the 
divergence from consideration4 ,it is   
assumed that the unexamined instances   
will resemble the examined1 an          
assumption for which abundant warrant   
may exist2 the theory of the calculation
 rests upon the supposition that there  
will be no difference between them1 and 
the practical error is insignificant    
simply because this difference is small4
  #f4 ,but the rule we are now about to 
discuss1 and which may be called the    
,rule of ,succession1 is of a very      

different kind4 ,it not only recognizes 
the fact that we are leaving the ground 
of past experience1 but takes the       
consequences of this divergence as the  
express subject of its calculation4 ,it 
professes to give a general rule for the
 measure of expectation that we should  
have of the reappearance of a phenomenon
 that has been already observed any     
number of times4 ,this rule is generally
 stated somewhat as follows3 ,to find   
the chance of the recurrence of an event
 already observed1 divide the number of 
times the event has been observed1      
increased by one1 by the same number    
increased by two4  #g4 ,it will be      
instructive to point out the origin of  
this rule2 if only to remind the reader 
of the necessity of keeping mathematical
 formul to their proper province1 and to
 show what astonishing conclusions are  
apt to be accepted on the supposed      
warrant of mathematics4 ,revert then to 
the example of ,inverse ,probability on 
p4 #ahb4 ,we saw that under certain     

assumptions1 it would follow that when a
 single white ball had been drawn from a
 bag known to contain #aj balls which   
were white or black1 the chance could be
 determined that there was only one     
white ball in it4 ,having done this we  
readily calculate 'directly' the chance 
that this white ball will be drawn next 
time4 ,similarly we can reckon the      
chances of there being two1 three1 c4 up
 to ten white balls in it1 and determine
 on each of these suppositions the      
chance of a white ball being drawn next 
time4 ,adding these together we have the
 answer to the question3--a white ball  
has been drawn once from a bag known to 
contain ten balls1 white or black2 what 
is the chance of a second time drawing a
 white ball8  ,so far only arithmetic is
 required4 ,for the next step we need   
higher mathematics1 and by its aid we   
solve this problem3--,a white ball has  
been drawn m times from a bag which     
contains any number1 we know not what1  
of balls each of which is white or      

black1 find the chance of the next      
drawing also yielding a white ball4 ,the
 answer is 7m  #a7_/7m  #b74  ,thus far 
mathematics4 ,then comes in the physical
 assumption that the universe may be    
likened to such a bag as the above1 in  
the sense that the above rule may be    
applied to solve this question3--an     
event has been observed to happen m     
times in a certain way1 find the chance 
that it will happen in that way next    
time4 ,laplace1 for instance1 has       
pointed out that at the date of the     
writing of his ,essai ,philosophique1   
the odds in favour of the sun's rising  
again 7on the old assumption as to the  
age of the world7 were #a1hbf1bad to #a4
 ,de ,morgan says that a man who        
standing on the bank of a river has seen
 ten ships pass by with flags should    
judge it to be #aa to #a that the next  
ship will also carry a flag4  #h4 ,it is
 hard to take such a rule as this       
seriously1 for there does not seem to be
 even that moderate confirmation of it  

which we shall find to hold good in the 
case of the application of abstract     
formul to the estimation of the evidence
 of witnesses4 ,if however its validity 
is to be discussed there appear to be   
two very distinct lines of enquiry along
 which we may be led4  7#a7 ,in the     
first place we may take it for what it  
professes to be1 and for what it is     
commonly understood to be1 viz4 a rule  
which assigns the measure of expectation
 we ought to entertain of the recurrence
 of the event under the circumstances in
 question4 ,of course1 on the view      
adopted in this work1 we insist on      
enquiring whether it is really true that
 on the average events do thus repeat   
their performance in accordance with    
this law4 ,thus tested1 no one surely   
would attempt to defend such a formula4 
,so far from past occurrence being a    
ground for belief in future recurrence1 
there are 7as will be more fully pointed
 out in the ,chapter on ,fallacies7     
plenty of cases in which the direct     

contrary holds good4 ,then again a rule 
of this kind is subject to the very     
serious perplexity to be explained in   
our next chapter1 arising out of the    
necessary arbitrariness of such inverse 
reference4 ,that is1 when an event has  
happened but a few times1 we have no    
certain guide2 and when it has happened 
but once1#b we have no guide whatever1  
as to the class of cases to which it is 
to be referred4 ,in the example above1  
about the flags1 why did we stop short  
at this notion simply1 instead of       
specifying the size1 shape1 c4 of the   
flags8  ,de ,morgan1 it must be         
remembered1 only accepts this rule in a 
qualified sense4 ,he regards it as      
furnishing a minimum value for the      
amount of our expectation4 ,he terms it 
the rule of probability of a pure       
induction1 and says of it1 ,the         
probabilities shown by the above rules  
are merely minima which may be augmented
 by other sources of knowledge4 ,that   
is1 he recognizes only those instances  

in which our belief in the ,uniformity  
of ,nature and in the existence of      
special laws of causation comes in to   
supplement that which arises from the   
mere frequency of past occurrence4 ,this
 however does not meet those cases in   
which past occurrence is a positive     
ground of disbelief in future           
recurrence4  #i4 7#b7 ,there is however 
another and very different view which   
might be taken of such a rule4 ,it is   
one1 an obscure recognition of which has
 very likely had much to do with the    
acceptance which the rule has received4 
,what we might suppose ourselves to be  
thus expressing is1--not the measure of 
rational expectation which might be held
 by minds sufficiently advanced to be   
able to classify and to draw conscious  
inferences1 but1--the law according to  
which the primitive elements of belief  
were started and developed4 ,of course  
such an interpretation as this would be 
equivalent to quitting the province of  
,logic altogether and crossing over into

 that of ,psychology2 but it would be a 
perfectly valid line of enquiry4 ,we    
should be attempting nothing more than a
 development of the researches of       
,fechner and his followers in           
psychophysical measurement4 ,only then  
we ought1 like them1 not to start with  
any analogy of a ballot box and its     
contents1 but to base our enquiry on    
careful determination of the actual     
mental phenomena experienced4 ,we know  
how the law has been determined in      
accordance with which the intensity of  
the feeling of light varies with that of
 its objective source4 ,we see how it is
 possible to measure the growth of      
memory according to the number of       
repetitions of a sentence or a          
succession of mere syllables4 ,in this  
latter case1 for instance1 we just try  
experiments1 and determine how much     
better a man can remember any utterances
 after eight hearings than after        
seven4#c  ,now this case furnishes a    
very close parallel to our supposed     

attempt to measure the increase of      
intensity of belief after repeated      
recurrence4 ,that is1 if it were        
possible to experiment in this order of 
mental phenomena1 we ought simply to    
repeat a phenomenon a certain number of 
times and then ascertain by actual      
introspection or by some simple test1   
how fast the belief was increasing4     
,thus viewed the problem seems to me a  
hopeless one4 ,the difficulties are     
serious enough1 when we are trying to   
measure our simple sensations1 of laying
 aside the effects of past training1 and
 of attempting1 as it were1 to leave the
 mind open and passive to mere reception
 of stimuli4 ,but if we were to attempt 
in this way to measure our belief these 
difficulties would become quite         
insuperable4 ,we can no more divest     
ourselves of past training here than we 
can of intelligence or thought4 ,i do   
not see how any one could possibly avoid
 classing the observed recurrences with 
others which he had experienced1 and of 

being thus guided by special analogies  
and inductions instead of trusting      
solely to ,de ,morgan's 'pure           
induction'4 ,the same considerations    
tend to rebut another form of defence   
for the rule in question4 ,it is urged1 
for instance1 that we may at least      
resort to it in those cases in which we 
are in entire ignorance as to the number
 and nature of the antecedents4 ,this is
 a position to which ,i can hardly      
conceive it possible that we should ever
 be reduced4 ,however remote or         
exceptional may be the phenomenon       
selected we may yet bring it into       
relation with some accepted             
generalizations and thus draw our       
conclusions from these rather than from 
purely  priori considerations4  #aj4    
,since then past acquisitions cannot be 
laid aside or allowed for1 the only     
remaining resource would be to          
experiment upon the infant mind4 ,one   
would not like to pronounce that any    
line of enquiry is impossible2 but the  

difficulties would certainly be         
enormous4 ,and interesting as the facts 
would be1 supposing that we had         
succeeded in securing them1 they would  
not be of the slightest importance in   
,logic4 ,however the question were      
settled3--whether1 for instance1 we     
proved that the sentiment or emotion of 
belief grew up slowly and gradually from
 a sort of zero point under the impress 
of repetition of experience2 or whether 
we proved that a single occurrence      
produced complete belief in the         
repetition of the event1 so that        
experience gradually untaught us and    
weakened our convictions2--in no case   
would the mature mind gain any aid as to
 what it ought to believe4  ,i cannot   
but think that some such view as this   
must occasionally underlie the          
acceptance which this rule has received4
 ,for instance1 ,laplace1 though        
unhesitatingly adopting it as a real1   
that is1 objective rule of inference1   
has gone into so much physiological and 

psychological matter towards the end of 
his discussion 7,essai philosophique7 as
 to suggest that what he had in view was
 the natural history of belief rather   
than its subsequent justification4      
,again1 the curious doctrine adopted by 
,jevons1 that the principles of         
,induction rest entirely upon the theory
 of ,probability1--a very different     
doctrine from that which is conveyed by 
saying that all knowledge of facts is   
probable only1 i4e4 not                 
necessary1--seems unintelligible except 
on some such interpretation4 ,we shall  
have more to say on this subject in our 
next chapter4 ,it will be enough here to
 remark that in our present reflective  
and rational stage we find that every   
inference in ,probability involves some 
appeal to1 or support from1 ,induction1 
but that it is impossible to base either
 upon the other4 ,however far back we   
try to push our way1 and however        
disposed we might be to account for our 
ultimate beliefs by ,association1 it    

seems to me that so long as we consider 
ourselves to be dealing with rules of   
inference we must still distinguish     
between ,induction and ,probability4    
#a4 ,john ,craig1 in his often named    
work1 ,theologi ,christian   ,principia 
,mathematica 7,lond4 #afii7 attempted   
something in this   direction when he   
proposed to solve such problems         
as3--,quando   evanescet probabilitas   
cujusvis ,histori1 cujus subjectum est  
transiens1 viv tantum voce transmiss1   
determinare4  #b4 ,when m  #a the       
fraction becomes #b_/c2 i4e4 the odds   
are #b to #a in   favour of recurrence4 
,and there are writers who accept this  
result4 ,for instance1 ,jevons          
7,principles of ,science p4 #beh7 says  
,thus on the first occasion on which a  
person sees a shark1 and   notices that 
it is accompanied by a little pilot     
fish1 the odds are   #b to #a that the  
next shark will be so accompanied4  ,to 
say nothing   of the fact that          
recognizing and naming the fish implies 

that they   have often been seen before1
 how many of the observed               
characteristics of that single 'event'  
are to be considered   essential8  ,must
 the pilot precede2 and at the same     
distance8 ,must   we consider the       
latitude1 the ocean1 the season1 the    
species of   shark1 as matter also of   
repetition on the next occasion8  and so
   on4 ,i cannot see how the ,inductive 
problem can be even intelligibly        
stated1 for quantitative purposes1 on   
the first occurrence of any   event4    
#c4 ,see in ,mind 7x4 #ded7 ,mr ,jacob's
 account of the researches of ,herr     
,ebbinghaus as described in his work    
,ueber das ,gedchtniss4     ,,chapter   
,,ix4   ,,induction ,,and ,,its         
,,connection ,,with ,,probability4  #a4 
,we were occupied1 during the last      
chapter1 with the examination of a rule1
 the object of which was to enable us to
 make inferences about instances as yet 
unexamined4 ,it was professedly1        
therefore1 a rule of an inductive       

character4 ,but1 in the form in which it
 is commonly expressed1 it was found to 
fail utterly4 ,it is reasonable         
therefore to enquire at this point      
whether ,probability is entirely a      
formal or deductive science1 or whether1
 on the other hand1 we are able1 by     
means of it1 to make valid inferences   
about instances as yet unexamined4 ,this
 question has been already in part      
answered by implication in the course of
 the last two chapters4 ,it is proposed 
in the present chapter to devote a      
fuller investigation to this subject1   
and to describe1 as minutely as limits  
will allow1 the nature of the connection
 between ,probability and ,induction4   
,we shall find it advisable for         
clearness of conception to commence our 
enquiry at a somewhat early stage4 ,we  
will travel over the ground1 however1 as
 rapidly as possible1 until we approach 
the boundary of what can properly be    
termed ,probability4  #b4 ,let us then  
conceive some one setting to work to    

investigate nature1 under its broadest  
aspect1 with the view of systematizing  
the facts of experience that are known1 
and thence 7in case he should find that 
this is possible7 discovering others    
which are at present unknown4 ,he       
observes a multitude of phenomena1      
physical and mental1 contemporary and   
successive4 ,he enquires what           
connections are there between them8 what
 rules can be found1 so that some of    
these things being observed ,i can infer
 others from them8 ,we suppose him1 let 
it be observed1 deliberately resolving  
to investigate the things themselves1   
and not to be turned aside by any prior 
enquiry as to there being laws under    
which the mind is compelled to judge of 
the things4 ,this may arise either from 
a disbelief in the existence of any     
independent and necessary mental laws1  
and a consequent conviction that the    
mind is perfectly competent to observe  
and believe anything that experience    
offers1 and should believe nothing else1

 or simply from a preference for        
investigations of the latter kind4 ,in  
other words1 we suppose him to reject   
,formal ,logic1 and to apply himself to 
a study of objective existences4  ,it   
must not for a moment be supposed that  
we are here doing more than conceiving a
 fictitious case for the purpose of more
 vividly setting before the reader the  
nature of the inductive process1 the    
assumptions it has to make1 and the     
character of the materials to which it  
is applied4 ,it is not psychologically  
possible that any one should come to the
 study of nature with all his mental    
faculties in full perfection1 but void  
of all materials of knowledge1 and free 
from any bias as to the uniformities    
which might be found to prevail around  
him4 ,in practice1 of course1 the form  
and the matter--the laws of belief or   
association1 and the objects to which   
they are applied--act and react upon one
 another1 and neither can exist in any  
but a low degree without presupposing   

the existence of the other4 ,but the    
supposition is perfectly legitimate for 
the purpose of calling attention to the 
requirements of such a system of ,logic1
 and is indeed nothing more than what   
has to be done at almost every step in  
psychological enquiry4#a  #c4 ,his task 
at first might be conceived to be a slow
 and tedious one4 ,it would consist of a
 gradual accumulation of individual     
instances1 as marked out from one       
another by various points of            
distinction1 and connected with one     
another by points of resemblance4 ,these
 would have to be respectively          
distinguished and associated in the     
mind1 and the consequent results would  
then be summed up in general            
propositions1 from which inferences     
could afterwards be drawn4 ,these       
inferences could1 of course1 contain no 
new facts1 they would only be           
repetitions of what he or others had    
previously observed4 ,all that we should
 have so far done would have been to    

make our classifications of things and  
then to appeal to them again4 ,we should
 therefore be keeping well within the   
province of ordinary logic1 the         
processes of which 7whatever their      
ultimate explanation7 may of course     
always be expressed1 in accordance with 
,aristotle's ,dictum1 as ways of        
determining whether or not we can show  
that one given class is included wholly 
or partly within another1 or excluded   
from it1 as the case may be4  #d4 ,but a
 very short course of observation would 
suggest the possibility of a wide       
extension of his information4           
,experience itself would soon detect    
that events were connected together in a
 regular way2 he would ascertain that   
there are 'laws of nature4' ,coming with
 no  priori necessity of believing in   
them1 he would soon find that as a      
matter of fact they do exist1 though he 
could not feel any certainty as to the  
extent of their prevalence4 ,the        
discovery of this arrangement in nature 

would at once alter the plan of his     
proceedings1 and set the tone to the    
whole range of his methods of           
investigation4 ,his main work now would 
be to find out by what means he could   
best discover these laws of nature4  ,an
 illustration may assist4 ,suppose ,i   
were engaged in breaking up a vast piece
 of rock1 say slate1 into small pieces4 
,i should begin by wearily working      
through it inch by inch4 ,but ,i should 
soon find the process completely changed
 owing to the existence of cleavage4 ,by
 this arrangement of things a very few  
blows would do the work--not1 as ,i     
might possibly have at first supposed1  
to the extent of a few inches--but right
 through the whole mass4 ,in other      
words1 by the process itself of cutting1
 as shown in experience1 and by nothing 
else1 a constitution would be detected  
in the things that would make that      
process vastly more easy and extensive4 
,such a discovery would of course change
 our tactics4 ,our principal object     

would thenceforth be to ascertain the   
extent and direction of this cleavage4  
,something resembling this is found in  
,induction4 ,the discovery of laws of   
nature enables the mind to dart with its
 inferences from a few facts completely 
through a whole class of objects1 and   
thus to acquire results the successive  
individual attainment of which would    
have involved long and wearisome        
investigation1 and would indeed in      
multitudes of instances have been out of
 the question4 ,we have no demonstrative
 proof that this state of things is     
universal2 but having found it prevail  
extensively1 we go on with the          
resolution at least to try for it       
everywhere else1 and we are not         
disappointed4 ,from propositions        
obtained in this way1 or rather from the
 original facts on which these          
propositions rest1 we can make new      
inferences1 not indeed with absolute    
certainty1 but with a degree of         
conviction that is of the utmost        

practical use4 ,we have gained the great
 step of being able to make trustworthy 
generalizations4 ,we conclude1 for      
instance1 not merely that ,john and     
,henry die1 but that all men die4  #e4  
,the above brief investigation contains1
 it is hoped1 a tolerably correct       
outline of the nature of the ,inductive 
inference1 as it presents itself in     
,material or ,scientific ,logic4 ,it    
involves the distinction drawn by ,mill1
 and with which the reader of his       
,system of ,logic will be familiar1     
between an inference drawn according to 
a formula and one drawn from a formula4 
,we do in reality make our inference    
from the data afforded by experience    
directly to the conclusion2 it is a mere
 arrangement of convenience to do so by 
passing through the generalization4 ,but
 it is one of such extreme convenience1 
and one so necessarily forced upon us   
when we are appealing to our own past   
experience or to that of others for the 
grounds of our conclusion1 that         

practically we find it the best plan to 
divide the process of inference into two
 parts4 ,the first part is concerned    
with establishing the generalization2   
the second 7which contains the rules of 
ordinary logic7 determines what         
conclusions can be drawn from this      
generalization4  #f4 ,we may now see our
 way to ascertaining the province of    
,probability and its relation to kindred
 sciences4 ,inductive ,logic gives rules
 for discovering such generalizations as
 those spoken of above1 and for testing 
their correctness4 ,if they are         
expressed in universal propositions it  
is the part of ordinary logic to        
determine what inferences can be made   
from and by them2 if1 on the other hand1
 they are expressed in proportional     
propositions1 that is1 propositions of  
the kind described in our first chapter1
 they are handed over to ,probability4  
,we find1 for example1 that three       
infants out of ten die in their first   
four years4 ,it belongs to ,induction to

 say whether we are justified in        
generalizing our observation into the   
assertion1 ,all infants die in that     
proportion4 ,when such a proposition is 
obtained1 whatever may be the value to  
be assigned to it1 we recognize in it a 
series of a familiar kind1 and it is at 
once claimed by ,probability4  ,in this 
latter case the division into two parts1
 the inductive and the ratiocinative1   
seems decidedly more than one of        
convenience2 it is indeed imperatively  
necessary for clearness of thought and  
cogency of treatment4 ,it is true that  
in almost every example that can be     
selected we shall find both of the above
 elements existing together and         
combining to determine the degree of our
 conviction1 but when we come to examine
 them closely it appears to me that the 
grounds of their cogency1 the kind of   
conviction they produce1 and            
consequently the rules which they give  
rise to1 are so entirely distinct that  
they cannot possibly be harmonized into 

a single consistent system4  ,the       
opinion therefore according to which    
certain ,inductive formul are regarded  
as composing a portion of ,probability1 
and which finds utterance in the ,rule  
of ,succession criticised in our last   
chapter1 cannot1 ,i think1 be           
maintained4 ,it would be more correct to
 say1 as stated above1 that ,induction  
is quite distinct from ,probability1 yet
 co-operates in almost all its          
inferences4 ,by ,induction we determine1
 for example1 whether1 and how far1 we  
can safely generalize the proposition   
that four men in ten live to be         
fifty-six2 supposing such a proposition 
to be safely generalized1 we hand it    
over to ,probability to say what sort of
 inferences can be deduced from it4  #g4
 ,so much then for the opinion which    
tends to regard pure ,induction as a    
subdivision of ,probability4 ,by the    
majority of philosophical and logical   
writers a widely different view has of  
course been entertained4 ,they are      

mostly disposed to distinguish these    
sciences very sharply from1 not to say  
to contrast them with1 one another2 the 
one being accepted as philosophical or  
logical1 and the other rejected as      
mathematical4 ,this may without offence 
be termed the popular prejudice against 
,probability4  ,a somewhat different    
view1 however1 must be noticed here1    
which1 by a sort of reaction against the
 latter1 seems even to go beyond the    
former2 and which occasionally finds    
expression in the statement that all    
inductive reasoning of every kind is    
merely a matter of ,probability4 ,two   
examples of this may be given4          
,beginning with the older authority1    
there is an often quoted saying by      
,butler at the commencement of his      
,analogy1 that 'probability is the very 
guide of life'2 a saying which seems    
frequently to be understood to signify  
that the rules or principles of         
,probability are thus all-prevalent when
 we are drawing conclusions in practical

 life4 ,judging by the drift of the     
context1 indeed1 this seems a fair      
interpretation of his meaning1 in so far
 of course as there could be said to be 
any such thing as a science of          
,probability in those days4 ,prof4      
,jevons1 in his ,principles of ,science 
7p4 #aig71 has expressed a somewhat     
similar view1 of course in a way more   
consistent with the principles of modern
 science1 physical and mathematical4 ,he
 says1 ,i am convinced that it is       
impossible to expound the methods of    
induction in a sound manner1 without    
resting them on the theory of           
,probability4 ,perfect knowledge alone  
can give certainty1 and in nature       
perfect knowledge would be infinite     
knowledge1 which is clearly beyond our  
capacities4 ,we have1 therefore1 to     
content ourselves with partial          
knowledge1--knowledge mingled with      
ignorance1 producing doubt4#b  #h4      
,there are two senses in which this     
disposition to merge the two sciences   

into one may be understood4 ,using the  
word ,probability in its vague popular  
signification1 nothing more may be      
intended than to call attention to the  
fact1 that in every case alike our      
conclusions are nothing more than       
'probable1' that is1 that they are not1 
and cannot be1 absolutely certain4 ,this
 must be fully admitted1 for of course  
no one acquainted with the complexity of
 physical and other evidence would      
seriously maintain that absolute ideal  
certainty can be attained in any branch 
of applied logic4 ,hypothetical         
certainty1 in abstract science1 may be  
possible1 but not absolute certainty in 
the domain of the concrete4 ,this has   
been already noticed in a former        
chapter1 where1 however1 it was pointed 
out that whatever justification may     
exist1 on the subjective view of logic1 
for regarding this common prevalence of 
absence of certainty as warranting us in
 fusing the sciences into one1 no such  
justification is admitted when we take  

the objective view4  #i4 ,what may be   
meant1 however1 is that the grounds of  
this absence of certainty are always of 
the same general character4 ,this       
argument1 if admitted1 would have real  
force1 and must therefore be briefly    
noticed4 ,we have seen abundantly that  
when we say of a conclusion within the  
strict province of ,probability1 that it
 is not certain1 all that we mean is    
that in some proportion of cases only   
will such conclusion be right1 in the   
other cases it will be wrong4 ,now when 
we say1 in reference to any inductive   
conclusion1 that we feel uncertain about
 its absolute cogency1 are we conscious 
of the same interpretation8 ,it seems to
 me that we are not4 ,it is indeed quite
 possible that on ultimate analysis it  
might be proved that experience of      
failure in the past employment of our   
methods of investigation was the main   
cause of our present want of perfect    
confidence in them4 ,but this1 as we    
have repeatedly insisted1 does not      

belong to the province of logical1 but  
to that of ,psychological enquiry4 ,it  
is surely not the case that we are1 as a
 rule1 consciously guided by such       
occasional or repeated instances of past
 failure4 ,in so far as they are at all 
influential1 they seem to do their work 
by infusing a vague want of confidence  
which cannot be referred to any         
statistical grounds for its             
justification1 at least not in a        
quantitative way4 ,part of our want of  
confidence is derived sympathetically   
from those who have investigated the    
matter more nearly at first hand4 ,here 
again1 analysis might detect that a     
given proportion of past failures lay at
 the root of the distrust1 but it does  
not show at the surface4 ,moreover1 one 
reason why we cannot feel perfectly     
certain about our inductions is1 that   
the memory has to be appealed to for    
some of our data2 and will any one      
assert that the only reason why we do   
not place absolute reliance on our      

memory of events long past is that we   
have been deceived in that way before8  
,in any other sense1 therefore1 than as 
a needful protest against attaching too 
great demonstrative force to the        
conclusions of ,inductive ,logic1 it    
seems decidedly misleading to speak of  
its reasonings as resting upon          
,probability4  #aj4 ,we may now see     
clearly the reasons for the limits      
within which causation#c is necessarily 
required1 but beyond which it is not    
needed4 ,to be able to generalize a     
formula so as to extend it from the     
observed to the unobserved1 it is       
clearly essential that there should be a
 certain permanence in the order of     
nature2 this permanence is one form of  
what is implied in the term causation4  
,if the circumstances under which men   
live and die remaining the same1 we did 
not feel warranted in inferring that    
four men out of ten would continue to   
live to fifty1 because in the case of   
those whom we had observed this         

proportion had hitherto done so1 it is  
clear that we should be admitting that  
the same antecedents need not be        
followed by the same consequents4 ,this 
uniformity being what the ,law of       
,causation asserts1 the truth of the law
 is clearly necessary to enable us to   
obtain our generalizations3 in other    
words1 it is necessary for the          
,inductive part of the process4 ,but it 
seems to be equally clear that causation
 is not necessary for that part of the  
process which belongs to ,probability4  
,provided only that the truth of our    
generalizations is secured to us1 in the
 way just mentioned1 what does it matter
 to us whether or not the individual    
members are subject to causation8 ,for  
it is not in reality about these        
individuals that we make inferences4 ,as
 this last point has been already fully 
treated in ,chapter ,,vi41 any further  
allusion to it need not be made here4   
#aa4 ,the above description1 or rather  
indication1 of the process of obtaining 

these generalizations must suffice for  
the present4 ,let us now turn and       
consider the means by which we are      
practically to make use of them when    
they are obtained4 ,the point which we  
had reached in the course of the        
investigations entered into in the sixth
 and seventh chapters was this3--,given 
a series of a certain kind1 we could    
draw inferences about the members which 
composed it2 inferences1 that is1 of a  
peculiar kind1 the value and meaning of 
which were fully discussed in their     
proper place4  ,we must now shift our   
point of view a little2 instead of      
starting1 as in the former chapters1    
with a determinate series supposed to be
 given to us1 let us assume that the    
individual only is given1 and that the  
work is imposed upon us of finding out  
the appropriate series4 ,how are we to  
set about the task8 ,in the former case 
our data were of this kind3--,eight out 
of ten men1 aged fifty1 will live eleven
 years more1 and we ascertained in what 

sense1 and with what certainty1 we could
 infer that1 say1 ,john ,smith1 aged    
fifty1 would live to sixty-one4  #ab4   
,let us then suppose1 instead1 that     
,john ,smith presents himself1 how      
should we in this case set about        
obtaining a series for him8 ,in other   
words1 how should we collect the        
appropriate statistics8 ,it should be   
borne in mind that when we are          
attempting to make real inferences about
 things as yet unknown1 it is in this   
form that the problem will practically  
present itself4  ,at first sight the    
answer to this question may seem to be  
obtained by a very simple process1 viz4 
by counting how many men of the age of  
,john ,smith1 respectively do and do not
 live for eleven years4 ,in reality     
however the process is far from being so
 simple as it appears4 ,for it must be  
remembered that each individual thing   
has not one distinct and appropriate    
class or group1 to which1 and to which  
alone1 it properly belongs4 ,we may     

indeed be practically in the habit of   
considering it under such a single      
aspect1 and it may therefore seem to us 
more familiar when it occupies a place  
in one series rather than in another2   
but such a practice is merely customary 
on our part1 not obligatory4 ,it is     
obvious that every individual thing or  
event has an indefinite number of       
properties or attributes observable in  
it1 and might therefore be considered as
 belonging to an indefinite number of   
different classes of things4 ,by        
belonging to any one class it of course 
becomes at the same time a member of all
 the higher classes1 the genera1 of     
which that class was a species4 ,but1   
moreover1 by virtue of each accidental  
attribute which it possesses1 it becomes
 a member of a class intersecting1 so to
 say1 some of the other classes4 ,john  
,smith is a consumptive man say1 and a  
native of a northern climate4 ,being a  
man he is of course included in the     
class of vertebrates1 also in that of   

animals1 as well as in any higher such  
classes that there may be4 ,the property
 of being consumptive refers him to     
another class1 narrower than any of the 
above2 whilst that of being born in a   
northern climate refers him to a new and
 distinct class1 not conterminous with  
any of the rest1 for there are things   
born in the north which are not men4    
#ac4 ,when therefore ,john ,smith       
presents himself to our notice without1 
so to say1 any particular label attached
 to him informing us under which of his 
various aspects he is to be viewed1 the 
process of thus referring him to a class
 becomes to a great extent arbitrary4   
,if he had been indicated to us by a    
general name1 that1 of course1 would    
have been some clue2 for the name having
 a determinate connotation would specify
 at any rate a fixed group of attributes
 within which our selection was to be   
confined4 ,but names and attributes     
being connected together1 we are here   
supposed to be just as much in ignorance

 what name he is to be called by1 as    
what group out of all his innumerable   
attributes is to be taken account of2   
for to tell us one of these things would
 be precisely the same in effect as to  
tell us the other4 ,in saying that it is
 thus arbitrary under which class he is 
placed1 we mean1 of course1 that there  
are no logical grounds of decision2 the 
selection must be determined by some    
extraneous considerations4 ,mere        
inspection of the individual would      
simply show us that he could equally be 
referred to an indefinite number of     
classes1 but would in itself give no    
inducement to prefer1 for our special   
purpose1 one of these classes to        
another4  ,this variety of classes to   
which the individual may be referred    
owing to his possession of a            
multiplicity of attributes1 has an      
important bearing on the process of     
inference which was indicated in the    
earlier sections of this chapter1 and   
which we must now examine in more       

special reference to our particular     
subject4  #ad4 ,it will serve to bring  
out more clearly the nature of some of  
those peculiarities of the step which we
 are now about to take in the case of   
,probability1 if we first examine the   
form which the corresponding step       
assumes in the case of ordinary ,logic4 
,suppose then that we wished to         
ascertain whether a certain ,john       
,smith1 a man of thirty1 who is amongst 
other things a resident in ,india1 and  
distinctly affected with cancer1 will   
continue to survive there for twenty    
years longer4 ,the terms in which the   
man is thus introduced to us refer him  
to different classes in the way already 
indicated4 ,corresponding to these      
classes there will be a number of       
propositions which have been obtained by
 previous observations and inductions1  
and which we may therefore assume to be 
available and ready at hand when we want
 to make use of them4 ,let us conceive  
them to be such as these                

following3--,some men live to fifty2    
some ,indian residents live to fifty2 no
 man suffering thus from cancer lives   
for five years4 ,from the first and     
second of these premises nothing        
whatever can be inferred1 for they are  
both#d particular propositions1 and     
therefore lead to no conclusion in this 
case4 ,the third answers our enquiry    
decisively4  ,to the logical reader it  
will hardly be necessary to point out   
that the process here under             
consideration is that of finding middle 
terms which shall serve to connect the  
subject and predicate of our conclusion4
 ,this subject and predicate in the case
 in question1 are the individual before 
us and his death within the stated      
period4 ,regarded by themselves there is
 nothing in common between them1 and    
therefore no link by which they may be  
connected or disconnected with each     
other4 ,the various classes above       
referred to are a set of such middle    
terms1 and the propositions belonging to

 them are a corresponding set of major  
premises4 ,by the help of any one of    
them we are enabled1 under suitable     
circumstances1 to connect together the  
subject and predicate of the conclusion1
 that is1 to infer whether the man will 
or will not live twenty years4  #ae4    
,now in the performance of such a       
logical process there are two           
considerations to which the reader's    
attention must for a moment be directed4
 ,they are simple enough in this case1  
but will need careful explanation in the
 corresponding case in ,probability4 ,in
 the first place1 it is clear that      
whenever we can make any inference at   
all1 we can do so with absolute         
certainty4 ,logic1 within its own       
domain1 knows nothing of hesitation or  
doubt4 ,if the middle term is           
appropriate it serves to connect the    
extremes in such a way as to preclude   
all uncertainty about the conclusion2 if
 it is not1 there is so far an end of   
the matter3 no conclusion can be drawn1 

and we are therefore left where we were4
 ,assuming our premises to be correct1  
we either know our conclusion for       
certain1 or we know nothing whatever    
about it4 ,in the second place1 it      
should be noticed that none of the      
possible alternatives in the shape of   
such major premises as those given above
 can ever contradict any of the others1 
or be at all inconsistent with them4    
,regarded as isolated propositions1     
there is of course nothing to secure    
such harmony2 they have very different  
predicates1 and may seem quite out of   
each other's reach for either support or
 opposition4 ,but by means of the other 
premise they are in each case brought   
into relation with one another1 and the 
general interests of truth and          
consistency prevent them therefore from 
contradicting one another4 ,as isolated 
propositions it might have been the case
 that all men live to fifty1 and that no
 ,indian residents do so1 but having    
recognised that some men are residents  

in ,india1 we see at once that these    
premises are inconsistent1 and therefore
 that one or other of them must be      
rejected4 ,in all applied logic this    
necessity of avoiding self-contradiction
 is so obvious and imperious that no one
 would think it necessary to lay down   
the formal postulate that all such      
possible major premises are to be       
mutually consistent4 ,to suppose that   
this postulate is not complied with1    
would be in effect to make two or more  
contradictory assumptions about matters 
of fact4  #af4 ,but now observe the     
difference when we attempt to take the  
corresponding step in ,probability4 ,for
 ordinary propositions1 universal or    
particular1 substitute statistical      
propositions of what we have been in the
 habit of calling the 'proportional'    
kind4 ,in other words1 instead of asking
 whether the man will live for twenty   
years1 let us ask whether he will live  
for one year8 ,we shall be unable to    
find any universal propositions which   

will cover the case1 but we may without 
difficulty obtain an abundance of       
appropriate proportional ones4 ,they    
will be of the following                
description3--,of men aged #cj1 #ih in  
#ajj live another year2 of residents in 
,india a smaller proportion survive1 let
 us for example say #ij in #ajj2 of men 
suffering from cancer a smaller         
proportion still1 let us say #bj in     
#ajj4  ,now in both of the respects to  
which attention has just been drawn1    
propositions of this kind offer a marked
 contrast with those last considered4   
,in the first place1 they do not1 like  
ordinary propositions1 either assert    
unequivocally yes or no1 or else refuse 
to open their lips2 but they give       
instead a sort of qualified or          
hesitating answer concerning the        
individuals included in them4 ,this is  
of course nothing more than the familiar
 characteristic of what may be called   
'probability propositions4' ,but it     
leads up to1 and indeed renders         

possible1 the second and more important 
point2 viz4 that these various answers1 
though they cannot directly and formally
 contradict each other 7this their      
nature as proportional propositions1    
will not as a rule permit71 may yet1 in 
a way which will now have to be pointed 
out1 be found to be more or less in     
conflict with each other4  ,hence it    
follows that in the attempt to draw a   
conclusion from premises of the kind in 
question1 we may be placed in a position
 of some perplexity2 but it is a        
perplexity which may present itself in  
two forms1 a mild and an aggravated     
form4 ,we will notice them in turn4     
#ag4 ,the mild form occurs when the     
different classes to which the          
individual case may be appropriately    
referred are successively included one  
within another2 for here our sets of    
statistics1 though leading to different 
results1 will not often be found to be  
very seriously at variance with one     
another4 ,all that comes of it is that  

as we ascend in the scale by appealing  
to higher and higher genera1 the        
statistics grow continually less        
appropriate to the particular case in   
point1 and such information therefore as
 they afford becomes gradually less     
explicit and accurate4  ,the question   
that we originally wanted to determine1 
be it remembered1 is whether ,john      
,smith will die within one year4 ,but   
all knowledge of this fact being        
unattainable1 owing to the absence of   
suitable inductions1 we felt justified  
7with the explanation1 and under the    
restrictions mentioned in ,chap ,,vi471 
in substituting1 as the only available  
equivalent for such individual          
knowledge1 the answer to the following  
statistical enquiry1 ,what proportion of
 men in his circumstances die8  #ah4    
,but then at once there begins to arise 
some doubt and ambiguity as to what     
exactly is to be understood by his      
circumstances4 ,we may know very well   
what these circumstances are in         

themselves1 and yet be in perplexity as 
to how many of them we ought to take    
into account when endeavouring to       
estimate his fate4 ,we might            
conceivably1 for a beginning1 choose to 
confine our attention to those          
properties only which he has in common  
with all animals4 ,if so1 and statistics
 on the subject were attainable1 they   
would presumably be of some such        
character as this1 ,ninety-nine animals 
out of a hundred die within a year4     
,unusual as such a reference would be1  
we should1 logically speaking1 be doing 
nothing more than taking a wider class  
than the one we were accustomed to4     
,similarly we might1 if we pleased1 take
 our stand at the class of vertebrates1 
or at that of mammalia1 if zoologists   
were able to give us the requisite      
information4 ,of course we reject these 
wide classes and prefer a narrower one4 
,if asked why we reject them1 the       
natural answer is that they are so      
general1 and resemble the particular    

case before us in so few points1 that we
 should be exceedingly likely to go     
astray in trusting to them4 ,though     
accuracy cannot be insured1 we may at   
least avoid any needless exaggeration of
 the relative number and magnitude of   
our errors4  #ai4 ,the above answer is  
quite valid2 but whilst cautioning us   
against appealing to too wide a class1  
it seems to suggest that we cannot go   
wrong in the opposite direction1 that is
 in taking too narrow a class4 ,and yet 
we do avoid any such extremes4 ,john    
,smith is not only an ,englishman2 he   
may also be a native of such a part of  
,england1 be living in such a           
,presidency1 and so on4 ,an indefinite  
number of such additional               
characteristics might be brought out    
into notice1 many of which at any rate  
have some bearing upon the question of  
vitality4 ,why do we reject any         
consideration of these narrower classes8
 ,we do reject them1 but it is for what 
may be termed a practical rather than a 

theoretical reason4 ,as was explained in
 the first chapters1 it is essential    
that our series should contain a        
considerable number of terms if they are
 to be of any service to us4 ,now many  
of the attributes of any individual are 
so rare that to take them into account  
would be at variance with the           
fundamental assumption of our science1  
viz4 that we are properly concerned only
 with the averages of large numbers4    
,the more special and minute our        
statistics the better1 provided only    
that we can get enough of them1 and so  
make up the requisite large number of   
instances4 ,this is1 however1 impossible
 in many cases4 ,we are therefore       
obliged to neglect one attribute after  
another1 and so to enlarge the contents 
of our class2 at the avowed risk of     
somewhat increased variety and          
unsuitability in the members of it1 for 
at each step of this kind we diverge    
more and more from the sort of instances
 that we really want4 ,we continue to do

 so1 until we no longer gain more in    
quantity than we lose in quality4 ,we   
finally take our stand at the point     
where we first obtain statistics drawn  
from a sufficiently large range of      
observation to secure the requisite     
degree of stability and uniformity4     
#bj4 ,in such an example as the one just
 mentioned1 where one of the successive 
classes--man--is a well-defined natural 
kind or species1 there is such a        
complete break in each direction at this
 point1 that every one is prompted to   
take his stand here4 ,on the one hand1  
no enquirer would ever think of         
introducing any reference to the higher 
classes with fewer attributes1 such as  
animal or organized being3 and on the   
other hand1 the inferior classes1       
created by our taking notice of his     
employment or place of residence1 c41 do
 not as a rule differ sufficiently in   
their characteristics from the class man
 to make it worth our while to attend to
 them4  ,now and then indeed these      

characteristics do rise into importance1
 and whenever this is the case we       
concentrate our attention upon the class
 to which they correspond1 that is1 the 
class which is marked off by their      
presence4 ,thus1 for instance1 the      
quality of consumptiveness separates any
 one off so widely from the majority of 
his fellow-men in all questions         
pertaining to mortality1 that statistics
 about the lives of consumptive men     
differ materially from those which refer
 to men in general4 ,and we see the     
result2 if a consumptive man can effect 
an insurance at all1 he must do it for a
 much higher premium1 calculated upon   
his special circumstances4 ,in other    
words1 the attribute is sufficiently    
important to mark off a fresh class or  
series4 ,so with insurance against      
accident4 ,it is not indeed attempted to
 make a special rate of insurance for   
the members of each separate trade1 but 
the differences of risk to which they   
are liable oblige us to take such facts 

to some degree into account4 ,hence1    
trades are roughly divided into two or  
three classes1 such as the ordinary1 the
 hazardous1 and the extra-hazardous1    
each having to pay its own rate of      
premium4  #ba4 ,where one or other of   
the classes thus corresponds to natural 
kinds1 or involves distinctions of      
co-ordinate importance with those of    
natural kinds1 the process is not       
difficult2 there is almost always some  
one of these classes which is so        
universally recognised to be the        
appropriate one1 that most persons are  
quite unaware of there being any        
necessity for a process of selection4   
,except in the cases where a man has a  
sickly constitution1 or follows a       
dangerous employment1 we seldom have    
occasion to collect statistics for him  
from any class but that of men in       
general of his age in the country4      
,when1 however1 these successive classes
 are not ready marked out for us by     
nature1 and thence arranged in easily   

distinguishable groups1 the process is  
more obviously arbitrary4 ,suppose we   
were considering the chance of a man's  
house being burnt down1 with what       
collection of attributes should we rest 
content in this instance8 ,should we    
include all kinds of buildings1 or only 
dwelling-houses1 or confine ourselves to
 those where there is much wood1 or     
those which have stoves8 ,all these     
attributes1 and a multitude of others   
may be present1 and1 if so1 they are all
 circumstances which help to modify our 
judgment4 ,we must be guided here by the
 statistics which we happen to be able  
to obtain in sufficient numbers4 ,here  
again1 rough distinctions of this kind  
are practically drawn in ,insurance     
,offices1 by dividing risks into        
ordinary1 hazardous1 and                
extra-hazardous4 ,we examine our case1  
refer it to one or other of these       
classes1 and then form our judgment upon
 its prospects by the statistics        
appropriate to its class4  #bb4 ,so much

 for what may be called the mild form in
 which the ambiguity occurs2 but there  
is an aggravated form in which it may   
show itself1 and which at first sight   
seems to place us in far greater        
perplexity4  ,suppose that the different
 classes mentioned above are not        
included successively one within the    
other4 ,we may then be quite at a loss  
which of the statistical tables to      
employ4 ,let us assume1 for example1    
that nine out of ten ,englishmen are    
injured by residence in ,madeira1 but   
that nine out of ten consumptive persons
 are benefited by such a residence4     
,these statistics1 though fanciful1 are 
conceivable and perfectly compatible4   
,john ,smith is a consumptive           
,englishman2 are we to recommend a visit
 to ,madeira in his case or not8 ,in    
other words1 what inferences are we to  
draw about the probability of his death8
 ,both of the statistical tables apply  
to his case1 but they would lead us to  
directly contradictory conclusions4     

,this does not mean1 of course1         
contradictory precisely in the logical  
sense of that word1 for one of these    
propositions does not assert that an    
event must happen and the other deny    
that it must2 but contradictory in the  
sense that one would cause us in some   
considerable degree to believe what the 
other would cause us in some            
considerable degree to disbelieve4 ,this
 refers1 of course1 to the individual   
events2 the statistics are by           
supposition in no degree contradictory4 
,without further data1 therefore1 we can
 come to no decision4  #bc4             
,practically1 of course1 if we were     
forced to a decision with only these    
data before us1 we should make our      
choice by the consideration that the    
state of a man's lungs has probably more
 to do with his health than the place of
 his birth has2 that is1 we should      
conclude that the duration of life of   
consumptive ,englishmen corresponds much
 more closely with that of consumptive  

persons in general than with that of    
their healthy countrymen4 ,but this is1 
of course1 to import empirical          
considerations into the question4 ,the  
data1 as they are given to us1 and if we
 confine ourselves to them1 leave us in 
absolute uncertainty upon the point4 ,it
 may be that the consumptive ,englishmen
 almost all die when transported into   
the other climate2 it may be that they  
almost all recover4 ,if they die1 this  
is in obvious accordance with the first 
set of statistics2 it will be found in  
accordance with the second set through  
the fact of the foreign consumptives    
profiting by the change of climate in   
more than what might be termed their due
 proportion4 ,a similar explanation will
 apply to the other alternative1 viz4 to
 the supposition that the consumptive   
,englishmen mostly recover4 ,the problem
 is1 therefore1 left absolutely         
indeterminate1 for we cannot here appeal
 to any general rule so simple and so   
obviously applicable as that which1 in a

 former case1 recommended us always to  
prefer the more special statistics1 when
 sufficiently extensive1 to those which 
are wider and more general4 ,we have no 
means here of knowing whether one set is
 more special than the other4  ,and in  
this no difficulty can be found1 so long
 as we confine ourselves to a just view 
of the subject4 ,let me again recall to 
the reader's mind what our present      
position is2 we have substituted for    
knowledge of the individual 7finding    
that unattainable7 a knowledge of what  
occurs in the average of similar cases4 
,this step had to be taken the moment   
the problem was handed over to          
,probability4 ,but the conception of    
similarity in the cases introduces us to
 a perplexity2 we manage indeed to evade
 it in many instances1 but here it is   
inevitably forced upon our notice4      
,there are here two aspects of this     
similarity1 and they introduce us to two
 distinct averages4 ,two assertions are 
made as to what happens in the long run1

 and both of these assertions1 by       
supposition1 are verified4 ,of their    
truth there need be no doubt1 for both  
were supposed to be obtained from       
experience4  #bd4 ,it may perhaps be    
supposed that such an example as this is
 a reductio ad absurdum of the principle
 upon which ,life and other ,insurances 
are founded4 ,but a moment's            
consideration will show that this is    
quite a mistake1 and that the principle 
of insurance is just as applicable to   
examples of this kind as to any other4  
,an office need find no difficulty in   
the case supposed4 ,they might 7for a   
reason to be mentioned presently1 they  
probably would not7 insure the          
individual without inconsistency at a   
rate determined by either average4 ,they
 might say to him1 ,you are an          
,englishman4 ,out of the multitude of   
,english who come to us nine in ten die 
if they go to ,madeira4 ,we will insure 
you at a rate assigned by these         
statistics1 knowing that in the long run

 all will come right so far as we are   
concerned4 ,you are also consumptive1 it
 is true1 and we do not know what       
proportion of the ,english are          
consumptive1 nor what proportion of     
,english consumptives die in ,madeira4  
,but this does not really matter for our
 purpose4 ,the formula1 nine in ten die1
 is in reality calculated by taking into
 account these unknown proportions2 for1
 though we do not know them in          
themselves1 statistics tell us all that 
we care to know about their results4 ,in
 other words1 whatever unknown elements 
may exist1 must1 in regard to all the   
effects which they can produce1 have    
been already taken into account1 so that
 our ignorance about them cannot in the 
least degree invalidate such conclusions
 as we are able to draw4 ,and this is   
sufficient for our purpose4 ,but        
precisely the same language might be    
held to him if he presented himself as a
 consumptive man2 that is to say1 the   
office could safely carry on its        

proceedings upon either alternative4    
,this would1 of course1 be a very       
imperfect state for the matter to be    
left in4 ,the only rational plan would  
be to isolate the case of consumptive   
,englishmen1 so as to make a separate   
calculation for their circumstances4    
,this calculation would then at once    
supersede all other tables so far as    
they were concerned2 for though1 in the 
end1 it could not arrogate to itself any
 superiority over the others1 it would  
in the mean time be marked by fewer and 
slighter aberrations from the truth4    
#be4 ,the real reason why the ,insurance
 office could not long work on the above
 terms is of a very different kind from 
that which some readers might           
contemplate1 and belongs to a class of  
considerations which have been much     
neglected in the attempts to construct  
sciences of the different branches of   
human conduct4 ,it is nothing else than 
that annoying contingency to which      
prophets since the time of ,jonah have  

been subject1 of uttering suicidal      
prophecies2 of publishing conclusions   
which are perfectly certain when every  
condition and cause but one have been   
taken into account1 that one being the  
effect of the prophecy itself upon those
 to whom it refers4  ,in our example    
above1 the office 7in so far as the     
particular cases in ,madeira are        
concerned7 would get on very well until 
the consumptive ,englishmen in question 
found out what much better terms they   
could make by announcing themselves as  
consumptives1 and paying the premium    
appropriate to that class1 instead of   
announcing themselves as ,englishmen4   
,but if they did this they would of     
course be disturbing the statistics4    
,the tables were based upon the         
assumption that a certain fixed         
proportion 7it does not matter what     
proportion7 of the ,english lives would 
continue to be consumptive lives1 which1
 under the supposed circumstances1 would
 probably soon cease to be true4 ,when  

it is said that nine ,englishmen out of 
ten die in ,madeira1 it is meant that of
 those who come to the office1 as the   
phrase is1 at random1 or in their fair  
proportions1 nine-tenths die4 ,the      
consumptives are supposed to go there   
just like red-haired men1 or poets1 or  
any other special class4 ,or they might 
go in any proportions greater or less   
than those of other classes1 so long as 
they adhered to the same proportion     
throughout4 ,the tables are then        
calculated on the continuance of this   
state of things2 the practical          
contradiction is in supposing such a    
state of things to continue after the   
people had once had a look at the       
tables4 ,if we merely make the          
assumption that the publication of these
 tables made no such alteration in the  
conduct of those to whom it referred1 no
 hitch of this kind need occur4  #bf4   
,the assumptions here made1 as has been 
said1 are not in any way contradictory1 
but they need some explanation4 ,it will

 readily be seen that1 taken together1  
they are inconsistent with the          
supposition that each of these classes  
is homogeneous1 that is1 that the       
statistical proportions which hold of   
the whole of either of them will also   
hold of any portion of them which we may
 take4 ,there are certain individuals   
7viz4 the consumptive ,englishmen7 who  
belong to each class1 and of course the 
two different sets of statistics cannot 
both be true of them taken by           
themselves4 ,they might coincide in     
their characteristics with either class1
 but not with both2 probably in most    
practical cases they will coincide with 
neither1 but be of a somewhat           
intermediate character4 ,now when it is 
said of any such heterogeneous body     
that1 say1 nine-tenths die1 what is     
meant 7or rather implied7 is that the   
class might be broken up into smaller   
subdivisions of a more homogeneous      
character1 in some of which1 of course1 
more than nine-tenths die1 whilst in    

others less1 the differences depending  
upon their character1 constitution1     
profession1 c42 the number of such      
divisions and the amount of their       
divergence from one another being       
perhaps very considerable4  ,now when we
 speak of either class as a whole and   
say that nine-tenths die1 the most      
natural and soundest meaning is that    
that would be the proportion if all     
without exception went abroad1 or 7what 
comes to the same thing7 if each of     
these various subdivisions was          
represented in fair proportion to its   
numbers4 ,or it might only be meant that
 they go in some other proportion1      
depending upon their tastes1 pursuits1  
and so on4 ,but whatever meaning be     
adopted one condition is necessary1 viz4
 that the proportion of each class that 
went at the time the statistics were    
drawn up must be adhered to throughout4 
,when the class is homogeneous this is  
not needed1 but when it is heterogeneous
 the statistics would be interfered with

 unless this condition were secured4    
,we are here supposed to have two sets  
of statistics1 one for the ,english and 
one for the consumptives1 so that the   
consumptive ,english are in a sense     
counted twice over4 ,if their mortality 
is of an intermediate amount1 therefore1
 they serve to keep down the mortality  
of one class and to keep up that of the 
other4 ,if the statistics are supposed  
to be exhaustive1 by referring to the   
whole of each class1 it follows that    
actually the same individuals must be   
counted each time2 but if               
representatives only of each class are  
taken1 the same individuals need not be 
inserted in each set of tables4  #bg4   
,when therefore they come to insure 7our
 remarks are still confined to our      
supposed ,madeira case71 we have some   
,english consumptives counted as        
,english1 and paying the high rate2 and 
others counted as consumptives and      
paying the low rate4 ,logically indeed  
we may suppose them all entered in each 

class1 and paying therefore each rate4  
,what we have said above is that any    
individual may be conceived to present  
himself for either of these classes4    
,conceive that some one else pays his   
premium for him1 so that it is a matter 
of indifference to him personally at    
which rate he insures1 and there is     
nothing to prevent some of the class 7or
 for that matter all7 going to one      
class1 and others 7or all again7 going  
to the other class4  ,so long therefore 
as we make the logically possible though
 practically absurd supposition that    
some men will continue to pay a higher  
rate than they need1 there is nothing to
 prevent the ,english consumptives 7some
 or all7 from insuring in each category 
and paying its appropriate premium4 ,as 
soon as they gave any thought to the    
matter1 of course they would1 in the    
case supposed1 all prefer to insure as  
consumptives4 ,but their doing this     
would disturb each set of statistics4   
,the ,english mortality in ,madeira     

would instantly become heavier1 so far  
as the ,insurance company was concerned1
 by the loss of all their best lives2   
whilst the consumptive statistics       
7unless all the ,english consumptives   
had already been taken for insurance7   
would be in the same way deteriorated4#e
 ,a slight readjustment therefore of    
each scale of insurance would then be   
needed2 this is the disturbance         
mentioned just above4 ,it must be       
clearly understood1 however1 that it is 
not our original statistics which have  
proved to be inconsistent1 but simply   
that there were practical obstacles to  
carrying out a system of insurance upon 
them4  #bh4 ,examples subject to the    
difficulty now under consideration will 
doubtless seem perplexing to the student
 unacquainted with the subject4 ,they   
are difficult to reconcile with any     
other view of the science than that     
insisted on throughout this ,essay1 viz4
 that we are only concerned with        
averages4 ,it will perhaps be urged that

 there are two different values of the  
man's life in these cases1 and that they
 cannot both be true4 ,why not8 ,the    
'value' of his life is simply the number
 of years to which men in his           
circumstances do1 on the average1       
attain2 we have the man set before us   
under two different circumstances2 what 
wonder1 therefore1 that these should    
offer different averages8 ,in such an   
objection it is forgotten that we have  
had to substitute for the unattainable  
result about the individual1 the really 
attainable result about a set of men as 
much like him as possible4 ,the         
difficulty and apparent contradiction   
only arise when people will try to find 
some justification for their belief in  
the individual case4 ,what can we       
possibly conclude1 it may be asked1     
about this particular man ,john ,smith's
 prospects when we are thus offered two 
different values for his life8 ,nothing 
whatever1 it must be replied2 nor could 
we in reality draw a conclusion1 be it  

remembered1 in the former case1 when we 
were practically confined to one set of 
statistics4 ,there also we had what we  
called the 'value' of his life1 and     
since we only knew of one such value1 we
 came to regard it as in some sense     
appropriate to him as an individual4    
,here1 on the other hand1 we have two   
values1 belonging to different series1  
and as these values are really different
 it may be complained that they are     
discordant1 but such a complaint can    
only be made when we do what we have no 
right to do1 viz4 assign a value to the 
individual which shall admit of         
individual justification4  #bi4 ,is it  
then perfectly arbitrary what series or 
class of instances we select by which to
 judge8 ,by no means2 it has been stated
 repeatedly that in choosing a series1  
we must seek for one the members of     
which shall resemble our individual in  
as many of his attributes as possible1  
subject only to the restriction that it 
must be a sufficiently extensive series4

 ,what is meant is1 that in the above   
case1 where we have two series1 we      
cannot fairly call them contradictory2  
the only valid charge is one of         
incompleteness or insufficiency for     
their purpose1 a charge which applies in
 exactly the same sense1 be it          
remembered1 to all statistics which     
comprise genera unnecessarily wider than
 the species with which we are          
concerned4 ,the only difference between 
the two different classes of cases is1  
that in the one instance we are on a    
path which we know will lead at the     
last1 through many errors1 towards the  
truth 7in the sense in which truth can  
be attained here71 and we took it for   
want of a better4 ,in the other instance
 we have two such paths1 perfectly      
different paths1 either of which however
 will lead us towards the truth as      
before4 ,contradiction can only seem to 
arise when it is attempted to justify   
each separate step on our paths1 as well
 as their ultimate tendency4  ,still it 

cannot be denied that these objections  
are a serious drawback to the           
completeness and validity of any        
anticipations which are merely founded  
upon statistical frequency1 at any rate 
in an early stage of experience1 when   
but few statistics have been collected4 
,such knowledge as ,probability can give
 is not in any individual case of a high
 order1 being subject to the            
characteristic infirmity of repeated    
error2 but even when measured by its own
 standard it commences at a very low    
stage of proficiency4 ,the errors are   
then relatively very numerous and large 
compared with what they may ultimately  
be reduced to4  #cj4 ,here as elsewhere 
there is a continuous process of        
specialization going on4 ,the needs of a
 gradually widening experience are      
perpetually calling upon us to subdivide
 classes which are found to be too      
heterogeneous4 ,sometimes the only      
complaint that has to be made is that   
the class to which we are obliged to    

refer is found to be somewhat too broad 
to suit our purpose1 and that it might  
be subdivided with convenience4 ,this is
 the case1 as has been shown above1 when
 an ,insurance office finds that its    
increasing business makes it possible   
and desirable to separate off the men   
who follow some particular trades from  
the rest of their fellow-countrymen4    
,similarly in every other department in 
which statistics are made use of4 ,this 
increased demand for specificness leads1
 in fact1 as naturally in this          
direction1 as does the progress of      
civilization to the subdivision of      
trades in any town or country4 ,so in   
reference to the other kind of          
perplexity mentioned above4 ,nothing is 
more common in those sciences or        
practical arts1 in which deduction is   
but little available1 and where in      
consequence our knowledge is for the    
most part of the empirical kind1 than to
 meet with suggestions which point more 
or less directly in contrary directions4

 ,whenever some new substance is        
discovered or brought into more general 
use1 those who have to deal with it must
 be familiar with such a state of       
things4 ,the medical man who has to     
employ a new drug may often find himself
 confronted by the two distinct         
recommendations1 that on the one hand it
 should be employed for certain         
diseases1 and that on the other hand it 
should not be tried on certain          
constitutions4 ,a man with such a       
constitution1 but suffering from such a 
disease1 presents himself2 which        
recommendation is the doctor to follow8 
,he feels at once obliged to set to work
 to collect narrower and more special   
statistics1 in order to escape from such
 an ambiguity4  #ca4 ,in this and a     
multitude of analogous cases afforded by
 the more practical arts it is not of   
course necessary that numerical data    
should be quoted and appealed to2 it is 
sufficient that the judgment is more or 
less consciously determined by them4    

,all that is necessary to make the      
examples appropriate is that we should  
admit that in their case statistical    
data are our ultimate appeal in the     
present state of knowledge4 ,of course  
if the empirical laws can be resolved   
into their component causes we may      
appeal to direct deduction1 and in this 
case the employment of statistics1 and  
consequently the use of the theory of   
,probability1 may be superseded4  ,in   
this direction therefore1 as time       
proceeds1 the advance of statistical    
refinement by the incessant subdivision 
of classes to meet the developing wants 
of man is plain enough4 ,but if we      
glance backwards to a more primitive    
stage1 we shall soon see in what a very 
imperfect state the operation commences4
 ,at this early stage1 however1         
,probability and ,induction are so      
closely connected together as to be very
 apt to be merged into one1 or at any   
rate to have their functions confounded4
  #cb4 ,since the generalization of our 

statistics is found to belong to        
,induction1 this process of             
generalization may be regarded as prior 
to1 or at least independent of1         
,probability4 ,we have1 moreover1       
already discussed 7in ,chapter ,,vi47   
the step corresponding to what are      
termed immediate inferences1 and 7in    
,chapter ,,vii47 that corresponding to  
syllogistic inferences4 ,our present    
position therefore is that in which we  
may consider ourselves in possession of 
any number of generalizations1 but wish 
to employ them so as to make inferences 
about a given individual2 just as in one
 department of common logic we are      
engaged in finding middle terms to      
establish the desired conclusion4 ,in   
this latter case the process is found to
 be extremely simple1 no accumulation of
 different middle terms being able to   
lead to any real ambiguity or           
contradiction4 ,in ,probability1        
however1 the case is different4 ,here1  
if we attempt to draw inferences about  

the individual case before us1 as often 
is attempted--in the ,rule of           
,succession for example--we shall       
encounter the full force of this        
ambiguity and contradiction4 ,treat the 
question1 however1 fairly1 and all      
difficulty disappears4 ,our inference   
really is not about the individuals as  
individuals1 but about series or        
successions of them4 ,we wished to know 
whether ,john ,smith will die within the
 year2 this1 however1 cannot be known4  
,but ,john ,smith1 by the possession of 
many attributes1 belongs to many        
different series4 ,the multiplicity of  
middle terms1 therefore1 is what ought  
to be expected4 ,we can know whether a  
succession of men1 residents in ,india1 
consumptives1 c4 die within a year4 ,we 
may make our selection1 therefore1      
amongst these1 and in the long run the  
belief and consequent conduct of        
ourselves and other persons 7as         
described in ,chapter ,,vi47 will become
 capable of justification4 ,with regard 

to choosing one of these series rather  
than another1 we have two opposing      
principles of guidance4 ,on the one     
hand1 the more special the series the   
better2 for1 though not more right in   
the end1 we shall thus be more nearly   
right all along4 ,but1 on the other     
hand1 if we try to make the series too  
special1 we shall generally meet the    
practical objection arising from        
insufficient statistics4   #a4 ,some of 
my readers may be familiar with a very  
striking digression   in ,buffon's      
,natural ,history 7,natural ,hist4 of   
,man1 ,,viii471 in   which he supposes  
the first man in full possession of his 
faculties1   but with all his experience
 to gain1 and speculates on the gradual 
acquisition of his knowledge4 ,whatever 
may be thought of his   particular      
conclusions the passage is very         
interesting and   suggestive to any     
student of ,psychology4  #b4 ,see also  
,dugald ,stewart 7,ed4 by ,hamilton2    
,,vii4 pp4 #aae-aai74  #c4 ,required    

that is for purposes of logical         
inference within the   limits of        
,probability2 it is not intended to     
imply any doubts as to   its actual     
universal prevalence1 or its            
all-importance for   scientific         
purposes4 ,the subject is more fully    
discussed in a future   chapter4  #d4   
,as particular propositions they are    
both of course identical in   form4 ,the
 fact that the 'some' in the former     
corresponds to a   larger proportion    
than in the latter1 is a distinction    
alien to pure   ,logic4  #e4 ,the reason
 is obvious4 ,the healthiest ,english   
lives in ,madeira   7viz4 the           
consumptive ones7 have now ceased to be 
reckoned as   ,english2 whereas the     
worst consumptive lives there 7viz4 the 
,english7 are now increased in relative 
numbers4     ,,chapter ,x4   ,,chance   
,,as ,,opposed ,,to ,,causation ,,and   
,,design4  #a4 ,the remarks in the      
previous chapter will have served to    
clear the way for an enquiry which      

probably excites more popular interest  
than any other within the range of our  
subject1 viz4 the determination whether 
such and such events are to be          
attributed to ,chance on the one hand1  
or to ,causation or ,design on the      
other4 ,as the principal difficulty     
seems to arise from the ambiguity with  
which the problem is generally conceived
 and stated1 owing to the extreme       
generality of the conceptions involved1 
it becomes necessary to distinguish     
clearly between the several distinct    
issues which are apt to be involved4    
,i4 ,there is1 to begin with1 a very old
 objection1 founded on the assumption   
which our science is supposed to make of
 the existence of ,chance4 ,the         
objection against chance is of course   
many centuries older than the ,theory of
 ,probability2 and as it seems a nearly 
obsolete objection at the present day we
 need not pause long for its            
consideration4 ,if we spelt the word    
with a capital ,c1 and maintained that  

it was representative of some distinct  
creative or administrative agency1 we   
should presumably be guilty of some form
 of ,manicheism4 ,but the only rational 
meaning of the objection would appear to
 be that the principles of the science  
compel us to assume that events 7some   
events1 only1 that is7 happen without   
causes1 and are thereby removed from the
 customary control of the ,deity4 ,as   
repeatedly pointed out already this is  
altogether a mistake4 ,the science of   
,probability makes no assumption        
whatever about the way in which events  
are brought about1 whether by causation 
or without it4 ,all that we undertake to
 do is to establish and explain a body  
of rules which are applicable to classes
 of cases in which we do not or cannot  
make inferences about the individuals4  
,the objection therefore must be        
somewhat differently stated1 and appears
 finally to reduce itself to this3--that
 the assumptions upon which the science 
of ,probability rests1 are not          

inconsistent with a disbelief in        
causation within certain limits2        
causation being of course understood    
simply in the sense of regular sequence4
 ,so stated the objection seems         
perfectly valid1 or rather the facts on 
which it is based must be admitted2     
though what connection there would be   
between such lack of causation and      
absence of ,divine superintendence ,i   
quite fail to see4  ,as this            
,theological objection died away the men
 of physical science1 and those who     
sympathized with them1 began to enforce 
the same protest2 and similar cautions  
are still to be found from time to time 
in modern treatises4 ,hume1 for         
instance1 in his short essay on         
,probability1 commences with the remark1
 though there be no such thing as chance
 in the world1 our ignorance of the real
 cause of any event has the same        
influence on the understanding1 c4 ,de  
,morgan indeed goes so far as to declare
 that the foundations of the theory of  

,probability have ceased to exist in the
 mind that has formed the conception1   
that anything ever did happen or will   
happen without some particular reason   
why it should have been precisely what  
it was and not anything else4#a ,similar
 remarks might be quoted from ,laplace  
and others4  #b4 ,in the particular form
 of the controversy above referred to1  
and which is mostly found in the region 
of the natural and physical sciences1   
the contention that chance and causation
 are irreconcileable occupies rather a  
defensive position2 the main fact       
insisted on being that1 whenever in     
these subjects we may happen to be      
ignorant of the details we have no      
warrant for assuming as a consequence   
that the details are uncaused4 ,but this
 supposed irreconcileability is         
sometimes urged in a much more          
aggressive spirit in reference to social
 enquiries4 ,here the attempt is often  
made to prove causation in the details1 
from the known and admitted regularity  

in the averages4 ,a considerable amount 
of controversy was excited some years   
ago upon this topic1 in great part      
originated by the vigorous and outspoken
 support of the necessitarian side by   
,buckle in his ,history of              
,civilization4  ,it should be remarked  
that in these cases the attempt is      
sometimes made as it were to startle the
 reader into acquiescence by the        
singularity of the examples chosen4     
,instances are selected which1 though   
they possess no greater logical value1  
are1 if one may so express it1          
emotionally more effective4 ,every      
reader of ,buckle's ,history1 for       
instance1 will remember the stress which
 he laid upon the observed fact1 that   
the number of suicides in ,london       
remains about the same1 year by year2   
and he may remember also the sort of    
panic with which the promulgation of    
this fact was accompanied in many       
quarters4 ,so too the way in which      
,laplace notices that the number of     

undirected letters annually sent to the 
,post ,office remains about the same1   
and the comments of ,dugald ,stewart    
upon this particular uniformity1 seem to
 imply that they regarded this instance 
as more remarkable than many analogous  
ones taken from other quarters4  ,that  
there is a certain foundation of truth  
in the reasonings in support of which   
the above examples are advanced1 cannot 
be denied1 but their authors appear to  
me very much to overrate the sort of    
opposition that exists between the      
theory of ,chances and the doctrine of  
,causation4 ,as regards first that wider
 conception of order or regularity which
 we have termed uniformity1 anything    
which might be called objective chance  
would certainly be at variance with this
 in one respect4 ,in ,probability       
ultimate regularity is always           
postulated2 in tossing a die1 if not    
merely the individual throws were       
uncertain in their results1 but even the
 average also1 owing to the nature of   

the die1 or the number of the marks upon
 it1 being arbitrarily interfered with1 
of course no kind of science would      
attempt to take any account of it4  #c4 
,so much must undoubtedly be granted2   
but must the same admission be made as  
regards the succession of the individual
 events8 ,can causation1 in the sense of
 invariable succession 7for we are here 
shifting on to this narrower ground71 be
 denied1 not indeed without suspicion of
 scientific heterodoxy1 but at any rate 
without throwing uncertainty upon the   
foundations of ,probability8 ,de        
,morgan1 as we have seen1 strongly      
maintains that this cannot be so4 ,i    
find myself unable to agree with him    
here1 but this disagreement springs not 
so much from differences of detail1 as  
from those of the point of view in which
 we regard the science4 ,he always      
appears to incline to the opinion that  
the individual judgment in probability  
is to admit of justification2 that when 
we say1 for instance1 that the odds in  

favour of some event are three to two1  
that we can explain and justify our     
statement without any necessary         
reference to a series or class of such  
events4 ,it is not easy to see how this 
can be done in any case1 but the        
obstacles would doubtless be greater    
even than they are1 if knowledge of the 
individual event were not merely        
unattained1 but1 owing to the absence of
 any causal connection1 essentially     
unattainable4 ,on the theory adopted in 
this work we simply postulate ignorance 
of the details1 but it is not regarded  
as of any importance on what sort of    
grounds this ignorance is based4 ,it may
 be that knowledge is out of the        
question from the nature of the case1   
the causative link1 so to say1 being    
missing4 ,it may be that such links are 
known to exist1 but that either we      
cannot ascertain them1 or should find it
 troublesome to do so4 ,it is the fact  
of this ignorance that makes us appeal  
to the theory of ,probability1 the      

grounds of it are of no importance4  #d4
 ,on the view here adopted we are       
concerned only with averages1 or with   
the single event as deduced from an     
average and conceived to form one of a  
series4 ,we start with the assumption1  
grounded on experience1 that there is   
uniformity in this average1 and1 so long
 as this is secured to us1 we can afford
 to be perfectly indifferent to the     
fate1 as regards causation1 of the      
individuals which compose the average4  
,the question then assumes the following
 form3--,is this assumption1 of average 
regularity in the aggregate1            
inconsistent with the admission of what 
may be termed causeless irregularity in 
the details8 ,it does not seem to me    
that it would be at all easy to prove   
that this is so4 ,as a matter of fact   
the two beliefs have constantly         
co-existed in the same minds4 ,this may 
not count for much1 but it suggests that
 if there be a contradiction between    
them it is by no means palpable and     

obvious4 ,millions1 for instance1 have  
believed in the general uniformity of   
the seasons taken one with another1 who 
certainly did not believe in1 and would 
very likely have been ready distinctly  
to deny1 the existence of necessary     
sequences in the various phenomena which
 compose what we call a season4 ,so with
 cards and dice2 almost every gambler   
must have recognized that judgment and  
foresight are of use in the long run1   
but writers on chance seem to think that
 gamblers need a good deal of reasoning 
to convince them that each separate     
throw is in its nature essentially      
predictable4  #e4 ,in its application to
 moral and social subjects1 what gives  
this controversy its main interest is   
its real or supposed bearing upon the   
vexed question of the freedom of the    
will2 for in this region ,causation1 and
 ,fatalism or ,necessitarianism1 are    
regarded as one and the same thing4     
,here1 as in the last case1 that wide   
and somewhat vague kind of regularity   

that we have called ,uniformity1 must be
 admitted as a notorious fact4          
,statistics have put it out of the power
 of any reasonably informed person to   
feel any hesitation upon this point4    
,some idea has already been gained1 in  
the earlier chapters1 of the nature and 
amount of the evidence which might be   
furnished of this fact1 and any quantity
 more might be supplied from the works  
of professed writers upon the subject4  
,if1 therefore1 ,free-will be so        
interpreted as to imply such essential  
irregularity as defies prediction both  
in the average1 and also in the single  
case1 then the negation of free-will    
follows1 not as a remote logical        
consequence1 but as an obvious inference
 from indisputable facts of experience4 
,few persons1 however1 would go so far  
as to interpret it in this sense4 ,all  
that troubles them is the fear that     
somehow this general regularity may be  
found to carry with it causation1       
certainly in the sense of regular       

invariable sequence1 and probably also  
with the further association of         
compulsion4 ,rejecting the latter       
association as utterly unphilosophical1 
,i cannot even see that the former      
consequence can be admitted as really   
proved1 though it doubtless gains some  
confirmation from this source4  #f4 ,the
 nature of the argument against         
free-will1 drawn from statistics1 at    
least in the form in which it is very   
commonly expressed1 seems to me         
exceedingly defective4 ,the antecedents 
and consequents1 in the case of our     
volitions1 must clearly be supposed to  
be very nearly immediately in           
succession1 if anything approaching to  
causation is to be established3 whereas 
in statistical enquiries the data are   
often widely separate1 if indeed they do
 not apply merely to single groups of   
actions or results4 ,for instance1 in   
the case of the misdirected letters1    
what it is attempted to prove is that   
each writer was so much the 'victim of  

circumstances' 7to use a common but     
misleading expression7 that he could not
 have done otherwise than he did under  
his circumstances4 ,but really no       
accumulation of figures to prove that   
the number of such letters remains the  
same year by year1 can have much bearing
 upon this doctrine1 even though they   
were accompanied by corresponding       
figures which should connect the        
forgetfulness thus indicated with some  
other characteristics in the writers4   
,so with the number of suicides4 ,if    
#bej people do1 or lately did1 annually 
put an end to themselves in ,london1 the
 fact1 as it thus stands by itself1 may 
be one of importance to the             
philanthropist and statesman1 but it    
needs bringing into much closer relation
 with psychological elements if it is to
 convince us that the actions of men are
 always instances of inflexible order4  
,in fact1 instead of having secured our 
,a and ,b here in closest intimacy of   
succession to one another1--to employ   

the symbolic notation commonly used in  
works on ,inductive ,logic to illustrate
 the causal connection1--we find them   
separated by a considerable interval2   
often indeed we merely have an ,a or a  
,b by itself4  #g4 ,again1 another      
deficiency in such reasoning seems to be
 the laying undue weight upon the mere  
regularity or persistency of the        
statistics4 ,these may lead to very     
important results1 but they are not     
exactly what is wanted for the purpose  
of proving anything against the freedom 
of the will2 it is not indeed easy to   
see what connection this has with such  
facts as that the annual number of      
thefts or of suicides remains at pretty 
nearly the same figure4 ,statistical    
uniformity seems to me to establish     
nothing else1 at least directly1 in the 
case of human actions1 than it does in  
that of physical characteristics4 ,take 
but one instance1 that of the           
misdirected letters4 ,we were already   
aware that the height1 weight1 chest    

measurement1 and so on1 of a large      
number of persons preserved a tolerably 
regular average amidst innumerable      
deflections1 and we were prepared by    
analogy to anticipate the same          
regularity in their mental              
characteristics4 ,all that we gain1 by  
counting the numbers of letters which   
are posted without addresses1 is a      
certain amount of direct evidence that  
this is the case4 ,just as observations 
of the former kind had already shown    
that statistics of the strength and     
stature of the human body grouped       
themselves about a mean1 so do those of 
the latter that a similar state of      
things prevails in respect of the       
readiness and general trustworthiness of
 the memory4 ,the evidence is not so    
direct and conclusive in the latter     
case1 for the memory is not singled out 
and subjected to measurement by itself1 
but is taken in combination with        
innumerable other influencing           
circumstances4 ,still there can be      

little doubt that the statistics tell on
 the whole in this direction1 and that  
by duly varying and extending them they 
may obtain considerable probative force4
  ,the fact is that ,probability has    
nothing more to do with ,natural        
,theology1 either in its favour or      
against it1 than the general principles 
of ,logic or ,induction have4 ,it is    
simply a body of rules for drawing      
inferences about classes of events which
 are distinguished by a certain quality4
 ,the believer in a ,deity will1 by the 
study of nature1 be led to form an      
opinion about ,his works1 and so to a   
certain extent about ,his attributes4   
,but it is surely unreasonable to       
propose that he should abandon his      
belief because the sequence of          
events1--not1 observe1 their general    
tendency towards happiness or misery1   
good or evil1--is brought about in a way
 different from what he had expected2   
whether it be by displaying order where 
he had expected irregularity1 or by     

involving the machinery of secondary    
causes where he had expected immediate  
agency4  #h4 ,it is both amusing and    
instructive to consider what very       
different feelings might have been      
excited in our minds by this            
co-existence of1 what may be called1    
ignorance of individuals and knowledge  
of aggregates1 if they had presented    
themselves to our observation in a      
reverse order4 ,being utterly unable to 
make assured predictions about a single 
life1 or the conduct of individuals1    
people are sometimes startled1 and      
occasionally even dismayed1 at the      
unexpected discovery that such          
predictions can be confidently made when
 we are speaking of large numbers4 ,and 
so some are prompted to exclaim1 ,this  
is denying ,providence6 it is utter     
,fatalism6 ,but let us assume1 for a    
moment1 that our familiarity with the   
subject had been experienced1 in the    
first instance1 in reference to the     
aggregates instead of the individual    

lives4 ,it is difficult1 perhaps1 to    
carry out such a supposition completely2
 though we may readily conceive         
something approaching to it in the case 
of an ignorant clerk in a ,life         
,assurance ,office1 who had never       
thought of life1 except as having such a
 'value' at such an age1 and who had    
hardly estimated it except in the form  
of averages4 ,might we not suppose him1 
in some moment of reflectiveness1 being 
astonished and dismayed at the sudden   
realization of the utter uncertainty in 
which the single life is involved8 ,and 
might not his exclamation in turn be1   
,why this is denying ,providence6 ,it is
 utter chaos and chance6 ,a belief in a 
,creator and ,administrator of the world
 is not confined to any particular      
assumption about the nature of the      
immediate sequence of events1 but those 
who have been accustomed hitherto to    
regard the events under one of the      
aspects above referred to1 will often   
for a time feel at a loss how to connect

 them with the other4  #i4 ,so far we   
have been touching on a very general    
question2 viz4 the relation of the      
fundamental postulates of ,probability  
to the conception of ,order or          
,uniformity in the world1 physical or   
moral4 ,the difficulties which thence   
arise are mainly theological1           
metaphysical or psychological4 ,what we 
must now consider are problems of a more
 detailed or logical character4 ,they   
are prominently these two2 7#a7 the     
distinction between chance arrangement  
and causal arrangement in physical      
phenomena2 and 7#b7 the distinction     
between chance arrangement and designed 
arrangement where we are supposed to be 
contemplating rational agency as acting 
on one side at least4  ,,ii4 ,the first 
of these questions raises the antithesis
 between chance and causation1 not as a 
general characteristic pervading all    
phenomena1 but in reference to some     
specified occurrence3--,is this a case  
of chance or not8 ,the most strenuous   

supporters of the universal prevalence  
of causation and order admit that the   
question is a relevant one1 and they    
must therefore be supposed to have some 
rule for testing the answers to it4     
,suppose1 for instance1 a man is seized 
with a fit in a house where he has gone 
to dine1 and dies there2 and some one   
remarks that that was the very house in 
which he was born4 ,we begin to wonder  
if this was an odd coincidence and      
nothing more4 ,but if our informant goes
 on to tell us that the house was an old
 family one1 and was occupied by the    
brother of the deceased1 we should feel 
at once that these facts put the matter 
in a rather different light4 ,or again1 
as ,cournot suggests1 if we hear that   
two brothers have been killed in battle 
on the same day1 it makes a great       
difference in our estimation of the case
 whether they were killed fighting in   
the same engagement or whether one fell 
in the north of ,france and the other in
 the south4 ,the latter we should at    

once class with mere coincidences1      
whereas the former might admit of       
explanation4  #aj4 ,the problem1 as thus
 conceived1 seems to be one rather of   
,inductive ,logic than of ,probability1 
because there is not the slightest      
attempt to calculate chances4 ,but it   
deserves some notice here4 ,of course no
 accurate thinker who was under the sway
 of modern physical notions would for a 
moment doubt that each of the two       
elements in question had its own 'cause'
 behind it1 from which 7assuming perfect
 knowledge7 it might have been          
confidently inferred4 ,no more would he 
doubt1 ,i apprehend1 that if we could   
take a sufficiently minute and          
comprehensive view1 and penetrate       
sufficiently far back into the past1 we 
should reach a stage at which 7again    
assuming perfect knowledge7 the         
co-existence of the two events could    
equally have been foreseen4 ,the        
employment of the word casual therefore 
does not imply any rejection of a cause2

 but it does nevertheless correspond to 
a distinction of some practical         
importance4 ,we call a coincidence      
casual1 ,i apprehend1 when we mean to   
imply that no knowledge of one of the   
two elements1 which we can suppose to be
 practically attainable1 would enable us
 to expect the other4 ,we know of no    
generalization which covers them both1  
except of course such as are taken for  
granted to be inoperative4 ,in such an  
application it seems that the word      
'casual' is not used in antithesis to   
'causal' or to 'designed'1 but rather to
 that broader conception of order or    
regularity to which ,i should apply the 
term ,uniformity4 ,the casual           
coincidence is one which we cannot bring
 under any special generalization2      
certain1 probable1 or even plausible4   
,a slightly different way of expressing 
this distinction is to regard these     
'mere coincidences' as being simply     
cases in point of independent events1 in
 the sense in which independence was    

described in a former chapter4 ,we saw  
that any two events1 ,a and ,b1 were so 
described when each happens with        
precisely the same relative statistical 
frequency whether the other happens or  
not4 ,this state of things seems to hold
 good of the successions of heads and   
tails in tossing coins1 as in that of   
male and female births in a town1 or    
that of the digits in many mathematical 
tables4 ,thus we suppose that when men  
are picked up in the street and taken   
into a house to die1 there will not be  
in the long run any preferential        
selection for or against the house in   
which they were born4 ,and all that we  
necessarily mean to claim when we deny  
of such an occurrence1 in any particular
 case1 that it is a mere coincidence1 is
 that that particular case must be taken
 out of the common list and transferred 
to one in which there is some such      
preferential selection4  #aa4 ,,iii4    
,the next problem is a somewhat more    
intricate one1 and will therefore       

require rather careful subdivision4 ,it 
involves the antithesis between ,chance 
and ,design4 ,that is1 we are not now   
7as in the preceding case7 considering  
objects in their physical aspect alone1 
and taking account only of the relative 
frequency of their co-existence or      
sequence2 but we are considering the    
agency by which they are produced1 and  
we are enquiring whether that agency    
trusted to what we call chance1 or      
whether it employed what we call design4
  ,the reader must clearly understand   
that we are not now discussing the mere 
question of fact whether a certain      
assigned arrangement is what we call a  
chance one4 ,this1 as was fully pointed 
out in the fourth chapter1 can be       
settled by mere inspection1 provided the
 materials are extensive enough4 ,what  
we are now proposing to do is to carry  
on the enquiry from the point at which  
we then had to leave it off1 by solving 
the question1 ,given a certain          
arrangement1 is it more likely that this

 was produced by design1 or by some of  
the methods commonly called chance      
methods8 ,the distinction will be       
obvious if we revert to the succession  
of figures which constitute the ratio 4 
,as ,i have said1 this arrangement1     
regarded as a mere succession of digits1
 appears to fulfil perfectly the        
characteristics of a chance arrangement4
 ,if we were to omit the first four or  
five digits1 which are familiar to most 
of us1 we might safely defy any one to  
whom it was shown to say that it was not
 got at by simply drawing figures from a
 bag4 ,he might look at it for his whole
 life without detecting that it was     
anything but the result of such a chance
 selection4 ,and rightly so1 because    
regarded as a mere arrangement it is a  
chance one3 it fulfils all the          
requirements of such an arrangement4#b  
,the question we are now proceeding to  
discuss is this3 ,given any such        
arrangement how are we to determine the 
process by which it was arrived at8  ,we

 are supposed to have some event before 
us which might have been produced in    
either of two alternative ways1 i4e4 by 
chance or by some kind of deliberate    
design2 and we are asked to determine   
the odds in favour of one or other of   
these alternatives4 ,it is therefore a  
problem in ,inverse ,probability and is 
liable to all the difficulties to which 
problems of this class are apt to be    
exposed4  #ab4 ,for the theoretic       
solution of such a question we require  
the two following data3--  7#a7 ,the    
relative frequency of the two classes of
 agencies1 viz4 that which is to act in 
a chance way and that which is to act   
designedly4  7#b7 ,the probability that 
each of these agencies1 if it were the  
really operative one1 would produce the 
event in question4  ,the latter of these
 data can generally be secured without  
any difficulty4 ,the determination of   
the various contingencies on the chance 
hypothesis ought not1 if the example    
were a suitable one1 to offer any other 

than arithmetical difficulties4 ,and as 
regards the design alternative1 it is   
generally taken for granted that if this
 had been operative it would certainly  
have produced the result aimed at4 ,for 
instance1 if ten pence are found on a   
table1 all with head uppermost1 and it  
be asked whether chance or design had   
been at work here2 we feel no difficulty
 up to a certain point4 ,had the pence  
been tossed we should have got ten heads
 only once in #ajbd throws2 but had they
 been placed designedly the result would
 have been achieved with certainty4     
,but the other postulate1 viz4 that of  
the relative prevalence of these two    
classes of agencies1 opens up a far more
 serious class of difficulties4 ,cases  
can be found no doubt1 though they are  
not very frequent1 in which this        
question can be answered approximately1 
and then there is no further trouble4   
,for instance1 if in a school class-list
 ,i were to see the four names ,brown1  
,jones1 ,robinson1 ,smith1 standing in  

this order1 it might occur to me to     
enquire whether this arrangement were   
alphabetical or one of merit4 ,in our   
enlarged sense of the terms this is     
equivalent to chance and design as the  
alternatives2 for1 since the initial    
letter of a boy's name has no known     
connection with his attainments1 the    
successive arrangement of these letters 
on any other than the alphabetical plan 
will display the random features1 just  
as we found to be the case with the     
digits of an incommensurable magnitude4 
,the odds are #bc to #a against #d names
 coming undesignedly in alphabetical    
order2 they are equivalent to certainty 
in favour of their doing so if this     
order had been designed4 ,as regards the
 relative frequency of the two kinds of 
orders in school examinations ,i do not 
know that statistics are at hand1 though
 they could easily be procured if       
necessary1 but it is pretty certain that
 the majority adopt the order of merit4 
,put for hypothesis the proportion as   

high as #i to #a1 and it would still be 
found more likely than not that in the  
case in question the order was really an
 alphabetical one4  #ac4 ,but in the    
vast majority of cases we have no such  
statistics at hand1 and then we find    
ourselves exposed to very serious       
ambiguities4 ,these may be divided into 
two distinct classes1 the nature of     
which will best be seen by the          
discussion of examples4  ,in the first  
place we are especially liable to the   
drawback already described in a former  
chapter as rendering mere statistics so 
untrustworthy1 which consists in the    
fact that the proportions are so apt to 
be disturbed almost from moment to      
moment by the possession of fresh hints 
or information4 ,we saw for instance why
 it was that statistics of mortality    
were so very unserviceable in the midst 
of a disease or in the crisis of a      
battle4 ,suppose now that on coming into
 a room ,i see on the table ten coins   
lying face uppermost1 and am asked what 

was the likelihood that the arrangement 
was brought about by design4 ,everything
 turns upon special knowledge of the    
circumstances of the case4 ,who had been
 in the room8 ,were they children1 or   
coin-collectors1 or persons who might   
have been supposed to have indulged in  
tossing for sport or for gambling       
purposes8 ,were the coins new or old    
ones8 a distinction of this kind would  
be very pertinent when we were          
considering the existence of any motive 
for arranging them the same way         
uppermost4 ,and so on2 we feel that our 
statistics are at the mercy of any      
momentary fragment of information4  #ad4
 ,but there is another consideration    
besides this4 ,not only should we be    
thus influenced by what may be called   
external circumstances of a general     
kind1 such as the character and position
 of the agents1 we should also be       
influenced by what we supposed to be the
 conventional#c estimate with which this
 or that particular chance arrangement  

was then regarded4 ,thus from time to   
time as new games of cards become       
popular new combinations acquire        
significance2 and therefore when the    
question of design takes the form of    
possible cheating a knowledge of the    
current estimate of such combinations   
becomes exceedingly important4  #ae4    
,the full significance of these         
difficulties will best be apprehended by
 the discussion of a case which is not  
fictitious or invented for the purpose1 
but which has actually given rise to    
serious dispute4 ,some years ago ,prof4 
,piazzi ,smyth published a work#d upon  
the great pyramid of ,ghizeh1 the       
general object of which was to show that
 that building contained1 in its        
magnitude1 proportions and contents1 a  
number of almost imperishable natural   
standards of length1 volume1 c4 ,amongst
 other things it was determined that the
 value of  was accurately 7the degree of
 accuracy is not1 ,i think1 assigned7   
indicated by the ratio of the sides to  

the height4 ,the contention was that    
this result could not be accidental but 
must have been designed4  ,as regards   
the estimation of the value of the      
chance hypothesis the calculation is not
 quite so clear as in the case of dice  
or cards4 ,we cannot indeed suppose     
that1 for a given length of base1 any   
height can be equally possible4 ,we must
 limit ourselves to a certain range     
here2 for if too high the building would
 be insecure1 and if too low it would be
 ridiculous4 ,again1 we must decide to  
how close an approximation the          
measurements are made4 ,if they are     
guaranteed to the hundredth of an inch  
the coincidence would be of a quite     
different order from one where the      
guarantee extended only to an inch4     
,suppose that this has been decided1 and
 that we have ascertained that out of   
#aj1jjj possible heights for a pyramid  
of given base just that one has been    
selected which would most nearly yield  
the ratio of the radius to the          

circumference of a circle4  ,the        
remaining consideration would be the    
relative frequency of the 'design'      
alternative1--what is called its  priori
 probability1--that is1 the relative    
frequency with which such builders can  
be supposed to have aimed at that ratio2
 with the obvious implied assumption    
that if they did aim at it they would   
certainly secure it4 ,considering our   
extreme ignorance of the attainments of 
the builders it is obvious that no      
attempt at numerical appreciation is    
here possible4 ,if indeed the 'design'  
was interpreted to mean conscious       
resolve to produce that ratio1 instead  
of mere resolve to employ some method   
which happened to produce it1 few       
persons would feel much hesitation4 ,not
 only do we feel tolerably certain that 
the builders did not know the value of 1
 except in the rude way in which all    
artificers must know it2 but we can see 
no rational motive1 if they did know it1
 which should induce them to perpetuate 

it in their building4 ,if1 however1 to  
adopt an ingenious suggestion1#e we     
suppose that the builder may have       
proceeded in the following fashion1 the 
matter assumes a different aspect4      
,suppose that having decided on the     
height of his pyramid he drew a circle  
with that as radius3 that1 laying down a
 cord along the line of this circle1 he 
drew this cord out into a square1 which 
square marked the base of the building4 
,hardly any simpler means could be      
devised in a comparatively rude age2 and
 it is obvious that the circumference of
 the base1 being equal to the length of 
the cord1 would bear exactly the        
admitted ratio to the height4 ,in other 
words1 the exact attainment of a        
geometric value does not imply a        
knowledge of that ratio1 but merely of  
some method which involves and displays 
it4 ,a teredo can bore1 as well as any  
of us1 a hole which displays the        
geometric properties of a circle1 but we
 do not credit it with corresponding    

knowledge4  ,as before said1 all        
numerical appreciation of the likelihood
 of the design alternative is out of the
 question4 ,but1 if the precision is    
equal to what ,mr ,smyth claimed1 ,i    
suppose that most persons 7with the     
above suggestion before them7 will think
 it somewhat more likely that the       
coincidence was not a chance one4  #af4 
,there still remains a serious1 and     
highly interesting speculative          
consideration4 ,in the above argument we
 took it for granted1 in calculating the
 chance alternative1 that only one of   
the #aj1jjj possible values was         
favourable2 that is1 we took it for     
granted that the ratio  was the only one
 whose claims1 so to say1 were before   
the court4 ,but it is clear that if we  
had obtained just double this ratio the 
result would have been of similar       
significance1 for it would have been    
simply the ratio of the circumference to
 the diameter4 ,in fact1 ,mr ,smyth's   
selected ratio1--the height to twice the

 breadth of the base as compared with   
the diameter to the circumference1--is  
obviously only one of a plurality of    
ratios4 ,again2 if the measured results 
had shown that the ratio of the height  
to one side of the base was #a 3 sqrt#b 
7i4e4 that of a side to a diagonal of a 
square7 or #a 3 sqrt#c 7i4e4 that of a  
side to a diagonal of a cube7 would not 
such results equally show evidence of   
design8 ,proceeding in this way1 we     
might suggest one known mathematical    
ratio after another until most of the   
#aj1jjj supposed possible values had    
been taken into account4 ,we might then 
argue thus3 since almost every possible 
height of the pyramid would correspond  
to some mathematical ratio1 a builder1  
ignorant of them all alike1 would be not
 at all unlikely to stumble upon one or 
other of them3 why then attribute design
 to him in one case rather than another8
  #ag4 ,the answer to this objection has
 been already hinted at4 ,everything    
turns upon the conventional estimate of 

one result as compared with another4    
,revert1 for simplicity to the coins4   
,ten heads is just as likely as         
alternate heads and tails1 or five heads
 followed by five tails2 or1 in fact1 as
 any one of the remaining #ajbc possible
 cases4 ,but universal convention has   
picked out a run of ten as being        
remarkable4 ,here1 of course1 the       
convention seems a very natural and     
indeed inevitable one1 but in other     
cases it is wholly arbitrary4 ,for      
instance1 in cards1 queen of spades and 
knave of diamonds is exactly as uncommon
 as any other such pair3 moreover1 till 
bezique was introduced it offered       
presumably no superior interest over any
 other specified pair4 ,but during the  
time when that game was very popular    
this combination was brought into the   
category of coincidences in which       
interest was felt2 and1 given dishonesty
 amongst the players1 its chance of     
being designed stood at once on a much  
better footing4#f  ,returning then to   

the pyramid1 we see that in balancing   
the claims of chance and design we must1
 in fairness to the latter1 reckon to   
its account several other values as well
 as that of 1 e4g4 sqrt#b and sqrt#c1   
and a few more such simple and familiar 
ratios1 as well as some of their        
multiples4 ,but though the number of    
such values which might be reckoned1 on 
the ground that they are actually known 
to us1 is infinite1 yet the number that 
ought to be reckoned1 on the ground that
 they could have been familiar to the   
builders of a pyramid1 are very few4    
,the order of probability for or against
 the existence of design will not       
therefore be seriously altered here by  
such considerations4#g  #ah4 ,up to this
 point it will be observed that what we 
have been balancing against each other  
are two forms of agency1--of human      
agency1 that is1--one acting through    
chance1 and the other by direct design4 
,in this case we know where we are1 for 
we can thoroughly understand agency of  

this kind4 ,the problem is indeed but   
seldom numerically soluble1 and in most 
cases not soluble at all1 but it is at  
any rate capable of being clearly       
stated4 ,we know the kind of answer to  
be expected and the reasons which would 
serve to determine it1 if they were     
attainable4  ,the next stage in the     
enquiry would be that of balancing      
ordinary human chance agency            
against1--,i will not call it direct    
spiritualist agency1 for that would be  
narrowing the hypothesis                
unnecessarily1--but against all other   
possible causes4 ,some of the           
investigations of the ,society for      
,psychical ,research will furnish an    
admirable illustration of what is       
intended by this statement4 ,there is a 
full discussion of these applications in
 a recent essay by ,mr ,f4 ,y4          
,edgeworth2#h but as his account of the 
matter is connected with other          
calculations and diagrams ,i can only   
quote it in part4 ,but ,i am in         

substantial agreement with him4  ,it is 
recorded that #ahcc guesses were made by
 a 'percipient' as to the suit of cards 
which the 'agent' had fixed upon4 ,the  
number of successful guesses was #eaj1  
considerably above #deh1 the number     
which1 as being the quarter of #ahcc1   
would1 on the supposition of pure       
chance1 be more likely than any other   
number4 ,now1 by the ,law of ,error1 we 
are able approximately to determine the 
probability of such an excess occurring 
by chance4 ,it is equal to the extremity
 of the tail of a probability-curve such
 as those we   have already had occasion
 to examine4444 ,the proportion of this 
extremity of the tail to the whole body 
is #j4jjc to #a4 ,that fraction1 then1  
is the probability of a chance shot     
striking that extremity of the tail2 the
 probability that1 if the guessing were 
governed by pure chance1 a number of    
successful guesses equal or greater than
 #eaj would occur3 odds1 that is1 of    
about #ccb to #a against such           

occurrence4  #ai4 ,mr ,edgeworth holds1 
as strongly as ,i do1 that for purposes 
of calculation1 in any strict sense of  
the word1 we ought to have some         
determination of the data on the        
non-chance side of the hypothesis4 ,we  
ought to know its relative frequency of 
occurrence1 and the relative frequency  
with which it attains its aims4 ,i am   
also in agreement with him that what    
that other cause may be1--whether some  
trick1 or unconscious illusion1 or      
thought-transference of the sort which  
is vindicated by the investigators--it  
is for common-sense and ordinary ,logic 
to consider4  ,i am in agreement        
therefore with those who think that     
though we cannot form a quantitative    
opinion we can in certain cases form a  
tolerably decisive one4 ,of course if we
 allow the last word to the supporters  
of the chance hypothesis we can never   
reach proof1 for it will always be open 
to them to revise and re-fix the        
antecedent probability of the counter   

hypothesis4 ,what we may fairly require 
is that those who deny the chance       
explanation should assign some sort of  
minimum value to the probability of     
occurrence on the other supposition1 and
 we can then try to surmount this by    
increasing the rarity of the actually   
produced phenomenon on the chance       
hypothesis4 ,if1 for instance1 they     
declare that in their estimation the    
odds against any other than the chance  
agency being at work are greater than   
#ccb to #a1 we must try to secure a yet 
uncommoner occurrence than that in      
question4 ,if the supporters of         
thought-transference have the courage of
 their convictions1--as they most       
assuredly have1--they would not shrink  
from accepting this test4 ,i am inclined
 to think that even at present1 on such 
evidence as that above1 the probability 
that the results were got at by ordinary
 guessing is very small4  #bj4 ,the     
problems discussed in the preceding     
sections are at least intelligible even 

if they are not always resolvable4 ,but 
before finishing this chapter we must   
take notice of some speculations upon   
this part of the subject which do not   
seem to keep quite within the limits of 
what is intelligible4 ,take for instance
 the question discussed by ,arbuthnott  
7in a paper in the ,phil4 ,transactions1
 ,vol4 ,,xxvii47 under the title ,an    
,argument for ,divine ,providence1 taken
 from the constant ,regularity observed 
in the birth of both sexes4 ,had his    
argument been of the ordinary           
teleological kind2 that is1 had he      
simply maintained that the existent     
ratio of approximate equality1 with a   
six per cent4 surplusage of males1 was a
 beneficent one1 there would have been  
nothing here to object against4 ,but    
what he contemplated was just such a    
balance of alternate hypotheses between 
chance and design as we are here        
considering4 ,his conclusion in his own 
words is1 it is art1 not chance1 that   
governs4  ,it is difficult to render    

such an argument precise without        
rendering it simply ridiculous4         
,strictly understood it can surely bear 
only one of two interpretations4 ,on the
 one hand we may be personifying        
,chance3 regarding it as an agent which 
must be reckoned with as being quite    
capable of having produced man1 or at   
any rate having arranged the proportion 
of the sexes4 ,and then the decision    
must be drawn1 as between this agent and
 the ,creator1 which of the two produced
 the existent arrangement4 ,if so1 and  
,chance be defined as any agent which   
produces a chance or random arrangement1
 ,i am afraid there can be little doubt 
that it was this agent that was at work 
in the case in question4 ,the           
arrangement of male and female births   
presents1 so far as we can see1 one of  
the most perfect examples of chance3    
there is ultimate uniformity emerging   
out of individual irregularity3 all the 
'runs' or successions of each           
alternative are duly represented3 the   

fact of1 say1 five sons having been     
already born in a family does not seem  
to have any certain effect in           
diminishing the likelihood of the next  
being a son1 and so on4 ,such a nearly  
perfect instance of 'independent events'
 is comparatively very rare in physical 
phenomena4 ,it is all that we can claim 
from a chance arrangement4#i ,the only  
other interpretation ,i can see is to   
suggest that there was but one agent who
 might1 like any one of us1 have either 
tossed up or designed1 and we have to   
ascertain which course he probably      
adopted in the case in question4 ,here  
too1 if we are to judge of his mode of  
action by the tests we should apply to  
any work of our own1 it would certainly 
look very much as if he had adopted some
 scheme of tossing4  #ba4 ,the simple   
fact is that any rational attempt to    
decide between chance and design as     
agencies must be confined to the case of
 finite intelligences4 ,one of the      
important determining elements here1 as 

we have seen1 is the state of knowledge 
of the agent1 and the conventional      
estimate entertained about this or that 
particular arrangement2 and these can be
 appreciated only when we are dealing   
with beings like ourselves4  ,for       
instance1 to return to that much debated
 question about the arrangement of the  
stars1 there can hardly be any doubt    
that what ,mitchell1--who started the   
discussion1--had in view was the        
decision between ,chance and ,design4   
,he says 7,trans4 ,roy4 ,soc4 #agfg7    
,the argument ,i intend to make use     
of444 is of that kind which infers      
either design or some general law from a
 general analogy and from the greatness 
of the odds against things having been  
in the present situation if it was not  
owing to some such cause4 ,and he       
concludes that had the stars been       
scattered by mere chance as it might    
happen there would be odds of near      
#ejj1jjj to #a that no six stars out of 
that number #aejj1 scattered at random  

in the whole heavens1 would be within so
 small a distance from each other as the
 ,pleiades are4 ,under any such         
interpretation the controversy seems to 
me to be idle4 ,i do not for a moment   
dispute that there is some force in the 
ordinary teleological argument which    
seeks to trace signs of goodness and    
wisdom in the general tendency of       
things4 ,but what do we possibly        
understand about the nature of creation1
 or the designs of the ,creator1 which  
should enable us to decide about the    
likelihood of his putting the stars in  
one shape rather than in another1 or    
which should allow any significance to  
mere chance as contrasted with his      
supposed all-pervading agency8  #bb4    
,reduced to intelligible terms the two  
following questions seem to me to emerge
 from the controversy3--  7,i47 ,the    
stars being distributed through space1  
some of them would of course be nearly  
in a straight line behind others when   
looked at from our planet4 ,supposing   

that they were tolerably uniformly      
distributed1 we could calculate about   
how many of them would thus be seen in  
apparent close proximity to one another4
 ,the question is then put1 ,are there  
more of them near to each other1 two and
 two1 than such calculation would       
account for8 ,the answer is that there  
are many more4 ,so far as ,i can see the
 only direct inference that can be drawn
 from this is that they are not         
uniformly distributed1 but have a       
tendency to go in pairs4 ,this1 however1
 is a perfectly sound and reasonable    
application of the theory4 ,any further 
conclusions1 such as that these pairs of
 stars will form systems1 as it were1 to
 themselves1 revolving about one        
another1 and for all practical purposes 
unaffected by the rest of the sidereal  
system1 are of course derived from      
astronomical considerations4#aj         
,probability confines itself to the     
simple answer that the distribution is  
not uniform2 it cannot pretend to say   

whether1 and by what physical process1  
these binary systems of stars have been 
'caused'4#aa  #bc4 7,,ii47 ,the second  
question is this1 ,does the distribution
 of the stars1 after allowing for the   
case of the binary stars just mentioned1
 resemble that which would be produced  
by human agency sprinkling things 'at   
random'8 7,we are speaking1 of course1  
of their distribution as it appears to  
us1 on the visible heavens1 for this is 
nearly all that we can observe2 but if  
they extend beyond the telescopic range 
in every direction1 this would lead to  
practically much the same discussion as 
if we considered their actual           
arrangement in space47 ,we have fully   
discussed1 in a former chapter1 the     
meaning of 'randomness4' ,applying it to
 the case before us1 the question       
becomes this1 ,is the distribution      
tolerably uniform on the whole1 but with
 innumerable individual deflections8    
,that is1 when we compare large areas1  
are the ratios of the number of stars in

 each equal area approximately equal1   
whilst1 as we compare smaller and       
smaller areas1 do the relative numbers  
become more and more irregular8 ,with   
certain exceptions1 such as that of the 
,milky ,way and other nebular clusters1 
this seems to be pretty much the case1  
at any rate as regards the bulk of the  
stars4#ab  ,all further questions3 the  
decision1 for instance1 for or against  
any form of the ,nebular ,hypothesis3   
or1 admitting this1 the decision whether
 such and such parts of the visible     
heavens have sprung from the same       
nebula1 must be left to ,astronomy to   
adjudicate4   ,,note ,,on ,,the         
,,proportions ,,of ,,the ,,sexes4  ,the 
following remarks were rather too long  
for convenient insertion on p4 #bei1 and
 are therefore appended here4  ,the     
'random' character of male and female   
births has generally been rested almost 
entirely on statistics of place and     
time4 ,but what is more wanted1 surely1 
is the proportion displayed when we     

compare a number of families4 ,this     
seems so obvious that ,i cannot but     
suppose that the investigation must have
 been already made somewhere1 though ,i 
have not found any trace of it in the   
most likely quarters4 ,thus ,prof4      
,lexis 7,massenerscheinungen7 when      
supporting his view that the proportion 
between the sexes at birth is almost the
 only instance known to him1 in natural 
phenomena1 of true normal dispersion    
about a mean1 rests his conclusions on  
the ordinary statistics of the registers
 of different countries4  ,it certainly 
needs proof that the same               
characteristics will hold good when the 
family is taken as the unit1 especially 
as some theories 7e4g4 that of ,sadler7 
would imply that 'runs' of boys or girls
 would be proportionally commoner than  
pure chance would assign4 ,lexis has    
shown that this is most markedly the    
case with twins3 i4e41 to use an        
obviously intelligible notation1 7,m for
 male1 ,f for female71 that ,m4,m4 and  

,f4,f4 are very much commoner in        
proportion than ,m4,f4  ,i have         
collected statistics including over     
#ac1jjj male and female births1 arranged
 in families of four and upwards4 ,they 
were taken from the pedigrees in the    
,herald's ,visitations1 and therefore   
represent as a rule a somewhat select   
class1 viz4 the families of the eldest  
sons of ,english country gentlemen in   
the sixteenth century4 ,they are not    
sufficiently extensive yet for          
publication1 but ,i give a summary of   
the results to indicate their tendency  
so far4 ,the upper line of figures in   
each case gives the observed results3   
i4e4 in the case of a family of four1   
the numbers which had four male1 three  
male and one female1 two male and two   
female1 and so on4 ,the lower line gives
 the calculated results2 i4e4 the       
corresponding numbers which would have  
been obtained had batches of ,m4s and   
,f4s been drawn from a bag in which they
 were mixed in the ratio assigned by the

 total observed numbers for those       
families4  #eab families of #d2  m#d    
m#c f   m#b f#b   mf#c   f#d yielding   
#ha   #adh     #afa     #ih    #bd      
7observed47 #aahh ,m4 3 #hfj ,f4     #eg
   #afh     #ahd     #hh    #ae         
7calculated47  #eab families of #e2  m#e
   m#d f   m#c f#b   m#b f#c   mf#d     
f#e yielding             #ej    #hb     
#afa      #adc     #fa    #ae  7obs47   
#adjb ,m4 3 #aaeh ,f4    #be   #ajc     
#agb      #adc     #ei    #aj  7calc47  
#eab families of #f2  m#f   m#e f   m#d 
f#b   m#c f#c   m#b f#d   mf#e   f#f    
yielding             #cj    #dh     #aae
      #adf      #abf     #dj    #g      
7obs47 #afab ,m4 3 #adfj ,f4    #aj     
#ef     #acc      #aei      #ajh     #da
    #e  7calc47  ,the numbers for the   
larger families are as yet too small to 
be worth giving1 but they show the same 
tendency4 ,it will be seen that in every
 case the observed central values are   
less than the calculated2 and that the  
observed extreme values are much greater

 than the calculated4 ,the results seem 
to suggest 7so far7 that a family cannot
 be likened to a chance drawing of the  
requisite number from one bag4 ,a better
 analogy would be to suppose two bags1  
one with ,m4s in excess and the other   
with ,f4s in less excess1 and that some 
persons draw from one and some from the 
other4 ,but fuller statistics are       
needed4  ,it will be observed that the  
total excess of male births is large4   
,this may arise from undue omission of  
females2 but ,i have carefully confined 
myself to the two or three last         
generations1 in each pedigree1 for      
greater security4   #a4 ,essay on       
,probabilities1 p4 #aad4  #b4 ,doubts   
have been expressed about the truly     
random character of the   digits in this
 case 7v4 ,de ,morgan1 ,budget of       
,paradoxes1 p4 #bia71   and ,jevons has 
gone so far as to ask 7,principles of   
,science1   p4 #ebi71 ,why should the   
value of 1 when expressed to a great    
number of figures1 contain the digit #g 

much less frequently than any   other   
digit6  ,i do not quite understand what 
this means4 ,if such a   question were  
asked in relation to any unusual        
divergence from the    priori chance in 
a case of throwing dice1 say1 we should 
probably substitute for it the          
following1 as being more appropriate    
to our science3--,assign the degree of  
improbability of the event in           
question2 i4e4 its statistical rarity4  
,and we should then proceed   to judge1 
in the way indicated in the text1       
whether this   improbability gave rise  
to any grounds of suspicion4    ,the    
calculation is simple4 ,the actual      
number of #g's1 in the #gjh   digits1 is
 #ec3 whilst the fair average would be  
#ga4 ,the question is1   ,what is the   
chance of such a departure from the     
average in #gjh   turns8  ,by the usual 
methods of calculation 7v4 ,galloway on 
,probability7 the chances against an    
excess or defect of #ah are   about #dd 
3 #a1 in respect of any specified digit4

 ,but of course what   we want to decide
 are the chances against some one of the
 ten   showing this divergence4 ,this ,i
 estimate as being approximately        
determined by the fraction 7#dd_/de7aj1 
viz4 #j4h4 ,this represents   odds of   
only about #d 3 #a against such an      
occurrence1 which is   nothing          
remarkable4 ,as a matter of fact several
 digits in the two   other magnitudes   
which ,mr ,shanks had calculated to the 
same length1   viz4 ,tan-#a #a_/e and   
,tan-#a #a_/bci1 show the same          
divergencies   7v4 ,proc4 ,roy4 ,soc4   
xxi4 #cai74    ,i may call attention    
here to a point which should have been  
noticed   in the chapter on ,randomness4
 ,we must be cautious when we decide    
upon the random character by mere       
inspection4 ,it is very instructive     
here to compare the digits in  with     
those within the 'period' of a          
circulating decimal of very long period4
 ,that of #a  #gfii1 which   yields the 
full period of #gfih figures1 was       

calculated some years   ago by two      
,cambridge graduates 7,mr ,lunn and ,mr 
,suffield71 and   privately printed4 ,if
 we confine our examination to a portion
 of the   succession the random         
character seems plausible2 i4e4 the     
digits1   and their various             
combinations1 come out in nearly1 but   
not exactly1   equal numbers4 ,so if we 
take batches of #aj2 the averages hover 
nicely about #de4 ,but if we took the   
whole period which   'circulates1' we   
should find these characteristics       
overdone1 and the   random character    
would disappear4 ,that is1 instead of a 
merely   ultimate approximation to      
equality we should have 7as far as this 
is   possible7 an absolute attainment of
 it4  #c4 ,of course this conventional  
estimate is nothing different in kind   
from that which may attach to any order 
or succession4 ,ten heads   in          
succession is intrinsically or          
objectively indistinguishable in        
character from alternate heads and      

tails1 or seven heads and three   tails1
 c4 ,its distinction only consists in   
its almost universal   acceptance as    
remarkable4  #d4 ,our ,inheritance in   
the ,great ,pyramid1 ,ed4 ,,iii4 #ahgg4 
#e4 ,made in ,nature 7,jan4 #bd1 #ahgh7 
by ,mr ,j4 ,g4 ,jackson4 ,it must be    
remarked that ,mr ,smyth's alternative  
statement of his case leads up   to that
 explanation3--,the vertical height of  
the great pyramid is   the radius of a  
theoretical circle the length of whose  
curved   circumference is exactly equal 
to the sum of the lengths of the four   
straight sides of the actual and        
practical square base4 ,as regards   the
 alternatives of chance and design1     
here1 it must be remembered   in justice
 to ,mr ,smyth's argument that the      
antithesis he admits to   chance is not 
human1 but divine design4  #f4 ,see     
,cournot1 ,essai sur les fondements de  
nos connaissances4   ,vol4 ,i4 p4 #ga4  
#g4 ,it deserves notice that            
considerations of this kind have found  

their way into the ,law ,courts though  
of course without any attempt   at      
numerical valuation4 ,thus1 in the      
celebrated ,de ,ros trial1 in so   far  
as the evidence was indirect1 one main  
ground of suspicion seems   to have been
 that ,lord ,de ,ros1 when dealing at   
whist1 obtained far   more court cards  
than chance could be expected to assign 
him2 and   that in consequence his      
average gains for several years in      
succession were unusually large4 ,the   
counsel for the defence urged   that    
still larger gains had been secured by  
other players without   suspicion of    
unfairness1--7,i cannot find that it was
 explained over   how large an area of  
experience these instances had been     
sought2 nor   how far the magnitude of  
the stakes1 as distinguished from the   
number of successes1 accounted for that 
of the actual gains71--and   that large 
allowance must be made for skill where  
the actual gains   were computed4 7,see 
the ,times' report1 ,feb4 #aa1 #ahcg47  

#h4 ,metretike4 ,at the end of this     
volume will be found a useful list   of 
a number of other publications by the   
same author on allied   topics4  #i4    
,that is1 if we look simply to          
statistical results1 as ,arbuthnott     
did1 and as we should do if we were     
examining the tosses of a   penny4 ,if  
the remarkable theory of ,dr ,dsing     
7,die ,regulierung des                  
,geschlechts-verhltnisses444 ,jena1     
#ahhd7 be confirmed1 the matter   would 
assume a somewhat different aspect4 ,he 
attempts to show1 both   on             
physiological grounds1 and by analysis  
of statistics referring to   men and    
animals1 that there is a decidedly      
compensatory process at   work4 ,that   
is1 if for any cause either sex attains 
a preponderance1   agencies are at once 
set in motion which tend to redress the 
balance4 ,this is a modification and    
improvement of the older theory1   that 
the relative age of the parents has     
something to do with the   sex of the   

offspring4    ,quetelet 7,letters1 p4   
#fa7 has attempted to prove a           
proposition   about the succession of   
male and female births by certain       
experiments supposed to be tried upon an
 urn with black and white   balls in it4
 ,but this is going too far4 7,see the  
note at the end of   this chapter47     
#aj4 ,it is precisely analogous to the  
conclusion that the flowers of   the    
daisies 7as distinguished from the      
plants1 v4 p4 #aji7 are not             
distributed at random1 but have a       
tendency to go in groups of two or      
more4 ,mere observation shows this3 and 
then1 from our knowledge of   the growth
 of plants we may infer that these      
little groups spring   from the same    
root4  #aa4 ,in this discussion1 writers
 often speak of the probability of a    
physical connection between these double
 stars4 ,the phrase seems   misleading1 
for on the usual hypothesis of universal
 gravitation   all stars are physically 
connected1 by gravitation4 ,it is       

therefore better1 as above1 to make it  
simply a question of relative           
proximity1 and to leave it to astronomy 
to infer what follows from   unusual    
proximity4  #ab4 ,professor ,forbes in  
the paper in the ,philosophical         
,magazine   already referred to 7,ch4   
,,vii4 #ah7 gave several diagrams to    
show   what were the actual arrangements
 of a random distribution4 ,he          
scattered peas over a chess-board1 and  
then counted the number which   rested  
on each square4 ,his figures seem to    
show that the general   appearance of   
the stars is much the same as that      
produced by such a   plan of scattering4
    ,some recent investigations by ,mr  
,r4 ,a4 ,proctor seem to show1          
however1 that there are at least two    
exceptions to this tolerably   uniform  
distribution4 7#a7 ,he has ascertained  
that the stars are   decidedly more     
thickly aggregated in the ,milky ,way   
than   elsewhere4 ,so far as this is to 
be relied on the argument is the   same 

as in the case of the double stars2 it  
tends to prove that the   proximity of  
the stars in the ,milky ,way is not     
merely apparent1 but   actual4 7#b7 ,he 
has ascertained that there are two large
 areas1 in   the ,north and ,south      
hemispheres1 in which the stars are much
 more   thickly aggregated than         
elsewhere4 ,here1 it seems to me1       
,probability proves nothing3 we are     
simply denying that the   distribution  
is uniform4 ,what may follow in the way 
of inferences   as to the physical      
process of causation by which the stars 
have been   disposed is a question for  
the ,astronomer4 ,see ,mr ,proctor's    
,essays   on ,astronomy1 p4 #big4 ,also 
a series of ,essays in ,the ,universe   
and   the coming ,transits4             
,,chapter ,,xi4   ,,on ,,certain        
,,consequences ,,of ,,the ,,objective   
,,treatment ,,of ,a ,,science ,,of      
,,inference4#a  #a4 ,in the previous    
edition a large part of this chapter was
 devoted to   the general consideration 

of the distinction between a ,material  
and   a ,conceptualist view of ,logic4  
,i have omitted most of this here1 as   
also a large part of a chapter devoted  
to the detailed discussion of   the ,law
 of ,causation1 as ,i hope before very  
long to express my   opinions on these  
subjects more fully1 and more           
appropriately1 in a   treatise on the   
general principles of ,inductive ,logic4
   #a4 ,students of ,logic are familiar 
with that broad distinction between the 
two methods of treatment to which the   
names of ,material and ,conceptualist   
may be applied4 ,the distinction was one
 which had been gradually growing up    
under other names before it was         
emphasized1 and treated as a distinction
 within the field of ,logic proper1 by  
the publication of ,mill's well known   
work4 ,no one1 for instance1 can read   
,whewell's treatises on ,induction1 or  
,herschel's ,discourse1 without seeing  
that they are treating of much the same 
subject-matter1 and regarding it in much

 the same way1 as that which ,mill      
discussed under the name of ,logic1     
though they were not disposed to give it
 that name4 ,that is1 these writers     
throughout took it for granted that what
 they had to do was to systematise the  
facts of nature in their objective form1
 and under their widest possible        
treatment1 and to expound the principal 
modes of inference and the principal    
practical aids in the investigation of  
these modes of inference1 which reason  
could suggest and which experience could
 justify4 ,what ,mill did was to bring  
these methods into close relation with  
such portions of the old scholastic     
,logic as he felt able to retain1 to    
work them out into much fuller detail1  
to systematize them by giving them a    
certain philosophical and psychological 
foundation1--and to entitle the result  
,logic4  ,the practical treatment of a  
science will seldom correspond closely  
to the ideal which its supporters       
propose to themselves1 and still        

seldomer to that which its antagonists  
insist upon demanding from the          
supporters4 ,if we were to take our     
account of the distinction between the  
two views of ,logic expounded           
respectively by ,hamilton and by ,mill1 
from ,mill and ,hamilton respectively1  
we should certainly not find it easy to 
bring them under one common definition4 
,by such a test1 the material ,logic    
would be regarded as nothing more than a
 somewhat arbitrary selection from the  
domain of ,physical ,science in general1
 and the conceptualist ,logic nothing   
more than a somewhat arbitrary selection
 from the domain of ,psychology4 ,the   
former would omit all consideration of  
the laws of thought and the latter all  
consideration of the truth or falsehood 
of our conclusions4  ,of course1 in     
practice1 such extremes as these are    
soon seen to be avoidable1 and in spite 
of all controversial exaggerations the  
expounders of the opposite views do     
contrive to retain a large area of      

speculation in common4 ,i do not propose
 here to examine in detail the          
restrictions by which this accommodation
 is brought about1 or the very real and 
important distinctions of method1 aim1  
tests1 and limits which in spite of all 
approach to agreement are still found to
 subsist4 ,to attempt this would be to  
open up rather too wide an enquiry to be
 suitable in a treatise on one          
subdivision only of the general science 
of ,inference4  #b4 ,one subdivision of 
this enquiry is however really forced   
upon our notice4 ,it does become        
important to consider the restrictions  
to which the ultra-material account of  
the province of ,logic has to be        
subjected1 because we shall thus have   
our attention drawn to an aspect of the 
subject which1 slight and fleeting as it
 is within the region of ,induction     
becomes very prominent and comparatively
 permanent in that of ,probability4     
,according to this ultra-material view1 
,inductive ,logic would generally be    

considered to have nothing to do with   
anything but objective facts3 its duty  
is to start from facts and to confine   
itself to such methods as will yield    
nothing but facts4 ,what is doubtful it 
either establishes or it lets alone for 
the present1 what is unattainable it    
rejects1 and in this way it proceeds to 
build up by slow accretion a vast fabric
 of certain knowledge4  ,but of course  
all this is supposed to be done by human
 minds1 and therefore if we enquire     
whether notions or concepts1--call them 
what we will1--have no place in such a  
scheme it must necessarily be admitted  
that they have some place4 ,the facts   
which form our starting point must be   
grasped by an intelligent being before  
inference can be built upon them2 and   
the 'facts' which form the conclusion   
have often1 at any rate for some time1  
no place anywhere else than in the mind 
of man4 ,but no one can read ,mill's    
treatise1 for instance1 without noticing
 how slight is his reference to this    

aspect of the question4 ,he remarks1 in 
almost contemptuous indifference1 that  
the man who digs must of course have a  
notion of the ground he digs and of the 
spade he puts into it1 but he evidently 
considers that these 'notions' need not 
much more occupy the attention of the   
speculative logician1 in so far as his  
mere inferences are concerned1 than they
 occupy that of the husbandman4  #c4 ,it
 must be admitted that there is some    
warrant for this omission of all        
reference to the subjective side of     
inference so long as we are dealing with
 ,inductive ,logic4 ,the inductive      
discoverer is of course in a very       
different position4 ,if he is worthy of 
the name his mind at every moment will  
be teeming with notions which he would  
be as far as any one from calling facts3
 he is busy making them such to the best
 of his power4 ,but the logician who    
follows in his steps1 and whose business
 it is to explain and justify what his  
leader has discovered1 is rather apt to 

overlook this mental or uncertain stage4
 ,what he mostly deals in are the       
'complete inductions' and 'well-grounded
 generalizations' and so forth1 or the  
exploded errors which contradict them3  
the prisoners and the corpses           
respectively1 which the real discoverer 
leaves on the field behind him whilst he
 presses on to complete his victory4    
,the whole method of                    
science1--expository as contrasted with 
militant1--is to emphasize the          
distinction between fact and non-fact1  
and to treat of little else but these   
two4 ,in other words a treatise on      
,inductive ,logic can be written without
 any occasion being found to define what
 is meant by a notion or concept1 or    
even to employ such terms4  #d4 ,and    
yet1 when we come to look more closely1 
signs may be detected even within the   
field of ,inductive ,logic1 of an       
occasional breaking down of the sharp   
distinction in question2 we may meet now
 and then with entities 7to use the     

widest term attainable7 in reference to 
which it would be hard to say that they 
are either facts or conceptions4 ,for   
instance1 ,inductive ,logic has often   
occasion to make use of ,hypotheses3 to 
which of the above two classes are these
 to be referred8 ,they do not seem in   
strictness to belong to either2 nor are 
they1 as will presently be pointed out1 
by any means a solitary instance of the 
kind4  ,it is true that within the      
province of ,inductive ,logic these     
hypotheses do not give much trouble on  
this score4 ,however vague may be the   
form in which they first present        
themselves to the philosopher's mind1   
they have not much business to come     
before us in our capacity of logicians  
until they are well on their way1 so to 
say1 towards becoming facts3 until they 
are beginning to harden into that firm  
tangible shape in which they will       
eventually appear4 ,we generally have   
some such recommendations given to us as
 that our hypotheses shall be           

well-grounded and reasonable4 ,this     
seems only another way of telling us    
that however freely the philosopher may 
make his guesses in the privacy of his  
own study1 he had better not bring them 
out into public until they can with fair
 propriety be termed facts1 even though 
the name be given with some             
qualification1 as by terming them       
'probable facts4' ,the reason1          
therefore1 why we do not take much      
account of this intermediate state in   
the hypothesis1 when we are dealing with
 the inductive processes1 is that here  
at any rate it plays only a temporary   
part2 its appearance in that guise is   
but very fugitive4 ,if the hypothesis be
 a sound one1 it will soon take its     
place as an admitted fact2 if not1 it   
will soon be rejected altogether4 ,its  
state as a hypothesis is not a normal   
one1 and therefore we have not much     
occasion to scrutinize its              
characteristics4 ,in so saying1 it must 
of course be understood that we are     

speaking as inductive logicians2 the    
philosopher in his workshop ought1 as   
already remarked1 to be familiar enough 
with the hypothesis in every stage of   
its existence from its origin2 but the  
logician's duty is different1 dealing as
 he does with proof rather than with the
 processes of original investigation and
 discovery4  ,we might indeed even go   
further1 and say that in many cases the 
hypothesis does not present itself to   
the reader1 that is to the recipient of 
the knowledge1 until it has ceased to   
deserve that name at all4 ,it may be    
first suggested to him along with the   
proof which establishes it1 he not      
having had occasion to think of it      
before4 ,it thus comes at a single step 
out of the obscurity of the unknown into
 the full possession of its rights as a 
fact1 skipping practically the          
intermediate or hypothetical stage      
altogether4 ,the original investigator  
himself may have long pondered over it1 
and kept it present to his mind1 in this

 its dubious stage1 but finally have    
given it to the world with that amount  
of evidence which raises it at once in  
the minds of others to the level of     
commonly accepted facts4  ,still this   
doubtful stage exists in every          
hypothesis1 though for logical purposes1
 and to most minds1 it exists in a very 
fugitive way only4 ,when attention has  
been directed to it1 it may be also     
detected elsewhere in ,logic4 ,take the 
case1 for instance1 of the reference of 
names4 ,mill gives the examples of the  
sun1 and a battle1 as distinguished from
 the ideas of them which we1 or         
children1 may entertain4 ,here the      
distinction is plain and obvious enough4
 ,but if1 on the other hand1 we take the
 case of things whose existence is      
doubtful or disputed1 the difficulty    
above mentioned begins to show itself4  
,the case of merely extinct things1 or  
such as have not yet come into          
existence1 offers indeed no trouble1    
since of course actually present        

existence is not necessary to constitute
 a fact4 ,the usual distinction may even
 be retained also in the case of        
mythical existences4 ,centaur and       
,griffin have as universally recognised 
a significance amongst the poets1       
painters1 and heralds as lion and       
leopard have4 ,hence we may claim1 even 
here1 that our conceptions shall be     
'truthful1' 'consistent with fact1' and 
so on1 by which we mean that they are to
 be in accordance with universal        
convention upon such subjects4          
,necessary and universal accordance is  
sometimes claimed to be all that is     
meant by 'objective1' and since         
universal accordance is attainable in   
the case of the notoriously fictitious1 
our fundamental distinction between fact
 and conception1 and our determination  
that our terms shall refer to what is   
objective rather than to what is        
subjective1 may with some degree of     
strain be still conceived to be tenable 
even here4  #e4 ,but when we come to the

 case of disputed phenomena the         
difficulty re-emerges4 ,a supposed      
planet or new mineral1 a doubtful fact  
in history1 a disputed theological      
doctrine1 are but a few examples out of 
many that might be offered4 ,what some  
persons strenuously assert1 others as   
strenuously deny1 and whatever hope     
there may be of speedy agreement in the 
case of physical phenomena1 experience  
shows that there is not much prospect of
 this in the case of those which are    
moral and historical1 to say nothing of 
theological4 ,so long as those who are  
in agreement confine their intercourse  
to themselves1 their 'facts' are        
accepted as such1 but as soon as they   
come to communicate with others all     
distinction between fact and conception 
is lost at once1 the 'facts' of one     
party being mere groundless             
'conceptions' to their opponents4 ,there
 is therefore1 ,i think1 in these cases 
a real difficulty in carrying out       
distinctly and consistently the account 

which the ,materialist logician offers  
as to the reference of names4 ,it need  
hardly be pointed out that what thus    
applies to names or terms applies       
equally to propositions in which        
particular or general statements are    
made involving names4  #f4 ,but when we 
step into ,probability1 and treat this  
from the same material or ,phenomenal   
point of view1 we can no longer neglect 
the question which is thus presented to 
us4 ,the difficulty cannot here be      
rejected1 as referring to what is merely
 temporary or occasional4 ,the          
intermediate condition between          
conjecture and fact1 so far from being  
temporary or occasional only1 is here   
normal4 ,it is just the condition which 
is specially characteristic of          
,probability4 ,hence it follows that    
however decidedly we may reject the     
,conceptualist theory we cannot         
altogether reject the use of            
,conceptualist language4 ,if we can     
prove that a given man will die next    

year1 or attain sufficiently near to    
proof to leave us practically certain on
 the point1 we may speak of his death as
 a 7future7 fact4 ,but if we merely     
contemplate his death as probable8 ,this
 is the sort of inference1 or substitute
 for inference1 with which ,probability 
is specially concerned4 ,we may1 if we  
so please1 speak of 'probable facts1'   
but if we examine the meaning of the    
words we may find them not merely       
obscure1 but self-contradictory4        
,doubtless there are facts here1 in the 
fullest sense of the term1 namely the   
statistics upon which our opinion is    
ultimately based1 for these are known   
and admitted by all who have looked into
 the matter4 ,the same language may also
 be applied to that extension of these  
statistics by induction which is        
involved in the assertion that similar  
statistics will be found to prevail     
elsewhere1 for these also may rightfully
 claim universal acceptance4 ,but these 
statements1 as was abundantly shown in  

the earlier chapters1 stand on a very   
different footing from a statement      
concerning the individual event2 the    
establishment and discussion of the     
former belong by rights to ,induction1  
and only the latter to ,probability4    
#g4 ,it is true that for want of        
appropriate terms to express such things
 we are often induced1 indeed compelled1
 to apply the same name of 'facts' to   
such individual contingencies4 ,we      
should not1 for instance1 hesitate to   
speak of the fact of the man dying being
 probable1 possible1 unlikely1 or       
whatever it might be4 ,but ,i cannot    
help regarding such expressions as a    
strictly incorrect usage arising out of 
a deficiency of appropriate technical   
terms4 ,it is doubtless certain that one
 or other of the two alternatives must  
happen1 but this alternative certainty  
is not the subject of our contemplation2
 what we have before us is the single   
alternative1 which is notoriously       
uncertain4 ,it is this1 and this only1  

which is at present under notice1 and   
whose occurrence has to be estimated4   
,we have surely no right to dignify this
 with the name of a fact1 under any     
qualifications1 when the opposite       
alternative has claims1 not perhaps     
actually equal to1 but at any rate not  
much inferior to its own4 ,such         
language1 as already remarked1 may be   
quite right in ,inductive logic1 where  
we are only concerned with conjectures  
of such a high degree of likelihood that
 their non-occurrence need not be taken 
into practical account1 and which are   
moreover regarded as merely temporary4  
,but in ,probability the conjecture may 
have any degree of likelihood about it2 
it may be just as likely as the other   
alternative1 nay it may be much less    
likely4 ,in these latter cases1 for     
instance1 if the chances are very much  
against the man's death1 it is surely an
 abuse of language to speak of the      
'fact' of his dying1 even though we     
qualify it by declaring it to be highly 

improbable4 ,the subject-matter         
essential to ,probability being the     
uncertain1 we can never with propriety  
employ upon it language which in its    
original and correct application is only
 appropriate to what is actually or     
approximately certain4  #h4 ,it should  
be remembered also that this state of   
things1 thus characteristic of          
,probability1 is permanent there4 ,so   
long as they remain under the treatment 
of that science our conjectures1 or     
whatever we like to call them1 never    
develop into facts4 ,i calculate1 for   
instance1 the chance that a die will    
give ace1 or that a man will live beyond
 a certain age4 ,such an approximation  
to knowledge as is thus acquired is as  
much as we can ever afterwards hope to  
get1 unless we resort to other methods  
of enquiry4 ,we do not1 as in           
,induction1 feel ourselves on the brink 
of some experimental or other proof     
which at any moment may raise it into   
certainty4 ,it is nothing but a         

conjecture of a certain degree of       
strength1 and such it will ever remain1 
so long as ,probability is left to deal 
with it4 ,if anything more is ever to be
 made out of it we must appeal to direct
 experience1 or to some kind of         
inductive proof4 ,as we have so often   
said1 individual facts can never be     
determined here1 but merely ultimate    
tendencies and averages of many events4 
,i may1 indeed1 by a second appeal to   
,probability improve the character of my
 conjecture1 through being able to refer
 it to a narrower and better class of   
statistics2 but its essential nature    
remains throughout what it was4  ,it    
appears to me therefore that the account
 of the ,materialist view of logic      
indicated at the commencement of this   
chapter1 though substantially sound1    
needs some slight reconsideration and   
re-statement4 ,it answers admirably so  
far as ordinary ,induction is concerned1
 but needs some revision if it is to be 
equally applicable to that wider view of

 the nature and processes of acquiring  
knowledge wherein the science of logic  
is considered to involve ,probability   
also as well as ,induction4  #i4        
,briefly then it is this4 ,we regard the
 scientific thinker1 whether he be the  
original investigator who discovers1 or 
the logician who analyses and describes 
the proofs that may be offered1 as      
surrounded by a world of objective      
phenomena extending indefinitely both   
ways in time1 and in every direction in 
space4 ,most of them are1 and always    
will remain1 unknown4 ,if we speak of   
them as facts we mean that they are     
potential objects of human knowledge1   
that under appropriate circumstances men
 could come to determinate and final    
agreement about them4 ,the scientific or
 material logician has to superintend   
the process of converting as much as    
possible of these unknown phenomena into
 what are known1 of aggregating them1 as
 we have said above1 about the nucleus  
of certain data which experience and    

observation had to start with4 ,in so   
doing his principal resources are the   
,methods of ,induction1 of which        
something has been said in a former     
chapter2 another resource is found in   
the ,theory of ,probability1 and another
 in ,deduction4  ,now1 however such     
language may be objected to as savouring
 of ,conceptualism1 ,i can see no better
 compendious way of describing these    
processes than by saying that we are    
engaged in getting at conceptions of    
these external phenomena1 and as far as 
possible converting these conceptions   
into facts4 ,what is the natural history
 of 'facts' if we trace them back to    
their origin8 ,they first come into     
being as mere guesses or conjectures1 as
 contemplated possibilities whose       
correspondence with reality is either   
altogether disbelieved or regarded as   
entirely doubtful4 ,in this stage1 of   
course1 their contrast with facts is    
sharp enough4 ,how they arise it does   
not belong to ,logic but to ,psychology 

to say4 ,logic indeed has little or     
nothing to do with them whilst they are 
in this form4 ,everyone is busy all his 
life in entertaining such guesses upon  
various subjects1 the superiority of the
 philosopher over the common man being  
mainly found in the quality of his      
guesses1 and in the skill and           
persistence with which he sifts and     
examines them4 ,in the next stage they  
mostly go by the name of theories or    
hypotheses1 when they are comprehensive 
in their scope1 or are in any way on a  
scale of grandeur and importance3 when  
however they are of a trivial kind1 or  
refer to details1 we really have no     
distinctive or appropriate name for     
them1 and must be content therefore to  
call them 'conceptions4' ,through this  
stage they flit with great rapidity in  
,inductive ,logic2 often the logician   
keeps them back until their evidence is 
so strong that they come before the     
world at once in the full dignity of    
facts4 ,hence1 as already remarked1 this

 stage of their career is not much dwelt
 upon in ,logic4 ,but the whole business
 of ,probability is to discuss and      
estimate them at this point4            
,consequently1 so far as this science is
 concerned1 the explanation of the      
,material logician as to the reference  
of names and propositions has to be     
modified4  #aj4 ,the best way therefore 
of describing our position in           
,probability is as follows3--,we are    
entertaining a conception of some event1
 past1 present1 or future4 ,from the    
nature of the case this conception is   
all that can be actually entertained by 
the mind4 ,in its present condition it  
would be incorrect to call it a fact1   
though we would willingly1 if we could1 
convert it into such by making certain  
of it one way or the other4 ,but so long
 as our conclusions are to be effected  
by considerations of ,probability only1 
we cannot do this4 ,the utmost we can do
 is to estimate or evaluate it4 ,the    
whole function of ,probability is to    

give rules for so doing4 ,by means of   
reference to statistics or by direct    
deduction1 as the case may be1 we are   
enabled to say how much this conception 
is to be believed1 that is in what      
proportion out of the total number of   
cases we shall be right in so doing4    
,our position1 therefore1 in these cases
 seems distinctly that of entertaining a
 conception1 and the process of         
inference is that of ascertaining to    
what extent we are justified in adding  
this conception to the already received 
body of truth and fact4  ,so long1 then1
 as we are confined to ,probability     
these conceptions remain such4 ,but if  
we turn to ,induction we see that they  
are meant to go a step further4 ,their  
final stage is not reached until they   
have ripened into facts1 and so taken   
their place amongst uncontested truths4 
,this is their final destination in     
,logic1 and our task is not accomplished
 until they have reached it4  #aa4 ,such
 language as this in which we speak of  

our position in ,probability as being   
that of entertaining a conception1 and  
being occupied in determining what      
degree of belief is to be assigned to   
it1 may savour of ,conceptualism1 but is
 in spirit perfectly different from it4 
,our ultimate reference is always to    
facts4 ,we start from them as our data1 
and reach them again eventually in our  
results whenever it is possible4 ,in    
,probability1 of course1 we cannot do   
this in the individual result1 but even 
then 7as shown in ,ch4 ,,vi47 we always 
justify our conclusions by appeal to    
facts1 viz4 to what happens in the long 
run4  ,the discussion which has been    
thus given to this part of the subject  
may seem somewhat tedious1 but it was so
 obviously forced upon us when          
considering the distinction between the 
two main views of ,logic1 that it was   
impossible to pass it over without fear 
of misapprehension and confusion4       
,moreover1 as will be seen in the course
 of the next chapter1 several important 

conclusions could not have been properly
 explained and justified without first  
taking pains to make this part of our   
ground perfectly plain and satisfactory4
       ,,chapter ,,xii4   ,,consequences
 ,,of ,,the ,,foregoing ,,distinctions4 
#a4 ,we are now in a position to explain
 and justify some important conclusions 
which1 if not direct consequences of the
 distinctions laid down in the last     
chapter1 will at any rate be more       
readily appreciated and accepted after  
that exposition4  ,in the first place1  
it will be seen that in ,probability    
time has nothing to do with the         
question2 in other words1 it does not   
matter whether the event1 whose         
probability we are discussing1 be past1 
present1 or future4 ,the problem before 
us1 in its simplest form1 is            
this3--,statistics 7extended by         
,induction1 and practically often gained
 by ,deduction7 inform us that a certain
 event has happened1 does happen1 or    
will happen1 in a certain way in a      

certain proportion of cases4 ,we form a 
conception of that event1 and regard it 
as possible2 but we want to do more2 we 
want to know how much we ought to expect
 it 7under the explanations given in a  
former chapter about quantity of        
belief74 ,there is therefore a sort of  
relative futurity about the event1      
inasmuch as our knowledge of the fact1  
and therefore our justification or      
otherwise of the correctness of our     
surmise1 almost necessarily comes after 
the surmise was formed2 but the futurity
 is only relative4 ,the evidence by     
which the question is to be settled may 
not be forthcoming yet1 or we may have  
it by us but only consult it afterwards4
 ,it is from the fact of the futurity   
being1 as above described1 only         
relative1 that ,i have preferred to     
speak of the conception of the event    
rather than of the anticipation of it4  
,the latter term1 which in some respects
 would have seemed more intelligible and
 appropriate1 is open to the objection1 

that it does rather1 in popular         
estimation1 convey the notion of an     
absolute as opposed to a relative       
futurity4  #b4 ,for example2 a die is   
thrown4 ,once in six times it gives ace2
 if therefore we assume1 without        
examination1 that the throw is ace1 we  
shall be right once in six times4 ,in so
 doing we may1 according to the usual   
plan1 go forwards in time2 that is1 form
 our opinion about the throw beforehand1
 when no one can tell what it will be4  
,or we might go backwards2 that is1 form
 an opinion about dice that had been    
cast on some occasion in time past1 and 
then correct our opinion by the         
testimony of some one who had been a    
witness of the throws4 ,in either case  
the mental operation is precisely the   
same2 an opinion formed merely on       
statistical grounds is afterwards       
corrected by specific evidence4 ,the    
opinion may have been formed upon a     
past1 present1 or future event2 the     
evidence which corrects it afterwards   

may be our own eyesight1 or the         
testimony of others1 or any kind of     
inference2 by the evidence is merely    
meant such subsequent examination of the
 case as is assumed to set the matter at
 rest4 ,it is quite possible1 of course1
 that this specific evidence should     
never be forthcoming2 the conception in 
that case remains as a conception1 and  
never obtains that degree of conviction 
which qualifies it to be regarded as a  
'fact4' ,this is clearly the case with  
all past throws of dice the results of  
which do not happen to have been        
recorded4  ,in discussing games of      
chance there are obvious advantages in  
confining ourselves to what is really1  
as well as relatively1 future1 for in   
that case direct information concerning 
the contemplated result being           
impossible1 all persons are on precisely
 the same footing of comparative        
ignorance1 and must form their opinion  
entirely from the known or inferred     
frequency of occurrence of the event in 

question4 ,on the other hand1 if the    
event be passed1 there is almost always 
evidence of some kind and of some value1
 however slight1 to inform us what the  
event really was2 if this evidence is   
not actually at hand1 we can generally1 
by waiting a little1 obtain something   
that shall be at least of some use to us
 in forming our opinion4 ,practically   
therefore we generally confine          
ourselves1 in anticipations of this     
kind1 to what is really future1 and so  
in popular estimation futurity becomes  
indissolubly associated with            
probability4  #c4 ,there is however an  
error closely connected with the above  
view of the subject1 or at least an     
inaccuracy of expression which is       
constantly liable to lead to error1     
which has found wide acceptance1 and has
 been sanctioned by writers of the      
greatest authority4 ,for instance1 both 
,butler1 in his ,analogy1 and ,mill1    
have drawn attention1 under one form of 
expression or another1 to the           

distinction between improbability before
 the event and improbability after the  
event1 which they consider to be        
perfectly different things4 ,that this  
phraseology indicates a distinction of  
importance cannot be denied1 but it     
seems to me that the language in which  
it is often expressed requires to be    
amended4  ,butler's remarks on this     
subject occur in his ,analogy1 in the   
chapter on miracles4 ,admitting that    
there is a strong presumption against   
miracles 7his equivalent for the        
ordinary expression1 an 'improbability  
before the event'7 he strives to obtain 
assent for them by showing that other   
events1 which also have a strong        
presumption against them1 are received  
on what is in reality very slight       
evidence4 ,he says1 ,there is a very    
strong presumption against common       
speculative truths1 and against the most
 ordinary facts1 before the proof of    
them2 which yet is overcome by almost   
any proof4 ,there is a presumption of   

millions to one against the story of    
,csar1 or of any other man4 ,for1       
suppose a number of common facts so and 
so circumstanced1 of which one had no   
kind of proof1 should happen to come    
into one's thoughts1 every one would    
without any possible doubt conclude them
 to be false4 ,and the like may be said 
of a single common fact4  #d4 ,these    
remarks have been a good deal           
criticized1 and they certainly seem to  
me misleading and obscure in their      
reference4 ,if one may judge by the     
context1 and by another passage in which
 the same argument is afterwards        
referred to1#a it would certainly appear
 that ,butler drew no distinction       
between miraculous accounts1 and other  
accounts which1 to use any of the       
various expressions in common use1 are  
unlikely or improbable or have a        
presumption against them2 and concluded 
that since some of the latter were      
instantly accepted upon somewhat        
mediocre testimony1 it was altogether   

irrational to reject the former when    
similarly or better supported4#b ,this  
subject will come again under our       
notice1 and demand fuller discussion1 in
 the chapter on the ,credibility of     
extraordinary stories4 ,it will suffice 
here to remark that1 however            
satisfactory such a view of the matter  
might be to some theologians1 no        
antagonist of miracles would for a      
moment accept it4 ,he would naturally   
object that1 instead of the miraculous  
element being 7as ,butler considers7 a  
small additional presumption against the
 narrative1 it involved the events in a 
totally distinct class of incredibility2
 that it multiplied1 rather than merely 
added to1 the difficulties and          
objections in the way of accepting the  
account4  ,mill's remarks 7,logic1 ,bk4 
,,iii4 ch4 ,,xxv4 #d7 are of a different
 character4 ,discussing the grounds of  
disbelief he speaks of people making the
 mistake of overlooking the distinction 
between 7what may be called7            

improbability before the fact1 and      
improbability after it1 two different   
properties1 the latter of which is      
always a ground of disbelief1 the former
 not always4 ,he instances the throwing 
of a die4 ,it is improbable beforehand  
that it should turn up ace1 and yet     
afterwards1 there is no reason for      
disbelieving it if any credible witness 
asserts it4 ,so again1 the chances are  
greatly against ,a4 ,b4's dying1 yet if 
any one tells us that he died yesterday 
we believe it4  #e4 ,that there is some 
difficulty about such problems as these 
must be admitted4 ,the fact that so many
 people find them a source of           
perplexity1 and that such various       
explanations are offered to solve the   
perplexity1 are a sufficient proof of   
this4#c ,the considerations of the last 
chapter1 however1 over-technical and    
even scholastic as some of the language 
in which it was expressed may have      
seemed to the reader1 will ,i hope guide
 us to a more satisfactory way of       

regarding the matter4  ,when we speak of
 an improbable event1 it must be        
remembered that1 objectively considered1
 an event can only be more or less rare2
 the extreme degree of rarity being of  
course that in which the event does not 
occur at all4 ,now1 as was shown in the 
last chapter1 our position1 when forming
 judgments of the time in question1 is  
that of entertaining a conception or    
conjecture 7call it what we will71 and  
assigning a certain weight of           
trustworthiness to it4 ,the real        
distinction1 therefore1 between the two 
classes of examples respectively1 which 
are adduced both by ,butler and by      
,mill1 consists in the way in which     
those conceptions are obtained2 they    
being obtained in one case by the       
process of guessing1 and in the other by
 that of giving heed to the reports of  
witnesses4  #f4 ,take ,butler's instance
 first4 ,in the 'presumption before the 
proof' we have represented to us a man  
thinking of the story of ,csar1 that is1

 making a guess about certain historical
 events without any definite grounds for
 it1 and then speculating as to what    
value is to be attached to the          
probability of its truth4 ,such a guess 
is of course1 as he says1 concluded to  
be false4 ,but what does he understand  
by the 'presumption after the proof'8   
,that a story not adopted at random1 but
 actually suggested and supported by    
witnesses1 should be true4 ,the latter  
might be accepted1 whilst the former    
would undoubtedly be rejected2 but all  
that this proves1 or rather illustrates1
 is that the testimony of almost any    
witness is in most cases vastly better  
than a mere guess4#d ,we may in both    
cases alike speak of 'the event' if we  
will2 in fact1 as was admitted in the   
last chapter1 common language will not  
readily lend itself to any other way of 
speaking4 ,but it should be clearly     
understood that1 phrase it how we will1 
what is really present to the man's     
mind1 and what is to have its probable  

value assigned to it1 is the conception 
of an event1 in the sense in which that 
expression has already been explained4  
,and surely no two conceptions can have 
a much more important distinction put   
between them than that which is involved
 in supposing one to rest on a mere     
guess1 and the other on the report of a 
witness4 ,precisely the same remarks    
apply to the example given by ,mill4    
,before ,a4 ,b4's death our opinion upon
 the subject was nothing but a guess of 
our own founded upon life statistics2   
after his death it was founded upon the 
evidence of some one who presumably had 
tolerable opportunities of knowing what 
the facts really were4  #g4 ,that the   
distinction before us has no essential  
connection whatever with time is indeed 
obvious on a moment's consideration4    
,conceive for a moment that some one had
 opportunities of knowing whether ,a4   
,b4 would die or not4 ,if he told us    
that ,a4 ,b4 would die to-morrow1 we    
should in that case be just as ready to 

believe him as when he tells us that ,a4
 ,b4 has died4 ,if we continued to feel 
any doubt about the statement 7supposing
 always that we had full confidence     
about his veracity in matters into which
 he had duly enquired71 it would be     
because we thought that in his case1 as 
in ours1 it was equivalent to a guess1  
and nothing more4 ,so with the event    
when past1 the fact of its being past   
makes no difference whatever2 until the 
credible witness informs us of what he  
knows to have occurred1 we should doubt 
it if it happened to come into our      
minds1 just as much as if it were       
future4  ,the distinction1 therefore1   
between probability before the event and
 probability after the event seems to   
resolve itself simply into this2--before
 the event we often have no better means
 of information than to appeal to       
statistics in some form or other1 and so
 to guess amongst the various possible  
alternatives2 after the event the guess 
may most commonly be improved or        

superseded by appeal to specific        
evidence1 in the shape of testimony or  
observation4 ,hence1 naturally1 our     
estimate in the latter case is commonly 
of much more value4 ,but if these       
characteristics were anyhow inverted2   
if1 that is1 we were to confine         
ourselves to guessing about the past1   
and if we could find any additional     
evidence about the future1 the          
respective values of the different      
estimates would also be inverted4 ,the  
difference between these values has no  
necessary connection with time1 but     
depends entirely upon the different     
grounds upon which our conception or    
conjecture about the event in question  
rests4  #h4 ,the following imaginary    
example will serve to bring out the     
point indicated above4 ,conceive a      
people with very short memories1 and who
 preserved no kind of record to         
perpetuate their hold upon the events   
which happened amongst them4#e ,the     
whole region of the past would then be  

to them what much of the future is to   
us2 viz4 a region of guesses and        
conjectures1 one in reference to which  
they could only judge upon general      
considerations of probability1 rather   
than by direct and specific evidence4   
,but conceive also that they had amongst
 them a race of prophets who could      
succeed in foretelling the future with  
as near an approach to accuracy and     
trustworthiness as our various          
histories1 and biographies1 and         
recollections1 can attain in respect to 
the past4 ,the present and usual        
functions of direct evidence or         
testimony1 and of probability1 would    
then be simply inverted2 and so in      
consequence would the present accidental
 characteristics of improbability before
 and after the event4 ,it would then be 
the latter which would by comparison be 
regarded as 'not always a ground of     
disbelief1' whereas in the case of the  
former we should then have it maintained
 that it always was so4  #i4 ,the origin

 of the mistake just discussed is worth 
enquiring into4 ,i take it to be as     
follows4 ,it is often the case1 as above
 remarked1 when we are speculating about
 a future event1 and almost always the  
case when that future event is taken    
from a game of chance1 that all persons 
are in precisely the same condition of  
ignorance in respect to it4 ,the limit  
of available information is confined to 
statistics1 and amounts to the knowledge
 that the unknown event must assume some
 one of various alternative forms4 ,the 
conjecture1 therefore1 of any one man   
about it is as valuable as that of any  
other4 ,but in regard to the past the   
case is very different4 ,here we are not
 in the habit of relying upon           
statistical information4 ,hence the     
conjectures of different men are of     
extremely different values2 in the case 
of many they amount to what we call     
positive knowledge4 ,this puts a broad  
distinction1 in popular estimation1     
between what may be called the objective

 certainty of the past and of the       
future1 a distinction1 however1 which   
from the standing-point of a science of 
inference ought to have no existence4   
,in consequence of this1 when we apply  
to the past and the future respectively 
the somewhat ambiguous expression 'the  
chance of the event1' it commonly comes 
to bear very different significations4  
,applied to the future it bears its     
proper meaning1 namely1 the value to be 
assigned to a conjecture upon           
statistical grounds4 ,it does so1       
because in this case hardly any one has 
more to judge by than such conjectures4 
,but applied to the past it shifts its  
meaning1 owing to the fact that whereas 
some men have conjectures only1 others  
have positive knowledge4 ,by the chance 
of the event is now often meant1 not the
 value to be assigned to a conjecture   
founded on statistics1 but to such a    
conjecture derived from and enforced by 
any body else's conjecture1 that is by  
his knowledge and his testimony4  #aj4  

,there is a class of cases in apparent  
opposition to some of the statements in 
this chapter1 but which will be found1  
when examined closely1 decidedly to     
confirm them4 ,i am walking1 say1 in a  
remote part of the country1 and suddenly
 meet with a friend4 ,at this ,i am     
naturally surprised4 ,yet if the view be
 correct that we cannot properly speak  
about events in themselves being        
probable or improbable1 but only say    
this of our conjectures about them1 how 
do we explain this8 ,we had formed no   
conjecture beforehand1 for we were not  
thinking about anything of the kind1 but
 yet few would fail to feel surprise at 
such an incident4  ,the reply might     
fairly be made that we had formed such  
anticipations tacitly4 ,on any such     
occasion every one unconsciously divides
 things into those which are known to   
him and those which are not4 ,during a  
considerable previous period a countless
 number of persons had met us1 and all  
fallen into the list of the unknown to  

us4 ,there was nothing to remind us of  
having formed the anticipation or       
distinction at all1 until it was        
suddenly called out into vivid          
consciousness by the exceptional event4 
,the words which we should instinctively
 use in our surprise seem to show       
this3--',who would have thought of      
seeing you here8' viz4 ,who would have  
given any weight to the latent thought  
if it had been called out into          
consciousness beforehand8 ,we put our   
words into the past tense1 showing that 
we have had the distinction lurking in  
our minds all the time4 ,we always have 
a multitude of such ready-made classes  
of events in our minds1 and when a thing
 happens to fall into one of those      
classes which are very small we cannot  
help noticing the fact4  ,or suppose ,i 
am one of a regiment into which a shot  
flies1 and it strikes me1 and me only4  
,at this ,i am surprised1 and why8 ,our 
common language will guide us to the    
reason4 ',how strange that it should    

just have hit me of all men6' ,we are   
thinking of the very natural two-fold   
division of mankind into1 ourselves1 and
 everybody else2 our surprise is again1 
as it were1 retrospective1 and in       
reference to this division4 ,no         
anticipation was distinctly formed1     
because we did not think beforehand of  
the event1 but the event1 when it has   
happened1 is at once assigned to its    
appropriate class4  #aa4 ,this view is  
confirmed by the following              
considerations4 ,tell the story to a    
friend1 and he will be a little         
surprised1 but less so than we were1 his
 division in this particular case       
being1--his friends 7of whom we are but 
one71 and the rest of mankind4 ,it is   
not a necessary division1 but it is the 
one which will be most likely suggested 
to him4  ,tell it again to a perfect    
stranger1 and his division being        
different 7viz4 we falling into the     
majority7 we shall fail to make him     
perceive that there is anything at all  

remarkable in the event4  ,it is not of 
course attempted in these remarks to    
justify our surprise in every case in   
which it exists4 ,different persons     
might be differently affected in the    
cases supposed1 and the examples are    
therefore given mainly for illustration4
 ,still on principles already discussed 
7,ch4 ,,vi4 #cb7 we might expect to find
 something like a general justification 
of the amount of surprise4  #ab4 ,the   
answer commonly given in these cases is 
confined to attempting to show that the 
surprise should not arise1 rather than  
to explaining how it does arise4 ,it    
takes the following form1--',you have no
 right to be surprised1 for nothing     
remarkable has really occurred4 ,if this
 particular thing had not happened      
something equally improbable must4 ,if  
the shot had not hit you or your friend1
 it must have hit some one else who was 
priori as unlikely to be hit4'  ,for one
 thing this answer does not explain the 
fact that almost every one is surprised 

in such cases1 and surprised somewhat in
 the different proportions mentioned    
above4 ,moreover it has the inherent    
unsatisfactoriness of admitting that    
something improbable has really         
happened1 but getting over the          
difficulty by saying that all the other 
alternatives were equally improbable4 ,a
 natural inference from this is that    
there is a class of things1 in          
themselves really improbable1 which can 
yet be established upon very slight     
evidence4 ,butler accepted this         
inference1 and worked it out to the     
strange conclusion given above4 ,mill   
attempts to avoid it by the             
consideration of the very different     
values to be assigned to improbability  
before and after the event4 ,some       
further discussion of this point will be
 found in the chapter on ,fallacies1 and
 in that on the ,credibility of         
,extraordinary ,stories4  #ac4 ,in      
connection with the subject at present  
under discussion we will now take notice

 of a distinction which we shall often  
find insisted on in works on            
,probability1 but to which apparently   
needless importance has been attached4  
,it is frequently said that probability 
is relative1 in the sense that it has a 
different value to different persons    
according to their respective           
information upon the subject in         
question4 ,for example1 two persons1 ,a 
and ,b1 are going to draw a ball from a 
bag containing #d balls3 ,a knows that  
the balls are black and white1 but does 
not know more2 ,b knows that three are  
black and one white4 ,it would be said  
that the probability of a white ball to 
,a is #a_/b1 and to ,b #a_/d4  ,when    
however we regard the subject from the  
material standing point1 there really   
does not seem to me much more in this   
than the principle1 equally true in     
every other science1 that our inferences
 will vary according to the data we     
assume4 ,we might on logical grounds    
with almost equal propriety speak of the

 area of a field or the height of a     
mountain being relative1 and therefore  
having one value to one person and      
another to another4 ,the real meaning of
 the example cited above is this3 ,a    
supposes that he is choosing white at   
random out of a series which in the long
 run would give white and black equally 
often2 ,b supposes that he is choosing  
white out of a series which in the long 
run would give three black to one white4
 ,by the application1 therefore1 of a   
precisely similar rule they draw        
different conclusions2 but so they would
 under the same circumstances in any    
other science4 ,if two men are measuring
 the height of a mountain1 and one      
supposes his base to be #ajjj feet1     
whilst the other takes it to be #ajja1  
they would of course form different     
opinions about the height4 ,the science 
of mensuration is not supposed to have  
anything to do with the truth of the    
data1 but assumes them to have been     
correctly taken2 why should not this be 

equally the case with ,probability1     
making of course due allowance for the  
peculiar character of the data with     
which it is concerned8  #ad4 ,this view 
of the relativeness of probability is   
connected1 as it appears to me1 with the
 subjective view of the science1 and is 
indeed characteristic of it4 ,it seems a
 fair illustration of the weak side of  
that view1 that it should lead us to lay
 any stress on such an expression4 ,as  
was fully explained in the last chapter1
 in proportion as we work out the       
,conceptualist principle we are led away
 from the fundamental question of the   
material logic1 viz4 ,is our belief     
actually correct1 or not8 and1 if the   
former1 to what extent and degree is it 
correct8 ,we are directed rather to ask1
 ,what belief does any one as a matter  
of fact hold8 ,and1 since the belief    
thus entertained naturally varies       
according to the circumstances and other
 sources of information of the person in
 question1 its relativeness comes to be 

admitted as inevitable1 or at least it  
is not to be wondered at if such should 
be the case4  ,on our view of           
,probability1 therefore1 its            
'relativeness' in any given case is a   
misleading expression1 and it will be   
found much preferable to speak of the   
effect produced by variations in the    
nature and amount of the data which we  
have before us4 ,now it must be admitted
 that there are frequently cases in our 
science in which such variations are    
peculiarly likely to be found4 ,for     
instance1 ,i am expecting a friend who  
is a passenger in an ocean steamer4     
,there are a hundred passengers on      
board1 and the crew also numbers a      
hundred4 ,i read in the papers that one 
person was lost by falling overboard2 my
 anticipation that it was my friend who 
was lost is but small1 of course4 ,on   
turning to another paper1 ,i see that   
the man who was lost was a passenger1   
not one of the crew2 my slight anxiety  
is at once doubled4 ,but another account

 adds that it was an ,englishman1 and on
 that line at that season the ,english  
passengers are known to be few2 ,i at   
once begin to entertain decided fears4  
,and so on1 every trifling bit of       
information instantly affecting my      
expectations4  #ae4 ,now since it is    
peculiarly characteristic of            
,probability1 as distinguished from     
,induction1 to be thus at the mercy1 so 
to say1 of every little fact that may be
 floating about when we are in the act  
of forming our opinion1 what can be the 
harm 7it may be urged7 of expressing    
this state of things by terming our     
state of expectation relative8  ,there  
seem to me to be two objections4 ,in the
 first place1 as just mentioned1 we are 
induced to reject such an expression on 
grounds of consistency4 ,it is          
inconsistent with the general spirit and
 treatment of the subject hitherto      
adopted1 and tends to divorce           
,probability from ,inductive logic      
instead of regarding them as cognate    

sciences4 ,we are aiming at truth1 as   
far as that goal can be reached by our  
road1 and therefore we dislike to regard
 our conclusions as relative in any     
other sense than that in which truth    
itself may be said to be relative4  ,in 
the second place1 this condition of     
unstable assent1 this constant liability
 to have our judgment affected1 to any  
degree and at any moment1 by the        
accession of new knowledge1 though      
doubtless characteristic of             
,probability1 does not seem to me       
characteristic of it in its sounder and 
more legitimate applications4 ,it seems 
rather appropriate to a precipitate     
judgment formed in accordance with the  
rules1 than a strict example of their   
natural employment4 ,such precipitate   
judgments may occur in the case of      
ordinary deductive conclusions4 ,in the 
practical exigencies of life we are     
constantly in the habit of forming a    
hasty opinion with nearly full          
confidence1 at any rate temporarily1    

upon the strength of evidence which we  
must well know at the time cannot be    
final4 ,we wait a short time1 and       
something else turns up which induces us
 to alter our opinion1 perhaps to       
reverse it4 ,here our conclusions may   
have been perfectly sound under the     
given circumstances1 that is1 they may  
be such as every one else would have    
drawn who was bound to make up his mind 
upon the data before us1 and they are   
unquestionably 'relative' judgments in  
the sense now under discussion4 ,and    
yet1 ,i think1 every one would shrink   
from so terming them who wished         
systematically to carry out the view    
that ,logic was to be regarded as an    
organon of truth4  #af4 ,in the examples
 of ,probability which we have hitherto 
employed1 we have for the most part     
assumed that there was a certain body of
 statistics set before us on which our  
conclusion was to rest4 ,it was assumed1
 on the one hand1 that no direct        
specific evidence could be got1 so that 

the judgment was really to be one of    
,probability1 and to rest on these      
statistics2 in other words1 that nothing
 better than them was available for us4 
,but it was equally assumed1 on the     
other hand1 that these statistics were  
open to the observation of every one1 so
 that we need not have to put up with   
anything inferior to them in forming our
 opinion4 ,in other words1 we have been 
assuming that here1 as in the case of   
most other sciences1 those who have to  
draw a conclusion start from the same   
footing of opportunity and information4 
,this1 for instance1 clearly is or ought
 to be the case when we are concerned   
with games of chance2 ignorance or      
misapprehension of the common data is   
never contemplated there4 ,so with the  
statistics of life1 or other insurance3 
so long as our judgment is to be        
accurate 7after its fashion7 or         
justifiable1 the common tables of       
mortality are all that any one has to go
 by4  #ag4 ,it is true that in the case 

of a man's prospect of death we should  
each qualify our judgment by what we    
knew or reasonably supposed as to his   
health1 habits1 profession1 and so on1  
and should thus arrive at varying       
estimates4 ,but no one could justify his
 own estimate without appealing         
explicitly or implicitly to the         
statistical grounds on which he had     
relied1 and if these were not previously
 available to other persons1 he must now
 set them before their notice4 ,in other
 words1 the judgments we entertain1 here
 as elsewhere1 are only relative so long
 as we rest them on grounds peculiar to 
ourselves4 ,the process of              
justification1 which ,i consider to be  
essential to logic1 has a tendency to   
correct such individualities of         
judgment1 and to set all observers on   
the same basis as regards their data4   
,it is better therefore to regard the   
conclusions of ,probability as being    
absolute and objective1 in the same     
sense as1 though doubtless in a far less

 degree than1 they are in ,induction4   
,fully admitting that our conclusions   
will in many cases vary exceedingly from
 time to time by fresh accessions of    
knowledge1 it is preferable to regard   
such fluctuations of assent as partaking
 of the nature of precipitate judgments1
 founded on special statistics1 instead 
of depending only on those which are    
common to all observers4 ,in calling    
such judgments precipitate it is not    
implied that there is any blame in      
entertaining them1 but simply that1 for 
one reason or another1 we have been     
induced to form them without waiting for
 the possession of the full amount of   
evidence1 statistical or otherwise1     
which might ultimately be looked for4   
,this explanation will suit the facts   
equally well1 and is more consistent    
with the general philosophical position 
maintained in this work4   #a4 ,is it   
not self-evident that internal          
improbabilities of all kinds   weaken   
external proof8 ,doubtless1 but to what 

practical purpose can   this be alleged 
here1 when it has been proved before1   
that real   internal improbabilities1   
which rise even to moral certainty1 are 
overcome by the most ordinary testimony4
 ,part ,,ii4 ch4 ,,iii4  #b4 ,miracles  
must not be compared to common natural  
events2 or to   events which1 though    
uncommon1 are similar to what we daily  
experience2 but to the extraordinary    
phenomena of nature4 ,and then   the    
comparison will be between the          
presumption against miracles1 and   the 
presumption against such uncommon       
appearances1 suppose as   comets14444   
,part ,,ii4 ch4 ,,ii4  #c4 ,for         
instance1 ,sir ,j4 ,f4 ,stephen explains
 it by drawing a   distinction between  
chances and probabilities1 which he says
 that   ,butler has confused together2  
the objection that very ordinary   proof
 will overcome a presumption of millions
 to one is based upon a   confusion     
between probabilities and chances4 ,the 
probability of an   event is its        

capability of being proved4 ,its chance 
is the numerical   proportion between   
the number of possible cases--supposed  
to be   equally favourable--favourable  
to its occurrence2 and the number of    
possible cases unfavourable to its      
occurrence 7,general view of the        
,criminal ,law of ,england1 p4 #bee74   
,donkin1 again 7,phil4 ,magazine1       
,june1 #ahea71 employs the terms        
improbability and incredibility to      
mark the same distinction4  #d4 ,in the 
extreme case of the witness himself     
merely guessing1 or   being as          
untrustworthy as if he merely guessed1  
the two stories will   of course stand  
on precisely the same footing4 ,this    
case will be   noticed again in ,chapter
 ,,xvii4 ,it may be remarked that there 
are   several subtleties here which     
cannot be adequately noticed without    
some previous investigation into the    
question of the credibility of          
witnesses4  #e4 ,according to ,dante1   
something resembling this prevailed     

amongst the   occupants of the ,inferno4
 ,the cardinals and others whom he there
   meets are able to give information   
about many events which were   yet to   
happen upon earth1 but they had to ask  
it for many events   which actually had 
happened4     ,,chapter ,,xiii4   ,,on  
,,the ,,conception ,,and ,,treatment    
,,of ,,modality4  #a4 ,the reader who   
knows anything of the scholastic ,logic 
will have perceived before now that we  
have been touching in a variety of      
places upon that most thorny and        
repulsive of districts in the logical   
territory2--modality4 ,it will be       
advisable1 however1 to put together1    
somewhat more definitely1 what has to be
 said upon the subject4 ,i propose1     
therefore1 to devote this chapter to a  
brief account of the principal varieties
 of treatment which the modals have     
received at the hands of professed      
logicians4  ,it must be remarked at the 
outset that the sense in which modality 
and modal propositions have been at     

various times understood1 is by no means
 fixed and invariably the same4 ,this   
diversity of view has arisen partly from
 corresponding differences in the view  
taken of the province and nature of     
logic1 and partly from differences in   
the philosophical and scientific        
opinions entertained as to the          
constitution and order of nature4 ,in   
later times1 moreover1 another very     
powerful agent in bringing about a      
change in the treatment of the subject  
must be recognized in the gradual and   
steady growth of the theory of          
,probability1 as worked out by the      
mathematicians from their own point of  
view4  #b4 ,in spite1 however1 of these 
differences of treatment1 there has     
always been some community of           
subject-matter in the discussions upon  
this topic4 ,there has almost always    
been some reference to quantity of      
belief2 enough perhaps to justify ,de   
,morgan's#a remark1 that ,probability   
was the unknown ,god whom the schoolmen 

ignorantly worshipped when they so dealt
 with this species of enunciation1 that 
it was said to be beyond human          
determination whether they most tortured
 the modals1 or the modals them4 ,but   
this reference to quantity of belief has
 sometimes been direct and immediate1   
sometimes indirect and arising out of   
the nature of the subject-matter of the 
proposition4 ,the fact is1 that that    
distinction between the purely          
subjective and purely objective views of
 logic1 which ,i have endeavoured to    
bring out into prominence in the        
eleventh chapter1 was not by any means  
clearly recognized in early times1 nor  
indeed before the time of ,kant1 and the
 view to be taken of modality naturally 
shared in the consequent confusion4     
,this will1 ,i hope1 be made clear in   
the course of the following chapter1    
which is intended to give a brief sketch
 of the principal different ways in     
which the modality of propositions has  
been treated in logic4 ,as it is not    

proposed to give anything like a regular
 history of the subject1 there will be  
no necessity to adhere to any strict    
sequence of time1 or to discuss the     
opinions of any writers1 except those   
who may be taken as representative of   
tolerably distinct views4 ,the outcome  
of such investigation will be1 ,i hope1 
to convince the reader 7if1 indeed1 he  
had not come to that conviction before71
 that the logicians1 after having had a 
long and fair trial1 have failed to make
 anything satisfactory out of this      
subject of the modals by their methods  
of enquiry and treatment2 and that it   
ought1 therefore1 to be banished        
entirely from that science1 and         
relegated to ,probability4  #c4 ,from   
the earliest study of the syllogistic   
process it was seen that1 complete as   
that process is within its own domain1  
the domain1 at any rate under its       
simplest treatment1 is a very limited   
one4 ,propositions of the pure          
form1--,all 7or some7 ,a is 7or is not7 

,b1--are found in practice to form but a
 small portion even of our categorical  
statements4 ,we are perpetually meeting 
with others which express the relation  
of ,b to ,a with various degrees of     
necessity or probability2 e4g4 ,a must  
be ,b1 ,a may be ,b2 or the effect of   
such facts upon our judgment1 e4g4 ,i am
 perfectly certain that ,a is ,b1 ,i    
think that ,a may be ,b2 with many      
others of a more or less similar type4  
,the question at once arises1 ,how are  
such propositions to be treated8 ,it    
does not seem to have occurred to the   
old logicians1 as to some of their      
successors in modern times1 simply to   
reject all consideration of this topic4 
,their faith in the truth and           
completeness of their system of         
inference was far too firm for them to  
suppose it possible that forms of       
proposition universally recognized as   
significant in popular speech1 and forms
 of inference universally recognized    
there as valid1 were to be omitted      

because they were inconvenient or       
complicated4  #d4 ,one very simple plan 
suggests itself1 and has indeed been    
repeatedly advocated1 viz4 just to      
transfer all that is characteristic of  
such propositions into that convenient  
receptacle for what is troublesome      
elsewhere1 the predicate4#b ,has not    
another so-called modality been thus got
 rid of8#c and has it not been attempted
 by the same device to abolish the      
distinctive characteristic of negative  
propositions1 viz4 by shifting the      
negative particle into the predicate8   
,it must be admitted that1 up to a      
certain point1 something may be done in 
this way4 ,given the reasoning1 ',those 
who take arsenic will probably die2 ,a  
has taken it1 therefore he will probably
 die2' it is easy to convert this into  
an ordinary syllogism of the pure type1 
by simply wording the major1 ',those who
 take arsenic are                       
people-who-will-probably-die1' when the 
conclusion follows in the same form1 ',a

 is one who-will-probably-die4' ,but    
this device will only carry us a very   
little way4 ,suppose that the minor     
premise also is of the same modal       
description1 e4g4 ',a has probably taken
 arsenic1' and it will be seen that we  
cannot relegate the modality here also  
to the predicate without being brought  
to a stop by finding that there are four
 terms in the syllogism4  ,but even if  
there were not this particular          
objection1 it does not appear that      
anything is to be gained in the way of  
intelligibility or method by such a     
device as the above4 ,for what is meant 
by a modal predicate1 by the predicate  
'probably mortal1' for instance1 in the 
proposition ',all poisonings by arsenic 
are probably mortal'8 ,if the analogy   
with ordinary pure propositions is to   
hold good1 it must be a predicate       
referring to the whole of the subject1  
for the subject is distributed4 ,but    
then we are at once launched into the   
difficulties discussed in a former      

chapter 7,ch4 ,,vi4 #ai-be71 when we    
attempt to justify or verify the        
application of the predicate4 ,we have  
to enquire 7at least on the view adopted
 in this work7 whether the application  
of the predicate 'probably mortal' to   
the whole of the subject1 really means  
at bottom anything else than that the   
predicate 'mortal' is to be applied to a
 portion 7more than half7 of the members
 denoted by the subject4 ,when the      
transference of the modality to the     
predicate raises such intricate         
questions as to the sense in which the  
predicate is to be interpreted1 there is
 surely nothing gained by the step4  #e4
 ,a second1 and more summary way of     
shelving all difficulties of the        
subject1 so far at least as logic1 or   
the writers upon logic1 are concerned1  
is found by simply denying that modality
 has any connection whatever with logic4
 ,this is the course adopted by many    
modern writers1 for instance1 by        
,hamilton and ,mansel1 in reference to  

whom one cannot help remarking that an  
unduly large portion of their logical   
writings seems occupied with telling us 
what does not belong to logic4 ,they    
justify their rejection on the ground   
that the mode belongs to the matter1 and
 must be determined by a consideration  
of the matter1 and therefore is         
extralogical4 ,to a certain extent ,i   
agree with their grounds of rejection1  
for 7as explained in ,chapter ,,vi47 it 
is not easy to see how the degree of    
modality of any proposition1 whether    
premise or conclusion1 can be justified 
without appeal to the matter4 ,but then 
questions of justification1 in any      
adequate sense of the term1 belong to a 
range of considerations somewhat alien  
to ,hamilton's and ,mansel's way of     
regarding the science4 ,the complete    
justification of our inferences is a    
matter which involves their truth or    
falsehood1 a point with which these     
writers do not much concern themselves1 
being only occupied with the consistency

 of our reasonings1 not with their      
conformity with fact4 ,were ,i speaking 
as a ,hamiltonian ,i should say that    
modality is formal rather than material1
 for though we cannot justify the degree
 of our belief of a proposition without 
appeal to the matter1 we can to a       
moderate degree of accuracy estimate it 
without any such appeal2 and this would 
seem to be quite enough to warrant its  
being regarded as formal4  ,it must be  
admitted that ,hamilton's account of the
 matter when he is recommending the     
rejection of the modals1 is not by any  
means clear and consistent4 ,he not only
 fails1 as already remarked1 to         
distinguish between the formal and the  
material 7in other words1 the true and  
the false7 modality2 but when treating  
of the former he fails to distinguish   
between the extremely diverse aspects of
 modality when viewed from the          
,aristotelian and the ,kantian          
stand-points4 ,of the amount and        
significance of this difference we shall

 speak presently1 but it may be just    
pointed out here that ,hamilton begins  
7,vol4 ,i4 p4 #beg7 by rejecting the    
modals on the ground that the           
distinctions between the necessary1 the 
contingent1 the possible1 and the       
impossible1 must be wholly rested on an 
appeal to the matter of the             
propositions1 in which he is1 ,i think1 
quite correct4 ,but then a little       
further on 7p4 #bfj71 in explaining 'the
 meaning of three terms which are used  
in relation to pure and modal           
propositions1' he gives the widely      
different ,kantian1 or three-fold       
division into the apodeictic1 the       
assertory1 and the problematic4 ,he does
 not take the precaution of pointing out
 to his hearers the very different      
general views of logic from which these 
two accounts of modality spring4#d  #f4 
,there is one kind of modal syllogism   
which it would seem unreasonable to     
reject on the ground of its not being   
formal1 and which we may notice in      

passing4 ,the premise ',any ,a is       
probably ,b1' is equivalent to ',most ,a
 are ,b4' ,now it is obvious that from  
two such premises as ',most ,a are ,b1' 
',most ,a are ,c1' we can deduce the    
consequence1 ',some ,c are ,b4' ,since  
this holds good whatever may be the     
nature of ,a1 ,b1 and ,c1 it is1        
according to ordinary usage of the term1
 a formal syllogism4 ,mansel1 however1  
refuses to admit that any such          
syllogisms belong to formal logic4 ,his 
reasons are given in a rather elaborate 
review#e and criticism of some of the   
logical works of ,de ,morgan1 to whom   
the introduction of 'numerically        
definite syllogisms' is mainly due4     
,mansel does not take the particular    
example given above1 as he is discussing
 a somewhat more comprehensive algebraic
 form4 ,he examines it in a special     
numerical example3#f--ah out of #ba ,ys 
are ,x2 #ae out of #ba ,ys are ,z2 the  
conclusion that #ab ,zs are ,x is       
rejected from formal logic on the ground

 that the arithmetical judgment involved
 is synthetical1 not analytical1 and    
rests upon an intuition of quantity4 ,we
 cannot enter upon any examination of   
these reasons here2 but it may merely be
 remarked that his criticism demands the
 acceptance of the ,kantian doctrines as
 to the nature of arithmetical          
judgments1 and that it would be better  
to base the rejection not on the ground 
that the syllogism is not formal1 but on
 the ground that it is not analytical4  
#g4 ,there is another and practical way 
of getting rid of the perplexities of   
modal reasoning which must be noticed   
here4 ,it is the resource of ordinary   
reasoners rather than the decision of   
professed logicians1#g and1 like the    
first method of evasion already pointed 
out in this chapter1 is of very partial 
application4 ,it consists in treating   
the premises1 during the process of     
reasoning1 as if they were pure1 and    
then reintroducing the modality into the
 conclusion1 as a sort of qualification 

of its full certainty4 ,when each of the
 premises is nearly certain1 or when    
from any cause we are not concerned with
 the extent of their departure from full
 certainty1 this rough expedient will   
answer well enough4 ,it is1 ,i          
apprehend1 the process which passes     
through the minds of most persons in    
such cases1 in so far as they reason    
consciously4 ,they would1 presumably1 in
 such an example as that previously     
given 7#d71 proceed as if the premises  
that 'those who take arsenic will die1' 
and that 'the man in question has taken 
it1' were quite true1 instead of being  
only probably true1 and they would      
consequently draw the conclusion that   
'he would die4' ,but bearing in mind    
that the premises are not certain1 they 
would remember that the conclusion was  
only to be held with a qualified assent4
 ,this they would express quite         
correctly1 if the mere nature and not   
the degree of that assent is taken into 
account1 by saying that 'he is likely to

 die4' ,in this case the modality is    
rejected temporarily from the premises  
to be reintroduced into the conclusion4 
,it is obvious that such a process as   
this is of a very rough and imperfect   
kind4 ,it does1 in fact1 omit from      
accurate consideration just the one     
point now under discussion4 ,it takes no
 account of the varying shades of       
expression by which the degree of       
departure from perfect conviction is    
indicated1 which is of course the very  
thing with which modality is intended to
 occupy itself4 ,at best1 therefore1 it 
could only claim to be an extremely rude
 way of deciding questions1 the accurate
 and scientific methods of treating     
which are demanded of us4  #h4 ,in any  
employment of applied logic we have of  
course to go through such a process as  
that just mentioned4 ,outside of pure   
mathematics it can hardly ever be the   
case that the premises from which we    
reason are held with absolute           
conviction4 ,hence there must be a lapse

 from absolute conviction in the        
conclusion4 ,but we reason on the       
hypothesis that the premises are true1  
and any trifling defection from         
certainty1 of which we may be conscious1
 is mentally reserved as a qualification
 to the conclusion4 ,but such           
considerations as these belong rather to
 ordinary applied logic2 they amount to 
nothing more than a caution or hint to  
be borne in mind when the rules of the  
syllogism1 or of induction1 are applied 
in practice4 ,when1 however1 we are     
treating of modality1 the extent of the 
defection from full certainty is        
supposed to be sufficiently great for   
our language to indicate and appreciate 
it4 ,what we then want is of course a   
scientific discussion of the principles 
in accordance with which this departure 
is to be measured and expressed1 both in
 our premises and in our conclusion4    
,such a plan therefore for treating     
modality1 as the one under discussion1  
is just as much a banishment of it from 

the field of real logical enquiry1 as if
 we had determined avowedly to reject it
 from consideration4  #i4 ,before       
proceeding to a discussion of the       
various ways in which modality may be   
treated by those who admit it into      
logic1 something must be said to clear  
up a possible source of confusion in    
this part of the subject4 ,in the cases 
with which we have hitherto been mostly 
concerned1 in the earlier chapters of   
this work1 the characteristic of        
modality 7for in this chapter we may    
with propriety use this logical term7   
has generally been found in singular and
 particular propositions4 ,it presented 
itself when we had to judge of          
individual cases from a knowledge of the
 average1 and was an expression of the  
fact that the proposition relating to   
these individuals referred to a portion 
only of the whole class from which the  
average was taken4 ,given that of men of
 fifty-five1 three out of five will die 
in the course of twenty years1 we have  

had to do with propositions of the vague
 form1 ',it is probable that ,,ab 7of   
that age7 will die1' or of the more     
precise form1 ',it is three to two that 
,,ab will die1' within the specified    
time4 ,here the modal proposition       
naturally presents itself in the form of
 a singular or particular proposition4  
#aj4 ,but when we turn to ordinary logic
 we may find universal propositions     
spoken of as modal4 ,this must mostly be
 the case with those which are termed   
necessary or impossible1 but it may also
 be the case with the probable4 ,we may 
meet with the form ',all ,x is probably 
,y4' ,adopting the same explanation here
 as has been throughout adopted in      
analogous cases1 we must say that what  
is meant by the modality of such a      
proposition is the proportional number  
of times in which the universal         
proposition would be correctly made4    
,and in this there is1 so far1 no       
difficulty4 ,the only difference is that
 whereas the justification of the       

former1 viz4 the particular or          
individual kind of modal1 was obtainable
 within the limits of the universal     
proposition which included it1 the      
justification of the modality of a      
universal proposition has to be sought  
in a group or succession of other       
propositions4 ,the proposition has to be
 referred to some group of similar ones 
and we have to consider the proportion  
of cases in which it will be true4 ,but 
this distinction is not at all          
fundamental4  ,it is quite true that    
universal propositions from their nature
 are much less likely than individual   
ones to be justified1 in practice1 by   
such appeal4 ,but1 as has been already  
frequently pointed out1 we are not      
concerned with the way in which our     
propositions are practically obtained1  
nor with the way in which men might find
 it most natural to test them2 but with 
that ultimate justification to which we 
appeal in the last resort1 and which has
 been abundantly shown to be of a       

statistical character4 ,when1 therefore1
 we say that 'it is probable that all ,x
 is ,y1' what we mean is1 that in more  
than half the cases we come across we   
should be right in so judging1 and in   
less than half the cases we should be   
wrong4  #aa4 ,it is at this step that   
the possible ambiguity is encountered4  
,when we talk of the chance that ,all ,x
 is ,y1 we contemplate or imply the     
complementary chance that it is not so4 
,now this latter alternative is not free
 from ambiguity4 ,it might happen1 for  
instance1 in the cases of failure1 that 
no ,x is ,y1 or it might happen that    
some ,x1 only1 is not ,y2 for both of   
these suppositions contradict the       
original proposition1 and are therefore 
instances of its failure4 ,in practice1 
no doubt1 we should have various        
recognized rules and inductions to fall 
back upon in order to decide between    
these alternatives1 though1 of course1  
the appeal to them would be in          
strictness extralogical4 ,but the mere  

existence of such an ambiguity1 and the 
fact that it can only be cleared up by  
appeal to the subject-matter1 are in    
themselves no real difficulty in the    
application of the conception of        
modality to universal propositions as   
well as to individual ones4  #ab4       
,having noticed some of the ways in     
which the introduction of modality into 
logic has been evaded or rejected1 we   
must now enter into a brief account of  
its treatment by those who have more or 
less deliberately admitted its claims to
 acceptance4  ,the first enquiry will   
be1 ,what opinions have been held as to 
the nature of modality8 that is1 ,is it 
primarily an affection of the matter of 
the proposition1 and1 if not1 what is it
 exactly8 ,in reference to this enquiry 
it appears to me1 as already remarked1  
that amongst the earlier logicians no   
such clear and consistent distinction   
between the subjective and objective    
views of logic as is now commonly       
maintained1 can be detected4#h ,the     

result of this appears in their         
treatment of modality4 ,this always had 
some reference to the subjective side of
 the proposition1 viz4 in this case to  
the nature or quantity of the belief    
with which it was entertained2 but it is
 equally clear that this characteristic 
was not estimated at first hand1 so to  
say1 and in itself1 but rather from a   
consideration of the matter determining 
what it should be4 ,the commonly        
accepted scholastic or ,aristotelian    
division1 for instance1 is into the     
necessary1 the contingent1 the possible1
 and the impossible4 ,this is clearly a 
division according to the matter almost 
entirely1 for on the purely mental side 
the necessary and the impossible would  
be just the same2 one implying full     
conviction of the truth of a            
proposition1 and the other of that of   
its contradictory4 ,so too1 on the same 
side1 it would not be easy to           
distinguish between the contingent and  
the possible4 ,on the view in question1 

therefore1 the modality of a proposition
 was determined by a reference to the   
nature of the subject-matter4 ,in some  
propositions the nature of the          
subject-matter decided that the         
predicate was necessarily joined to the 
subject2 in others that it was          
impossible that they should be joined2  
and so on4  #ac4 ,the artificial        
character of such a four-fold division  
will be too obvious to modern minds for 
it to be necessary to criticize it4 ,a  
very slight study of nature and         
consequent appreciation of inductive    
evidence suffice to convince us that    
those uniformities upon which all       
connections of phenomena1 whether called
 necessary or contingent1 depend1 demand
 extremely profound and extensive       
enquiry2 that they admit of no such     
simple division into clearly marked     
groups2 and that1 therefore1 the pure   
logician had better not meddle with     
them4#i  ,the following extract from    
,grote's ,aristotle 7,vol4 ,i4 p4 #aib7 

will serve to show the origin of this   
four-fold division1 its conformity with 
the science of the day1 and consequently
 its utter want of conformity with that 
of our own time3--,the distinction of   
,problematical and ,necessary           
,propositions corresponds1 in the mind  
of ,aristotle1 to that capital and      
characteristic doctrine of his ,ontology
 and ,physics1 already touched on in    
this chapter4 ,he thought1 as we have   
seen1 that in the vast circumferential  
region of the ,kosmos1 from the outer   
sidereal sphere down to the lunar       
sphere1 celestial substance was a       
necessary existence and energy1         
sempiternal and uniform in its rotations
 and influence2 and that through its    
beneficent influence1 pervading the     
concavity between the lunar sphere and  
the terrestrial centre 7which included  
the four elements with their compounds7 
there prevailed a regularizing tendency 
called ,nature2 modified1 however1 and  
partly counteracted by independent and  

irregular forces called ,spontaneity and
 ,chance1 essentially unknowable and    
unpredictable4 ,the irregular sequences 
thus named by ,aristotle were the       
objective correlate of the              
,problematical ,proposition in ,logic4  
,in these sublunary sequences1 as to    
future time1 may or may not1 was all    
that could be attained1 even by the     
highest knowledge2 certainty1 either of 
affirmation or negation1 was out of the 
question4 ,on the other hand1 the       
necessary and uniform energies of the   
celestial substance1 formed the         
objective correlate of the ,necessary   
,proposition in ,logic2 this substance  
was not merely an existence1 but an     
existence necessary and unchangeable444 
he considers the ,problematical         
,proposition in ,logic to be not purely 
subjective1 as an expression of the     
speaker's ignorance1 but something more1
 namely1 to correlate with an objective 
essentially unknowable to all4  #ad4    
,even after this philosophy began to    

pass away1 the divisions of modality    
originally founded upon it might have   
proved1 as ,de ,morgan has remarked1#aj 
of considerable service in medival      
times4 ,as he says1 people were much    
more frequently required to decide in   
one way or the other upon a single      
testimony1 without there being a        
sufficiency of specific knowledge to    
test the statements made4 ,the old      
logician did not know but that any day  
of the week might bring from ,cathay or 
,tartary an account of men who ran on   
four wheels of flesh and blood1 or grew 
planted in the ground1 like ,polydorus  
in the neid1 as well evidenced as a     
great many nearly as marvellous stories4
 ,hence1 in default of better           
inductions1 it might have been          
convenient to make rough classifications
 of the facts which were and which were 
not to be accepted on testimony 7the    
necessary1 the impossible1 c471 and to  
employ these provisional inductions     
7which is all we should now regard them7

 as testing the stories which reached   
him4 ,propositions belonging to the     
class of the impossible might be        
regarded as having an antecedent        
presumption against them so great as to 
prevail over almost any testimony worth 
taking account of1 and so on4  #ae4 ,but
 this old four-fold division of modals  
continued to be accepted and perpetuated
 by the logicians long after all        
philosophical justification for it had  
passed away4 ,so far as ,i have been    
able to ascertain1 scarcely any logician
 of repute or popularity before ,kant1  
was bold enough to make any important   
change in the way of regarding them4#aa 
,even the ,port-,royal ,logic1 founded  
as it is on ,cartesianism1 repeats the  
traditional statements1 though with     
extreme brevity4 ,this adherence to the 
old forms led1 it need not be remarked1 
to considerable inconsistency and       
confusion in many cases4 ,these forms   
were founded1 as we have seen1 on an    
objective view of the province of logic1

 and this view was by no means rigidly  
carried out in many cases4 ,in fact it  
was beginning to be abandoned1 to an    
extent and in directions which we have  
not opportunity here to discuss1 before 
the influence of ,kant was felt4 ,many1 
for instance1 added to the list of the  
four1 by including the true and the     
false2 occasionally also the probable1  
the supposed1 and the certain were      
added4 ,this seems to show some tendency
 towards abandoning the objective for   
the subjective view1 or at least        
indicates a hesitation between them4    
#af4 ,with ,kant's view of modality     
almost every one is familiar4 ,he       
divides judgments1 under this head1 into
 the apodeictic1 the assertory1 and the 
problematic4 ,we shall have to say      
something about the number and mutual   
relations of these divisions presently2 
we are now only concerned with the      
general view which they carry out4 ,in  
this respect it will be obvious at once 
what a complete change of position has  

been reached4 ,the 'necessary' and the  
'impossible' demanded an appeal to the  
matter of a proposition in order to     
recognize them2 the 'apodeictic' and the
 'assertory'1 on the other hand1 may be 
true of almost any matter1 for they     
demand nothing but an appeal to our     
consciousness in order to distinguish   
between them4 ,moreover1 the distinction
 between the assertory and the          
problematic is so entirely subjective   
and personal1 that it may vary not only 
between one person and another1 but in  
the case of the same person at different
 times4 ,what one man knows to be true1 
another may happen to be in doubt about4
 ,the apodeictic judgment is one which  
we not only accept1 but which we find   
ourselves unable to reverse in thought2 
the assertory is simply accepted2 the   
problematic is one about which we feel  
in doubt4  ,this way of looking at the  
matter is the necessary outcome of the  
conceptualist or ,kantian view of logic4
 ,it has been followed by many          

logicians1 not only by those who may be 
called followers of ,kant1 but by almost
 all who have felt his influence4       
,ueberweg1 for instance1 who is         
altogether at issue with ,kant on some  
fundamental points1 adopts it4  #ag4    
,the next question to be discussed is1  
,how many subdivisions of modality are  
to be recognized8 ,the ,aristotelian or 
scholastic logicians1 as we have seen1  
adopted a four-fold division4 ,the exact
 relations of some of these to each     
other1 especially the possible and the  
contingent1 is an extremely obscure     
point1 and one about which the          
commentators are by no means agreed4    
,as1 however1 it seems tolerably clear  
that it was not consciously intended by 
the use of these four terms to exhibit a
 graduated scale of intensity of        
conviction1 their correspondence with   
the province of modern probability is   
but slight1 and the discussion of them1 
therefore1 becomes rather a matter of   
special or antiquarian interest4 ,de    

,morgan1 indeed 7,formal ,logic1 p4     
#bcb71 says that the schoolmen          
understood by contingent more likely    
than not1 and by possible less likely   
than not4 ,i do not know on what        
authority this statement rests1 but it  
credits them with a much nearer approach
 to the modern views of probability than
 one would have expected1 and decidedly 
nearer than that of most of their       
successors4#ab ,the general conclusion  
at which ,i have arrived1 after a       
reasonable amount of investigation1 is  
that there were two prevalent views on  
the subject4 ,some 7e4g4 ,burgersdyck1  
,bk4 ,i4 ch4 #cb7 admitted that there   
were at bottom only two kinds of        
modality2 the contingent and the        
possible being equipollent1 as also the 
necessary and the impossible1 provided  
the one asserts and the other denies4   
,this is the view to which those would  
naturally be led who looked mainly to   
the nature of the subject-matter4 ,on   
the other hand1 those who looked mainly 

at the form of expression1 would be led 
by the analogy of the four forms of     
proposition1 and the necessity that each
 of them should stand in definite       
opposition to each other1 to insist upon
 a distinction between the four         
modals4#ac ,they1 therefore1 endeavoured
 to introduce a distinction by          
maintaining 7e4g4 ,crackanthorpe1 ,bk4  
,,iii4 ch4 #aa7 that the contingent is  
that which now is but may not be1 and   
the possible that which now is not but  
may be4 ,a few appear to have made the  
distinction correspondent to that       
between the physically and the logically
 possible4  #ah4 ,when we get to the    
,kantian division we have reached much  
clearer ground4 ,the meaning of each of 
these terms is quite explicit1 and it is
 also beyond doubt that they have a more
 definite tendency in the direction of  
assigning a graduated scale of          
conviction4 ,so long as they are        
regarded from a metaphysical rather than
 a logical standing point1 there is much

 to be said in their favour4 ,if we use 
introspection merely1 confining         
ourselves to a study of the judgments   
themselves1 to the exclusion of the     
grounds on which they rest1 there       
certainly does seem a clear and         
well-marked distinction between         
judgments which we cannot even conceive 
to be reversed in thought2 those which  
we could reverse1 but which we accept as
 true2 and those which we merely        
entertain as possible4  ,regarded1      
however1 as a logical division1 ,kant's 
arrangement seems to me of very little  
service4 ,for such logical purposes     
indeed1 as we are now concerned with1 it
 really seems to resolve itself into a  
two-fold division4 ,the distinction     
between the apodeictic and the assertory
 will be admitted1 ,i presume1 even by  
those who accept the metaphysical or    
psychological theory upon which it      
rests1 to be a difference which         
concerns1 not the quantity of belief    
with which the judgments are            

entertained1 but rather the violence    
which would have to be done to the mind 
by the attempt to upset them4 ,each is  
fully believed1 but the one can1 and the
 other cannot1 be controverted4 ,the    
belief with which an assertory judgment 
is entertained is full belief1 else it  
would not differ from the problematic2  
and therefore in regard to the quantity 
of belief1 as distinguished from the    
quality or character of it1 there is no 
difference between it and the           
apodeictic4 ,it is as though1 to offer  
an illustration1 the index had been     
already moved to the top of the scale in
 the assertory judgment1 and all that   
was done to convert this into an        
apodeictic one1 was to clamp it there4  
,the only logical difference which then 
remains is that between problematic and 
assertory1 the former comprehending all 
the judgments as to the truth of which  
we have any degree of doubt1 and the    
latter those of which we have no doubt4 
,the whole range of the former1         

therefore1 with which ,probability is   
appropriately occupied1 is thrown       
undivided into a single compartment4 ,we
 can hardly speak of a 'division' where 
one class includes everything up to the 
boundary line1 and the other is confined
 to that boundary line4 ,practically1   
therefore1 on this view1 modality1 as   
the mathematical student of ,probability
 would expect to find it1 as completely 
disappears as if it were intended to    
reject it4  #ai4 ,by less consistent and
 systematic thinkers1 and by those in   
whom ingenuity was an over prominent    
feature1 a variety of other arrangements
 have been accepted or proposed4 ,there 
is1 of course1 some justification for   
such attempts in the laudable desire to 
bring our logical forms into better     
harmony with ordinary thought and       
language4 ,in practice1 as was pointed  
out in an earlier chapter1 every one    
recognizes a great variety of modal     
forms1 such as 'likely1' 'very likely1' 
'almost certainly1' and so on almost    

without limit in each direction4 ,it was
 doubtless supposed that1 by neglecting 
to make use of technical equivalents for
 some of these forms1 we should lose our
 logical control over certain possible  
kinds of inference1 and so far fall     
short even of the precision of ordinary 
thought4  ,with regard to such          
additional forms1 it appears to me that 
all those which have been introduced by 
writers who were uninfluenced by the    
,theory of ,probability1 have done      
little else than create additional      
confusion1 as such writers do not       
attempt to marshal their terms in order1
 or to ascertain their mutual relations4
 ,omitting1 of course1 forms obviously  
of material modality1 we have already   
mentioned the true and the false2 the   
probable1 the supposed1 and the certain4
 ,these subdivisions seem to have       
reached their climax at a very early    
stage in ,occam 7,prantl1 ,,iii4 #chj71 
who held that a proposition might be    
modally affected by being 'vera1 scita1 

falsa1 ignota1 scripta1 prolata1        
concepta1 credita1 opinata1 dubitata4'  
#bj4 ,since the growth of the science of
 ,probability1 logicians have had better
 opportunities of knowing what they had 
to aim at2 and1 though it cannot be said
 that their attempts have been really   
successful1 these are at any rate a     
decided improvement upon those of their 
predecessors4 ,dr ,thomson1#ad for      
instance1 gives a nine-fold division4   
,he says that1 arranging the degrees of 
modality in an ascending scale1 we find 
that a judgment may be either possible1 
doubtful1 probable1 morally certain for 
the thinker himself1 morally certain for
 a class or school1 morally certain for 
all1 physically certain with a limit1   
physically certain without limitation1  
and mathematically certain4 ,many other 
divisions might doubtless be mentioned1 
but1 as every mathematician will        
recognize1 the attempt to secure any    
general agreement in such a matter of   
arrangement is quite hopeless4 ,it is   

here that the beneficial influence of   
the mathematical theory of ,probability 
is to be gratefully acknowledged4 ,as   
soon as this came to be studied it must 
have been perceived that in attempting  
to mark off clearly from one another    
certain gradations of belief1 we should 
be seeking for breaches in a continuous 
magnitude4 ,in the advance from a slight
 presumption to a strong presumption1   
and from that to moral certainty1 we are
 making a gradual ascent1 in the course 
of which there are no natural           
halting-places4 ,the proof of this      
continuity need not be entered upon     
here1 for the materials for it will have
 been gathered from almost every chapter
 of this work4 ,the reader need merely  
be reminded that the grounds of our     
belief1 in all cases which admit of     
number and measurement1 are clearly seen
 to be of this description2 and that    
therefore unless the belief itself is to
 be divorced from the grounds on which  
it rests1 what thus holds as to their   

characteristics must hold also as to its
 own4  ,it follows1 therefore1 that     
modality in the old sense of the word1  
wherein an attempt was made to obtain   
certain natural divisions in the scale  
of conviction1 must be finally          
abandoned4 ,all that it endeavoured to  
do can now be done incomparably better  
by the theory of ,probability1 with its 
numerical scale which admits of         
indefinite subdivision4 ,none of the old
 systems of division can be regarded as 
a really natural one2 those which admit 
but few divisions being found to leave  
the whole range of the probable in one  
unbroken class1 and those which adopt   
many divisions lapsing into unavoidable 
vagueness and uncertainty4  #ba4        
,corresponding to the distinction       
between pure and modal propositions1 but
 even more complicated and              
unsatisfactory in its treatment1 was    
that between pure and modal syllogisms4 
,the thing discussed in the case of the 
latter was1 of course1 the effect       

produced upon the conclusion in respect 
of modality1 by the modal affection of  
one or both premises4 ,it is only when  
we reach such considerations as these   
that we are at all getting on to the    
ground appropriate to ,probability2 but 
it is obvious that very little could be 
done with such rude materials1 and the  
inherent clumsiness and complication of 
the whole modal system come out very    
clearly here4 ,it was in reference      
probably to this complication that some 
of the bitter sayings#ae of the         
schoolmen and others which have been    
recorded1 were uttered4  ,aristotle has 
given an intricate investigation of this
 subject1 and his followers naturally   
were led along a similar track4 ,it     
would be quite foreign to my purpose in 
the slight sketch in this chapter to    
attempt to give any account of these    
enquiries1 even were ,i competent to do 
so2 for1 as has been pointed out1 the   
connection between the ,aristotelian    
modals and the modern view of the nature

 of ,probability1 though real1 is       
exceedingly slight4 ,it need only be    
remarked that what was complicated      
enough with four modals to be taken     
account of1 grows intricate beyond all  
endurance when such as the 'probable'   
and the 'true' and the 'false' have also
 to be assigned a place in the list4    
,the following examples#af will show the
 kind of discussions with which the     
logicians exercised themselves4         
',whether1 with one premise certain1 and
 the other probable1 a certain          
conclusion may be inferred'3 ',whether1 
from the impossible1 the necessary can  
be inferred'2 ',whether1 with one       
premise necessary and the other de      
inesse1 the conclusion is necessary'1   
and so on1 endlessly4  #bb4 ,on the     
,kantian view of modality the discussion
 of such kinds of syllogisms becomes at 
once decidedly more simple 7for here but
 three modes are recognized71 and also  
somewhat more closely connected with    
strict ,probability1 7for the modes are 

more nearly of the nature of gradations 
of conviction74 ,but1 on the other hand1
 there is less justification for their  
introduction1 as logicians might really 
be expected to know that what they are  
aiming to effect by their clumsy        
contrivances is the very thing which    
,probability can carry out to the       
highest desired degree of accuracy4 ,the
 former methods are as coarse and       
inaccurate1 compared with the latter1 as
 were the roughest measurements of      
,babylonian night-watchers compared with
 the refined calculations of the modern 
astronomer4 ,it is indeed only some of  
the general adherents of the ,kantian   
,logic who enter upon any such          
considerations as these2 some1 such as  
,hamilton and ,mansel1 entirely reject  
them1 as we have seen4 ,by those who do 
treat of the subject1 such conclusions  
as the following are laid down2 that    
when both premises are apodeictic the   
conclusion will be the same2 so when    
both are assertory or problematic4 ,if  

one is apodeictic and the other         
assertory1 the latter1 or 'weaker1' is  
all that is to be admitted for the      
conclusion2 and so on4 ,the ,english    
reader will find some account of these  
rules in ,ueberweg's ,logic4#ag  #bc4   
,but although those modals1 regarded as 
instruments of accurate thought1 have   
been thus superseded by the precise     
arithmetical expressions of             
,probability1 the question still remains
 whether what may be termed our popular 
modal expressions could not be improved 
and adapted to more accurate use4 ,it is
 true that the attempt to separate them 
from one another by any fundamental     
distinctions is futile1 for the         
magnitude of which they take cognizance 
is1 as we have remarked1 continuous2 but
 considering the enormous importance of 
accurate terminology1 and of recognizing
 numerical distinctions wherever        
possible1 it would be a real advance if 
any agreement could be arrived at with  
regard to the use of modal expressions4 

,we have already noticed 7,ch4 ,,ii4    
#af7 some suggestions by ,mr ,galton as 
to the possibility of a natural system  
of classification1 resting upon the     
regularity with which most kinds of     
magnitudes tend to group themselves     
about a mean4 ,it might be proposed1 for
 instance1 that we should agree to apply
 the term 'good' to the first quarter1  
measuring from the best downwards2      
'indifferent' to the middle half1 and   
'bad' to the last quarter4 ,there seems 
no reason why a similarly improved      
terminology should not some day be      
introduced into the ordinary modal      
language of common life4 ,it might be   
agreed1 for instance1 that 'very        
improbable' should as far as possible be
 confined to those events which had odds
 of 7say7 more than #ii to #a against   
them2 and so on1 with other similar     
expressions4 ,there would1 no doubt1 be 
difficulties in the way1 for in all     
applications of classification we have  
to surmount the two-fold obstacles which

 lie in the way1 firstly 7to use ,kant's
 expression7 of the faculty of making   
rules1 and secondly of that of          
subsumption under rules4 ,that is to    
say1 even if we had agreed upon our     
classes1 there would still be much doubt
 and dispute1 in the case of things     
which did not readily lend themselves to
 be counted or measured1 as to whether  
the odds were more or less than the     
assigned quantity4  ,it is true that    
when we know the odds for or against an 
event1 we can always state them         
explicitly without the necessity of     
first agreeing as to the usage of terms 
which shall imply them4 ,but there would
 often be circumlocution and pedantry in
 so doing1 and as long as modal terms   
are in practical use it would seem that 
there could be no harm1 and might be    
great good1 in arriving at some         
agreement as to the degree of           
probability which they should be        
generally understood to indicate4       
,bentham1 as is well known1 in despair  

of ever obtaining anything accurate out 
of the language of common life on this  
subject1 was in favour of a direct      
appeal to the numerical standard4 ,he   
proposed the employment1 in judicial    
trials1 of an instrument1 graduated from
 #j to #aj1 on which scale the witness  
was to be asked to indicate the degree  
of his belief of the facts to which he  
testified3 similarly the judge might    
express the force with which he held his
 conclusion4 ,the use of such a         
numerical scale1 however1 was to be     
optional only1 not compulsory1 as       
,bentham admitted that many persons     
might feel at a loss thus to measure the
 degree of their belief4 7,rationale of 
,judicial ,evidence1 ,bk4 ,i41 ,ch4     
,,vi47  #bd4 ,throughout this chapter we
 have regarded the modals as the nearest
 counterpart to modern ,probability     
which was afforded by the old systems of
 logic4 ,the reason for so regarding    
them is1 that they represented some     
slight attempt1 rude as it was1 to      

recognize and measure certain gradations
 in the degree of our conviction1 and to
 examine the bearing of such            
considerations upon our logical         
inferences4  ,but although it is amongst
 the modals that the germs of the       
methods of ,probability are thus to be  
sought2 the true subject-matter of our  
science1 that is1 the classes of objects
 with which it is most appropriately    
concerned1 are rather represented by    
another part of the scholastic logic4   
,this was the branch commonly called    
,dialectic1 in the old sense of that    
term4 ,dialectic1 according to          
,aristotle1 seems to have been a sort of
 sister art to ,rhetoric4 ,it was       
concerned with syllogisms differing in  
no way from demonstrative syllogisms1   
except that their premises were probable
 instead of certain4 ,premises of this  
kind he termed topics1 and the          
syllogisms which dealt with them        
enthymemes4 ,they were said to start    
from 'signs and likelihoods' rather than

 from axioms4#ah  #be4 ,the terms in    
which such reasonings are commonly      
described sound very much like those    
applicable to ,probability1 as we now   
understand it4 ,when we hear of         
likelihood1 and of probable syllogisms1 
our first impression might be that the  
inferences involved would be of a       
similar character4#ai ,this1 however1   
would be erroneous4 ,in the first place 
the province of this ,dialectic was much
 too wide1 for it covered in addition   
the whole field of what we should now   
term ,scientific or ,material           
,induction4 ,the distinctive            
characteristic of the dialectic premises
 was their want of certainty1 and of    
such uncertain premises ,probability 7as
 ,i have frequently insisted7 takes     
account of one class only1 ,induction   
concerning itself with another class4   
,again1 not the slightest attempt was   
made to enter upon the enquiry1 ,how    
uncertain are the premises8 ,it is only 
when this is attempted that we can be   

considered to enter upon the field of   
,probability1 and it is because1 after a
 rude fashion1 the modals attempted to  
grapple with this problem1 that we have 
regarded them as in any way occupied    
with our special subject-matter4  #bf4  
,amongst the older logics with which ,i 
have made any acquaintance1 that of     
,crackanthorpe gives the fullest        
discussion upon this subject4 ,he       
divides his treatment of the syllogism  
into two parts1 occupied respectively   
with the 'demonstrative' and the        
'probable' syllogism4 ,to the latter a  
whole book is devoted4 ,in this the     
nature and consequences of thirteen     
different 'loci'#bj are investigated1   
though it is not very clear in what     
sense they can every one of them be     
regarded as being 'probable4'  ,it is   
doubtless true1 that if the old         
logicians had been in possession of such
 premises as modern ,probability is     
concerned with1 and had adhered to their
 own way of treating them1 they would   

have had to place them amongst such     
loci1 and thus to make the consideration
 of them a part of their ,dialectic4    
,but inasmuch as there does not seem to 
have been the slightest attempt on their
 part to do more here than recognize the
 fact of the premises being probable2   
that is1 since it was not attempted to  
measure their probability and that of   
the conclusion1 ,i cannot but regard    
this part of ,logic as having only the  
very slightest relation to ,probability 
as now conceived4 ,it seems to me little
 more than one of the ways 7described at
 the commencement of this chapter7 by   
which the problem of ,modality is not   
indeed rejected1 but practically evaded4
  #bg4 ,as ,logic is not the only       
science which is directly and           
prominently occupied with questions     
about belief and evidence1 so the       
difficulties which have arisen there    
have been by no means unknown elsewhere4
 ,in respect of the modals1 this seems  
to have been manifestly the case in     

,jurisprudence4 ,some remarks1          
therefore1 may be conveniently made here
 upon this application of the subject1  
though of course with the brevity       
suitable on the part of a layman who has
 to touch upon professional topics4     
,recall for a moment what are the       
essentials of modality4 ,these ,i       
understand to be the attempt to mark off
 from one another1 without any resort to
 numerical notation1 varying degrees of 
conviction or belief1 and to determine  
the consequent effect of premises1 thus 
affected1 upon our conclusions4         
,moreover1 as we cannot construct or    
retain a scale of any kind without      
employing a standard from and by which  
to measure it1 the attainment and       
recognition of a standard of certainty1 
or of one of the other degrees of       
conviction1 is almost inseparably       
involved in the same enquiry4 ,in this  
sense of the term1 modal difficulties   
have certainly shown themselves in the  
department of ,law4 ,there have been    

similar attempts here1 encountered by   
similar difficulties1 to come to some   
definite agreement as to a scale of     
arrangement of the degrees of our       
assent4 ,it is of course much more      
practicable to secure such agreement in 
the case of a special science1 confined 
more or less to the experts1 than in    
subjects into which all classes of      
outsiders have almost equal right of    
entry4 ,the range of application under  
the former circumstances is narrower1   
and the professional experts have       
acquired habits and traditions by which 
the standards may be retained in        
considerable integrity4 ,it does not    
appear1 however1 according to all       
accounts1 as if any very striking       
success had been attained in this       
direction by the lawyers4  #bh4 ,the    
difficulty in its scientific1 or        
strictly jurisprudential shape1 seems to
 have shown itself principally in the   
attempt to arrange legal evidence into  
classes in respect of the degree of its 

cogency4 ,this1 ,i understand1 was the  
case in the ,roman law1 and in some of  
the continental systems of jurisprudence
 which took their rise from the ,roman  
law4 ,the direct evidence of so many    
witnesses was plena probatio4 ,then came
 minus plena probatio1 then semiplen    
major and semiplen minor2 and by adding 
together a certain number of            
half-proofs--for instance1 by the       
production of a tradesman's             
account-books1 plus his supplementary   
oath--full proof might be made out4 ,it 
was on this principle that torture was  
employed to obtain a confession4 ,the   
confession was evidence suppletory to   
the circumstances which were held to    
justify its employment4#ba  ,according  
to ,bentham1#bb the corresponding scale 
in the ,english school was3--,positive  
proof1 ,violent presumption4 ,probable  
presumption1 ,light or ,rash            
presumption4 ,though admitted by        
,blackstone and others1 ,i understand   
that these divisions are not at all     

generally accepted at the present day4  
#bi4 ,in the above we are reminded      
rather of modal syllogisms4 ,the        
principal practical form in which the   
difficulty underlying the simple modal  
propositions presents itself1 is in the 
attempt to obtain some criterion of     
judicial certainty4 ,by 'certainty' here
 we mean1 of course1 not what the       
metaphysicians term apodeictic1#bc for  
that can seldom or never be secured in  
practical affairs1 but such a degree of 
conviction1 short of this1 as every     
reasonable person will feel to be       
sufficient for all his wants4 ,here     
again1 one would think1 the quest must  
appear1 to accurate thinkers1 an utterly
 hopeless one2 an effort to discover    
natural breaks in a continuous          
magnitude4 ,there cannot indeed be the  
least doubt that1 amongst limited       
classes of keen and practised           
intellects1 a standard of certainty1 as 
of everything else1 might be retained   
and handed down with considerable       

accuracy3 this is possible in matters of
 taste and opinion where personal       
peculiarities of judgment are far more  
liable to cause disagreement and        
confusion4 ,but then such a consensus is
 almost entirely an affair of tact and  
custom2 whereas what is wanted in the   
case in question is some criterion to   
which the comparatively uninitiated may 
be able to appeal4 ,the standard1       
therefore1 must not merely be retained  
by recollection1 but be generally       
recognizable by its characteristics4 ,if
 such a criterion could be secured1 its 
importance could hardly be overrated4   
,but so far as one may judge from the   
speeches of counsel1 the charges of     
judges1 and the verdicts of juries1     
nothing really deserving the name is    
ever attained4  #cj4 ,the nearest       
approach1 perhaps1 to a recognized      
standard is to be found in the frequent 
assurance that juries are not bound to  
convict only in case they have no doubt 
of the guilt of the accused2 for the    

absolute exclusion of all doubt1 the    
utter impossibility of suggesting any   
counter hypothesis which this assumes1  
is unattainable in human affairs4 ,but1 
it is frequently said1 they are to      
convict if they have no 'reasonable     
doubt1' no such doubt1 that is1 as would
 be 'a hindrance to acting in the       
important affairs of life4' ,as a       
caution against seeking after           
unattainable certainty1 such advice may 
be very useful2 but it need hardly be   
remarked that the certainty upon which  
we act in the important affairs of life 
is no fixed standard1 but varies        
exceedingly according to the nature of  
those affairs4 ,the greater the reward  
at stake1 the greater the risk we are   
prepared to run1 and conversely4 ,hardly
 any degree of certainty can exist1 upon
 the security of which we should not be 
prepared to act under appropriate       
circumstances4#bd  ,some writers indeed 
altogether deny that any standard1 in   
the common sense of the word1 either is1

 or ought to be1 aimed at in legal      
proceedings4 ,for instance1 ,sir ,j4 ,f4
 ,stephen1 in his work on ,english      
,criminal ,law1#be after noticing and   
rejecting such standards as that last   
indicated1 comes to the conclusion that 
the only standard recognized by our law 
is that which induces juries to         
convict3--,what is judicial proof8 ,that
 which being permitted by law to be     
given in evidence1 induces twelve men1  
chosen according to the ,jury ,act1 to  
say that1 having heard it1 their minds  
are satisfied of the truth of the       
proposition which it affirms4 ,they may 
be prejudiced1 they may be timid1 they  
may be rash1 they may be ignorant2 but  
the oath1 the number1 and the property  
qualification1 are intended1 as far as  
possible1 to neutralize these           
disadvantages1 and answer precisely to  
the conditions imposed upon standards of
 value or length4 7p4 #bfc47  ,to admit 
this is much about the same thing as to 
abandon such a standard as unattainable4

 ,evidence which induces a jury to      
convict may doubtless be a standard to  
me and others of what we ought to       
consider 'reasonably certain1' provided 
of course that the various juries are   
tolerably uniform in their conclusions4 
,but it clearly cannot be proposed as a 
standard to the juries themselves2 if   
their decisions are to be consistent and
 uniform1 they want some external       
indication to guide them4 ,when a man is
 asking1 ,how certain ought ,i to feel8 
to give such an answer as the above is1 
surely1 merely telling him that he is to
 be as certain as he is4 ,if1 indeed1   
juries composed a close profession1 they
 might1 as was said above1 retain a     
traditional standard4 ,but being1 as    
they are1 a selection from the ordinary 
lay public1 their own decisions in the  
past can hardly be held up to them as a 
direction what they are to do in future4
  #ca4 ,it would appear therefore that  
we may fairly say that the ,english law1
 at any rate1 definitely rejects the    

main assumption upon which the logical  
doctrine of modality and its legal      
counterpart are based3 the assumption1  
namely1 that different grades of        
conviction can be marked off from one   
another with sufficient accuracy for us 
to be able to refer individual cases to 
their corresponding classes4 ,and that  
with regard to the collateral question  
of fixing a standard of certainty1 it   
will go no further than pronouncing1 or 
implying1 that we are to be content with
 nothing short of1 but need not go      
beyond1 'reasonable certainty4'  ,this  
is a statement of the standard1 with    
which the logician and scientific man   
can easily quarrel2 and they may with   
much reason maintain that it has not the
 slightest claim to accuracy1 even if it
 had one to strict intelligibility4 ,if 
a man wishes to know whether his present
 degree of certainty is reasonable1     
whither is he to appeal8 ,he can        
scarcely compare his mental state with  
that which is experienced in 'the       

important affairs of life1' for these1  
as already remarked1 would indicate no  
fixed value4 ,at the same time1 one     
cannot suppose that such an expression  
is destitute of all signification4      
,people would not continue to use       
language1 especially in matters of      
paramount importance and interest1      
without meaning something by it4 ,we are
 driven therefore to conclude that      
'reasonable certainty' does in a rude   
sort of way represent a traditional     
standard to which it is attempted to    
adhere4 ,as already remarked1 this is   
perfectly practicable in the case of any
 class of professional men1 and         
therefore not altogether impossible in  
the case of those who are often and     
closely brought into connection with    
such a class4 ,though it is hard to     
believe that any such expressions1 when 
used for purposes of ordinary life1     
attain at all near enough to any        
conventional standard to be worth       
discussion2 yet in the special case of a

 jury1 acting under the direct influence
 of a judge1 it seems quite possible    
that their deliberate assertion that    
they are 'fully convinced' may reach    
somewhat more nearly to a tolerably     
fixed standard than ordinary outsiders  
would at first think likely4  #cb4 ,are 
there then any means by which we could  
ascertain what this standard is2 in     
other words1 by which we could determine
 what is the real worth1 in respect of  
accuracy1 of this 'reasonable certainty'
 which the juries are supposed to       
secure8 ,in the absence of authoritative
 declarations upon the subject1 the     
student of ,logic and ,probability would
 naturally resort to two means1 with a  
momentary notice of which we will       
conclude this enquiry4  ,the first of   
these would aim at determining the      
standard of judicial certainty          
indirectly1 by simply determining the   
statistical frequency with which the    
decisions 7say7 of a jury were found to 
be correct4 ,this may seem to be a      

hopeless task2 and so indeed it is1 but 
not so much on any theoretic            
insufficiency of the determining        
elements as on account of the numerous  
arbitrary assumptions which attach to   
most of the problems which deal with the
 probability of testimony and judgments4
 ,it is not necessary for this purpose  
that we should have an infallible       
superior court which revised the        
decisions of the one under              
consideration2#bf it is sufficient if a 
large number of ordinary representative 
cases are submitted to a court          
consisting even of exactly similar      
materials to the one whose decisions we 
wish to test4 ,provided always that we  
make the monstrous assumption that the  
judgments of men about matters which    
deeply affect them are 'independent' in 
the sense in which the tosses of pence  
are independent1 then the statistics of 
mere agreement and disagreement will    
serve our purpose4 ,we might be able to 
say1 for instance1 that a jury of a     

given number1 deciding by a given       
majority1 were right nine times out of  
ten in their verdict4 ,conclusions of   
this kind1 in reference to the ,french  
courts1 are what ,poisson has attempted 
at the end of his great work on the     
,probability of ,judgments2 though ,i do
 not suppose that he attached much      
numerical accuracy to his results4  ,a  
scarcely more hopeful means would be    
found by a reference to certain cases of
 legal 'presumptions4' ,a 'conclusive   
presumption' is defined as              
follows3--,conclusive1 or as they are   
elsewhere termed imperative or absolute 
presumptions of law1 are rules          
determining the quantity of evidence    
requisite for the support of any        
particular averment which is not        
permitted to be overcome by any proof   
that the fact is otherwise4#bg ,a large 
number of such presumptions will be     
found described in the text-books1 but  
they seem to refer to matters far too   
vague1 for the most part1 to admit of   

any reduction to statistical frequency  
of occurrence4 ,it is indeed maintained 
by some authorities that any assignment 
of degree of ,probability is not their  
present object1 but that they are simply
 meant to exclude the troublesome delays
 that would ensue if everything were    
considered open to doubt and question4  
,moreover1 even if they did assign a    
degree of certainty this would rather be
 an indication of what legislators or   
judges thought reasonable than of what  
was so considered by the juries         
themselves4  ,there are indeed          
presumptions as to the time after which 
a man1 if not heard of1 is supposed to  
be dead 7capable of disproof1 of course1
 by his reappearance74 ,if this time    
varied with the age of the man in       
question1 we should at once have some   
such standard as we desire1 for a       
reference to the ,life tables would fix 
his probable duration of life1 and so   
determine indirectly the measure of     
probability which satisfied the law4    

,but this is not the case2 the period   
chosen is entirely irrespective of age4 
,the nearest case in point 7and that    
does not amount to much7 which ,i have  
been able to ascertain is that of the   
age after which it has been presumed    
that a woman was incapable of bearing   
children4 ,this was the age of #ec4 ,a  
certain approach to a statistical       
assignment of the chances in this case  
is to be found in ,quetelet's ,physique 
,sociale 7,vol4 ,i4 p4 #ahd1 note74     
,according to the authorities which he  
there quotes it would seem that in about
 one birth in #eejj the mother was of   
the age of #ej or upwards4 ,this does   
not quite assign the degree of what may 
be called the  priori chance against the
 occurrence of a birth at that age1     
because the fact of having commenced a  
family at an early age represents some  
diminution of the probability of        
continuing it into later life4 ,but it  
serves to give some indication of what  
may be called the odds against such an  

event4  ,it need not be remarked that   
any such clues as these to the measure  
of judicial certainty are far too slight
 to be of any real value4 ,they only    
deserve passing notice as a possible    
logical solution of the problem in      
question1 or rather as an indication of 
the mode in which1 in theory1 such a    
solution would have to be sought1 were  
the ,english law1 on those subjects1 a  
perfectly consistent scheme of          
scientific evidence4 ,this is the mode  
in which one would1 under those         
circumstances1 attempt to extract from  
its proceedings an admission of the     
exact measure of that standard of       
certainty which it adopted1 but which it
 declined openly to enunciate4   #a4    
,formal ,logic1 p4 #bcb4  #b4 ,this     
appears to be the purport of some       
statements in a very   confused passage 
in ,whately's ,logic 7,bk4 ,,ii41 ch4   
,,iv4 #a74 ,a   modal proposition may be
 stated as a pure one by attaching the  
mode   to one of the terms1 and the     

proposition will in all respects fall   
under the foregoing rules2444 ',it is   
probable that all knowledge is          
useful2' 'probably useful' is here the  
predicate4 ,he draws   apparently no    
such distinction as that between the    
true and false   modality referred to in
 the next note4 ,what is really         
surprising is   that even ,hamilton puts
 the two 7the true and the false        
modality7   upon the same footing4 ,in  
regard to these the former the case is  
precisely the same2 the mode is merely a
 part of the predicate4   ,logic1 ,i4   
#beg4  #c4 ,i allude of course to such  
examples as ',a killed ,b unjustly1' in 
which the killing of ,b by ,a was       
sometimes said to be asserted not       
simply but with a modification4         
7,hamilton's ,logic1 ,i4 #bef47 ,it is  
obvious that the modification in such   
cases is by rights merely a   part of   
the predicate1 there being no formal    
distinction between ',a   is the killer 
of ,b' and ',a is the unjust killer of  

,b4' ,indeed some   logicians who were  
too conservative to reject the generic  
name of   modality in this application  
adopted the common expedient of         
introducing a specific distinction which
 did away with its meaning1   terming   
the spurious kind 'material modality'   
and the genuine kind   'formal          
modality'4 ,the former included all the 
cases in which the   modification       
belonged by right either to the         
predicate or to the   subject2 the      
latter was reserved for the cases in    
which the   modification affected the   
real conjunction of the predicate with  
the   subject4 7,keckermann1 ,systema   
,logic1 ,lib4 ,,ii4 ch4 #c47 ,it was1 ,i
   believe1 a common scholastic         
distinction4    ,for some account of the
 dispute as to whether the negative     
particle   was to be considered to      
belong to the copula or to the          
predicate1   see ,hamilton's ,logic1 ,i4
 #bec4  #d4 ,he has also given a short  
discussion of the subject elsewhere     

7,discussions1 ,ed4 ,,ii4 p4 #gjb71 in  
which a somewhat different view   is    
taken4 ,the modes are indeed here       
admitted into logic1 but only   in so   
far as they fall by subdivision under   
the relation of genus   and species1    
which is of course tantamount to their  
entire   rejection2 for they then differ
 in no essential way from any other     
examples of that relation4  #e4         
,letters1 ,lectures and ,reviews1 p4    
#fa4 ,elsewhere in the review   7p4 #de7
 he gives what appears to me a somewhat 
different decision4  #f4 ,it must be    
remembered that this is not one of the  
proportional   propositions with which  
we have been concerned in previous      
chapters3   it is meant that there are  
exactly #ba ,ys1 of which just #ah are  
,x1   not that on the average #ah out of
 #ba may be so regarded4  #g4 ,i        
consider however1 as ,i have said       
further on 7p4 #cbj71 that the          
treatment in the older logics of        
,probable syllogisms1 and ,dialectic    

syllogisms1 came to somewhat the same   
thing as this1 though they   looked at  
the matter from a different point of    
view1 and expressed   it in very        
different language4  #h4 ,the           
distinction is however by no means      
entirely neglected4 ,thus   ,smiglecius1
 when discussing the modal affections of
 certainty and   necessity1 says1       
certitudo ad cognitionem spectat3       
necessitas vero   est in re             
7,disputationes2 ,disp4 ,,xiii41 ,qust4 
,,xii474  #i4 ,it may be remarked that  
,whately 7,logic1 ,bk4 ,,ii4 ch4 ,,ii4  
#b7   speaks of necessary1 impossible   
and contingent matter1 without any      
apparent suspicion that they belong     
entirely to an obsolete point of   view4
  #aj4 ,formal ,logic1 p4 #bcc4  #aa4   
,the subject was sometimes altogether   
omitted1 as by ,wolf4 ,he says   a good 
deal however about probable propositions
 and syllogisms1 and1   like ,leibnitz  
before him1 looked forward to a logica  
probabilium   as something new and      

desirable4 ,i imagine that he had been  
influenced by the writers on ,chances1  
as of the few who had already   treated 
that subject nearly all the most        
important are referred to   in one      
passage 7,philosophia ,rationalis sive  
,logica1 #eic74    ,lambert stands quite
 apart4 ,in this respect1 as in most    
others where   mathematical conceptions 
and symbols are involved1 his logical   
attitude is thoroughly unconventional4  
,see1 for instance1 his   chapter ',von 
dem ,wahrscheinlichen'1 in his ,neues   
,organon4  #ab4 ,i cannot find the      
slightest authority for the statement in
 the   elaborate history of ,logic by   
,prantl4  #ac4 ,hi quatuor modi magnam  
censeri solent analogiam habere cum     
quadruplici propositionum in quantitate 
et qualitate varietate   7,wallis's     
,instit4 ,logic4 ,bk4 ,,ii4 ch4 #h74    
#ad4 ,laws of ,thought1 #aah4  #ae4     
,haud scio magis ne doctrinam modalium  
scholastici exercuerint1   quam ea illos
 vexarit4 ,certe usque adeo sudatum hic 

fuit1 ut   dicterio locus sit datus2 ,de
 modalibus non gustabit asinus4         
,keckermann1 ,syst4 ,log4 ,bk4 ,,ii4 ch4
 #c4  #af4 ,smiglecii ,disputationes1   
,ingolstadt1 #afah4    ,see also        
,prantl's ,geschichte der ,logik 7under 
,occam and   ,buridan7 for accounts of  
the excessive complication which the    
subtlety of those learned schoolmen     
evolved out of such suitable            
materials4  #ag4 ,translation by ,t4 ,m4
 ,lindsay1 p4 #dci4  #ah4 ,the  and     
themselves are propositions2 the former 
stating a general probability1 the      
latter a fact1 which is known   to be an
 indication1 more or less certain1 of   
the truth of some   further statement1  
whether of a single fact1 or of a       
general   belief4 ,the former is a      
general proposition1 nearly1 though not 
quite1 universal2 as 'most men who envy 
hate'2 the latter is a   singular       
proposition1 which however is not       
regarded as a sign1   except relatively 
to some other proposition1 which it is  

supposed   may by inferred from it4     
7,mansel's ,aldrich2 ,appendix ,f1 where
 an   account will be found of the      
,aristotelian enthymeme1 and dialectic  
syllogism4 ,also1 of course1 ,grote's   
,aristotle1 ,topics and   elsewhere47   
#ai4 ,nam in hoc etiam differt          
demonstratio1 sen demonstrativa         
argumentatio1  probabili1 quia in ill   
tam conclusio quam prmiss   necessari   
sunt2 in probabili autem argumentatione 
sicut conclusio   ut probabilis infertur
 ita prmiss ut probabiles afferuntur    
7,crackanthorpe1 ,bk4 ,v41 ,ch4 #a72    
almost the words with which ,de         
,morgan distinguishes between logic and 
probability in a passage   already cited
 7see ,ch4 ,,vi4 #c74    ,perhaps it was
 a development of some such view as this
 that ,leibnitz   looked forward to4    
,j'ai dit plus d'une fois qu'il faudrait
 une   nouvelle espce de ,logique1 qui  
traiteroit des degrs de   ,probabilit1  
puisqu',aristote dans ses ,topiques n'a 
rien moins fait   que cela 7,nouveaux   

essais1 ,lib4 ,,iv4 ch4 ,,xvi74 ,it is  
possible1   indeed1 that he had in his  
mind more what we now understand by the 
mathematical theory of ,probability1 but
 in the infancy of a science   it is of 
course hard to say whether any          
particular subject is   definitely      
contemplated or not4 ,leibnitz 7as      
,todhunter has shown in   his history7  
took the greatest interest in such      
chance problems as   had yet been       
discussed4  #bj4 ,by loci were          
understood certain general classes of   
premises4 ,they stood1 in fact1 to the  
major premise in somewhat the   same    
relation that the ,category or          
,predicament did to the   term4         
,crackanthorpe says of them1 sed duci a 
loco probabiliter   arguendi1 hoc vere  
proprium est ,argumentationis           
probabilis2 et in   hoc a               
,demonstratione differt1 quia           
,demonstrator utitur solummodo   quatuor
 ,locis eisque necessariis4444 ,prter   
hos autem1 ex quibus   quoque           

probabiliter arguere licet1 sunt multo  
plures ,loci arguendi   probabiliter2 ut
 a ,genere1 a ,specie1 ab ,adjuncto1 ab 
,oppositis1 et   similia 7,logica1 ,lib4
 ,v41 ch4 ,,ii474  #ba4 ,stephen's      
,general ,view of the ,criminal ,law of 
,england1 p4 #bda4  #bb4 ,rationale of  
,judicial ,evidence2 ,bk4 ,i4 ch4 ,,vi4 
#bc4 ,though this is claimed by some    
,kantian logicians2--,nie darf an       
einem angeblichen ,verbrecher die       
gesetzliche ,strafe vollzogen   werden1 
bevor er nicht selbst das ,verbrechen   
eingestanden4 ,denn wenn   auch alle    
,zeugnisse und die brigen ,anzeigen     
wider ihn wren1 so   bleibt doch das    
,gegentheil immer mglich 7,krug1        
,denklehre1 #aca74  #bd4 ,as ,mr ,c4 ,j4
 ,monro puts it3 ,suppose that a man is 
suspected of   murdering his daughter4  
,evidence which would not convict him   
before   an ordinary jury might make a  
grand jury find a true bill2 evidence   
which would not do this might make a    
coroner's jury bring in a   verdict     

against him2 evidence which would not do
 this would very   often prevent a      
,chancery judge from appointing the man 
guardian to a   ward of the court2      
evidence which would not affect the     
judge's mind   might make a father think
 twice on his death-bed before he       
appointed   the man guardian to his     
daughter4  #be4 ,the portions of this   
work which treat of the nature of proof 
in   general1 and of judicial proof in  
particular1 are well worth reading   by 
every logical student4 ,it appears to   
me1 however1 that the author   goes much
 too far in the direction of regarding  
proof as subjective1   that is as what  
does satisfy people1 rather than as what
 should   satisfy them4 ,he compares the
 legislative standard of certainty with 
that of value2 this latter is declared  
to be a certain weight of   gold1       
irrespective of the rarity or commonness
 of that metal4 ,so   with certainty2 if
 people grow more credulous the         
intrinsic value of   the standard will  

vary4  #bf4 ,the question will be more  
fully discussed in a future chapter1 but
   a few words may be inserted here by  
way of indication4 ,reduce the   case to
 the simplest possible elements by      
supposing only two judges   or courts1  
of the same average correctness of      
decision4 ,let this be   indicated by x4
 ,then the chance of their agreeing is  
x#b  7#a - x7#b1 for they agree if both 
are right or both wrong4   ,if the      
statistical frequency of this agreement 
is known1 that is1 the   frequency with 
which the first judgment is confirmed by
 the second1   we have the means of     
determining x4  #bg4 ,taylor on         
,evidence3 the latter part of the       
extract does not seem   very clear4     
,,chapter ,,xiv4   ,,fallacies4  #a4 ,in
 works on ,logic a chapter is generally 
devoted to the discussion of ,fallacies1
 that is1 to the description and        
classification of the different ways in 
which the rules of ,logic may be        
transgressed4 ,the analogy of           

,probability to ,logic is sufficiently  
close to make it advisable to adopt the 
same plan here4 ,in describing his own  
opinions an author is1 of course1       
perpetually obliged to describe and     
criticise those of others which he      
considers erroneous4 ,but some of the   
most widely spread errors find no       
supporters worth mentioning1 and exist  
only in vague popular misapprehension4  
,it will be found the best arrangement1 
therefore1 at the risk of occasional    
repetition1 to collect a few of the     
errors that occur most frequently1 and  
as far as possible to trace them to     
their sources2 but it will hardly be    
worth the trouble to attempt any regular
 system of arrangement and              
classification4 ,we shall mainly confine
 ourselves1 in accordance with the      
special province of this work1 to       
problems which involve questions of     
logical interest1 or to those which     
refer to the application of ,probability
 to moral and social science4 ,we shall 

avoid the discussion of isolated        
problems in games of chance and skill   
except when some error of principle     
seems to be involved in them4  #b4 7,i47
 ,one of the most fertile sources of    
error and confusion upon the subject has
 been already several times alluded to1 
and in part discussed in a previous     
chapter4 ,this consists in choosing the 
class to which to refer an event1 and   
therefore judging of the rarity of the  
event and the consequent improbability  
of foretelling it1 after it has         
happened1 and then transferring the     
impressions we experience to a supposed 
contemplation of the event beforehand4  
,the process in itself is perfectly     
legitimate 7however unnecessary it may  
be71 since time does not in strictness  
enter at all into questions of          
,probability4 ,no error therefore need  
arise in this way1 if we were careful as
 to the class which we thus selected2   
but such carefulness is often neglected4
  ,an illustration may afford help here4

 ,a man once pointed to a small target  
chalked upon a door1 the target having a
 bullet hole through the centre of it1  
and surprised some spectators by        
declaring that he had fired that shot   
from an old fowling-piece at a distance 
of a hundred yards1 ,his statement was  
true enough1 but he suppressed a rather 
important fact4 ,the shot had really    
been aimed in a general way at the      
barn-door1 and had hit it2 the target   
was afterwards chalked round the spot   
where the bullet struck4 ,a deception   
analogous to this is1 ,i think1 often   
practised unconsciously in other        
matters4 ,we judge of events on a       
similar principle1 feeling and          
expressing surprise in an equally       
unreasonable way1 and deciding as to    
their occurrence on grounds which are   
really merely a subsequent adjunct of   
our own4 ,butler's remarks about 'the   
story of ,csar1' discussed already in   
the twelfth chapter1 are of this        
character4 ,he selects a series of      

events from history1 and then imagines a
 person guessing them correctly who at  
the time had not the history before him4
 ,as ,i have already pointed out1 it is 
one thing to be unlikely to guess an    
event rightly without specific evidence2
 it is another and very different thing 
to appreciate the truth of a story which
 is founded partly or entirely upon     
evidence4 ,but it is a great mistake to 
transfer to one of these ways of viewing
 the matter the mental impressions which
 properly belong to the other4 ,it is   
like drawing the target afterwards1 and 
then being surprised to find that the   
shot lies in the centre of it4  #c4 ,one
 aspect of this fallacy has been already
 discussed1 but it will serve to clear  
up difficulties which are often felt    
upon the subject if we reexamine the    
question under a somewhat more general  
form4  ,in the class of examples under  
discussion we are generally presented   
with an individual which is not indeed  
definitely referred to a class1 but in  

regard to which we have no great        
difficulty in choosing the appropriate  
class4 ,now suppose we were             
contemplating such an event as the      
throwing of sixes with a pair of dice   
four times running4 ,such a throw would 
be termed a very unlikely event1 as the 
odds against its happening would be #cf 
#cf  #cf  #cf - #a to #a or #afgifae to 
#a4 ,the meaning of these phrases1 as   
has been abundantly pointed out1 is     
simply that the event in question occurs
 very rarely2 that1 stated with         
numerical accuracy1 it occurs once in   
#afgifaf times4  #d4 ,but now let us    
make the assumption that the throw has  
actually occurred2 let us put ourselves 
into the position of contemplating sixes
 four times running when it is known or 
reported that this throw has happened4  
,the same phrase1 namely that the event 
is a very unlikely one1 will often be   
used in relation to it1 but we shall    
find that this phrase may be employed to
 indicate1 on one occasion or another1  

extremely different meanings4  7#a7     
,there is1 firstly1 the most correct    
meaning4 ,the event1 it is true1 has    
happened1 and we know what it is1 and   
therefore1 we have not really any       
occasion to resort to the rules of      
,probability2 but we can nevertheless   
conceive ourselves as being in the      
position of a person who does not know1 
and who has only ,probability to appeal 
to4 ,by calling the chances #afgifae to 
#a against the throw we then mean to    
imply the fact1 that inasmuch as such a 
throw occurs only once in #afgifaf      
times1 our guess1 were we to guess1     
would be correct only once in the same  
number of times2 provided1 that is1 that
 it is a fair guess1 based simply on    
these statistical grounds4  #e4 7#b7    
,but there is a second and very         
different conception sometimes          
introduced1 especially when the event in
 question is supposed to be known1 not  
as above by the evidence of our         
experience1 but by the report of a      

witness4 ,we may then mean by the       
'chances against the event' 7as was     
pointed out in ,chapter ,,xii47 not the 
proportional number of times we should  
be right in guessing the event1 but the 
proportional number of times the witness
 will be right in reporting it4 ,the    
bases of our inference are here shifted 
on to new ground4 ,in the former case   
the statistics were the throws and their
 respective frequency1 now they are the 
witnesses' statements and their         
respective truthfulness4  #f4 7#c7 ,but 
there is yet another meaning sometimes  
intended to be conveyed when persons    
talk of the chances against such an     
event as the throw in question4 ,they   
may mean--not1 ,here is an event1 how   
often should ,i have guessed it8--nor1  
,here is a report1 how often will it be 
correct8--but something different from  
either1 namely1 ,here is an event1 how  
often will it be found to be produced by
 some one particular kind of cause8     
,when1 for example1 a man hears of dice 

giving the same throw several times     
running1 and speaks of this as very     
extraordinary1 we shall often find that 
he is not merely thinking of the        
improbability of his guess being right1 
or of the report being true1 but1 that  
along with this1 he is introducing the  
question of the throw having been       
produced by fair dice4 ,there is1 of    
course1 no reason whatever why such a   
question as this should not also be     
referred to ,probability1 provided      
always that we could find the           
appropriate statistics by which to      
judge4 ,these statistics would be       
composed1 not of throws of the          
particular dice1 nor of reports of the  
particular witness1 but of the occasions
 on which such a throw as the one in    
question respectively had1 and had not1 
been produced fairly4 ,the objection to 
entering upon this view of the question 
would be that no such statistics are    
obtainable1 and that if they were1 we   
should prefer to form our opinion 7on   

principles to be described in ,chapter  
,,xvi47 from the special circumstances  
of the case rather than from an appeal  
to the average4  #g4 ,the reader will   
easily be able to supply examples in    
illustration of the distinctions just   
given2 we will briefly examine but one4 
,i hide a banknote in a certain book in 
a large library1 and leave the room4 ,a 
person tells me that1 after ,i went out1
 a stranger came in1 walked straight up 
to that particular book1 and took it    
away with him4 ,many people on hearing  
this account would reply1 ,how extremely
 improbable6 ,on analysing the phrase1  
,i think we shall find that certainly   
two1 and possibly all three1 of the     
above meanings might be involved in this
 exclamation4 7#a7 ,what may be meant is
 this1--,assuming that the report is    
true1 and the stranger innocent1 a rare 
event has occurred4 ,many books might   
have been thus taken without that       
particular one being selected4 ,i should
 not therefore have expected the event1 

and when it has happened ,i am          
surprised4 ,now a man has a perfect     
right to be surprised1 but he has no    
logical right 7so long as we confine    
ourselves to this view7 to make his     
surprise a ground for disbelieving the  
event4 ,to do this is to fall into the  
fallacy described at the commencement of
 this chapter4 ,the fact of my not      
having been likely to have guessed a    
thing beforehand is no reason in itself 
for doubting it when ,i am informed of  
it4 7#b7 ,or ,i may stop short of the   
events reported1 and apply the rules of 
,probability to the report itself4 ,if  
so1 what ,i mean is that such a story as
 this now before me is of a kind very   
generally false1 and that ,i cannot     
therefore attach much credit to it now4 
7#c7 ,or ,i may accept the truth of the 
report1 but doubt the fact of the       
stranger having taken the book at       
random4 ,if so1 what ,i mean is1 that of
 men who take books in the way          
described1 only a small proportion will 

be found to have taken them really at   
random2 the majority will do so because 
they had by some means ascertained1 or  
come to suspect1 what there was inside  
the book4  ,each of the above three     
meanings is a possible and a legitimate 
meaning4 ,the only requisite is that we 
should be careful to ascertain which of 
them is present to the mind1 so as to   
select the appropriate statistics4 ,the 
first makes in itself the most          
legitimate use of ,probability2 the     
drawback being that at the time in      
question the functions of ,probability  
are superseded by the event being       
otherwise known4 ,the second or third1  
therefore1 is the more likely meaning to
 be present to the mind1 for in these   
cases ,probability1 if it could be      
practically made use of1 would1 at the  
time in question1 be a means of drawing 
really important inferences4 ,the       
drawbacks are the difficulty of finding 
such statistics1 and the extreme        
disturbing influence upon these         

statistics of the circumstances of the  
special case4  #h4 7,,ii47 ,closely     
connected with the tendency just        
mentioned is that which prompts us to   
confound a true chance selection with   
one which is more or less picked4 ,when 
we are dealing with familiar objects in 
a concrete way1 especially when the     
greater rarity corresponds to           
superiority of quality1 almost every one
 has learnt to recognize the            
distinction4 ,no one1 for instance1 on  
observing a fine body of troops in a    
foreign town1 but would be prompted to  
ask whether they came from an average   
regiment or from one that was picked4   
,when however the distinction refers to 
unfamiliar objects1 and especially when 
only comparative rarity seems to be     
involved1 the fallacy may assume a      
rather subtle and misleading form1 and  
seems to deserve special notice by the  
consideration of a few examples4        
,sometimes the result is not so much an 
actual fallacy as a slight misreckoning 

of the order of probability of the event
 under consideration4 ,for instance1 in 
the ,pyramid question1 we saw that it   
made some difference whether we         
considered that  alone was to be taken  
into account or whether we put this     
constant into a class with a small      
number of other similar ones4 ,in       
deciding1 however1 whether or not there 
is anything remarkable in the actual    
falling short of the representation of  
the number #g in the evaluation of  7v4 
p4 #bdh7 the whole question turns upon  
considerations of this kind4 ,the only  
enquiry raised is whether there is      
anything remarkable in this departure   
from the mean1 and the answer depends   
upon whether we suppose that we are     
referring to a predetermined digit1 or  
to whatever digit of the ten happens to 
be most above or below the average4 ,or1
 take the case raised by ,cournot       
7,exposition de la ,thorie des ,chances1
 #ajb1 #aad71 that a certain deviation  
from the mean in the case of            

,departmental returns of the proportion 
between male and female births is       
significant and indicative of a         
difference in kind1 provided that we    
select at random a single ,french       
,department2 but that the same deviation
 may be accidental if it is the maximum 
of the respective returns for several   
,departments4#a ,the answer may be given
 one way or the other according as we   
bear this consideration in mind4  #i4   
,we are peculiarly liable to be misled  
in this way when we are endeavouring to 
determine the cause of some phenomenon1 
by mere statistics1 in entire ignorance 
as to the direction in which the cause  
should be expected4 ,in such cases an   
ingenious person who chooses to look    
about over a large field can never fail 
to hit upon an explanation which is     
plausible in the sense that it fits in  
with the hitherto observed facts4 ,with 
a tithe of the trouble which ,mr ,piazzi
 ,smyth expended upon the measurement of
 the great pyramid1 ,i think ,i would   

undertake to find plausible intimations 
of several of the important constants   
and standards which he discovered there1
 in the dimensions of the desk at which 
,i am writing4 ,the oddest instance of  
this sort of conclusion is perhaps to be
 found in the researches of a writer who
 has discovered#b that there is a       
connection of a striking kind between   
the respective successes of the ,oxford 
and the ,cambridge boat in the annual   
race1 and the greater and less frequency
 of sun-spots4  ,of course our usual    
practical resource in such cases is to  
make appeal to our previous knowledge of
 the subject in question1 which enables 
us to reject as absurd a great number of
 hypotheses which can nevertheless make 
a fair show when they are allowed to    
rest upon a limited amount of adroitly  
selected instances4 ,but it must be     
remembered that if any theory chooses to
 appeal to statistics1 to statistics it 
must be suffered to go for judgment4    
,even the boat race theory could be     

established 7if sound7 on this ground   
alone4 ,that is1 if it really could be  
shown that experience in the long run   
confirmed the preponderance of successes
 on one side or the other according to  
the relative frequency of the sun-spots1
 we should have to accept the fact that 
the two classes of events were not      
really independent4 ,one of the two1    
whichever it may be1 must be suspected  
of causing or influencing the other2 or 
both must be caused or influenced by    
some common circumstances4  #aj4        
7,,iii47 ,the fallacy described at the  
commencement of this chapter arose from 
determining to judge of an observed or  
reported event by the rules of          
,probability1 but employing a wrong set 
of statistics in the process of judging4
 ,another fallacy1 closely connected    
with this1 arises from the practice of  
taking some only of the characteristics 
of such an event1 and arbitrarily       
confining to these the appeal to        
,probability4 ,suppose ,i toss up twelve

 pence and find that eleven of them give
 heads4 ,many persons on witnessing such
 an occurrence would experience a       
feeling which they would express by the 
remark1 ,how near that was to getting   
all heads6 ,and if any thing very       
important were staked on the throw they 
would be much excited at the occurrence4
 ,but in what sense were we near to     
twelve8 ,there is a not uncommon error1 
,i apprehend1 which consists in         
unconsciously regarding the eleven heads
 as a thing which is already somehow    
secured1 so that one might as it were   
keep them1 and then take our chance for 
securing the remaining one4 ,the eleven 
are mentally set aside1 looked upon as  
certain 7for they have already          
happened71 and we then introduce the    
notion of chance merely for the twelfth4
 ,but this twelfth1 having also         
happened1 has no better claim to such a 
distinction than any of the others4 ,if 
we will introduce the notion of chance  
in the case of the one that gave tail we

 must do the same in the case of all the
 others as well4 ,in other words1 if the
 tosser be dissatisfied at the          
appearance of the one tail1 and wish to 
cancel it and try his luck again1 he    
must toss up the whole lot of pence     
again fairly together4 ,in this case1 of
 course1 so far from his having a better
 prospect for the next throw he may     
think himself in very good luck if he   
makes again as good a throw as the one  
he rejected4 ,what he is doing is       
confounding this case with that in which
 the throws are really successive4 ,if  
eleven heads have been tossed up in     
turn1 we are of course within an even   
chance of getting a twelfth2 but the    
circumstances are quite different in the
 instance proposed4  #aa4 ,in the above 
example the error is transparent4 ,but  
in forming a judgment upon matters of   
greater complexity than dice and pence1 
especially in the case of what are      
called 'narrow escapes1' a mistake of an
 analogous kind is1 ,i apprehend1 far   

from uncommon4 ,a person1 for example1  
who has just experienced a narrow escape
 will often be filled with surprise and 
anxiety amounting almost to terror4 ,the
 event being past1 these feelings are1  
at the time1 in strictness              
inappropriate4 ,if1 as is quite         
possible1 they are merely instinctive1  
or the result of association1 they do   
not fall within the province of any kind
 of ,logic4 ,if1 however1 as seems more 
likely1 they partially arise from a     
supposed transference of ourselves into 
that point of past time at which the    
event was just about to happen1 and the 
production by imagination of the        
feelings we should then expect to       
experience1 this process partakes of the
 nature of an inference1 and can be     
right or wrong4 ,in other words1 the    
alarm may be proportionate or           
disproportionate to the amount of danger
 that might fairly have been reckoned   
upon in such a hypothetical             
anticipation4 ,if the supposed transfer 

were completely carried out1 there would
 be no fallacy2 but it is often very    
incompletely done1 some of the component
 parts of the event being supposed to be
 determined or 'arranged' 7to use a     
sporting phrase7 in the form in which we
 now know that they actually have       
happened1 and only the remaining ones   
being fairly contemplated as future     
chances4  ,a man1 for example1 is out   
with a friend1 whose rifle goes off by  
accident1 and the bullet passes through 
his hat4 ,he trembles with anxiety at   
thinking what might have happened1 and  
perhaps remarks1 ',how very near ,i was 
to being killed6' ,now we may safely    
assume that he means something more than
 that a shot passed very close to him4  
,he has some vague idea that1 as he     
would probably say1 'his chance of being
 killed then was very great4' ,his      
surprise and terror may be in great part
 physical and instinctive1 arising      
simply from the knowledge that the shot 
had passed very near him4 ,but his      

mental state may be analysed1 and we    
shall then most likely find1 at bottom1 
a fallacy of the kind described above4  
,to speak or think of chance in         
connection with the incident1 is to     
refer the particular incident to a class
 of incidents of a similar character1   
and then to consider the comparative    
frequency with which the contemplated   
result ensues4 ,now the series which we 
may suppose to be most naturally        
selected in this case is one composed of
 shooting excursions with his friend2 up
 to this point the proceedings are      
assumed to be designed1 beyond it only1 
in the subsequent event1 was there      
accident4 ,once in a thousand times     
perhaps on such occasions the gun will  
go off accidentally2 one in a thousand  
only of those discharges will be        
directed near his friend's head4 ,if we 
will make the accident a matter of      
,probability1 we ought by rights in this
 way 7to adopt the language of the first
 example71 to 'toss up again' fairly4   

,but we do not do this2 we seem to      
assume for certain that the shot goes   
within an inch of our heads1 detach that
 from the notion of chance at all1 and  
then begin to introduce this notion     
again for possible deflections from that
 saving inch4  #ab4 7,,iv47 ,we will now
 notice a fallacy connected with the    
subjects of betting and gambling4 ,many 
or most of the popular misapprehensions 
on this subject imply such utter        
ignorance and confusion as to the       
foundations of the science that it would
 be needless to discuss them here4 ,the 
following however is of a far more      
plausible kind1 and has been a source of
 perplexity to persons of considerable  
acuteness4  ,the case1 put into the     
simplest form1 is as follows4#c ,suppose
 that a person ,a is playing against ,b1
 ,b being either another individual or a
 group of individuals1 say a gambling   
bank4 ,they begin by tossing for a      
shilling1 and ,a maintains that he is in
 possession of a device which will      

insure his winning4 ,if he does win on  
the first occasion he has clearly gained
 his point so far4 ,if he loses1 he     
stakes next time two shillings instead  
of one4 ,the result of course is that if
 he wins on the second occasion he      
replaces his former loss1 and is left   
with one shilling profit as well4 ,so he
 goes on1 doubling his stake after every
 loss1 with the obvious result that on  
the first occasion of success he makes  
good all his previous losses1 and is    
left with a shilling over4 ,but such an 
occasion must come sooner or later1 by  
the assumptions of chance on which the  
game is founded4 ,hence it follows that 
he can insure1 sooner or later1 being   
left a final winner4 ,moreover he may   
win to any amount2 firstly from the     
obvious consideration that he might make
 his initial stake as large as he       
pleased1 a hundred pounds1 for instance1
 instead of a shilling2 and secondly1   
because what he has done once he may do 
again4 ,he may put his shilling by1 and 

have a second spell of play1 long or    
short as the case may be1 with the same 
termination to it4 ,accordingly by mere 
persistency he may accumulate any sum of
 money he pleases1 in apparent defiance 
of all that is meant by luck4  #ac4 ,i  
have classed this opinion among         
fallacies1 as the present is the most   
convenient opportunity of discussing it1
 though in strictness it should rather  
be termed a paradox1 since the          
conclusion is perfectly sound4 ,the only
 fallacy consists in regarding such a   
way of obtaining the result as          
mysterious4 ,on the contrary1 there is  
nothing more easy than to insure        
ultimate success under the given        
conditions4 ,the point is worth enquiry1
 from the principles it involves1 and   
because the answers commonly given do   
not quite meet the difficulty4 ,it is   
sometimes urged1 for instance1 that no  
bank would or does allow the speculator 
to choose at will the amount of his     
stake1 but puts a limit to the amount   

for which it will consent to play4 ,this
 is quite true1 but is of course no     
answer to the hypothetical enquiry      
before us1 which assumes that such a    
state of things is allowed4 ,again1 it  
has been urged that the possibility in  
question turns entirely upon the fact   
that credit must be supposed to be      
given1 for otherwise the fortune of the 
player may not hold out until his turn  
of luck arrives3--that1 in fact1 sooner 
or later1 if he goes on long enough1 his
 fortune will not hold out long enough1 
and all his gains will be swept away4   
,it is quite true that credit is a      
condition of success1 but it is in no   
sense the cause4 ,we may suppose both   
parties to agree at the outset that     
there shall be no payments until the    
game be ended1 ,a having the right to   
decide when it shall be considered to be
 ended4 ,it still remains true that     
whereas in ordinary gambling1 i4e4 with 
fixed or haphazard stakes1 ,a could not 
ensure winning eventually to any extent1

 he can do so if he adopt such a scheme 
as the one in question4 ,and this is the
 state of things which seems to call for
 explanation4  #ad4 ,what causes        
perplexity here is the supposed fact    
that in some mysterious way certainty   
has been conjured out of uncertainty2   
that in a game where the detailed events
 are utterly inscrutable1 and where the 
average1 by supposition1 shows no       
preference for either side1 one party is
 nevertheless succeeding somehow in     
steadily drawing the luck his own way4  
,it looks as if it were a parallel case 
with that of a man who should succeed by
 some device in permanently securing    
more than half of the tosses with a     
penny which was nevertheless to be      
regarded as a perfectly fair one4  ,this
 is quite a mistake4 ,the real fact is  
that ,a does not expose his gains to    
chance at all2 all that he so exposes is
 the number of times he has to wait     
until he gains4 ,put such a case as     
this4 ,i offer to give a man any sum of 

money he chooses to mention provided he 
will at once give it back again to me   
with one pound more4 ,it does not need  
much acuteness to see that it is a      
matter of indifference to me whether he 
chooses to mention one pound1 or ten1 or
 a hundred4 ,now suppose that instead of
 leaving it to his choice which of these
 sums is to be selected each time1 the  
two parties agree to leave it to chance4
 ,let them1 for instance1 draw a number 
out of a bag each time1 and let that be 
the sum which ,a gives to ,b under the  
prescribed conditions4 ,the case is not 
altered4 ,a still gains his pound each  
time1 for the introduction of the       
element of chance has not in any way    
touched this4 ,all that it does is to   
make this pound the result of an        
uncertain subtraction1 sometimes #aj    
minus #i1 sometimes #ej minus #di1 and  
so on4 ,it is these numbers only1 not   
their difference1 which he submits to   
luck1 and this is of no consequence     
whatever4  ,to suggest to any individual

 or company that they should consent to 
go on playing upon such terms as these  
would be too barefaced a proposal4 ,and 
yet the case in question is identical in
 principle1 and almost identical in     
form1 with this4 ,to offer to give a man
 any sum he likes to name provided he   
gives you back again that same sum plus 
one1 and to offer him any number of     
terms he pleases of the series #a1 #b1  
#d1 #h1 #af1 c41 provided you have the  
next term of the set1 are equivalent4   
,the only difference is that in the     
latter case the result is attained with 
somewhat more of arithmetical parade4   
,similarly equivalent are the processes 
in case we prefer to leave it to chance1
 instead of to choice1 to decide what   
sum or what number of terms shall be    
fixed upon4 ,this latter is what is     
really done in the case in question4 ,a 
man who consents to go on doubling his  
stake every time he wins1 is leaving    
nothing else to chance than the         
determination of the particular number  

of terms of such a geometrical series   
which shall be allowed to pass before he
 stops4  #ae4 ,it may be added that     
there is no special virtue in the       
particular series in question1 viz4 that
 in accordance with which the stake is  
doubled each time4 ,all that is needed  
is that the last term of the series     
should more than balance all the        
preceding ones4 ,any other series which 
increased faster than this geometrical  
one1 would answer the purpose as well or
 better4 ,nor is it necessary1 again1   
that the game should be an even or      
'fair' one4 ,chance1 be it remembered1  
affects nothing here but the number of  
terms to which the series attains on    
each occasion1 its final result being   
always arithmetically fixed4 ,when a    
penny is tossed up it is only on one of 
every two occasions that the series runs
 to more than two terms1 and so his     
fixed gains come in pretty regularly4   
,but unless he was playing for a limited
 time only1 it would not affect him if  

the series ran to two hundred terms2 it 
would merely take him somewhat longer to
 win his stakes4 ,a man might safely1   
for instance1 continue to lay an even   
bet that he would get the single prize  
in a lottery of a thousand tickets1     
provided he thus doubled1 or more than  
doubled1 his stake each time1 and       
unlimited credit was given4  #af4 ,so   
regarded1 the problem is simple enough1 
but there are two points in it to which 
attention may conveniently be directed4 
,in the first place1 it serves very     
pointedly to remind us of the           
distinction between a series of events  
7in this case the tosses of the penny7  
which really are subjects of chance1 and
 our conduct founded upon these events1 
which may or may not be so subject4#d   
,it is quite possible that this latter  
may be so contrived as to be in many    
respects a matter of absolute           
certainty1--a consideration1 ,i presume1
 familiar enough to professional betting
 men4 ,why is the ordinary way of       

betting on the throws of a penny fair to
 both parties8 ,because a 'fair' series 
is 'fairly' treated4 ,the heads and     
tails occur at random1 but on an average
 equally often1 and the stakes are      
either fixed or also arranged at random4
 ,if a man backs heads every time for   
the same amount1 he will of course in   
the long run neither win nor lose4      
,neither will he if he varies the stake 
every time1 provided he does not vary it
 in such a way as to make its amount    
dependent on the fact of his having won 
or lost the time before4 ,but he may1 if
 he pleases1 and the other party        
consents1 so arrange his stakes 7as in  
the case in question7 that ,chance1 if  
one might so express it1 does not get a 
fair chance4 ,here the human elements of
 choice and design have been so brought 
to bear upon a series of events which1  
regarded by themselves1 exhibit nothing 
but the physical characteristics of     
chance1 that the latter elements        
disappear1 and we get a result which is 

arithmetically certain4 ,other analogous
 instances might be suggested1 but the  
one before us has the merit of most     
ingeniously disguising the actual       
process4  #ag4 ,the meaning of the      
remark just made will be better seen by 
a comparison with the following case4   
,it has been attempted#e to explain the 
preponderance of male births over female
 by assuming that the chances of the two
 are equal1 but that the general desire 
to have a male heir tends to induce many
 unions to persist until the occurrence 
of this event1 and no longer4 ,it is    
supposed that in this way there would be
 a slight preponderance of families     
which consisted of one son only1 or of  
two sons and one daughter1 and so forth4
  ,this is quite fallacious 7as had been
 noticed by ,laplace1 in his ,essai72   
and there could not be a better instance
 chosen than this to show just what we  
can do and what we cannot do in the way 
of altering the luck in a real          
chance-succession of events4 ,to suppose

 that the number of actual births could 
be influenced in the way in question is 
exactly the same thing as to suppose    
that a number of gamblers could increase
 the ratio of heads to tails1 to        
something over one-half1 by each handing
 the coin to his neighbour as soon as he
 had thrown a head3 that they have only 
to leave off as soon as head has        
appeared2 an absurdity which we need not
 pause to explain at this stage4 ,the   
essential point about the ',martingale' 
is that1 whereas the occurrence of the  
events on which the stakes are laid is  
unaffected1 the stakes themselves can be
 so adjusted as to make the luck swing  
one way4  #ah4 ,in the second place1    
this example brings before us what has  
had to be so often mentioned already1   
namely1 that the series of ,probability 
are in strictness supposed to be        
interminable4 ,if therefore we allow    
either party to call upon us to stop1   
especially at a point which just happens
 to suit him1 we may get results        

decidedly opposed to the integrity of   
the theory4 ,in the case before us it is
 a necessary stipulation for ,a that he 
may be allowed to leave off when he     
wishes1 that is at one of the points at 
which the throw is in his favour4       
,without this stipulation he may be left
 a loser to any amount4  ,introduce the 
supposition that one party may          
arbitrarily call for a stoppage when it 
suits him and refuse to permit it       
sooner1 and almost any system of what   
would be otherwise fair play may be     
converted into a very one-sided         
arrangement4 ,indeed1 in the case in    
question1 ,a need not adopt this device 
of doubling the stakes every time he    
loses4 ,he may play with a fixed stake1 
and nevertheless insure that one party  
shall win any assigned sum1 assuming    
that the game is even and that he is    
permitted to play on credit4  #ai4 7,v47
 ,a common mistake is to assume that a  
very unlikely thing will not happen at  
all4 ,it is a mistake which1 when thus  

stated in words1 is too obvious to be   
committed1 for the meaning of an        
unlikely thing is one that happens at   
rare intervals2 if it were not assumed  
that the event would happen sometimes it
 would not be called unlikely1 but      
impossible4 ,this is an error which     
could scarcely occur except in vague    
popular misapprehension1 and is so      
abundantly refuted in works on          
,probability1 that it need only be      
touched upon briefly here4 ,it follows  
of course1 from our definition of       
,probability1 that to speak of a very   
rare combination of events as one that  
is 'sure never to happen1' is to use    
language incorrectly4 ,such a phrase may
 pass current as a loose popular        
exaggeration1 but in strictness it      
involves a contradiction4 ,the truth    
about such rare events cannot be better 
described than in the following         
quotation from ,de ,morgan3#f--  ,it is 
said that no person ever does arrive at 
such extremely improbable cases as the  

one just cited drawing the same ball    
five times running out of a bag         
containing twenty balls4 ,that a given  
individual should never throw an ace    
twelve times running on a single die1 is
 by far the most likely2 indeed1 so     
remote are the chances of such an event 
in any twelve trials 7more than         
#b1jjj1jjj1jjj to #a against it7 that it
 is unlikely the experience of any given
 country1 in any given century1 should  
furnish it4 ,but let us stop for a      
moment1 and ask ourselves to what this  
argument applies4 ,a person who rarely  
touches dice will hardly believe that   
doublets sometimes occur three times    
running2 one who handles them frequently
 knows that such is sometimes the fact4 
,every very practised user of those     
implements has seen still rarer         
sequences4 ,now suppose that a society  
of persons had thrown the dice so often 
as to secure a run of six aces observed 
and recorded1 the preceding argument    
would still be used against twelve4 ,and

 if another society had practised long  
enough to see twelve aces following each
 other1 they might still employ the same
 method of doubting as to a run of      
twenty-four2 and so on1 ad infinitum4   
,the power of imagining cases which     
contain long combinations so much       
exceeds that of exhibiting and arranging
 them1 that it is easy to assign a      
telegraph which should make a separate  
signal for every grain of sand in a     
globe as large as the visible universe1 
upon the hypothesis of the most         
space-penetrating astronomer4 ,the      
fallacy of the preceding objection lies 
in supposing events in number beyond our
 experience1 composed entirely of       
sequences such as fall within our       
experience4 ,it makes the past          
necessarily contain the whole1 as to the
 quality of its components2 and judges  
by samples4 ,now the least cautious     
buyer of grain requires to examine a    
handful before he judges of a bushel1   
and a bushel before he judges of a load4

 ,but relatively to such enormous       
numbers of combinations as are          
frequently proposed1 our experience does
 not deserve the title of a handful as  
compared with a bushel1 or even of a    
single grain4  #bj4 ,the origin of this 
inveterate mistake is not difficult to  
be accounted for4 ,it arises1 no doubt1 
from the exigencies of our practical    
life4 ,no man can bear in mind every    
contingency to which he may be exposed4 
,if therefore we are ever to do anything
 at all in the world1 a large number of 
the rarer contingencies must be left    
entirely out of account4 ,and the       
necessity of this oblivion is           
strengthened by the shortness of our    
life4 ,mathematically speaking1 it would
 be said to be certain that any one who 
lives long enough will be bitten by a   
mad dog1 for the event is not an        
impossible1 but only an improbable one1 
and must therefore come to pass in time4
 ,but this and an indefinite number of  
other disagreeable contingencies have on

 most occasions to be entirely ignored  
in practice1 and thence they come almost
 necessarily to drop equally out of our 
thought and expectation4 ,and when the  
event is one in itself of no importance1
 like a rare throw of the dice1 a great 
effort of imagination may be required1  
on the part of persons not accustomed to
 abstract mathematical calculation1 to  
enable them to realize the throw as     
being even possible4  ,attempts have    
sometimes been made to estimate what    
extremity of unlikelihood ought to be   
considered as equivalent to this        
practical zero point of belief4 ,in so  
far as such attempts are carried out by 
logicians1 or by those who are unwilling
 to resort to mathematical valuation of 
chances1 they must be regarded as merely
 a special form of the modal            
difficulties discussed in the last      
chapter1 and need not therefore be      
reconsidered here2 but a word or two may
 be added concerning the views of some  
who have looked at the matter from the  

mathematician's point of view4  ,the    
principal of these is perhaps ,buffon4  
,he has arrived at the estimate         
7,arithmtique ,morale ,,viii47 that this
 practical zero is equivalent to a      
chance of #a_/aj1jjj4 ,the grounds for  
selecting this fraction are found in the
 fact that1 according to the tables of  
mortality accessible to him1 it         
represents the chance of a man of #ef   
dying in the course of the next day4    
,but since no man under common          
circumstances takes the chance into the 
slightest consideration1 it follows that
 it is practically estimated as having  
no value4  ,it is obvious that this     
result is almost entirely arbitrary1 and
 in fact his reasons cannot be regarded 
as anything more than a slender         
justification from experience for       
adopting a conveniently simple fraction2
 a justification however which would    
apparently have been equally available  
in the case of any other fractions lying
 within wide limits of the one          

selected4#g  #ba4 ,there is one         
particular form of this error1 which1   
from the importance occasionally        
attached to it1 deserves perhaps more   
special examination4 ,as stated above1  
there can be no doubt that1 however     
unlikely an event may be1 if we 7loosely
 speaking7 vary the circumstances       
sufficiently1 or if1 in other words1 we 
keep on trying long enough1 we shall    
meet with such an event at last4 ,if we 
toss up a pair of dice a few times we   
shall get doublets2 if we try longer    
with three we shall get triplets1 and so
 on4 ,however unusual the event may be1 
even were it sixes a thousand times     
running1 it will come some time or other
 if we have only patience and vitality  
enough4 ,now apply this result to the   
letters of the alphabet4 ,suppose that  
one letter at a time is drawn from a bag
 which contains them all1 and is then   
replaced4 ,if the letters were written  
down one after another as they occurred1
 it would commonly be expected that they

 would be found to make mere nonsense1  
and would never arrange themselves into 
the words of any language known to men4 
,no more they would in general1 but it  
is a commonly accepted result of the    
theory1 and one which we may assume the 
reader to be ready to admit without     
further discussion1 that1 if the process
 were continued long enough1 words      
making sense would appear2 nay more1    
that any book we chose to               
mention1--,milton's ,paradise ,lost or  
the plays of ,shakespeare1 for          
example1--would be produced in this way 
at last4 ,it would take more days than  
we have space in this volume to         
represent in figures1 to make tolerably 
certain of obtaining the former of these
 works by thus drawing letters out of a 
bag1 but the desired result would be    
obtained at length4#h ,now many people  
have not unnaturally thought it         
derogatory to genius to suggest that its
 productions could have also been       
obtained by chance1 whilst others have  

gone on to argue1 ,if this be the case1 
might not the world itself in this      
manner have been produced by chance8    
#bb4 ,we will begin with the            
comparatively simple1 determinate1 and  
intelligible problem of the possible    
production of the works of a great human
 genius by chance4 ,with regard to this 
possibility1 it may be a consolation to 
some timid minds to be reminded that the
 power of producing the works of a      
,shakespeare1 in time1 is not confined  
to consummate genius and to mere chance4
 ,there is a third alternative1 viz4    
that of purely mechanical procedure4    
,any one1 down almost to an idiot1 might
 do it1 if he took sufficient time about
 the task4 ,for suppose that the        
required number of letters were procured
 and arranged1 not by chance1 but       
designedly1 and according to rules      
suggested by the theory of permutations3
 the letters of the alphabet and the    
number of them to be employed being     
finite1 every order in which they could 

occur would come in its due turn1 and   
therefore every thing which can be      
expressed in language would be arrived  
at some time or other4  ,there is really
 nothing that need shock any one in such
 a result4 ,its possibility arises from 
the following cause4 ,the number of     
letters1 and therefore of words1 at our 
disposal is limited2 whatever therefore 
we may desire to express in language    
necessarily becomes subject to          
corresponding limitation4 ,the possible 
variations of thought are literally     
infinite1 so are those of spoken        
language 7by intonation of the voice1   
c472 but when we come to words there is 
a limitation1 the nature of which is    
distinctly conceivable by the mind1     
though the restriction is one that in   
practice will never be appreciable1     
owing to the fact that the number of    
combinations which may be produced is so
 enormous as to surpass all power of the
 imagination to realize4#i ,the answer  
therefore is plain1 and it is one that  

will apply to many other cases as well1 
that to put a finite limit upon the     
number of ways in which a thing can be  
done1 is to determine that any one who  
is able and willing to try long enough  
shall succeed in doing it4 ,if a great  
genius condescends to perform it under  
these circumstances1 he must submit to  
the possibility of having his claims    
rivalled or disputed by the chance-man  
and idiot4 ,if ,shakespeare were limited
 to the use of eight or nine assigned   
words1 the time within which the latter 
agents might claim equality with him    
would not be very great4 ,as it is1     
having had the range of the ,english    
language at his disposal1 his reputation
 is not in danger of being assailed by  
any such methods4  #bc4 ,the case of the
 possible production of the world by    
chance leads us into an altogether      
different region of discussion4 ,we are 
not here dealing with figures the nature
 and use of which are within the fair   
powers of the understanding1 however the

 imagination may break down in          
attempting to realize the smallest      
fraction of their full significance4    
,the understanding itself is wandering  
out of its proper province1 for the     
conditions of the problem cannot be     
assigned4 ,when we draw letters out of a
 bag we know very well what we are      
doing2 but what is really meant by      
producing a world by chance8 ,by analogy
 of the former case1 we may assume that 
some kind of agent is                   
presupposed2--perhaps therefore the     
following supposition is less absurd    
than any other4 ,imagine some being1 not
 a ,creator but a sort of ,demiurgus1   
who has had a quantity of materials put 
into his hands1 and he assigns them     
their collocations and their laws of    
action1 blindly and at haphazard3 what  
are the odds that such a world as we    
actually experience should have been    
brought about in this way8  ,if it were 
worth while seriously to set about      
answering such a question1 and if some  

one would furnish us with the number of 
the letters of such an alphabet1 and the
 length of the work to be written with  
them1 we could proceed to indicate the  
result4 ,but so much as this may surely 
be affirmed about it2--that1 far from   
merely finding the length of this small 
volume insufficient for containing the  
figures in which the adverse odds would 
be given1 all the paper which the world 
has hitherto produced would be used up  
before we had got far on our way in     
writing them down4  #bd4 ,the most      
seductive form in which the difficulty  
about the occurrence of very rare events
 generally presents itself is probably  
this4 ',you admit 7some persons will be 
disposed to say7 that such an event may 
sometimes happen2 nay1 that it does     
sometimes happen in the infinite course 
of time4 ,how then am ,i to know that   
this occasion is not one of these       
possible occurrences8' ,to this1 one    
answer only can be given1--the same     
which must always be given where        

statistics and probability are          
concerned1--',the present may be such an
 occasion1 but it is inconceivably      
unlikely that it should be one4 ,amongst
 countless billions of times in which   
you1 and such as you1 urge this1 one    
person only will be justified2 and it is
 not likely that you are that one1 or   
that this is that occasion4'  #be4      
,there is another form of this practical
 inability to distinguish between one   
high number and another in the          
estimation of chances1 which deserves   
passing notice from its importance in   
arguments about heredity4 ,people will  
often urge an objection to the doctrine 
that qualities1 mental and bodily1 are  
transmitted from the parents to the     
offspring1 on the ground that there are 
a multitude of instances to the         
contrary1 in fact a great majority of   
such instances4 ,to raise this objection
 implies an utter want of appreciation  
of the very great odds which possibly   
may exist1 and which the argument in    

support of heredity implies do exist    
against any given person being          
distinguished for intellectual or other 
eminence4 ,this is doubtless partly a   
matter of definition1 depending upon the
 degree of rarity which we consider to  
be implied by eminence2 but taking any  
reasonable sense of the term1 we shall  
readily see that a very great proportion
 of failures may still leave an enormous
 preponderance of evidence in favour of 
the heredity doctrine4 ,take1 for       
instance1 that degree of eminence which 
is implied by being one of four         
thousand4 ,this is a considerable       
distinction1 though1 since there are    
about two thousand such persons to be   
found amongst the total adult male      
population of ,great ,britain1 it is far
 from implying any conspicuous genius4  
,now suppose that in examining the cases
 of a large number of the children of   
such persons1 we had found that #aii out
 of #bjj of them failed to reach the    
same distinction4 ,many persons would   

conclude that this was pretty conclusive
 evidence against any hereditary        
transmission4 ,to be able to adduce only
 one favourable1 as against #aii hostile
 instances1 would to them represent the 
entire break-down of any such theory4   
,the error1 of course1 is obvious       
enough1 and one which1 with the figures 
thus before him1 hardly any one could   
fail to avoid4 ,but if one may judge    
from common conversation and other such 
sources of information1 it is found in  
practice exceedingly difficult          
adequately to retain the conviction that
 even though only one in #bjj instances 
were favourable1 this would represent   
odds of about #bj to #a in favour of the
 theory4 ,if hereditary transmission did
 not prevail1 only one in #djjj sons    
would thus rival their fathers2 but we  
find actually1 let us say 7we are of    
course taking imaginary proportions     
here71 that one in #bjj does4 ,hence1 if
 the statistics are large enough to be  
satisfactory1 there has been some       

influence at work which has improved the
 chances of mere coincidence in the     
ratio of #bj to #a4 ,we are in fact so  
little able to realise the meaning of   
very large numbers1--that is1 to retain 
the ratios in the mind1 where large     
numbers are concerned1--that unless we  
repeatedly check ourselves by           
arithmetical considerations we are too  
apt to treat and estimate all beyond    
certain limits as equally vast and      
vague4  #bf4 7,,vi47 ,in discussing the 
nature of the connexion between         
,probability and ,induction1 we examined
 the claims of a rule commonly given for
 inferring the probability that an event
 which had been repeatedly observed     
would recur again4 ,i endeavoured to    
show that all attempts to obtain and    
prove such a rule were necessarily      
futile2 if these reasons were conclusive
 the employment of such a rule must of  
course be regarded as fallacious4 ,a few
 examples may conveniently be added     
here1 tending to show how instead of    

there being merely a single rule of     
succession we might better divide the   
possible forms into three classes4  7#a7
 ,in some cases when a thing has been   
observed to happen several times it     
becomes in consequence more likely that 
the thing should happen again4 ,this    
agrees with the ordinary form of the    
rule1 and is probably the case of most  
frequent occurrence4 ,the necessary     
vagueness of expression when we talk of 
the 'happening of a thing' makes it     
quite impossible to tolerate the rule in
 this general form1 but if we specialize
 it a little we shall find it assume a  
more familiar shape4 ,if1 for example1  
we have observed two or more properties 
to be frequently associated together in 
a succession of individuals1 we shall   
conclude with some force that they will 
be found to be so connected in future4  
,the strength of our conviction however 
will depend not merely on the number of 
observed coincidences1 but on far more  
complicated considerations2 for a       

discussion of which the reader must be  
referred to regular treatises on        
,inductive evidence4 ,or again1 if we   
have observed one of two events succeed 
the other several times1 the occurrence 
of the former will excite in most cases 
some degree of expectation of the       
latter4 ,as before1 however1 the degree 
of our expectation is not to be assigned
 by any simple formula2 it will depend  
in part upon the supposed intimacy with 
which the events are connected4 ,to     
attempt to lay down definite rules upon 
the subject would lead to a discussion  
upon laws of causation1 and the         
circumstances under which their         
existence may be inferred1 and therefore
 any further consideration of the matter
 must be abandoned here4  #bg4 7#b7 ,or1
 secondly1 the past recurrence may in   
itself give no valid grounds for        
inference about the future2 this is the 
case which most properly belongs to     
,probability4#aj ,that it does so belong
 will be easily seen if we bear in mind 

the fundamental conception of the       
science4 ,we are there introduced to a  
series1--for purposes of inference an   
indefinitely extended series1--of terms1
 about the details of which1            
information1 except on certain points1  
is not given2 our knowledge being       
confined to the statistical fact1 that1 
say1 one in ten of them has some        
attribute which we will call ,x4        
,suppose now that five of these terms in
 succession have been ,x1 what hint does
 this give about the sixth being also an
 ,x8 ,clearly none at all2 this past    
fact tells us nothing2 the formula for  
our inference is still precisely what it
 was before1 that one in ten being ,x it
 is one to nine that the next term is   
,x4 ,and however many terms in          
succession had been of one kind1        
precisely the same formula would still  
be given4  #bh4 ,the way in which events
 will justify the answer given by this  
formula is often misunderstood4 ,for the
 benefit therefore of those unacquainted

 with some of the conceptions familiar  
to mathematicians1 a few words of       
explanation may be added4 ,suppose then 
that we have had ,x twelve times in     
succession4 ,this is clearly an         
anomalous state of things4 ,to suppose  
anything like this continuing to occur  
would be obviously in opposition to the 
statistics1 which assert that in the    
long run only one in ten is ,x4 ,but how
 is this anomaly got over8 ,in other    
words1 how do we obviate the conclusion 
that ,x's must occur more frequently    
than once in ten times1 after such a    
long succession of them as we have now  
had8 ,many people seem to believe that  
there must be a diminution of ,x's      
afterwards to counterbalance their past 
preponderance4 ,this however would be   
quite a mistake2 the proportion in which
 they occur in future must remain the   
same throughout2 it cannot be altered if
 we are to adhere to our statistical    
formula4 ,the fact is that the          
rectification of the exceptional        

disturbance in the proportion will be   
brought about simply by the continual   
influx of fresh terms in the series4    
,these will in the long run neutralize  
the disturbance1 not by any special     
adaptation1 as it were1 for the purpose1
 but by the mere weight of their        
overwhelming numbers4 ,at every stage   
therefore1 in the succession1 whatever  
might have been the number and nature of
 the preceding terms1 it will still be  
true to say that one in ten of the terms
 will be an ,x4  ,if we had to do only  
with a finite number of terms1 however  
large that number might be1 such a      
disturbance as we have spoken of would1 
it is true1 need a special alteration in
 the subsequent proportions to          
neutralize its effects4 ,but when we    
have to do with an infinite number of   
terms1 this is not the case2 the 'limit'
 of the series1 which is what we then   
have to deal with1 is unaffected by     
these temporary disturbances4 ,in the   
continued progress of the series we     

shall find1 as a matter of fact1 more   
and more of such disturbances1 and these
 of a more and more exceptional         
character4 ,but whatever the point we   
may occupy at any time1 if we look      
forward or backward into the indefinite 
extension of the series1 we shall still 
see that the ultimate limit to the      
proportion in which its terms are       
arranged remains the same2 and it is    
with this limit1 as above mentioned1    
that we are concerned in the strict     
rules of ,probability4  ,the most       
familiar example1 perhaps1 of this kind 
is that of tossing up a penny4 ,suppose 
we have had four heads in succession2   
people#aa have tolerably realized by now
 that 'head the fifth time' is still an 
even chance1 as 'head' was each time    
before1 and will be ever after4 ,the    
preceding paragraph explains how it is  
that these occasional disturbances in   
the average become neutralized in the   
long run4  #bi4 7#c7 ,there are other   
cases which1 though rare1 are by no     

means unknown1 in which such an         
inference as that obtained from the     
,rule of ,succession would be the direct
 reverse of the truth4 ,the oftener a   
thing happens1 it may be1 the more      
unlikely it is to happen again4 ,this is
 the case whenever we are drawing things
 from a limited source 7as balls from a 
bag without replacing them71 or whenever
 the act of repetition itself tends to  
prevent the succession 7as in giving    
false alarms74  ,i am quite ready to    
admit that we believe the results       
described in the last two classes on the
 strength of some such general          
,inductive rule1 or rather principle1 as
 that involved in the first4 ,but it    
would be a great error to confound this 
with an admission of the validity of the
 rule in each special instance4 ,we are 
speaking about the application of the   
rule to individual cases1 or classes of 
cases2 this is quite a distinct thing1  
as was pointed out in a previous        
chapter1 from giving the grounds on     

which we rest the rule itself4 ,if a man
 were to lay it down as a universal     
rule1 that the testimony of all persons 
was to be believed1 and we adduced an   
instance of a man having lied1 it would 
not be considered that he saved his rule
 by showing that we believed that it was
 a lie on the word of other persons4    
,but it is perfectly consistent to give 
as a merely general1 but not universal1 
rule1 that the testimony of men is      
credible2 then to separate off a second 
class of men whose word is not to be    
trusted1 and finally1 if any one wants  
to know our ground for the second rule1 
to rest it upon the first4 ,if we were  
speaking of necessary laws1 such a      
conflict as this would be as hopeless as
 the old ',cretan' puzzle in logic2 but 
in instances of ,inductive and          
,analogical extension it is perfectly   
harmless4  #cj4 ,a familiar example will
 serve to bring out the three different 
possible conclusions mentioned above4   
,we have observed it rain on ten        

successive days4 ,a and ,b conclude     
respectively for and against rain on the
 eleventh day2 ,c maintains that the    
past rain affords no data whatever for  
an opinion4 ,which is right8 ,we really 
cannot determine  priori4 ,an appeal    
must be made to direct observation1 or  
means must be found for deciding on     
independent grounds to which class we   
are to refer the instance4 ,if1 for     
example1 it were known that every       
country produces its own rain1 we should
 choose the third rule1 for it would be 
a case of drawing from a limited supply4
 ,if again we had reasons to believe    
that the rain for our country might be  
produced anywhere on the globe1 we      
should probably conclude that the past  
rainfall threw no light whatever on the 
prospect of a continuance of wet        
weather1 and therefore take the second4 
,or if1 finally1 we knew that rain came 
in long spells or seasons1 as in the    
tropics1 then the occurrence of ten wet 
days in succession would make us believe

 that we had entered on one of these    
seasons1 and that therefore the next day
 would probably resemble the preceding  
ten4  ,since then all these forms of    
such an ,inductive rule are possible1   
and we have often no  priori grounds for
 preferring one to another1 it would    
seem to be unreasonable to attempt to   
establish any universal formula of      
anticipation4 ,all that we can do is to 
ascertain what are the circumstances    
under which one or other of these rules 
is1 as a matter of fact1 found to be    
applicable1 and to make use of it under 
those circumstances4  #ca4 7,,vii47 ,in 
the cases discussed in 7,v47 the almost 
infinitely small chances with which we  
were concerned were rightly neglected   
from all practical consideration1       
however proper it might be1 on          
speculative grounds1 to keep our minds  
open to their actual existence4 ,but it 
has often occurred to me that there is a
 common error in neglecting to take them
 into account when they may1 though     

individually small1 make up for their   
minuteness by their number4 ,as the     
mathematician would express it1 they may
 occasionally be capable of being       
integrated into a finite or even        
considerable magnitude4  ,for instance1 
we may be confronted with a difficulty  
out of which there appears to be only   
one appreciably possible mode of escape4
 ,the attempt is made to force us into  
accepting this1 however great the odds  
apparently are against it1 on the ground
 that improbable as it may seem1 it is  
at any rate vastly more probable than   
any of the others4 ,i can quite admit   
that1 on practical grounds1 we may often
 find it reasonable to adopt this       
course2 for we can only act on one      
supposition1 and we naturally and       
rightly choose1 out of a quantity of    
improbabilities1 the least improbable4  
,but when we are not forced to act1 no  
such decisive preference is demanded of 
us4 ,it is then perfectly reasonable to 
refuse assent to the proposed           

explanation2 even to say distinctly that
 we do not believe it1 and at the same  
time to decline1 at present1 to accept  
any other explanation4 ,we remain1 in   
fact1 in a state of suspense of         
judgment1 a state perfectly right and   
reasonable so long as no action         
demanding a specific choice is forced   
upon us4 ,one alternative may be        
decidedly probable as compared with any 
other individually1 but decidedly       
improbable as compared with all others  
collectively4 ,this in itself is        
intelligible enough2 what people often  
fail to see is that there is no         
necessary contradiction between saying  
and feeling this1 and yet being prepared
 vigorously to act1 when action is      
forced upon us1 as though this          
alternative were really the true one4   
#cb4 ,to take a specific instance1 this 
way of regarding the matter has often   
occurred to me in disputes upon         
',spiritualist' manifestations4 ,assent 
is urged upon us because1 it is said1 no

 other possible solution can be         
suggested4 ,it may be quite true that   
apparently overwhelming difficulties may
 lie as against each separate           
alternative solution2 but is it always  
sufficiently realized how numerous such 
solutions may be8 ,no matter that each  
individually may be almost incredible3  
they ought all to be massed together and
 thrown into the scale against the      
proffered solution1 when the only       
question asked is1 ,are we to accept    
this solution8 ,there is no unfairness  
in such a course4 ,we are perfectly     
ready to adopt the same plan against any
 other individual alternative1 whenever 
any person takes to claiming this as the
 solution of the difficulty4 ,we are    
looking at the matter from a purely     
logical point of view1 and are quite    
willing1 so far1 to place every         
solution1 spiritualist or otherwise1    
upon the same footing4 ,the partisans of
 every alternative are in somewhat the  
same position as the members of a       

deliberative assembly1 in which no one  
will support the motion of any other    
member4 ,every one can aid effectively  
in rejecting every other motion1 but no 
one can succeed in passing his own4     
,pressure of urgent necessity may       
possibly force them out of this state of
 practical inaction1 by1 so to say1     
breaking through the opposition at some 
point of least resistance2 but unless   
aided by some such pressure they are    
left in a state of hopeless dead-lock4  
#cc4 ,assuming that the spiritualistic  
solution admits of1 and is to receive1  
scientific treatment1 this1 it seems to 
me1 is the conclusion to which one might
 sometimes be led in the face of the    
evidence offered4 ,we might have to say 
to every individual explanation1 ,it is 
incredible1 ,i cannot accept it2 and    
unless circumstances should 7which it is
 hardly possible that they should7 force
 us to a hasty decision1--a decision1   
remember1 which need indicate no        
preference of the judgment beyond what  

is just sufficient to turn the scale in 
its favour as against any other single  
alternative1--we leave the matter thus  
in abeyance4 ,it will very likely be    
urged that one of the explanations      
7assuming that all the possible ones had
 been included7 must be true2 this we   
readily admit4 ,it will probably also be
 urged that 7on the often-quoted        
principle of ,butler7 we ought forthwith
 to accept the one which1 as compared   
with the others1 is the most plausible1 
whatever its absolute worth may be4     
,this seems distinctly an error4 ,to say
 that such and such an explanation is   
the one we should accept1 if            
circumstances compelled us to anticipate
 our decision1 is quite compatible with 
its present rejection4 ,the only        
rational position surely is that of     
admitting that the truth is somewhere   
amongst the various alternatives1 but   
confessing plainly that we have no such 
preference for one over another as to   
permit our saying anything else than    

that we disbelieve each one of them4    
#cd4 7,,viii47 ,the very common fallacy 
of 'judging by the event1' as it is     
generally termed1 deserves passing      
notice here1 as it clearly belongs to   
,probability rather than to ,logic2     
though its nature is so obvious to those
 who have grasped the general principles
 of our science1 that a very few words  
of remark will suffice4 ,in one sense   
every proposition must consent to be    
judged by the event1 since this is      
merely1 in other words1 submitting it to
 the test of experience4 ,but there is  
the widest difference between the test  
appropriate to a universal proposition  
and that appropriate to a merely        
proportional or statistical one4 ,the   
former is subverted by a single         
exception2 the latter not merely admits 
exceptions1 but implies them4 ,nothing1 
however1 is more common than to blame   
advice 7in others7 because it has       
happened to turn out unfortunately1 or  
to claim credit for it 7in oneself7     

because it has happened to succeed4 ,of 
course if the conclusion was avowedly   
one of a probable kind we must be       
prepared with complacency to accept a   
hostile event1 or even a succession of  
them2 it is not until the succession    
shows a disposition to continue over    
long that suspicion and doubt should    
arise1 and then only by a comparison of 
the degree of the assigned probability1 
and the magnitude of the departure from 
it which experience exhibits4 ,for any  
single failure the reply must be1 'the  
advice was sound' 7supposing1 that is1  
that it was to be justified in the long 
run71 'and ,i shall offer it again under
 the same circumstances4'  #ce4 ,the    
distinction drawn in the above instance 
deserves careful consideration2 for     
owing to the wide difference between the
 kind of propositions dealt with in     
,probability and in ordinary ,logic1 and
 the consequent difference in the nature
 of the proof offered1 it is quite      
possible for arguments of the same      

general appearance to be valid in the   
former and fallacious in the latter1 and
 conversely4  ,for instance1 take the   
well-known fallacy which consists in    
simply converting a universal           
affirmative1 i4e4 in passing from ,all  
,a is ,b to ,all ,b is ,a4 ,when1 as in 
common ,logic1 the conclusion is to be  
as certain as the premise1 there is not 
a word to be said for such a step4 ,but 
if we look at the process with the more 
indulgent eye of ,induction or          
,probability we see that a very fair    
case may sometimes be made out for it4  
,the mere fact that ',some ,b is ,a'    
raises a certain presumption that any   
particular ,b taken at random will be an
 ,a4 ,there is some reason1 at any rate1
 for the belief1 though in the absence  
of statistics as to the relative        
frequency of ,a and ,b we are unable to 
assign a value to this belief4 ,i       
suspect that there may be many cases in 
which a man has inferred that some      
particular ,b is an ,a on the ground    

that ,all ,a is ,b1 who might justly    
plead in his behalf that he never meant 
it to be a necessary1 but only a        
probable inference4 ,the same remarks   
will of course apply also to the logical
 fallacy of ,undistributed ,middle4     
,now for a case of the opposite kind1   
i4e4 one in which ,probability fails us1
 whereas the circumstances seem closely 
analogous to those in which ordinary    
inference would be able to make a stand4
 ,suppose that ,i know that one letter  
in a million is lost when in charge of  
the post4 ,i write to a friend and get  
no answer4 ,have ,i any reason to       
suppose that the fault lies with him8   
,here is an event 7viz4 the loss of the 
letter7 which has certainly happened2   
and we suppose that1 of the only two    
causes to which it can be assigned1 the 
'value1' i4e4 statistical frequency1 of 
one is accurately assigned1 does it not 
seem natural to suppose that something  
can be inferred as to the likelihood    
that the other cause had been operative8

 ,to say that nothing can be known about
 its adequacy under these circumstances 
looks at first sight like asserting that
 an equation in which there is only one 
unknown term is theoretically insoluble4
  ,as examples of this kind have been   
amply discussed in the chapter upon     
,inverse rules of ,probability ,i need  
do no more here than remind the reader  
that no conclusion whatever can be drawn
 as to the likelihood that the fault lay
 with my friend rather than with the    
,post ,office4 ,unless we either know1  
or make some assumption about1 the      
frequency with which he neglects to     
answer the letters he receives1 the     
problem remains insoluble4  ,the reason 
why the apparent analogy1 indicated     
above1 to an equation with only one     
unknown quantity1 fails to hold good1 is
 that for the purposes of ,probability  
there are really two unknown quantities4
 ,what we deal with are proportional or 
statistical propositions4 ,now we are   
only told that in the instance in       

question the letter was lost1 not that  
they were found to be lost in such and  
such a proportion of cases4 ,had this   
latter information been given to us we  
should really have had but one unknown  
quantity to determine1 viz4 the relative
 frequency with which my correspondent  
neglects to answer his letters1 and we  
could then have determined this with the
 greatest ease4   #a4 ,discussed by ,mr 
,f4 ,y4 ,edgeworth1 in the ,phil4 ,mag4 
for ,april1   #ahhg4  #b4 ,journal of   
the ,statistical ,soc4 7,vol4 ,,xlii4 p4
 #cbh7 ,dare one   suspect a joke8  #c4 
,it appears to have been long known to  
gamblers under the name of   the        
,martingale4 ,there is a paper by       
,babbage 7,trans4 of ,royal   ,soc4 of  
,edinburgh1 for #ahbc7 which discusses  
certain points   connected with it1 but 
scarcely touches on the subject of the  
sections which follow4  #d4 ,attention  
will be further directed to this        
distinction in the   chapter on         
,insurance and ,gambling4  #e4 ,as by   

,prvost in the ,bibliothque ,universelle
 de ,genve1   ,oct4 #ahbi4 ,the         
explanation is noted1 and apparently    
accepted1 by   ,quetelet 7,physique     
,sociale1 ,i4 #aga74  #f4 ,essay on     
,probabilities1 p4 #abf4  #g4 ,this     
theoretical or absolute neglect of what 
is very rare must not   be confused with
 the practical neglect sometimes        
recommended by   astronomical and other 
observers4 ,a criterion1 known as       
,chauvenet's1   for indicating the      
limits of such rejection will be found  
described   in ,mr ,merriman's ,least   
,squares 7p4 #aff74 ,but this rests on  
the   understanding that a smaller      
balance of error would thus result in   
the long run4 ,the very rare event is   
deliberately rejected1 not   overlooked4
  #h4 ,the process of calculation may be
 readily indicated4 ,there are1   say1  
about #cej1jjj letters in the work in   
question4 ,since any of the   #bf       
letters of the alphabet may be drawn    
each time1 the possible   number of     

combinations would be #bfcej1jjj2 a     
number which1 as may   easily be        
inferred from a table of logarithms1    
would demand for its   expression nearly
 #ejj1jjj figures4 ,only one of these   
combinations is   favourable1 if we     
reject variations of spelling4 ,hence   
unity divided   by this number would    
represent the chance of getting the     
desired   result by successive random   
selection of the required number of     
#cej1jjj letters4    ,if this chance is 
thought too small1 and any one asks how 
often the   above random selection must 
be repeated in order to give him odds of
   #b to #a in favour of success1 this  
also can be easily shown4 ,if the       
chance of an event on each occasion is  
#a_/n1 the chance of getting it   once  
at least in n trials is #a - 77n -      
#a7_/n7n2 for we shall do   this unless 
we fail n times running4 ,when 7as in   
the case in   question7 n is very large1
 this may be shown algebraically to be  
equivalent to odds of about #b to #a4   

,that is1 when we have drawn the        
requisite quantity of letters a number  
of times equal to the   inconceivably   
great number above represented1 it is   
still only #b to   #a that we shall have
 secured what we want3--and then we have
 to   recognize it4  #i4 ,the longest   
life which could reasonably be          
attributed to any   language would of   
course dwindle into utter insignificance
 in the   face of such periods of time  
as are being here arithmetically        
contemplated4  #aj4 ,we are here        
assuming of course that the ultimate    
limit to which   our average tends is   
known1 either from knowledge of the     
causes or   from previous extensive     
experience4 ,we are assuming that e4g4  
the   die is known to be a fair one2 if 
this is not known but a possible   bias 
has to be inferred from its observed    
performances1 the case   falls under the
 former head4  #aa4 ,except indeed the  
gamblers4 ,according to a gambling      
acquaintance   whom ,houdin1 the        

conjurer1 describes himself as having   
met at ,spa1   the oftener a particular 
combination has occurred the more       
certain   it is that it will not be     
repeated at the next coup3 this is the  
groundwork of all theories of           
probabilities and is termed the         
maturity of chances 7,card-sharping     
exposed1 p4 #he74     ,,chapter ,,xv4   
,,insurance ,,and ,,gambling4  #a4 ,if  
the reader will recall to mind the      
fundamental postulate of the ,science of
 ,probability1 established and explained
 in the first few chapters1 and so      
abundantly illustrated since1 he will   
readily recognize that the two opposite 
characteristics of individual           
irregularity and average regularity will
 naturally be differently estimated by  
different minds4 ,to some persons the   
elements of uncertainty may be so       
painful1 either in themselves or in     
their consequences1 that they are       
anxious to adopt some means of          
diminishing them4 ,to others the        

ultimate regularity of life1 at any rate
 within certain departments1 its        
monotony as they consider it1 may be so 
wearisome that they equally wish to     
effect some alteration and improvement  
in its characteristics4 ,we shall       
discuss briefly these mental tendencies1
 and the most simple and obvious modes  
of satisfying them4  ,to some persons1  
as we have said1 the world is all too   
full of change and irregularity and     
consequent uncertainty4 ,civilization   
has done much to diminish these         
characteristics in certain directions1  
but it has unquestionably aggravated    
them in other directions1 and it might  
not be very easy to say with certainty  
in which of these respects its operation
 has been1 at present1 on the whole most
 effective4 ,the diminution of          
irregularity is exemplified1 amongst    
other things1 in the case of the staple 
products which supply our necessary food
 and clothing4 ,with respect to them1   
famine and scarcity are by comparison   

almost unknown now1 at any rate in      
tolerably civilized communities4 ,as a  
consequence of this1 and of the vast    
improvements in the means of            
transporting goods and conveying        
intelligence1 the fluctuations in the   
price of such articles are much less    
than they once were4 ,in other          
directions1 however1 the reverse has    
been the case4 ,fashion1 for instance1  
now induces so many people in every     
large community simultaneously to desire
 the same thing1 that great fluctuations
 in value may ensue4 ,moreover a whole  
group of causes 7to enter upon any      
discussion of which would be to trench  
upon the ground of ,political ,economy7 
combine to produce great and frequent   
variations in matters concerning credit 
and the currency1 which formerly had no 
existence4 ,bankruptcy1 for instance1 is
 from the nature of the case1 almost    
wholly a creation of modern times4 ,we  
will not attempt to strike any balance  
between these opposite results of modern

 civilization1 beyond remarking that in 
matters of prime importance the actual  
uncertainties have been probably on the 
whole diminished1 whereas in those which
 affect the pocket rather than the life1
 they have been rather increased4 ,it   
might also be argued with some          
plausibility that in cases where the    
actual uncertainties have not become    
greater1 they have for all practical    
purposes done so1 by their consequences 
frequently becoming more serious1 or by 
our estimate of these consequences      
becoming higher4  #b4 ,however the above
 question1 as to the ultimate balance of
 gain or loss1 should be decided1 there 
can be no doubt that many persons find  
the present amount of uncertainty in    
some of the affairs of life greater than
 suits their taste4 ,how are they to    
diminish it8 ,something of course may be
 done1 as regards the individual cases1 
by prudence and foresight4 ,our houses  
may be built with a view not to take    
fire so readily1 or precautions may be  

taken that there shall be fire-engines  
at hand4 ,in the warding off of death   
from disease and accident1 something may
 be done by every one who chooses to    
live prudently4 ,precautions of the     
above kind1 however1 do not introduce   
any questions of ,probability4 ,these   
latter considerations only come in when 
we begin to invoke the regularity of the
 average to save us from the            
irregularities of the details4 ,we      
cannot1 it is true1 remove the          
uncertainty in itself1 but we can so act
 that the consequences of that          
uncertainty shall be less to us1 or to  
those in whom we are interested4 ,take  
the case of ,life ,insurance4 ,a        
professional man who has nothing but the
 income he earns to depend upon1 knows  
that the whole of that income may vanish
 in a moment by his death4 ,this is a   
state of things which he cannot prevent2
 and if he were the only one in such a  
position1 or were unable or unwilling to
 combine with his fellow-men1 there     

would be nothing more to be done in the 
matter except to live within his income 
as much as possible1 and so leave a     
margin of savings4  #c4 ,there is       
however an easy mode of escape for him4 
,all that he has to do is to agree with 
a number of others1 who are in the same 
position as himself1 to make up1 so to  
say1 a common purse4 ,they may resolve  
that those of their number who live to  
work beyond the average length of life  
shall contribute to support the families
 of those who die earlier4 ,if a few    
only concurred in such a resolution they
 would not gain very much1 for they     
would still be removed by but a slight  
step from that uncertainty which they   
are seeking to escape4 ,what is         
essential is that a considerable number 
should thus combine so as to get the    
benefit of that comparative regularity  
which the average1 as is well known1    
almost always tends to exhibit4  #d4    
,the above simple considerations really 
contain the essence of all insurance4   

,such points as the fact that the       
agreement for indemnity extends only to 
a certain definite sum of money2 and    
that instead of calling for an          
occasional general contribution at the  
time of the death of each member they   
substitute a fixed annual premium1 out  
of the proceeds of which the payment is 
to be made1 are merely accidents of     
convenience and arrangement4 ,insurance 
is simply equivalent to a mutual        
contract amongst those who dread the    
consequences of the uncertainty of their
 life or employment1 that they will     
employ the aggregate regularity to      
neutralize as far as possible the       
individual irregularity4 ,they know that
 for every one who gains by such a      
contract another will lose as much2 or  
if one gains a great deal many must have
 lost a little4 ,they know also that    
hardly any of their number can expect to
 find the arrangement a 'fair' one1 in  
the sense that they just get back again 
what they have paid in premiums1 after  

deducting the necessary expenses of     
management2 but they deliberately prefer
 this state of things4 ,they consist of 
a body of persons who think it decidedly
 better to leave behind them a          
comparatively fixed fortune1 rather than
 one which is extremely uncertain in    
amount2 although they are perfectly     
aware that1 owing to the unavoidable    
expenses of managing the affairs of such
 a society1 the comparatively fixed sum1
 so to be left1 will be a trifle less   
than the average fortunes which would   
have been left had no such system of    
insurance been adopted4  ,as this is not
 a regular treatise upon ,insurance no  
more need be said upon the exact nature 
of such societies1 beyond pointing out  
that they are of various different      
kinds4 ,sometimes they really are what  
we have compared them with1 viz4 mutual 
agreements amongst a group of persons to
 make up each other's losses to a       
certain extent4 ,into this category fall
 the ,mutual ,insurance ,societies1     

,benefit ,societies1 ,trades ,unions 7in
 respect of some of their functions71   
together with innumerable other         
societies which go by various names4    
,sometimes they are companies worked by 
proprietors or shareholders for a       
profit1 like any other industrial       
enterprise4 ,this is the case1 ,i       
believe1 with the majority of the       
ordinary ,life ,insurance ,societies4   
,sometimes1 again1 it is the ,state     
which undertakes the management1 as in  
the case of our ,post ,office ,insurance
 business4  #e4 ,it is clear that there 
is no necessary limit to the range of   
application of this principle4#a ,it is 
quite conceivable that the majority of  
the inhabitants of some nation might be 
so enamoured of security that they      
should devise a grand insurance society 
to cover almost every concern in life4  
,they could not indeed abolish          
uncertainty1 for the conditions of life 
are very far from permitting this1 but  
they could without much difficulty get  

rid of the worst of the consequences of 
it4 ,they might determine to insure not 
merely their lives1 houses1 ships1 and  
other things in respect of which sudden 
and total loss is possible1 but also to 
insure their business2 in the sense of  
avoiding not only bankruptcy1 but even  
casual bad years1 on the same principle 
of commutation4 ,unfamiliar as such an  
aim may appear when introduced in this  
language1 it is nevertheless one which  
under a name of suspicious import to the
 conservative classes has had a good    
deal of attention directed to it4 ,it is
 really scarcely anything else than     
,communism1 which might indeed be       
defined as a universal and compulsory#b 
insurance society which is to take      
account of all departments of business1 
and1 in some at least of its forms1 to  
invade the province of social and       
domestic life as well4  ,although       
nothing so comprehensive as this is     
likely to be practically carried out on 
any very large scale1 it deserves notice

 that the principle itself is steadily  
spreading in every direction in matters 
of detail4 ,it is1 for instance1 the    
great complaint against ,trades ,unions 
that they too often seek to secure these
 results in respect of the equalization 
of the workmen's wages1 thus insuring to
 some degree against incompetence1 as   
they rightly and wisely do against      
illness and loss of work4 ,again1 there 
is the ,tradesman's ,mutual ,protection 
,society1 which insures against the     
occasional loss entailed by the         
necessity of having to conduct          
prosecutions at law4 ,there are         
societies in many towns for the         
prosecution of petty thefts1 with the   
object of escaping the same uncertain   
and perhaps serious loss4 ,amongst      
instances of insurance for the people   
rather than by them1 there is of course 
the giant example of the ,english ,poor 
,law1 in which the resemblance to an    
initial ,communistic system becomes very
 marked4 ,the poor are insured against  

loss of work arising not only from      
illness and old age1 but from any cause 
except wilful idleness4 ,they do not1 it
 is true1 pay the whole premium1 but    
since they mostly bear some portion of  
the burden of municipal and county      
taxation they must certainly be         
considered as paying a part of the      
premium4 ,in some branches also of the  
public and private services the system  
is adopted of deducting a percentage    
from the wage or salary1 for the purpose
 of a semi-compulsory insurance against 
death1 illness or superannuation4  #f4  
,closely connected with ,insurance1 as  
an application of ,probability1 though  
of course by contrast1 stands ,gambling4
 ,though we cannot1 in strictness1 term 
either of these practices the converse  
of the other1 it seems nevertheless     
correct to say that they spring from    
opposite mental tendencies4 ,some       
persons1 as has been said1 find life too
 monotonous for their taste1 or rather  
the region of what can be predicted with

 certainty is too large and predominant 
in their estimation4 ,they can easily   
adopt two courses for securing the      
changes they desire4 ,they may1 for one 
thing1 aggravate and intensify the      
results of events which are             
comparatively incapable of prevision1   
these events not being in themselves of 
sufficient importance to excite any     
strong emotions4 ,the most obvious way  
of doing this is by betting upon them4  
,or again1 they may invent games or     
other pursuits1 the individual          
contingencies of which are entirely     
removed from all possible human         
prevision1 and then make heavy money    
consequences depend upon these          
contingencies4 ,this is gambling proper1
 carried on mostly by means of cards and
 dice and the roulette4  ,the gambling  
spirit1 as we have said1 seeks for the  
excitement of uncertainty and variety4  
,when therefore people make a long      
continued practice of playing1          
especially if the stakes for which they 

play are moderate in comparison with    
their fortune1 this uncertainty from the
 nature of the case begins to diminish4 
,the thoroughly practised gambler1 if he
 possesses more than usual skill 7in    
games where skill counts for something71
 must be regarded as a man following a  
profession1 though a profession for the 
most part of a risky and exciting kind1 
to say nothing of its ignoble and often 
dishonest character4 ,if1 on the other  
hand1 his skill is below the average1 or
 the game is one in which skill does not
 tell and the odds are slightly in      
favour of his antagonist1 as in the     
gaming tables1 one light in which he can
 be regarded is that of a man who is    
following a favourite amusement2 if this
 amusement involves a constant annual   
outlay on his part1 that is nothing more
 than what has to be said of most other 
amusements4  #g4 ,we cannot1 of course1 
give such a rational explanation as the 
above in every case4 ,there are plenty  
of novices1 and plenty of fanatics1 who 

go on steadily losing in the full       
conviction that they will eventually    
come out winners4 ,but it is hard to    
believe that such ignorance1 or such    
intellectual twist1 can really be so    
widely prevalent as would be requisite  
to constitute them the rule rather than 
the exception4 ,there must surely be    
some very general impulse which is      
gratified by such resources1 and it is  
not easy to see what else this can be   
than a love of that variety and         
consequent excitement which can only be 
found in perfection where exact         
prevision is impossible4  ,it is of     
course very difficult to make any       
generalization here as to the           
comparative prevalence of various       
motives amongst mankind2 but when one   
considers what is the difference which  
most quiet ordinary whist players feel  
between a game for 'love' and one in    
which there is a small stake1 one cannot
 but assign a high value to the         
influence of a wish to emphasize the    

excitement of loss and gain4  ,i would  
not for a moment underrate the practical
 dangers which are found to attend the  
practice of gambling4 ,it is remarked   
that the gambler1 if he continues to    
play for a long time1 is under an almost
 irresistible impulse to increase his   
stakes1 and so re-introduce the element 
of uncertainty4 ,it is in fact this     
tendency to be thus led on1 which makes 
the principal danger and mischief of the
 practice4 ,risk and uncertainty are    
still such normal characteristics of    
even civilized life1 that the mere      
extension of such tendencies into new   
fields does not in itself offer any very
 alarming prospect4 ,it is only to be   
deprecated in so far as there is a      
danger1 which experience shows to be no 
trifling one1 that the fascination found
 in the pursuit should lead men into    
following it up into excessive          
lengths4#c  #h4 ,the above general      
treatment of ,gambling and ,insurance   
seems to me the only rational and sound 

principle of division2--namely1 that on 
which the different practices which1    
under various names1 are known as       
gambling or insurance1 are arranged in  
accordance with the spirit of which they
 are the outcome1 and therefore of the  
results which they are designed to      
secure4 ,if we were to attempt to judge 
and arrange them according to the names 
which they currently bear1 we should    
find ourselves led to no kind of        
systematic division whatever2 the fact  
being that since they all alike involve1
 as their essential characteristic1     
payments and receipts1 one or both of   
which are necessarily uncertain in their
 date or amount1 the names may often be 
interchanged4  ,for instance1 a lottery 
and an ordinary insurance society       
against accident1 if we merely look to  
the processes performed in them1 are to 
all intents and purposes identical4 ,in 
each alike there is a small payment     
which is certain in amount1 and a great 
receipt which is uncertain in amount4 ,a

 great many persons pay the small       
premium1 whereas a few only of their    
number obtain a prize1 the rest getting 
no return whatever for their outlay4 ,in
 each case alike1 also1 the aggregate   
receipts and losses are intended to     
balance each other1 after allowing for  
the profits of those who carry on the   
undertaking4 ,but of course when we take
 into account the occasions upon which  
the insurers get their prizes1 we see   
that there is all the difference in the 
world between receiving them at         
haphazard1 as in a lottery1 and         
receiving them as a partial set-off to a
 broken limb or injured constitution1 as
 in the insurance society4  ,again1 the 
language of betting may be easily made  
to cover almost every kind of insurance4
 ,indeed ,de ,morgan has described life 
insurance as a bet which the individual 
makes with the company1 that he will not
 live beyond a certain age4 ,if he dies 
young1 he is pecuniarily a gainer1 if he
 dies late he is a loser4#d ,here1 too1 

though the expression is technically    
quite correct 7since any such deliberate
 risk of money1 upon an unproductive    
venture1 may fall under the definition  
of a bet71 there is the broadest        
distinction between betting with no     
other view whatever than that of risking
 money1 and betting with the view of    
diminishing risk and loss as much as    
possible4 ,in fact1 if the language of  
sporting life is to be introduced into  
the matter1 we ought1 ,i presume1 to    
speak of the insurer as 'hedging'       
against his death4  #i4 ,again1 in      
,tontines we have a system of what is   
often called ,insurance1 and in certain 
points rightly so1 but which is to all  
intents and purposes simply and         
absolutely a gambling transaction4 ,they
 have been entirely abandoned1 ,i       
believe1 for some time1 but were once   
rather popular1 especially in ,france4  
,on this plan the ,state1 or whatever   
society manages the business1 does not  
gain anything until the last member of  

the ,tontine is dead4 ,as the number of 
the survivors diminishes1 the same      
sum-total of annuities still continues  
to be paid amongst them1 as long as any 
are left alive1 so that each receives a 
gradually increasing sum4 ,hence those  
who die early1 instead of receiving the 
most1 as on the ordinary plan1 receive  
the least2 for at the death of each     
member the annuity ceases absolutely1 so
 far as he and his relations are        
concerned4 ,the whole affair therefore  
is to all intents and purposes a        
gigantic system of betting1 to see which
 can live the longest2 the ,state being 
the common stake-holder1 and receiving a
 heavy commission for its               
superintendence1 this commission being  
naturally its sole motive for           
encouraging such a transaction4 ,it is  
recorded of one of the ,french          
,tontines#e that a widow of #ig was     
left1 as the last survivor1 to receive  
an annuity of #gc1ejj livres during the 
rest of the life which she could manage 

to drag on after that age2--she having  
originally subscribed a single sum of   
#cjj livres only4 ,it is obvious that   
such a system as this1 though it may    
sometimes go by the name of insurance1  
is utterly opposed to the spirit of true
 insurance1 since it tends to aggravate 
existing inequalities of fortune instead
 of to mitigate them4 ,the insurer here 
bets that he will die old2 in ordinary  
insurance he bets that he will die      
young4  ,again1 to take one final       
instance1 common opinion often regards  
the bank or company which keeps a rouge 
et noir table1 and the individuals who  
risk their money at it1 as being both   
alike engaged in gambling4 ,so they may 
be1 technically1 but for all practical  
purposes such a bank is as sure and safe
 a business as that of any ordinary     
insurance society1 and probably far     
steadier in its receipts than the       
majority of ordinary trades in a        
manufacturing or commercial city4 ,the  
bank goes in for many and small         

transactions1 in proportion to its      
capital2 their customers1 very often1 in
 proportion to their incomes go in for  
very heavy transactions4 ,that the      
former comes out a gainer year after    
year depends1 of course1 upon the fact  
that the tables are notoriously slightly
 in their favour4 ,but the steadiness of
 these gains when compared with the     
unsteadiness of the individual losses   
depends simply upon1--in fact1 is merely
 an illustration of1--the one great     
permanent contrast which lies at the    
basis of all reasoning in ,probability4 
#aj4 ,we have so far regarded ,insurance
 and ,gambling as being each the product
 of a natural impulse1 and as having    
each1 if we look merely to experience1 a
 great mass of human judgment in its    
favour4 ,the popular moral judgment1    
however1 which applauds the one and     
condemns the other rests in great part  
upon an assumption1 which has doubtless 
much truth in it1 but which is often    
interpreted with an absoluteness which  

leads to error in each direction2--the  
duty of insurance being too peremptorily
 urged upon every one1 and the practice 
of gambling too universally regarded as 
involving a sacrifice of real           
self-interest1 as being in fact little  
better than a persistent blunder4 ,the  
assumption in question seems to be      
extracted from the acknowledged         
advantages of insurance1 and then       
invoked to condemn the practice of      
gambling4 ,but in so doing the fact does
 not seem to be sufficiently recognized 
that the latter practice1 if we merely  
look to the extent and antiquity of the 
tacit vote of mankind in its favour1    
might surely claim to carry the day4    
,it is of course obvious that in all    
cases with which we are concerned1 the  
aggregate wealth is unaltered2 money    
being merely transferred from one person
 to another4 ,the loss of one is        
precisely equivalent to the gain of     
another4 ,at least this is the          
approximation to the truth with which we

 find it convenient to start4#f ,now if 
the happiness which is yielded by wealth
 were always in direct proportion to its
 amount1 it is not easy to see why      
insurance should be advocated or        
gambling condemned4 ,in the case of the 
latter this is obvious enough4 ,i have  
lost #ej1 say1 but others 7one or more  
as the case may be7 have gained it1 and 
the increase of their happiness would   
exactly balance the diminution of mine4 
,in the case of ,insurance there is a   
slight complication1 arising from the   
fact that the falling in of the policy  
does not happen at random 7otherwise1 as
 already pointed out1 it would be simply
 a lottery71 but is made contingent upon
 some kind of loss1 which it is intended
 as far as possible to balance4 ,i      
insure myself on a railway journey1     
break my leg in an accident1 and1 having
 paid threepence for my ticket1 receive 
say #bjj compensation from the insurance
 company4 ,the same remarks1 however1   
apply here2 the happiness ,i acquire by 

this #bjj would only just balance the   
aggregate loss of the #af1jjj who have  
paid their threepences and received no  
return for them1 were happiness always  
directly proportional to wealth4  #aa4  
,the practice of ,insurance does not1 ,i
 think1 give rise to many questions of  
theoretic interest1 and need not        
therefore detain us longer4 ,the fact is
 that it has hardly yet been applied    
sufficiently long and widely1 or to     
matters which admit of sufficiently     
accurate statistical treatment1 except  
in one department4 ,this1 of course1 is 
,life ,insurance2 but the subject is one
 which requires constant attention to   
details of statistics1 and is 7rightly7 
mainly carried out in strict accordance 
with routine4 ,as an illustration of    
this we need merely refer to the works  
of ,de ,morgan1--a professional actuary 
as well as a writer on the theory of    
,probability1--who has found but little 
opportunity to aid his speculative      
treatment of ,probability by examples   

drawn from this class of considerations4
  ,with ,gambling it is otherwise4 ,not 
only have a variety of interesting      
single problems been discussed 7of which
 the ,petersburg problem is the best    
known7 but several speculative questions
 of considerable importance have been   
raised4 ,one of these concerns the      
disadvantages of the practice of        
gambling4 ,there have been a number of  
writers who1 not content with dwelling  
upon the obvious moral and indirect     
mischief which results1 in the shape of 
over-excitement1 consequent greed1      
withdrawal from the steady business     
habits which alone insure prosperity in 
the long run1 diversion of wealth into  
dishonest hands1 c41 have endeavoured to
 demonstrate the necessary loss caused  
by the practice4  #ab4 ,these attempts  
may be divided into two classes4 ,there 
are 7#a7 those which appeal to merely   
numerical considerations1 and 7#b7 those
 which introduce what is called the     
'moral' as distinguished from the       

mathematical value of a future          
contingency4  7#a7 ,for instance1 an    
ingenious attempt has been made by ,mr  
,whitworth to prove that gambling is    
necessarily disadvantageous on purely   
mathematical grounds4  ,when two persons
 play against each other one of the two 
must be ruined sooner or later1 even    
though the game be a fair one1 supposing
 that they go on playing long enough2   
the one with the smaller income having  
of course the worst chance of being the 
lucky survivor4 ,if one of them has a   
finite1 and the other an infinite       
income1 it must clearly be the former   
who will be the ultimate sufferer if    
they go on long enough4 ,it is then     
maintained that this is in fact every   
individual gambler's position1 no one is
 restricted to gambling with one single 
opponent2 the speculator deals with the 
public at large1 with a world whose     
resources are practically unlimited4    
,there is a prospect that his operations
 may terminate to his own disadvantage1 

through his having nothing more to      
stake2 but there is no prospect that it 
will terminate to his advantage through 
the exhaustion of the resources of the  
world4 ,every one who gambles is        
carrying on an unequal warfare3 he is   
ranged with a restricted capital against
 an adversary whose means are           
infinite4#g  ,in the above argument it  
is surely overlooked that the           
adversaries against whom he plays are   
not one body with a common purse1 like  
the bank in a gambling establishment4   
,each of these adversaries is in exactly
 the same position as he himself is1 and
 a precisely similar proof might be     
employed to show that each of them must 
be eventually ruined which is of course 
a reduction to absurdity4 ,gambling can 
only transfer money from one player to  
another1 and therefore none of it can be
 actually lost4  #ac4 ,what really      
becomes of the money1 when they play to 
extremity1 is not difficult to see4     
,first suppose a limited number of      

players4 ,if they go on long enough1 the
 money will at last all find its way    
into the pocket of some one of their    
number4 ,if their fortunes were         
originally equal1 each stands the same  
chance of being the lucky survivor2 in  
which case we cannot assert1 on any     
numerical grounds1 that the prospect of 
the play is disadvantageous to any one  
of them4 ,if their fortunes were        
unequal1 the one who had the largest sum
 to begin with can be shown to have the 
best chance1 according to some          
assignable law1 of being left the final 
winner2 in which case it must be just as
 advantageous for him1 as it was        
disadvantageous for his less wealthy    
competitors4  ,when1 instead of a       
limited number of players1 we suppose an
 unlimited number1 each as he is ruined 
retiring from the table and letting     
another come in1 the results are more   
complicated1 but their general tendency 
can be readily distinguished4 ,if we    
supposed that no one retired except when

 he was ruined1 we should have a state  
of things in which all the old players  
were growing gradually richer4 ,in this 
case the prospect before the new comers 
would steadily grow worse and worse1 for
 their chance of winning against such   
rich opponents would be exceedingly     
small4 ,but as this is an unreasonable  
supposition1 we ought rather to assume  
that not only do the ruined victims     
retire1 but also that those who have    
gained fortunes of a certain amount     
retire also1 so that the aggregate and  
average wealth of the gambling body     
remains pretty steady4 ,what chance any 
given player has of being ruined1 and   
how long he may expect to hold out      
before being ruined1 will depend of     
course upon the initial incomes of the  
players1 the rules of the game1 the     
stakes for which they play1 and other   
considerations4 ,but it is clear that   
for all that is lost by one1 a precisely
 equal sum must be gained by others1 and
 that therefore any particular gambler  

can only be cautioned beforehand that   
his conduct is not to be recommended1 by
 appealing to some such suppositions as 
those already mentioned in a former     
section4  #ad4 ,as an additional        
justification of this view the reader   
may observe that the state of things in 
the last example is one which1 expressed
 in somewhat different language and with
 a slight alteration of circumstances1  
is being incessantly carried on upon a  
gigantic scale upon every side of us4   
,call it the competition of merchants   
and traders in a commercial country1 and
 the general results are familiar       
enough4 ,it is true that in so far as   
skill comes into the question1 they are 
not properly gamblers2 but in so far as 
chance and risk do1 they may be fairly  
so termed1 and in many branches of      
business this must necessarily be the   
case to a very considerable extent4     
,whenever business is carried on in a   
reckless way1 the comparison is on      
general grounds fair enough4 ,in each   

case alike we find some retiring ruined1
 and some making their fortunes2 and in 
each case alike also the chances1       
coeteris paribus1 lie with those who    
have the largest fortunes4 ,every one   
is1 in a sense1 struggling against the  
collective commercial world1 but since  
each of his competitors is doing the    
same1 we clearly could not caution any  
of them 7except indeed the poorer ones7 
that their efforts must finally end in  
disadvantage4  #ae4 ,if we wish to see  
this result displayed in its most       
decisive form we may find a good analogy
 in a very different class of events1   
viz4 in the fate of surnames4 ,we are   
all gamblers in this respect1 and the   
game is carried out to the last farthing
 with a rigour unknown at ,newmarket or 
,monte ,carlo4 ,in its complete         
treatment the subject is a very         
intricate one1#h but a simple example   
will serve to display the general       
tendency4 ,suppose a colony comprising  
#ajjj couples of different surnames1 and

 suppose that each of these has four    
children who grow up to marry4          
,approximately1 one in #af of these     
families will consist of girls only2 and
 therefore1 under ordinary conventions1 
about #fb of the names will have        
disappeared for ever after the next     
generation4 ,four again out of #af will 
have but one boy1 each of whom will of  
course be in the same position as his   
father1 viz4 the sole representative of 
his name4 ,accordingly in the next      
generation one in #af of these names    
will again drop out1 and so the process 
continues4 ,the number which disappears 
in each successive generation becomes   
smaller1 as the stability of the        
survivors becomes greater owing to their
 larger numbers4 ,but there is no check 
to the process4  #af4 ,the analogy here 
is a very close one1 the names which    
thus disappear corresponding to the     
gamblers who retire ruined and those    
which increase in number corresponding  
to the lucky winners4 ,the ultimate goal

 in each case alike1--of course an      
exceedingly remote one1--is the         
exclusive survival of one at the expense
 of all the others4 ,that one surname   
does thus drop out after another must   
have struck every one who has made any  
enquiry into family genealogy1 and      
various fanciful accounts have been     
given by those unfamiliar with the      
theory of probability4 ,what is often   
apt to be overlooked is the extreme     
slightness of what may be termed the    
turn of the tables in favour of the     
survival at each generation4 ,in the    
above numerical example we have made an 
extravagantly favourable supposition1 by
 assuming that the population doubles at
 every generation4 ,in an old and       
thickly populated country where the     
numbers increase very slowly1 we should 
be much nearer the mark in assuming that
 the average effective family1--that is1
 the average number of children who live
 to marry1--was only two4 ,in this case 
every family which was represented at   

any time by but a single male would have
 but three chances in four of surviving 
extinction1 and of course the process of
 thinning out would be a more rapid one4
  #ag4 ,the most interesting class of   
attempts to prove the disadvantages of  
gambling appeal to what is technically  
called 'moral expectation' as           
distinguished from 'mathematical        
expectation4' ,the latter may be defined
 simply as the average money value of   
the venture in question2 that is1 it is 
the product of the amount to be gained  
7or lost7 and the chance of gaining 7or 
losing7 it4 ,for instance1 if ,i bet    
four to one in sovereigns against the   
occurrence of ace with a single die     
there would be1 on the average of many  
throws1 a loss of four pounds against a 
gain of five pounds on each set of six  
occurrences2 i4e4 there would be an     
average gain of three shillings and     
fourpence on each throw4 ,this is called
 the true or mathematical expectation4  
,the so-called 'moral expectation'1 on  

the other hand1 is the subjective value 
of this mathematical expectation4 ,that 
is1 instead of reckoning a money fortune
 in the ordinary way1 as what it is1 the
 attempt is made to reckon it at what it
 is felt to be4 ,the elements of        
computation therefore become1 not pounds
 and shillings1 but sums of pleasure    
enjoyed actually or in prospect4        
,accordingly when reckoning the present 
value of a future gain1 we must now     
multiply1 not the objective but the     
subjective value1 by the chance we have 
of securing that gain4  ,with regard to 
the exact relation of this moral fortune
 to the physical various more or less   
arbitrary assumptions have been made4   
,one writer 7,buffon7 considers that the
 moral value of any given sum varies    
inversely with the total wealth of the  
person who gains it4 ,another 7,d4      
,bernoulli7 starting from a different   
assumption1 which we shall presently    
have to notice more particularly1 makes 
the moral value of a fortune vary as the

 logarithm of its actual amount4#i ,a   
third 7,cramer7 makes it vary with the  
square root of the amount4  #ah4        
,historically1 these proposals have     
sprung from the wish to reconcile the   
conclusions of the ,petersburg problem  
with the dictates of practical common   
sense2 for1 by substituting the moral   
for the physical estimate the total     
value of the expectation could be       
reduced to a finite sum4 ,on this ground
 therefore such proposals have no great 
interest1 for1 as we have seen1 there is
 no serious difficulty in the problem   
when rightly understood4  ,these same   
proposals however have been employed in 
order to prove that gambling is         
necessarily disadvantageous1 and this to
 both parties4 ,take1 for instance1     
,bernoulli's supposition4 ,it can be    
readily shown that if two persons each  
with a sum of #ej to start with choose  
to risk1 say1 #aj upon an even wager    
there will be a loss of happiness as a  
result2 for the pleasure gained by the  

possessor of #fj will not be equal to   
that which is lost by the man who leaves
 off with #dj4aj  #ai4 ,this is the form
 of argument commonly adopted2 but1 as  
it stands1 it does not seem conclusive4 
,it may surely be replied that all which
 is thus proved is that inequality is   
bad1 on the ground that two fortunes of 
#ej are better than one of #fj and one  
of #dj4 ,conceive for instance that the 
original fortunes had been #fj and #dj  
respectively1 the event may result in an
 increase of happiness2 for this will   
certainly be the case if the richer man 
loses and the fortunes are thus         
equalized4 ,this is quite true2 and we  
are therefore obliged to show1--what can
 be very easily shown1--that if the     
other alternative had taken place and   
the two fortunes had been made still    
more unequal 7viz4 #fe and #ce          
respectively7 the happiness thus lost   
would more than balance what would have 
been gained by the equalization4 ,and   
since these two suppositions are equally

 likely there will be a loss in the long
 run4  ,the consideration just adduced  
seems however to show that the common   
way of stating the conclusion is rather 
misleading2 and that1 on the assumption 
in question as to the law of dependence 
of happiness on wealth1 it really is the
 case that the effective element in     
rendering gambling disadvantageous is   
its tendency to the increase of the     
inequality in the distribution of       
wealth4  #bj4 ,this raises two          
questions1 one of some speculative      
interest in connection with our subject1
 and the other of supreme importance in 
the conduct of life4 ,the first is this3
 quite apart from any particular        
assumption which we make about moral    
fortunes or laws of variation of        
happiness1 is it the fact that gambling 
tends to increase the existing          
inequalities of wealth8 ,theoretically  
there is no doubt that this is so4 ,take
 the simplest case and suppose two      
people tossing for a pound4 ,if their   

fortunes were equal to begin with there 
must be resultant inequality4 ,if they  
were unequal there is an even chance of 
the inequality being increased or       
diminished2 but since the increase is   
proportionally greater than the         
decrease1 the final result remains of   
the same kind as when the fortunes were 
equal4#aa ,taking a more general view   
the same conclusion underlies all our   
reasoning as to the averages of large   
numbers1 viz4 that the resultant        
divergencies increase absolutely        
7however they diminish relatively7 as   
the numbers become greater4 ,and of     
course we refer to these absolute       
divergencies when we are talking of the 
distribution of wealth4  #ba4 ,this is  
the theoretic conclusion4 ,how far the  
actual practice of gambling introduces  
counteracting agencies must be left to  
the determination of those who are      
competent to pronounce4 ,so far as      
outsiders are authorised to judge from  
what they read in the newspapers and    

other public sources of information1 it 
would appear that these counteracting   
agencies are very considerable1 and that
 in consequence it is a rather insecure 
argument to advance against gambling4   
,many a large fortune has notoriously   
been squandered on the race-course or in
 gambling saloons1 and most certainly a 
large portion1 if not the major part1   
has gone to swell the incomes of many   
who were by comparison poor4 ,but the   
solution of this question must clearly  
be left to those who have better        
opportunities of knowing the facts than 
is to be expected on the part of writers
 on ,probability4  #bb4 ,the general    
conclusion to be drawn is that those who
 invoked this principle of moral fortune
 as an argument against gambling were   
really raising a much more intricate and
 far-reaching problem than they were    
aware of4 ,what they were at work upon  
was the question1 ,what is the          
distribution of wealth which tends to   
secure the maximum of happiness8 ,is    

this best secured by equality or        
inequality8 ,had they really followed   
out the doctrine on which their         
denunciation of gambling was founded    
they ought to have adopted the          
,socialist's ideal as being distinctly  
that which tends to increase happiness4 
,and they ought to have brought under   
the same disapprobation which they      
expressed against gambling all those    
tendencies of modern civilized life     
which work in the same direction4 ,for  
instance2 keen competition1 speculative 
operations1 extended facilities of      
credit1 mechanical inventions1          
enlargement of business operations into 
vast firms3--all these1 and other       
similar tendencies too numerous to      
mention here1 have had some influence in
 the way of adding to existing          
inequalities4 ,they are1 or have been1  
in consequence denounced by socialists3 
are we honestly to bring them to this   
test in order to ascertain whether or   
not they are to be condemned8 ,the      

reader who wishes to see what sort of   
problems this assumption of 'moral      
fortune' ought to introduce may be      
recommended to read ,mr ,f4 ,y4         
,edgeworth's ,mathematical ,psychics1   
the only work with which ,i am          
acquainted which treats of these        
questions4   #a4 ,the question of the   
advisability of inoculation against the 
small-pox1 which gave rise to much      
discussion amongst the writers on       
,probability during the last century1 is
 a case in point of the same            
principles applied to a very different  
kind of instance4 ,the loss   against   
which the insurance was directed was    
death by small-pox1 the   premium paid  
was the illness and other inconvenience1
 and the very   small risk of death1    
from the inoculation4 ,the disputes     
which thence   arose amongst writers on 
the subject involved the same           
difficulties   as to the balance between
 certain moderate loss and contingent   
great   loss4 ,in the seventeenth       

century it seems to have been an        
occasional   practice1 before a journey 
into the ,mediterranean1 to insure      
against   capture by ,moorish pirates1  
with a view to secure having the ransom 
paid4 7,see1 for an account of some     
extraordinary developments of the       
insurance principle1 ,walford's         
,insurance ,guide and ,handbook4 ,it is 
not written in a very scientific spirit1
 but it contains much   information on  
all matters connected with insurance47  
#b4 ,all that is meant by the above     
comparison is that the ideal aimed   at 
by ,communism is similar to that of     
,insurance4 ,if we look at the          
processes by which it would be carried  
out1 and the means for   enforcing it1  
the matter would of course assume a very
 different   aspect4 ,similarly with the
 action of ,trades ,unionism referred to
 in   the next paragraph4  #c4 ,one of  
the best discussions that ,i have       
recently seen on these   subjects1 by a 
writer at once thoroughly competent and 

well   informed1 is in ,mr ,proctor's   
,chance and ,luck4 ,it appears to me    
however that he runs into an extreme in 
his denunciation not of the   folly but 
of the dishonesty of all gambling4      
,surely also it is a   strained use of  
language to speak of all lotteries as   
'unfair' and   even 'swindling' on the  
ground that the sum-total of what they  
distribute in prizes is less than that  
of what they receive in   payments4 ,the
 difference1 in respect of information  
deliberately   withheld and false       
reports wilfully spread1 between most of
 the   lotteries that have been         
supported1 and the bubble companies     
which   justly deserve the name of      
swindles1 ought to prevent the same name
   being applied to both4  #d4 ,a fire  
insurance is a simple bet between the   
office and the party1   and a life      
insurance is a collection of wagers4    
,there is something   of the principle  
of a wager in every transaction in which
 the   results of a future event are to 

bring gain or loss4 ,penny   ,cyclopdia1
 under the head of ,wager4  #e4         
,encyclopdie ,methodique1 under the head
 of ,tontines4  #f4 ,of course1 if we   
introduce considerations of ,political  
,economy1   corrections will have to be 
made4 ,for one thing1 every ,insurance  
,office is1 as ,de ,morgan repeatedly   
insists1 a ,savings ,bank as well   as  
an ,insurance ,office4 ,the ,office     
invests the premiums1 and can           
therefore afford to pay a larger sum    
than would otherwise be the   case4     
,again1 in the case of gambling1 a large
 loss of capital by any   one will      
almost necessarily involve an actual    
destruction of wealth2   to say nothing 
of the fact that1 practically1 gambling 
often causes   a constant transfer of   
wealth from productive to unproductive  
purposes4  #g4 ,choice and ,chance1 ,ed4
 ,,ii4 p4 #bjh4  #h4 ,it was1 ,i        
believe1 first treated as a serious     
problem by ,mr ,galton4   7,see the     
,journal ,anthrop4 ,inst4 ,vol4 ,,iv4   

#ahge1 where a complete   mathematical  
solution is indicated by ,mr ,h4 ,w4    
,watson47  #i4 ,bernoulli himself does  
not seem to have based his conclusions  
upon   actual experience4 ,but it is a  
noteworthy fact that the assumption     
with which he starts1 viz4 that the     
subjective value of any small           
increment 7dx7 is inversely proportional
 to the sum then possessed   7x71 and   
which leads at once to the logarithmic  
law above mentioned1   is identical with
 one which is now familiar enough to    
every   psychologist4 ,it is what is    
commonly called ,fechner's ,law1 which  
he   has established by aid of an       
enormous amount of careful experiment   
in the case of a number of our simple   
sensations4 ,but ,i do not   believe    
that he has made any claim that such a  
law holds good in the   far more        
intricate dependence of happiness upon  
wealth4  #aj4 ,the formula expressive of
 this moral happiness is c log x_/a2    
where   x stands for the physical       

fortune possessed at the time1 and a for
   that small value of it at which      
happiness is supposed to disappear3   c 
being an arbitrary constant4 ,let two   
persons1 whose fortune is x1   risk y on
 an even bet4 ,then the balance1 as     
regards happiness1 must   be drawn      
between   c log x_/a    and    #a_/b;c  
log 7x  y7_/a  #a_/b;c log 7x - y7_/a1  
or    log x#b and log7x  y77x - y71   or
    x#b and x#b - y#b1   the former of  
which is necessarily the greater4  #aa4 
,this may be seen more clearly as       
follows4 ,suppose two pair of           
gamblers1 each pair consisting of men   
possessing #ej and #cj   respectively4  
,now if we suppose the richer man to win
 in one case   and the poorer in the    
other these two results will be a fair  
representation of the average2 for there
 are only two alternatives   and these  
will be equally frequent in the long    
run4 ,it is obvious   that we have had  
two fortunes of #ej and two of #cj      
converted into   one of #bj1 two of #dj1

 and one of #fj4 ,and this is clearly an
   increase of inequality4     ,,chapter
 ,,xvi4   ,,the ,,application ,,of      
,,probability ,,to ,,testimony4  #a4 ,on
 the principles which have been adopted 
in this work1 it becomes questionable   
whether several classes of problems     
which may seem to have acquired a       
prescriptive right to admission1 will   
not have to be excluded from the science
 of ,probability4 ,the most important1  
perhaps1 of these refer to what is      
commonly called the credibility of      
testimony1 estimated either at first    
hand and directly1 or as influencing a  
juryman1 and so reaching us through his 
sagacity and trustworthiness4 ,almost   
every treatise upon the science contains
 a discussion of the principles         
according to which credit is to be      
attached to combinations of the reports 
of witnesses of various degrees of      
trustworthiness1 or the verdicts of     
juries consisting of larger or smaller  
numbers4 ,a great modern mathematician1 

,poisson1 has written an elaborate      
treatise expressly upon this subject2   
whilst a considerable portion of the    
works of ,laplace1 ,de ,morgan1 and     
others1 is devoted to an examination of 
similar enquiries4 ,it would be         
presumptuous to differ from such        
authorities as these1 except upon the   
strongest grounds2 but ,i confess that  
the extraordinary ingenuity and         
mathematical ability which have been    
devoted to these problems1 considered as
 questions in ,probability1 fails to    
convince me that they ought to have been
 so considered4 ,the following are the  
principal grounds for this opinion4  #b4
 ,it will be remembered that in the     
course of the chapter on ,induction we  
entered into a detailed investigation of
 the process demanded of us when1       
instead of the appropriate propositions 
from which the inference was to be made 
being set before us1 the individual     
presented himself1 and the task was     
imposed upon us of selecting the        

requisite groups or series to which to  
refer him4 ,in other words1 instead of  
calculating the chance of an event from 
determinate conditions of frequency of  
its occurrence 7these being either      
obtained by direct experience1 or       
deductively inferred7 we have to select 
the conditions of frequency out of a    
plurality of more or less suitable ones4
 ,when the problem is presented to us at
 such a stage as this1 we may of course 
assume that the preliminary process of  
obtaining the statistics which are      
extended into the proportional          
propositions has been already performed2
 we may suppose therefore that we are   
already in possession of a quantity of  
such propositions1 our principal        
remaining doubt being as to which of    
them we should then employ4 ,this       
selection was shown to be to a certain  
extent arbitrary2 for1 owing to the fact
 of the individual possessing a large   
number of different properties1 he      
became in consequence a member of       

different series or groups1 which might 
present different averages4 ,we must now
 examine1 somewhat more fully than we   
did before1 the practical conditions    
under which any difficulty arising from 
this source ceases to be of importance4 
#c4 ,one condition of this kind is very 
simple and obvious4 ,it is that the     
different statistics with which we are  
presented should not in reality offer   
materially different results1 ,if1 for  
instance1 we were enquiring into the    
probability of a man aged forty dying   
within the year1 we might if we pleased 
take into account the fact of his having
 red hair1 or his having been born in a 
certain county or town4 ,each of these  
circumstances would serve to specialize 
the individual1 and therefore to        
restrict the limits of the statistics   
which were applicable to his case4 ,but 
the consideration of such qualities as  
these would either leave the average    
precisely as it was1 or produce such an 
unimportant alteration in it as no one  

would think of taking into account4     
,though we could hardly say with        
certainty of any conceivable            
characteristic that it has absolutely no
 bearing on the result1 we may still    
feel very confident that the bearing of 
such characteristics as these is utterly
 insignificant4 ,of course in the       
extreme case of the things most         
perfectly suited to the ,calculus of    
,probability1 viz4 games of pure chance1
 these subsidiary characteristics are   
quite irrelevant4 ,any further          
particulars about the characteristics of
 the cards in a really fair pack1 beyond
 those which are familiar to all the    
players1 would convey no information    
whatever about the result4  ,or again2  
although the different sets of          
statistics may not as above give almost 
identical results1 yet they may do what 
practically comes to very much the same 
thing1 that is1 arrange themselves into 
a small number of groups1 all of the    
statistics in any one group practically 

coinciding in their results4 ,if for    
example a consumptive man desired to    
insure his life1 there would be a marked
 difference in the statistics according 
as we took his peculiar state of health 
into account or not4 ,we should here    
have two sets of statistics1 so clearly 
marked off from one another that they   
might almost rank with the distinctions 
of natural kinds1 and which would in    
consequence offer decidedly different   
results4 ,if we were to specialize still
 further1 by taking into account        
insignificant qualities like those      
mentioned in the last paragraph1 we     
might indeed get more limited sets of   
statistics applicable to persons still  
more closely resembling the individual  
in question1 but these would not differ 
sufficiently in their results to make it
 worth our while to do so4 ,in other    
words1 the different propositions which 
are applicable to the case in point     
arrange themselves into a limited number
 of groups1 which1 and which only1 need 

be taken into account2 whence the range 
of choice amongst them is very much     
diminished in practice4  #d4 ,the       
reasons for the conditions above        
described are not difficult to detect4  
,where these conditions exist the       
process of selecting a series or class  
to which to refer any individual is very
 simple1 and the selection is1 for the  
particular purposes of inference1 final4
 ,in any case of insurance1 for example1
 the question we have to decide is of   
the very simple kind2 ,is ,a4 ,b4 a man 
of a certain age8 ,if so one in fifty in
 his circumstances will die in the      
course of the year4 ,if any further     
questions have to be decided they would 
be of the following description4 ,is ,a4
 ,b4 a healthy man8 ,does he follow a   
dangerous trade8 ,but here too the      
classes in question are but few1 and the
 limits by which they are bounded are   
tolerably precise2 so that the reference
 of an individual to one or other of    
them is easy4 ,and when we have once    

chosen our class we remain untroubled by
 any further considerations2 for since  
no other statistics are supposed to     
offer a materially different average1 we
 have no occasion to take account of any
 other properties than those already    
noticed4  ,the case of games of chance1 
already referred to1 offers of course an
 instance of these conditions in an     
almost ideal state of perfection2 the   
same circumstances which fit them so    
eminently for the purposes of fair      
gambling1 fitting them equally to become
 examples in ,probability4 ,when a die  
is to be thrown1 all persons alike stand
 on precisely the same footing of       
knowledge and of ignorance about the    
result2 the only data to which any one  
could appeal being that each face turns 
up on an average once in six times4  #e4
 ,let us now examine how far the above  
conditions are fulfilled in the case of 
problems which discuss what is called   
the credibility of testimony4 ,the      
following would be a fair specimen of   

one of the elementary enquiries out of  
which these problems are                
composed2--,here is a statement made by 
a witness who lies once in ten times1   
what am ,i to conclude about its truth8 
,objections might fairly be raised      
against the possibility of thus         
assigning a man his place upon a        
graduated scale of mendacity4 ,this     
however we will pass over1 and will     
assume that the witness goes about the  
world bearing stamped somehow on his    
face the appropriate class to which he  
belongs1 and consequently1 the degree of
 credit to which he has a claim on such 
general grounds4 ,but there are other   
and stronger reasons against the        
admissibility of this class of problems4
  #f4 ,that which has been described in 
the previous sections as the            
'individual' which had to be assigned to
 an appropriate class or series of      
statistics is1 of course1 in this case1 
a statement4 ,in the particular instance
 in question this individual statement  

is already assigned to a class1 that    
namely of statements made by a witness  
of a given degree of veracity2 but it is
 clearly optional with us whether or not
 we choose to confine our attention to  
this class in forming our judgment2 at  
least it would be optional whenever we  
were practically called on to form an   
opinion4 ,but in the case of this       
statement1 as in that of the mortality  
of the man whose insurance we were      
discussing1 there are a multitude of    
other properties observable1 besides the
 one which is supposed to mark the given
 class4 ,just as in the latter there    
were 7besides his age71 the place of his
 birth1 the nature of his occupation1   
and so on2 so in the former there are   
7besides its being a statement by a     
certain kind of witness71 the fact of   
its being uttered at a certain time and 
place and under certain circumstances4  
,at the time the statement is made all  
these qualities or attributes of the    
statement are present to us1 and we     

clearly have a right to take into       
account as many of them as we please4   
,now the question at present before us  
seems to be simply this2--,are the      
considerations1 which we might thus     
introduce1 as immaterial to the result  
in the case of the truth of a statement 
of a witness1 as the corresponding      
considerations are in the case of the   
insurance of a life8 ,there can surely  
be no hesitation in the reply to such a 
question4 ,under ordinary circumstances 
we soon know all that we can know about 
the conditions which determine us in    
judging of the prospect of a man's      
death1 and we therefore rest content    
with general statistics of mortality2   
but no one who heard a witness speak    
would think of simply appealing to his  
figure of veracity1 even supposing that 
this had been authoritatively           
communicated to us4 ,the circumstances  
under which the statement is made       
instead of being insignificant1 are of  
overwhelming importance4 ,the appearance

 of the witness1 the tone of his voice1 
the fact of his having objects to gain1 
together with a countless multitude of  
other circumstances which would         
gradually come to light as we reflect   
upon the matter1 would make any sensible
 man discard the assigned average from  
his consideration4 ,he would1 in fact1  
no more think of judging in this way    
than he would of appealing to the       
,carlisle or ,northampton tables of     
mortality to determine the probable     
length of life of a soldier who was     
already in the midst of a battle4  #g4  
,it cannot be replied that under these  
circumstances we still refer the witness
 to a class1 and judge of his veracity  
by an average of a more limited kind2   
that we infer1 for example1 that of men 
who look and act like him under such    
circumstances1 a much larger proportion1
 say nine-tenths1 are found to lie4     
,there is no appeal to a class in this  
way at all1 there is no immediate       
reference to statistics of any kind     

whatever2 at least none which we are    
conscious of using at the time1 or to   
which we should think of resorting for  
justification afterwards4 ,the decision 
seems to depend upon the quickness of   
the observer's senses and of his        
apprehension generally4  ,statistics    
about the veracity of witnesses seem in 
fact to be permanently as inappropriate 
as all other statistics occasionally may
 be4 ,we may know accurately the        
percentage of recoveries after          
amputation of the leg2 but what surgeon 
would think of forming his judgment     
solely by such tables when he had a case
 before him8 ,we need not deny1 of      
course1 that the opinion he might form  
about the patient's prospects of        
recovery might ultimately rest upon the 
proportions of deaths and recoveries he 
might have previously witnessed4 ,but if
 this were the case1 these data are     
lying1 as one may say1 obscurely in the 
background4 ,he does not appeal to them 
directly and immediately in forming his 

judgment4 ,there has been a far more    
important intermediate process of       
apprehension and estimation of what is  
essential to the case and what is not4  
,sharp senses1 memory1 judgment1 and    
practical sagacity have had to be called
 into play1 and there is not therefore  
the same direct conscious and sole      
appeal to statistics that there was     
before4 ,the surgeon may have in his    
mind two or three instances in which the
 operation performed was equally severe1
 but in which the patient's constitution
 was different2 the latter element      
therefore has to be properly allowed    
for4 ,there may be other instances in   
which the constitution was similar1 but 
the operation more severe2 and so on4   
,hence1 although the ultimate appeal may
 be to the statistics1 it is not so     
directly2 their value has to be         
estimated through the somewhat hazy     
medium of our judgment and memory1 which
 places them under a very different     
aspect4  #h4 ,any one who knows anything

 of the game of whist may supply an     
apposite example of the distinction here
 insisted on1 by recalling to mind the  
alteration in the nature of our         
inferences as the game progresses4 ,at  
the commencement of the game our sole   
appeal is rightfully made to the theory 
of ,probability4 ,all the rules upon    
which each player acts1 and therefore   
upon which he infers that the others    
will act1 rest upon the observed        
frequency 7or rather upon the frequency 
which calculation assures us will be    
observed7 with which such and such      
combinations of cards are found to      
occur4 ,why are we told1 if we have more
 than four trumps1 to lead them out at  
once8 ,because we are convinced1 on pure
 grounds of probability1 capable of     
being stated in the strictest           
statistical form1 that in a majority of 
instances we shall draw our opponent's  
trumps1 and therefore be left with the  
command4 ,similarly with every other    
rule which is recognized in the early   

part of the play4  ,but as the play     
progresses all this is changed1 and     
towards its conclusion there is but     
little reliance upon any rules which    
either we or others could base upon     
statistical frequency of occurrence1    
observed or inferred4 ,a multitude of   
other considerations have come in2 we   
begin to be influenced partly by our    
knowledge of the character and practice 
of our partner and opponents2 partly by 
a rapid combination of a multitude of   
judgments1 founded upon our observation 
of the actual course of play1 the       
grounds of which we could hardly realize
 or describe at the time and which may  
have been forgotten since4 ,that is1 the
 particular combination of cards1 now   
before us1 does not readily fall into   
any well-marked class to which alone it 
can reasonably be referred by every one 
who has the facts before him4  #i4 ,a   
criticism somewhat resembling the above 
has been given by ,mill 7,logic1 ,bk4   
,,iii4 ,chap4 ,,xviii4 #c7 upon the     

applicability of the theory of          
,probability to the credibility of      
witnesses4 ,but he has added other      
reasons which do not appear to me to be 
equally valid2 he says common sense     
would dictate that it is impossible to  
strike a general average of the         
veracity1 and other qualifications for  
true testimony1 of mankind or any class 
of them2 and if it were possible1 such  
an average would be no guide1 the       
credibility of almost every witness     
being either below or above the average1
 ,the latter objection would however    
apply with equal force to estimating the
 length of a man's life from tables of  
mortality2 for the credibility of       
different witnesses can scarcely have a 
wider range of variation than the length
 of different lives4 ,if statistics of  
credibility could be obtained1 and could
 be conveniently appealed to when they  
were obtained1 they might furnish us in 
the long run with as accurate inferences
 as any other statistics of the same    

general description4 ,these statistics  
would however in practice naturally and 
rightly be neglected1 because there can 
hardly fail to be circumstances in each 
individual statement which would more   
appropriately refer it to some new class
 depending on different statistics1 and 
affording a far better chance of our    
being right in that particular case4 ,in
 most instances of the kind in question1
 indeed1 such a change is thus produced 
in the mode of formation of our opinion1
 that1 as already pointed out1 the      
mental operation ceases to be in any    
proper sense founded on appeal to       
statistics4#a  #aj4 ,the ,chance        
problems which are concerned with       
testimony are not altogether confined to
 such instances as those hitherto       
referred to4 ,though we must1 as it     
appears to me1 reject all attempts to   
estimate the credibility of any         
particular witness1 or to refer him to  
any assigned class in respect of his    
trustworthiness1 and consequently       

abandon as unsuitable any of the        
numerous problems which start from such 
data as 'a witness who is wrong once in 
ten times1' yet it does not follow that 
testimony may not to a slight extent be 
treated by our science in a somewhat    
different manner4 ,we may be quite      
unable to estimate1 except in the       
roughest possible way1 the veracity of  
any particular witness1 and yet it may  
be possible to form some kind of opinion
 upon the veracity of certain classes of
 witnesses2 to say1 for instance1 that  
,europeans are superior in this way to  
,orientals4 ,so we might attempt to     
explain why1 and to what extent1 an     
opinion in which the judgments of ten   
persons1 say jurors1 concur1 is superior
 to one in which five only concur4      
,something may also be done towards     
laying down the principles in accordance
 with which we are to decide whether1   
and why1 extraordinary stories deserve  
less credence than ordinary ones1 even  
if we cannot arrive at any precise and  

definite decision upon the point4 ,this 
last question is further discussed in   
the course of the next chapter4  #aa4   
,the change of view in accordance with  
which it follows that questions of the  
kind just mentioned need not be entirely
 rejected from scientific consideration1
 presents itself in other directions    
also4 ,it has1 for instance1 been       
already pointed out that the individual 
characteristics of any sick man's       
disease would be quite sufficiently     
important in most cases to prevent any  
surgeon from judging about his recovery 
by a genuine and direct appeal to       
statistics1 however such considerations 
might indirectly operate upon his       
judgment4 ,but if an opinion had to be  
formed about a considerable number of   
cases1 say in a large hospital1         
statistics might again come prominently 
into play1 and be rightly recognized as 
the principal source of appeal4 ,we     
should feel able to compare one         
hospital1 or one method of treatment1   

with another4 ,the ground of the        
difference is obvious4 ,it arises from  
the fact that the characteristics of the
 individuals1 which made us so ready to 
desert the average when we had to judge 
of them separately1 do not produce the  
same disturbance when were have to judge
 about a group of cases4 ,the averages  
then become the most secure and         
available ground on which to form an    
opinion1 and therefore ,probability     
again becomes applicable4  ,but although
 some resort to ,probability may be     
admitted in such cases as these1 it     
nevertheless does not appear to me that 
they can ever be regarded as            
particularly appropriate examples to    
illustrate the methods and resources of 
the theory4 ,indeed it is scarcely      
possible to resist the conviction that  
the refinements of mathematical         
calculation have here been pushed to    
lengths utterly unjustifiable1 when we  
bear in mind the impossibility of       
obtaining any corresponding degree of   

accuracy and precision in the data from 
which we have to start4 ,to cite but one
 instance4 ,it would be hard to find a  
case in which love of consistency has   
prevailed over common sense to such an  
extent as in the admission of the       
conclusion that it is unimportant what  
are the numbers for and against a       
particular statement1 provided the      
actual majority is the same4 ,that is1  
the unanimous judgment of a jury of     
eight is to count for the same as a     
majority of ten to two in a jury of     
twelve4 ,and yet this conclusion is     
admitted by ,poisson4 ,the assumptions  
under which it follows will be indicated
 in the course of the next chapter4     
,again1 perfect independence amongst the
 witnesses or jurors is an almost       
necessary postulate4 ,but where can this
 be secured8 ,to say nothing of direct  
collusion1 human beings are in almost   
all instances greatly under the         
influence of sympathy in forming their  
opinions4 ,this influence1 under the    

various names of political bias1 class  
prejudice1 local feeling1 and so on1    
always exists to a sufficient degree to 
induce a cautious person to make many of
 those individual corrections which we  
saw to be necessary when we were        
estimating the trustworthiness1 in any  
given case1 of a single witness2 that   
is1 they are sufficient to destroy much1
 if not all1 of the confidence with     
which we resort to statistics and       
averages in forming our judgment4 ,since
 then this ,essay is mainly devoted to  
explaining and establishing the general 
principles of the science of            
,probability1 we may very fairly be     
excused from any further treatment of   
this subject1 beyond the brief          
discussions which are given in the next 
chapter4   #a4 ,it may be remarked also 
that there is another reason which tends
 to   dissuade us from appealing to     
principles of ,probability in the       
majority of the cases where testimony   
has to be estimated4 ,it often1         

perhaps usually happens1 that we are not
 absolutely forced to come   to a       
decision2 at least so far as the        
acquitting of an accused   person may be
 considered as avoiding a decision4 ,it 
may be of much   greater importance to  
us to attain not merely truth on the    
average1   but truth in each individual 
instance1 so that we had rather not     
form an opinion at all than form one of 
which we can only say in its            
justification that it will tend to lead 
us right in the long run4     ,,chapter 
,,xvii4   ,,on ,,the ,,credibility ,,of 
,,extraordinary ,,stories4  #a4 ,it is  
now time to recur for fuller            
investigation to an enquiry which has   
been already briefly touched upon more  
than once2 that is1 the validity of     
testimony to establish1 as it is        
frequently expressed1 an otherwise      
improbable story4 ,it will be remembered
 that in a previous chapter 7the        
twelfth7 we devoted some examination to 
an assertion by ,butler1 which seemed to

 be to some extent countenanced by      
,mill1 that a great improbability before
 the proof might become but a very small
 improbability after the proof4 ,in     
opposition to this it was pointed out   
that the different estimates which we   
undoubtedly formed of the credibility of
 the examples adduced1 had nothing to do
 with the fact of the event being past  
or future1 but arose from a very        
different cause2 that the conception of 
the event which we entertain at the     
moment 7which is all that is then and   
there actually present to us1 and as to 
the correctness of which as a           
representation of facts we have to make 
up our minds7 comes before us in two    
very different ways4 ,in one instance it
 was a mere guess of our own which we   
knew from statistics would be right in a
 certain proportion of cases2 in the    
other instance it was the assertion of a
 witness1 and therefore the appeal was  
not now primarily to statistics of the  
event1 but to the trustworthiness of the

 witness4 ,the conception1 or 'event' if
 we will so term it1 had in fact passed 
out of the category of guesses 7on      
statistical grounds71 into that of      
assertions 7most likely resting on some 
specific evidence71 and would therefore 
be naturally regarded in a very         
different light4  #b4 ,but it may seem  
as if this principle would lead us to   
somewhat startling conclusions4 ,for1 by
 transferring the appeal from the       
frequency with which the event occurs to
 the trustworthiness of the witness who 
makes the assertion1 is it not implied  
that the probability or improbability of
 an assertion depends solely upon the   
veracity of the witness8 ,if so1 ought  
not any story whatever to be believed   
when it is asserted by a truthful       
person8  ,in order to settle this       
question we must look a little more     
closely into the circumstances under    
which such testimony is commonly        
presented to us4 ,as it is of course    
necessary1 for clearness of exposition1 

to take a numerical example1 let us     
suppose that a given statement is made  
by a witness who1 on the whole and in   
the long run1 is right in what he says  
nine times out of ten4#a ,here then is  
an average given to us1 an average      
veracity that is1 which includes all the
 particular statements which the witness
 has made or will make4  #c4 ,now it has
 been abundantly shown in a former      
chapter 7,ch4 ,,ix4 #ad-cb7 that the    
mere fact of a particular average having
 been assigned1 is no reason for our    
being forced invariably to adhere to it1
 even in those cases in which our most  
natural and appropriate ground of       
judgment is found in an appeal to       
statistics and averages4 ,the general   
average may constantly have to be       
corrected in order to meet more         
accurately the circumstances of         
particular cases4 ,in statistics of     
mortality1 for instance2 instead of     
resorting to the wider tables furnished 
by people in general of a given age1 we 

often prefer the narrower tables        
furnished by men of a particular        
profession1 abode1 or mode of life4 ,the
 reader may however be conveniently     
reminded here that in so doing we must  
not suppose that we are able1 by any    
such device1 in any special or peculiar 
way to secure truth4 ,the general       
average1 if persistently adhered to     
throughout a sufficiently wide and      
varied experience1 would in the long run
 tend to give us the truth2 all the     
advantage which the more special        
averages can secure for us is to give us
 the same tendency to the truth with    
fewer and slighter aberrations4  #d4    
,returning then to our witness1 we know 
that if we have a very great many       
statements from him upon all possible   
subjects1 we may feel convinced that in 
nine out of ten of these he will tell us
 the truth1 and that in the tenth case  
he will go wrong4 ,this is nothing more 
than a matter of definition or          
consistency4 ,but cannot we do better   

than thus rely upon his general average8
 ,cannot we1 in almost any given case1  
specialize it by attending to various   
characteristic circumstances in the     
nature of the statement which he makes2 
just as we specialize his prospects of  
mortality by attending to circumstances 
in his constitution or mode of life8    
,undoubtedly we may do this2 and in any 
of the practical contingencies of life1 
supposing that we were at all guided by 
considerations of this nature1 we should
 act very foolishly if we did not adopt 
some such plan4 ,two methods of thus    
correcting the average may be suggested3
 one of them being that which practical 
sagacity would be most likely to employ1
 the other that which is almost         
universally adopted by writers on       
,probability4 ,the former attempts to   
make the correction by the following    
considerations3 instead of relying upon 
the witness' general average1 we assign 
to it a sort of conjectural correction  
to meet the case before us1 founded on  

our experience or observation2 that is1 
we appeal to experience to establish    
that stories of such and such a kind are
 more or less likely to be true1 as the 
case may be1 than stories in general4   
,the other proceeds upon a different and
 somewhat more methodical plan4 ,it is  
here endeavoured to show1 by an analysis
 of the nature and number of the sources
 of error in the cases in question1 that
 such and such kinds of stories must be 
more or less likely to be correctly     
reported1 and this in certain numerical 
proportions4  #e4 ,before proceeding to 
a discussion of these methods a         
distinction must be pointed out to which
 writers upon the subject have not      
always attended1 or at any rate to which
 they have not generally sufficiently   
directed their readers' attention4#b    
,there are1 broadly speaking1 two       
different ways in which we may suppose  
testimony to be given4 ,it may1 in the  
first place1 take the form of a reply to
 an alternative question1 a question1   

that is1 framed to be answered by yes or
 no4 ,here1 of course1 the possible     
answers are mutually contradictory1 so  
that if one of them is not correct the  
other must be so3--,has ,a happened1 yes
 or no8 ,the common mode of illustrating
 this kind of testimony numerically is  
by supposing a lottery with a prize and 
blanks1 or a bag of balls of two colours
 only1 the witness knowing that there   
are only two1 or at any rate being      
confined to naming one or other of them4
 ,if they are black and white1 and he   
errs when black is drawn1 he must say   
'white1' ,the reason for the prominence 
assigned to examples of this class is1  
probably1 that they correspond to the   
very important case of verdicts of      
juries2 juries being supposed to have   
nothing else to do than to say 'guilty' 
or 'not guilty4'  ,on the other hand1   
the testimony may take the form of a    
more original statement or piece of     
information4 ,instead of saying1 ,did ,a
 happen8 we may ask1 ,what happened8    

,here if the witness speaks truth he    
must be supposed1 as before1 to have but
 one way of doing so2 for the occurrence
 of some specific event was of course   
contemplated4 ,but if he errs he has    
many ways of going wrong1 possibly an   
infinite number4 ,ordinarily however his
 possible false statements are assumed  
to be limited in number1 as must        
generally be more or less the result in 
practice4 ,this case is represented     
numerically by supposing the balls in   
the bag not to be of two colours only1  
but to be all distinct from each other2 
say by their being all numbered         
successively4 ,it may of course be      
objected that a large number of the     
statements that are made in the world   
are not in any way answers to questions1
 either of the alternative or of the    
open kind4 ,for instance1 a man simply  
asserts that he has drawn the seven of  
spades from a pack of cards2 and we do  
not know perhaps whether he had been    
asked ',has that card been drawn8' or   

',what card has been drawn8' or indeed  
whether he had been asked anything at   
all4 ,still more might this be so in the
 case of any ordinary historical        
statement4  ,this objection is quite to 
the point1 and must be recognized as    
constituting an additional difficulty4  
,all that we can do is to endeavour1 as 
best we may1 to ascertain1 from the     
circumstances of the case1 what number  
of alternatives the witness may be      
supposed to have had before him4 ,when  
he simply testifies to some matter well 
known to be in dispute1 and does not go 
much into detail1 we may fairly consider
 that there were practically only the   
two alternatives before him of saying   
'yes' or 'no4' ,when1 on the other hand1
 he tells a story of a more original    
kind1 or 7what comes to much the same   
thing7 goes into details1 we must regard
 him as having a wide comparative range 
of alternatives before him4  ,these two 
classes of examples1 viz4 that of the   
black and white balls1 in which only one

 form of error is possible1 and the     
numbered balls1 in which there may be   
many forms of error1 are the only two   
which we need notice4 ,in practice it   
would seem that they may gradually merge
 into each other1 according to the      
varying ways in which we choose to frame
 our question4 ,besides asking1 ,did you
 see ,a strike ,b8 and1 ,what did you   
see8 we may introduce any number of     
intermediate leading questions1 as1     
,what did ,a do8 ,what did he do to ,b8 
and so on4 ,in this way we may gradually
 narrow the possible openings to wrong  
statement1 and so approach to the direct
 alternative question4 ,but it is clear 
that all these cases may be represented 
numerically by a supposed diminution in 
the number of the balls which are thus  
distinguished from each other4  #f4 ,of 
the two plans mentioned in #d we will   
begin with the latter1 as it is the only
 methodical and scientific one which has
 been proposed4 ,suppose that there is a
 bag with #ajjj balls1 only one of which

 is white1 the rest being all black4 ,a 
ball is drawn at random1 and our witness
 whose veracity is #i_/aj reports that  
the white ball was drawn4 ,take a great 
many of his statements upon this        
particular subject1 say #aj1jjj2 that   
is1 suppose that #aj1jjj balls having   
been successively drawn out of this bag1
 or bags of exactly the same kind1 he   
makes his report in each case4 ,his     
#aj1jjj statements being taken as a fair
 sample of his general average1 we shall
 find1 by supposition1 that #i out of   
every #aj of them are true and the      
remaining one false4 ,what will be the  
nature of these false statements8 ,under
 the circumstances in question1 he      
having only one way of going wrong1 the 
answer is easy4 ,in the #aj1jjj drawings
 the white ball would come out #aj      
times1 and therefore be rightly asserted
 #i times1 whilst on the one of these   
occasions on which he goes wrong he has 
nothing to say but 'black4' ,so with the
 #iiij occasions on which black is      

drawn2 he is right and says black on    
#hiia of them1 and is wrong and         
therefore says white on #iii of them4   
,on the whole1 therefore1 we conclude   
that out of every #ajjh times on which  
he says that white is drawn he is wrong 
#iii times and right only #i times4     
,that is1 his special veracity1 as we   
may term it1 for cases of this          
description1 has been reduced from      
#i_/aj to #i_/ajjh4 ,as it would        
commonly be expressed1 the latter       
fraction represents the chance that this
 particular statement of his is true4#c 
#g4 ,we will now take the case in which 
the witness has many ways of going      
wrong1 instead of merely one4 ,suppose  
that the balls were all numbered1 from  
#a to #a1jjj1 and the witness knows this
 fact4 ,a ball is drawn1 and he tells me
 that it was numbered #be1 what are the 
odds that he is right8 ,proceeding as   
before1 in #aj1jjj drawings this ball   
would be obtained #aj times1 and        
correctly named #i times4 ,but on the   

#iiij occasions on which it was not     
drawn there would be a difference1 for  
the witness has now many openings for   
error before him4 ,it is1 however1      
generally considered reasonable to      
assume that his errors will all take the
 form of announcing wrong numbers2 and  
that1 there being no apparent reason why
 he should choose one number rather than
 another1 he will be likely to announce 
all the wrong ones equally often4 ,hence
 his #iii errors1 instead of all leading
 him now back again to one spot1 will be
 uniformly spread over as many distinct 
ways of going wrong4 ,on one only of    
these occasions1 therefore1 will he     
mention #be as having been drawn4 ,it   
follows therefore that out of every #aj 
times that he names #be he is right #i  
times2 so that in this case his average 
or general truthfulness applies equally 
well to the special case in point4  #h4 
,with regard to the truth of these      
conclusions1 it must of course be       
admitted that if we grant the validity  

of the assumptions about the limits     
within which the blundering or mendacity
 of the witness are confined1 and the   
complete impartiality with which his    
answers are disposed within those       
limits1 the reasoning is perfectly      
sound4 ,but are not these assumptions   
extremely arbitrary1 that is1 are not   
our lotteries and bags of balls rendered
 perfectly precise in many respects in  
which1 in ordinary life1 the conditions 
supposed to correspond to them are so   
vague and uncertain that no such method 
of reasoning becomes practically        
available8 ,suppose that a person whom  
,i have long known1 and of whose measure
 of veracity and judgment ,i may be     
supposed therefore to have acquired some
 knowledge1 informs me that there is    
something to my advantage if ,i choose  
to go to certain trouble or expense in  
order to secure it4 ,as regards the     
general veracity of the witness1 then1  
there is no difficulty2 we suppose that 
this is determined for us4 ,but as      

regards his story1 difficulty and       
vagueness emerge at every point4 ,what  
is the number of balls in the bag here8 
,what in fact are the nature and        
contents of the bag out of which we     
suppose the drawing to have been made8  
,it does not seem that the materials for
 any rational judgment exist here4 ,but 
if we are to get at any such amended    
figure of veracity as those attained in 
the above example1 these questions must 
necessarily be answered with some degree
 of accuracy2 for the main point of the 
method consists in determining how often
 the event must be considered not to    
happen1 and thence inferring how often  
the witness will be led wrongly to      
assert that it has happened4  ,it is not
 of course denied that considerations of
 the kind in question have some         
influence upon our decision1 but only   
that this influence could under any     
ordinary circumstances be submitted to  
numerical determination4 ,we are        
doubtless liable to have information    

given to us that we have come in for    
some kind of fortune1 for instance1 when
 no such good luck has really befallen  
us2 and this not once only but          
repeatedly4 ,but who can give the       
faintest intimation of the nature and   
number of the occasions on which1 a     
blank being thus really drawn1 a prize  
will nevertheless be falsely announced8 
,it appears to me therefore that        
numerical results of any practical value
 can seldom1 if ever1 be looked for from
 this method of procedure4  #i4 ,our    
conclusion in the case of the lottery1  
or1 what comes to the same thing1 in the
 case of the bag with black and white   
balls1 has been questioned or objected  
to#d on the ground that it is contrary  
to all experience to suppose that the   
testimony of a moderately good witness  
could be so enormously depreciated under
 such circumstances4 ,i should prefer to
 base the objection on the ground that  
experience scarcely ever presents such  
circumstances as those supposed2 but if 

we postulate their existence the given  
conclusion seems correct enough4 ,assume
 that a man is merely required to say   
yes or no2 assume also a group or       
succession of cases in which no should  
rightly be said very much oftener than  
yes4 ,then1 assuming almost any general 
truthfulness of the witness1 we may     
easily suppose the rightful occasions   
for denial to be so much the more       
frequent that a majority of his         
affirmative answers will actually occur 
as false 'noes' rather than as correct  
'ayes4' ,this of course lowers the      
average value of his 'ayes1' and renders
 them comparatively untrustworthy4      
,consider the following example4 ,i have
 a gardener whom ,i trust as to all     
ordinary matters of fact4 ,if he were to
 tell me some morning that my dog had   
run away ,i should fully believe him4   
,he tells me however that the dog has   
gone mad4 ,surely ,i should accept the  
statement with much hesitation1 and on  
the grounds indicated above4 ,it is not 

that he is more likely to be wrong when 
the dog is mad2 but that experience     
shows that there are other complaints   
7e4g4 fits7 which are far more common   
than madness1 and that most of the      
assertions of madness are erroneous     
assertions referring to these4 ,this    
seems a somewhat parallel case to that  
in which we find that most of the       
assertions that a white ball had been   
drawn are really false assertions       
referring to the drawing of a black     
ball4 ,practically ,i do not think that 
any one would feel a difficulty in thus 
exorbitantly discounting some particular
 assertion of a witness whom in most    
other respects he fully trusted4  #aj4  
,there is one particular case which has 
been regarded as a difficulty in the way
 of this treatment of the problem1 but  
which seems to me to be a decided       
confirmation of it2 always1 be it       
understood1 within the very narrow and  
artificial limits to which we must      
suppose ourselves to be confined4 ,this 

is the case of a witness whose veracity 
is just one-half2 that is1 one who1 when
 a mere yes or no is demanded of him1 is
 as often wrong as right4 ,in the case  
of any other assigned degree of veracity
 it is extremely difficult to get       
anything approaching to a confirmation  
from practical judgment and experience4 
,we are not accustomed to estimate the  
merits of witnesses in this way1 and    
hardly appreciate what is meant by his  
numerical degree of truthfulness4 ,but  
as regards the man whose veracity is    
one-half1 we are 7as ,mr ,c4 ,j4 ,monro 
has very ingeniously suggested7 only too
 well acquainted with such witnesses1   
though under a somewhat different name2 
for this is really nothing else than the
 case of a person confidently answering 
a question about a subject-matter of    
which he knows nothing1 and can         
therefore only give a mere guess4  ,now 
in the case of the lottery with one     
prize1 when the witness whose veracity  
is one-half tells us that we have gained

 the prize1 we find on calculation that 
his testimony goes for absolutely       
nothing2 the chances that we have got   
the prize are just the same as they     
would be if he had never opened his     
lips1 viz4 #a_/ajjj4 ,but clearly this  
is what ought to be the result1 for the 
witness who knows nothing about the     
matter leaves it exactly as he found it4
 ,he is indeed1 in strictness1 scarcely 
a witness at all2 for the natural       
function of a witness is to examine the 
matter1 and so to add confirmation1 more
 or less1 according to his judgment and 
probity1 but at any rate to offer an    
improvement upon the mere guesser4 ,if1 
however1 we will give heed to his mere  
guess we are doing just the same thing  
as if we were to guess ourselves1 in    
which case of course the odds that we   
are right are simply measured by the    
frequency of occurrence of the events4  
,we cannot quite so readily apply the   
same rule to the other case1 namely to  
that of the numbered balls1 for there   

the witness who is right every other    
time may really be a very fair1 or even 
excellent1 witness4 ,if he has many ways
 of going wrong1 and yet is right in    
half his statements1 it is clear that he
 must have taken some degree of care1   
and cannot have merely guessed4 ,in a   
case of yes or no1 any one can be right 
every other time1 but it is different   
where truth is single and error is      
manifold4 ,to represent the case of a   
simply worthless witness when there were
 #ajjj balls and the drawing of one     
assigned ball was in question1 we should
 have to put his figure of veracity at  
#a_/ajjj4 ,if this were done we should  
of course get a similar result4  #aa4   
,it deserves notice therefore that the  
figure of veracity1 or fraction         
representing the general truthfulness of
 a witness1 is in a way relative1 not   
absolute2 that is1 it depends upon1 and 
varies with1 the general character of   
the answer which he is supposed to give4
 ,two witnesses of equal intrinsic      

veracity and worth1 one of whom confined
 himself to saying yes and no1 whilst   
the other ventured to make more original
 assertions1 would be represented by    
different fractions2 the former having  
set himself a much easier task than the 
latter4 ,the real caution and           
truthfulness of the witness are only one
 factor1 therefore1 in his actual figure
 of veracity2 the other factor consists 
of the nature of his assertions1 as just
 pointed out4 ,the ordinary plan        
therefore1 in such problems1 of         
assigning an average truthfulness to the
 witness1 and accepting this alike in   
the case of each of the two kinds of    
answers1 though convenient1 seems       
scarcely sound4 ,this consideration     
would however be of much more importance
 were not the discussions upon the      
subject mainly concerned with only one  
description of answer1 namely that of   
the 'yes or no' kind4  #ab4 ,so much for
 the methodical way of treating such a  
problem4 ,the way in which it would be  

taken in hand by those who had made no  
study of ,probability is very different4
 ,it would1 ,i apprehend1 strike them as
 follows4 ,they would say to themselves1
 ,here is a story related by a witness  
who tells the truth1 say1 nine times out
 of ten4 ,but it is a story of a kind   
which experience shows to be very       
generally made untruly1 say #ii times   
out of #ajj4 ,having then these opposite
 inducements to belief1 they would      
attempt in some way to strike a balance 
between them4 ,nothing in the nature of 
a strict rule could be given to enable  
them to decide how they might escape out
 of the difficulty4 ,probably1 in so far
 as they did not judge at haphazard1    
they would be guided by still further   
resort to experience1 or unconscious    
recollections of its previous teachings1
 in order to settle which of the two    
opposing inductions was better entitled 
to carry the day in the particular case 
before them4 ,the reader will readily   
see that any general solution of the    

problem1 when thus presented1 is        
impossible4 ,it is simply the now       
familiar case 7,chap4 ,,ix4 #ad-cb7 of  
an individual which belongs equally to  
two distinct1 or even1 in respect of    
their characteristics1 opposing classes4
 ,we cannot decide off-hand to which of 
the two its characteristics most        
naturally and rightly refer it4 ,a fresh
 induction is needed in order to settle 
this point4  #ac4 ,rules have indeed    
been suggested by various writers in    
order to extricate us from the          
difficulty4 ,the controversy about      
miracles has probably been the most     
fertile occasion for suggestions of this
 kind on one side or the other4 ,it is  
to this controversy1 presumably1 that   
the phrase is due1 so often employed in 
discussions upon similar subjects1 'a   
contest of opposite improbabilities4'   
,what is meant by such an expression is 
clearly this3 that in forming a judgment
 upon the truth of certain assertions we
 may find that they are comprised in two

 very distinct classes1 so that1        
according as we regarded them as        
belonging to one or the other of these  
distinct classes1 our opinion as to     
their truth would be very different4    
,such an assertion belongs to one class1
 of course1 by its being a statement of 
a particular witness1 or kind of        
witness2 it belongs to the other by its 
being a particular kind of story1 one of
 what is called an improbable nature4   
,its belonging to the former class is so
 far favourable to its truth1 its       
belonging to the latter is so far       
hostile to its truth4 ,it seems to be   
assumed1 in speaking of a contest of    
opposite improbabilities1 that when     
these different sources of conviction   
co-exist together1 they would each in   
some way retain their probative force so
 as to produce a contest1 ending        
generally in a victory to one or other  
of them4 ,hume1 for instance1 speaks of 
our deducting one probability from the  
other1 and apportioning our belief to   

the remainder4#e ,thomson1 in his ,laws 
of ,thought1 speaks of one probability  
as entirely superseding the other4  #ad4
 ,it does not appear to me that the     
slightest philosophical value can be    
attached to any such rules as these4    
,they doubtless may1 and indeed will1   
hold in individual cases1 but they      
cannot lay claim to any generality4     
,even the notion of a contest1 as any   
necessary ingredient in the case1 must  
be laid aside4 ,for let us refer again  
to the way in which the perplexity      
arises1 and we shall readily see1 as has
 just been remarked1 that it is nothing 
more than a particular exemplification  
of a difficulty which has already been  
recognized as incapable of solution by  
any general  priori method of treatment4
 ,all that we are supposed to have      
before us is a statement4 ,on this      
occasion it is made by a witness who    
lies1 say1 once in ten times in the long
 run2 that is1 who mostly tells the     
truth4 ,but on the other hand1 it is a  

statement which experience1 derived from
 a variety of witnesses on various      
occasions1 assures us is mostly false2  
stated numerically it is found1 let us  
suppose1 to be false #ii times in a     
hundred4  ,now1 as was shown in the     
chapter on ,induction1 we are thus      
brought to a complete dead lock4 ,our   
science offers no principles by which we
 can form an opinion1 or attempt to     
decide the matter one way or the other2 
for1 as we found1 there are an          
indefinite number of conclusions which  
are all equally possible4 ,for instance1
 all the witness' extraordinary         
assertions may be true1 or they may all 
be false1 or they may be divided into   
the true and the false in any proportion
 whatever4 ,having gone so far in our   
appeal to statistics as to recognize    
that the witness is generally right1 but
 that his story is generally false1 we  
cannot stop there4 ,we ought to make    
still further appeal to experience1 and 
ascertain how it stands with regard to  

his stories when they are of that       
particular nature3 or rather1 for this  
would be to make a needlessly narrow    
reference1 how it stands with regard to 
stories of that kind when advanced by   
witnesses of his general character1     
position1 sympathies1 and so on4#f  #ae4
 ,that extraordinary stories are in many
 cases1 probably in a great majority of 
cases1 less trustworthy than others must
 be fully admitted4 ,that is1 if we were
 to make two distinct classes of such   
stories respectively1 we should find    
that the same witness1 or similar       
witnesses1 were proportionally more     
often wrong when asserting the former   
than when asserting the latter4 ,but it 
does not by any means appear to me that 
this must always be the case4 ,we may   
well conceive1 for instance1 that with  
some people the mere fact of the story  
being of a very unusual character may   
make them more careful in what they     
state1 so as actually to add to their   
veracity4 ,if this were so we might be  

ready to accept their extraordinary     
stories with even more readiness than   
their ordinary ones4  ,such a           
supposition as that just made does not  
seem to me by any means forced4 ,put    
such a case as this3 let us suppose that
 two persons1 one of them a man of      
merely ordinary probity and             
intelligence1 the other a scientific    
naturalist1 make a statement about some 
common event4 ,we believe them both4    
,let them now each report some          
extraordinary lusus natur or monstrosity
 which they profess to have seen4 ,most 
persons1 we may presume1 would receive  
the statement of the naturalist in this 
latter case almost as readily as in the 
former3 whereas when the same story came
 from the unscientific observer it would
 be received with considerable          
hesitation4 ,whence arises the          
difference8 ,from the conviction that   
the naturalist will be far more careful1
 and therefore to the full as accurate1 
in matters of this kind as in those of  

the most ordinary description1 whereas  
with the other man we feel by no means  
the same confidence4 ,even if any one is
 not prepared to go this length1 he will
 probably admit that the difference of  
credit which he would attach to the two 
kinds of story1 respectively1 when they 
came from the naturalist1 would be much 
less than what it would be when they    
came from the other man4  #af4 ,whilst  
we are on this part of the subject1 it  
must be pointed out that there is       
considerable ambiguity and consequent   
confusion about the use of the term 'an 
extraordinary story4' ,within the       
province of pure ,probability it ought  
to mean simply a story which asserts an 
unusual event4 ,at least this is the    
view which has been adopted and         
maintained1 it is hoped consistently1   
throughout this work4 ,so long as we    
adhere to this sense we know precisely  
what we mean by the term4 ,it has a     
purely objective reference2 it simply   
connotes a very low degree of relative  

statistical frequency1 actual or        
prospective4 ,out of a great number of  
events we suppose a selection of some   
particular kind to be contemplated1     
which occurs relatively very seldom1 and
 this is termed an unusual or           
extraordinary event4 ,it follows1 as was
 abundantly shown in a former chapter1  
that owing to the rarity of the event we
 are very little disposed to expect its 
occurrence in any given case4 ,our guess
 about it1 in case we thus anticipated  
it1 would very seldom be justified1 and 
we are therefore apt to be much         
surprised when it does occur4 ,this1 ,i 
take it1 is the only legitimate sense of
 'extraordinary' so far as ,probability 
is concerned4  ,but there is another and
 very different use of the word1 which  
belongs to ,induction1 or rather to the 
science of evidence in general1 more    
than to that limited portion of it      
termed ,probability4 ,in this sense the 
'extraordinary1' and still more the     
'improbable1' event is not merely one of

 extreme statistical rarity1 which we   
could not expect to guess aright1 but   
which on moderate evidence we may pretty
 readily accept2 it is rather one which 
possesses1 so to say1 an actual         
evidence-resisting power4 ,it may be    
something which affects the credibility 
of the witness at the fountain-head1    
which makes1 that is1 his statements    
upon such a subject essentially inferior
 to those on other subjects4 ,this is   
the case1 for instance1 with anything   
which excites his prejudices or passions
 or superstitions4 ,in these cases it   
would seem unreasonable to attempt to   
estimate the credibility of the witness 
by calculating 7as in #f7 how often his 
errors would mislead us through his     
having been wrongly brought to an       
affirmation instead of adhering         
correctly to a negation4 ,we should     
rather be disposed to put our correction
 on the witness' average veracity at    
once4  #ag4 ,in true ,probability1 as   
has just been remarked1 every event has 

its own definitely recognizable degree  
of frequency of occurrence4 ,it may be  
excessively rare1 rare to any extreme we
 like to postulate1 but still every one 
who understands and admits the data upon
 which its occurrence depends will be   
able to appreciate within what range of 
experience it may be expected to present
 itself4 ,we do not expect it in any    
individual case1 nor within any brief   
range1 but we do confidently expect it  
within an adequately extensive range4   
,how therefore can miraculous stories be
 similarly taken account of1 when the   
disputants1 on one side at least1 are   
not prepared to admit their actual      
occurrence anywhere or at any time8 ,how
 can any arrangement of bags and balls1 
or other mechanical or numerical        
illustrations of unlikely events1 be    
admitted as fairly illustrative of      
miraculous occurrences1 or indeed of    
many of those which come under the      
designation of 'very extraordinary' or  
'highly improbable'8 ,those who contest 

the occurrence of a particular miracle1 
as reported by this or that narrator1 do
 not admit that miracles are to be      
confidently expected sooner or later4   
,it is not a question as to whether what
 must happen sometimes has happened some
 particular time1 and therefore no      
illustration of the kind can be regarded
 as apposite4  ,how unsuitable these    
merely rare events1 however excessive   
their rarity may be1 are as examples of 
miraculous events1 will be evident from 
a single consideration4 ,no one1 ,i     
presume1 who admitted the occasional    
occurrence of an exceedingly unusual    
combination1 would be in much doubt if  
he considered that he had actually seen 
it himself4#g ,on the other hand1 few   
men of any really scientific turn would 
readily accept a miracle even if it     
appeared to happen under their very     
eyes4 ,they might be staggered at the   
time1 but they would probably soon come 
to discredit it afterwards1 or so       
explain it as to evacuate it of all that

 is meant by miraculous4  #ah4 ,it      
appears to me therefore1 on the whole1  
that very little can be made of these   
problems of testimony in the way in     
which it is generally intended that they
 should be treated2 that is1 in         
obtaining specific rules for the        
estimation of the testimony under any   
given circumstances4 ,assuming that the 
veracity of the witness can be measured1
 we encounter the real difficulty in the
 utter impossibility of determining the 
limits within which the failures of the 
event in question are to be considered  
to lie1 and the degree of explicitness  
with which the witness is supposed to   
answer the enquiry addressed to him2    
both of these being characteristics of  
which it is necessary to have a         
numerical estimate before we can        
consider ourselves in possession of the 
requisite data4  ,since therefore the   
practical resource of most persons1 viz4
 that of putting a direct and immediate 
correction1 of course of a somewhat     

conjectural nature1 upon the general    
trustworthiness of the witness1 by a    
consideration of the nature of the      
circumstances under which his statement 
is made1 is essentially unscientific and
 irreducible to rule2 it really seems to
 me that there is something to be said  
in favour of the simple plan of trusting
 in all cases alike to the witness'     
general veracity4#h ,that is1 whether   
his story is ordinary or extraordinary1 
we may resolve to put it on the same    
footing of credibility1 provided of     
course that the event is fully          
recognized as one which does or may     
occasionally happen4 ,it is true that we
 shall thus go constantly astray1 and   
may do so to a great extent1 so that if 
there were any rational and precise     
method of specializing his              
trustworthiness1 according to the nature
 of his story1 we should be on much     
firmer ground4 ,but at least we may thus
 know what to expect on the average4    
,provided we have a sufficient number   

and variety of statements from him1 and 
always take them at the same constant   
rate or degree of trustworthiness1 we   
may succeed in balancing and correcting 
our conduct in the long run so as to    
avoid any ruinous error4  #ai4 ,a few   
words may now be added about the        
combination of testimony4 ,no new       
principles are introduced here1 though  
the consequent complication is naturally
 greater4 ,let us suppose two witnesses1
 the veracity of each being #i_/aj4 ,now
 suppose #ajj statements made by the    
pair2 according to the plan of          
proceeding adopted before1 we should    
have them both right #ha times and both 
wrong once1 in the remaining #ah cases  
one being right and the other wrong4    
,but since they are both supposed to    
give the same account1 what we have to  
compare together are the number of      
occasions on which they agree and are   
right1 and the total number on which    
they agree whether right or wrong4 ,the 
ratio of the former to the latter is the

 fraction which expresses the           
trustworthiness of their combination of 
testimony in the case in question4  ,in 
attempting to decide this point the only
 difficulty is in determining how often 
they will be found to agree when they   
are both wrong1 for clearly they must   
agree when they are both right4 ,this   
enquiry turns of course upon the number 
of ways in which they can succeed in    
going wrong4 ,suppose first the case of 
a simple yes or no 7as in #f71 and take 
the same example1 of a bag with #ajjj   
balls1 in which one only is white4      
,proceeding as before1 we should find   
that out of #ajj1jjj drawings 7the      
number required in order to obtain a    
complete cycle of all possible          
occurrences1 as well as of all possible 
reports about them7 the two witnesses   
agree in a correct report of the        
appearance of white in #ha1 and agree in
 a wrong report of it in #iii4 ,the     
,probability therefore of the story when
 so attested is #ha_/ajhj2 the fact     

therefore of two such witnesses of equal
 veracity having concurred makes the    
report nearly #i times as likely as when
 it rested upon the authority of only   
one of them4#i  #bj4 ,when however the  
witnesses have many ways of going wrong1
 the fact of their agreeing makes the   
report far more likely to be true4 ,for 
instance1 in the case of the #ajjj      
numbered balls1 it is very unlikely that
 when they both mistake the number they 
should 7without collusion7 happen to    
make the same misstatement4 ,whereas1 in
 the last case1 every combined          
misstatement necessarily led them both  
to the assertion that the event in      
question had happened1 we should now    
find that only once in #iii  #iii times 
would they both be led to assert that   
some given number 7say1 as before1 #be7 
had been drawn4 ,the odds in favour of  
the event in fact now become            
#hjiai_/hjibj1 which are enormously     
greater than when there was only one    
witness4  ,it appears therefore that    

when two1 and of course still more when 
many1 witnesses agree in a statement in 
a matter about which they might make    
many and various errors1 the combination
 of their favourable testimony adds     
enormously to the likelihood of the     
event2 provided always that there is no 
chance of collusion4 ,and in the extreme
 case of the opportunities for error    
being1 as they well may be1 practically 
infinite in number1 such combination    
would produce almost perfect certainty4 
,but then this condition1 viz4 absence  
of collusion1 very seldom can be        
secured4 ,practically our main source of
 error and suspicion is in the possible 
existence of some kind of collusion4    
,since we can seldom entirely get rid of
 this danger1 and when it exists it can 
never be submitted to numerical         
calculation1 it appears to me that      
combination of testimony1 in regard to  
detailed accounts1 is yet more unfitted 
for consideration in ,probability than  
even that of single testimony4  #ba4    

,the impossibility of any adequate or   
even appropriate consideration of the   
credibility of miraculous stories by the
 rules of ,probability has been already 
noticed in #ag4 ,but1 since the grounds 
of this impossibility are often very    
insufficiently appreciated1 a few pages 
may conveniently be added here with a   
view to enforcing this point4 ,if it be 
regarded as a digression1 the importance
 of the subject and the persistency with
 which various writers have at one time 
or another attempted to treat it by the 
rules of our science must be the excuse 
for entering upon it4  ,a necessary     
preliminary will be to decide upon some 
definition of a miracle4 ,it will1 we   
may suppose1 be admitted by most persons
 that in calling a miracle 'a suspension
 of a law of causation1' we are giving  
what1 though it may not amount to an    
adequate definition1 is at least true as
 a description4 ,it is true1 though it  
may not be the whole truth4 ,whatever   
else the miracle may be1 this is its    

physical aspect3 this is the point at   
which it comes into contact with the    
subject-matter of science4 ,if it were  
not considered that any suspension of   
causation were involved1 the event would
 be regarded merely as an ordinary one  
to which some special significance was  
attached1 that is1 as a type or symbol  
rather than a miracle4 ,it is this      
aspect moreover of the miracle which is 
now exposed to the main brunt of the    
attack1 and in support of which         
therefore the defence has generally been
 carried on4  ,now it is obvious that   
this1 like most other definitions or    
descriptions1 makes some assumption as  
to matters of fact1 and involves        
something of a theory4 ,the assumption  
clearly is1 that laws of causation      
prevail universally1 or almost          
universally1 throughout nature1 so that 
infractions of them are marked and      
exceptional4 ,this assumption is made1  
but it does not appear that anything    
more than this is necessarily required2 

that is1 there is nothing which need    
necessarily make us side with either of 
the two principal schools which are     
divided as to the nature of these laws  
of causation4 ,the definition will serve
 equally well whether we understand by  
law nothing more than uniformity of     
antecedent and consequent1 or whether we
 assert that there is some deeper and   
more mysterious tie between the events  
than mere sequence4 ,the use of the term
 'causation' in this minimum of         
signification is common to both schools1
 though the one might consider it       
inadequate2 we may speak1 therefore1 of 
'suspensions of causation' without      
committing ourselves to either4  #bb4   
,it should be observed that the aspect  
of the question suggested by this       
definition is one from which we can     
hardly escape4 ,attempts indeed have    
been sometimes made to avoid the        
necessity of any assumption as to the   
universal prevalence of law and order in
 nature1 by defining a miracle from a   

different point of view4 ,a miracle may 
be called1 for instance1 'an immediate  
exertion of creative power1' 'a sign of 
a revelation1' or1 still more vaguely1  
an 'extraordinary event4' ,but nothing  
would be gained by adopting any such    
definitions as these4 ,however they     
might satisfy the theologian1 the       
student of physical science would not   
rest content with them for a moment4 ,he
 would at once assert his own belief1   
and that of other scientific men1 in the
 existence of universal law1 and enquire
 what was the connection of the         
definition with this doctrine4 ,an      
answer would imperatively be demanded to
 the question1 ,does the miracle1 as you
 have described it1 imply an infraction 
of one of these laws1 or does it not8   
,and an answer must be given1 unless    
indeed we reject his assumption by      
denying our belief in the existence of  
this universal law1 in which case of    
course we put ourselves out of the pale 
of argument with him4 ,the necessity of 

having to recognize this fact is growing
 upon men day by day1 with the increased
 study of physical science4 ,and since  
this aspect of the question has to be   
met some time or other1 it is as well to
 place it in the front4 ,the difficulty1
 in its scientific form1 is of course a 
modern one1 for the doctrine out of     
which it arises is modern4 ,but it is   
only one instance1 out of many that     
might be mentioned1 in which the growth 
of some philosophical conception has    
gradually affected the nature of the    
dispute1 and at last shifted the        
position of the battle-ground1 in some  
discussion with which it might not at   
first have appeared to have any         
connection whatever4  #bc4 ,so far our  
path is plain4 ,up to this point        
disciples of very different schools may 
advance together2 for in laying down the
 above doctrine we have carefully       
abstained from implying or admitting    
that it contains the whole truth4 ,but  
from this point two paths branch out    

before us1 paths as different from each 
other in their character1 origin1 and   
direction1 as can well be conceived4 ,as
 this enquiry is only a digression1 we  
may confine ourselves to stating briefly
 what seem to be the characteristics of 
each1 without attempting to give the    
arguments which might be used in their  
support4  7,i47 ,on the one hand1 we may
 assume that this principle of causation
 is the ultimate one4 ,by so terming it1
 we do not mean that it is one from     
which we consciously start in our       
investigations1 as we do from the axioms
 of geometry1 but rather that it is the 
final result towards which we find      
ourselves drawn by a study of nature4   
,finding that1 throughout the scope of  
our enquiries1 event follows event in   
never-failing uniformity1 and finding   
moreover 7some might add7 that this     
experience is supported or even demanded
 by a tendency or law of our nature 7it 
does not matter here how we describe    
it71 we may come to regard this as the  

one fundamental principle on which all  
our enquiries should rest4  7,,ii47 ,or1
 on the other hand1 we may admit a class
 of principles of a very different kind4
 ,allowing that there is this uniformity
 so far as our experience extends1 we   
may yet admit what can hardly be        
otherwise described than by calling it a
 ,superintending ,providence1 that is1 a
 ,scheme or ,order1 in reference to     
which ,design may be predicated without 
using merely metaphorical language4 ,to 
adopt an aptly chosen distinction1 it is
 not to be understood as over-ruling    
events1 but rather as underlying them4  
#bd4 ,now it is quite clear that        
according as we come to the discussion  
of any particular miracle or            
extraordinary story under one or other  
of these prepossessions1 the question of
 its credibility will assume a very     
different aspect4 ,it is sometimes      
overlooked that although a difference   
about facts is one of the conditions of 
a bon fide argument1 a difference which 

reaches to ultimate principles is fatal 
to all argument4 ,the possibility of    
present conflict is banished in such a  
case as absolutely as that of future    
concord4 ,a large amount of popular     
literature on the subject of miracles   
seems to labour under this defect4      
,arguments are stated and examined for  
and against the credibility of          
miraculous stories without the          
disputants appearing to have any        
adequate conception of the chasm which  
separates one side from the other4  #be4
 ,the following illustration may serve  
in some degree to show the sort of      
inconsistency of which we are speaking4 
,a sailor reports that in some remote   
coral island of the ,pacific1 on which  
he had landed by himself1 he had found a
 number of stones on the beach disposed 
in the exact form of a cross4 ,now if we
 conceive a debate to arise about the   
truth of his story1 in which it is      
attempted to decide the matter simply by
 considerations about the validity of   

testimony1 without introducing the      
question of the existence of            
inhabitants1 and the nature of their    
customs1 we shall have some notion of   
the unsatisfactory nature of many of the
 current arguments about miracles4 ,all 
illustrations of this subject are       
imperfect1 but a case like this1 in     
which a supposed trace of human agency  
is detected interfering with the orderly
 sequence of other and non-intelligent  
natural causes1 is as much to the point 
as any illustration can be4 ,the thing  
omitted here from the discussion is     
clearly the one important thing4 ,if we 
suppose that there is no inhabitant1 we 
shall probably disbelieve the story1 or 
consider it to be grossly exaggerated4  
,if we suppose that there are           
inhabitants1 the question is at once    
resolved into a different and somewhat  
more intricate one4 ,the credibility of 
the witness is not the only element1 but
 we should necessarily have to take into
 consideration the character of the     

supposed inhabitants1 and the object of 
such an action on their part4  #bf4     
,considerations of this character are   
doubtless often introduced into the     
discussion1 but it appears to me that   
they are introduced to a very inadequate
 extent4 ,it is often urged1 after      
,paley1 ',once believe in a ,god1 and   
miracles are not incredible4' ,such an  
admission surely demands some           
modification and extension4 ,it should  
rather be stated thus1 ,believe in a    
,god whose working may be traced        
throughout the whole moral and physical 
world4 ,it amounts1 in fact1 to         
this2--,admit that there may be a design
 which we can trace somehow or other in 
the course of things2 admit that we are 
not wholly confined to tracing the      
connection of events1 or following out  
their effects1 but that we can form some
 idea1 feeble and imperfect though it   
be1 of a scheme4#aj ,paley's advice     
sounds too much like saying1 ,admit that
 there are fairies1 and we can account  

for our cups being cracked4 ,the        
admission is not to be made in so       
off-hand a manner4 ,to any one labouring
 under the difficulty we are speaking   
of1 this belief in a ,god almost out of 
any constant relation to nature1 whom we
 then imagine to occasionally manifest  
himself in a perhaps irregular manner1  
is altogether impossible4 ,the only form
 under which belief in the ,deity can   
gain entrance into his mind is as the   
controlling ,spirit of an infinite and  
orderly system4 ,in fact1 it appears to 
me1 paradoxical as the suggestion may   
appear1 that it might even be more easy 
for a person thoroughly imbued with the 
spirit of ,inductive science1 though an 
atheist1 to believe in a miracle which  
formed a part of a vast system1 than for
 such a person1 as a theist1 to accept  
an isolated miracle4  #bg4 ,it is       
therefore with great prudence that      
,hume1 and others after him1 have       
practically insisted on commencing with 
a discussion of the credibility of the  

single miracle1 treating the question as
 though the ,christian ,revelation could
 be adequately regarded as a succession 
of such events4 ,as well might one      
consider the living body to be          
represented by the aggregate of the     
limbs which compose it4 ,what is to be  
complained of in so many popular        
discussions on the subject is the entire
 absence of any recognition of the      
different ground on which the attackers 
and defenders of miracles are so often  
really standing4 ,proofs and            
illustrations are produced in endless   
number1 which involving1 as they almost 
all do in the mind of the disputants on 
one side at least1 that very principle  
of causation1 the absence of which in   
the case in question they are intended  
to establish1 they fail in the single   
essential point4 ,to attempt to induce  
any one to disbelieve in the existence  
of physical causation1 in a given       
instance1 by means of illustrations     
which to him seem only additional       

examples of the principle in question1  
is like trying to make a dam1 in order  
to stop the flow of a river1 by         
shovelling in snow4 ,such illustrations 
are plentiful in times of controversy1  
but being in reality only modified forms
 of that which they are applied to      
counteract1 they change their shape at  
their first contact with the            
disbeliever's mind1 and only help to    
swell the flood which they were intended
 to check4   #a4 ,reasons were given in 
the last chapter against the propriety  
of   applying the rules of ,probability 
with any strictness to such   examples  
as these4 ,but although all approach to 
numerical accuracy   is unattainable1 we
 do undoubtedly recognize in ordinary   
life a   distinction between the        
credibility of one witness and another2 
such   a rough practical distinction    
will be quite sufficient for the        
purposes of this chapter4 ,for          
convenience1 and to illustrate the      
theory1 the examples are best stated in 

a numerical form1 but it is   not       
intended thereby to imply that any such 
accuracy is really   attainable in      
practice4  #b4 ,i must plead guilty to  
this charge myself1 in the first edition
 of   this work4 ,the result was to make
 the treatment of this part of the      
subject obscure and imperfect1 and in   
some respects erroneous4  #c4 ,the      
generalized algebraical form of this    
result is as follows4 ,let   p be the   
priori probability of an event1 and x be
 the   credibility of the witness4      
,then1 if he asserts that the event     
happened1 the probability that it really
 did happen is   px_/7px  7#a - p77#a - 
x772 whilst if he asserts that it did   
not   happen the probability that it did
 happen is   p7#a - x7_/7p7#a - x7  7#a 
- p7x74    ,in illustration of some     
remarks to be presently made1 the reader
   will notice that on making either of 
these expressions  p1 we   obtain in    
each case x  #a_/b4 ,that is1 a witness 
whose veracity  #a_/b   leaves the      

priori probability of an event 7of this 
kind7   unaffected4    ,if1 on the other
 hand1 we make these expressions equal  
to x and #a - x   respectively1 we      
obtain in each case p  #a_/b4 ,that is1 
when an event   7of this kind7 is as    
likely to happen as not1 the ordinary   
veracity   of the witness in respect of 
it remains unaffected4  #d4 ,todhunter's
 ,history1 p4 #djj4 ,philosophical      
,magazine1 ,july1 #ahfd4  #e4 ,when     
therefore these two kinds of experience 
are contrary1 we have   nothing to do   
but subtract the one from the other1 and
 embrace an   opinion1 either on one    
side or the other1 with that assurance  
which   arises from the remainder4      
7,essay on ,miracles47  #f4             
,considerations of this kind have indeed
 been introduced into the   mathematical
 treatment of the subject4 ,the common  
algebraical   solution of the problem in
 #e 7to begin with the simplest case7 is
   of course as follows4 ,let p be the  
antecedent probability of the   event1  

and t the measure of the truthfulness of
 the witness2 then   the chance of his  
statement being true is pt_/7pt  7#a -  
p77#a - t774   ,this supposes him to lie
 as much when the event does not happen 
as   when it does4 ,but we may meet the 
cases supposed in the text by   assuming
 that t' is the measure of his veracity 
when the event does   not happen1 so    
that the above formula becomes pt_/7pt  
7#a - p77#a - t'774   ,here t' and t    
measure respectively his trustworthiness
 in usual and   unusual events4 ,as a   
formal solution this certainly meets the
   objections stated above in #ad and   
#ae4 ,the determination however   of t' 
would demand1 as ,i have remarked1      
continually renewed appeal   to         
experience4 ,in any case the practical  
methods which would be   adopted1 if any
 plans of the kind indicated above were 
resorted to1   seem to me to differ very
 much from that adopted by the          
mathematicians1 in their spirit and     
plan4  #g4 ,laplace1 for instance       

7,essai1 ed4 #ahbe1 p4 #adi71 says that 
if we   saw #ajj dies 7known of course  
to be fair ones7 all give the same      
face1 we should be bewildered at the    
time1 and need confirmation   from      
others1 but that1 after due examination1
 no one would feel   obliged to         
postulate hallucination in the matter4  
,but the chance of   this occurrence is 
represented by a fraction whose         
numerator is #a1   and denominator      
contains #gg figures1 and is therefore  
utterly   inappreciable by the          
imagination4 ,it must be admitted1      
though1 that   there is something       
hypothetical about such an example1 for 
we could   not really know that the dies
 were fair with a confidence even       
distantly approaching such prodigious   
odds4 ,in other words1 it is   difficult
 here to keep apart those different     
aspects of the question   discussed in  
,chap4 ,,xiv4 #bh-cc4  #h4 ,in the first
 edition this was stated1 as it now     
seems to me1 in   decidedly too         

unqualified a manner4 ,it must be       
remembered1 however1   that 7as was     
shown in #g7 this plan is really the    
best theoretical   one which can be     
adopted in certain cases4  #i4 ,it is on
 this principle that the remarkable     
conclusion mentioned on   p4 #dje is    
based4 ,suppose an event whose          
probability is p2 and that1   of a      
number of witnesses of the same veracity
 7y71 m assert that it   happened1 and n
 deny this4 ,generalizing the           
arithmetical reasoning   given above we 
see that the chance of the event being  
asserted   varies as    pym 7#a - y7n   
7#a - p7 yn 7#a - y7m2    7viz4 as the  
chance that the event happens1 and that 
m are right and   n are wrong2 plus the 
chance that it does not happen1 and that
 n   are right and m are wrong74 ,and   
the chance of its being rightly         
asserted as pym 7#a - y7n4 ,therefore   
the chance that when we   have an       
assertion before us it is a true one is 
pym 7#a - y7n                           

----------------------------------------
------   pym 7#a - y7n  7#a - p7 yn 7#a 
- y7m1    which is equal to             
pym-n   --------------------------------
   pym-n  7#a - p7 7#a - y7m-n4    ,but 
this last expression represents the     
probability of an assertion   which is  
unanimously supported by m - n such     
witnesses4  #aj4 ,the stress which      
,butler lays upon this notion of a      
scheme is1 ,i   think1 one great merit  
of his ,analogy4     ,,chapter ,,xviii4 
,,the ,,nature ,,and ,,use ,,of ,,an    
,,average1 ,,and ,,on ,,the ,,different 
,,kinds ,,of ,,average4#a  #a4 ,there is
 much need of some good account1        
accessible to the ordinary   ,english   
reader1 of the nature and properties of 
the principal kinds   of ,mean4 ,the    
common text-books of ,algebra suggest   
that there are   only three such1 viz4  
the arithmetical1 the geometrical and   
the   harmonical3--thus including two   
with which the statistician has   little
 or nothing to do1 and excluding two or 

more with which he   should have a great
 deal to do4 ,the best three references 
,i can give   the reader are the        
following4 7#a7 ,the article ,moyenne in
 the   ,dictionnaire des ,sciences      
,mdicales1 by ,dr ,bertillon4 ,this is  
written somewhat from the ,quetelet     
point of view4 7#b7 ,a paper by         
,fechner in the ,abhandlungen d4 ,math4 
phys4 ,classe   d4 ,kn4 ,schs4          
,gesellschaft d4 ,wiss4 #ahgh2 pp4      
#a-gf4 ,this contains   a very          
interesting discussion1 especially for  
the statistician1 of a   number of      
different kinds of mean4 ,his account of
 the median is   remarkably full and    
valuable4 ,but little mathematical      
knowledge is   demanded4 7#c7 ,a paper  
by ,mr ,f4 ,y4 ,edgeworth in the        
,camb4 ,phil4 ,trans4 for #ahhe1        
entitled ,observations and              
,statistics4 ,this demands some         
mathematical knowledge4 ,instead of     
dealing1 as such investigations         
generally do1 with only one ,law of     

,error and with only one kind of mean1  
it covers a wide field of               
investigation4   #a4 ,we have had such  
frequent occasion to refer to averages1 
and to the kind of uniformity which they
 are apt to display in contrast with    
individual objects or events1 that it   
will now be convenient to discuss       
somewhat more minutely what are the     
different kinds of available average1   
and what exactly are the functions they 
perform4  ,the first vague notion of an 
average1 as we now understand it1 seems 
to me to involve little more than that  
of a something intermediate to a number 
of objects4 ,the objects must of course 
resemble each other in certain respects1
 otherwise we should not think of       
classing them together2 and they must   
also differ in certain respects1        
otherwise we should not distinguish     
between them4 ,what the average does for
 us1 under this primitive form1 is to   
enable us conveniently to retain the    
group together as a whole4 ,that is1 it 

furnishes a sort of representative value
 of the quantitative aspect of the      
things in question1 which will serve for
 certain purposes to take the place of  
any single member of the group4  ,it    
would seem then that the first dawn of  
the conception which science reduces to 
accuracy under the designation of an    
average or mean1 and then proceeds to   
subdivide into various distinct species 
of means1 presents itself as performing 
some of the functions of a general name4
 ,for what is the main use of a general 
name8 ,it is to reduce a plurality of   
objects to unity2 to group a number of  
things together by reference to some    
qualities which they possess in common4 
,the ordinary general name rests upon a 
considerable variety of attributes1     
mostly of a qualitative character1      
whereas the average1 in so far as it    
serves the same sort of purpose1 rests  
rather upon a single quantitative       
attribute4 ,it directs attention to a   
certain kind and degree of magnitude4   

,when the grazier says of his sheep that
 'one with another they will fetch about
 #ej shillings1' or the farmer buys a   
lot of poles which 'run to about #aj    
feet1' it is true that they are not     
strictly using the equivalent of either 
a general or a collective name4 ,but    
they are coming very near to such use1  
in picking out a sort of type or        
specimen of the magnitude to which      
attention is to be directed1 and in     
classing the whole group by its         
resemblance to this type4 ,the grazier  
is thinking of his sheep3 not in a      
merely general sense1 as sheep1 and     
therefore under that name or conception1
 but as sheep of a certain approximate  
money value4 ,some will be more1 some   
less1 but they are all near enough to   
the assigned value to be conveniently   
classed together as if by a name4 ,many 
of our rough quantitative designations  
seem to be of this kind1 as when we     
speak of 'eight-day clocks' or          
'twelve-stone men1' c42 unless of course

 we intend 7as we sometimes do in these 
cases7 to assign a maximum or minimum   
value4 ,it is not indeed easy to see how
 else we could readily convey a merely  
general notion of the quantitative      
aspect of things1 except by selecting a 
type as above1 or by assigning certain  
limits within which the things are      
supposed to lie4  #b4 ,so far there is  
not necessarily any idea introduced of  
comparison1--of comparison1 that is1 of 
one group with another1--by aid of such 
an average4 ,as soon as we begin to     
think of this we have to be more precise
 in saying what we mean by an average4  
,we can easily see that the number of   
possible kinds of average1 in the sense 
of intermediate values1 is very great2  
is1 in fact1 indefinitely great4 ,out of
 the general conception of an           
intermediate value1 obtained by some    
treatment of the original magnitudes1 we
 can elicit as many subdivisions as we  
please1 by various modes of treatment4  
,there are however only three or four   

which for our purposes need be taken    
into account4  7#a7 ,in the first place 
there is the arithmetical average or    
mean4 ,the rule for obtaining this is   
very simple3 add all the magnitudes     
together1 and divide the sum by their   
number4 ,this is the only kind of       
average with which the unscientific mind
 is thoroughly familiar4 ,but we must   
not let this simplicity and familiarity 
blind us to the fact that there are     
definite reasons for the employment of  
this average1 and that it is therefore  
appropriate only in definite            
circumstances4 ,the reason why it       
affords a safe and accurate intermediate
 value for the actual divergent values1 
is that for many of the ordinary        
purposes of life1 such as purchase and  
sale1 we come to exactly the same       
result1 whether we take account of those
 existent divergences1 or suppose all   
the objects equated to their average4   
,what the grazier must be understood to 
mean1 if he wishes to be accurate1 by   

saying that the average price of his    
sheep is #ej shillings1 is1 that so far 
as that flock is concerned 7and so far  
as he is concerned71 it comes to exactly
 the same thing1 whether they are each  
sold at different prices1 or are all    
sold at the 'average' price4            
,accordingly1 when he compares his sales
 of one year with those of another2 when
 he says that last year the sheep       
averaged #dh shillings against the #ej  
of this year2 the employment of this    
representative or average value is a    
great simplification1 and is perfectly  
accurate for the purpose in question4   
#c4 7#b7 ,now consider this case4 ,a    
certain population is found to have     
doubled itself in #ajj years3 can we    
talk of an 'average' increase here of #a
 per cent4 annually8 ,the circumstances 
are not quite the same as in the former 
case1 but the analogy is sufficiently   
close for our purpose4 ,the answer is   
decidedly1 ,no4 ,if #ajj articles of any
 kind are sold for #ajj1 we say that the

 average price is #a4 ,by this we mean  
that the total amount is the same       
whether the entire lot are sold for     
#ajj1 or whether we split the lot up    
into individuals and sell each of these 
for #a4 ,the average price here is a    
convenient fictitious substitute1 which 
can be applied for each individual      
without altering the aggregate total4   
,if therefore the question be1 ,will a  
supposed increase of #a p4 c4 in each of
 the #ajj years be equivalent to a total
 increase to double the original amount8
 we are proposing a closely analogous   
question4 ,and the answer1 as just      
remarked1 must be in the negative4 ,an  
annual increase of #a p4 c4 continued   
for #ajj years will more than double the
 total2 it will multiply it by about    
#b4g4 ,the true annual increment        
required is measured by sqrt#ajjb2 that 
is1 the population may be said to have  
increased 'on the average' #j4g p4 c4   
annually4  ,we are thus directed to the 
second kind of average discussed in the 

ordinary text-books of algebra1 viz4 the
 geometrical4 ,when only two quantities 
are concerned1 with a single            
intermediate value between them1 the    
geometrical mean constituting this last 
is best described as the mean           
proportional between the two former4    
,thus1 since #c 3 sqrt#ae 33 sqrt#ae 3  
#e1 sqrt#ae is the geometrical mean     
between #c and #e4 ,when a number of    
geometrical means have to be interposed 
between two quantities1 they are to be  
so chosen that every term in the entire 
succession shall bear the same constant 
ratio to its predecessor4 ,thus1 in the 
example in the last paragraph1 #ii      
intermediate steps were to be interposed
 between #a and #b1 with the condition  
that the #ajj ratios thus produced were 
to be all equal4  ,it would seem        
therefore that wherever accurate        
quantitative results are concerned1 the 
selection of the appropriate kind of    
average must depend upon the answer to  
the question1 ,what particular          

intermediate value may be safely        
substituted for the actual variety of   
values1 so far as the precise object in 
view is concerned8 ,this is an aspect of
 the subject which will have to be more 
fully considered in the next chapter4   
,but it may safely be laid down that for
 purposes of general comparison1 where  
accurate numerical relations are not    
required1 almost any kind of            
intermediate value will answer our      
purpose1 provided we adhere to the same 
throughout4 ,thus1 if we want to compare
 the statures of the inhabitants of     
different counties or districts in      
,england1 or of ,englishmen generally   
with those of ,frenchmen1 or to         
ascertain whether the stature of some   
particular class or district is         
increasing or diminishing1 it really    
does not seem to matter what sort of    
average we select provided1 of course1  
that we adhere to the same throughout   
our investigations4 ,a very large amount
 of the work performed by averages is of

 this merely comparative or             
non-quantitative description2 or1 at any
 rate1 nothing more than this is really 
required4 ,this being so1 we should     
naturally resort to the arithmetical    
average2 partly because1 having been    
long in the field1 it is universally    
understood and appealed to1 and partly  
because it happens to be remarkably     
simple and easy to calculate4  #d4 ,the 
arithmetical mean is for most ordinary  
purposes the simplest and best4 ,indeed1
 when we are dealing with a small number
 of somewhat artificially selected      
magnitudes1 it is the only mean which   
any one would think of employing4 ,we   
should not1 for instance1 apply any     
other method to the results of a few    
dozen measurements of lengths or        
estimates of prices4  ,when1 however1 we
 come to consider the results of a very 
large number of measurements of the kind
 which can be grouped together into some
 sort of 'probability curve' we begin to
 find that there is more than one       

alternative before us4 ,begin by        
recurring to the familiar curve         
represented on p4 #bi2 or1 better still1
 to the initial form of it represented  
in the next chapter 7p4 #dgf74 ,we see  
that there are three different ways in  
which we may describe the vertex of the 
curve4 ,we may call it the position of  
the maximum ordinate2 or that of the    
centre of the curve2 or 7as will be seen
 hereafter7 the point to which the      
arithmetical average of all the         
different values of the variable        
magnitude directs us4 ,these three are  
all distinct ways of describing a       
position2 but when we are dealing with a
 symmetrical curve at all resembling the
 binomial or exponential form they all  
three coincide in giving the same       
result3 as they obviously do in the case
 in question4  ,as soon1 however1 as we 
come to consider the case of            
asymmetrical1 or lop-sided curves1 the  
indications given by these three methods
 will be as a rule quite distinct2 and  

therefore the two former of these       
deserve brief notice as representing    
different kinds of means from the       
arithmetical or ordinary one4 ,we shall 
see that there is something about each  
of them which recommends it to common   
sense as being in some way natural and  
appropriate4  #e4 7#c7 ,the first of    
these selects from amongst the various  
different magnitudes that particular one
 which is most frequently represented4  
,it has not acquired any technical      
designation1#a except in so far as it is
 referred to1 by its graphical          
representation1 as the maximum ordinate 
method4 ,but ,i suspect that some appeal
 to such a mean or standard is really   
far from uncommon1 and that if we could 
draw out into clearness the conceptions 
latent in the judgments of the          
comparatively uncultivated1 we should   
find that there were various classes of 
cases in which this mean was naturally  
employed4 ,suppose1 for instance1 that  
there was a fishery in which the fish   

varied very much in size but in which   
the commonest size was somewhat near the
 largest or the smallest4 ,if the men   
were in the habit of selling their fish 
by weight1 it is probable that they     
would before long begin to acquire some 
kind of notion of what is meant by the  
arithmetical mean or average1 and would 
perceive that this was the most         
appropriate test4 ,but if the fish were 
sorted into sizes1 and sold by numbers  
in each of these sizes1 ,i suspect that 
this appeal to a maximum ordinate would 
begin to take the place of the other4   
,that is1 the most numerous class would 
come to be selected as a sort of type by
 which to compare the same fishery at   
one time and another1 or one fishery    
with others4 ,there is also1 as we shall
 see in the next chapter1 some          
scientific ground for the preference of 
this kind of mean in peculiar cases2    
viz4 where the quantities with which we 
deal are true 'errors1' in the estimate 
of some magnitude1 and where also it is 

of much more importance to be exactly   
right1 or very nearly right1 than to    
have merely a low average of error4  #f4
 7#d7 ,the remaining kind of mean is    
that which is now coming to be called   
the median4 ,it is one with which the   
writings of ,mr ,galton have done so    
much to familiarize statisticians1 and  
is best described as follows4 ,conceive 
all the objects in question to be       
marshalled in the order of their        
magnitude2 or1 what comes to the same   
thing1 conceive them sorted into a      
number of equally numerous classes2 then
 the middle one of the row1 or the      
middle one in the middle class1 will be 
the median4 ,i do not think that this   
kind of mean is at all generally        
recognized at present1 but if ,mr       
,galton's scheme of natural measurement 
by what he calls per-centiles should    
come to be generally adopted1 such a    
test would become an important one4     
,there are some conspicuous advantages  
about this kind of mean4 ,for one thing1

 in most statistical enquiries1 it is   
far the simplest to calculate2 and1 what
 is more1 the process of determining it 
serves also to assign another important 
element to be presently noticed1 viz4   
the 'probable error4' ,then again1 as   
,fechner notes1 whereas in the          
arithmetical mean a few exceptional and 
extreme values will often cause         
perplexity by their comparative         
preponderance1 in the case of the median
 7where their number only and not their 
extreme magnitude is taken into account7
 the importance of such disturbance is  
diminished4  #g4 ,a simple illustration 
will serve to indicate how these three  
kinds of mean coalesce into one when we 
are dealing with symmetrical ,laws of   
,error1 but become quite distinct as    
soon as we come to consider those which 
are unsymmetrical4  ,figure3 ,various   
definitions of the mean4  ,suppose that1
 in measuring a magnitude along ,,obdc1 
where the extreme limits are ,,ob and   
,,oc1 the law of error is represented by

 the triangle ,,bac3 the length ,,od    
will be at once the arithmetical mean1  
the median1 and the most frequent       
length3 its frequency being represented 
by the maximum ordinate ,,ad4 ,but now  
suppose1 on the other hand1 that the    
extreme lengths are ,,od and ,,oc1 and  
that the triangle ,,adc represents the  
law of error4 ,the most frequent length 
will be the same as before1 ,,od1 marked
 by the maximum ordinate ,,ad4 ,but the 
mean value will now be ,,ox1 where ,,dx 
#a_/c ,,dc2 and the median will be ,,oy1
 where ,,dy  7#a - #a_/sqrt#b7 ,,dc4    
,another example1 taken from natural    
phenomena1 may be found in the heights  
of the barometer as taken at the same   
hour on successive days4 ,so far as     
#dheg of these may be regarded as       
furnishing a sufficiently stable basis  
of experience1 it certainly seems that  
the resulting curve of frequency is     
asymmetrical4 ,the mean height here was 
found to be #bi4ih3 the median was      
#cj4ja3 the most frequent height was    

#cj4je4 ,the close approximation amongst
 these is an indication that the        
asymmetry is slight4#b  #h4 ,it must be 
clearly understood that the average1 of 
whatever kind it may be1 from the mere  
fact of its being a single substitute   
for an actual plurality of observed     
values1 must let slip a considerable    
amount of information4 ,in fact it is   
only introduced for economy4 ,it may    
entail no loss when used for some one   
assigned purpose1 as in our example     
about the sheep2 but for purposes in    
general it cannot possibly take the     
place of the original diversity1 by     
yielding all the information which they 
contained4 ,if all this is to be        
retained we must resort to some other   
method4 ,practically we generally do one
 of two things3 either 7#a7 we put all  
the figures down in statistical tables1 
or 7#b7 we appeal to a diagram4 ,this   
last plan is convenient when the data   
are very numerous1 or when we wish to   
display or to discover the nature of the

 law of facility under which they range4
  ,the mere assignment of an average    
lets drop nearly all of this1 confining 
itself to the indication of an          
intermediate value4 ,it gives a middle  
point of some kind1 but says nothing    
whatever as to how the original         
magnitudes were grouped about this      
point4 ,for instance1 whether two       
magnitudes had been respectively #be and
 #bg1 or #ae and #cg1 they would yield  
the same arithmetical average of #bf4   
#i4 ,to break off at this stage would   
clearly be to leave the problem in a    
very imperfect condition4 ,we therefore 
naturally seek for some simple test     
which shall indicate how closely the    
separate results were grouped about     
their average1 so as to recover some    
part of the information which had been  
let slip4  ,if any one were approaching 
this problem entirely anew1--that is1 if
 he had no knowledge of the mathematical
 exigencies which attend the theory of  
,least ,squares1--,i apprehend that     

there is but one way in which he would  
set about the business4 ,he would say1  
,the average which we have already      
obtained gave us a rough indication1 by 
assigning an intermediate point amongst 
the original magnitudes4 ,if we want to 
supplement this by a rough indication as
 to how near together these magnitudes  
lie1 the best way will be to treat their
 departures from the mean 7what are     
technically called the errors7 in       
precisely the same way1 viz4 by         
assigning their average4 ,suppose there 
are #ac men whose heights vary by equal 
differences from #e feet to #f feet1 we 
should say that their average height was
 #ff inches1 and their average departure
 from this average was #c-c_/ac inches4 
,looked at from this point of view we   
should then proceed to try how each of  
the above-named averages would answer   
the purpose4 ,two of them1--viz4 the    
arithmetical mean and the median1--will 
answer perfectly2 and1 as we shall      
immediately see1 are frequently used for

 the purpose4 ,so too we could1 if we   
pleased1 employ the geometrical mean1   
though such employment would be tedious1
 owing to the difficulty of calculation4
 ,the 'maximum ordinate' clearly would  
not answer1 since it would generally 7v4
 the diagram on p4 #ddc7 refer us back  
again to the average already obtained1  
and therefore give no information4  ,the
 only point here about which any doubt  
could arise concerns what is called in  
algebra the sign of the errors4 ,two    
equal and opposite errors1 added        
algebraically1 would cancel each other4 
,but when1 as here1 we are regarding the
 errors as substantive quantities1 to be
 considered on their own account1 we    
attend only to their real magnitude1 and
 then these equal and opposite errors   
are to be put upon exactly the same     
footing4  #aj4 ,of the various means    
already discussed1 two1 as just         
remarked1 are in common use4 ,one of    
these is familiarly known1 in           
astronomical and other calculations1 as 

the ',mean ,error1' and is so absolutely
 an application of the same principle of
 the arithmetical mean to the errors1   
that has been already applied to the    
original magnitudes1 that it needs no   
further explanation4 ,thus in the       
example in the last section the mean of 
the heights was #ff inches1 the mean of 
the errors was #c-c_/ac inches4  ,the   
other is the ,median1 though here it is 
always known under another name1 i4e4 as
 the ',probable ,error'2--a technical   
and decidedly misleading term4 ,it is   
briefly defined as that error which we  
are as likely to exceed as to fall short
 of3 otherwise phrased1 if we were to   
arrange all the errors in the order of  
their magnitude1 it corresponds to that 
one of them which just bisects the row4 
,it is therefore the 'median' error3 or1
 if we arrange all the magnitudes in    
successive order1 and divide them into  
four equally numerous classes1--what ,mr
 ,galton calls 'quartiles1'--the first  
and third of the consequent divisions   

will mark the limits of the 'probable   
error' on each side1 whilst the middle  
one will mark the 'median4' ,this       
median1 as was remarked1 coincides1 in  
symmetrical curves1 with the            
arithmetical mean4  ,it is best to stand
 by accepted nomenclature1 but the      
reader must understand that such an     
error is not in any strict sense        
'probable4' ,it is indeed highly        
improbable that in any particular       
instance we should happen to get just   
this error3 in fact1 if we chose to be  
precise and to regard it as one exact   
magnitude out of an infinite number1 it 
would be infinitely unlikely that we    
should hit upon it4 ,nor can it be said 
to be probable that we shall be within  
this limit of the truth1 for1 by        
definition1 we are just as likely to    
exceed as to fall short4 ,as already    
remarked 7see note on p4 #dda71 the     
'maximum ordinate' would have the best  
right to be regarded as indicating the  
really most probable value4  #aa4 7#e7  

,the error of mean square4 ,as          
previously suggested1 the plan which    
would naturally be adopted by any one   
who had no concern with the higher      
mathematics of the subject1 would be to 
take the 'mean error' for the purpose of
 the indication in view4 ,but a very    
different kind of average is generally  
adopted in practice to serve as a test  
of the amount of divergence or          
dispersion4 ,suppose that we have the   
magnitudes x#a1 x#b1 444 xn2 their      
ordinary average is #a_/n 7x#a  x#b  444
  xn71 and their 'errors' are the       
differences between this and x#a1 x#b1  
444 xn4 ,call these errors e#a1 e#b1 444
 en1 then the arithmetical mean of these
 errors 7irrespective of sign7 is #a_/n 
7e#a  e#b  444  en74 ,the ,error of     
,mean ,square1#c on the other hand1 is  
the square root of #a_/n 7e#ab  e#bb    
444  en#b74  ,the reasons for employing 
this latter kind of average in          
preference to any of the others will be 
indicated in the following chapter4 ,at 

present we are concerned only with the  
general logical nature of an average1   
and it is therefore sufficient to point 
out that any such intermediate value    
will answer the purpose of giving a     
rough and summary indication of the     
degree of closeness of approximation    
which our various measures display to   
each other and to their common average4 
,if we were to speak respectively of the
 'first' and the 'second average1' we   
might say that the former of these      
assigns a rough single substitute for   
the plurality of original values1 whilst
 the latter gives a similar rough       
estimate of the degree of their         
departure from the former4  #ab4 ,so far
 we have only been considering the      
general nature of an average1 and the   
principal kinds of average practically  
in use4 ,we must now enquire more       
particularly what are the principal     
purposes for which averages are         
employed4  ,in this respect the first   
thing we have to do is to raise doubts  

in the reader's mind on a subject on    
which he perhaps has not hitherto felt  
the slightest doubt4 ,every one is more 
or less familiar with the practice of   
appealing to an average in order to     
secure accuracy4 ,but distinctly what we
 begin by doing is to sacrifice         
accuracy2 for in place of the plurality 
of actual results we get a single result
 which very possibly does not agree with
 any one of them4 ,if ,i find the       
temperature in different parts of a room
 to be different1 but say that the      
average temperature is #fa1 there may   
perhaps be but few parts of the room    
where this exact temperature is         
realized4 ,and if ,i say that the       
average stature of a certain small group
 of men is #fh inches1 it is probable   
that no one of them will present        
precisely this height4  ,the principal  
way in which accuracy can be thus       
secured is when what we are really      
aiming at is not the magnitudes before  
us but something else of which they are 

an indication4 ,if they are themselves  
'inaccurate1'--we shall see presently   
that this needs some explanation1--then 
the single average1 which in itself     
agrees perhaps with none of them1 may be
 much more nearly what we are actually  
in want of4 ,we shall find it convenient
 to subdivide this view of the subject  
into two parts2 by considering first    
those cases in which quantitative       
considerations enter but slightly1 and  
in which no determination of the        
particular ,law of ,error involved is   
demanded1 and secondly those in which   
such determination cannot be avoided4   
,the latter are only noticed in passing 
here1 as a separate chapter is reserved 
for their fuller consideration4  #ac4   
,the process1 as a practical one1 is    
familiar enough to almost everybody who 
has to work with measures of any kind4  
,suppose1 for instance1 that ,i am      
measuring any object with a brass rod   
which1 as we know1 expands and contracts
 according to the temperature4 ,the     

results will vary slightly1 being       
sometimes a little too great and        
sometimes a little too small4 ,all these
 variations are physical facts1 and if  
what we were concerned with was the     
properties of brass they would be the   
one important fact for us4 ,but when we 
are concerned with the length of the    
object measured1 these facts become     
superfluous and misleading4 ,what we    
want to do is to escape their influence1
 and this we are enabled to effect by   
taking their 7arithmetical7 average1    
provided only they are as often in      
excess as in defect4#d ,for this purpose
 all that is necessary is that equal    
excesses and defects should be equally  
prevalent4 ,it is not necessary to know 
what is the law of variation1 or even to
 be assured that it is of one particular
 kind4 ,provided only that it is in the 
language of the diagram on p4 #bi1      
symmetrical1 then the arithmetical      
average of a suitable and suitably      
varied number of measurements will be   

free from this source of disturbance4   
,and what holds good of this cause of   
variation will hold good of all others  
which obey the same general conditions4 
,in fact the equal prevalence of equal  
and opposite errors seems to be the sole
 and sufficient justification of the    
familiar process of taking the average  
in order to secure accuracy4  #ad4 ,we  
must now make the distinction to which  
attention requires so often to be drawn 
in these subjects between the cases in  
which there respectively is1 and is not1
 some objective magnitude aimed at3 a   
distinction which the common use of the 
same word errors is so apt to obscure4  
,when we talked1 in the case of the     
brass rod1 of excesses and defects being
 equal1 we meant exactly what we said1  
viz4 that for every case in which the   
'true' length 7i4e4 that determined by  
the authorized standard7 is exceeded by 
a given fraction of an inch1 there will 
be a corresponding case in which there  
is an equal defect4  ,on the other hand1

 when there is no such fixed objective  
standard of reference1 it would appear  
that all that we mean by equal excesses 
and defects is permanent symmetry of    
arrangement4 ,in the case of the        
measuring rod we were able to start with
 something which existed1 so to say1    
before its variations2 but in many cases
 any starting point which we can find is
 solely determined by the average4      
,suppose1 for instance1 we take a great 
number of observations of the height of 
the barometer at a certain place1 at all
 times and seasons and in all weathers1 
we should generally consider that the   
average of all these showed the 'true'  
height for that place4 ,what we really  
mean is that the height at any moment is
 determined partly 7and principally7 by 
the height of the column of air above   
it1 but partly also by a number of other
 agencies such as local temperature1    
moisture1 wind1 c4 ,these are sometimes 
more and sometimes less effective1 but  
their range being tolerably constant1   

and their distribution through this     
range being tolerably symmetrical1 the  
average of one large batch of           
observations will be almost exactly the 
same as that of any other4 ,this        
constancy of the average is its truth4  
,i am quite aware that we find it       
difficult not to suppose that there must
 be something more than this constancy1 
but we are probably apt to be misled by 
the analogy of the other class of cases1
 viz4 those in which we are really      
aiming at some sort of mark4  #ae4 ,as  
regards the practical methods available 
for determining the various kinds of    
average there is very little to be said2
 as the arithmetical rules are simple   
and definite1 and involve nothing more  
than the inevitable drudgery attendant  
upon dealing with long rows of figures4 
,perhaps the most important contribution
 to this part of the subject is         
furnished by ,mr ,galton's suggestion to
 substitute the median for the mean1 and
 thus to elicit the average with        

sufficient accuracy by the mere act of  
grouping a number of objects together4  
,thus he has given an ingenious         
suggestion for obtaining the average    
height of a number of men without the   
trouble and risk of measuring them all4 
,a barbarian chief might often be       
induced to marshall his men in the order
 of their heights1 or in that of the    
popular estimate of their skill in any  
capacity2 but it would require some     
apparatus and a great deal of time to   
measure each man separately1 even       
supposing it possible to overcome the   
usually strong repugnance of uncivilized
 people to any such proceeding 7,phil4  
,mag4 ,jan4 #ahge74 ,that is1 it being  
known from wide experience that the     
heights of any tolerably homogeneous set
 of men are apt to group themselves     
symmetrically1--the condition for the   
coincidence of the three principal kinds
 of mean1--the middle man of a row thus 
arranged in order will represent the    
mean or average man1 and him we may     

subject to measurement4 ,moreover1 since
 the intermediate heights are much more 
thickly represented than the extreme    
ones1 a moderate error in the selection 
of the central man of a long row will   
only entail a very small error in the   
selection of the corresponding height4  
#af4 ,we can now conveniently recur to a
 subject which has been already noticed 
in a former chapter1 viz4 the attempt   
which is sometimes made to establish a  
distinction between an average and a    
mean4 ,it has been proposed to confine  
the former term to the cases in which we
 are dealing with a fictitious result of
 our own construction1 that is1 with a  
mere arithmetical deduction from the    
observed magnitudes1 and to apply the   
latter to cases in which there is       
supposed to be some objective magnitude 
peculiarly representative of the        
average4  ,recur to the three principal 
classes1 of things appropriate to       
,probability1 which were sketched out in
 ,ch4 ,,ii4 #d4 ,the first of these     

comprised the results of games of       
chance4 ,toss a die ten times3 the total
 number of pips on the upper side may   
vary from ten up to sixty4 ,suppose it  
to be thirty4 ,we then say that the     
average of this batch of ten is three4  
,take another set of ten throws1 and we 
may get another average1 say four4      
,there is clearly nothing objective     
peculiarly corresponding in any way to  
these averages4 ,no doubt if we go on   
long enough we shall find that the      
averages tend to centre about #c4e3 we  
then call this the average1 or the      
'probable' number of points2 and this   
ultimate average might have been pretty 
constantly asserted beforehand from our 
knowledge of the constitution of a die4 
,it has however no other truth or       
reality about it of the nature of a     
type3 it is simply the limit towards    
which the averages tend4  ,the next     
class is that occupied by the members of
 most natural groups of objects1        
especially as regards the               

characteristics of natural species4     
,somewhat similar remarks may be        
repeated here4 ,there is very frequently
 a 'limit' towards which the averages of
 increasing numbers of individuals tend 
to approach2 and there is certainly some
 temptation to regard this limit as     
being a sort of type which all had been 
intended to resemble as closely as      
possible4 ,but when we looked closer1 we
 found that this view could scarcely be 
justified2 all which could be safely    
asserted was that this type represented1
 for the time being1 the most numerous  
specimens1 or those which under existing
 conditions could most easily be        
produced4  ,the remaining class stands  
on a somewhat different ground4 ,when we
 make a succession of more or less      
successful attempts of any kind1 we get 
a corresponding series of deviations    
from the mark at which we aimed4 ,these 
we may treat arithmetically1 and obtain 
their averages1 just as in the former   
cases4 ,these averages are fictions1    

that is to say1 they are artificial     
deductions of our own which need not    
necessarily have anything objective     
corresponding to them4 ,in fact1 if they
 be averages of a few only they most    
probably will not have anything thus    
corresponding to them4 ,anything        
answering to a type can only be sought  
in the 'limit' towards which they       
ultimately tend1 for this limit         
coincides with the fixed point or object
 aimed at4  #ag4 ,fully admitting the   
great value and interest of ,quetelet's 
work in this direction1--he was         
certainly the first to direct public    
attention to the fact that so many      
classes of natural objects display the  
same characteristic property1--it       
nevertheless does not seem desirable to 
attempt to mark such a distinction by   
any special use of these technical      
terms4 ,the objections are principally  
the two following4  ,in the first place1
 a single antithesis1 like this between 
an average and a mean1 appears to       

suggest a very much simpler state of    
things than is actually found to exist  
in nature4 ,a reference to the three    
classes of things just mentioned1 and a 
consideration of the wide range and     
diversity included in each of them1 will
 serve to remind us not only of the very
 gradual and insensible advance from    
what is thus regarded as 'fictitious' to
 what is claimed as 'real2' but also of 
the important fact that whereas the     
'real type' may be of a fluctuating and 
evanescent character1 the 'fiction' may 
7as in games of chance7 be apparently   
fixed for ever4 ,provided only that the 
conditions of production remain stable1 
averages of large numbers will always   
practically present much the same       
general characteristics4 ,the far more  
important distinction lies between the  
average of a few1 with its fluctuating  
values and very imperfect and occasional
 attainment of its ultimate goal1 and   
the average of many and its gradually   
close approximation to its ultimate     

value3 i4e4 to its objective point of   
aim if there happen to be such4  ,then1 
again1 the considerations adduced in    
this chapter will show that within the  
field of the average itself there is far
 more variety than ,quetelet seems to   
have recognized4 ,he did not indeed     
quite ignore this variety1 but he       
practically confined himself almost     
entirely to those symmetrical           
arrangements in which three of the      
principal means coalesce into one4 ,we  
should find it difficult to carry out   
his distinction in less simple cases4   
,for instance1 when there is some degree
 of asymmetry1 it is the 'maximum       
ordinate' which would have to be        
considered as a 'mean' to the exclusion 
of the others2 for no appeal to an      
arithmetical average would guide us to  
this point1 which however is to be      
regarded1 if any can be so regarded1 as 
marking out the position of the ultimate
 type4  #ah4 ,we have several times     
pointed out that it is a characteristic 

of the things with which ,probability is
 concerned to present1 in the long run1 
a continually intensifying uniformity4  
,and this has been frequently described 
as what happens 'on the average4' ,now  
an objection may very possibly be raised
 against regarding an arrangement of    
things by virtue of which order thus    
emerges out of disorder as deserving any
 special notice1 on the ground that from
 the nature of the arithmetical average 
it could not possibly be otherwise4 ,the
 process by which an average is         
obtained1 it may be urged1 insures this 
tendency to equalization amongst the    
magnitudes with which it deals4 ,for    
instance1 let there be a party of ten   
men1 of whom four are tall and four are 
short1 and take the average of any five 
of them4 ,since this number cannot be   
made up of tall men only1 or of short   
men only1 it stands to reason that the  
averages cannot differ so much amongst  
themselves as the single measures can4  
,is not then the equalizing process1 it 

may be asked1 which is observable on    
increasing the range of our             
observations1 one which can be shown to 
follow from necessary laws of           
arithmetic1 and one therefore which     
might be asserted  priori8  ,whatever   
force there may be in the above         
objection arises principally from the   
limitations of the example selected1 in 
which the number chosen was so large a  
proportion of the total as to exclude   
the bare possibility of only extreme    
cases being contained within it4 ,as    
much confusion is often felt here       
between what is necessary and what is   
matter of experience1 it will be well to
 look at an example somewhat more       
closely1 in order to determine exactly  
what are the really necessary           
consequences of the averaging process4  
#ai4 ,suppose then that we take ten     
digits at random from a table 7say7 of  
logarithms4 ,unless in the highly       
unlikely case of our having happened    
upon the same digit ten times running1  

the average of the ten must be          
intermediate between the possible       
extremes4 ,every conception of an       
average of any sort not merely involves1
 but actually means1 the taking of      
something intermediate between the      
extremes4 ,the average therefore of the 
ten must lie closer to #d4e 7the average
 of the extremes7 than did some of the  
single digits4  ,now suppose we take    
#ajjj such digits instead of #aj4 ,we   
can say nothing more about the larger   
number1 with demonstrative certainty1   
than we could before about the smaller4 
,if they were unequal to begin with     
7i4e4 if they were not all the same7    
then the average must be intermediate1  
but more than this cannot be proved     
arithmetically4 ,by comparison with such
 purely arithmetical considerations     
there is what may be called a physical  
fact underlying our confidence in the   
growing stability of the average of the 
larger number4 ,it is that the          
constituent elements from which the     

average is deduced will themselves      
betray a growing uniformity3--that the  
proportions in which the different      
digits come out will become more and    
more nearly equal as we take larger     
numbers of them4 ,if the proportions in 
which the #ajjj digits were distributed 
were the same as those of the #aj the   
averages would be the same4 ,it is      
obvious therefore that the arithmetical 
process of obtaining an average goes a  
very little way towards securing the    
striking kind of uniformity which we    
find to be actually presented4  #bj4    
,there is another way in which the same 
thing may be put4 ,it is sometimes said 
that whatever may have been the         
arrangement of the original elements the
 process of continual averaging will    
necessarily produce the peculiar        
binomial or exponential law of          
arrangement4 ,this statement is         
perfectly true 7with certain safeguards7
 but it is not in any way opposed to    
what has been said above4 ,let us take  

for consideration the example above     
referred to4 ,the arrangement of the    
individual digits in the long run is the
 simplest possible4 ,it would be        
represented1 in a diagram1 not by a     
curve but by a finite straight line1 for
 each digit occurs about as often as any
 other1 and this exhausts all the       
'arrangement' that can be detected4     
,now1 when we consider the results of   
taking averages of ten such digits1 we  
see at once that there is an opening for
 a more extensive arrangement4 ,the     
totals may range from #j up to #ajj1 and
 therefore the average will have #ajj   
values from #j to #i2 and what we find  
is that the frequency of these numbers  
is determined according to the          
,binomial#e or ,exponential ,law4 ,the  
most frequent result is the true mean1  
viz4 #d4e1 and from this they diminish  
in each direction towards #j and #aj1   
which will each occur but once 7on the  
average7 in #ajaj occasions4  ,the      
explanation here is of the same kind as 

in the former case4 ,the resultant      
arrangement1 so far as the averages are 
concerned1 is only 'necessary' in the   
sense that it is a necessary result of  
certain physical assumptions or         
experiences4 ,if all the digits tend to 
occur with equal frequency1 and if they 
are 'independent' 7i4e4 if each is      
associated indifferently with every     
other71 then it is an arithmetical      
consequence that the averages when      
arranged in respect of their magnitude  
and prevalence will display the ,law of 
,facility above indicated4 ,experience1 
so far as it can be appealed to1 shows  
that the true randomness of the         
selection of the digits1--i4e4 their    
equally frequent recurrence1 and the    
impartiality of their combination1--is  
very fairly secured in practice4        
,accordingly the theoretic deduction    
that whatever may have been the original
 ,law of ,facility of the individual    
results we shall always find the        
familiar ,exponential ,law asserting    

itself as the law of the averages1 is   
fairly justified by experience in such a
 case4  ,the further discussion of      
certain corrections and refinements is  
reserved to the following chapter4  #ba4
 ,in regard to the three kinds of       
average employed to test the amount of  
dispersion1--i4e4 the mean error1 the   
probable error1 and the error of mean   
square1--two important considerations   
must be borne in mind4 ,they will both  
recur for fuller discussion and         
justification in the course of the next 
chapter1 when we come to touch upon the 
,method of ,least ,squares1 but their   
significance for logical purposes is so 
great that they ought not to be entirely
 passed by at present4  7#a7 ,in the    
first place1 then1 it must be remarked  
that in order to know what in any case  
is the real value of an error we ought  
in strictness to know what is the       
position of the limit or ultimate       
average1 for the amount of an error is  
always theoretically measured from this 

point4 ,but this is information which we
 do not always possess4 ,recurring once 
more to the three principal classes of  
events with which we are concerned1 we  
can readily see that in the case of     
games of chance we mostly do possess    
this knowledge4 ,instead of appealing to
 experience to ascertain the limit1 we  
practically deduce it by simple         
mechanical or arithmetical              
considerations1 and then the 'error' in 
any individual case or group of cases is
 obviously found by comparing the       
results thus obtained with that which   
theory informs us would ultimately be   
obtained in the long run4 ,in the case  
of deliberate efforts at an aim 7the    
third class7 we may or may not know     
accurately the value or position of this
 aim4 ,in astronomical observations we  
do not know it1 and the method of ,least
 ,squares is a method for helping us to 
ascertain it as well as we can2 in such 
experimental results as firing at a mark
 we do know it1 and may thus test the   

nature and amount of our failure by     
direct experience4 ,in the remaining    
case1 namely that of what we have termed
 natural kinds or groups of things1 not 
only do we not know the ultimate limit1 
but its existence is always at least    
doubtful1 and in many cases may be      
confidently denied4 ,where it does      
exist1 that is1 where the type seems for
 all practical purposes permanently     
fixed1 we can only ascertain it by a    
laborious resort to statistics4 ,having 
done this1 we may then test by it the   
results of observations on a small      
scale4 ,for instance1 if we find that   
the ultimate proportion of male to      
female births is about #ajf to #ajj1 we 
may then compare the statistics of some 
particular district or town and speak of
 the consequent 'error1' viz4 the       
departure1 in that particular and       
special district1 from the general      
average4  ,what we have therefore to do 
in the vast majority of practical cases 
is to take the average of a finite      

number of measurements or               
observations1--of all those1 in fact1   
which we have in hand1--and take this as
 our starting point in order to measure 
the errors4 ,the errors in fact are not 
known for certain but only probably     
calculated4 ,this however is not so much
 of a theoretic defect as it may seem at
 first sight2 for inasmuch as we seldom 
have to employ these methods1--for      
purposes of calculation1 that is1 as    
distinguished from mere                 
illustration1--except for the purpose of
 discovering what the ultimate average  
is1 it would be a sort of petitio       
principii to assume that we had already 
secured it4 ,but it is worth while      
considering whether it is desirable to  
employ one and the same term for        
'errors' known to be such1 and whose    
amount can be assigned with certainty1  
and for 'errors' which are only probably
 such and whose amount can be only      
probably assigned4 ,in fact it has been 
proposed#f to employ the two terms      

'error' and 'residual' respectively to  
distinguish between the magnitudes thus 
determined1 that is1 between the        
7generally unknown7 actual error and the
 observed error4  #bb4 7#b7 ,the other  
point involves the question to what     
extent either of the first two tests    
7pp4 #ddf1g7 of the closeness with which
 the various results have grouped       
themselves about their average is       
trustworthy or complete4 ,the answer is 
that they are necessarily incomplete4   
,no single estimate or magnitude can    
possibly give us an adequate account of 
a number of various magnitudes4 ,the    
point is a very important one2 and is   
not1 ,i think1 sufficiently attended to1
 the consequence being1 as we shall see 
hereafter1 that it is far too summarily 
assumed that a method which yields the  
result with the least 'error of mean    
square' must necessarily be the best    
result for all purposes4 ,it is not     
however by any means clear that a test  
which answers best for one purpose must 

do so for all4  ,it must be clearly     
understood that each of these tests is  
an 'average1' and that every average    
necessarily rejects a mass of varied    
detail by substituting for it a single  
result4 ,we had1 say1 a lot of statures3
 so many of #fj inches1 so many of #fa1 
c4 ,we replace these by an 'average' of 
#fh1 and thereby drop a mass of         
information4 ,a portion of this we then 
seek to recover by reconsidering the    
'errors' or departures of these statures
 from their average4 ,as before1        
however1 instead of giving the full     
details we substitute an average of the 
errors4 ,the only difference is that    
instead of taking the same kind of      
average 7i4e4 the arithmetical7 we often
 prefer to adopt the one called the     
'error of mean square4'  #bc4 ,a        
question may be raised here which is of 
sufficient importance to deserve a short
 consideration4 ,when we have got a set 
of measurements before us1 why is it    
generally held to be sufficient simply  

to assign3 7#a7 the mean value2 and 7#b7
 the mean departure from this mean8 ,the
 answer is1 of course1 partly given by  
the fact that we are only supposed to be
 in want of a rough approximation3 but  
there is more to be said than this4 ,a  
further justification is to be found in 
the fact that we assume that we need    
only contemplate the possibility of a   
single ,law of ,error1 or at any rate   
that the departures from the familiar   
,law will be but trifling4 ,in other    
words1 if we recur to the figure on p4  
#bi1 we assume that there are only two  
unknown quantities or disposable        
constants to be assigned2 viz4 first1   
the position of the centre1 and1        
secondly1 the degree of eccentricity1 if
 one may so term it1 of the curve4 ,the 
determination of the mean value directly
 and at once assigns the former1 and the
 determination of the mean error 7in    
either of the ways referred to already7 
indirectly assigns the latter by        
confining us to one alone of the        

possible curves indicated in the figure4
  ,except for the assumption of one such
 ,law of ,error the determination of the
 mean error would give but a slight     
intimation of the sort of outline of our
 ,curve of ,facility4 ,we might then    
have found it convenient to adopt some  
plan of successive approximation1 by    
adding a third or fourth 'mean4' ,just  
as we assign the mean value of the      
magnitude1 and its mean departure from  
this mean2 so we might take this mean   
error 7however determined7 as a fresh   
starting point1 and assign the mean     
departure from it4 ,if the point were   
worth further discussion we might easily
 illustrate by means of a diagram the   
sort of successive approximations which 
such indications would yield as to the  
ultimate form of the ,curve of ,facility
 or ,law of ,error4  99        99       
99        99        99  ,as this volume 
is written mainly for those who take an 
interest in the logical questions       
involved1 rather than as an introduction

 to the actual processes of calculation1
 mathematical details have been         
throughout avoided as much as possible4 
,for this reason comparatively few      
references have been made to the        
exponential equation of the ,law of     
,error1 or to the corresponding         
',probability integral1' tables of which
 are given in several handbooks on the  
subject4 ,there are two points however  
in connection with these particular     
topics as to which difficulties are1 or 
should be1 felt by so many students that
 some notice may be taken of them here  
7#a7 ,in regard to the ordinary         
algebraical expression for the law of   
error1 viz4 y  7h_/sqrt7 e-h#bx#b1 it   
will have been observed that ,i have    
always spoken of y as being proportional
 to the number of errors of the         
particular magnitude x4 ,it would hardly
 be correct to say1 absolutely1 that y  
represents that number1 because of      
course the actual number of errors of   
any precise magnitude1 where continuity 

of possibility is assumed1 must be      
indefinitely small4 ,if therefore we    
want to pass from the continuous to the 
discrete1 by ascertaining the actual    
number of errors between two consecutive
 divisions of our scale1 when1 as usual 
in measurements1 all within certain     
limits are referred to some one precise 
point1 we must modify our formula4 ,in  
accordance with the usual differential  
notation1 we must say that the number of
 errors falling into one subdivision    
7dx7 of our scale is dx 7h_/sqrt7 e-h#b 
x#b1 where dx is a 7small7 unit of      
length1 in which both h-#a and x must be
 measured4  ,the difficulty felt by most
 students is in applying the formula to 
actual statistics1 in other words in    
putting in the correct units4 ,to take  
an actual numerical example1 suppose    
that #adfj men have been measured in    
regard to their height true to the      
nearest inch1 and let it be known that  
the modulus here is #c4f inches4 ,then  
dx  #a 7inch72 h-#a  #c4f inches4 ,now  

sum 7h_/sqrt7 e-h#bx#b dx  #a2 that is1 
the sum of all the consecutive possible 
values is equal to unity4 ,when         
therefore we want the sum1 as here1 to  
be #adfj1 we must express the formula   
thus2--y  #adfj_/7sqrt  #c4f7           
e-7x_/#c4f7b1 or y  #bbh e-7x_/#c4f7b4  
,here x stands for the number of inches 
measured from the central or mean       
height1 and y stands for the number of  
men referred to that height in our      
statistical table4 7,the values of e-t#b
 for successive values of t are given in
 the handbooks47  ,for illustration ,i  
give the calculated numbers by this     
formula for values of x from #j to #h   
inches1 with the actual numbers observed
 in the ,cambridge measurements recently
 set on foot by ,mr ,galton4  inches    
calculated observed x  #j     y  #bbh   
#bca x  #a     y  #bab     #bah x  #b   
y  #aff     #agj x  #c     y  #aaa      
#aaj x  #d     y   #hb      #ff x  #e   
y   #cb      #ca x  #f     y   #aa      
#aj x  #g     y    #d       #f x  #h    

y    #a       #c  ,here the average     
height was #fi inches3 dx1 as stated1   
#a inch4 ,by saying1 'put x  #j1' we    
mean1 calculate the number of men who   
are assigned to #fi inches2 i4e4 who    
fall between #fh4e and #fi4e4 ,by       
saying1 'put x  #d1' we mean1 calculate 
the number who are assigned to #fe or to
 #gc2 i4e4 who lie between #fd4e and    
#fe4e1 or between #gb4e and #gc4e4 ,the 
observed results1 it will be seen1 keep 
pretty close to the calculated3 in the  
case of the former the means of equal   
and opposite divergences from the mean  
have been taken1 the actual results not 
being always the same in opposite       
directions4  7#b7 ,the other point      
concerns the interpretation of the      
familiar probability integral1          
7#b_/sqrt7 integral#jt e-t#b dt4 ,every 
one who has calculated the chance of an 
event1 by the help of the tables of this
 integral given in so many handbooks1   
knows that if we assign any numerical   
value to t1 the corresponding value of  

the above expression assigns the chance 
that an error taken at random shall lie 
within that same limit1 viz4 t4 ,thus   
put t  #a4e1 and we have the result     
#j4if2 that is1 only #d per cent4 of the
 errors will exceed 'one and a half4'   
,but when we ask1 'one and a half' what8
 the answer would not always be very    
ready4 ,as usual1 the main difficulty of
 the beginner is not to manipulate the  
formul1 but to be quite clear about his 
units4  ,it will be seen at once that   
this case differs from the preceding in 
that we cannot now choose our unit as we
 please4 ,where1 as here1 there is only 
one variable 7t71 if we were allowed to 
select our own unit1 the inch1 foot1 or 
whatever it might be1 we might get quite
 different results4 ,accordingly some   
comparatively natural unit must have    
been chosen for us in which we are bound
 to reckon1 just as in the circular     
measurement of an angle as distinguished
 from that by degrees4  ,the answer is  
that the unit here is the modulus1 and  

that to put 't  #a4e' is to say1        
'suppose the error half as great again  
as the modulus'2 the modulus itself     
being an error of a certain assignable  
magnitude depending upon the nature of  
the measurements or observations in     
question4 ,we shall see this better if  
we put the integral in the form         
7#b_/sqrt7 integral#j;hx e-h#b x#b      
d7hx72 which is precisely equivalent1   
since the value of a definite integral  
is independent of the particular        
variable employed4 ,here hx is the same 
as x 3 #a_/;h2 i4e4 it is the ratio of x
 to #a_/;h1 or x measured in terms of   
#a_/;h4 ,but #a_/;h is the modulus in   
the equation 7y  7h_/sqrt7 e-h#b x#b7   
for the law of error4 ,in other words   
the numerical value of an error in this 
formula1 is the number of times1 whole  
or fractional1 which it contains the    
modulus4   #a4 ,this kind of mean is    
called by ,fechner and others the       
dichteste   ,werth4 ,the most           
appropriate appeal to it that ,i have   

seen is by   ,prof4 ,lexis              
7,massenerscheinungen1 p4 #db7 where he 
shows that it   indicates clearly a sort
 of normal length of human life1 of     
about #gj   years2 a result which is    
almost entirely masked when we appeal to
   the arithmetical average4    ,this   
mean ought to be called the 'probable'  
value 7a name however   in possession of
 another7 on the ground that it         
indicates the point   of likeliest      
occurrence2 i4e4 if we compare all the  
indefinitely   small and equal units of 
variation1 the one corresponding to this
   will tend to be most frequently      
represented4  #b4 ,a diagram            
illustrative of this number of results  
was given in   ,nature 7,sept4 #a1      
#ahhg74 ,in calculating1 as above1 the  
different   means1 ,i may remark that   
the original results were given to three
   decimal places2 but1 in classing     
them1 only one place was   noted4 ,that 
is1 #bi4i includes all values between   
#bi4ijj and #bi4iii4   ,thus the value  

most frequently entered in my tables was
 #cj4j1 but on   the usual principles of
 interpolation this is reckoned as      
#cj4je4  #c4 ,there is some ambiguity in
 the phraseology in use here4 ,thus     
,airy   commonly uses the expression    
',error of ,mean ,square' to represent1 
as   here1 sqrt7sum e#b7_/n4 ,galloway  
commonly speaks of the ',mean   ,square 
of the ,errors' to represent 7sum       
e#b7_/n4 ,i shall adhere to   the former
 usage and represent it briefly by      
,e4,m4,s4 ,still more   unfortunate 7to 
my thinking7 is the employment1 by ,mr  
,merriman and   others1 of the          
expression ',mean ,error1' 7widely in   
use in its more   natural               
signification17 as the equivalent of    
this ,e4,m4,s4    ,the technical term   
',fluctuation' is applied by ,mr ,f4 ,y4
 ,edgeworth to   the expression 7#b sum 
e#b7_/n4  #d4 ,practically1 of course1  
we should allow for the expansion or    
contraction4 ,but for purposes of       
logical explanation we may              

conveniently take this variation as a   
specimen of one of those   disturbances 
which may be neutralised by resort to an
 average4  #e4 ,more strictly           
multinomial3 the relative frequency of  
the   different numbers being indicated 
by the coefficients of the powers   of x
 in the development of    7#a  x  x#b   
444  x#i7aj4  #f4 ,by ,mr ,merriman1 in 
his work on ,least ,squares4            
,,chapter ,,xix4   ,,the ,,theory ,,of  
,,the ,,average ,,as ,a ,,means ,,of    
,,approximation ,,to ,,the ,,truth4  #a4
 ,in the last chapter we were occupied  
with the ,average mainly under its      
qualitative rather than its quantitative
 aspect4 ,that is1 we discussed its     
general nature1 its principal varieties1
 and the main uses to which it could be 
put in ordinary life or in reasoning    
processes which did not claim to be very
 exact4 ,it is now time to enter more   
minutely into the specific question of  
the employment of the average in the way
 peculiarly appropriate to ,probability4

 ,that is1 we must be supposed to have a
 certain number of measurements1--in the
 widest sense of that term1--placed     
before us1 and to be prepared to answer 
such questions as2 ,why do we take their
 average8 ,with what degree of          
confidence8 ,must we in all cases take  
the average1 and1 if so1 one always of  
the same kind8  ,the subject upon which 
we are thus entering is one which1 under
 its most general theoretic treatment1  
has perhaps given rise to more profound 
investigation1 to a greater variety of  
opinion1 and in consequence to a more   
extensive history and literature1 than  
any other single problem within the     
range of mathematics4#a ,but1 in spite  
of this1 the main logical principles    
underlying the methods and processes in 
question are not1 ,i apprehend1         
particularly difficult to grasp3 though1
 owing to the extremely technical style 
of treatment adopted even in            
comparatively elementary discussions of 
the subject1 it is far from easy for    

those who have but a moderate command of
 mathematical resources to disentangle  
these principles from the symbols in    
which they are clothed4 ,the present    
chapter contains an attempt to remove   
these difficulties1 so far as a general 
comprehension of the subject is         
concerned4 ,as the treatment thus       
adopted involves a considerable number  
of subdivisions1 the reader will        
probably find it convenient to refer    
back occasionally to the table of       
contents at the commencement of this    
volume4  #b4 ,the subject1 in the form  
in which we shall discuss it1 will be   
narrowed to the consideration of the    
average1 on account of the comparative  
simplicity and very wide prevalence of  
this aspect of the problem4 ,the problem
 is however very commonly referred to1  
even in non-mathematical treatises1 as  
the ,rule or ,method of ,least ,squares2
 the fact being that1 in such cases as  
we shall be concerned with1 the ,rule of
 ,least ,squares resolves itself into   

the simpler and more familiar process of
 taking the arithmetical average4 ,a    
very simple example1--one given by      
,herschel1--will explain the general    
nature of the task under a slightly     
wider treatment1 and will serve to      
justify the familiar designation4       
,suppose that a man had been firing for 
some time with a pistol at a small mark1
 say a wafer on a wall4 ,we may take it 
for granted that the shot-marks would   
tend to group themselves about the wafer
 as a centre1 with a density varying in 
some way inversely with the distance    
from the centre4 ,but now suppose that  
the wafer which marked the centre was   
removed1 so that we could see nothing   
but the surface of the wall spotted with
 the shot-marks2 and that we were asked 
to guess the position of the wafer4 ,had
 there been only one shot1 common sense 
would suggest our assuming 7of course   
very precariously7 that this marked the 
real centre4 ,had there been two1 common
 sense would suggest our taking the     

mid-point between them4 ,but if three or
 more were involved1 common sense would 
be at a loss4 ,it would feel that some  
intermediate point ought to be selected1
 but would not see its way to a more    
precise determination1 because its      
familiar reliance1--the arithmetical    
average1--does not seem at hand here4   
,the rule in question tells us how to   
proceed4 ,it directs us to select that  
point which will render the sum of the  
squares of all the distances of the     
various shot-marks from it the least    
possible4#b  ,this is merely by way of  
illustration1 and to justify the        
familiar designation of the rule4 ,the  
sort of cases with which we shall be    
exclusively occupied are those          
comparatively simple ones in which only 
linear magnitude1 or some quality which 
can be adequately represented by linear 
magnitude1 is the object under          
consideration4 ,in respect of these the 
,rule of ,least ,squares reduces itself 
to the process of taking the average1 in

 the most familiar sense of that term1  
viz4 the arithmetical mean2 and a single
 ,law of ,error1 or its graphical       
equivalent1 a ,curve of ,facility1 will 
suffice accurately to indicate the      
comparative frequency of the different  
amounts of the one variable magnitude   
involved4  #c4 ,we may conveniently here
 again call attention to a misconception
 or confusion which has been already    
noticed in a former chapter4 ,it is that
 of confounding the ,law of ,error with 
the ,method of ,least ,squares4 ,these  
are things of an entirely distinct kind4
 ,the former is of the nature of a      
physical fact1 and its production is one
 which in many cases is entirely beyond 
our control4 ,the latter1--or any       
simplified application of it1 such as   
the arithmetical average1--is no law    
whatever in the physical sense4 ,it is  
rather a precept or rule for our        
guidance4 ,the ,law states1 in any given
 case1 how the errors tend to occur in  
respect of their magnitude and          

frequency4 ,the ,method directs us how  
to treat these errors when any number of
 them are presented to us4 ,no doubt    
there is a relation between the two1 as 
will be pointed out in the course of the
 following pages2 but there is nothing  
really to prevent us from using the same
 method for different laws of error1 or 
different methods for the same law4 ,in 
so doing1 the question of distinct right
 and wrong would seldom be involved1 but
 rather one of more or less propriety4  
#d4 ,the reader must understand1--as was
 implied in the illustration about the  
pistol shots1--that the ultimate problem
 before us is an inverse one4 ,that is1 
we are supposed to have a moderate      
number of 'errors' before us and we are 
to undertake to say whereabouts is the  
centre from which they diverge4 ,this   
resembles the determination of a cause  
from the observation of an effect4 ,but1
 as mostly happens in inverse problems1 
we must commence with the consideration 
of the direct problem4 ,in other words1 

so far as concerns the case before us1  
we shall have to begin by supposing that
 the ultimate object of our aim1--that  
is1 the true centre of our curve of     
frequency1--is already known to us3 in  
which case all that remains to be done  
is to study the consequences of taking  
averages of the magnitudes which        
constitute the errors4  #e4 ,we shall1  
for the present1 confine our remarks to 
what must be regarded as the typical    
case where considerations of            
,probability are concerned2 viz4 that in
 which the law of arrangement or        
development is of the ,binomial kind4   
,the nature of this law was explained in
 ,chap4 ,,ii41 where it was shown that  
the frequency of the respective numbers 
of occurrences was regulated in         
accordance with the magnitude of the    
successive terms of the expansion of the
 binomial 7#a  #a7n4 ,it was also       
pointed out that when n becomes very    
great1 that is1 when the number of      
influencing circumstances is very large1

 and their relative individual influence
 correspondingly small1 the form assumed
 by a curve drawn through the summits of
 ordinates representing these successive
 terms of the binomial tends towards    
that assigned by the equation y         
,ae-h#bx#b4  ,for all practical purposes
 therefore we may talk indifferently of 
the ,binomial or ,exponential law2 if   
only on the ground that the arrangement 
of the actual phenomena on one or other 
of these two schemes would soon become  
indistinguishable when the numbers      
involved are large4 ,but there is       
another ground than this4 ,even when the
 phenomena themselves represent a       
continuous magnitude1 our measurements  
of them1--which are all with which we   
can deal1--are discontinuous4 ,suppose  
we had before us the accurate heights of
 a million adult men4 ,for all practical
 purposes these would represent the     
variations of a continuous magnitude1   
for the differences between two         
successive magnitudes1 especially near  

the mean1 would be inappreciably small4 
,but our tables will probably represent 
them only to the nearest inch4 ,we have 
so many assigned as #fi inches2 so many 
as #gj2 and so on4 ,the tabular         
statement in fact is of much the same   
character as if we were assigning the   
number of 'heads' in a toss of a handful
 of pence2 that is1 as if we were       
dealing with discontinuous numbers on   
the binomial1 rather than with a        
continuous magnitude on the exponential 
arrangement4  #f4 ,confining ourselves  
then1 for the present1 to this general  
head1 of the binomial or exponential    
law1 we must distinguish two separate   
cases in respect of the knowledge we may
 possess as to the generating           
circumstances of the variable           
magnitudes4  7#a7 ,there is1 first1 the 
case in which the conditions of the     
problem are determinable  priori3 that  
is1 where we are able to say1 prior to  
specific experience1 how frequently each
 combination will occur in the long run4

 ,in this case the main or ultimate     
object for which we are supposing that  
the average is employed1--i4e4 that of  
discovering the true mean value1--is    
superseded4 ,we are able to say what the
 mean or central value in the long run  
will be2 and therefore there is no      
occasion to set about determining it1   
with some trouble and uncertainty1 from 
a small number of observations4 ,still  
it is necessary to discuss this case    
carefully1 because its assumption is a  
necessary link in the reasoning in other
 cases4  ,this comparatively  priori    
knowledge may present itself in two     
different degrees as respects its       
completeness4 ,in the first place it    
may1 so far as the circumstances in     
question are concerned1 be absolutely   
complete4 ,consider the results when a  
handful of ten pence is repeatedly      
tossed up4 ,we know precisely what the  
mean value is here1 viz4 equal division 
of heads and tails3 we know also the    
chance of six heads and four tails1 and 

so on4 ,that is1 if we had to plot out a
 diagram showing the relative frequency 
of each combination1 we could do so     
without appealing to experience4 ,we    
could draw the appropriate binomial     
curve from the generating conditions    
given in the statement of the problem4  
,but now consider the results of firing 
at a target consisting of a long and    
narrow strip1 of which one point is     
marked as the centre of aim4#c ,here    
7assuming that there are no causes at   
work to produce permanent bias7 we know 
that this centre will correspond to the 
mean value4 ,and we know also1 in a     
general way1 that the dispersion on each
 side of this will follow a binomial    
law4 ,but if we attempted to plot out   
the proportions1 as in the preceding    
case1 by erecting ordinates which should
 represent each degree of frequency as  
we receded further from the mean1 we    
should find that we could not do so4    
,fresh data must be given or inferred4  
,a good marksman and a bad marksman will

 both distribute their shot according to
 the same general law2 but the rapidity 
with which the shots thin off as we     
recede from the centre will be different
 in the two cases4 ,another 'constant'  
is demanded before the curve of         
frequency could be correctly traced out4
  #g4 7#b7 ,the second division1 to be  
next considered1 corresponds for all    
logical purposes to the first4 ,it      
comprises the cases in which though we  
have no  priori knowledge as to the     
situation about which the values will   
tend to cluster in the long run1 yet we 
have sufficient experience at hand to   
assign it with practical certainty4     
,consider for instance the tables of    
human stature4 ,these are often very    
extensive1 including tens or hundreds of
 thousands4 ,in such cases the mean or  
central value is determinable with just 
as great certainty as by any  priori    
rule4 ,that is1 if we took another      
hundred thousand measurements from the  
same class of population1 we should feel

 secure that the average would not be   
altered by any magnitude which our      
measuring instruments could practically 
appreciate4  #h4 ,but the mere          
assignment of the mean or central value 
does not here1 any more than in the     
preceding case1 give us all that we want
 to know4 ,it might so happen that the  
mean height of two populations was the  
same1 but that the law of dispersion    
about that mean was very different3 so  
that a man who in one series was an     
exceptional giant or dwarf should1 in   
the other1 be in no wise remarkable4    
,to explain the process of thus         
determining the actual magnitude of the 
dispersion would demand too much        
mathematical detail2 but some indication
 may be given4 ,what we have to do is to
 determine the constant h in the        
equation#d y  7h_/sqrt7 e-h#b x#b4 ,in  
technical language1 what we have to do  
is to determine the modulus of this     
equation4 ,the quantity #a_/;h in the   
above expression is called the modulus4 

,it measures the degree of contraction  
or dispersion about the mean indicated  
by this equation4 ,when it is large the 
dispersion is considerable2 that is the 
magnitudes are not closely crowded up   
towards the centre1 when it is small    
they are thus crowded up4 ,the smaller  
the modulus in the curve representing   
the thickness with which the shot-marks 
clustered about the centre of the       
target1 the better the marksman4  #i4   
,there are several ways of determining  
the modulus4 ,in the first of the cases 
discussed above1 where our theoretical  
knowledge is complete1 we are able to   
calculate it  priori from our knowledge 
of the chances4 ,we should naturally    
adopt this plan if we were tossing up a 
large handful of pence4  ,the usual     
posteriori plan1 when we have the       
measurements of the magnitudes or       
observations before us1 is this3--,take 
the mean square of the errors1 and      
double this2 the result gives the square
 of the modulus4 ,suppose1 for instance1

 that we had the five magnitudes1 #d1   
#e1 #f1 #g1 #h4 ,the mean of these is   
#f3 the 'errors' are respectively #b1   
#a1 #j1 #a1 #b4 ,therefore the 'modulus 
squared' is equal to #aj_/e2 i4e4 the   
modulus is sqrt#b4 ,had the magnitudes  
been #b1 #d1 #f1 #h1 #aj2 representing  
the same mean 7#f7 as before1 but       
displaying a greater dispersion about   
it1 the modulus would have been larger1 
viz4 sqrt#h instead of sqrt#b4  ,mr     
,galton's method is more of a graphical 
nature4 ,it is described in a paper on  
,statistics by ,intercomparison 7,phil4 
,mag4 #ahge71 and elsewhere4 ,it may be 
indicated as follows4 ,suppose that we  
were dealing with a large number of     
measurements of human stature1 and      
conceive that all the persons in        
question were marshalled in the order of
 their height4 ,select the average      
height1 as marked by the central man of 
the row4 ,suppose him to be #fi inches4 
,then raise 7or depress7 the scale from 
this point until it stands at such a    

height as just to include one half of   
the men above 7or below7 the mean4 7,in 
practice this would be found to require 
about #a4ga inches3 that is1 one quarter
 of any large group of such men will    
fall between #fi and #gj4ga inches47    
,divide this number by #j4dgfi and we   
have the modulus4 ,in the case in       
question it would be equal to about #c4f
 inches4  ,under the assumption with    
which we start1 viz4 that the law of    
error displays itself in the familiar   
binomial form1 or in some form          
approximating to this1 the three methods
 indicated above will coincide in their 
result4 ,where there is any doubt on    
this head1 or where we do not feel able 
to calculate beforehand what will be the
 rate of dispersion1 we must adopt the  
second plan of determining the modulus4 
,this is the only universally applicable
 mode of calculation3 in fact that it   
should yield the modulus is a truth of  
definition2 for in determining the error
 of mean square we are really doing     

nothing else than determining the       
modulus1 as was pointed out in the last 
chapter4  #aj4 ,the position then which 
we have now reached is this4 ,taking it 
for granted that the ,law of ,error will
 fall into the symbolic form expressed  
by the equation y  7h_/sqrt7 e-h#b x#b1 
we have rules at hand by which h may be 
determined4 ,we therefore1 for the      
purposes in question1 know all about the
 curve of frequency3 we can trace it out
 on paper3 given one value1--say the    
central one1--we can determine any other
 value at any distance from this4 ,that 
is1 knowing how many men in a million1  
say1 are #fi inches high1 we can        
determine without direct observation how
 many will be #fg1 #fh1 #gj1 #ga1 and so
 on4  ,we can now adequately discuss the
 principal question of logical interest 
before us2 viz4 why do we take averages 
or means8 ,what is the exact nature and 
amount of the advantage gained by so    
doing8 ,the advanced student would of   
course prefer to work out the answers to

 these questions by appealing at once to
 the ,law of ,error in its ultimate or  
exponential form4 ,but ,i feel convinced
 that the best method for those who wish
 to gain a clear conception of the      
logical nature of the process involved1 
is to begin by treating it as a question
 of combinations such as we are familiar
 with in elementary algebra2 in other   
words to take a finite number of errors 
and to see what comes of averaging      
these4 ,we can then proceed to work out 
arithmetically the results of combining 
two or more of the errors together so as
 to get a new series1 not contenting    
ourselves with the general character    
merely of the new law of error1 but     
actually calculating what it is in the  
given case4 ,for the sake of simplicity 
we will not take a series with a very   
large number of terms in it1 but it will
 be well to have enough of them to      
secure that our law of error shall      
roughly approximate in its form to the  
standard or exponential law4  ,for this 

purpose the law of error or divergence  
given by supposing our effort to be     
affected by ten causes1 each of which   
produces an equal error1 but which error
 is equally likely to be positive and   
negative 7or1 as it might perhaps be    
expressed1 'ten equal and indifferently 
additive and subtractive causes'7 will  
suffice4 ,this is the lowest number     
formed according to the ,binomial law1  
which will furnish to the eye a fair    
indication of the limiting or           
,exponential law4#e ,the whole number of
 possible cases here is #baj or #ajbd2  
that is1 this is the number required to 
exhibit not only all the cases which can
 occur 7for there are but eleven really 
distinct cases71 but also the relative  
frequency with which each of these cases
 occurs in the long run4 ,of this total1
 #beb will be situated at the mean1     
representing the 'true' result1 or that 
given when five of the causes of        
disturbance just neutralize the other   
five4 ,again1 #baj will be at what we   

will call one unit's distance from the  
mean1 or that given by six causes       
combining against four2 and so on2 until
 at the extreme distance of five places 
from the mean we get but one result1    
since in only one case out of the #ajbd 
will all the causes combine together in 
the same direction4 ,the set of #ajbd   
efforts is therefore a fair             
representation of the distribution of an
 infinite number of such efforts4 ,a    
graphical representation of the         
arrangement is given here4  ,figure3    
,binomial distribution for the tenth    
power4  #aa4 ,this representing a       
complete set of single observations or  
efforts1 what will be the number and    
arrangement in the corresponding set of 
combined or reduced observations1 say of
 two together8 ,with regard to the      
number we must bear in mind that this is
 not a case of the combinations of      
things which cannot be repeated2 for any
 given error1 say the extreme one at ,f1
 can obviously be repeated twice        

running4 ,such a repetition would be a  
piece of very bad luck no doubt1 but    
being possible it must have its place in
 the set4 ,now the possible number of   
ways of combining #ajbd things two      
together1 where the same thing may be   
repeated twice running1 is #ajbd  #ajbd 
or #ajdhegf4 ,this then is the number in
 a complete cycle of the results taken  
two and two together4  #ab4 ,so much for
 their number2 now for their arrangement
 or distribution4 ,what we have to      
ascertain is1 firstly1 how many times   
each possible pair of observations will 
present itself2 and1 secondly1 where the
 new results1 obtained from the         
combination of each pair1 are to be     
placed4 ,with regard to the first of    
these enquiries2--it will be readily    
seen that on one occasion we shall have 
,f repeated twice2 on #bj occasions we  
shall have ,f combined with ,e 7for ,f  
coming first we may have it followed by 
any one of the #aj at ,e1 or any one of 
these may be followed by ,f72 ,e can be 

repeated in #aj  #aj1 or #ajj ways1 and 
so on4  ,now for the position of each of
 these reduced observations1 the        
relative frequency of whose component   
elements has thus been pointed out4     
,this is easy to determine1 for when we 
take two errors there is 7as was seen7  
scarcely any other mode of treatment    
than that of selecting the mid-point    
between them2 this mid-point of course  
becoming identical with each of them    
when the two happen to coincide4 ,it    
will be seen therefore that ,f will     
recur once on the new arrangement1 viz4 
by its being repeated twice on the old  
one4 ,g midway between ,e and ,f1 will  
be given #bj times4 ,e1 on our new      
arrangement1 can be got at in two ways1 
viz4 by its being repeated twice 7which 
will happen #ajj times71 and by its     
being obtained as the mid-point between 
,d and ,f 7which will happen #ij times74
 ,hence ,e will occur #aij times        
altogether4  ,the reader who chooses to 
take the trouble may work out the       

frequency of all possible occurrences in
 this way1 and if the object were simply
 to illustrate the principle in         
accordance with which they occur1 this  
might be the best way of proceeding4    
,but as he may soon be able to observe1 
and as the mathematician would at once  
be able to prove1 the new 'law of       
facility of error' can be got at more   
quickly deductively1 viz4 by taking the 
successive terms of the expansion of 7#a
  #a7bj4 ,they are given1 below the     
line1 in the figure on p4 #dgf4  #ac4   
,there are two apparent obstacles to any
 direct comparison between the          
distribution of the old set of simple   
observations1 and the new set of        
combined or reduced ones4 ,in the first 
place1 the number of the latter is much 
greater4 ,this1 however1 is readily met 
by reducing them both to the same scale1
 that is by making the same total number
 of each4 ,in the second place1 half of 
the new positions have no               
representatives amongst the old1 viz4   

those which occur midway between ,f and 
,e1 ,e and ,d1 and so on4 ,this can be  
met by the usual plan of interpolation1 
viz4 by filling in such gaps by         
estimating what would have been the     
number at the missing points1 on the    
same scale1 had they been occupied4     
,draw a curve through the vertices of   
the ordinates at ,a1 ,b1 ,c1 c41 and the
 lengths of the ordinates at the        
intermediate points will very fairly    
represent the corresponding frequency of
 the errors of those magnitudes         
respectively4 ,when the gaps are thus   
filled up1 and the numbers thus reduced 
to the same scale1 we have a perfectly  
fair basis of comparison4 7,see figure  
on next page47  ,similarly we might     
proceed to group or 'reduce' three      
observations1 or any greater number4    
,the number of possible groupings       
naturally becomes very much larger1     
being 7#ajbd7c when they are taken three
 together4 ,as soon as we get to three  
or more observations1 we have 7as       

already pointed out7 a variety of       
possible modes of treatment or          
reduction1 of which that of taking the  
arithmetical mean is but one4  #ad4 ,the
 following figure is intended to        
illustrate the nature of the advantage  
secured by thus taking the arithmetical 
mean of several observations4  ,the     
curve ,,abcd represents the arrangement 
of a given number of 'errors' supposed  
to be disposed according to the binomial
 law already mentioned1 when the angles 
have been smoothed off by drawing a     
curve through them4 ,a',,cd' represents 
the similar arrangement of the same     
number when given not as simple errors1 
but as averages of pairs of errors4     
,a,,bd1 again1 represents the similar   
arrangement obtained as averages of     
errors taken three together4 ,they are  
drawn as carefully to scale as the small
 size of the figure permits4  ,figure3  
,gaussian distributions for two or three
 observations4  #ae4 ,a glance at the   
above figure will explain to the reader1

 better than any verbal description1 the
 full significance of the statement that
 the result of combining two or more    
measurements or observations together   
and taking the average of them1 instead 
of stopping short at the single         
elements1 is to make large errors       
comparatively more scarce4 ,the         
advantage is of the same general        
description as that of fishing in a lake
 where1 of the same number of fish1     
there are more big and fewer little ones
 than in another water3 of dipping in a 
bag where of the same number of coins   
there are more sovereigns and fewer     
shillings2 and so on4 ,the extreme      
importance1 however1 of obtaining a     
perfectly clear conception of the       
subject may render it desirable to work 
this out a little more fully in detail4 
,for one thing1 then1 it must be clearly
 understood that the result of a set of 
'averages' of errors is nothing else    
than another set of 'errors1' ,no device
 can make the attainment of the true    

result certain1--to suppose the contrary
 would be to misconceive the very       
foundations of ,probability1--no device 
even can obviate the possibility of     
being actually worse off as the result  
of our labour4 ,the average of two1     
three1 or any larger number of single   
results1 may give a worse result1 i4e4  
one further from the ultimate average1  
than was given by the first observation 
we made4 ,we must simply fall back upon 
the justification that big deviations   
are rendered scarcer in the long run4   
,again2 it may be pointed out that      
though1 in the above investigation1 we  
have spoken only of the arithmetical    
average as commonly understood and      
employed1 the same general results would
 be obtained by resorting to almost any 
symmetrical and regular mode of         
combining our observations or errors4   
,the two main features of the regularity
 displayed by the ,binomial ,law of     
facility were 7#a7 ultimate symmetry    
about the central or true result1 and   

7#b7 increasing relative frequency as   
this centre was approached4 ,a very     
little consideration will show that it  
is no peculiar prerogative of the       
arithmetical mean to retain the former  
of these and to increase the latter4 ,in
 saying this1 however1 a distinction    
must be attended to for which it will be
 convenient to refer to a figure4  #af4 
,suppose that ,o1 in the line ,d',,od1  
was the point aimed at by any series of 
measurements2 or1 what comes to the same
 thing for our present purpose1 was the 
ultimate average of all the measurements
 made4 ,what we mean by a symmetrical   
arrangement of the values in regard to  
,o1 is that for every error ,,ob1 there 
shall be in the long run a precisely    
corresponding opposite one ,,ob'2 so    
that when we erect the ordinate ,,bq1   
indicating the frequency with which ,b  
is yielded1 we must erect an equal one  
,b',q'4 ,accordingly the two halves of  
the curve on each side of ,p1 viz4 ,,pq 
and ,,pq' are precisely alike4  ,figure3

 ,symmetry of a distribution4  ,it then 
readily follows that the secondary      
curve1 viz4 that marking the law of     
frequency of the averages of two or more
 simple errors1 will also be            
symmetrical4 ,consider any three points 
,b1 ,c1 ,d3 to these correspond another 
three ,b'1 ,c'1 ,d'4 ,it is obvious     
therefore that any regular and          
symmetrical mode of dealing with all the
 groups1 of which ,,bcd is a sample1    
will result in symmetrical arrangement  
about the centre ,o4 ,the ordinary      
familiar arithmetical average is but one
 out of many such modes4 ,one way of    
describing it is by saying that the     
average of ,b1 ,c1 ,d1 is assigned by   
choosing a point such that the sum of   
the squares of its distances from ,b1   
,c1 ,d1 is a minimum4 ,but we might have
 selected a point such that the cubes1  
or the fourth powers1 or any higher     
powers should be a minimum4 ,these would
 all yield curves resembling in a       
general way the dotted line in our      

figure4 ,of course there would be       
insuperable practical objections to any 
such courses as these2 for the labour of
 calculation would be enormous1 and the 
results so far from being better would  
be worse than those afforded by the     
employment of the ordinary average4 ,but
 so far as concerns the general         
principle of dealing with discordant and
 erroneous results1 it must be          
remembered that the familiar average is 
but one out of innumerable possible     
resources1 all of which would yield the 
same sort of help4  #ag4 ,once more4 ,we
 saw that a resort to the average had   
the effect of 'humping up' our curve    
more towards the centre1 expressive of  
the fact that the errors of averages are
 of a better1 i4e4 smaller kind4 ,but it
 must be noticed that exactly the same  
characteristics will follow1 as a       
general rule1 from any other such mode  
of dealing with the individual errors4  
,no strict proof of this fact can be    
given here1 but a reference to one of   

the familiar results of taking          
combinations of things will show whence 
this tendency arises4 ,extreme results1 
as yielded by an average of any kind1   
can only be got in one way1 viz4 by     
repetitions of extremes in the          
individuals from which the averages were
 obtained4 ,but intermediate results can
 be got at in two ways1 viz4 either by  
intermediate individuals1 or by         
combinations of individuals in opposite 
directions4 ,in the case of the         
,binomial ,law of ,error this tendency  
to thicken towards the centre was       
already strongly predominant in the     
individual values before we took them in
 hand for our average2 but owing to this
 characteristic of combinations we may  
lay it down 7broadly speaking7 that any 
sort of average applied to any sort of  
law of distribution will give a result  
which bears the same general relation to
 the individual values that the dotted  
lines above bear to the black line4#f   
#ah4 ,this being so1 the speculative    

advantages of one method of combining1  
or averaging1 or reducing1 our          
observations1 over another              
method1--irrespective1 that is1 of the  
practical conveniences in carrying them 
out1--will consist solely in the degree 
of rapidity with which it tends thus to 
cluster the result about the centre4 ,we
 shall have to subject this merit to a  
somewhat further analysis1 but for the  
present purpose it will suffice to say  
that if one kind of average gave the    
higher dotted line in the figure on p4  
#dgi and another gave the lower dotted  
line1 we should say that the former was 
the better one4 ,the advantage is of the
 same general kind as that which is     
furnished in algebraical calculation1 by
 a series which converges rapidly       
towards the true value as compared with 
one which converges slowly4 ,we can do  
the work sooner or later by the aid of  
either2 but we get nearer the truth by  
the same amount of labour1 or get as    
near by a less amount of labour1 on one 

plan than on the other4  ,as we are here
 considering the case in which the      
individual observations are supposed to 
be grouped in accordance with the       
,binomial ,law1 it will suffice to say  
that in this case there is no doubt that
 the arithmetical average is not only   
the simplest and easiest to deal with1  
but is also the best in the above sense 
of the term4 ,and since this ,binomial  
,law1 or something approximating to it1 
is of very wide prevalence1 a strong    
prim facie case is made out for the     
general employment of the familiar      
average4  #ai4 ,the analysis of a few   
pages back carried the results of the   
averaging process as far as could be    
conveniently done by the help of mere   
arithmetic4 ,to go further we must      
appeal to higher mathematics1 but the   
following indication of the sort of     
results obtained will suffice for our   
present purpose4 ,after all1 the        
successive steps1 though demanding      
intricate reasoning for their proof1 are

 nothing more than generalizations of   
processes which could be established by 
simple arithmetic4#g ,briefly1 what we  
do is this3--  7#a7 ,we first extend the
 proof from the binomial form1 with its 
finite number of elements1 to the       
limiting or exponential form4 ,instead  
of confining ourselves to a small number
 of discrete errors1 we then recognize  
the possibility of any number of errors 
of any magnitude whatever4  7#b7 ,in the
 next place1 instead of confining       
ourselves to the consideration of an    
average of two or three only1--already1 
as we have seen1 a tedious piece of     
arithmetic1--we calculate the result of 
an average of any number1 n4 ,the actual
 result is extremely simple4 ,if the    
modulus of the single errors is c1 that 
of the average of n of these will be c  
sqrtn4  7#c7 ,finally we draw similar   
conclusions in reference to the sum or  
difference of two averages of any       
numbers4 ,suppose1 for instance1 that m 
errors were first taken and averaged1   

and then n similarly taken and averaged4
 ,these averages will be nearly1 but not
 quite1 equal4 ,their sum or            
difference1--these1 of course1 are      
indistinguishable in the end1 since     
positive and negative errors are        
supposed to be equal and opposite1--will
 itself be an 'error'1 every magnitude  
of which will have a certain assignable 
probability or facility of occurrence4  
,what we do is to assign the modulus of 
these errors4 ,the actual result again  
is simple4 ,if c had been the modulus of
 the single errors1 that of the sum or  
difference of the averages of m and n of
 them will be c sqrt#a_/m  #a_/n4  #bj4 
,so far1 the problem under investigation
 has been of a direct kind4 ,we have    
supposed that the ultimate mean value or
 central position has been given to us2 
either  priori 7as in many games of     
chance71 or from more immediate physical
 considerations 7as in aiming at a      
mark71 or from extensive statistics 7as 
in tables of human stature74 ,in all    

such cases therefore the main           
desideratum is already taken for        
granted1 and it may reasonably be asked 
what remains to be done4 ,the answers   
are various4 ,for one thing we may want 
to estimate the value of an average of  
many when compared with an average of a 
few4 ,suppose that one man has collected
 statistics including #ajjj instances1  
and another has collected #djjj similar 
instances4 ,common sense can recognize  
that the latter are better than the     
former2 but it has no idea how much     
better they are4 ,here1 as elsewhere1   
quantitative precision is the privilege 
of science4 ,the answer we receive from 
this quarter is that1 in the long run1  
the modulus1--and with this the probable
 error1 the mean error1 and the error of
 mean square1 which all vary in         
proportion1--diminishes inversely as the
 square root of the number of           
measurements or observations4 7,this    
follows from the second of the above    
formul47 ,accordingly the probable error

 of the more extensive statistics here  
is one half that of the less extensive4 
,take another instance4 ,observation    
shows that the mean height of #b1cae    
criminals differs from the mean height  
of #h1ehe members of the general adult  
population by about two inches 7v4      
,edgeworth1 ,methods of ,statistics3    
,stat4 ,soc4 ,journ4 #ahhe74 ,as before1
 common sense would feel little doubt   
that such a difference was significant1 
but it could give no numerical estimate 
of the significance4 ,appealing to      
science1 we see that this is an         
illustration of the third of the above  
formul4 ,what we really want to know is 
the odds against the averages of two    
large batches differing by an assigned  
amount3 in this case by an amount       
equalling twenty-five times the modulus 
of the variable quantity4 ,the odds     
against this are many billions to one4  
#ba4 ,the number of direct problems     
which will thus admit of solution is    
very great1 but we must confine         

ourselves here to the main inverse      
problem to which the foregoing          
discussion is a preliminary4 ,it is     
this4 ,given a few only of one of these 
groups of measurements or observations2 
what can we do with these1 in the way of
 determining that mean about which they 
would ultimately be found to cluster8   
,given a large number of them1 they     
would betray the position of their      
ultimate centre with constantly         
increasing certainty3 but we are now    
supposing that there are only a few of  
them at hand1 say half a dozen1 and that
 we have no power at present to add to  
the number4  ,in other                  
words1--expressing ourselves by the aid 
of graphical illustration1 which is     
perhaps the best method for the novice  
and for the logical student1--in the    
direct problem we merely have to draw   
the curve of frequency from a knowledge 
of its determining elements2 viz4 the   
position of the centre1 and the         
numerical value of the modulus4 ,in the 

inverse problem1 on the other hand1 we  
have three elements at least1 to        
determine4 ,for not only must we1 7#a71 
as before1 determine whereabouts the    
centre may be assumed to lie2 and 7#b71 
as before1 determine the value of the   
modulus or degree of dispersion about   
this centre4 ,this does not complete our
 knowledge4 ,since neither of these two 
elements is assigned with certainty1 we 
want what is always required in the     
,theory of ,chances1 viz4 some estimate 
of their probable truth4 ,that is1 after
 making the best assignment we can as to
 the value of these elements1 we want   
also to assign numerically the 'probable
 error' committed in such assignment4   
,nothing more than this can be attained 
in ,probability1 but nothing less than  
this should be set before us4  #bb4 7#a7
 ,as regards the first of these         
questions1 the answer is very simple4   
,whether the number of measurements or  
observations be few or many1 we must    
make the assumption that their average  

is the point we want2 that is1 that the 
average of the few will coincide with   
the ultimate average4 ,this is the best1
 in fact the only assumption we can     
make4 ,we should adopt this plan1 of    
course1 in the extreme case of there    
being only one value before us1 by just 
taking that one2 and our confidence     
increases slowly with the number of     
values before us4 ,the only difference  
therefore here between knowledge resting
 upon such data1 and knowledge resting  
upon complete data1 lies not in the     
result obtained but in the confidence   
with which we entertain it4  #bc4 7#b7  
,as regards the second question1 viz4   
the determination of the modulus or     
degree of dispersion about the mean1    
much the same may be said4 ,that is1 we 
adopt the same rule for the             
determination of the ,e4,m4,s4 7error of
 mean square7 by which the modulus is   
assigned1 as we should adopt if we      
possessed full ,information4 ,or rather 
we are confined to one of the rules     

given on p4 #dgc1 viz4 the second1 for  
by supposition we have neither the      
priori knowledge which would be able to 
supply the first1 nor a sufficient      
number of observations to justify the   
third4 ,that is1 we reckon the errors1  
measured from the average1 and calculate
 their mean square3 twice this is equal 
to the square of the modulus of the     
probable curve of facility4#h  #bd4 7#c7
 ,the third question demands for its    
solution somewhat advanced mathematics2 
but the results can be indicated without
 much difficulty4 ,a popular way of     
stating our requirement would be to say 
that we want to know how likely it is   
that the mean of the few1 which we have 
thus accepted1 shall coincide with the  
true mean4 ,but this would be to speak  
loosely1 for the chances are of course  
indefinitely great against such precise 
coincidence4 ,what we really do is to   
assign the 'probable error'2 that is1 to
 assign a limit which it is as likely as
 not that the discrepancy between the   

inferred mean and the true mean should  
exceed4#i ,to take a numerical example3 
suppose we had made several measurements
 of a wall with a tape1 and that the    
average of these was #aej feet4 ,the    
scrupulous surveyor would give us this  
result1 with some such correction as    
this added1--'probable error #c inches'4
 ,all that this means is that we may    
assume that the true value is #aej feet1
 with a confidence that in half the     
cases 7of this description7 in which we 
did so1 we should really be within three
 inches of the truth4  ,the expression  
for this probable error is a simple     
multiple of the modulus3 it is the      
modulus multiplied by #j4dgfi4444 ,that 
it should be some function of the       
modulus1 or ,e4,m4,s41 seems plausible  
enough2 for the greater the errors1--in 
other words the wider the observed      
discrepancy amongst our                 
measurements1--the less must be the     
confidence we can feel in the accuracy  
of our determination of the mean4 ,but1 

of course1 without mathematics we should
 be quite unable to attempt any         
numerical assignment4  #be4 ,the general
 conclusion therefore is that the       
determination of the curve of           
facility1--and therefore ultimately of  
every conclusion which rests upon a     
knowledge of this curve1--where only a  
few observations are available1 is of   
just the same kind as where an infinity 
are available4 ,the rules for obtaining 
it are the same1 but the confidence with
 which it can be accepted is less4  ,the
 knowledge1 therefore1 obtainable by an 
average of a small number of            
measurements of any kind1 hardly differs
 except in degree from that which would 
be attainable by an indefinitely        
extensive series of them4 ,we know the  
same sort of facts1 only we are less    
certain about them4 ,but1 on the other  
hand1 the knowledge yielded by an       
average even of a small number differs  
in kind from that which is yielded by a 
single measurement4 ,revert to our      

marksman1 whose bullseye is supposed to 
have been afterwards removed4 ,if he had
 fired only a single shot1 not only     
should we be less certain of the point  
he had aimed at1 but we should have no  
means whatever of guessing at the       
quality of his shooting1 or of inferring
 in consequence anything about the      
probable remoteness of the next shot    
from that which had gone before4 ,but   
directly we have a plurality of shots   
before us1 we not merely feel more      
confident as to whereabouts the centre  
of aim was1 but we also gain some       
knowledge as to how the future shots    
will cluster about the spot thus        
indicated4 ,the quality of his shooting 
begins at once to be betrayed by the    
results4  #bf4 ,thus far we have been   
supposing the ,law of ,facility to be of
 the ,binomial type4 ,there are several 
reasons for discussing this at such     
comparative length4 ,for one thing it is
 the only type which1--or something     
approximately resembling which1--is     

actually prevalent over a wide range of 
phenomena4 ,then again1 in spite of its 
apparent intricacy1 it is really one of 
the simplest to deal with2 owing to the 
fact that every curve of facility       
derived from it by taking averages      
simply repeats the same type again4 ,the
 curve of the average only differs from 
that of the single elements in having a 
smaller modulus2 and its modulus is     
smaller in a ratio which is exceedingly 
easy to give4 ,if that of the one is c1 
that of the other 7derived by averaging 
n single elements7 is c_/sqrtn4  ,but   
for understanding the theory of averages
 we must consider other cases as well4  
,take then one which is intrinsically as
 simple as it possibly can be1 viz4 that
 in which all values within certain     
assigned limits are equally probable4   
,this is a case familiar enough in      
abstract ,probability1 though1 as just  
remarked1 not so common in natural      
phenomena4 ,it is the state of things   
when we act at random directly upon the 

objects of choice2#aj as when1 for      
instance1 we choose digits at random out
 of a table of logarithms4  ,the reader 
who likes to do so can without much     
labour work out the result of taking an 
average of two or three results by      
proceeding in exactly the same way which
 we adopted on p4 #dgf4 ,the 'curve of  
facility' with which we have to start in
 this case has become of course simply a
 finite straight line4 ,treating the    
question as one of simple combinations1 
we may divide the line into a number of 
equal parts1 by equidistant points2 and 
then proceed to take these two and two  
together in every possible way1 as we   
did in the case discussed some pages    
back4  ,if we did so1 what we should    
find would be this4 ,when an average of 
two is taken1 the 'curve of facility' of
 the average becomes a triangle with the
 initial straight line for base2 so that
 the ultimate mean or central point     
becomes the likeliest result even with  
this commencement of the averaging      

process4 ,if we were to take averages of
 three1 four1 and so on1 what we should 
find would be that the ,binomial law    
begins to display itself here4 ,the     
familiar bell shape of the exponential  
curve would be more and more closely    
approximated to1 until we obtained      
something quite indistinguishable from  
it4  #bg4 ,the conclusion therefore is  
that when we are dealing with averages  
involving a considerable number it is   
not necessary1 in general1 to presuppose
 the binomial law of distribution in our
 original data4 ,the law of arrangement 
of what we may call the derived curve1  
viz4 that corresponding to the averages1
 will not be appreciably affected       
thereby4 ,accordingly we seem to be     
justified in bringing to bear all the   
same apparatus of calculation as in the 
former case4 ,we take the initial       
average as the probable position of the 
true centre or ultimate average3 we     
estimate the probability that we are    
within an assignable distance of the    

truth in so doing by calculating the    
'error of mean square'2 and we appeal to
 this same element to determine the     
modulus1 i4e4 the amount of contraction 
or dispersion1 of our derived curve of  
facility4  ,the same general            
considerations will apply to most other 
kinds of ,law of ,facility4 ,broadly    
speaking1--we shall come to the         
examination of certain exceptions       
immediately1--whatever may have been the
 primitive arrangement 7i4e4 that of the
 single results7 the arrangement of the 
derived results 7i4e4 that of the       
averages7 will be more crowded up       
towards the centre4 ,this follows from  
the characteristic of combinations      
already noticed1 viz4 that extreme      
values can only be got at by a          
repetition of several extremes1 whereas 
intermediate values can be got at either
 by repetition of intermediates or      
through the counteraction of opposite   
extremes4 ,provided the original        
distribution be symmetrical about the   

centre1 and provided the limits of      
possible error be finite1 or if         
infinite1 that the falling off of       
frequency as we recede from the mean be 
very rapid1 then the results of taking  
averages will be better than those of   
trusting to single results4  #bh4 ,we   
will now take notice of an exceptional  
case4 ,we shall do so1 not because it is
 one which can often actually occur1 but
 because the consideration of it will   
force us to ask ourselves with some     
minuteness what we mean in the above    
instances by calling the results of the 
averages 'better' than those of the     
individual values4 ,a diagram will bring
 home to us the point of the difficulty 
better than any verbal or symbolic      
description4  ,figure3 ,distribution for
 two samples from a non-,gaussian       
distribution4  ,the black line          
represents a ,law of ,error easily      
stated in words1 and one which1 as we   
shall subsequently see1 can be conceived
 as occurring in practice4 ,it          

represents a state of things under which
 up to a certain distance from ,o1 on   
each side1 viz4 to ,a and ,b1 the       
probability of an error diminishes      
uniformly with the distance from ,o2    
whilst beyond these points1 up to ,e and
 ,f1 the probability of error remains   
constant4 ,the dotted line represents   
the resultant ,law of ,error obtained by
 taking the average of the former two   
and two together4 ,now is the latter    
'better' than the former8  ,under it1   
certainly1 great errors are less        
frequent and intermediate ones more     
frequent2 but then on the other hand the
 small errors are less frequent3 is this
 state of things on the whole an        
improvement or not8 ,this requires us to
 reconsider the whole question4  #bi4   
,in all the cases discussed in the      
previous sections the superiority of the
 curve of averages over that of the     
single results showed itself at every   
point4 ,the big errors were scarcer and 
the small errors were commoner2 it was  

only just at one intermediate point that
 the two were on terms of equality1 and 
this point was not supposed to possess  
any particular significance or          
importance4 ,accordingly we had no      
occasion to analyse the various cases   
included under the general relation4 ,it
 was enough to say that one was better  
than the other1 and it was sufficient   
for all purposes to take the 'modulus'  
as the measure of this superiority4 ,in 
fact we are quite safe in simply saying 
that the average of those average       
results is better than that of the      
individual ones4  ,when however we      
proceed in what ,hume calls the sifting 
humour1 and enquire why it is sufficient
 thus to trust to the average2 we find1 
in addition to the considerations       
hitherto advanced1 that some postulate  
was required as to the consequences of  
the errors we incur4 ,it involved an    
estimate of what is sometimes called the
 'detriment' of an error4 ,it seemed to 
take for granted that large and small   

errors all stand upon the same general  
footing of being mischievous in their   
consequences1 but that their evil       
effects increase in a greater ratio than
 that of their own magnitude4  #cj4     
,suppose1 for comparison1 a case in     
which the importance of an error is     
directly proportional to its magnitude  
7of course we suppose positive and      
negative errors to balance each other in
 the long run73 it does not appear that 
any advantage would be gained by taking 
averages4 ,something of this sort may be
 considered to prevail in cases of mere 
purchase and sale4 ,suppose that any one
 had to buy a very large number of yards
 of cloth at a constant price per yard3 
that he had to do this1 say1 five times 
a day for many days in succession4 ,and 
conceive that the measurement of the    
cloth was roughly estimated on each     
separate occasion1 with resultant errors
 which are as likely to be in excess as 
in defect4 ,would it make the slightest 
difference to him whether he paid       

separately for each piece2 or whether   
the five estimated lengths were added   
together1 their average taken1 and he   
were charged with this average price for
 each piece8 ,in the latter case the    
errors which will be made in the        
estimation of each piece will of course 
be less in the long run than they would 
be in the former3 will this be of any   
consequence8 ,the answer surely is that 
it will not make the slightest          
difference to either party in the       
bargain4 ,in the long run1 since the    
same parties are concerned1 it will not 
matter whether the intermediate errors  
have been small or large4  ,of course   
nothing of this sort can be regarded as 
the general rule4 ,in almost every case 
in which we have to make measurements we
 shall find that large errors are much  
more mischievous than small ones1 that  
is1 mischievous in a greater ratio than 
that of their mere magnitude4 ,even in  
purchase and sale1 where different      
purchasers are concerned1 this must be  

so1 for the pleasure of him who is      
overserved will hardly equal the pain of
 him who is underserved4 ,and in many   
cases of scientific measurement large   
errors may be simply fatal1 in the sense
 that if there were no reasonable       
prospect of avoiding them we should not 
care to undertake the measurement at    
all4  #ca4 ,if we were only concerned   
with practical considerations we might  
stop at this point2 but if we want to   
realize the full logical import of      
average-taking as a means to this       
particular end1 viz4 of estimating some 
assigned magnitude1 we must look more   
closely into such an exceptional case as
 that which was indicated in the figure 
on p4 #dic4 ,what we there assumed was a
 state of things in reference to which  
extremely small errors were very        
frequent1 but that when once we got     
beyond a certain small range all other  
errors1 within considerable limits1 were
 equally likely4  ,it is not difficult  
to imagine an example which will aptly  

illustrate the case in point3 at worst  
it may seem a little far-fetched4       
,conceive then that some firm in        
,england received a hurried order to    
supply a portion of a machine1 say a    
steam-engine1 to customers at a distant 
place2 and that it was absolutely       
essential that the work should be true  
to the tenth of an inch for it to be of 
any use4 ,but conceive also that two    
specifications had been sent1 resting on
 different measurements1 in one of which
 the length of the requisite piece was  
described as sixty and in the other     
sixty-one inches4 ,on the assumption of 
any ordinary law of error1 whether of   
the binomial type or not1 there can be  
no doubt that the firm would make the   
best of a very bad job by constructing a
 piece of #fj inches and a half3 i4e4   
they would have a better chance of being
 within the requisite tenth of an inch  
by so doing1 than by taking either of   
the two specifications at random and    
constructing it accurately to this4 ,but

 if the law were of the kind indicated  
in our diagram1#aa then it seems equally
 certain that they would be less likely 
to be within the requisite narrow margin
 by so doing4 ,as a mere question of    
probability1--that is1 if such estimates
 were acted upon again and again1--there
 would be fewer failures encountered by 
simply choosing one of the conflicting  
measurements at random and working      
exactly to this1 than by trusting to the
 average of the two4  ,this suggests    
some further reflections as to the      
taking of averages4 ,we will turn now to
 another exceptional case1 but one      
involving somewhat different            
considerations than those which have    
been just discussed4 ,as before1 it may 
be most conveniently introduced by      
commencing with an example4  #cb4       
,suppose then that two scouts were sent 
to take the calibre of a gun in a       
hostile fort1--we may conceive that the 
fort was to be occupied next day1 and   
used against the enemy1 and that it was 

important to have a supply of shot or   
shell1--and that the result is that one 
of them reports the calibre to be #h    
inches and the other #i4 ,would it be   
wise to assume that the mean of these   
two1 viz4 #h-a_/b inches1 was a likelier
 value than either separately8  ,the    
answer seems to be this4 ,if we have    
reason to suppose that the possible     
calibres partake of the nature of a     
continuous magnitude1--i4e4 that all    
values1 with certain limits1 are to be  
considered as admissible1 7an assumption
 which we always make in our ordinary   
inverse step from an observation or     
magnitude to the thing observed or      
measured7--then we should be justified  
in selecting the average as the likelier
 value4 ,but if1 on the other hand1 we  
had reason to suppose that whole inches 
are always or generally preferred1 as is
 in fact the case now with heavy guns1  
we should do better to take1 even at    
hazard1 one of the two estimates set    
before us1 and trust this alone instead 

of taking an average of the two4  #cc4  
,the principle upon which we act here   
may be stated thus4 ,just as in the     
direct process of calculating or        
displaying the 'errors'1 whether in an  
algebraic formula or in a diagram1 we   
generally assume that their possibility 
is continuous1 i4e4 that all            
intermediate values are possible2 so1 in
 the inverse process of determining the 
probable position of the original from  
the known value of two or more errors1  
we assume that that position is capable 
of falling at any point whatever between
 certain limits4 ,in such an example as 
the above1 where we know or suspect a   
discontinuity of that possibility of    
position1 the value of the average may  
be entirely destroyed4  ,in the above   
example we were supposed to know that   
the calibre of the guns was likely to   
run in ,english inches or in some other 
recognized units4 ,but if the battery   
were in ,china or ,japan1 and we knew   
nothing of the standards of length in   

use there1 we could no longer appeal to 
this principle4 ,it is doubtless highly 
probable that those calibres are not of 
the nature of continuously varying      
magnitudes2 but in an entire ignorance  
of the standards actually adopted1 we   
are to all intents and purposes in the  
same position as if they were of that   
continuous nature4 ,when this is so the 
objections to trusting to the average   
would no longer hold good1 and if we had
 only one opportunity1 or a very few    
opportunities1 we should do best to     
adhere to the customary practice4  #cd4 
,when however we are able to collect and
 compare a large number of measurements 
of various objects1 this consideration  
of the probable discontinuity of the    
objects we thus measure1--that is1 their
 tendency to assume some one or other of
 a finite number of distinct magnitudes1
 instead of showing an equal readiness  
to adapt themselves to all intermediate 
values1--again assumes importance4 ,in  
fact1 given a sufficient number of      

measurable objects1 we can actually     
deduce with much probability the        
standard according to which the things  
in question were made4  ,this is the    
problem which ,mr ,flinders ,petrie has 
attacked with so much acuteness and     
industry in his work on ,inductive      
,metrology1 a work which1 merely on the 
ground of its speculative interest1 may 
well be commended to the student of     
,probability4 ,the main principles on   
which the reasoning is based are these  
two3--7#a7 that all artificers are prone
 to construct their works according to  
round numbers1 or simple fractions1 of  
their units of measurement2 and 7#b7    
that1 aiming to secure this1 they will  
stray from it in tolerable accordance   
with the law of error4 ,the result of   
these two assumptions is that if we     
collect a very large number of          
measurements of the different parts and 
proportions of some ancient             
building1--say an ,egyptian             
temple1--whilst no assignable length is 

likely to be permanently unrepresented1 
yet we find a marked tendency for the   
measurements to cluster about certain   
determinate points in our own1 or any   
other standard scale of measurement4    
,these points mark the length of the    
standard1 or of some multiple or        
submultiple of the standard1 employed by
 the old builders4 ,it need hardly be   
said that there are a multitude of      
practical considerations to be taken    
into account before this method can be  
expected to give trustworthy results1   
but the leading principles upon which it
 rests are comparatively simple4  #ce4  
,the case just considered is really     
nothing else than the recurrence1 under 
a different application1 of one which   
occupied our attention at a very early  
stage4 ,we noticed 7,chap4 ,,ii47 the   
possibility of a curve of facility which
 instead of having a single vertex like 
that corresponding to the common law of 
error1 should display two humps or      
vertices4 ,it can readily be shown that 

this problem of the measurements of     
ancient buildings1 is nothing more than 
the reopening of the same question1 in a
 slightly more complex form1 in         
reference to the question of the        
functions of an average4  ,take a simple
 example4 ,suppose an instance in which 
great errors1 of a certain approximate  
magnitude1 are distinctly more likely to
 be committed than small ones1 so that  
the curve of facility1 instead of rising
 into one peak towards the centre1 as in
 that of the familiar law of error1     
shows a depression or valley there4     
,imagine1 in fact1 two binomial curves1 
with a short interval between their     
centres4 ,now if we were to calculate   
the result of taking averages here we   
should find that this at once tends to  
fill up the valley2 and if we went on   
long enough1 that is1 if we kept on     
taking averages of sufficiently large   
numbers1 a peak would begin to arise in 
the centre4 ,in fact the familiar single
 binomial curve would begin to make its 

appearance4  #cf4 ,the question then at 
once suggests itself1 ought we to do    
this8 ,shall we give the average free   
play to perform its allotted function of
 thus crowding things up towards the    
centre8 ,to answer this question we must
 introduce a distinction4 ,if that      
peculiar double-peaked curve had been1  
as it conceivably might1 a true         
error-curve1--that is1 if it had        
represented the divergences actually    
made in aiming at the real centre1--the 
result would be just what we should     
want4 ,it would furnish an instance of  
the advantages to be gained by taking   
averages even in circumstances which    
were originally unfavourable4 ,it is not
 difficult to suggest an appropriate    
illustration4 ,suppose a man firing at a
 mark from some sheltered spot1 but such
 that the range crossed a broad exposed 
valley up or down which a strong wind   
was generally blowing4 ,if the          
shot-marks were observed we should find 
them clustering about two centres to the

 right and left of the bullseye4 ,and if
 the results were plotted out in a curve
 they would yield such a double-peaked  
curve as we have described4 ,but if the 
winds were equally strong and prevalent 
in opposite directions1 we should find  
that the averaging process redressed the
 consequent disturbance4  ,if however   
the curve represented1 as it is         
decidedly more likely to do1 some       
outcome of natural phenomena in which   
there was1 so to say1 a real double aim 
on the part of nature1 it would be      
otherwise4 ,take1 for instance1 the     
results of measuring a large number of  
people who belonged to two very         
heterogeneous races4 ,the curve of      
facility would here be of the kind      
indicated on p4 #de1 and if the numbers 
of the two commingled races were equal  
it would display a pair of twin peaks4  
,again the question arises1 'ought' we  
to involve the whole range within the   
scope of a single average8  ,the answer 
is that the obligation depends upon the 

purpose we have in view4 ,if we want to 
compare that heterogeneous race1 as a   
whole1 with some other1 or with itself  
at some other time1 we shall do well to 
average without analysis4 ,all          
statistics of population1 as we have    
already seen 7v4 p4 #dg71 are forced to 
neglect a multitude of discriminating   
characteristics of the kind in question4
 ,but if our object were to interpret   
the causes of this abnormal error-curve 
we should do well to break up the       
statistics into corresponding parts1 and
 subject these to analysis separately4  
,similarly with the measurements of the 
ancient buildings4 ,in this case if all 
our various 'errors' were thrown        
together into one group of statistics we
 should find that the resultant curve of
 facility displayed1 not two peaks only1
 but a succession of them2 and these of 
various magnitudes1 corresponding to the
 frequency of occurrence of each        
particular measurement4 ,we might take  
an average of the whole1 but hardly any 

rational purpose could be subserved in  
so doing2 whereas each separate point of
 maximum frequency of occurrence has    
something significant to teach us4  #cg4
 ,one other peculiar case may be noticed
 in conclusion4 ,suppose a distinctly   
asymmetrical1 or lop-sided curve of     
facility1 such as this3--  ,figure3 ,an 
asymmetric 7lop-sided7 distribution4    
,laws of error1 of which this is a      
graphical representation1 are1 ,i       
apprehend1 far from uncommon4 ,the curve
 in question1 is1 in fact1 but a slight 
exaggeration of that of barometrical    
heights as referred to in the last      
chapter2 when it was explained that in  
such cases the mean1 the median1 and the
 maximum ordinate would show a mutual   
divergence4 ,the doubt here is not1 as  
in the preceding instances1 whether or  
not a single average should be taken1   
but rather what kind of average should  
be selected4 ,as before1 the answer must
 depend upon the special purpose we have
 in view4 ,for all ordinary purposes of 

comparison between one time or place and
 another1 any average will answer1 and  
we should therefore naturally take the  
arithmetical1 as the most familiar1 or  
the median1 as the simplest4  #ch4      
,cases might however arise under which  
other kinds of average could justify    
themselves1 with a momentary notice of  
which we may now conclude4 ,suppose1 for
 instance1 that the question involved   
here were one of desirability of        
climate4 ,the ordinary mean1 depending  
as it does so largely upon the number   
and magnitude of extreme values1 might  
very reasonably be considered a less    
appropriate test than that of judging   
simply by the relatively most frequent  
value3 in other words1 by the maximum   
ordinate4 ,and various other points of  
view can be suggested in respect of     
which this particular value would be the
 most suitable and significant4  ,in the
 foregoing case1 viz4 that of the       
weather curve1 there was no objective or
 'true' value aimed at4 ,but a curve    

closely resembling this would be        
representative of that particular class 
of estimates indicated by ,mr ,galton1  
and for which1 as he has pointed out1   
the geometrical mean becomes the only   
appropriate one4 ,in this case the curve
 of facility ends abruptly at ,o3 it    
resembles a much foreshortened          
modification of the common exponential  
form4 ,its characteristics have been    
discussed in the paper by ,dr           
,macalister already referred to1 but any
 attempt to examine its properties here 
would lead us into far too intricate    
details4  #ci4 ,the general conclusion  
from all this seems quite in accordance 
with the nature and functions of an     
average as pointed out in the last      
chapter4 ,every average1 it was urged1  
is but a single representative          
intermediate value substituted for a    
plurality of actual values4 ,it must    
accordingly let slip the bulk of the    
information involved in these latter4   
,occasionally1 as in most ordinary      

measurements1 the one thing which it    
represents is obviously the thing we are
 in want of2 and then the only question 
can be1 which mean will most accord with
 the 'true' value we are seeking4 ,but  
when1 as may happen in most of the      
common applications of statistics1 there
 is really no 'true value' of an        
objective kind behind the phenomena1 the
 problem may branch out in various      
directions4 ,we may have a variety of   
purposes to work out1 and these may     
demand some discrimination as regards   
the average most appropriate for them4  
,whenever therefore we have any doubt   
whether the familiar arithmetical       
average is suitable for the purpose in  
hand we must first decide precisely what
 that purpose is4   #a4 ,mr ,mansfield  
,merriman published in #ahgg 7,trans4 of
 the ,connecticut   ,acad47 a list of   
#djh writings on the subject of ,least  
,squares4  #b4 ,in other words1 we are  
to take the centre of gravity of the    
shot-marks1 regarding them as all of    

equal weight4 ,this is1 in   reality1   
the 'average' of all the marks1 as the  
elementary   geometrical construction   
for obtaining the centre of gravity of a
   system of points will show2 but it is
 not familiarly so regarded4 ,of        
course1 when we are dealing with such   
cases as occur in ,mensuration1   where 
we have to combine or reconcile three or
 more inconsistent   equations1 some    
such rule as that of ,least ,squares    
becomes   imperative4 ,no taking of an  
average will get us out of the          
difficulty4  #c4 ,the only reason for   
supposing this exceptional shape is to  
secure   simplicity4 ,the ordinary      
target1 allowing errors in two          
dimensions1   would yield slightly more 
complicated results4  #d4 ,when first   
referred to1 the general form of this   
equation was   given 7v4 p4 #bi74 ,the  
special form here assigned1 in which    
7h_/sqrt7 is substituted for ,a1 is     
commonly employed in   ,probability1    
because the integral of y dx1 between   

infinity and   -infinity1 becomes equal 
to unity4 ,that is1 the sum of all the  
mutually exclusive possibilities is     
represented1 as usual1 by   unity4 ,in  
this form of expression h is a quantity 
of the order   x-#a2 for hx is to be a  
numerical quantity1 standing as it does 
as   an index4 ,the modulus1 being the  
reciprocal of this1 is of the same      
order of quantities as the errors       
themselves4 ,in fact1 if we   multiply  
it by #j4dgfi444 we have the so-called  
'probable error4'  #e4 ,see1 for the    
explanation of this1 and of the         
graphical method of   illustrating it1  
the note on p4 #bi4  #f4 ,broadly       
speaking1 we may say that the above     
remarks hold good of   any law of       
frequency of error in which there are   
actual limits1   however wide1 to the   
possible magnitude of an error4 ,if     
there are no   limits to the possible   
errors1 this characteristic of an       
average to   heap its results up towards
 the centre will depend upon            

circumstances4 ,when1 as in the         
exponential curve1 the approximation    
to the base1 as asymptote1 is           
exceedingly rapid1--that is1 when the   
extreme errors are relatively very      
few1--it still holds good4 ,but if   we 
were to take as our law of facility such
 an equation as   y  _/7#a  x#b71 7as   
hinted by ,de ,morgan and noted by ,mr  
,edgeworth3   ,camb4 ,phil4 ,trans4 vol4
 ,x4 p4 #ahd1 and vol4 ,,xiv4 p4 #afj7  
it does   not hold good4 ,the result of 
averaging is to diminish the tendency   
to cluster towards the centre4  #g4 ,the
 reader will find the proofs of these   
and other similar formul   in ,galloway 
on ,probability1 and in ,airy on        
,errors4  #h4 ,the formula commonly used
 for the ,e4,m4,s4 in this case is      
7sum e#b7_/7n - #a7 and not 7sum        
e#b7_/n4 ,the difference is trifling1   
unless n be small2 the justification has
 been offered for it that   since the   
sum of the squares measured from the    
true centre is a   minimum 7that centre 

being the ultimate arithmetical mean7   
the sum   of the squares measured from  
the somewhat incorrectly assigned       
centre will be somewhat larger4  #i4 ,it
 appears to me that in strict logical   
propriety we should like to   know the  
probable error committed in both the    
assignments of the   preceding two      
sections4 ,but the profound             
mathematicians who have   discussed this
 question1 and who alone are competent  
to treat it1   have mostly written with 
the practical wants of ,astronomy in    
view2   and for this purpose it is      
sufficient to take account of the one   
great desideratum1 viz4 the true values 
sought4 ,accordingly the only   rules   
commonly given refer to the probable    
error of the mean4  #aj4 i4e4 as        
distinguished from acting upon them     
indirectly4 ,this   latter proceeding1  
as explained in the chapter on          
,randomness1 may   result in giving a   
non-uniform distribution4  #aa4 ,there  
is no difficulty in conceiving          

circumstances under which a   law very  
closely resembling this would prevail4  
,suppose1 e4g41 that   one of the two   
measurements had been made by a careful 
and skilled   mechanic1 and the other by
 a man who to save himself trouble had  
put   in the estimate at random 7within 
certain limits71--the firm having   a   
knowledge of this fact but being of     
course unable to assign the   two to    
their authors1--we should get very much 
such a ,law of ,error   as is supposed  
above4     ,,index4  ,accidents #cdb    
,airy1 ,g4 ,b4 #ddg1 #dhd               
,anticipations1 tacit #bhg  ,arbuthnott 
#beh  ,aristotle #bje1 #cjg  ,average   
arithmetical #dcg   geometrical #dci    
median #ddb   consequences of #dhb      
necessary results of #deg   uses of     
#dci1 #dhi  ,babbage #cdc  ,bags and    
balls #ahj1 #daa  ,belief   correctness 
of #abe1 #aca1 #agh   gradations of #aci
   growth of #aii   language of #adc    
measurement of #aai1 #abe1 #adf         
quantity of #acc   test of #adj1 #adi1  

#bid   undue #abi   vagueness of #abg   
,bentham #cai1 #cbc  ,bernoulli #ia1    
#aag1 #chi  ,bertillon #dce  ,births1   
male and female #ij1 #beh1 #bfc  ,boat  
race1 ,oxford and ,cambridge #cci       
,boole #ahc  ,buckle #bcg  ,buffon #aec1
 #bje1 #ceb1 #chi  ,burgersdyck #caa    
,butler #bji1 #bha1 #ccc1 #cff          
,carlisle ,tables #afi  ,casual1 meaning
 of #bde  ,causation   need of #bcg     
proof of #bdd  ,centre of gravity #dfg  
,certainty1 in ,law #cbd   reasonable   
#cbg   hypothetical #baj  ,chance   and 
,causation #bdd     ,creation #beh      
,design #bef     ,genius #cec   neglect 
of small #cfc   selections #cch         
,chauvenet #ceb  ,classification1       
numerical scheme of #dh  ,coincidences  
#bde  ,combinations and ,permutations   
#hg  ,communism #cge1 #cib              
,conceptualism #bge  ,conflict of       
chances #dah  ,consumptives1 insurance  
of #bbg  ,cournot #bde1 #bee1 #cch      
,crackanthorpe #cab1 #cbj  ,craig1 ,j4  
#aib  ,crofton1 ,m4 ,w4 #fa1 #aja1 #ajd 

,dante #bhe  ,deflection   causes of #eg
   from aim #ch  ,de ,morgan #hc1 #ajf1 
#aai1 #abb1 #ace1 #agg1 #agi1 #aig1     
#bcf1 #bdg1 #bif1 #cjh1 #cej1 #cgi1     
#chb1 #dhc  ,de ,ros trial #bee         
,digits1 random #aaa1 #aad              
,discontinuity #aaf  ,distribution1     
random #ajf  ,diagrams #bi1 #de1 #aah1  
#ddc1 #dgf1 #dha1 #dic1 #eja  ,dialectic
 #cjb1 #cbj  ,donkin #abc1 #ahh1 #bhc   
,duration of life #ae1 #dda  ,dsing #bei
  ,ebbinghaus #aii  ,edgeworth1 ,f4 ,y4 
#cd1 #aai1 #bef1 #cci1 #cic1 #dce1 #dhc 
,ellis1 ,l4 #i  ,epidemics #fb  ,error1 
law of #bi   asymmetrical #cd1 #dda1    
#ddc   binomial #cg1 #deg1 #dfi1 #dhj   
geometrical #cd1 #ejb   heterogeneous   
#de   production of #cf  ,error   mean  
#ddf   probable #ddf1 #dgb1 #dhh   of   
mean square #ddg1 #dhh  ,escapes1 narrow
 #cda  ,expectation1 moral #chh         
,experience and probability #gd         
,exponential curve #bi  ,extraordinary  
sense of #aei1 #dbc   stories #djg1 #dba
  ,fallacies in ,logic and ,probability 

#cfg  ,fatalism #bdc  ,fechner #cd1     
#chi1 #dce1 #dda  ,fluctuation #ddh     
unlimited #gc  ,forbes1 ,j4 ,d4 #ahh1   
#bfb  ,formal ,logic #abc  ,formal and  
,material treatment #hf  ,free will #bdj
  ,galloway #bdh1 #ddh1 #dhd  ,galton1  
,f4 #cc1 #ej1 #gj1 #cah1 #ddb1 #dea1    
#dgc1 #ejb  ,gambling   and ,insurance  
#cgj   disadvantage of #chd   final     
results of #che1 #cia  ,godfray1 ,h4 #ii
  ,grote1 ,g4 #cjg  ,guy #f  ,hamilton1 
,w4 #bff1 #big  ,happiness1 human #chb  
,heads and ,tails #gg  ,heredity #ej1   
#ceg  ,herschel #cj1 #dff  ,houdin #cfa 
,hume #bcf1 #dai1 #dcc  ,hypotheses #bfh
  ,immediate inferences #aba            
,independent events #age1 #bdf          
,induction   and ,probability #aid1     
#bja1 #bjh1 #bcc1 #ceh   difficulty of  
#bac   pure #bjj  ,inequality of wealth 
#chb  ,inference1 rules of #afg         
,inoculation #cgd  ,insurance           
justification of #adi   difficulties of 
#bba   life #aea   peculiar case #bbd   
theory of #cgb   varieties of #cgd      

,inverse probability #agi1 #aif1 #bdi   
,irregularity1 absolute and relative #f 
,jacobs1 ,j4 #aii  ,jackson1 ,j4 ,g4    
#bec  ,jevons #cg1 #hc1 #acf1 #aih1     
#bja1 #bji1 #bdg  ,kant #caj1 #cag      
,keckermann #bih1 #caf  ,kinds1 natural 
#ee  ,krug #cbd  ,lambert #cji          
,language of ,chance #aei  ,laplace #hi1
 #abj1 #aig1 #bcg1 #dbd  ,law   absence 
of #aja   empirical #afj   of causation 
#bjf  ,least squares #da1 #dfg          
,leibnitz #cji1 #cbj  ,letters   lost   
#afb1 #cfh   misdirected #fg1 #bcg1 #bda
  ,lexis1 ,w4 #bfc1 #dda  ,limit        
conception of #ah1 #aji1 #afd   of      
possible fluctuation #cb  ,lines1 random
 #aac  ,lister's method #ahg  ,lotteries
 #abh  ,lunn1 ,j4 ,r4 #bdh  ,likely1    
equally #gg1 #ahc  ,mc,alister1 ,d4 #cd1
 #ahg1 #ejb  ,mansel1 ,h4 ,l4 #bii1     
#cja1 #cbj  ,martingale #cdc  ,material 
and ,formal ,logic #bfe  ,maximum       
ordinate #dda1 #dee  ,measurement of    
,belief #aai   ,memory #aib  ,mental    
qualities1 measurement of #di           

,merriman1 ,m4 #ceb1 #ddh1 #dfj1 #dfe   
,mill1 ,j4 ,s4 #aca1 #bjg1 #bff1 #bhb1  
#djb  ,milton1 chance production of #cec
  ,miracles #dbh  ,michell1 ,j4 #bfj    
,modality #bie   divisions of #cjg      
false #big   formal #bih   in ,law #cai 
,modulus #dfd1 #dgb1 #dhd  ,monro1 ,c4  
,j4 #cbe1 #daf  ,names1 reference of    
#bgj  ,nations1 comparison of #ea       
,natural ,kinds #ee1 #fc1 #ga           
,necessary and impossible matter #caj   
,objects and agencies #ec  ,occam #cad  
,paley #dcc  ,penny1 tosses of #add     
,petrie1 ,f4 #dih  ,petersburg ,problem 
#ai1 #aed  ,poisson #dje  ,prantl #caa  
,presumption1 legal #cbi  ,prvost #cdh  
,probability   definition of #afe       
relative #bij   integral #dfc  ,probable
   facts #bfi   value #dda   error #ddf1
 #dgb  ,problem1 ,three point #ajd      
,proctor1 ,r4 ,a4 #bfb1 #cgh            
,prophecies1 suicidal #bbf  ,providence 
#hi1 #dca  ,propositions1 proportional  
#b  ,psychical research #bef  ,pyramid1 
the great #bea  1 digits in #aaa1 #bdg  

,quartiles #ddf  ,quetelet #bc1 #cj1    
#dc1 #ia1 #bei1 #ccj1 #cdh1 #ded        
,randomness   etymology of #if   in     
firing #ih   proof of #ajg  ,rare events
 #cdi  ,realism #ib  ,reason1 sufficient
 #hb  ,residuals #dfj  ,roberts1 ,c4 #be
  ,rod   broken at random #ih   thrown  
at random #ajc  ,rules   ,inductive and 
,deductive #agf   of ,succession #aia   
conflict of #bbb   plurality of #bag    
,series   definite proportions in #aa   
fixed and variable #af   ideal #ie      
peculiar #ab  ,shanks #bdh  ,skeat1 ,w4 
,w4 #if  ,smiglecius #cjf1 #caf  ,smyth1
 ,p4 #bea  ,socialism #cib              
,spiritualism #cfe  ,stars1 random      
arrangement of #ajh1 #bfj  ,statistics  
by ,intercomparison #dgc   unconscious  
appeal to #djj  ,statistical ,journal #f
  ,stature   human #be1 #dga   ,french  
and ,english #dd  ,stephen1 ,j4 ,f4     
#bhb1 #cbc1 #cbf  ,stewart1 ,d4 #bji1   
#bcg  ,subjective and objective terms   
#afj  ,succession   long #cfj   ,rule of
 #aij1 #cfb  ,suffield1 ,g4 #bdh        

,suicides #fg1 #bcg  ,surnames1         
extinction of #chg  ,surprise1 emotion  
of #aeg  ,syllogisms1 pure and modal    
#caf  ,taylor #cbi  ,testimony   single 
#daa   combined #dbf   two kinds of #dji
   worthless #daf  ,thomson1 ,w4 #aec1  
#cad1 #dai  ,time   influence of #aia   
in ,probability #bgi  ,todhunter #dae   
,tontines #chj  ,triangle1 random #ajc  
,tucker1 ,a4 #abg  ,types   existence of
 #db1 #fj1 #dec   fixed and fluctuating 
#fd1 #ic  ,ueberweg #caa  ,uncertainty  
in life #cgj  ,uniformity #bdj  ,units  
of calculation #dfd  ,voluntary agency  
#fe1 #fh1 #he  ,watford #cgd  ,wallis1  
,j4 #cab  ,watson1 ,h4 ,w4 #chg         
,whately #big1 #cjg  ,whist #dja        
,whitworth1 ,w4 ,a4 #hg1 #ahc1 #chd     
,wilson1 ,j4 ,m4 #ajd  ,witnesses1      
independent #dje  ,wolf #cji  ,woolhouse
 #aja  ,,cambridge3 ,,printed ,,by      
,c4,j4 ,,clay1 ,m4,a4 ,,and ,,sons1 ,,at
 ,,the ,,university ,,press4