,,the ,,organisation                    
,,of ,,thought                          
,,the ,,organisation                    
,,of ,,thought                          
,,educational ,,and ,,scientific        
,,by                      ,a4 ,n4       
,,whitehead1 ,sc4,d41 ,f4,r4,s4         
,,fellow ,,of ,,trinity ,,college1      
,,cambridge1 ,,and ,,professor ,,of     
,,applied ,,mathematics ,,at ,,the      
,,imperial ,,college ,,of               
,,science ,,and ,,technology            
,,london                                
,,williams ,,and ,,norgate              
#ad ,,henrietta ,,street1 ,,covent      
,,garden1 ,w4,c4#b                      
#aiag                     ,the rights of
 translation are reserved               
,,preface      ,,the discourses included
 in this volume have been delivered as  
addresses  on various occasions which   
are duly noted2 the only exception is   
,the ,anatomy of ,some ,scientific      
,ideas1 which is now published  for the 
first time4 ,these discourses fall into 

two sections1 the first  five chapters  
deal with education1 and the remaining  
three embody  discussions on certain    
points arising in the philosophy of     
science4  ,but a common line of         
reflection extends through the whole1   
and the two  sections influence each    
other4    ,i have left in each chapter  
the reference to the particular occasion
 of  its first production1 and ,i have  
not sought for a verbal consistency     
covering perplexity4 ,but the various   
parts of the book were in fact  composed
 with express reference to each other1  
so as to form one whole4    ,i have to  
thank the ,syndics of the ,cambridge    
,university ,press for  permission to   
republish the contents of ,chapter ,v4  
,imperial ,college of ,science and      
,technology1  ,april #aiag4             
,,table ,,of ,,contents      ,,chap4    
,,page     ,,preface                    
v     ,i4 ,,the ,,aims ,,of             
,,education--,a ,,plea ,,for ,,reform   
#a   7,presidential ,address to the     

,mathematical ,association1   ,january1 
#aiaf47     ,,ii4 ,,technical           
,,education ,,and ,,its ,,relation ,,to 
,,science ,,and ,,literature            
#bi   7,presidential ,address to the    
,mathematical ,association1   ,january1 
#aiag47     ,,iii4 ,a ,,polytechnic ,,in
 ,,war-,,time                           
#eh   7,address at the ,prize           
,distribution1 ,borough ,polytechnic    
,institute1 ,southwark1 ,february1      
#aiag47     ,,iv4 ,,the ,,mathematical  
,,curriculum                            
#fi   7,presidential ,address to the    
,london ,branch of the   ,mathematical  
,association1 #aiab47     ,v4 ,,the     
,,principles ,,of ,,mathematics ,,in    
,,relation   ,,to ,,elementary          
,,teaching                              
#ib   7,paper read in the ,educational  
,section of the ,international          
,congress of ,mathematicians1           
,cambridge1   #aiab47     ,,vi4 ,,the   
,,organisation ,,of ,,thought           
#aje   7,presidential ,address to       

,section ,a1 ,british ,association1     
,newcastle1 #aiaf2 also read            
subsequently before the   ,aristotelian 
,society47     ,,vii4 ,,the ,,anatomy   
,,of ,,some ,,scientific ,,ideas        
#acd     ,,viii4 ,,space1 ,,time1 ,,and 
,,relativity                            
#aia   7,paper read to ,section ,a at   
the ,manchester ,meeting of   ,section  
,a1 #aiae2 also1 with the appended      
,commentary1   read subsequently before 
the ,aristotelian   ,society47          
,,organisation ,,of ,,thought           
,,chapter ,i                   ,,the    
,,aims ,,of ,,education--,a ,,plea ,,for
 ,,reform       7,presidential ,address 
to the ,mathematical ,association1      
,january1                               
#aiaf7      ,,when ,i had the honour of 
being made ,president of the            
,mathematical  ,association1 ,i did not 
foresee the unusual responsibility which
 it  entailed4 ,it was my intention to  
take as the theme of a presidential     
address the consideration of some aspect

 of those special subjects to  which my 
own researches have principally been    
directed4 ,events have  forced me to    
abandon that intention4 ,it is useless  
to discuss abstract  questions in the   
midst of dominant practical             
preoccupation4 ,we cannot  disregard the
 present crisis in ,european            
civilisation4 ,it affects every         
function of life4 ,in the harder        
struggle for existence which lies       
before the nation1 all departments of   
national effort will be reviewed  for   
judgment4 ,the mere necessity for       
economy in resources will provoke  this 
reformation4    ,we are concerned with  
education4 ,this ,association1 so rich  
in its  membership of educationalists1  
with the conception of reform as the    
very reason of its being1 is among those
 bodies which must take the  lead in    
guiding that educational reconstruction 
which by a sociological  law follows    
every social revolution4 ,we do not want
 impracticable  ideals1 only to be      

realised beyond the clouds in           
,some wild1 weird clime1      ,out of   
,space1 and out of ,time4    ,we require
 to know what is possible now in        
,england1 a nation conscious  of its    
high achievements1 and of great         
failures1 shaken to its  foundations1   
distrustful of the old ways1 and        
dreading fantastic  novelties4    ,i    
will take my courage in both hands1 and 
put before you an outline of            
educational principles4 ,what ,i am     
going to say is of course entirely      
without your authority1 and does not    
pledge or prejudge any action of  the   
,association4 ,we are primarily         
concerned only with the intellectual    
side of education1 and1 as              
mathematicians1 are naturally concerned 
to  illustrate details more particularly
 by reference to mathematics4 ,thus     
much to explain deliberate omissions in 
what follows4    ,consider now the      
general and special education of two    
types of boys1  namely those in         

secondary schools who in after life must
 form the  professional and directing   
classes in commerce1 industry1 and      
public  administration1 and again those 
in junior technical schools and later   
in advanced continuation classes1 who   
are going to form the class of  skilled 
artisans and foremen of workshops4      
,these two sets compose the  educated   
strength of the nation4 ,we must form no
 ideals which include  less than these  
entire classes within their scope4 ,what
 ,i shall say1  will in phraseology     
apply more directly to the secondary    
schools1 but  with unessential changes  
it will apply equally to the other      
group4    ,what is the first commandment
 to be obeyed in any educational scheme8
  ,it is this3 ,do not teach too many   
subjects4 ,the second command is this3  
,what you teach1 teach thoroughly4 ,the 
devil in the scholastic world  has      
assumed the form of a general education 
consisting of scraps of a  large number 
of disconnected subjects2 and1 with the 

artfulness of the  serpent1 he has      
entrenched himself behind the           
matriculation examination  of the       
,university of ,london1 with a wire     
entanglement formed by the  ,oxford and 
,cambridge schools' examination4        
,culture is activity of thought1 and    
receptiveness to beauty1 and humane     
feeling4 ,scraps of information have    
nothing to do with it4 ,a merely        
well-informed man is the most useless   
bore on ,god's earth4 ,what we  should  
aim at producing is men who possess both
 culture and expert  knowledge in some  
special direction4 ,their expert        
knowledge will give  them the ground to 
start from1 and their culture will lead 
them as deep  as philosophy and as high 
as art4 ,we have to remember that the   
valuable  intellectual development is   
self-development1 and that it mostly    
takes  place between the ages of sixteen
 and thirty4 ,as to training1 the most  
important part is given by mothers      
before the age of twelve4 ,a saying  due

 to ,archbishop ,temple illustrates my  
meaning4 ,surprise was expressed  at the
 success in after-life of a man1 who as 
a boy at ,rugby had been  somewhat      
undistinguished4 ,he answered1 ,it is   
not what they are at  eighteen1 it is   
what they become afterwards that        
matters4    ,in training a child to     
activity of thought1 above all things we
 must  beware of what ,i will call inert
 ideas--that is to say1 ideas that  are 
merely received into the mind without   
being utilised1 or tested1 or  thrown   
into fresh combinations4    ,in the     
history of education1 the most striking 
phenomenon is that  schools of learning1
 which at one epoch are alive with a    
ferment of  genius1 in a succeeding     
generation exhibit merely pedantry and  
routine4  ,the reason is1 that they are 
overladen with inert ideas4 ,education  
with inert ideas is not only useless3 it
 is1 above all things1                  
harmful--,corruptio optimi1 pessima4    
,except at rare intervals  of           

intellectual ferment1 education in the  
past has been radically  infected with  
inert ideas4 ,that is the reason why    
uneducated clever  women1 who have seen 
much of the world1 are in middle life so
 much  the most cultured part of the    
community4 ,they have been saved from   
this horrible burden of inert ideas4    
,every intellectual revolution  which   
has ever stirred humanity into greatness
 has been a passionate  protest against 
inert ideas4 ,then1 alas1 with pathetic 
ignorance of  human psychology1 it has  
proceeded by some educational scheme to 
bind  humanity afresh with inert ideas  
of its own fashioning4    ,let us now   
ask how in our system of education we   
are to guard against  this mental dry   
rot4 ,we recur to our two educational   
commandments1  ,do not teach too many   
subjects1 and again1 ,what you teach1   
teach  thoroughly4    ,the result of    
teaching small parts of a large number  
of subjects is the  passive reception of
 disconnected ideas1 not illumined with 

any spark  of vitality4 ,let the main   
ideas which are introduced into a       
child's  education be few and important1
 and let them be thrown into every      
combination possible4 ,the child should 
make them his own1 and should           
understand their application here and   
now in the circumstances of his  actual 
life4 ,from the very beginning of his   
education1 the child should  experience 
the joy of discovery4 ,the discovery    
which he has to make1  is that general  
ideas give an understanding of that     
stream of events  which pours through   
his life1 which is his life4 ,by        
understanding ,i  mean more than a mere 
logical analysis1 though that is        
included4 ,i mean  understanding in the 
sense in which it is used in the ,french
 proverb1  ,to understand all1 is to    
forgive all4 ,pedants sneer at an       
education  which is useful4 ,but if     
education is not useful1 what is it8 ,is
 it a  talent1 to be hidden away in a   
napkin8 ,of course1 education should be 

useful1 whatever your aim in life4 ,it  
was useful to ,saint ,augustine  and it 
was useful to ,napoleon4 ,it is useful1 
because understanding is  useful4    ,i 
pass lightly over that understanding    
which should be given by the  literary  
side of education4 ,it is not peculiarly
 the function of this  ,association to  
consider it4 ,nor do ,i wish to be      
supposed to pronounce  on the relative  
merits of a classical or a modern       
curriculum4 ,i would  only remark that  
the understanding which we want is an   
understanding  of an insistent present4 
,the only use of a knowledge of the past
  is to equip us for the present4 ,no   
more deadly harm can be done to  young  
minds than by depreciation of the       
present4 ,the present contains  all that
 there is4 ,it is holy ground2 for it is
 the past1 and it is  the future4 ,at   
the same time it must be observed that  
an age is no  less past if it existed   
two hundred years ago than if it existed
 two  thousand years ago4 ,do not be    

deceived by the pedantry of dates4 ,the 
ages of ,shakespeare and of ,molire are 
no less past than are the ages  of      
,sophocles and of ,virgil4 ,the         
communion of saints is a great and      
inspiring assemblage1 but it has only   
one possible hall of meeting1  and that 
is1 the present2 and the mere lapse of  
time through which any  particular group
 of saints must travel to reach that    
meeting-place1  makes very little       
difference4    ,passing now to the      
scientific and logical side of          
education1 we  remember that here also  
ideas which are not utilised are        
positively  harmful4 ,by utilising an   
idea1 ,i mean relating it to that       
stream1  compounded of sense            
perceptions1 feelings1 hopes1 desires1  
and of  mental activities relating      
thought to thought1 which forms our     
life4  ,i can imagine a set of beings   
which might fortify their souls by      
passively reviewing disconnected ideas4 
,humanity is not built that  way--except

 perhaps some editors of newspapers4    
,in scientific training1 the first thing
 to do with an idea is to prove  it4    
,but allow me for one moment to extend  
the meaning of prove2 ,i  mean--to prove
 its worth4 ,now an idea is not worth   
much unless the  propositions in which  
it is embodied are true4 ,accordingly an
 essential  part of the proof of an idea
 is the proof1 either by experiment or  
by  logic1 of the truth of the          
propositions4 ,but it is not essential  
that  this proof of the truth should    
constitute the first introduction to    
the idea4 ,after all1 its assertion by  
the authority of respectable  teachers  
is sufficient evidence to begin with4   
,in our first contact  with a set of    
propositions1 we commence by            
appreciating their  importance4 ,that is
 what we all do in after-life4 ,we do   
not attempt1  in the strict sense1 to   
prove or to disprove anything1 unless   
its  importance makes it worthy of that 
honour4 ,these two processes of  proof1 

in the narrow sense1 and of appreciation
 do not require a rigid  separation in  
time4 ,both can be proceeded with nearly
 concurrently4 ,but  in so far as either
 process must have the priority1 it     
should be that  of appreciation by use4 
,furthermore1 we should not endeavour to
 use propositions in isolation4         
,emphatically ,i do not mean1 a neat    
little set of experiments to  illustrate
 ,proposition ,i and then the proof of  
,proposition ,i1 a neat  little set of  
experiments to illustrate ,proposition  
,,ii and then the  proof of ,proposition
 ,,ii1 and so on to the end of the book4
 ,nothing  could be more boring4        
,inter-related truths are utilised en   
bloc1  and the various propositions are 
employed in any order1 and with any     
reiteration4 ,choose some important     
applications of your theoretical        
subject2 and study them concurrently    
with the systematic theoretical         
exposition4 ,keep the theoretical       
exposition short and simple1 but  let it

 be strict and rigid so far as it goes4 
,it should not be too  long for it      
easily to be known with thoroughness and
 accuracy4 ,the  consequences of a      
plethora of half-digested theoretical   
knowledge are  deplorable4 ,also the    
theory should not be muddled up with the
 practice4  ,the child should have no   
doubt when it is proving and when it is 
utilising4 ,my point is that what is    
proved should be utilised1 and that     
what is utilised should--so far as is   
practicable--be proved4 ,i am far  from 
asserting that proof and utilisation are
 the same thing4    ,at this point of my
 discourse1 ,i can most directly carry  
forward  my argument in the outward form
 of a digression4 ,we are only just     
realising that the art and science of   
education require a genius and  a study 
of their own2 and that this genius and  
this science are more  than a bare      
knowledge of some branch of science or  
of literature4 ,this  truth was         
partially perceived in the past         

generation2 and headmasters1  somewhat  
crudely1 were apt to supersede learning 
in their colleagues  by requiring       
left-hand bowling and a taste for       
football4 ,but culture  is more than    
cricket1 and more than football1 and    
more than extent of  knowledge4         
,education is the acquisition of the art
 of the utilisation of  knowledge4 ,this
 is an art very difficult to impart4    
,whenever a  text-book is written of    
real educational worth1 you may be quite
  certain that some reviewer will say   
that it will be difficult to  teach from
 it4 ,of course it will be difficult to 
teach from it4 ,if it  were easy1 the   
book ought to be burned2 for it cannot  
be educational4  ,in education1 as      
elsewhere1 the broad primrose path leads
 to a nasty  place4 ,this evil path is  
represented by a book or a set of       
lectures  which will practically enable 
the student to learn by heart all the   
questions likely to be asked at the next
 external examination4 ,and ,i  may say 

in passing that no educational system is
 possible unless every  question        
directly asked of a pupil at any        
examination is either framed  or        
modified by the actual teacher of that  
pupil in that subject4 ,the  external   
assessor may report on the curriculum or
 on the performance  of the pupils1 but 
never should be allowed to ask the pupil
 a question  which has not been strictly
 supervised by the actual teacher1 or   
at least inspired by a long conference  
with him4 ,there are a few  exceptions  
to this rule1 but they are exceptions1  
and could easily be  allowed for under  
the general rule4    ,we now return to  
my previous point1 that theoretical     
ideas should  always find important     
applications within the pupil's         
curriculum4 ,this  is not an easy       
doctrine to apply1 but a very hard one4 
,it contains  within itself the problem 
of keeping knowledge alive1 of          
preventing it  from becoming inert1     
which is the central problem of all     

education4    ,the best procedure will  
depend on several factors1 none of which
 can  be neglected1 namely1 the genius  
of the teacher1 the intellectual type   
of the pupils1 their prospects in life1 
the opportunities offered by  the       
immediate surroundings of the school1   
and allied factors of this  sort4 ,it is
 for this reason that the uniform       
external examination  is so deadly4 ,we 
do not denounce it because we are       
cranks1 and like  denouncing established
 things4 ,we are not so childish4 ,also1
 of course1  such examinations have     
their use in testing slackness4 ,our    
reason of  dislike is very definite and 
very practical4 ,it kills the best part 
of  culture4 ,when you analyse in the   
light of experience the central task  of
 education1 you find that its successful
 accomplishment depends on a  delicate  
adjustment of many variable factors4    
,the reason is that we are  dealing with
 human minds1 and not with dead matter4 
,the evocation of  curiosity1 of        

judgment1 of the power of mastering a   
complicated tangle  of circumstances1   
the use of theory in giving foresight in
 special  cases--all these powers are   
not to be imparted by a set rule        
embodied  in one schedule of examination
 subjects4    ,i appeal to you1 as      
practical teachers4 ,with good          
discipline1 it is  always possible to   
pump into the minds of a class a certain
 quantity of  inert knowledge4 ,you take
 a text-book and make them learn it4 ,so
 far1  so good4 ,the child then knows   
how to solve a quadratic equation4 ,but 
what is the point of teaching a child to
 solve a quadratic equation8  ,there is 
a traditional answer to this question4  
,it runs thus3 ,the  mind is an         
instrument1 you first sharpen it1 and   
then use it2 the  acquisition of the    
power of solving a quadratic equation is
 part of  the process of sharpening the 
mind4 ,now there is just enough truth in
  this answer to have made it live      
through the ages4 ,but for all its      

half-truth1 it embodies a radical error 
which bids fair to stifle the  genius of
 the modern world4 ,i do not know who   
was first responsible  for this analogy 
of the mind to a dead instrument4 ,for  
aught ,i know1  it may have been one of 
the seven wise men of ,greece1 or a     
committee  of the whole lot of them4    
,whoever was the originator1 there can  
be no  doubt of the authority which it  
has acquired by the continuous approval 
which it has received from eminent      
persons4 ,but whatever its weight of    
authority1 whatever the high approval   
which it can quote1 ,i have no          
hesitation in denouncing it as one of   
the most fatal1 erroneous1 and          
dangerous conceptions ever introduced   
into the theory of education4 ,the  mind
 is never passive2 it is a perpetual    
activity1 delicate1 receptive1          
responsive to stimulus4 ,you cannot     
postpone its life until you have        
sharpened it4 ,whatever interest        
attaches to your subject-matter1 must   

be evoked here and now2 whatever powers 
you are strengthening in the  pupil1    
must be exercised here and now2 whatever
 possibilities of mental  life your     
teaching should impart1 must be         
exhibited here and now4 ,that  is the   
golden rule of education1 and a very    
difficult rule to follow4    ,the       
difficulty is just this3 the            
apprehension of general ideas1          
intellectual habits of mind1 and        
pleasurable interest in mental          
achievement can be evoked by no form of 
words1 however accurately  adjusted4    
,all practical teachers know that       
education is a patient  process of the  
mastery of details1 minute by minute1   
hour by hour1 day  by day4 ,there is no 
royal road to learning through an airy  
path of  brilliant generalisations4     
,there is a proverb about the difficulty
 of  seeing the wood because of the     
trees4 ,that difficulty is exactly the  
point which ,i am enforcing4 ,the       
problem of education is to make the     

pupil see the wood by means of the      
trees4    ,the solution which ,i am     
urging1 is to eradicate the fatal       
disconnection  of subjects which kills  
the vitality of our modern curriculum4  
,there is only one subject-matter for   
education1 and that is ,life  in all its
 manifestations4 ,instead of this single
 unity1 we offer  children--,algebra1   
from which nothing follows2 ,geometry1  
from which  nothing follows2 ,science1  
from which nothing follows2 ,history1   
from  which nothing follows2 a ,couple  
of ,languages1 never mastered2 and      
lastly1 most dreary of all1 ,literature1
 represented by plays of  ,shakespeare1 
with philological notes and short       
analyses of plot and  character to be in
 substance committed to memory4 ,can    
such a list be  said to represent ,life1
 as it is known in the midst of the     
living of  it8 ,the best that can be    
said of it is1 that it is a rapid table 
of  contents which a deity might run    
over in his mind while he was thinking  

of creating a world1 and had not yet    
determined how to put it together4      
,let us now return to quadratic         
equations4 ,we still have on hand the   
unanswered question4 ,why should        
children be taught their solution8      
,unless quadratic equations fit into a  
connected curriculum1 of course  there  
is no reason to teach anything about    
them4 ,furthermore1 extensive  as should
 be the place of mathematics in a       
complete culture1 ,i am a  little       
doubtful whether for many types of boys 
algebraic solutions of  quadratic       
equations do not lie on the specialist  
side of mathematics4  ,i may here remind
 you that as yet ,i have not said       
anything of the  psychology or the      
content of the specialism1 which is so  
necessary a  part of an ideal education4
 ,but all that is an evasion of our real
  question1 and ,i merely state it in   
order to avoid being misunderstood  in  
my answer4    ,quadratic equations are  
part of algebra1 and algebra is the     

intellectual instrument which has been  
created for rendering clear  the        
quantitative aspects of the world4      
,there is no getting out of  it4        
,through and through the world is       
infected with quantity4 ,to talk  sense1
 is to talk in quantities4 ,it is no use
 saying that the nation  is large1--,how
 large8 ,it is no use saying that radium
 is scarce1--,how  scarce8 ,you cannot  
evade quantity4 ,you may fly to poetry  
and to music1  and quantity and number  
will face you in your rhythms and your  
octaves4  ,elegant intellects which     
despise the theory of quantity1 are but 
half  developed4 ,they are more to be   
pitied than blamed4 ,the scraps of      
gibberish1 which in their school-days   
were taught to them in the name  of     
algebra1 deserve some contempt4    ,this
 question of the degeneration of algebra
 into gibberish1 both in  word and in   
fact1 affords a pathetic instance of the
 uselessness of  reforming educational  
schedules without a clear conception of 

the  attributes which you wish to evoke 
in the living minds of the children4  ,a
 few years ago there was an outcry that 
school algebra was in need  of reform1  
but there was a general agreement that  
graphs would put  everything right4 ,so 
all sorts of things were extruded1 and  
graphs were  introduced4 ,so far as ,i  
can see1 with no sort of idea behind    
them1  but just graphs4 ,now every      
examination paper has one or two        
questions  on graphs4 ,personally1 ,i am
 an enthusiastic adherent of graphs4    
,but ,i  wonder whether as yet we have  
gained very much4 ,you cannot put life  
into any schedule of general education  
unless you succeed in exhibiting  its   
relation to some essential              
characteristic of all intelligent or    
emotional perception4 ,it is a hard     
saying1 but it is true2 and ,i do  not  
see how to make it any easier4 ,in      
making these little formal  alterations 
you are beaten by the very nature of    
things4 ,you are pitted  against too    

skilful an adversary1 who will see to it
 that the pea is  always under the other
 thimble4    ,reformation must begin at 
the other end4 ,first1 you must make up 
your  mind as to those quantitative     
aspects of the world which are simple   
enough to be introduced into general    
education2 then a schedule of  algebra  
should be framed which will about find  
its exemplification in  these           
applications4 ,we need not fear for our 
pet graphs1 they will  be there in      
plenty when we once begin to treat      
algebra as a serious  means of studying 
the world4 ,some of the simplest        
applications will be  found in the      
quantities which occur in the simplest  
study of society4  ,the curves of       
history are more vivid and more         
informing than the dry  catalogues of   
names and dates which comprise the      
greater part of  that arid school study4
 ,what purpose is effected by a         
catalogue of  undistinguished kings and 
queens8 ,tom1 ,dick1 or ,harry1 they are

 all  dead4 ,general resurrections are  
failures1 and are better postponed4     
,the quantitative flux of the forces of 
modern society are capable  of very     
simple exhibition4 ,meanwhile1 the idea 
of the variable1 of  the function1 of   
rate of change1 of equations and their  
solution1 of  elimination1 are being    
studied as an abstract science for their
 own  sake4 ,not1 of course1 in the     
pompous phrases with which ,i am        
alluding  to them here1 but with that   
iteration of simple special cases proper
 to  teaching4    ,if this course be    
followed1 the route from ,chaucer to the
 ,black ,death1  from the ,black ,death 
to modern ,labour troubles1 will connect
 the tales  of the medival pilgrims with
 the abstract science of algebra1 both  
yielding diverse aspects of that single 
theme1 ,life4 ,i know what most  of you 
are thinking at this point4 ,it is that 
the exact course which  ,i have sketched
 out is not the particular one which you
 would have  chosen1 or even see how to 

work4 ,i quite agree4 ,i am not claiming
 that  ,i could do it myself4 ,but your 
objection is the precise reason why a   
common external examination system is   
fatal to education4 ,the process  of    
exhibiting the applications of knowledge
 must1 for its success1  essentially    
depend on the character of the pupils   
and the genius of the  teacher4 ,of     
course ,i have left out the easiest     
applications with which  most of us are 
more at home4 ,i mean the quantitative  
sides of sciences1  such as mechanics   
and physics4    ,my meaning can be      
illustrated by looking more closely into
 a special  case of this type of        
application4 ,in my rough catalogue of  
the sort  of subjects which should form 
the schedule for algebra1 ,i mentioned  
,elimination4 ,it was not put there by  
accident1 for it covers a very          
important body of thought4    ,in the   
first place1 there is the abstract      
process of algebraic  elimination for   
suitable simple cases4 ,the pupil       

acquires a firm grasp  of this by the   
process1 inevitable in education1 of    
working an adequate  number of examples4
 ,again1 there are the graphical        
solutions of the  same problem4 ,then we
 consider the significance in the       
external world4  ,we consider the       
velocity1 time1 space1 acceleration     
diagrams4 ,we take  uniform             
acceleration2 we eliminate t between    
v  u  t1 and s  ut  t#b1    and         
eliminate s between    v#b  u#b  #bs1   
and s  ut  t#b4    ,then we remember    
that constant acceleration is a very    
special case1  and we consider graphical
 solutions or empirically given         
variations of  v or of 4 ,in preference1
 we use those empirical formul  which   
occur in the pupil's experimental work4 
,we compare the strong and  weak points 
of the algebraic and graphical          
solutions4    ,again1 in the same       
connection we plot the statistics of    
social  phenomena against the time4 ,we 
then eliminate the time between suitable

  pairs4 ,we can speculate how far we   
have exhibited a real casual            
connection1 or how far a mere temporal  
coincidence4 ,we notice that  we might  
have plotted against the time one set of
 statistics for one  country and another
 set for another country1 and thus1 with
 suitable  choice of subjects1 have     
obtained graphs which certainly         
exhibited mere  coincidence4 ,also other
 graphs exhibit obvious casual          
connections4 ,we  wonder how to         
discriminate4 ,and so are drawn on as   
far as we will4    ,but in considering  
this description1 ,i must beg you to    
remember what ,i  have been insisting on
 above4 ,in the first place1 one train  
of thought  will not suit all groups of 
children4 ,for example1 ,i should expect
 that  artisan children will want       
something more concrete and1 in a sense1
  swifter than ,i have set down here4   
,perhaps ,i am wrong1 but that is  what 
,i should guess4 ,in the second place1  
,i am not contemplating one  beautiful  

lecture stimulating1 once and for all1  
an admiring class4  ,that is not the way
 in which education proceeds4 ,no2 all  
the time the  pupils are hard at work   
solving examples1 drawing graphs1 and   
making  experiments1 until they have a  
thorough hold on the whole subject4  ,i 
am describing the interspersed          
explanations1 the directions which      
should be given to their thoughts4 ,the 
pupils have got to be made to  feel that
 they are studying something1 and are   
not merely executing  intellectual      
minuets4    ,in this connection the     
excellence of some of the most recent   
text-books  on elementary algebra       
emanating from members of this          
,association1  should create an epoch in
 the teaching of the subject4           
,finally1 if you are teaching pupils for
 some general examination1  the problem 
of sound teaching is greatly            
complicated4 ,have you ever  noticed the
 zig-zag moulding round a ,norman arch8 
,the ancient work is  beautiful1 the    

modern work is hideous4 ,the reason is1 
that the modern  work is done to exact  
measure1 the ancient work is varied     
according  to the idiosyncrasy of the   
workman4 ,here it is crowded1 and there 
it  is expanded4 ,now the essence of    
getting pupils through examinations  is 
to give equal weight to all parts of the
 schedule4 ,but mankind is  naturally   
specialist4 ,one man sees a whole       
subject1 where another can  find only a 
few detached examples4 ,i know that it  
seems contradictory  to allow for       
specialism in a curriculum especially   
designed for a  broad culture4 ,without 
contradictions the world would be       
simpler1 and  perhaps duller4 ,but ,i am
 certain that in education wherever you 
exclude  specialism you destroy life4   
,we now come to the other great branch  
of a general mathematical  education1   
namely ,geometry4 ,the same principles  
apply4 ,the theoretical  part should be 
clear-cut1 rigid1 short1 and important4 
,every  proposition not absolutely      

necessary to exhibit the main connection
  of ideas should be cut out1 but the   
great fundamental ideas should  be all  
there4 ,no omission of concepts1 such as
 those of ,similarity  and ,proportion4 
,we must remember that1 owing to the aid
 rendered by  the visual presence of a  
figure1 ,geometry is a field of         
unequalled  excellence for the exercise 
of the deductive faculties of reasoning4
  ,then1 of course1 there follows       
,geometrical ,drawing1 with its training
  for the hand and eye4    ,but1 like   
,algebra1 ,geometry and ,geometrical    
,drawing must be extended  beyond the   
mere circle of geometrical ideas4 ,in an
 industrial  neighbourhood1 machinery   
and workshop practice form the          
appropriate  extension4 ,for example1 in
 the ,london ,polytechnics this has been
  achieved with conspicuous success4    
,for many secondary schools ,i suggest  
that surveying and maps are the natural 
applications4 ,in particular1           
plane-table surveying should lead pupils

 to a vivid apprehension of the         
immediate application of geometric      
truths4 ,simple drawing apparatus1 a    
surveyor's chain1 and a surveyor's      
compass1 should enable the pupils  to   
rise from the survey and mensuration of 
a field to the construction  of the map 
of a small district4 ,the best education
 is to be found  in gaining the utmost  
information from the simplest apparatus4
 ,the  provision of elaborate           
instruments is greatly to be deprecated4
 ,to  have constructed the map of a     
small district1 to have considered its  
roads1 its contours1 its geology1 its   
climate1 its relation to other          
districts1 the effects on the status of 
its inhabitants1 will teach  more       
history and geography than any knowledge
 of ,perkin ,warbeck or  of ,behren's   
,straits4 ,i mean not a nebulous lecture
 on the subject1  but a serious         
investigation in which the real facts   
are definitely  ascertained by the aid  
of accurate theoretical knowledge4 ,a   

typical  mathematical problem should be3
 ,survey such and such a field1 draw a  
plan of it to such and such a scale1 and
 find the area4 ,it would be  quite a   
good procedure to impart the necessary  
geometrical propositions  without their 
proofs4 ,then1 concurrently in the same 
term1 the proofs  of the propositions   
would be learnt while the survey was    
being made4    ,fortunately1 the        
specialist side of education presents an
 easier  problem than does the provision
 of a general culture4 ,for this there  
are many reasons4 ,one is that many of  
the principles of procedure  to be      
observed are the same in both cases1 and
 it is unnecessary  to recapitulate4    
,another reason is that specialist      
training takes  place--or should take   
place--at a more advanced stage of the  
pupil's  course1 and thus there is      
easier material to work upon4 ,but      
undoubtedly  the chief reason is that   
the specialist study is normally a study
  of peculiar interest to the student4  

,he is studying it because1 for  some   
reason1 he wants to know it4 ,this makes
 all the difference4  ,the general      
culture is designed to foster an        
activity of mind2 the  specialist course
 utilises this activity4 ,but it does   
not do to lay too  much stress on these 
neat antitheses4 ,as we have already    
seen1 in the  general course foci of    
special interest will arise2 and        
similarly in  the special study1 the    
external connections of the subject drag
 thought  outwards4    ,again1 there is 
not one course of study which merely    
gives general  culture1 and another     
which gives special knowledge4 ,the     
subjects  pursued for the sake of a     
general education are special subjects  
specially studied2 and1 on the other    
hand1 one of the ways of  encouraging   
general mental activity is to foster a  
special devotion4  ,you may not divide  
the seamless coat of learning4 ,what    
education has  to impart is an intimate 
sense for the power of ideas1 for the   

beauty  of ideas1 and for the structure 
of ideas1 together with a particular    
body of knowledge which has peculiar    
reference to the life of the being      
possessing it4    ,the appreciation of  
the structure of ideas is that side of a
 cultured  mind which can only grow     
under the influence of a special study4 
,i  mean that eye for the whole         
chess-board1 for the bearing of one set 
of  ideas on another4 ,nothing but a    
special study can give any appreciation 
for the exact formulation of general    
ideas1 for their relations when         
formulated1 for their service in the    
comprehension of life4 ,a mind so       
disciplined should be both more abstract
 and more concrete4 ,it has been        
trained in the comprehension of abstract
 thought and in the analysis of  facts4 
,finally1 there should grow the most    
austere of all mental qualities2 ,i     
mean the sense for style4 ,it is an     
sthetic sense1 based on admiration  for 
the direct attainment of a foreseen end1

 simply and without waste4  ,style in   
art1 style in literature1 style in      
science1 style in logic1  style in      
practical execution have fundamentally  
the same sthetic  qualities1 namely1    
attainment and restraint4 ,the love of a
 subject in  itself and for itself1     
where it is not the sleepy pleasure of  
pacing a  mental quarter-deck1 is the   
love of style as manifested in that     
study4    ,here we are brought back to  
the position from which we started1  the
 utility of education4 ,style1 in its   
finest sense1 is the last  acquirement  
of the educated mind2 it is also the    
most useful4 ,it  pervades the whole    
being4 ,the administrator with a sense  
for style1  hates waste2 the engineer   
with a sense for style1 economises his  
material2 the artisan with a sense for  
style1 prefers good work4 ,style  is the
 ultimate morality of mind4    ,but     
above style1 and above knowledge1 there 
is something1 a vague  shape like fate  
above the ,greek gods4 ,that something  

is ,power4 ,style  is the fashioning of 
power1 the restraining of power4 ,but1  
after all1  the power of attainment of  
the desired end is fundamental4 ,the    
first  thing is to get there4 ,do not   
bother about your style1 but solve your 
problem1 justify the ways of ,god to    
man1 administer your province1 or  do   
whatever else is set before you4        
,where1 then1 does style help8 ,in this1
 with style the end is attained  without
 side issues1 without raising           
undesirable inflammations4 ,with  style 
you attain your end and nothing but your
 end4 ,with style the  effect of your   
activity is calculable1 and foresight is
 the last gift  of gods to men4 ,with   
style your power is increased1 for your 
mind is  not distracted with            
irrelevancies1 and you are more likely  
to attain  your object4 ,now style is   
the exclusive privilege of the expert4  
,whoever heard of the style of an       
amateur painter1 of the style of an     
amateur poet8 ,style is always the      

product of specialist study1 the        
peculiar contribution of specialism to  
culture4    ,english education in its   
present phase suffers from a lack of    
definite  aim1 and from an external     
machinery which kills its vitality4     
,hitherto  in this address ,i have been 
considering the aims which should govern
  education4 ,in this respect ,england  
halts between two opinions4 ,it has  not
 decided whether to produce amateurs or 
experts4 ,the profound change  in the   
world which the nineteenth century has  
produced is that the  growth of         
knowledge has given foresight4 ,the     
amateur is essentially a  man with      
appreciation and with immense           
versatility in mastering a given        
routine4 ,but he lacks the foresight    
which comes from special knowledge4     
,the object of this address is to       
suggest how to produce the expert       
without loss of the essential virtues of
 the amateur4 ,the machinery  of our    
secondary education is rigid where it   

should be yielding1  and lax where it   
should be rigid4 ,every school is bound 
on pain of  extinction to train its boys
 for a small set of definite            
examinations4  ,no headmaster has a free
 hand to develop his general education  
or his  specialist studies in accordance
 with the opportunities of his school1  
which are created by its staff1 its     
environment1 its class of boys1 and  its
 endowments4 ,i suggest that no system  
of external tests which aims  primarily 
at examining individual scholars can    
result in anything but  educational     
waste4    ,primarily it is the schools  
and not the scholars which should be    
inspected4 ,each school should grant its
 own leaving certificates1 based  on its
 own curriculum4 ,the standards of these
 schools should be sampled  and         
corrected4 ,but the first requisite for 
educational reform is the  school as a  
unit1 with its approved curriculum based
 on its own needs1  and evolved by its  
own staff4 ,if we fail to secure that1  

we simply fall  from one formalism into 
another1 from one dung-hill of inert    
ideas into  another4    ,in stating that
 the school is the true educational unit
 in any national  system for the        
safe-guarding of efficiency1 ,i have    
conceived the  alternative system as    
being the external examination of the   
individual  scholar4 ,but every ,scylla 
is faced by its ,charybdis--or1 in more 
homely  language1 there is a ditch on   
both sides of the road4 ,it will be     
equally fatal to education if we fall   
into the hands of a supervising         
department which is under the impression
 that it can divide all schools  into   
two or three rigid categories1 each type
 being forced to adopt  a rigid         
curriculum4 ,when ,i say that the school
 is the educational  unit1 ,i mean      
exactly what ,i say1 no larger unit1 no 
smaller unit4  ,each school must have   
the claim to be considered in relation  
to its  special circumstances4 ,the     
classifying of schools for some purposes

 is  necessary4 ,but no absolutely rigid
 curriculum1 not modified by its own    
staff1 should be permissible4 ,exactly  
the same principles apply1 with  the    
proper modifications1 to universities   
and to technical colleges4    ,when one 
considers in its length and in its      
breadth the importance  of this question
 of the education of a nation's young1  
the broken  lives1 the defeated hopes1  
the national failures1 which result from
 the  frivolous inertia with which it is
 treated1 it is difficult to restrain   
within oneself a savage rage4 ,in the   
conditions of modern life the ride  is  
absolute1 the race which does not value 
trained intelligence is  doomed4 ,not   
all your heroism1 not all your social   
charm1 not all your  wit1 not all your  
victories on land or at sea1 can move   
back the finger  of fate4 ,to-day we    
maintain ourselves4 ,to-morrow science  
will have  moved forward yet one more   
step1 and there will be no appeal from  
the  judgment which will then be        

pronounced on the uneducated4    ,we can
 be content with no less than the old   
summary of educational  ideal which has 
been current at any time from the dawn  
of our  civilisation4 ,the essence of   
education is that it be religious4      
,pray1 what is religious education8     
,a religious education is an education  
which inculcates duty and  reverence4   
,duty arises from our potential control 
over the course  of events4 ,where      
attainable knowledge could have changed 
the issue1  ignorance has the guilt of  
vice4 ,and the foundation of reverence  
is  this perception1 that the present   
holds within itself the complete sum  of
 existence1 backwards and forwards1 that
 whole amplitude of time1  which is     
eternity4                               
,,chapter ,,ii        ,,technical       
,,education ,,and ,,its ,,relation ,,to 
,,science ,,and ,,literature            
7,presidential ,address to the          
,mathematical ,association1 ,january1   
#aiag7      ,,the subject of this       

address is ,technical ,education4 ,i    
wish to examine  its essential nature   
and also its relation to a liberal      
education4 ,such  an inquiry may help us
 to realise the conditions for the      
successful  working of a national system
 of technical training4 ,it is also a   
very  burning question among            
mathematical teachers2 for mathematics  
is  included in most technological      
courses4    ,now it is unpractical to   
plunge into such a discussion without   
framing  in our own minds the best ideal
 towards which we desire to work1       
however modestly we may frame our hopes 
as to the result which in the  near     
future is likely to be achieved4        
,people are shy of ideals2 and          
accordingly we find a formulation of the
  ideal state of mankind placed by a    
modern dramatist#a in the mouth of  a   
mad priest3 ,in my dreams it is a       
country where the ,state is the  ,church
 and the ,church the people3 three in   
one and one in three4 ,it is  a         

commonwealth in which work is play and  
play is life3 three in one and  one in  
three4 ,it is a temple in which the     
priest is the worshipper and  the       
worshipper the worshipped3 three in one 
and one in three4 ,it is a  godhead in  
which all life is human and all humanity
 divine3 three in  one and one in three4
 ,it is1 in short1 the dream of a       
madman4    ,now the part of this speech 
to which ,i would direct attention is   
embodied in the phrase1 ,it is a        
commonwealth in which work is play  and 
play is life4 ,this is the ideal of     
technical education4 ,it sounds  very   
mystical when we confront it with the   
actual facts1 the toiling  millions1    
tired1 discontented1 mentally           
indifferent1 and then the  employers----
 ,i am not undertaking a social         
analysis1 but ,i shall carry  you with  
me when ,i admit that the present facts 
of society are a long  way off this     
ideal4 ,furthermore1 we are agreed that 
an employer who  conducted his workshop 

on the principle that work should be    
play  would be ruined in a week4    ,the
 curse that has been laid on humanity1  
in fable and in fact1 is1  that by the  
sweat of its brow shall it live4 ,but   
reason and moral  intuition have seen in
 this curse the foundation for advance4 
,the early  ,benedictine monks rejoiced 
in their labours because they conceived 
themselves as thereby made              
fellow-workers with ,christ4            
,stripped of its theological trappings1 
the essential idea remains1  that work  
should be transfused with intellectual  
and moral vision and  thereby turned    
into a joy1 triumphing over its         
weariness and its pain4  ,each of us    
will re-state this abstract formulation 
in a more concrete  shape in accordance 
with his private outlook4 ,state it how 
you like1  so long as you do not lose   
the main point in your details4 ,however
 you  phrase it1 it remains the sole    
real hope of toiling humanity2 and it   
is in the hands of technical teachers1  

and of those who control their  spheres 
of activity1 so to mould the nation that
 daily it may pass to  its labours in   
the spirit of the monks of old4    ,the 
immediate need of the nation is a large 
supply of skilled workmen1  of men with 
inventive genius1 and of employers alert
 in the development  of new ideas4      
,there is one--and only one--way to     
obtain these admirable results4  ,it is 
by producing workmen1 men of science1   
and employers who enjoy  their work4    
,view the matter practically in the     
light of our knowledge  of average human
 nature4 ,is it likely that a tired1    
bored workman1  however skilful his     
hands1 will produce a large output of   
first-class  work8 ,he will limit his   
production1 will scamp his work1 and be 
an  adept at evading inspection2 he will
 be slow in adapting himself to  new    
methods2 he will be a focus of          
discontent1 full of unpractical         
revolutionary ideas1 controlled by no   
sympathetic apprehension of the  real   

working of trade conditions4 ,if1 in the
 troubled times which may  be before us1
 you wish appreciably to increase the   
chance of some  savage upheaval1        
introduce widespread technical education
 and ignore  the ,benedictine ideal4    
,society will then get what it deserves4
    ,again1 inventive genius requires   
pleasurable mental activity as a        
condition for its vigorous exercise4    
,necessity is the mother of  invention  
is a silly proverb4 ,necessity is the   
mother of futile  dodges is much nearer 
to the truth4 ,the basis of the growth  
of modern  invention is science1 and    
science is almost wholly the outgrowth  
of  pleasurable intellectual curiosity4 
,the third class are the employers1 who 
are to be enterprising4 ,now  it is to  
be observed that it is the successful   
employers who are the  important people 
to get at1 the men with business        
connections all over  the world1 men who
 are already rich4 ,no doubt there will 
always be a  continuous process of rise 

and fall of businesses4 ,but it is      
futile  to expect flourishing trade1 if 
in the mass the successful houses of    
business are suffering from atrophy4    
,now if these men conceive their        
businesses as merely indifferent means  
for acquiring other disconnected        
opportunities of life1 they have no spur
 to alertness4 ,they are  already doing 
very well1 the mere momentum of their   
present business  engagements will carry
 them on for their time4 ,they are not  
at all  likely to bother themselves with
 the doubtful chances of new methods4   
,their real soul is in the other side of
 their life4 ,desire for money  will    
produce hard-fistedness and not         
enterprise4 ,there is much more  hope   
for humanity from manufacturers who     
enjoy their work than from  those who   
continue in irksome business with the   
object of founding  hospitals4          
,finally1 there can be no prospect of   
industrial peace so long as  masters and
 men in the mass conceive themselves as 

engaged in a  soulless operation of     
extracting money from the public4       
,enlarged views  of the work performed1 
and of the communal service thereby     
rendered1  can be the only basis on     
which to found sympathetic co-operation4
    ,the conclusion to be drawn from    
this discussion is1 that alike for      
masters and for men a technical or      
technological education1 which is to    
have any chance of satisfying the       
practical needs of the nation1 must  be 
conceived in a liberal spirit as a real 
intellectual enlightenment  in regard to
 principles applied and services        
rendered4 ,in such an  education        
geometry and poetry are as essential as 
turning lathes4    ,the mythical figure 
of ,plato may stand for modern liberal  
education  as does that of ,st4         
,benedict for technical education4 ,we  
need not  entangle ourselves in the     
qualifications necessary for a balanced 
representation of the actual thoughts of
 the actual men4 ,they are used  here as

 symbolic figures typical of            
antithetical notions4 ,we consider      
,plato in the light of the type of      
culture he now inspires4    ,in its     
essence a liberal education is an       
education for thought and  for sthetic  
appreciation4 ,it proceeds by imparting 
a knowledge of  the masterpieces of     
thought1 of imaginative literature1 and 
of art4  ,the action which it           
contemplates is command4 ,it is an      
aristocratic  education implying        
leisure4 ,this ,platonic ideal has      
rendered  imperishable services to      
,european civilisation4 ,it has         
encouraged art1  it has fostered that   
spirit of disinterested curiosity which 
is the  origin of science1 it has       
maintained the dignity of mind in the   
face of  material force1 a dignity which
 claims freedom of thought4 ,plato did  
not1 like ,st4 ,benedict1 bother himself
 to be a fellow-worker with his  slaves2
 but he must rank among the emancipators
 of mankind4 ,his type  of culture is   

the peculiar inspiration of the liberal 
aristocrat1 the  class from which       
,europe derives what ordered liberty it 
now possesses4  ,for centuries1 from    
,pope ,nicholas ,v to the schools of the
 ,jesuits1 and  from the ,jesuits to the
 modern headmasters of ,english public  
schools1  this educational ideal has had
 the strenuous support of the clergy4   
,for certain people it is a very good   
education4 ,it suits their type  of mind
 and the circumstances amid which their 
life is passed4 ,but  more has been     
claimed for it than this4 ,all education
 has been judged  adequate or defective 
according to its approximation to this  
sole type4    ,the essence of the type  
is a large discursive knowledge of the  
best  literature4 ,the ideal product of 
the type is the man who is acquainted   
with the best that has been written4 ,he
 will have acquired the chief           
languages1 he will have considered the  
histories of the rise and fall  of      
nations1 the poetic expression of human 

feeling1 and have read the  great dramas
 and novels4 ,he will also be well      
grounded in the chief  philosophies1 and
 have attentively read those philosophic
 authors who  are distinguished for     
lucidity of style4    ,it is obvious    
that1 except at the close of a long     
life1 he will not  have much time for   
anything else if any approximation is to
 be made to  the fulfilment of this     
programme4 ,one is reminded of the      
calculation  in a dialogue of ,lucian   
that1 before a man could be justified in
  practising any one of the current     
ethical systems1 he should have spent  a
 hundred and fifty years in examining   
their credentials4    ,such ideals are  
not for human beings4 ,what is meant by 
a liberal  culture is nothing so        
ambitious as a full acquaintance with   
the varied  literary expression of      
civilised mankind from ,asia to ,europe1
 and from  ,europe to ,america4 ,a small
 selection only is required2 but then1  
as  we are told1 it is a selection of   

the very best4 ,i have my doubts of  a  
selection which includes ,xenophon and  
omits ,confucius1 but then ,i  have read
 neither in the original4 ,the ambitious
 programme of a liberal  education      
really shrinks to a study of some       
fragments of literature  included in a  
couple of important languages4    ,but  
the expression of the human spirit is   
not confined to literature4  ,there are 
the other arts1 and there are the       
sciences4 ,also education  must pass    
beyond the passive reception of the     
ideas of others4 ,powers  of initiative 
must be strengthened4 ,unfortunately    
initiative does not  mean just one      
acquirement--there is initiative in     
thought1 initiative  in action1 and the 
imaginative initiative of art2 and these
 three  categories require many         
subdivisions4    ,the field of          
acquirement is large1 and the individual
 so fleeting and  so fragmentary3       
classical scholars1 scientists1         
headmasters are alike  ignoramuses4     

,there is a curious illusion that a more
 complete culture was possible  when    
there was less to know4 ,surely the only
 gain was1 that it was more  possible to
 remain unconscious of ignorance4 ,it   
cannot have been a gain  to ,plato to   
have read neither ,shakespeare1 nor     
,newton1 nor ,darwin4  ,the achievements
 of a liberal education have in recent  
times not been  worsened4 ,the change is
 that its pretensions have been found   
out4    ,my point is1 that no course of 
study can claim any position of ideal   
completeness4 ,nor are the omitted      
factors of subordinate importance4  ,the
 insistence in the ,platonic culture on 
disinterested intellectual  appreciation
 is a psychological error4 ,action and  
our implication in  the transition of   
events amid the evitable bond of cause  
to effect  are fundamental4 ,an         
education which strives to divorce      
intellectual  or sthetic life from these
 fundamental facts carries with it the  
decadence of civilisation4 ,essentially 

culture should be for action1  and its  
effect should be to divest labour from  
the associations of  aimless toil4 ,art 
exists that we may know the deliverances
 of our  senses as good4 ,it heightens  
the sense-world4    ,disinterested      
scientific curiosity is a passion for an
 ordered  intellectual vision of the    
connection of events4 ,but the goal of  
such curiosity is the marriage of action
 to thought4 ,this essential            
intervention of action even in abstract 
science is often overlooked4 ,no  man of
 science wants merely to know4 ,he      
acquires knowledge to appease  his      
passion for discovery4 ,he does not     
discover in order to know1 he  knows in 
order to discover4 ,the pleasure which  
art and science can  give to toil is the
 enjoyment which arises from            
successfully directed  intention4 ,also 
it is the same pleasure which is yielded
 to the  scientist and to the artist4   
,the antithesis between a technical and 
a liberal education is  fallacious4     

,there can be no adequate technical     
education which is not  liberal1 and no 
liberal education which is not          
technical3 that is1 no  education which 
does not impart both technique and      
intellectual vision4  ,in simpler       
language1 education should turn out the 
pupil with something  he knows well and 
something he can do well4 ,this intimate
 union of  practice and theory aids     
both4 ,the intellect does not work best 
in a  vacuum4 ,the stimulation of       
creative impulse requires1 especially in
  the case of a child1 the quick        
transition to practice4 ,geometry and   
mechanics1 followed by workshop         
practice1 gain that reality without     
which mathematics is verbiage4    ,there
 are three main methods which are       
required in a national system of        
education1 namely1 the literary         
curriculum1 the scientific curriculum1  
the technical curriculum4 ,but each of  
these curricula should include  the     
other two4 ,what ,i mean is1 that every 

form of education should give  the pupil
 a technique1 a science1 an assortment  
of general ideas1 and  sthetic          
appreciation1 and that each of these    
sides of his training  should be        
illuminated by the others4 ,lack of     
time1 even for the most  favoured pupil1
 makes it impossible to develop fully   
each curriculum4  ,always there must be 
a dominant emphasis4 ,the most direct   
sthetic  training naturally falls in the
 technical curriculum in those cases    
when the training is that requisite for 
some art or artistic craft4 ,but  it is 
of high importance in both a literary   
and a scientific education4    ,the     
educational method of the literary      
curriculum is the study of  language1   
that is1 the study of our most habitual 
method of conveying  to others our      
states of mind4 ,the technique which    
should be acquired  is the technique of 
verbal expression1 the science is the   
study of the  structure of language and 
the analysis of the relations of        

language  to the states of mind         
conveyed4 ,furthermore1 the subtle      
relations of  language to feeling1 and  
the high development of the sense organs
  to which written and spoken words     
appeal1 lead to keen sthetic            
appreciations being aroused by the      
successful employment of language4      
,finally1 the wisdom of the world is    
preserved in the masterpieces of        
linguistic composition4    ,this        
curriculum has the merit of homogeneity4
 ,all its various parts  are            
co-ordinated and play into each other's 
hands4 ,we can hardly be  surprised that
 such a curriculum1 when once broadly   
established1 should  have claimed the   
position of the sole perfect type of    
education4 ,its  defect is unduly to    
emphasise the importance of language4   
,indeed the  varied importance of verbal
 expression is so overwhelming that its 
sober estimation is difficult4 ,recent  
generations have been witnessing  the   
retreat of literature1 and of literary  

forms of expression1 from  their        
position of unique importance in        
intellectual life4 ,in order  truly to  
become a servant and a minister of      
nature something more is  required than 
literary aptitudes4    ,a scientific    
education is primarily a training in the
 art of observing  natural phenomena1   
and in the knowledge and deduction of   
laws  concerning the sequence of such   
phenomena4 ,but here1 as in the case of 
a liberal education1 we are met by the  
limitations imposed by shortness  of    
time4 ,there are many types of natural  
phenomena1 and to each type  there      
corresponds a science with its peculiar 
modes of observation1  and its peculiar 
types of thought employed in the        
deduction of laws4 ,a  study of science 
in general is impossible in education1  
all that can be  achieved is the study  
of two or three allied sciences4 ,hence 
the charge  of narrow specialism urged  
against any education which is primarily
  scientific4 ,it is obvious that the   

charge is apt to be well-founded2  and  
it is worth considering how1 within the 
limits of a scientific  education and to
 the advantage of such an education1 the
 danger can be  avoided4    ,such a     
discussion requires the consideration of
 technical education4 ,a  technical     
education is in the main a training in  
the art of utilising  knowledge for the 
manufacture of material products4 ,such 
a training  emphasises manual skill1 and
 the co-ordinated action of hand and    
eye1 and judgment in the control of the 
process of construction4 ,but  judgment 
necessitates knowledge of those natural 
processes of which the  manufacture is  
the utilisation4 ,thus somewhere in     
technical training  an education in     
scientific knowledge is required4 ,if   
you minimise  the scientific side1 you  
will confine it to the scientific       
experts2  if you maximise it1 you will  
impart it in some measure to the men1   
and--what is of no less importance--to  
the directors and managers of  the      

businesses4    ,technical education is  
not necessarily allied exclusively to   
science  on its mental side4 ,it may be 
an education for an artist or for       
apprentices to an artistic craft4 ,in   
that case sthetic appreciation  will    
have to be cultivated in connection with
 it4    ,an evil side of the ,platonic  
culture has been its total neglect of   
technical education as an ingredient in 
the complete development  of ideal human
 beings4 ,this neglect has arisen from  
two disastrous  antitheses1 namely1 that
 between mind and body1 and that between
  thought and action4 ,i will here      
interject1 solely to avoid criticism1   
that ,i am well aware that the ,greeks  
highly valued physical beauty and       
physical activity4 ,they had1 however1  
that perverted sense of values  which is
 the nemesis of slave-owning4    ,i lay 
it down as an educational axiom that in 
teaching you will come  to grief as soon
 as you forget that your pupils have    
bodies4 ,this is  exactly the mistake of

 the post-renaissance ,platonic         
curriculum4 ,but  nature can be kept at 
bay by no pitchfork2 so in ,english     
education1  being expelled from the     
classroom1 she returned with a cap and  
bells in  the form of all-conquering    
athleticism4    ,the connections between
 intellectual activity and the body1    
though  diffused in every bodily        
feeling1 are focussed in the eyes1 the  
ears1  the voice1 and the hands4 ,there 
is a co-ordination of senses and        
thought1 and also a reciprocal influence
 between brain activity and  material   
creative activity4 ,in this reaction the
 hands are peculiarly  important4 ,it is
 a moot point whether the human hand    
created the human  brain1 or the brain  
created the hand4 ,certainly the        
connection is  intimate and reciprocal4 
,such deep-seated relations are not     
widely  atrophied by a few hundred years
 of disuse in exceptional families4     
,the disuse of hand-craft is a          
contributory cause to the brain-lethargy

  of aristocracies1 which is only       
mitigated by sport where the concurrent 
brain-activity is reduced to a minimum  
and the hand-craft lacks  subtlety4 ,the
 necessity for constant writing and     
vocal exposition is  some slight        
stimulus to the thought-power of the    
professional classes4  ,great readers1  
who exclude other activities1 are not   
distinguished by  subtlety of brain4    
,they tend to be timid conventional     
thinkers4 ,no  doubt this is partly due 
to their excessive knowledge outrunning 
their  powers of thought2 but it is     
partly due to the lack of brain-stimulus
  from the productive activities of hand
 or voice4    ,in estimating the        
importance of technical education we    
must rise above  the exclusive          
association of learning with            
book-learning4 ,first-hand  knowledge is
 the ultimate basis of intellectual     
life4 ,to a large  extent book-learning 
conveys second-hand information1 and as 
such can  never rise to the importance  

of immediate practice4 ,our goal is to  
see  the immediate event of our lives as
 instances of our general ideas4  ,what 
the learned world tends to offer is one 
second-hand scrap of  information       
illustrating ideas derived from another 
second-hand scrap  of information4 ,the 
second-handedness of the learned world  
is the  secret of its mediocrity4 ,it is
 tame because it has never been scared  
by facts4 ,the main importance of       
,francis ,bacon's influence does not    
lie in any peculiar theory of inductive 
reasoning which he happened to  express1
 but in the revolt against second-hand  
information of which he  was a leader4  
,the peculiar merit of a scientific     
education should be1 that it bases      
thought upon first-hand observation2 and
 the corresponding merit of a  technical
 education is1 that it follows our deep 
natural instinct to  translate thought  
into manual skill1 and manual activity  
into thought4    ,we are a ,mathematical
 ,association1 and it is natural to ask3

 ,where do  we come in8 ,we come in just
 at this point4    ,the thought which   
science evokes is logical thought4 ,now 
logic is of  two kinds3 the logic of    
discovery and the logic of the          
discovered4    ,the logic of discovery  
consists in the weighing of             
probabilities1 in  discarding details   
deemed to be irrelevant1 in divining the
 general  rules according to which      
events occur1 and in testing hypotheses 
by  devising suitable experiments4 ,this
 is inductive logic4    ,the logic of   
the discovered is the deduction of the  
special events  which1 under certain    
circumstances1 would happen in obedience
 to the  assumed laws of nature4 ,thus  
when the laws are discovered or assumed1
  their utilisation entirely depends on 
deductive logic4 ,without  deductive    
logic science would be entirely useless4
 ,it is merely a  barren game to ascend 
from the particular to the general1     
unless  afterwards we can reverse the   
process and descend from the general to 

the particular1 ascending and descending
 like the angels on ,jacob's  ladder4   
,when ,newton had divined the law of    
gravitation he at once  proceeded to    
calculate the earth's attractions on an 
apple at its  surface and on the moon4  
,we may note in passing that inductive  
logic  would be impossible without      
deductive logic4 ,thus ,newton's        
calculations  were an essential step in 
his inductive verification of the great 
law4    ,now mathematics is nothing else
 than the more complicated parts of  the
 art of deductive reasoning1 especially 
where it concerns number1  quantity1 and
 space4    ,in the teaching of science1 
the art of thought should be taught3    
namely1 the art of forming clear        
conceptions applying to first-hand      
experience1 the art of divining the     
general truths which apply1 the  art of 
testing divinations1 and the art of     
utilising general truths  by reasoning  
to more particular cases of some        
peculiar importance4  ,furthermore1 a   

power of scientific exposition is       
necessary1 so that the  relevant issues 
from a confused mass of ideas can be    
stated clearly1  with due emphasis on   
important points4    ,by the time a     
science1 or a small group of sciences1  
has been taught  thus amply1 with due   
regard to the general art of thought1 we
 have gone  a long way towards          
correcting the specialism of science4   
,the worst  of a scientific education   
based1 as necessarily must be the case1 
on  one or two particular branches of   
science1 is that the teachers under  the
 influence of the examination system are
 apt merely to stuff  their pupils with 
the narrow results of these special     
sciences4 ,it is  essential that the    
generality of the method be continually 
brought to  light and contrasted with   
the speciality of the particular        
application4  ,a man who only knows his 
own science1 as a routine peculiar to   
that  science1 does not even know that4 
,he has no fertility of thought1 no     

power of quickly seizing the bearing of 
alien ideas4 ,he will discover  nothing1
 and be stupid in practical             
applications4    ,this exhibition of the
 general in the particular is extremely 
difficult  to effect1 especially in the 
case of younger pupils4 ,the art of     
education is never easy4 ,to surmount   
its difficulties1 especially those  of  
elementary education1 is a task worthy  
of the highest genius4 ,it is  the      
training of human souls4                
,mathematics1 well taught1 should be the
 most powerful instrument  in gradually 
implanting this generality of idea4 ,the
 essence of  mathematics is perpetually 
to be discarding more special ideas in  
favour of more general ideas1 and       
special methods in favour of general    
methods4 ,we express the conditions of a
 special problem in the form of  an     
equation1 but that equation will serve  
for a hundred other problems1  scattered
 through diverse sciences4 ,the general 
reasoning is always  the powerful       

reasoning1 because deductive cogency is 
the property of  abstract form4         
,here1 again1 we must be careful4 ,we   
shall ruin mathematical education  if we
 use it merely to impress general       
truths4 ,the general ideas are  the     
means of connecting particular results4 
,after all1 it is the  concrete special 
cases which are important4 ,thus in the 
handling of  mathematics in your results
 you cannot be too concrete1 and in your
  methods you cannot be too general4    
,the essential course of reasoning  is  
to generalise what is particular1 and   
then to particularise what is  general4 
,without generality there is no         
reasoning1 without concreteness  there  
is no importance4    ,concreteness is   
the strength of technical education4 ,i 
would  remind you that truths which lack
 the highest generality are not         
necessarily concrete facts4 ,for        
example1 x  y  y  x is an  algebraic    
truth more general than #b  #b  #d4 ,but
 two and two make  four is itself a     

highly general proposition lacking any  
element of  concreteness4 ,to obtain a  
concrete proposition immediate intuition
 of a  truth concerning particular      
objects is requisite2 for example1 these
  two apples and those apples together  
make four apples is a concrete          
proposition1 if you have direct         
perception or immediate memory of the   
apples4    ,in order to obtain the full 
realisation of truths as applying1 and  
not  as empty formul1 there is no       
alternative to technical education4     
,mere  passive observation is not       
sufficient4 ,in creation only is there  
vivid  insight into the properties of   
the object thereby produced4 ,if you    
want to understand anything1 make it    
yourself1 is a sound rule4 ,your        
faculties will be alive1 your thoughts  
gain vividness by an immediate          
translation into acts4 ,your ideas gain 
that reality which comes from  seeing   
the limits of their application4    ,in 
elementary education this doctrine has  

long been put into practice4  ,young    
children are taught to familiarise      
themselves with shapes and  colours by  
simple manual operations of cutting out 
and of sorting4 ,but  good though this  
is1 it is not quite what ,i mean4 ,that 
is practical  experience before you     
think1 experience antecedent to thought 
in order  to create ideas1 a very       
excellent discipline4 ,but technical    
education  should be much more than     
that3 it is creative experience while   
you  think1 experience which realises   
your thought1 experience which teaches  
you to co-ordinate act and thought1     
experience leading you to associate     
thought with foresight and foresight    
with achievement4 ,technical  education 
gives theory1 and a shrewd insight as to
 where theory fails4    ,a technical    
education is not to be conceived as a   
maimed alternative  to the perfect      
,platonic culture3 namely1 as a         
defective training  unfortunately made  
necessary by cramped conditions of life4

 ,no  human being can attain to anything
 but fragmentary knowledge and a        
fragmentary training of his capacities4 
,there are1 however1 three main  roads  
along which we can proceed with good    
hope of advancing towards  the best     
balance of intellect and character3     
these are the way of  literary culture1 
the way of scientific culture1 the way  
of technical  culture4 ,no one of these 
methods can be exclusively followed     
without  grave loss of intellectual     
activity and of character4 ,but a mere  
mechanical mixture of the three         
curricula will produce bad results in   
the shape of scraps of information never
 interconnected or utilised4  ,we have  
already noted as one of the strong      
points of the traditional  literary     
culture that all its parts are          
co-ordinated4 ,the problem  of education
 is to retain the dominant emphasis1    
whether literary1  scientific1 or       
technical1 and without loss of          
co-ordination to infuse  into each way  

of education something of the other two4
    ,to make definite the problem of    
technical education fix attention on    
two ages3 one thirteen1 when elementary 
education ends2 and the other           
seventeen1 when technical education ends
 so far as it is compressed  within a   
school curriculum4 ,i am aware that for 
artisans in junior  technical schools a 
three-years' course would be more usual4
 ,on the  other hand1 for naval         
officers1 and for directing classes     
generally1  a longer time can be        
afforded4 ,we want to consider the      
principles  to govern a curriculum which
 shall land these children at the age of
  seventeen in the position of having   
technical skill useful to the           
community4    ,their technical manual   
training should start at thirteen1      
bearing a  modest proportion to the rest
 of their work1 and should increase in  
each year finally to attain to a        
substantial proportion4 ,above all      
things it should not be too specialised4

 ,workshop finish and workshop  dodges1 
adapted to one particular job1 should be
 taught in the  commercial workshop1 and
 should form no essential part of the   
school  course4 ,a properly trained     
worker would pick them up in no time4   
,in all  education the main cause of    
failure is staleness4 ,technical        
education  is doomed if we conceive it  
as a system for catching children young 
and  for giving them one highly         
specialised manual aptitude4 ,the nation
 has  need of a fluidity of labour1 not 
merely from place to place1 but also    
within reasonable limits of allied      
aptitudes1 from one special type of     
work to another special type4 ,i know   
that here ,i am on delicate ground1  and
 ,i am not claiming that men while they 
are specialising on one sort  of work   
should spasmodically be set to other    
kinds4 ,that is a question  of trade    
organisation with which educationalists 
have no concern4 ,i am  only asserting  
the principles that training should be  

broader than the  ultimate              
specialisation1 and that the resulting  
power of adaptation to  varying demands 
is advantageous to the workers1 to the  
employers1 and  to the nation4    ,in   
considering the intellectual side of the
 curriculum we must be  guided by the   
principle of the co-ordination of       
studies4 ,in general1  the intellectual 
studies most immediately related to     
manual training  will be some branches  
of science4 ,more than one branch will1 
in fact1  be concerned2 and even if that
 be not the case1 it is impossible to   
narrow down scientific study to a single
 thin line of thought4 ,it is  possible1
 however1 provided that we do not press 
the classification  too far1 roughly to 
classify technical pursuits according to
 the  dominant science involved4 ,we    
thus find a sixfold division1 namely1   
7#a7 ,geometrical techniques1 7#b7      
,mechanical techniques1 7#c7 ,physical  
techniques1 7#d7 ,chemical techniques1  
7#e7 ,biological techniques1 7#f7       

,techniques of commerce and of social   
service4    ,by this division1 it is    
meant that apart from auxiliary sciences
  some particular science requires      
emphasis in the training for most       
occupations4 ,we can1 for example1      
reckon carpentry1 ironmongery1  and many
 artistic crafts among geometrical      
techniques4 ,similarly  agriculture is a
 biological technique4 ,probably        
cookery1 if it includes  food catering1 
would fall midway between biological1   
physical1 and  chemical sciences1 though
 of this ,i am not sure4    ,the        
sciences associated with commerce and   
social service would be  partly algebra1
 including arithmetic and statistics1   
and partly  geography and history4 ,but 
this section is somewhat heterogeneous  
in  its scientific affinities4 ,anyhow  
the exact way in which technical        
pursuits are classified in relation to  
science is a detail4 ,the  essential    
point is1 that with some thought it is  
possible to find  scientific courses    

which illuminate most occupations4      
,furthermore1 the  problem is well      
understood1 and has been brilliantly    
solved in many of  the schools of       
technology and junior technical schools 
throughout the  country4    ,in passing 
from science to literature1 in our      
review of the  intellectual elements of 
technical education1 we note that many  
studies  hover between the two3 for     
example1 history and geography4 ,they   
are  both of them very essential in     
education1 provided that they are the   
right history and the right geography4  
,also books giving descriptive  accounts
 of general results1 and trains of      
thought in various sciences  fall in the
 same category4 ,such books should be   
partly historical  and partly expository
 of the main ideas which have finally   
arisen4  ,prof4 ,r4 ,a4 ,gregory's      
recent book1 ,discovery1 and the ,home  
,university ,library series illustrate  
my meaning4 ,their value in  education  
depends on their quality as mental      

stimulants4 ,they must not  be inflated 
with gas on the wonders of science1 and 
must be informed  with a broad outlook4 
,it is unfortunate that the literary    
element in education has rarely  been   
considered apart from grammatical study4
 ,the historical reason is1  that when  
the modern ,platonic curriculum was     
being formed ,latin and  ,greek were the
 sole keys which rendered great         
literature accessible4  ,but there is no
 necessary connection between literature
 and grammar4  ,the great age of ,greek 
literature was already past before the  
arrival  of the grammarians of          
,alexandria4 ,of all types of men to-day
 existing1  classical scholars are the  
most remote from the ,greeks of the     
,periclean  times4    ,mere literary    
knowledge is of slight importance4 ,the 
only thing that  matters is1 how it is  
known4 ,the facts related are nothing4  
,literature  only exists to express and 
develop that imaginative world which is 
our  life1 the kingdom which is within  

us4 ,it follows that the literary side  
of a technical education should consist 
in an effort to make the pupils  enjoy  
literature4 ,it does not matter what    
they know1 but the enjoyment  is vital4 
,the great ,english ,universities1 under
 whose direct authority  school-children
 are examined in plays of ,shakespeare1 
to the certain  destruction of their    
enjoyment1 should be prosecuted for     
soul-murder4    ,now there are two kinds
 of intellectual enjoyment3 the         
enjoyment of  creation1 and the         
enjoyment of relaxation4 ,they are not  
necessarily  separated4 ,a change of    
occupation may give the full tide of    
happiness  which comes from the         
concurrence of both forms of pleasure4  
,the  appreciation of literature is     
really creation4 ,the written word1 its 
music1 and its associations1 are only   
the stimuli4 ,the vision which  they    
evoke is our own doing4 ,no one1 no     
genius other than our own1  can make our
 own life live4 ,but except for those   

engaged in literary  occupations1       
literature is also a relaxation4 ,it    
gives exercise to that  other side which
 any occupation must suppress during the
 working hours4  ,art also has the same 
function in life as has literature4     
,to obtain the pleasure of relaxation   
requires no help4 ,the pleasure  is     
merely to cease doing4 ,some such pure  
relaxation is a necessary  condition of 
health4 ,its dangers are notorious1 and 
to the greater  part of the necessary   
relaxation nature has affixed1 not      
enjoyment1  but the oblivion of sleep4  
,creative enjoyment is the outcome of   
successful effort and requires help for 
its initiation4 ,such enjoyment  is     
necessary for high-speed work and for   
original achievement4    ,to speed up   
production with unrefreshed workmen is a
 disastrous  economic policy4 ,temporary
 success will be at the expense of the  
nation1 which1 for long years of their  
lives1 will have to support  worn-out   
artisans--unemployables4 ,equally       

disastrous is the  alternation of spasms
 of effort with periods of pure         
relaxation4  ,such periods are the      
seed-times of degeneration1 unless      
rigorously  curtailed4 ,the normal      
recreation should be change of activity1
  satisfying the cravings of instincts4 
,games afford such activity4 ,their     
disconnection emphasises the relaxation1
 but their excess leaves us  empty4     
,it is here that literature and art     
should play an essential part in  a     
healthily organised nation4 ,their      
services to economic production  would  
be only second to those of sleep or of  
food4 ,i am not now talking  of the     
training of an artist1 but of the use of
 art as a condition of  healthy life4   
,it is analogous to sunshine in the     
physical world4    ,when we have once   
rid our minds of the idea that knowledge
 is to  be exacted1 there is no especial
 difficulty or expense involved in      
helping the growth of artistic          
enjoyment4 ,all school-children could   

be sent at regular intervals to         
neighbouring theatres where suitable    
plays could be subsidised4 ,similarly   
for concerts and cinema films4          
,pictures are more doubtful in their    
popular attraction2 but interesting     
representations of scenes or ideas which
 the children have read about  would    
probably appeal4 ,the pupils themselves 
should be encouraged  in artistic       
efforts4 ,above all the art of reading  
aloud should be  cultivated4 ,the ,roger
 de ,coverley essays of ,addison are    
perfect  examples of readable prose4    
,art and literature have not merely an  
indirect effect on the main  energies of
 life4 ,directly1 they give vision4 ,the
 world spreads wide  beyond the         
deliverances of material sense1 with    
subtleties of reaction  and with pulses 
of emotion4 ,vision is the necessary    
antecedent to  control and to direction4
 ,in the contest of races which in its  
final  issues will be decided in the    
workshops and not on the battle-field1  

the victory will belong to those who are
 masters of stores of trained  nervous  
energy1 working under conditions        
favourable to growth4 ,one such         
essential condition is ,art4    ,if     
there had been time1 there are other    
things which ,i should like  to have    
said3 for example1 to advocate the      
inclusion of one foreign  language in   
all education4 ,from direct observation 
,i know this to be  possible for artisan
 children4 ,but enough has been put     
before you1  to make plain the          
principles with which we should         
undertake national  education4    ,in   
conclusion1 ,i recur to the thought of  
the ,benedictines1 who  saved for       
mankind the vanishing civilisation of   
the ancient world  by linking together  
knowledge1 labour1 and moral energy4    
,our danger  is to conceive practical   
affairs as the kingdom of evil1 in which
  success is only possible by the       
extrusion of ideal aims4 ,i believe     
that such a conception is a fallacy     

directly negatived by practical         
experience4 ,in education this error    
takes the form of a mean view of        
technical training4 ,our forefathers in 
the dark ages saved themselves  by      
embodying high ideals in great          
organisations4 ,it is our task1  without
 servile imitation1 boldly to exercise  
our creative energies1  remembering amid
 discouragements that the coldest hour  
immediately  precedes the dawn4         
,,footnotes3    ,footnote #a3 ,cf4      
,,bernard ,,shaw3 ,john ,bull's ,other  
,island4                                
,,chapter ,,iii                         
,a ,,polytechnic ,,in ,,war-,,time      
,address at the ,prize ,distribution1   
,borough ,polytechnic ,institute1       
,southwark1 #afth ,february1 #aiag      
,i ,,will commence by drawing your      
attention to some of the satisfactory   
features of the ,principal's report on  
the work of the ,institute during  the  
past year4 ,it has been a year of great 
difficulties4 ,some of our  staff are   

serving with the colours1 and our       
classes have been depleted4  ,but in    
spite of everything1 we have done very  
well4 ,first1 the average  result in the
 examinations has been good1            
surprisingly good in view of  the       
present circumstances4 ,the ,governors  
attach great importance to  the         
maintenance of a high average result2 it
 is the best single test  of efficiency4
 ,again1 our individual successes have  
been notable4 ,we  have gained--,i say  
we because we are all one in our        
pleasure at  these successes--we have   
gained two #hj ,l4,c4,c4 scholarships1  
nineteen  exhibitions1 in addition to a 
first-place1 and medals1 prizes and     
certificates4 ,all this is very         
satisfactory4 ,it tells of efficient    
teaching1 and of hard work and regular  
attendance on the part of the  students4
 ,we know that we are keeping up the    
standard of efficiency  which in the    
past has been a source of pride to every
 one connected  with this ,institute4   

,now all this good work does not come   
about by itself without any one  making 
an effort4 ,such a record requires our  
skilled staff of teachers  and          
organisers4 ,they have worked very hard 
during the last session  under great    
difficulties1 in order to create the    
successful result  which we are here to 
celebrate4 ,i know something about      
teaching4 ,it  is very exacting work1   
and can be made successful only by      
continual  devotion4 ,i am sure that ,i 
am voicing your feelings1 and ,i know   
that  ,i am expressing those of the     
,governors1 when ,i thank the ladies and
  gentlemen of the staff very heartily  
for their services during the last      
session4    ,prize-givings are always   
pleasant occasions4 ,we have come here  
to think  about our successes1 and to   
congratulate our students4 ,there is no 
more  satisfactory ,governors' ,meeting 
in the course of the year than when we  
meet on this occasion1 and face our     
friends and tell them how pleased  we   

are at the successful result of their   
hard work4 ,this evening ,i am  in a    
doubly happy position1 for my colleagues
 have asked me to be their  spokesman in
 tendering our good wishes to the       
prize-winners4 ,you have  worked hard   
and you have done well1 and ,i am sure  
that you all deserve  your successes2   
they are a pleasure not only to you1 but
 in your homes  and to your companions  
and fellow-students4    ,successful work
 here will enable you to acquire skill  
in your trades1  and thereby the better 
to earn your living4 ,earning a living  
is on the  average no bad test of       
service rendered to the community4 ,a   
man who  has made himself skilful in his
 trade and has done well for himself in 
his walk of life1 has in general good   
reason to believe that he is a  citizen 
who has benefited his country4 ,it is an
 evil day for a nation  when it loses   
respect for success in industry4    ,but
 if you steer your lives by the compass 
which points steadily to  the ,north    

,pole of personal success1 you will have
 missed your greatest  chances in life4 
,the genial climate is in the south4    
,what ,i mean is this3 you must make up 
your mind to find the best part  of your
 happiness in kindly helpful relations  
with others4 ,it should be  our ambition
 to leave our own small corner of the   
world a little tidier  and a little     
happier than when we entered it4 ,i am  
well aware that  this is an old story2  
but old stories are sometimes true1 and 
this is  the biggest truth in the whole 
world4 ,the warm kindly feelings are the
  happy feelings4 ,the fortunate people 
are those whose minds are filled  with  
thoughts in which they forget themselves
 and remember others4 ,it  is not true  
that nature is a mere scene of struggle 
in which every one  competes with his   
neighbour4 ,those communities thrive    
best and last  longest which are filled 
with a spirit of mutual help4    ,the   
future of the country lies with you4    
,the crown of your success  is the      

promise of future work1 often           
unrecognised work1 done under           
discouragement1 but done steadily and   
cheerfully4 ,it is on you that  the     
country depends for the maintenance and 
the growth of those ideals  without     
which a race withers4 ,do not be        
discouraged by difficulties  which seem 
unsurmountable4 ,the conditions of life 
which mould us all  are modified by our 
will1 by our energy1 and by the purity  
of our  intentions4    ,if we may judge 
of intensity of feeling by length of    
memory1 the  enjoyment of receiving a   
prize bites very deep4 ,across the space
 of  more than forty years1 before many 
of your parents were born1 or when  they
 were being carried about in long       
clothes1 ,i can remember1 as if it  were
 yesterday1 the occasion when ,i        
received my first prize at school4  ,i  
can see the medival school-room1 the    
headmaster1 and my companions4  ,perhaps
 some of you1 when a generation has     
passed by1 will remember the  scene     

to-night--this ,stanley ,gymnasium      
recalling the memory of ,miss  ,maude   
,stanley1 who devoted to our welfare so 
much of her energy and  her thought--the
 adjoining ,edric ,hall associated with 
the name of ,mr4  ,edric ,bayley1 the   
,father of the ,institute2 ,mr4 ,millis 
and ,miss ,smith1  the first ,principal 
and the first ,lady ,superintendent1 the
 architects  of our prosperity2 ,mr4    
,leonard ,spicer1 our ,chairman and     
member of a  family and of a firm known 
throughout the world1 and respected in  
proportion as they are known4 ,and the  
cause why to-night we are a small       
gathering is one more reason why this   
assembly can never slip from your       
memory4 ,we meet at a moment when       
,england stands in as deadly a peril as 
in any previous moment of her           
history--such peril as when the ,spanish
  ships of the ,armada rode in the      
,english ,channel1 or when ,napoleon    
watched our coast across the ,strait of 
,dover4 ,the present danger can be      

overcome only by the same courage as    
that which saved our freedom in  those  
former times4    ,therefore1 to-night1  
in recalling the activities of the      
various  sections of our society which  
form this great ,polytechnic ,institute1
  our thoughts go further afield4 ,they 
travel by land and by sea1 till  they   
bring before our minds the gallant band 
whom this ,institute has  sent to the   
,front--more than #hjj of our members   
are with the ,colours4  ,what our       
fighting men have done for us1 for the  
world in general1 and  for the future of
 ,england1 is so overwhelming that words
 cannot praise  them enough4 ,i will    
just say one thing to you3 ,when you    
read of great  deeds done in past times1
 of perils encountered1 of adventures1  
of  undaunted courage1 of patriotism1 of
 self-sacrifice1 of suffering  endured  
for noble cause1 you each can say--,i1  
too1 have known such  heroes2 they are  
among my countrymen1 they are among my  
fellow-workers1  they are among my      

fellow-students and companions1 they are
 among the  dear inmates of my home4    
,and for those who have fallen1 it is   
for us  to erect a monument sufficient  
to transmit to future ages the memory   
of their sacrifice4 ,for this purpose   
there is only one memorial which  can   
suffice1 namely1 the cause for which    
they died4 ,the greatness of  ,england1 
the future of ,england1 has been left by
 them to our keeping4  ,guard it well4  
,the greatness of a country is nothing  
else than the greatness of the  lives of
 the men and of the women who compose   
it4 ,do not look round and  think who   
ought to be great ,englishmen--be great 
yourselves--you are  the people to      
achieve it1 you who are sitting here    
to-night4 ,there can  be no substitute  
service for this purpose4 ,it is the    
collective energy  of the whole people  
that will be needed to fashion a new    
,england worthy  of the sufferings which
 for its sake have been endured4    ,a  
few days ago ,i asked a man who has     

worked in ,egypt for many years  under  
,lord ,kitchener1 what he would pick out
 as the best sign of ,lord  ,kitchener's
 greatness4 ,he answered1 whatever      
,kitchener set himself to  do1 thereby  
became important4 ,now that is the      
secret of it all--take  hold of your    
opportunities and make them important4  
,here we are in this ,borough           
,polytechnic4 ,what an opportunity it   
represents4 ,this ,institute is a centre
 for social meeting1 a centre  for      
recreation1 a centre for education1 a   
centre for discussion4 ,we  will not    
sacrifice any one of our sides4 ,they   
must all be part of the  greatness which
 we claim4 ,make them all first-rate4   
,consider first the social and          
recreative sides4 ,for heaven's sake    
don't  think that you must be dull in   
order to be great4 ,there is no finer   
test of a nation than the way in which  
it fashions its amusements4  ,three     
centuries ago after the ,armada we made 
a good start in ,southwark4             

,shakespeare had his theatre here and   
wrote his plays to be acted in  this    
borough4 ,he has walked these streets1  
and if you had met him in  ,westminster 
he would1 quite likely1 have told you   
that he was going  down to the          
,elephant4 ,and even now the            
performances given at the  ,old ,vic are
 among the best in ,london for the      
purpose of seeing his  plays properly   
acted4 ,what ,southwark has done for the
 drama1 she can  do for the other arts1 
by using this ,institute as the         
instrument for  her energies4 ,why      
should we not be a centre for artistic  
enterprise--,i  mean for our own art and
 our own enterprise1 thought of by      
ourselves  and enjoyed by ourselves and 
carried through by ourselves8 ,we shall 
not  always enjoy each others'          
creations1 but the great point is to    
make our  own efforts4 ,of course all   
efforts require preparation and stimulus
 and  knowledge of what others are      
doing4    ,at the present               

time--interrupted for the moment by the 
war--a great  revolution in the art of  
painting is in progress throughout the  
world4  ,its centres are ,paris and     
,italy and ,london and ,munich1 and its 
origin  in the far east1 in ,china and  
,japan4 ,there are two sides as in every
  revolution1 the ,conservatives and the
 ,revolutionists4 ,our own frescoes  in 
a neighbouring room represent an early  
stage of the movement in  ,london4 ,why 
should we not know all about it--obtain 
loans of pictures  which illustrate its 
phases and its cross currents1 and      
compare these  with examples of the old 
style8    ,but pictures are only one    
phase of art1 and not the sort of art   
which  we ourselves can produce most    
easily4 ,there are music1 dancing1      
recitation1 literature1 carving and     
modelling1 and the various  decorative  
arts1 such as embroidery1 bookbinding1  
dress-making and  upholstery4 ,this     
list1 incomplete as it is1 tells us two 
great  truths--you cannot separate art  

and recreation1 and you cannot  separate
 art and business4 ,the list includes   
items which we consider  as amusements1 
and items which we think of as business4
 ,we began with  dancing and ended with 
upholstery4 ,make them all beautiful4   
,beautiful things have dignity4 ,enjoy  
the rhythm of your dancing and  admire  
the beauty of your embroidery or your   
bookbinding4 ,in whatever  you do1 have 
an ideal of excellence4 ,any separation 
between art and  work is not only an    
error1 but it is very bad business4 ,our
 brave  allies1 the ,french1 have made  
,paris the art centre of the world4     
,they  have built up and maintain their 
large and lucrative trade in the        
decorative products of ,france1 mainly  
by reason of three qualities  which they
 possess4 ,in the first place1 they     
enjoy art themselves1 and  reverence it4
 ,in the second place1 they have a      
tremendous power of hard  work4 ,and in 
the third place1 every ,frenchman1 and  
still more every  ,frenchwoman1 have    

within them an immense fund of common   
sense4 ,the  threefold secret is1 ,love 
of ,art1 ,industry1 and ,common ,sense4 
,to make available our industry and     
common sense in the trades where  they  
are wanted1 rigorous training in schools
 of design and technique  are necessary4
 ,we have such departments here4 ,but   
all such training of  you will be a     
failure unless you yourselves enjoy art 
and beauty as  a natural recreation4 ,a 
technical school of training is like a  
deep1  narrow well1 sunk with careful   
labour to tap the underground river of  
water which flows below the surface of  
our natures4 ,but your well will  be dry
 unless the bright warm sun has first   
drawn up the vapour from  the wide      
ocean1 and the free untrammelled breezes
 have carried the  clouds hither and    
thither1 until at length they break1 as 
it were by  chance over the distant     
hills and soak the land with their      
downpour4    ,what ,i have said about   
art is a parable which applies to other 

occupations and other studies4 ,it is   
more than a parable2 it is the  literal 
truth4 ,the love of art is the love of  
excellence1 it is the  enjoyment of the 
triumph of design over the shapeless    
products of  chance forces4 ,an         
engineer1 who is worth his salt1 loves  
the beauty of  his machines1 shown in   
their adjustment of parts and in their  
swift1  smooth motions4 ,he loves also  
the sense of foresight and of insight   
which knowledge can give him4 ,people   
say that machinery and commerce  are    
driving beauty out of the modern world4 
,i do not believe it4 ,a new  beauty is 
being added1 a more intellectual beauty1
 appealing to the  understanding as much
 as to the eye4    ,the wonder of       
,london ever takes the mind with fresh  
astonishment4 ,the  city possesses      
parks1 and palaces1 and cathedrals4 ,but
 no other parts  of it surpass in wonder
 its houses of business and its         
workshops and  its factories4    ,in the
 next few years the future of the world 

will be decided for  centuries to come4 
,the battles of this war are only the   
first part of  the contest between      
races1 and between the ways of life for 
which  those races stand4 ,we believe   
that ,england1 with its various peoples 
and communities scattered in islands and
 continents beyond the seas1  stands for
 ways of life infinitely precious1 the  
way of humanity1 the  way of liberty1   
the way of self-government1 the way of  
good order based  on toleration and     
kindly feeling1 the way of peaceful     
industry4 ,the  final decision in this  
struggle will be found in the workshops 
of the  world4 ,it lies in your hands4  
,statesmen and emperors will only       
register  the results which you have    
achieved4 ,your weapons will be skill1  
and  energy1 and knowledge4 ,you will   
require a sane understanding of your    
own rights1 and a sane understanding of 
the rights and the difficulties  of     
other classes4 ,the greatness of        
,england will be your greatness1 and    

its success your success4    ,the       
arsenal for war is at ,woolwich4 ,this  
,polytechnic ,institute is an  arsenal  
for peace1 where you can find the       
weapons for the conquest of  your lives4
                                        
,,chapter ,,iv                          
,,the ,,mathematical ,,curriculum       
7,presidential ,address to the ,london  
,branch of the ,mathematical            
,association1 #aiab7      ,,the         
situation in regard to education at the 
present time cannot find  its parallel  
without going back for some centuries to
 the break-up of  the medival traditions
 of learning4 ,then1 as now1 the        
traditional  intellectual outlook1      
despite the authority which it had      
justly  acquired from its notable       
triumphs1 had grown to be too narrow for
 the  interests of mankind4 ,the result 
of this shifting of human interest  was 
a demand for a parallel shifting of the 
basis of education1 so  as to fit the   
pupils for the ideas which later in life

 would in fact  occupy their minds4 ,any
 serious fundamental change in the      
intellectual  outlook of human society  
must necessarily be followed by an      
educational  revolution4 ,it may be     
delayed for a generation by vested      
interests or  by the passionate         
attachment of some leaders of thought to
 the cycle  of ideas within which they  
received their own mental stimulus at an
  impressionable age4 ,but the law is   
inexorable that education to be  living 
and effective must be directed to       
informing pupils with those  ideas1 and 
to creating for them those capacities   
which will enable them  to appreciate   
the current thought of their epoch4     
,there is no such thing as a successful 
system of education in a vacuum1  that  
is to say1 a system which is divorced   
from immediate contact with  the        
existing intellectual atmosphere4       
,education which is not modern  shares  
the fate of all organic things which are
 kept too long4    ,but the blessed word

 modern does not really solve our       
difficulties4  ,what we mean is1        
relevant to modern thought1 either in   
the ideas  imparted or in the aptitudes 
produced4 ,something found out only     
yesterday may not really be modern in   
this sense4 ,it may belong to  some     
bygone system of thought prevalent in a 
previous age1 or1 what  is very much    
more likely1 it may be too recondite4   
,when we demand  that education should  
be relevant to modern thought1 we are   
referring  to thoughts broadly spread   
throughout cultivated society4 ,it is   
this  question of the unfitness of      
recondite subjects for use in general   
education which ,i wish to make the     
keynote of my address this afternoon4   
,it is in fact rather a delicate subject
 for us mathematicians4  ,outsiders are 
apt to accuse our subject of being      
recondite4 ,let us  grasp the nettle at 
once and frankly admit that in general  
opinion it  is the very typical example 
of reconditeness4 ,by this word ,i do   

not  mean difficulty1 but that the ideas
 involved are of highly special         
application1 and rarely influence       
thought4    ,this liability to          
reconditeness is the characteristic evil
 which is  apt to destroy the utility of
 mathematics in liberal education4 ,so  
far as it clings to the educational use 
of the subject1 so far we  must         
acquiesce in a miserably low level of   
mathematical attainment  among          
cultivated people in general4 ,i yield  
to no one in my anxiety to  increase the
 educational scope of mathematics4 ,the 
way to achieve this  end is not by a    
mere blind demand for more mathematics4 
,we must face  the real difficulty which
 obstructs its extended use4    ,is the 
subject recondite8 ,now1 viewed as a    
whole1 ,i think it is4  ,securus judicat
 orbis terrarum--the general judgment of
 mankind  is sure4    ,the subject as it
 exists in the minds and in the books of
 students of  mathematics is recondite4 
,it proceeds by deducing innumerable    

special results from general ideas1 each
 result more recondite than the         
preceding4 ,it is not my task this      
afternoon to defend mathematics as a    
subject for profound study4 ,it can very
 well take care of itself4 ,what  ,i    
want to emphasise is1 that the very     
reasons which make this science  a      
delight to its students are reasons     
which obstruct its use as an            
educational instrument--namely1 the     
boundless wealth of deductions from  the
 interplay of general theorems1 their   
complication1 their apparent  remoteness
 from the ideas from which the argument 
started1 the variety  of methods1 and   
their purely abstract character which   
brings1 as its  gift1 eternal truth4    
,of course1 all these characteristics   
are of priceless value to  students2 for
 ages they have fascinated some of the  
keenest intellects4  ,my only remark is 
that1 except for a highly selected      
class1 they are  fatal in education4    
,the pupils are bewildered by a         

multiplicity of  detail1 without        
apparent relevance either to great ideas
 or to ordinary  thoughts4 ,the         
extension of this sort of training in   
the direction  of acquiring more detail 
is the last measure to be desired in the
  interests of education4    ,the       
conclusion at which we arrive is1 that  
mathematics1 if it is  to be used in    
general education1 must be subjected to 
a rigorous  process of selection and    
adaptation4 ,i do not mean1 what is of  
course  obvious1 that however much time 
we devote to the subject the average    
pupil will not get very far4 ,but that1 
however limited the progress1  certain  
characteristics of the subject1 natural 
at any stage1 must be  rigorously       
excluded4 ,the science as presented to  
young pupils must  lose its aspect of   
reconditeness4 ,it must1 on the face of 
it1 deal  directly and simply with a few
 general ideas of far-reaching          
importance4    ,now1 in this matter of  
the reform of mathematical instruction1 

the  present generation of teachers may 
take a very legitimate pride in  its    
achievements4 ,it has shown immense     
energy in reform1 and has  accomplished 
more than would have been thought       
possible in so short a  time4 ,it is not
 always recognised how difficult is the 
task of changing  a well-established    
curriculum entrenched behind public     
examinations4    ,but for all that1     
great progress has been made1 and1 to   
put the matter  at its lowest1 the old  
dead tradition has been broken up4 ,i   
want to  indicate this afternoon the    
guiding idea which should direct our    
efforts at reconstruction4 ,i have      
already summed it up in a phrase1       
namely1 we must aim at the elimination  
of reconditeness from the  educational  
use of the subject4    ,our courses of  
instruction should be planned to        
illustrate simply a  succession of ideas
 of obvious importance4 ,all pretty     
divagations  should be rigorously       
excluded4 ,the goal to be aimed at is   

that the  pupil should acquire          
familiarity with abstract thought1      
should realise  how it applies to       
particular concrete circumstances1 and  
should know  how to apply general       
methods to its logical investigation4   
,with this  educational ideal nothing   
can be worse than the aimless accretion 
of theorems in our text-books1 which    
acquire their position merely  because  
the children can be made to learn them  
and examiners can set  neat questions on
 them4 ,the bookwork to be learnt should
 all be very  important as illustrating 
ideas4 ,the examples set--and let there 
be as many examples as teachers find    
necessary--should be direct             
illustrations of the theorems1 either by
 way of abstract particular  cases or by
 way of application to concrete         
phenomena4 ,here it is worth  remarking 
that it is quite useless to simplify the
 bookwork1 if the  examples set in      
examinations in fact require an extended
 knowledge of  recondite details4 ,there

 is a mistaken idea that problems test  
ability  and genius1 and that bookwork  
tests cram4 ,this is not my experience4 
,only boys who have been specially      
crammed for scholarships can ever  do a 
problem paper successfully4 ,bookwork   
properly set1 not in mere  snippets     
according to the usual bad plan1 is a   
far better test of  ability1 provided   
that it is supplemented by direct       
examples4 ,but this  is a digression on 
the bad influence of examinations on    
teaching4    ,the main ideas which lie  
at the base of mathematics are not at   
all  recondite4 ,they are abstract4 ,but
 one of the main objects of the         
inclusion of mathematics in a liberal   
education is to train the pupils  to    
handle abstract ideas4 ,the science     
constitutes the first large  group of   
abstract ideas which naturally occur to 
the mind in any  precise form4 ,for the 
purposes of education1 mathematics      
consists of  the relations of number1   
the relations of quantity1 and the      

relations  of space4 ,this is not a     
general definition of mathematics1      
which1 in  my opinion1 is a much more   
general science4 ,but we are now        
discussing  the use of mathematics in   
education4 ,these three groups of       
relations1  concerning number1 quantity1
 and space1 are interconnected4    ,now1
 in education we proceed from the       
particular to the general4              
,accordingly1 children should be taught 
the use of these ideas by  practice     
among simple examples4 ,my point is     
this3 ,the goal should be1  not an      
aimless accumulation of special         
mathematical theorems1 but the  final   
recognition that the preceding years of 
work have illustrated  those relations  
of number1 and of quantity1 and of      
space1 which are of  fundamental        
importance4 ,such a training should lie 
at the base of all  philosophical       
thought4 ,in fact elementary mathematics
 rightly conceived  would give just that
 philosophical discipline of which the  

ordinary  mind is capable4 ,but what at 
all costs we ought to avoid1 is the     
pointless accumulation of details4 ,as  
many examples as you like2 let  the     
children work at them for terms1 or for 
years4 ,but these examples  should be   
direct illustrations of the main ideas4 
,in this way1 and  this only1 can the   
fatal reconditeness be avoided4    ,i am
 not now speaking in particular of those
 who are to be professional             
mathematicians1 or of those who for     
professional reasons require a          
knowledge of certain mathematical       
details4 ,we are considering the        
liberal education of all students1      
including these two classes4 ,this      
general use of mathematics should be the
 simple study of a few general  truths1 
well illustrated by practical examples4 
,this study should  be conceived by     
itself1 and completely separated in idea
 from the  professional study mentioned 
above1 for which it would make a most   
excellent preparation4 ,its final stage 

should be the recognition of  the       
general truths which the work done has  
illustrated4 ,as far as ,i  can make    
out1 at present the final stage is the  
proof of some property  of circles      
connected with triangles4 ,such         
properties are immensely  interesting to
 mathematicians4 ,but are they not      
rather recondite1  and what is the      
precise relation of such theorems to the
 ideal of  a liberal education8 ,the end
 of all the grammatical studies of the  
student in classics is to read ,virgil  
and ,horace--the greatest thoughts  of  
the greatest men4 ,are we content1 when 
pleading for the adequate               
representation in education of our own  
science1 to say that the end of  a      
mathematical training is that the       
student should know the properties  of  
the nine-point circle8 ,i ask you       
frankly1 is it not rather a come  down8 
,this generation of mathematical        
teachers has done so much strenuous     
work in the way of reorganising         

mathematical instruction that there is  
no need to despair of its being able to 
elaborate a curriculum which  shall     
leave in the minds of the pupils        
something even nobler than the          
ambiguous case4    ,let us think how    
this final review1 closing the          
elementary course1  might be conducted  
for the more intelligent pupils4 ,partly
 no doubt  it requires a general        
oversight of the whole work done1       
considered  without undue detail so as  
to emphasise the general ideas used1 and
  their possibilities of importance when
 subjected to further study4 ,also  the 
analytical and geometrical ideas find   
immediate application in the  physical  
laboratory where a course of simple     
experimental mechanics  should have been
 worked through4 ,here the point of view
 is twofold1 the  physical ideas and the
 mathematical ideas illustrate each     
other4    ,the mathematical ideas are   
essential to the precise formulation of 
the  mechanical laws4 ,the idea of a    

precise law of nature1 the extent to    
which such laws are in fact verified in 
our experience1 and the rle of  abstract
 thought in their formulation1 then     
become practically apparent  to the     
pupil4 ,the whole topic of course       
requires detailed development  with full
 particular illustration1 and is not    
suggested as requiring  merely a few    
bare abstract statements4    ,it would1 
however1 be a grave error to put too    
much emphasis on the  mere process of   
direct explanation of the previous work 
by way of  final review4 ,my point is1  
that the latter end of the course should
  be so selected that in fact the       
general ideas underlying all the        
previous mathematical work should be    
brought into prominence4 ,this may  well
 be done by apparently entering on a new
 subject4 ,for example1  the ideas of   
quantity and the ideas of number are    
fundamental to  all precise thought4 ,in
 the previous stages they will not have 
been  sharply separated2 and children   

are1 rightly enough1 pushed on to       
algebra without too much bother and     
quantity4 ,but the more intelligent     
among them at the end of their          
curriculum would gain immensely by a    
careful consideration of those          
fundamental properties of quantity in   
general which lead to the introduction  
of numerical measurement4 ,this  is a   
topic which also has the advantage that 
the necessary books are  actually to    
hand4 ,euclid's fifth book is regarded  
by those qualified  to judge as one of  
the triumphs of ,greek mathematics4 ,it 
deals with  this very point4 ,nothing   
can be more characteristic of the       
hopelessly  illiberal character of the  
traditional mathematical education than 
the  fact that this book has always been
 omitted4 ,it deals with ideas1  and    
therefore was ostracised4 ,of course a  
careful selection of the  more important
 propositions and a careful revision of 
the argument are  required4 ,this also  
is to hand in the publications of my    

immediate  predecessor in the office of 
president1 ,prof4 ,hill4 ,furthermore1  
in ,sir  ,t4 ,l4 ,heath's complete      
edition of ,euclid1 there is a full     
commentary  embodying most of what has  
been said and thought on the point4     
,thus  it is perfectly easy for teachers
 to inform themselves generally  on the 
topic4 ,the whole book would not be     
wanted1 but just the few  propositions  
which embody the fundamental ideas4 ,the
 subject is not  fit for backward       
pupils2 but certainly it could be made  
interesting to  the more advanced class4
 ,there would be great scope for        
interesting  discussion as to the nature
 of quantity1 and the tests which we    
should  apply to ascertain when we are  
dealing with quantities4 ,the work      
would not be at all in the air1 but     
would be illustrated at every  stage by 
reference to actual examples of cases   
where the quantitative  character is    
absent1 or obscure1 or doubtful1 or     
evident4 ,temperature1  heat1           

electricity1 pleasure and pain1 mass and
 distance could all be  considered4     
,another idea which requires            
illustration is that of functionality4  
,a function in analysis is the          
counterpart of a law in the physical    
universe1 and of a curve in geometry4   
,children have studied the  relations of
 functions to curves from the first     
beginning of their  study of algebra1   
namely in drawing graphs4 ,of recent    
years there has  been a great reform in 
respect to graphs4 ,but at its present  
stage it  has either gone too far or not
 far enough4 ,it is not enough merely   
to draw a graph4 ,the idea behind the   
graph--like the man behind the  gun--is 
essential in order to make it effective4
 ,at present there is  some tendency    
merely to set the children to draw      
curves1 and there to  leave the whole   
question4    ,in the study of simple    
algebraic functions and of              
trigonometrical  functions we are       
initiating the study of the precise     

expression of  physical laws4 ,curves   
are another way of representing these   
laws4 ,the  simple fundamental          
laws--such as the inverse square and the
 direct  distance--should be passed     
under review1 and the applications of   
the  simple functions to express        
important concrete cases of physical    
laws  considered4 ,i cannot help        
thinking that the final review of this  
topic  might well take the form of a    
study of some of the main ideas of the  
differential calculus applied to simple 
curves4 ,there is nothing  particularly 
difficult about the conception of a rate
 of change2 and  the differentiation of 
a few powers of x1 such as x#b1  x#c1   
etc41 could easily be effected2 perhaps 
by the aid of  geometry even sin x and  
cos x could be differentiated4  ,if we  
once abandon our fatal habit of cramming
 the children with  theorems which they 
do not understand1 and will never use1  
there will  be plenty of time to        
concentrate their attention on really   

important  topics4 ,we can give them    
familiarity with conceptions which      
really  influence thought4    ,before   
leaving this topic of physical laws and 
mathematical functions1  there are other
 points to be noticed4 ,the fact that   
the precise law is  never really        
verified by observation in its full     
precision is capable  of easy           
illustration and of affording excellent 
examples4 ,again1  statistical laws1    
namely laws which are only satisfied on 
the average  by large numbers1 can      
easily be studied and illustrated4 ,in  
fact a  slight study of statistical     
methods and their application to social 
phenomena affords one of the simplest   
examples of the application of          
algebraic ideas4    ,another way in     
which the students' ideas can be        
generalised is by the  use of the       
,history of ,mathematics1 conceived not 
as a mere assemblage  of the dates and  
names of men1 but as an exposition of   
the general  current of thought which   

occasioned the subjects to be objects of
  interest at the time of their first   
elaboration4 ,the use of the ,history   
of ,mathematics is to be considered at a
 later stage of our proceedings  this   
afternoon4 ,accordingly ,i merely draw  
attention to it now1 to  point out that 
perhaps it is the very subject which may
 best obtain the  results for which ,i  
am pleading4    ,we have indicated two  
main topics1 namely general ideas of    
quantity  and of laws of nature1 which  
should be an object of study in the     
mathematical curriculum of a liberal    
education4 ,but there is another  side  
to mathematics which must not be        
overlooked4 ,it is the chief  instrument
 for discipline in logical method4      
,now1 what is logical method1 and how   
can any one be trained in it8           
,logical method is more than the mere   
knowledge of valid types of  reasoning  
and practice in the concentration of    
mind necessary to  follow them4 ,if it  
were only this1 it would still be very  

important2  for the human mind was not  
evolved in the bygone ages for the sake 
of  reasoning1 but merely to enable     
mankind with more art to hunt between   
meals for fresh food supplies4          
,accordingly few people can follow close
  reasoning without considerable        
practice4    ,more than this is wanted  
to make a good reasoner1 or even to     
enlighten  ordinary people with         
knowledge of what constitutes the       
essence of the  art4 ,the art of        
reasoning consists in getting hold of   
the subject at  the right end1 of       
seizing on the few general ideas which  
illuminate  the whole1 and of           
persistently marshalling all subsidiary 
facts round  them4 ,nobody can be a good
 reasoner unless by constant practice he
  has realised the importance of getting
 hold of the big ideas and of  hanging  
on to them like grim death4 ,for this   
sort of training geometry  is1 ,i think1
 better than algebra4 ,the field of     
thought of algebra is  rather obscure1  

whereas space is an obvious insistent   
thing evident to  all4 ,then the process
 of simplification1 or abstraction1 by  
which all  irrelevant properties of     
matter1 such as colour1 taste1 and      
weight1 are  put aside is an education  
in itself4 ,again1 the definitions and  
the  propositions assumed without proof 
illustrate the necessity of forming     
clear notions of the fundamental facts  
of the subject-matter and of the        
relations between them4 ,all this       
belongs to the mere prolegomena of the  
subject4 ,when we come to its           
development1 its excellence increases4  
,the  learner is not initially          
confronted with any symbolism which     
bothers  the memory by its rules1       
however simple they may be4 ,also1 from 
the  very beginning the reasoning1 if   
properly conducted1 is dominated by     
well-marked ideas which guide each stage
 of development4 ,accordingly  the      
essence of logical method receives      
immediate exemplification4    ,let us   

now put aside for the moment the        
limitations introduced by  the dullness 
of average pupils and the pressure on   
time due to other  subjects1 and        
consider what geometry has to offer in  
the way of a  liberal education4 ,i will
 indicate some stages in the subject1   
without meaning that necessarily they   
are to be studied in this  exclusive    
order4 ,the first stage is the study of 
congruence4  ,our perception of         
congruence is in practice dependent on  
our judgments  of the invariability of  
the intrinsic properties of bodies when 
their  external circumstances are       
varying4 ,but however it arises1        
congruence  is in essence the           
correlation of two regions of space1    
point by point1  so that all homologous 
distances and all homologous angles are 
equal4  ,it is to be noticed that the   
definition of the equality of lengths   
and  angles is their congruence1 and all
 tests of equality1 such as the use  of 
the yard measure1 are merely devices for

 making immediate judgments  of         
congruence easy4 ,i make these remarks  
to suggest that apart from the          
reasoning connected with it1 congruence1
 both as an example of a larger  and    
very far-reaching idea and also for its 
own sake1 is well worthy  of attentive  
consideration4 ,the propositions        
concerning it elucidate  the elementary 
properties of the triangle1 the         
parallelogram1 and the  circle1 and of  
the relations of two planes to each     
other4 ,it is very  desirable to        
restrict the proved propositions of this
 part within the  narrowest bounds1     
partly by assuming redundant axiomatic  
propositions1  and partly by introducing
 only those propositions of absolutely  
fundamental importance4    ,the second  
stage is the study of similarity4 ,this 
can be reduced to  three or four        
fundamental propositions4 ,similarity is
 an enlargement of  the idea of         
congruence1 and1 like that idea1 is     
another example of a  one-to-one        

correlation of points of spaces4 ,any   
extension of study of  this subject     
might well be in the direction of the   
investigation of one  or two simple     
properties of similar and similarly     
situated rectilinear  figures4 ,the     
whole subject receives its immediate    
applications in plans  and maps4 ,it is 
important1 however1 to remember that    
trigonometry is  really the method by   
which the main theorems are made        
available for use4    ,the third stage  
is the study of the elements of         
trigonometry4 ,this is  the study of the
 periodicity introduced by rotation and 
of properties  preserved in a           
correlation of similar figures4 ,here   
for the first time  we introduce a      
slight use of the algebraic analysis    
founded on the  study of number and     
quantity4 ,the importance of the        
periodic character  of the functions    
requires full illustration4 ,the        
simplest properties of  the functions   
are the only ones required for the      

solution of triangles1  and the         
consequent applications to surveying4   
,the wealth of formul1  often important 
in themselves1 but entirely useless for 
this type of  study1 which crowd our    
books should be rigorously excluded1    
except  so far as they are capable of   
being proved by the pupils as direct    
examples of the bookwork4    ,this      
question of the exclusion of formul is  
best illustrated by  considering this   
example of ,trigonometry1 though of     
course ,i may well  have hit on an      
unfortunate case in which my judgment is
 at fault4 ,a  great part of the        
educational advantage of the subject can
 be obtained  by confining study to     
,trigonometry of one angle and by       
exclusion of the  addition formul for   
the sine and cosine of the sum of two   
angles4 ,the  functions can be graphed1 
and the solution of triangles effected4 
,thus  the aspects of the science as    
7#a7 embodying analytically the         
immediate  results of some of the       

theorems deduced from congruence and    
similarity1  7#b7 as a solution of the  
main problem of surveying1 7#c7 as a    
study of  the fundamental functions     
required to express periodicity and wave
  motion1 will all be impressed on the  
pupils' minds both by bookwork and      
example4    ,if it be desired to extend 
this course1 the addition formul should 
be added4 ,but great care should be     
taken to exclude specialising  the      
pupils in the wealth of formul which    
comes in their train4 ,by  exclude is   
meant that the pupils should not have   
spent time or energy  in acquiring any  
facility in their deduction4 ,the       
teacher may find  it interesting to work
 a few such examples before a class4    
,but such  results are not among those  
which learners need retain4 ,also1 ,i   
would  exclude the whole subject of     
circumscribed and inscribed circles both
  from ,trigonometry and from the       
previous geometrical courses4 ,it is    
all very pretty1 but ,i do not          

understand what its function is in an   
elementary non-professional curriculum4 
,accordingly1 the actual bookwork of the
 subject is reduced to very  manageable 
proportions4 ,i was told the other day  
of an ,american college  where the      
students are expected to know by heart  
ninety formul or  results in            
,trigonometry alone4 ,we are not quite  
so bad as that4 ,in  fact1 in           
,trigonometry we have nearly approached 
the ideal here sketched  out as far as  
our elementary courses are concerned4   
,the fourth stage introduces ,analytical
 ,geometry4 ,the study of graphs in     
algebra has already employed the        
fundamental notions1 and all that is    
now required is a rigorously pruned     
course on the straight line1 the        
circle1 and the three types of conic    
sections1 defined by the forms of  their
 equations4 ,at this point there are two
 remarks to be made4 ,it is  often      
desirable to give our pupils            
mathematical information which we do    

not prove4 ,for example1 in co-ordinate 
geometry1 the reduction of the  general 
equation of the second degree is        
probably beyond the capacities  of most 
of the type of students whom we are     
considering4 ,but that need  not prevent
 us from explaining the fundamental     
position of conics1 as  exhausting the  
possible types of such curves4    ,the  
second remark is to advocate the entire 
sweeping away of  geometrical conics as 
a separate subject4 ,naturally1 on      
suitable  occasions the analysis of     
analytical geometry will be lightened by
  the use of direct deduction from some 
simple figure4 ,but geometrical  conics1
 as developed from the definition of a  
conic section by the  focus and         
directrix property1 suffers from glaring
 defects4 ,it is  hopelessly recondite4 
,the fundamental definition of a conic1 
,,sp  e   ,,pm1 usual in this subject at
 this stage1 is thoroughly bad4 ,it is  
very recondite1 and has no obvious      
importance4 ,why should such curves  be 

studied at all1 any more than those     
defined by an indefinite number  of     
other formul8 ,but when we have         
commenced the study of the ,cartesian   
methods1 the equations of the first and 
second degrees are naturally  the first 
things to think about4    ,in this ideal
 course of ,geometry1 the fifth stage is
 occupied with  the elements of         
,projective ,geometry4 ,the general     
ideas of cross ratio  and of projection 
are here fundamental4 ,projection is yet
 a more  general instance of that       
one-to-one correlation which we have    
already  considered under congruence and
 similarity4 ,here again we must avoid  
the danger of being led into a          
bewildering wealth of detail4    ,the   
intellectual idea which projective      
geometry is to illustrate is  the       
importance in reasoning of the          
correlation of all cases which  can be  
proved to possess in common certain     
identical properties4 ,the  preservation
 of the projective properties in        

projection is the one  important        
educational idea of the subject4 ,cross 
ratio only enters  as the fundamental   
metrical property which is preserved4   
,the few  propositions considered are   
selected to illustrate the two allied   
processes which are made possible by    
this procedure4 ,one is proof  by       
simplification4 ,here the simplification
 is psychological and not  logical--for 
the general case is logically the       
simplest4 ,what is meant  is3 ,proof by 
considering the case which is in fact   
the most familiar to  us1 or the easiest
 to think about4 ,the other procedure is
 the deduction  of particular cases from
 known general truths1 as soon as we    
have a  means of discovering such cases 
or a criterion for testing them4    ,the
 projective definition of conic sections
 and the identity of the  results       
obtained with the curves derived from   
the general equation  of the second     
degree are capable of simple exposition1
 but lie on  the border-line of the     

subject4 ,it is the sort of topic on    
which  information can be given1 and the
 proofs suppressed4    ,the course of   
geometry as here conceived in its       
complete ideal--and  ideals can never be
 realised--is not a long one4 ,the      
actual amount  of mathematical deduction
 at each stage in the form of bookwork  
is  very slight4 ,but much more         
explanation would be given1 the         
importance  of each proposition being   
illustrated by examples1 either worked  
out  or for students to work1 so        
selected as to indicate the fields of   
thought to which it applies4 ,by such a 
course the student would gain an        
analysis of the leading properties of   
space1 and of the chief methods  by     
which they are investigated4    ,the    
study of the elements of mathematics1   
conceived in this spirit1  would        
constitute a training in logical method 
together with an  acquisition of the    
precise ideas which lie at the base of  
the  scientific and philosophical       

investigations of the universe4 ,would  
it  be easy to continue the excellent   
reforms in mathematical instruction     
which this generation has already       
achieved1 so as to include in the       
curriculum this wider and more          
philosophic spirit8 ,frankly1 ,i think  
that this result would be very hard to  
achieve as the result of  single        
individual efforts4 ,for reasons which  
,i have already briefly  indicated1 all 
reforms in education are very difficult 
to effect4 ,but  the continued pressure 
of combined effort1 provided that the   
ideal is  really present in the minds of
 the mass of teachers1 can do much1 and 
effects in the end surprising           
modification4 ,gradually the requisite  
books get written1 still more gradually 
the examinations are reformed  so as to 
give weight to the less technical       
aspects of the subject1 and  then all   
recent experience has shown that the    
majority of teachers are  only too ready
 to welcome any practicable means of    

rescuing the subject  from the reproach 
of being a mechanical discipline4       
,,chapter ,v       ,,the ,,principles   
,,of ,,mathematics ,,in ,,relation ,,to 
,,elementary ,,teaching                 
7,international ,congress of            
,mathematicians1 ,cambridge1 ,august1   
#aiab7      ,,we are concerned not with 
the advanced teaching of a few          
specialist  mathematical students1 but  
with the mathematical education of the  
majority of boys in our secondary       
schools4 ,again these boys must be      
divided into two sections3 one section  
consists of those who desire to         
restrict their mathematical education2  
the other section is made up of  those  
who will require some mathematical      
training for their subsequent           
professional careers1 either in the form
 of definite mathematical  results or in
 the form of mathematically trained     
minds4    ,i shall call the latter of   
these two sections the mathematical     
section1  and the former the            

non-mathematical section4 ,but ,i must  
repeat that by  the mathematical section
 is meant that large number of boys who 
desire  to learn more than the minimum  
amount of mathematics4 ,furthermore1    
most  of my remarks about these sections
 of boys apply also to elementary       
classes of our ,university students4    
,the subject of this paper is the       
investigation of the place which  should
 be occupied by modern investigations   
respecting mathematical  principles in  
the education of both of these sections 
of school-boys4    ,to find a foothold  
from which to start the inquiry1 let us 
ask why the  non-mathematical section   
should be taught any mathematics at all 
beyond  the barest elements of          
arithmetic4 ,what are the qualities of  
mind which  a mathematical training is  
designed to produce when it is employed 
as  an element in a liberal education8  
,my answer1 which applies equally to    
both sections of students1 is  that     
there are two allied forms of mental    

discipline which should be  acquired by 
a well-designed mathematical course4    
,these two forms though  closely allied 
are perfectly distinct4    ,the first   
form of discipline is not in its essence
 logical at all4  ,it is the power of   
clearly grasping abstract ideas1 and of 
relating  them to particular            
circumstances4 ,in other words1 the     
first use of  mathematics is to         
strengthen the power of abstract        
thought4 ,i repeat  that in its essence 
this has nothing to do with logic1      
though as a  matter of fact a logical   
discipline is the best method of        
producing the  desired effect4 ,it is   
not the philosophical theory of abstract
 ideas  which is to be acquired1 but the
 habit and the power of using them4     
,there is one and only way of acquiring 
the habit and the power of using        
anything1 that is by the simple         
common-place method of habitually using 
it4 ,there is no other short cut4 ,if in
 education we desire to produce  a      

certain conformation of mind1 we must   
day by day1 and year by year1  accustom 
the students' minds to grow into the    
desired structural shape4  ,thus to     
teach the power of grasping abstract    
ideas and the habit of  using them1 we  
must select a group of such ideas1 which
 are important  and are also easy to    
think about because they are clear and  
definite4    ,the fundamental           
mathematical truths concerning geometry1
 ratio1  quantity1 and number1 satisfy  
these conditions as do no others4       
,hence1  the fundamental universal      
position held by mathematics as an      
element of  a liberal education4    ,but
 what are the fundamental mathematical  
truths concerning geometry1  quantity   
and number8    ,at this point we come to
 the great question of the relation     
between  the modern science of the      
principles of mathematics and a         
mathematical  education4    ,my answer  
to the question as to these fundamental 
mathematical truths  is1 that in any    

absolute sense there are none4 ,there is
 no unique  small body of independent   
primitive unproved propositions which   
are  the necessary starting points of   
all mathematical reasoning on these     
subjects4 ,in mathematical reasoning the
 only absolute necessary                
pre-suppositions are those which make   
logical deduction possible4  ,between   
these absolute logical truths and       
so-called fundamental truths  concerning
 geometry1 quantity and number1 there is
 a whole new world of  mathematical     
subjects concerning the logic of        
propositions1 of classes1  and of       
relations4    ,but this subject is too  
abstract to form an elementary training 
ground  in the difficult art of abstract
 thought4    ,it is for this reason that
 we have to make a compromise and start 
from  such obvious general ideas as     
naturally occur to all men when they    
perceive objects with their senses4     
,in geometry1 the ideas elaborated by   
the ,greeks and presented by ,euclid    

are1 roughly speaking1 those adapted for
 our purpose1 namely1 ideas  of volumes1
 surfaces1 lines1 of straightness and of
 curvature1 of  intersection and of     
congruence1 of greater and less1 of     
similarity1  shape1 and scale4 ,in fact1
 we use in education those general ideas
 of  spatial properties which must be   
habitually present in the mind of any   
one who is to observe the world of      
phenomena with understanding4    ,thus  
we come back to ,plato's opinion that   
for a liberal education1  geometry1 as  
he knew it1 is the queen of sciences4   
,in addition to geometry1 there remain  
the ideas of quantity1 ratio1  and of   
number4 ,this in practice means1        
elementary algebra4 ,here the  prominent
 ideas are those of any number1 in other
 words1 the use  of the familiar x1 y1  
z1 and of the dependence of variables on
  each other1 or otherwise1 the idea of 
functionality4 ,all this is to be       
gradually acquired by the continual use 
of the very simplest functions  which we

 can devise3 of linear functions1       
graphically represented  by straight    
lines2 of quadratic functions1          
graphically represented  by parabolas2  
and of those simple implicit functions1 
graphically  represented by conic       
sections4 ,thence1 with good fortune and
 a willing  class1 we can advance to the
 ideas of rates of increase1 still      
confining ourselves to the simplest     
possible cases4    ,i wish here         
emphatically to remind you that both in 
geometry and in  algebra a clear grasp  
of these general ideas is not what the  
pupil  starts from1 it is the goal at   
which he is to arrive4 ,the method of   
progression is continual practice in the
 consideration of the simplest          
particular cases1 and the goal is not   
philosophical analysis but the  power of
 use4    ,but how is he to practise     
himself in their use8 ,he cannot simply 
sit  down and think of the relation y  x
  #a1 he must employ  it in some simple 
obvious way4    ,this brings us to the  

second power of mind which is to be     
produced by  a mathematical training1   
namely1 the power of logical reasoning4 
,here  again1 it is not the knowledge of
 the philosophy of logic which it is    
essential to teach1 but the habit of    
thinking logically4 ,by logic1 ,i  mean 
deductive logic4    ,deductive logic is 
the science of certain relations1 such  
as  implication1 etc41 between general  
ideas4 ,when logic begins1 definite     
particular individual things have been  
banished4 ,i cannot relate  logically   
this thing to that thing1 for example   
this pen to that pen1  except by the    
indirect way of relating some general   
idea which applies  to this pen to some 
general idea which applies to that pen4 
,and the  individualities of the two    
pens are quite irrelevant to the logical
  process4 ,this process is entirely    
concerned with the two general ideas4   
,thus the practice of logic is a certain
 way of employing the mind  in the      
consideration of such ideas2 and an     

elementary mathematical  training is in 
fact nothing else but the logical use of
 the general  ideas respecting geometry 
and algebra which we have enumerated    
above4  ,it has therefore1 as ,i started
 this paper by stating1 a double        
advantage4 ,it makes the mind capable of
 abstract thought1 and it  achieves this
 object by training the mind in the most
 important kind of  abstract thought1   
namely1 deductive logic4    ,i may      
remind you that other choices of a type 
of abstract thought  might be made4 ,we 
might train the children to contemplate 
directly the  beauty of abstract moral  
ideas1 in the hope of making them       
religious  mystics4 ,the general        
practice of education has decided in    
favour of  logic1 as exemplified in     
elementary mathematics4    ,we have now 
to answer the further question1 what is 
the rle of  logical precision in the    
teaching of mathematics8 ,our general   
answer  to the implied question is      
obvious3 logical precision is one of the

  two objects of the teaching of        
mathematics1 and it is the only weapon  
by which the teaching of mathematics can
 achieve its other object4 ,to  teach   
mathematics is to teach logical         
precision4 ,a mathematical teacher  who 
has not taught that has taught nothing4 
,but having stated this thesis in this  
unqualified way1 its meaning must  be   
carefully explained2 for otherwise its  
real bearing on the problem  will be    
entirely misunderstood4    ,logical     
precision is the faculty to be acquired4
 ,it is the quality of  mind which it is
 the object of the teaching to impart4  
,thus the habit  of reading great       
literature is the goal at which a       
literary education  aims4 ,but we do not
 expect a child to start its first      
lesson by reading  for itself           
,shakespeare4 ,we recognise that reading
 is impossible till  the pupil has      
learnt its alphabet and can spell1 and  
then we start it  with books of one     
syllable4    ,in the same way1 a        

mathematical education should grow in   
logical  precision4 ,it is folly to     
expect the same careful logical analysis
 at  the commencement of the training as
 would be appropriate at the end4  ,it  
is an entire misconception of my thesis 
to construe it as meaning  that a       
mathematical training should assume in  
the pupil a power of  concentrated      
logical thought4 ,my thesis is in fact  
the exact opposite1  namely1 that this  
power cannot be assumed1 and has got to 
be acquired1  and that a mathematical   
training is nothing else than the       
process of  acquiring it4 ,my whole     
groundwork of assumption is that this   
power does  not initially exist in a    
fully developed state4 ,of course like  
every  other power which is acquired1 it
 must be developed gradually4    ,the   
various stages of development must be   
guided by the judgment and  the genius  
of the teacher4 ,but what is essential  
is1 that the teacher  should keep       
clearly in his mind that it is just this

 power of logical  precise reasoning    
which is the whole object of his        
efforts4 ,if his  pupils have in any    
measure gained this1 they have gained   
all4    ,we have not yet1 however1 fully
 considered this part of our subject4   
,logical precision is the full          
realisation of the steps of the         
argument4  ,but what are the steps of   
the argument8 ,the full statement of all
 the  steps is far too elaborate and    
difficult an operation to be introduced 
into the mathematical reasoning of an   
educational curriculum4 ,such a         
statement involves the introduction of  
abstract logical ideas which  are very  
difficult to grasp1 because there is so 
rarely any need to  make them explicit  
in ordinary thought4 ,they are therefore
 not a fit  subject-ground for an       
elementary education4    ,i do not think
 that it is possible to draw any        
theoretical line  between those logical 
steps which form a theoretically full   
logical  investigation1 and those which 

are full enough for most practical      
purposes1 including that of education4  
,the question is one of  psychology1 to 
be solved by a process of experiment4   
,the object to be  attained is to gain  
that amount of logical alertness which  
will enable  its possessors to detect   
fallacy and to know the types of sound  
logical  deduction4 ,the objects of     
going further are partly philosophical1 
and  also partly to lay bare abstract   
ideas whose investigation is in itself  
important4 ,but both these objects are  
foreign to education4    ,my opinion is1
 that1 on the whole1 the type of logical
 precision handed  down to us by the    
,greek mathematicians is1 roughly       
speaking1 what we  want4 ,in geometry1  
this means the sort of precision which  
we find in  ,euclid4 ,i do not mean that
 we should use his famous ,elements as  
a text-book1 nor that here and there a  
certain compression in his mode  of     
exposition is not advisable4 ,all this  
is mere detail4 ,what ,i do mean  is1   

that the sort of logical transition     
which he made explicit1 we  should make 
explicit1 and that the sort of          
transition which he omits1  we should   
omit4    ,i doubt1 however1 whether it  
is desirable to plunge the student into 
the full rigour of euclidean geometry   
without some mitigation4 ,it is  for    
this reason that the modern habit1 at   
least in ,england1 of laying  great     
stress in the initial stages on training
 the pupil in simple  constructions from
 numerical data is to be praised4 ,it   
means that after  a slight amount of    
reasoning on the euclidean basis of     
accuracy1 the  mind of the learner is   
relieved by doing the things in various 
special  cases1 and noting by rough     
measurements that the desired results   
are  actually attained4 ,it is          
important1 however1 that the            
measurements  be not mistaken for the   
proofs4 ,their object is to make the    
beginner  apprehend what the abstract   
ideas really mean4    ,again in algebra1

 the notation and the practical use of  
the symbols  should be acquired in the  
simplest cases1 and the more theoretical
  treatment of the symbolism reserved to
 a suitable later stage4 ,my  rule would
 be initially to learn the meaning of   
the ideas by a crude  practice in simple
 ways1 and to refine the logical        
procedure in  preparation for an advance
 to greater generality4 ,in fact the    
thesis of  my paper can be put in       
another way thus1 the object of a       
mathematical  education is1 to acquire  
the powers of analysis1 of              
generalisation1  and of reasoning4 ,the 
two processes of analysis and           
generalisation were  in my previous     
statement put together as the power of  
grasping abstract  ideas4    ,but in    
order to analyse and to generalise1 we  
must commence with the  crude material  
of ideas which are to be analysed and   
generalised4  ,accordingly it is a      
positive error in education to start    
with the  ultimate products of this     

process1 namely the ideas in their      
refined  analysed and generalised forms4
 ,we are thereby skipping an important  
part of the training1 which is to take  
the ideas as they actually exist  in the
 child's mind1 and to exercise the child
 in the difficult art of  civilising    
them and clothing them4    ,the         
schoolmaster is in fact a missionary1   
the savages are the ideas in  the       
child's mind2 and the missionary shirks 
his main task if he refuse  to risk his 
body among the cannibals4    ,at this   
point ,i should like to turn your       
attention to those pupils  forming the  
mathematical section4 ,there is an idea1
 widely prevalent1  that it is possible 
to teach mathematics of a relatively    
advanced  type--such as differential    
calculus1 for instance--in a way useful 
to  physicists and engineers without any
 attention to its logic or its  theory4 
,this seems to me to be a profound      
mistake4 ,it implies that a merely      
mechanical knowledge without            

understanding of ways of arriving at    
mathematical results is useful in       
applied science4 ,it is of no use       
whatever4 ,the results themselves can   
all be found stated in the  appropriate 
pocket books and in other elementary    
works of reference4 ,no  one when       
applying a result need bother himself as
 to why it is true4 ,he  accepts it and 
applies it4 ,what is of supreme         
importance in physics and  in           
engineering is a mathematically trained 
mind1 and such a mind can  only be      
acquired by a proper mathematical       
discipline4    ,i fully admit that the  
proper way to start such a subject as   
the  differential calculus is to plunge 
quickly into the use of the notation  in
 a few absurdly simple cases1 with a    
crude explanation of the idea  of rates 
of increase4 ,the notation as thus known
 can then be used by  the lecturers in  
the ,physical and ,engineering          
,laboratories4 ,but the  mathematical   
training of the applied scientists      

consists in making  these ideas precise 
and the proofs accurate4    ,i hope that
 the thesis of this paper respecting the
 position of logical  precision in the  
teaching of mathematics has been        
rendered plain4 ,the  habit of logical  
precision with its necessary            
concentration of thought  upon abstract 
ideas is not wholly possible in the     
initial stages of  learning4 ,it is the 
ideal at which the teacher should aim4  
,also logical  precision1 in the sense  
of logical explicitness1 is not an      
absolute  thing3 it is an affair of more
 or less4 ,accordingly the quantity of  
explicitness to be introduced at each   
stage of progress must depend  upon the 
practical judgment of the teacher4      
,lastly1 in a sense1 the  instructed    
mind is less explicit2 for it travels   
more quickly over a  well-remembered    
path1 and may save the trouble of       
putting into words  trains of thought   
which are very obvious to it4 ,but on   
the other  hand it atones for this      

rapidity by a concentration on every    
subtle  point where a fallacy can lurk4 
,the habit of logical precision is the  
instinct for the subtle difficulty4     
,,chapter ,,vi                          
,,the ,,organisation ,,of ,,thought     
7,presidential ,address to ,section ,a1 
,british ,association1 ,newcastle1      
#aiaf7      ,,the subject of this       
address is the organisation of thought1 
a topic  evidently capable of many      
diverse modes of treatment4 ,i intend   
more  particularly to give some account 
of that department of logical science   
with which some of my own studies have  
been connected4 ,but ,i am  anxious1 if 
,i can succeed in so doing1 to handle   
this account so as to  exhibit the      
relation with certain considerations    
which underlie general  scientific      
activities4    ,it is no accident that  
an age of science has developed into an 
age  of organisation4 ,organised thought
 is the basis of organised action4      
,organisation is the adjustment of      

diverse elements so that their  mutual  
relations may exhibit some predetermined
 quality4 ,an epic poem  is a triumph of
 organisation1 that is to say1 it is a  
triumph in the  unlikely event of its   
being a good epic poem4 ,it is the      
successful  organisation of             
multitudinous sounds of words1          
associations of words1  pictorial       
memories of diverse events and feelings 
ordinarily occurring  in life1 combined 
with a special narrative of great       
events3 the whole  so disposed as to    
excite emotions which1 as defined by    
,milton1 are  simple1 sensuous1 and     
passionate4 ,the number of successful   
epic poems  is commensurate1 or rather1 
is inversely commensurate1 with the     
obvious  difficulty of the task of      
organisation4    ,science is the        
organisation of thought4 ,but the       
example of the epic  poem warns us that 
science is not any organisation of      
thought4 ,it is  an organisation of a   
certain definite type which we will     

endeavour to  determine4    ,science is 
a river with two sources1 the practical 
source and the  theoretical source4 ,the
 practical source is the desire to      
direct our  actions to achieve          
predetermined ends4 ,for example1 the   
,british nation1  fighting for justice1 
turns to science1 which teaches it the  
importance  of compounds of nitrogen4   
,the theoretical source is the desire to
  understand4 ,now ,i am going to       
emphasise the importance of theory in   
science4 ,but to avoid misconception ,i 
most emphatically state that ,i  do not 
consider one source as in any sense     
nobler than the other1 or  intrinsically
 more interesting4 ,i cannot see why it 
is nobler to strive  to understand than 
to busy oneself with the right ordering 
of one's  actions4 ,both have their bad 
sides2 there are evil ends directing    
actions1 and there are ignoble          
curiosities of the understanding4       
,the importance1 even in practice1 of   
the theoretical side of science  arises 

from the fact that action must be       
immediate1 and takes place  under       
circumstances which are excessively     
complicated4 ,if we wait for  the       
necessities of action before we commence
 to arrange our ideas1 in  peace we     
shall have lost our trade1 and in war we
 shall have lost the  battle4 ,success  
in practice depends on theorists who1   
led by other  motives of exploration1   
have been there before1 and by some good
 chance  have hit upon the relevant     
ideas4 ,by a theorist ,i do not mean a  
man who  is up in the clouds1 but a man 
whose motive for thought is the desire  
to formulate correctly the rules        
according to which events occur4 ,a     
successful theorist should be           
excessively interested in immediate     
events1 otherwise he is not at all      
likely to formulate correctly  anything 
about them4 ,of course1 both sources of 
science exist in all  men4    ,now1 what
 is this thought organisation which we  
call science8 ,the  first aspect of     

modern science which struck thoughtful  
observers was  its inductive character4 
,the nature of induction1 its           
importance1 and  the rules of inductive 
logic have been considered by a long    
series of  thinkers1 especially ,english
 thinkers3 ,bacon1 ,herschel1 ,j4 ,s4   
,mill1  ,venn1 ,jevons1 and others4 ,i  
am not going to plunge into an analysis 
of the process of induction4 ,induction 
is the machinery and not the  product1  
and it is the product which ,i want to  
consider4 ,when we  understand the      
product we shall be in a stronger       
position to improve  the machinery4     
,first1 there is one point which it is  
necessary to emphasise4 ,there  is a    
tendency in analysing scientific        
processes to assume a given  assemblage 
of concepts applying to nature1 and to  
imagine that the  discovery of laws of  
nature consists in selecting by means of
 inductive  logic some one out of a     
definite set of possible alternative    
relations  which may hold between the   

things in nature answering to these     
obvious  concepts4 ,in a sense this     
assumption is fairly correct1 especially
  in regard to the earlier stages of    
science4 ,mankind found itself in       
possession of certain concepts          
respecting nature--for example1 the     
concept of fairly permanent material    
bodies--and proceeded to determine  laws
 which related the corresponding        
percepts in nature4 ,but the            
formulation of laws changed the         
concepts1 sometimes gently by an added  
precision1 sometimes violently4 ,at     
first this process was not much         
noticed1 or at least was felt to be a   
process curbed within narrow  bounds1   
not touching fundamental ideas4 ,at the 
stage where we now  are1 the formulation
 of the concepts can be seen to be as   
important  as the formulation of the    
empirical laws connecting the events in 
the  universe as thus conceived by us4  
,for example1 the concepts of life1  of 
heredity1 of a material body1 of a      

molecule1 of an atom1 of an  electron1  
of energy1 of space1 of time1 of        
quantity1 and of number4 ,i  am not     
dogmatising about the best way of       
getting such ideas straight4  ,certainly
 it will only be done by those who have 
devoted themselves to  a special study  
of the facts in question4 ,success is   
never absolute1  and progress in the    
right direction is the result of a slow1
 gradual  process of continual          
comparison of ideas with facts4 ,the    
criterion of  success is that we should 
be able to formulate empirical laws1    
that is1  statements of relations1      
connecting the various parts of the     
universe  as thus conceived1 laws with  
the property that we can interpret the  
actual events of our lives as being our 
fragmentary knowledge of this  conceived
 interrelated whole4    ,but1 for the   
purpose of science1 what is the actual  
world8 ,has science  to wait for the    
termination of the metaphysical debate  
till it can  determine its own          

subject-matter8 ,i suggest that science 
has a much  more homely starting-ground4
 ,its task is the discovery of the      
relations  which exist within that flux 
of perceptions1 sensations1 and emotions
  which forms our experience of life4   
,the panorama yielded by sight1  sound1 
taste1 smell1 touch1 and by more        
inchoate sensible feelings1  is the sole
 field of activity4 ,it is in this way  
that science is the  thought            
organisation of experience4 ,the most   
obvious aspect of this  field of actual 
experience is its disorderly character4 
,it is for  each person a continuum1    
fragmentary1 and with elements not      
clearly differentiated4 ,the comparison 
of the sensible experiences of  diverse 
people brings its own difficulties4 ,i  
insist on the radically  untidy1        
ill-adjusted character of the fields of 
actual experience from  which science   
starts4 ,to grasp this fundamental truth
 is the first  step in wisdom1 when     
constructing a philosophy of science4   

,this fact  is concealed by the         
influence of language1 moulded by       
science1 which  foists on us exact      
concepts as though they represented the 
immediate  deliverances of experience4  
,the result is1 that we imagine that we 
have  immediate experience of a world of
 perfectly defined objects implicated   
in perfectly defined events which1 as   
known to us by the direct  deliverance  
of our senses1 happen at exact instants 
of time1 in a space  formed by exact    
points1 without parts and without       
magnitude3 the neat1  trim1 tidy1 exact 
world which is the goal of scientific   
thought4    ,my contention is1 that this
 world is a world of ideas1 and that its
  internal relations are relations      
between abstract concepts1 and that  the
 elucidation of the precise connection  
between this world and the  feelings of 
actual experience is the fundamental    
question of scientific  philosophy4 ,the
 question which ,i am inviting you to   
consider is this3  ,how does exact      

thought apply to the fragmentary1 vague 
continua  of experience8 ,i am not      
saying that it does not apply3 quite the
  contrary4 ,but ,i want to know how it 
applies4 ,the solution ,i am  asking for
 is not a phrase1 however brilliant1 but
 a solid branch of  science1 constructed
 with slow patience1 showing in detail  
how the  correspondence is effected4    
,the first great steps in the           
organisation of thought were due        
exclusively to the practical source of  
scientific activity1 without  any       
admixture of theoretical impulse4 ,their
 slow accomplishment was  the cause and 
also the effect of the gradual evolution
 of moderately  rational beings4 ,i mean
 the formation of the concepts of       
definite  material objects1 of the      
determinate lapse of time1 of           
simultaneity1  of recurrence1 of        
definite relative position1 and of      
analogous  fundamental ideas1 according 
to which the flux of our experience is  
mentally arranged for handy reference3  

in fact1 the whole apparatus of         
commonsense thought4 ,consider in your  
mind some definite chair4 ,the  concept 
of that chair is simply the concept of  
all the interrelated  experiences       
connected with that chair--namely1 of   
the experience of the  folk who made it1
 of the folk who sold it1 of the folk   
who have seen  it or used it1 of the man
 who is now experiencing a comfortable  
sense  of support1 combined with our    
expectations of an analogous future1    
terminated finally by a different set of
 experiences when the chair  collapses  
and becomes firewood4 ,the formation of 
that type of concept  was a tremendous  
job1 and zoologists and geologists tell 
us that it  took many tens of millions  
of years4 ,i can well believe it4    ,i 
now emphasise two points4 ,in the first 
place1 science is rooted in  what ,i    
have just called the whole apparatus of 
commonsense thought4  ,that is the datum
 from which it starts1 and to which it  
must recur4 ,we may speculate1 if it    

amuses us1 of other beings in  other    
planets who have arranged analogous     
experiences according to  an entirely   
different conceptual code--namely1 who  
have directed  their chief attention to 
different relations between their       
various  experiences4 ,but the task is  
too complex1 too gigantic1 to be revised
  in its main outlines4 ,you may polish 
up commonsense1 you may contradict  it  
in detail1 you may surprise it4 ,but    
ultimately your whole task is to        
satisfy it4    ,in the second place1    
neither commonsense nor science can     
proceed with  their task of thought     
organisation without departing in some  
respect  from the strict consideration  
of what is actual in experience4 ,think 
again of the chair4 ,among the          
experiences upon which its concept is   
based1 ,i included our expectations of  
its future history4 ,i should  have gone
 further and included our imagination of
 all the possible  experiences which in 
ordinary language we should call        

perceptions of  the chair which might   
have occurred4 ,this is a difficult     
question1 and  ,i do not see my way     
through it4 ,but1 at present1 in the    
construction of  a theory of space and  
of time there seem insuperable          
difficulties if we  refuse to admit     
ideal experiences4    ,this imaginative 
perception of experiences1 which1 if    
they occurred1  would be coherent with  
our actual experiences1 seems           
fundamental in our  lives4 ,it is       
neither wholly arbitrary1 nor yet fully 
determined4 ,it  is a vague background  
which is only made in part definite by  
isolated  activities of thought4        
,consider1 for example1 our thoughts of 
the  unseen flora of ,brazil4    ,ideal 
experiences are closely connected with  
our imaginative  reproduction of the    
actual experiences of other people1 and 
also  with our almost inevitable        
conception of ourselves as receiving our
  impressions from an external complex  
reality beyond ourselves4 ,it  may be   

that an adequate analysis of every      
source and every type of  experience    
yields demonstrative proof of such a    
reality and of its  nature4 ,indeed1 it 
is hardly to be doubted that this is the
 case4 ,the  precise elucidation of this
 question is the problem of metaphysics4
 ,one  of the points which ,i am urging 
in this address is1 that the basis of   
science does not depend on the          
assumption of any of the conclusions of 
metaphysics2 but that both science and  
metaphysics start from the same  given  
groundwork of immediate experience1 and 
in the main proceed in  opposite        
directions on their diverse tasks4      
,for example1 metaphysics inquires how  
our perceptions of the chair  relate us 
to some true reality4 ,science gathers  
up these perceptions  into a determinate
 class1 adds to them ideal perceptions  
of analogous  sort1 which under         
assignable circumstances would be       
obtained1 and this  single concept of   
that set of perceptions is all that     

science needs2  unless indeed you prefer
 that thought find its origin in some   
legend of  those great twin brethren1   
the ,cock and ,bull4    ,my immediate   
problem is to inquire into the nature of
 the texture of  science4 ,science is   
essentially logical4 ,the nexus between 
its concepts  is a logical nexus1 and   
the grounds for its detailed assertions 
are logical grounds4 ,king ,james said1 
,no bishops1 no king4 ,with  greater    
confidence we can say1 ,no logic1 no    
science4 ,the reason for  the           
instinctive dislike which most men of   
science feel towards the  recognition of
 this truth is1 ,i think1 the barren    
failure of logical  theory during the   
past three or four centuries4 ,we may   
trace this  failure back to the worship 
of authority1 which in some respects    
increased in the learned world at the   
time of the ,renaissance4 ,mankind  then
 changed its authority1 and this fact   
temporally acted as an  emancipation4   
,but the main fact1 and we can find     

complaints#b of it  at the very         
commencement of the modern movement1 was
 the establishment  of a reverential    
attitude towards any statement made by a
 classical  author4 ,scholars became    
commentators on truths too fragile to   
bear  translation4 ,a science which     
hesitates to forget its founders is     
lost4  ,to this hesitation ,i ascribe   
the barrenness of logic4 ,another reason
  for distrust of logical theory and of 
mathematics is the belief that          
deductive reasoning can give you nothing
 new4 ,your conclusions are  contained  
in your premises1 which by hypothesis   
are known to you4    ,in the first place
 this last condemnation of logic        
neglects the  fragmentary1 disconnected 
character of human knowledge4 ,to know  
one  premise on ,monday1 and another    
premise on ,tuesday1 is useless to you  
on ,wednesday4 ,science is a permanent  
record of premises1 deductions1  and    
conclusions1 verified all along the line
 by its correspondence  with facts4     

,secondly1 it is untrue that when we    
know the premises we  also know the     
conclusions4 ,in arithmetic1 for        
example1 mankind are not  calculating   
boys4 ,any theory which proves that they
 are conversant with  the consequences  
of their assumptions must be wrong4 ,we 
can imagine  beings who possess such    
insight4 ,but we are not such creatures4
  ,both these answers are1 ,i think1    
true and relevant4 ,but they are  not   
satisfactory4 ,they are too much in the 
nature of bludgeons1  too external4 ,we 
want something more explanatory of the  
very real  difficulty which the question
 suggests4 ,in fact1 the true answer is 
embedded in the discussion of our main  
problem of the relation of logic  to    
natural science4    ,it will be         
necessary to sketch in broad outline    
some relevant features  of modern logic4
 ,in doing so ,i shall try to avoid the 
profound general  discussions and the   
minute technical classifications which  
occupy the  main part of traditional    

logic4 ,it is characteristic of a       
science in  its earlier stages--and     
logic has become fossilised in such a   
stage--to  be both ambitiously profound 
in its aims and trivial in its handling 
of  details4    ,we can discern four    
departments of logical theory4 ,by an   
analogy  which is not so very remote ,i 
will call these departments or sections 
the arithmetic section1 the algebraic   
section1 the section of                 
general-function theory1 the analytical 
section4 ,i do not mean that  arithmetic
 arises in the first section1 algebra in
 the second section1  and so on2 but the
 names are suggestive of certain        
qualities of  thought in each section   
which are reminiscent of analogous      
qualities  in arithmetic1 in algebra1 in
 the general theory of a mathematical   
function1 and in the mathematical       
analysis of the properties of           
particular functions4    ,the first     
section--namely1 the arithmetic         
stage--deals with the  relations of     

definite propositions to each other1    
just as arithmetic  deals with definite 
numbers4 ,consider any definite         
proposition2 call  it p4 ,we conceive   
that there is always another proposition
 which  is the direct contradictory to  
p2 call it not-p4 ,when we have  got two
 propositions1 p and q1 we can form     
derivative propositions  from them1 and 
from their contradictories4 ,we can say1
 ,at last one of  p or q is true1 and   
perhaps both4 ,let us call this         
proposition  p or q4 ,i may mention as  
an aside that one of the greatest living
  philosophers has stated that this use 
of the word or--namely1 p or  q in the  
sense that either or both may be        
true--makes him despair of  exact       
expression4 ,we must brave his wrath1   
which is unintelligible to  me4    ,we  
have thus got hold of four new          
propositions1 namely1 p or q1  and not-p
 or q1 and p or not-q1 and not-p or     
not-q4  ,call these the set of          
disjunctive derivatives4 ,there are1 so 

far1  in all eight propositions1 p1     
not-p1 q1 not-q1 and the four           
disjunctive derivatives4 ,any pair of   
these eight propositions can be  taken1 
and substituted for p and q in the      
foregoing treatment4  ,thus each pair   
yields eight propositions1 some of which
 may have been  obtained before4 ,by    
proceeding in this way we arrive at an  
unending set  of propositions of growing
 complexity1 ultimately derived from the
 two  original propositions p or q4 ,of 
course1 only a few are important4       
,similarly we can start from three      
propositions1 p1 q1 r1 or  from four    
propositions1 p1 q1 r1 s1 and so on4    
,any one of the  propositions of these  
aggregates may be true or false4 ,it has
 no other  alternative4 ,whichever it   
is1 true or false1 call it the          
truth-value  of the proposition4    ,the
 first section of logical inquiry is to 
settle what we know of the  truth-values
 of these propositions1 when we know the
 truth-values of  some of them4 ,the    

inquiry1 so far as it is worth while    
carrying it1  is not very abstruse1 and 
the best way of expressing its results  
is a  detail which ,i will not now      
consider4 ,this inquiry forms the       
arithmetic  stage4    ,the next section 
of logic is the algebraic stage4 ,now1  
the difference  between arithmetic and  
algebra is1 that in arithmetic definite 
numbers are considered1 and in algebra  
symbols--namely1 letters--are           
introduced which stand for any numbers4 
,the idea of a number is also  enlarged4
 ,these letters1 standing for any       
numbers1 are called sometimes  variables
 and sometimes parameters4 ,their       
essential characteristic is  that they  
are undetermined1 unless1 indeed1 the   
algebraic conditions  which they satisfy
 implicitly determine them4 ,then they  
are sometimes  called unknowns4 ,an     
algebraic formula with letters is a     
blank form4 ,it  becomes a determinate  
arithmetic statement when definite      
numbers are  substituted for the        

letters4 ,the importance of algebra is a
 tribute to  the study of form4         
,consider now the following             
proposition--     ,the specific heat of 
mercury is #jjcc4    ,this is a definite
 proposition which1 with certain        
limitations1 is  true4 ,but the         
truth-value of the proposition does not 
immediately  concern us4 ,instead of    
mercury put a mere letter which is the  
name of  some undetermined thing3 we    
get--     ,the specific heat of x is    
#jjcc4    ,this is not a proposition2 it
 has been called by ,russell a          
propositional function4 ,it is the      
logical analogy of an algebraic         
expression4 ,let us write 7x7 for any   
propositional function4    ,we could    
also generalise still further1 and say1 
,the specific heat of x is y4    ,we    
thus get another propositional function1
 ,f7x1 y71  of two arguments x and y1   
and so on for any number of  arguments4 
,now1 consider 7x74 ,there is the range 
of values of x1  for which 7x7 is a     

proposition1 true or false4 ,for values 
of  x outside this range1 7x7 is not a  
proposition at all1  and is neither true
 nor false4 ,it may have vague          
suggestions for us1  but it has no unit 
meaning of definite assertion4 ,for     
example1     ,the specific heat of water
 is #jjcc    is a proposition which is  
false2 and--     ,the specific heat of  
virtue is #jjcc    is1 ,i should        
imagine1 not a proposition at all2 so   
that it is neither  true nor false1     
though its component parts raise various
 associations  in our minds4 ,this range
 of values1 for which 7x7 has sense1 is 
called the type of the argument x4      
,but there is also a range of values of 
x for which 7x7  is a true proposition4 
,this is the class of those values of   
the  argument which satisfy 7x74 ,this  
class may have no  members1 or1 in the  
other extreme1 the class may be the     
whole type of  the arguments4    ,we    
thus conceive two general propositions  
respecting the indefinite  number of    

propositions which share in the same    
logical form1 that is1  which are values
 of the same propositional function4    
,one of these  propositions is1     7x7 
yields a true proposition for each value
 of x of   the proper type2    the other
 proposition is1     ,there is a value  
of x for which 7x7 is true4    ,given   
two1 or more1 propositional functions   
7x7 and 7x7  with the same argument x1  
we form derivative propositional        
functions1 namely1    7x7 or 7x71 7x7 or
 not-7x71    and so on with the         
contradictories1 obtaining1 as in the   
arithmetical  stage1 an unending        
aggregate of propositional functions4   
,also each  propositional function      
yields two general propositions4 ,the   
theory  of the interconnection between  
the truth-values of the general         
propositions arising from any such      
aggregate of propositional functions    
forms a simple and elegant chapter of   
mathematical logic4    ,in this         
algebraic section of logic the theory of

 types crops up1 as we  have already    
noted4 ,it cannot be neglected without  
the introduction of  error4 ,its theory 
has to be settled at least by some safe 
hypothesis1  even if it does not go to  
the philosophic basis of the question4  
,this  part of the subject is obscure   
and difficult1 and has not been finally 
elucidated1 though ,russell's brilliant 
work has opened out the subject4    ,the
 final impulse to modern logic comes    
from the independent discovery  of the  
importance of the logic variable by     
,frege and ,peano4 ,frege went  further 
than ,peano1 but by an unfortunate      
symbolism rendered his work  so obscure 
that no one fully recognised his meaning
 who had not found  it out for himself4 
,but the movement has a large history   
reaching back  to ,leibniz and even to  
,aristotle4 ,among ,english contributors
 are ,de  ,morgan1 ,boole1 and ,sir     
,alfred ,kempe2 their work is of the    
first rank4    ,the third logical       
section is the stage of general-function

 theory4  ,in logical language1 we      
perform in this stage the transition    
from  intension to extension1 and       
investigate the theory of denotation4   
,take  the propositional function1 7x74 
,there is the class1 or range  of values
 for x1 whose members satisfy 7x74 ,but 
the same  range may be the class whose  
members satisfy another propositional   
function 7x74 ,it is necessary to       
investigate how to indicate  the class  
by a way which is indifferent as between
 the various  propositional functions   
which are satisfied by any member of it1
 and of  it only4 ,what has to be done  
is to analyse the nature of propositions
  about a class--namely1 those          
propositions whose truth-values depend  
on  the class itself and not on the     
particular meaning by which the class   
is indicated4    ,furthermore1 there are
 propositions about alleged individuals 
indicated by descriptive phrases3 for   
example1 propositions about the  present
 ,king of ,england1 who does exist1 and 

the present ,emperor  of ,brazil1 who   
does not exist4 ,more complicated1 but  
analogous1  questions involving         
propositional functions of two variables
 involve  the notion of correlation1    
just as functions of one argument       
involve  classes4 ,similarly functions  
of three arguments yield three-cornered 
correlations1 and so on4 ,this logical  
section is one which ,russell has  made 
peculiarly his own by work which must   
always remain fundamental4  ,i have     
called this the section of functional   
theory1 because its  ideas are essential
 to the construction of logical denoting
 functions  which include as a special  
case ordinary mathematical functions1   
such  as sine1 logarithm1 etc4 ,in each 
of these three stages it will be        
necessary gradually to introduce an     
appropriate symbolism1 if we are to     
pass on to the fourth stage4    ,the    
fourth logical section1 the analytic    
stage1 is concerned with the            
investigation of the properties of      

special logical constructions1  that is1
 of classes and correlations of special 
sorts4 ,the whole of  mathematics is    
included here4 ,so the section is a     
large one4 ,in fact1  it is mathematics1
 neither more nor less1 but it includes 
an analysis  of mathematical ideas not  
hitherto included in the scope of that  
science1 nor1 indeed1 contemplated at   
all4 ,the essence of this stage is      
construction4 ,it is by means of        
suitable constructions that the great   
framework of applied mathematics1       
comprising the theories of number1      
quantity1 time1 and space1 is           
elaborated4    ,it is impossible1 even  
in brief outline1 to explain how        
mathematics  is developed from the      
concepts of class and correlation1      
including  many-cornered correlations1  
which are established in the third      
section4  ,i can only allude to the     
headings of the process1 which is fully 
developed in the work1 ,principia       
,mathematica1 by ,mr4 ,russell  and     

myself4 ,there are in this process of   
development seven special  sorts of     
correlations which are of peculiar      
interest4 ,the first sort  comprises    
one-to-many1 many-to-one1 and one-to-one
 correlations4  ,the second sort        
comprises serial relations1 that is1    
correlations  by which the members of   
some field are arranged in serial order1
 so  that1 in the sense defined by the  
relation1 any member of the field  is   
either before or after any other member4
 ,the third class comprises  inductive  
relations1 that is1 correlations on     
which the theory of  mathematical       
induction depends4 ,the fourth class    
comprises selective  relations1 which   
are required for the general theory of  
arithmetic  operations1 and elsewhere4  
,it is in connection with such relations
 that  the famous multiplicative axiom  
arises for consideration4 ,the fifth    
class comprises vector relations1 from  
which the theory of quantity  arises4   
,the sixth class comprises ratio        

relations1 which interconnect  number   
and quantity4 ,the seventh class        
comprises three-cornered and            
four-cornered relations which occur in  
geometry4    ,a bare enumeration of     
technical names1 such as the above1 is  
not very  illuminating1 though it may   
help to a comprehension of the          
demarcations  of the subject4 ,please   
remember that the names are technical   
names1  meant1 no doubt1 to be          
suggestive1 but used in strictly defined
  senses4 ,we have suffered much from   
critics who consider it sufficient  to  
criticise our procedure on the slender  
basis of a knowledge of  the dictionary 
meanings of such terms4 ,for example1 a 
one-to-one  correlation depends on the  
notion of a class with only one member1 
and  this notion is defined without     
appeal to the concept of the number one4
  ,the notion of diversity is all that  
is wanted4 ,thus the class  has  only   
one member1 if 7#a7 the class of values 
of x which satisfies  the propositional 

function1     x is not a member of 1    
is not the whole type of relevant values
 of x1 and if 7#b7 the  propositional   
function1     x and y are members of 1  
and x is diverse from   y    is false1  
whatever be the values of x and y in the
  relevant type4    ,analogous          
procedures are obviously possible for   
higher finite cardinal  members4 ,thus1 
step by step1 the whole cycle of current
 mathematical  ideas is capable of      
logical definition4 ,the process is     
detailed and  laborious1 and1 like all  
science1 knows nothing of a royal road  
of  airy phrases4 ,the essence of the   
process is1 first1 to construct  the    
notion in terms of the forms of         
propositions1 that is1 in terms  of the 
relevant propositional functions1 and   
secondly1 to prove the  fundamental     
truths which hold about the notion by   
reference to the  results obtained in   
the algebraic section of logic4    ,it  
will be seen that in this process the   
whole apparatus of special  indefinable 

mathematical concepts1 and special a    
priori  mathematical premises1          
respecting number1 quantity1 and space1 
has  vanished4 ,mathematics is merely an
 apparatus for analysing the  deductions
 which can be drawn from any particular 
premises1 supplied  by commonsense1 or  
by more refined scientific observation1 
so far as  these deductions depend on   
the forms of the propositions4          
,propositions  of certain forms are     
continually occurring in thought4 ,our  
existing  mathematics is the analysis of
 deductions which concern those forms   
and  in some way are important1 either  
from practical utility or theoretical   
interest4 ,here ,i am speaking of the   
science as it in fact exists4 ,a        
theoretical definition of mathematics   
must include in its scope any           
deductions depending on the mere forms  
of propositions4 ,but1 of course  no one
 would wish to develop that part of     
mathematics which in no sense  is of    
importance4    ,this hasty summary of   

logical ideas suggests some reflections4
 ,the  question arises1 ,how many forms 
of propositions are there8 ,the  answer 
is1 ,an unending number4 ,the reason for
 the supposed sterility  of logical     
science can thus be discerned4          
,aristotle founded the  science by      
conceiving the idea of the form of a    
proposition1 and by  conceiving         
deduction as taking place in virtue of  
the forms4 ,but he  confined            
propositions to four forms1 now named   
,a1 ,i1 ,e1 ,o4 ,so long as  logicians  
were obsessed by this unfortunate       
restriction1 real progress  was         
impossible4 ,again1 in their theory of  
form1 both ,aristotle and  subsequent   
logicians came very near to the theory  
of the logical  variable4 ,but to come  
very near to a true theory1 and to grasp
 its  precise application1 are two very 
different things1 as the history of     
science teaches us4 ,everything of      
importance has been said before by      
somebody who did not discover it4       

,again1 one reason why logical          
deductions are not obvious is1 that     
logical form is not a subject which     
ordinarily enters into thought4         
,commonsense deduction probably moves by
 blind instinct from concrete           
proposition to concrete proposition1    
guided by some habitual  association of 
ideas4 ,thus commonsense fails in the   
presence of a  wealth of material4    ,a
 more important question is the relation
 of induction1 based on  observation1 to
 deductive logic4 ,there is a tradition 
of opposition  between adherents of     
induction and of deduction4 ,in my view1
 it  would be just as sensible for the  
two ends of a worm to quarrel4  ,both   
observation and deduction are necessary 
for any knowledge worth  having4 ,we    
cannot get at an inductive law without  
having recourse to a  propositional     
function4 ,for example1 take the        
statement of observed  fact1     ,this  
body is mercury1 and its specific heat  
is #jjcc4    ,the propositional function

 is formed1     ,either x is not        
mercury1 or its specific heat is #jjcc4 
,the inductive law is the assumption of 
the truth of the general  proposition1  
that the above propositional function is
 true for every  value of x in the      
relevant type4    ,but it is objected   
that this process and its consequences  
are so simple  that an elaborate science
 is out of place4 ,in the same way1 a   
,british  sailor knows the salt sea when
 he sails over it4 ,what1 then1 is the  
use  of an elaborate chemical analysis  
of sea-water8 ,there is the general     
answer1 that you cannot know too much of
 methods which you always  employ2 and  
there is the special answer1 that       
logical forms and logical  implications 
are not so very simple1 and that the    
whole of mathematics  is evidence to    
this effect4    ,one great use of the   
study of logical method is not in the   
region of  elaborate deduction1 but to  
guide us in the study of the formation  
of  the main concepts of science4       

,consider geometry1 for example4 ,what  
are  the points which compose space8    
,euclid tells us that they are without  
parts and without magnitude4 ,but how is
 the notion of a point derived  from the
 sense-perceptions from which science   
starts8 ,certainly points  are not      
direct deliverances of the senses4 ,here
 and there we may see or  unpleasantly  
feel something suggestive of a point4   
,but this is a rare  phenomenon1 and    
certainly does not warrant the          
conception of space as  composed of     
points4 ,our knowledge of space         
properties is not based on  any         
observations of relations between       
points4 ,it arises from experience  of  
relations between bodies4 ,now a        
fundamental space-relation between      
bodies is that one body may be part of  
another4 ,we are tempted to  define the 
whole and part relation by saying that  
the points occupied  by the part are    
some of the points occupied by the      
whole4 ,but whole  and part being more  

fundamental than the notion of point1   
this  definition is really circular and 
vicious4    ,we accordingly ask whether 
any other definition of spatial whole   
and  part can be given4 ,i think that it
 can be done in this way1 though1  if ,i
 be mistaken1 it is unessential to my   
general argument4 ,we have  come to the 
conclusion that an extended body is     
nothing else than the  class of         
perception of it by all its percipients1
 actual or ideal4 ,of  course1 it is not
 any class of perceptions1 but a certain
 definite sort  of class which ,i have  
not defined here1 except by the vicious 
method  of saying that they are         
perceptions of body4 ,now1 the          
perceptions of a  part of a body are    
among the perceptions which compose the 
whole body4  ,thus two bodies a and b   
are both classes of perceptions2 and b  
is  part of a when the class which is b 
is contained in the class which  is a4  
,it immediately follows from the logical
 form of this definition  that if b is  

part of a1 and c is part of b1 then c is
 part of  a4 ,thus the relation whole to
 part is transitive4 ,again1 it will  be
 convenient to allow that a body is part
 of itself4 ,this is a mere  question of
 how you draw the definition4 ,with this
 understanding1 the  relation is        
reflexive4 ,finally1 if a is part of b1 
and b is part  of a1 then a and b must  
be identical4 ,these properties of whole
  and part are not fresh assumptions1   
they follow from the logical form  of   
our definition4    ,one assumption has  
to be made if we assume the ideal       
infinite  divisibility of space4        
,namely1 we assume that every class of  
perceptions which is an extended body   
contains other classes of  perceptions  
which are extended bodies diverse from  
itself4 ,this  assumption makes rather a
 large draft on the theory of ideal     
perceptions4 ,geometry vanishes unless  
in some form you make it4 ,the          
assumption is not peculiar to my        
exposition4    ,it is then possible to  

define what we mean by a point4 ,a point
 is the  class of extended objects      
which1 in ordinary language1 contain    
that  point4 ,the definition1 without   
presupposing the idea of a point1 is    
rather elaborate1 and ,i have not now   
time for its statement4    ,the         
advantage of introducing points into    
geometry is the simplicity  of the      
logical expression of their mutual      
relations4 ,for science1  simplicity of 
definition is of slight importance1 but 
simplicity of  mutual relations is      
essential4 ,another example of this law 
is the way  physicists and chemists have
 dissolved the simple idea of an        
extended  body1 say of a chair1 which a 
child understands1 into a bewildering   
notion of a complex dance of molecules  
and atoms and electrons and  waves of   
light4 ,they have thereby gained notions
 with simpler logical  relations4       
,space as thus conceived is the exact   
formulation of the properties of  the   
apparent space of the commonsense world 

of experience4 ,it is not  necessarily  
the best mode of conceiving the space of
 the physicist4  ,the one essential     
requisite is that the correspondence    
between the  commonsense world in its   
space and the physicists' world in its  
space  should be definite and           
reciprocal4    ,i will now break off the
 exposition of the function of logic in 
connection with the science of natural  
phenomena4 ,i have endeavoured  to      
exhibit it as the organising principle1 
analysing the derivation  of the        
concepts from the immediate phenomena1  
examining the structure  of the general 
propositions which are the assumed laws 
of nature1  establishing their relations
 to each other in respect to reciprocal 
implications1 deducing the phenomena we 
may expect under given  circumstances4  
,logic1 properly used1 does not shackle 
thought4 ,it gives freedom1 and  above  
all1 boldness4 ,illogical thought       
hesitates to draw conclusions1  because 
it never knows either what it means1 or 

what it assumes1 or  how far it trusts  
its own assumptions1 or what will be the
 effect of  any modification of         
assumptions4 ,also the mind untrained in
 that part  of constructive logic which 
is relevant to the subject in hand will 
be  ignorant of the sort of conclusions 
which follow from various sorts of      
assumptions1 and will be correspondingly
 dull in divining the inductive  laws4  
,the fundamental training in this       
relevant logic is1 undoubtedly1  to     
ponder with an active mind over the     
known facts of the case1  directly      
observed4 ,but where elaborate          
deductions are possible1 this  mental   
activity requires for its full exercise 
the direct study of the  abstract       
logical relations4 ,this is applied     
mathematics4    ,neither logic without  
observation1 nor observation without    
logic1 can  move one step in the        
formation of science4 ,we may conceive  
humanity  as engaged in an internecine  
conflict between youth and age4 ,youth  

is  not defined by years but by the     
creative impulse to make something4 ,the
  aged are those who1 before all things1
 desire not to make a mistake4  ,logic  
is the olive branch from the old to the 
young1 the wand which in  the hands of  
youth has the magic property of creating
 science4      ,,footnotes3    ,footnote
 #b3 ,e4g4 in #aeea by ,italian         
schoolmen2 cf4 ,sarpi's  ,history of the
 ,council of ,trent1 under that date4   
,,chapter ,,vii                         
,,the ,,anatomy ,,of ,,some ,,scientific
 ,,ideas                                
,i4 ,fact    ,,the characteristic of    
physical science is1 that it ignores all
  judgments of value3 for example1      
sthetic or moral judgments4 ,it is      
purely matter-of-fact1 and this is the  
sense in which we must interpret  the   
sonorous phrase1 ,man1 the servant and  
the minister of ,nature4    ,the sphere 
of thought which is thus left is even   
then too wide for  physical science4 ,it
 would include ,ontology1 namely1 the   

determination  of the nature of what    
truly exists2 in other words1           
,metaphysics4 ,from  an abstract point  
of view this exclusion of metaphysical  
inquiry  is a pity4 ,such an inquiry is 
a necessary critique of the worth of    
science1 to tell us what it all comes   
to4 ,the reasons for its careful        
separation from scientific thought are  
purely practical2 namely1  because we   
can agree about science--after due      
debate--whereas in  respect to          
metaphysics debate has hitherto         
accentuated disagreement4  ,these       
characteristics of science and          
metaphysics were unexpected in the      
early days of civilised thought4 ,the   
,greeks thought that metaphysics  was   
easier than physics1 and tended to      
deduce scientific principles  from a    
priori conceptions of the nature of     
things4 ,they were  restrained in this  
disastrous tendency by their vivid      
naturalism1 their  delight in first-hand
 perception4 ,medival ,europe shared the

 tendency  without the restraint4 ,it is
 possible that some distant generations 
may arrive at unanimous conclusions on  
ontological questions1 whereas          
scientific progress may have led to     
ingrained opposing veins of  thought    
which can neither be reconciled nor     
abandoned4 ,in such times  metaphysics  
and physical science will exchange their
 rles4 ,meanwhile  we must take the case
 as we find it4    ,but a problem       
remains4 ,how can mankind agree about   
science without  a preliminary          
determination of what really is8 ,the   
answer must be  found in an analysis of 
the facts which form the field of       
scientific  activity4 ,mankind          
perceives1 and finds itself thinking    
about its  perceptions4 ,it is the      
thought that matters and not that       
element of  perception which is not     
thought4 ,when the immediate judgment   
has been  formed--,hullo1 red6--it does 
not matter if we can imagine that in    
other  circumstances--in better         

circumstances1 perhaps--the judgment    
would  have been--,hullo1 blue6--or     
even--,hullo1 nothing6 ,for all intents 
and  purposes1 at the time it was red4  
,everything else is hypothetical        
reconstruction4 ,the field of physical  
science is composed of these  primary   
thoughts1 and of thoughts about these   
thoughts4    ,but--to avoid confusion--a
 false simplicity has been introduced   
above  into the example given of a      
primary perceptive thought4 ,hullo1 red6
  is not really a primary perceptive    
thought1 though it often is the  first  
thought which finds verbal expression   
even silently in the mind4  ,nothing is 
in isolation4 ,the perception of red is 
of a red object in  its relations to the
 whole content of the perceiving        
consciousness4    ,among the most easily
 analysed of such relations are the     
space  relations4 ,again the red object 
is in immediate perception nothing else 
than a red object4 ,it is better termed 
an object of redness4 ,thus a  better   

approximation to an immediate perceptive
 judgment is1 ,hullo1  object of redness
 there6 ,but1 of course1 in this        
formulation other  more complex         
relations are omitted4    ,this tendency
 towards a false simplicity in          
scientific analysis1 to  an excessive   
abstraction1 to an over-universalising  
of universals1  is derived from the     
earlier metaphysical stage4 ,it arises  
from the  implicit belief that we are   
endeavouring to qualify the real with   
appropriate adjectives4 ,in conformity  
with this tendency we think1  this real 
thing is red4 ,whereas our true goal is 
to make explicit  our perception of the 
apparent in terms of its relations4     
,what we  perceive is redness related to
 other apparents4 ,our object is the    
analysis of the relations4    ,one aim  
of science is the harmony of thought1   
that is1 to secure  that judgments which
 are logical contraries should not be   
thought-expressions of consciousness4   
,another aim is the extension of  such  

harmonised thought4    ,some thoughts   
arise directly from sense-presentation1 
and are part of  the state of           
consciousness which is perception4 ,such
 a thought is1 ,an  object of redness is
 there4 ,but in general the thought is  
not verbal1  but is a direct            
apprehension of qualities and relations 
within the  content of consciousness4   
,amid such thoughts there can be no lack
 of harmony4 ,for direct  apprehension  
is in its essence unique1 and it is     
impossible to  apprehend an object as   
both red and blue4 ,subsequently it may 
be judged  that if other elements of the
 consciousness had been different1 the  
apprehension would have been of a blue  
object4 ,then--under certain            
circumstances--the original apprehension
 will be called an error4 ,but  for all 
that the fact remains1 there was an     
apprehension of a red  object4    ,when 
we speak of sense-presentation1 we mean 
these primary thoughts  essentially     
involved in its perception4 ,but there  

are thoughts about  thoughts1 and       
thoughts derived from other thoughts4   
,these are secondary  thoughts4 ,at this
 point it is well explicitly to         
discriminate between  an actual         
thought-expression1 namely1 a judgment  
actually made1 and a  mere proposition  
which is a hypothetical                 
thought-expression1 namely1  an imagined
 possibility of thought-expression4     
,note that the actual  complete         
thought-content of the consciousness is 
explicitly neither  affirmed or denied4 
,it is just what is thought4 ,thus1 to  
think  two and two make four is distinct
 from affirming that two and two  make  
four4 ,in the first case the proposition
 is the thought-expression1  in the     
second case the affirmation of the      
proposition is the  thought-expression1 
and the proposition has been degraded to
 a mere  proposition1 namely1 to a      
hypothetical thought-expression which is
  reflected upon4    ,a distinction is  
sometimes made between facts and        

thoughts4 ,so far as  physical science  
is concerned1 the facts are thoughts1   
and thoughts  are facts4 ,namely1 the   
facts of sense-presentation as they     
affect  science are those elements in   
the immediate apprehensions which are   
thoughts4 ,also1 actual                 
thought-expressions1 primary or         
secondary1 are  the material facts which
 science interprets4    ,the distinction
 that facts are given1 but thoughts are 
free1 is  not absolute4 ,we can select  
and modify our sense-presentation1 so   
that facts--in the narrower sense of    
immediate apprehension of               
sense-presentation--are to some degree  
subject to volition4 ,again1 our  stream
 of thought-expression is only partially
 modified by explicit  volition4 ,we can
 choose our physical experience1 and we 
find ourselves  thinking2 namely1 on the
 one hand there is selection amid the   
dominant  necessity of sense1 and on the
 other hand1 the thought-content of     
consciousness 7so far as secondary      

thoughts are concerned7 is not  wholly  
constituted by the selection of will4   
,thus1 on the whole there is a large    
primary region of secondary  thought1 as
 well as of the primary thoughts of     
sense-presentation1  which is given in  
type4 ,that is the way in which we do   
think of things1  not wholly from any   
abstract necessity1 so far as we know1  
but because  we have inherited the      
method from an environment4 ,it is the  
way we find  ourselves thinking1 a way  
which can only be fundamentally laid    
aside by  an immense effort1 and then   
only for isolated short periods of time4
  ,this is what ,i have called the whole
 apparatus of commonsense thought4      
,it is this body of thought which is    
assumed in science4 ,it is a way  of    
thinking rather than a set of axioms4   
,it is1 in fact1 the set of  concepts   
which commonsense has found useful in   
sorting out human  experience4 ,it is   
modified in detail1 but assumed in      
gross4 ,the  explanations of science are

 directed to finding conceptions and    
propositions concerning nature which    
explain the importance of these  common 
sense notions4 ,for example1 a chair is 
a common sense notion1  molecules and   
electrons explain our vision of chairs4 
,now science aims at harmonising our    
reflective and derivative thoughts  with
 the primary thoughts involved in the   
immediate apprehension of               
sense-presentation4 ,it also aims at    
producing such derivative thoughts1     
logically knit together4 ,this is       
scientific theory2 and the harmony to   
be achieved is the agreement of theory  
with observation1 which is the          
apprehension of sense-presentation4     
,thus there is a twofold scientific aim3
 7#a7 the production of theory  which   
agrees with experience2 and 7#b7 the    
explanation of commonsense  concepts of 
nature1 at least in their main outlines4
 ,this explanation  consists in the     
preservation of the concepts in a       
scientific theory of  harmonised        

thought4    ,it is not asserted that    
this is what scientists in the past     
meant to  achieve1 or thought that they 
could achieve4 ,it is suggested as the  
actual result of scientific effort1 so  
far as that effort has had  any measure 
of success4 ,in short1 we are here      
discussing the natural  history of ideas
 and not volitions of scientists4       
,,ii4 ,objects    ,we perceive things in
 space4 ,for example1 among such things 
are dogs1  chairs1 curtains1 drops of   
water1 gusts of air1 flames1 rainbows1  
chimes of bells1 odours1 aches and      
pains4 ,there is a scientific           
explanation of the origin of these      
perceptions4 ,this explanation  is given
 in terms of molecules1 atoms1          
electrons1 and their mutual  relations1 
in particular of their space-relations1 
and waves of  disturbance of these      
space-relations which are propagated    
through  space4 ,the primary elements of
 the scientific explanation--molecules1 
etc4--are not the things directly       

perceived4 ,for example1 we do not      
perceive a wave of light2 the sensation 
of sight is the resultant  effect of the
 impact of millions of such waves       
through a stretch of  time4 ,thus the   
object directly perceived corresponds to
 a series of  events in the physical    
world1 events which are prolonged       
through  a stretch of time4 ,nor is it  
true that a perceived object always     
corresponds to the same group of        
molecules4 ,after a few years we        
recognise the same cat1 but we are      
thereby related to different  molecules4
    ,again1 neglecting for a moment the 
scientific explanation1 the  perceived  
object is largely the supposition of our
 imagination4 ,when  we recognised the  
cat1 we also recognised that it was glad
 to see  us4 ,but we merely heard its   
mewing1 saw it arch its back1 and felt  
it  rubbing itself against us4 ,we must 
distinguish1 therefore1 between the     
many direct objects of sense1 and the   
single indirect object of thought  which

 is the cat4    ,thus1 when we say that 
we perceived the cat and understood its 
feelings1 we mean that we heard a       
sense-object of sound1 that we saw a    
sense-object of sight1 that we felt a   
sense-object of touch1 and that  we     
thought of a cat and imagined its       
feelings4    ,sense-objects are         
correlated by time-relations and        
space-relations4  ,three simultaneous   
sense-objects which are also spatially  
coincident1  are combined by thought    
into the perception of one cat4 ,such   
combination of sense-objects is an      
instinctive immediate judgment in       
general without effort of reasoning4    
,sometimes only one sense-object  is    
present4 ,for example1 we hear mewing   
and say there must be a cat  in the     
room4 ,the transition from the          
sense-object to the cat has  then been  
made1 by deliberate ratiocination4 ,even
 the concurrence of  sense-objects may  
provoke such a self-conscious effort4   
,for example1 in  the dark we feel      

something1 and hear mewing from the same
 place1 and  think1 ,surely this is a   
cat4 ,sight is more bold2 when we see a 
cat1 we  do not think further4 ,we      
identify the sight with the cat1 whereas
 the  cat and the mew are separate4 ,but
 such immediate identification of  a    
sight object and an object of thought   
may lead to error2 the birds  pecked at 
the grapes of ,apelles4    ,a single    
sense-object is a complex entity4 ,the  
sight-object of a  tile on the hearth   
may remain unchanged as we watch it in a
 steady  light1 remaining ourselves     
unchanged in position4 ,even then it is 
prolonged in time1 and has parts in     
space4 ,also it is somewhat  arbitrarily
 distinguished from a larger whole of   
which it forms part4  ,but the glancing 
fire-light and a change in our position 
alters the  sight-object4 ,we judge that
 the tile thought-object remains        
unchanged4  ,the sight-object of the    
coal on the fire gradually modifies1    
though  within short intervals it       

remains unchanged4 ,we judge that the   
coal  thought-object is changing4 ,the  
flame is never the same1 and its shape  
is only vaguely distinguishable4    ,we 
conclude that a single self-identical   
sight-object is already a  phantasy of  
thought4 ,consider the unchanging       
sight-object of the tile1  as we remain 
still in a steady light4 ,now a         
sense-object perceived at  one time is a
 distinct object from a sense-object    
seen at another time4  ,thus the sight  
of the tile at noon is distinct from its
 sight at #ab4cj4  ,but there is no such
 thing as a sense-object at an instant4 
,as we stare  at the tile1 a minute1 or 
a second1 or a tenth of a second1 has   
flown  by3 essentially there is a       
duration4 ,there is a stream of sight1  
and  we can distinguish its parts4 ,but 
the parts also are streams1 and  it is  
only in thought that the stream         
separates into a succession of          
elements4 ,the stream may be steady as  
in the case of the unchanging           

sight-tile1 or may be turbulent as in   
the case of the glancing  sight-flame4  
,in either case a sight-object is some  
arbitrarily small  part of the stream4  
,again1 the stream which forms the      
succession of sight-tiles is merely a   
distinguishable part of the whole stream
 of sight-presentation4    ,so1 finally1
 we conceive ourselves each experiencing
 a complete  time-flux 7or stream7 of   
sense-presentation4 ,this stream is     
distinguishable into parts4 ,the grounds
 of distinction are differences  of     
sense--including within that term1      
differences of types of sense1  and     
differences of quality and of intensity 
within the same type  of sense--and     
differences of time-relations1 and      
differences of  space-relations4 ,also  
the parts are not mutually exclusive and
 exist in  unbounded variety4    ,the   
time-relation between the parts raises  
the questions of memory and             
recognition1 subjects too complex for   
discussion here4 ,one remark must  be   

made4 ,if it be admitted1 as stated     
above1 that we live in durations  and   
not in instants1 namely1 that the       
present essentially occupies  a stretch 
of time1 the distinction between memory 
and immediate  presentation cannot be   
quite fundamental2 for always we have   
with us  the fading present as it       
becomes the immediate past4 ,this region
 of our  consciousness is neither pure  
memory nor pure immediate presentation4 
,anyhow1 memory is also a presentation  
in consciousness4    ,another point is  
to be noted in connection with memory4  
,there is no  directly perceived        
time-relation between a present event   
and a past  event4 ,the present event is
 only related to the memory of the past 
event4 ,but the memory of a past event  
is itself a present element  in         
consciousness4 ,we assert the principle 
that directly perceived  relations can  
only exist between elements of          
consciousness1 both in  that present    
during which the perception occurs4 ,all

 other relations  between elements of   
perception are inferential              
constructions4 ,it thus  becomes        
necessary to explain how the time stream
 of events establishes  itself in       
thought1 and how the apparent world     
fails to collapse into  one single      
present4 ,the solution of the difficulty
 is arrived at  by observing that the   
present is itself a duration1 and       
therefore  includes directly perceived  
time-relations between events contained 
within it4 ,in other words we put the   
present on the same footing as  the past
 and the future in respect to the       
inclusion within it of  antecedent and  
succeeding events1 so that past1        
present1 and future  are in this respect
 exactly analogous ideas4 ,thus there   
will be two  events a and b1 both in the
 same present1 but the event  a will be 
directly perceived to precede the event 
b4  ,again time flows on1 and the event 
a fades into the past1 and  in the new  
present duration events b and c occur1  

event  b preceding event c1 also in the 
same present duration  there is the     
memory of the time-relation between a   
and b4  ,then by an inferential         
construction the event a in the past    
precedes the event c in the present4 ,by
 proceeding according to  this principle
 the time-relations between elements of 
consciousness1  not in the same present1
 are established4 ,the method of        
procedure  here explained is a first    
example of what we will call the        
,principle  of ,aggregation4 ,this is   
one of the fundamental principles of    
mental  construction according to which 
our conception of the external physical 
world is constructed4 ,other examples   
will later on be met with4    ,the      
space-relations between the parts are   
confused and fluctuating1  and in       
general lack determinate precision4 ,the
 master-key by which  we confine our    
attention to such parts as possess      
mutual relations  sufficiently simple   
for our intellects to consider is the   

principle of  convergence to simplicity 
with diminution of extent4 ,we will call
 it  the principle of convergence4 ,this
 principle extends throughout the  whole
 field of sense-presentation4    ,the   
first application of the principle      
occurs in respect to time4  ,the shorter
 the stretch of time1 the simpler are   
the aspects of the  sense-presentation  
contained within it4 ,the perplexing    
effects of  change are diminished and in
 many cases can be neglected4 ,nature   
has  restricted the acts of thought     
which endeavour to realise the content  
of the present1 to stretches of time    
sufficiently short to secure this       
static simplicity over the greater part 
of the sense-stream4    ,spatial        
relations become simplified within the  
approximately static  sense-world of the
 short time4 ,a further simplicity is   
gained by  partitioning this static     
world into parts of restricted          
space-content4  ,the various parts thus 
obtained have simpler mutual            

space-relations1  and again the         
principle of convergence holds4         
,finally1 the last simplicity is        
obtained by partitioning the  parts1    
already restricted as to space and time1
 into further parts  characterised by   
homogeneity in type of sense1 and       
homogeneity in  quality and intensity of
 sense4 ,these three processes of       
restriction  yield1 finally1 the        
sense-objects which have been mentioned 
above4 ,thus  the sense-object is the   
result of an active process of          
discrimination  made in virtue of the   
principle of convergence4 ,it is the    
result of  the quest for simplicity of  
relations within the complete stream of 
sense-presentation4    ,the             
thought-objects of perception are       
instances of a fundamental  law of      
nature1 the law of objective stability4 
,it is the law of the  coherence of     
sense-objects4 ,this law of stability   
has an application  to time and an      
application to space2 also it must be   

applied in  conjunction with that other 
law1 the principle of convergence to    
simplicity from which sense-objects are 
derived4    ,some composite partial     
streams of sense-presentation can be    
distinguished with the following        
characteristics3 7#a7 the               
time-succession of sense-objects1       
belonging to a single sense1 involved   
in any such a composite partial stream1 
is composed of very similar  objects    
whose modifications increase only       
gradually1 and thus forms  a homogeneous
 component stream within the composite  
stream2 7#b7 the  space-relations of    
those sense-objects 7of various senses7 
of such  a composite stream which are   
confined within any sufficiently short  
time are identical so far as they are   
definitely apprehended1 and thus  these 
various component streams1 each         
homogeneous1 cohere to form the  whole  
composite partial stream2 7#c7 there are
 other sense-presentations  occurring in
 association with that composite partial

 stream which  can be determined by     
rules derived from analogous composite  
partial  streams1 with other space and  
time relations1 provided that the       
analogy  be sufficiently close4 ,call   
these the associated                    
sense-presentations4  ,a partial stream 
of this sort1 viewed as a whole1 is here
 called a  first crude thought-object of
 perception4    ,for example1 we look at
 an orange for half a minute1 handle it1
 and  smell it1 note its position in the
 fruit-basket1 and then turn away4  ,the
 stream of sense-presentation of the    
orange during that half-minute  is a    
first crude thought-object of           
perception4 ,among the associated  sense
 presentations are those of the         
fruit-basket which we conceive as       
supporting the orange4    ,the essential
 ground of the association of           
sense-objects of  various types1        
perceived within one short duration1    
into a first  crude thought-object of   
perception is the coincidence of their  

space-relations1 that is1 in general an 
approximate coincidence of  such        
relations perhaps only vaguely          
apprehended4 ,thus coincident           
space-relations associate sense-objects 
into a first crude  thought-object1 and 
diverse space-relations dissociate      
sense-objects  from aggregation into a  
first crude thought-object4 ,in respect 
to  some groups of sense-objects the    
association may be an immediate         
judgment devoid of all inference1 so    
that the primary perceptual  thought is 
that of the first crude thought-object1 
and the separate  sense-objects are the 
result of reflective analysis acting on 
memory4  ,for example sense-objects of  
sight and sense-objects of touch are    
often thus primarily associated and only
 secondarily dissociated in  thought4   
,but sometimes the association is       
wavering and indeterminate1  for        
example1 that between the sound-object  
of the mew of the cat and  the          
sight-object of the cat4 ,thus to sum   

up1 the partial stream of               
sense-perceptions coalesces into that   
first crude thought-object of           
perception which is the momentary cat   
because the sense-perceptions  belonging
 to this stream are in the same place1  
but equally it would  be true to say    
that they are in the same place because 
they belong  to the same momentary cat4 
,this analysis of the complete stream of
  sense-presentation in any small       
present duration into a variety of      
first crude thought-objects only        
partially fits the facts2 for one       
reason because many sense-objects1 such 
as sound for instance1 have  vague and  
indeterminate space-relations1 for      
example vaguely those  space-relations  
which we associate with our organs of   
sense and also  vaguely those of the    
origin from which 7in the scientific    
explanation7  they proceed4    ,the     
procedure by which the orange of half a 
minute is elaborated into  the orange in
 the ordinary sense of the term involves

 in addition the  two principles of     
aggregation and of hypothetical         
sense-presentation4    ,the principle of
 aggregation1 as here employed1 takes   
the form that  many distinct first crude
 thought-objects of perception are      
conceived  as one thought-object of     
perception1 if the many partial streams 
forming these objects are sufficiently  
analogous1 if their times of  occurrence
 are distinct1 and if the associated    
sense-presentations are  sufficiently   
analogous4    ,for example1 after       
leaving the orange1 in five minutes we  
return4  ,a new first crude             
thought-object of perception presents   
itself to  us1 indistinguishable from   
the half-minute orange we previously    
experienced2 it is in the same          
fruit-basket4 ,we aggregate the two     
presentations of an orange into the same
 orange4 ,by such aggregations  we      
obtain second crude thought-objects of  
perception4 ,but however far  we can    
proceed with aggregation of this type1  

the orange is more than  that4 ,for     
example1 what do we mean when we say1   
,the orange is in the  cupboard1 if ,tom
 has not eaten it8    ,the world of     
present fact is more than a stream of   
sense-presentation4  ,we find ourselves 
with emotions1 volitions1 imaginations1 
conceptions1  and judgments4 ,no factor 
which enters into consciousness is by   
itself  or even can exist in isolation4 
,we are analysing certain relations     
between sense-presentation and other    
factors of consciousness4  ,hitherto we 
have taken into account merely the      
factors of concept and  judgment4       
,imagination is necessary to complete   
the orange1 namely1  the imagination of 
hypothetical sense-presentations4 ,it is
 beside  the point to argue whether we  
ought to have such imaginations1 or to  
discuss what are the metaphysical truths
 concerning reality to which  they      
correspond4 ,we are here only concerned 
with the fact that such  imaginations   
exist and essentially enter into the    

formation of the  concepts of the       
thought-objects of perception which are 
the first  data of science4 ,we conceive
 the orange as a permanent collection of
  sense-presentations existing as if    
they were an actual element in our      
consciousness1 which they are not4 ,the 
orange is thus conceived as  in the     
cupboard with its shape1 odour1 colour1 
and other qualities4  ,namely1 we       
imagine hypothetical possibilities of   
sense-presentation1  and conceive their 
want of actuality in our consciousness  
as immaterial  to their existence in    
fact4 ,the fact which is essential for  
science is  our conception2 its meaning 
in regard to the metaphysics of reality 
is  of no scientific importance1 so far 
as physical science is concerned4       
,the orange completed in this way is the
 thought-object of perception4    ,it   
must be remembered that the judgments   
and concepts arising in the  formation  
of thought-objects of perception are in 
the main instinctive  judgments1 and    

instinctive concepts1 and are not       
concepts and judgments  consciously     
sought for and consciously criticised   
before adoption4  ,their adoption is    
facilitated by and interwoven with the  
expectation of  the future in which the 
hypothetical passes into the actual1 and
 also  with the further judgment of the 
existence of other consciousnesses1  so 
that much that is hypothetical to one   
consciousness is judged to be  actual to
 others4    ,the thought-object of      
perception is1 in fact1 a device to make
 plain  to our reflective consciousness 
relations which hold within the         
complete stream of sense-presentation4  
,concerning the utility of this  weapon 
there can be no question2 it is the rock
 upon which the whole  structure of     
commonsense thought is erected4 ,but    
when we consider the  limits of its     
application the evidence is confused4 ,a
 great part of our  sense-presentation  
can be construed as perception of       
various persistent  thought-objects4    

,but hardly at any time can the         
sense-presentations  be construed wholly
 in that way4 ,sights lend themselves   
easily to  this construction1 but sight 
can be baffled3 for example1 consider   
reflections in looking-glasses1         
apparently bent sticks half in and half 
out of water1 rainbows1 brilliant       
patches of light which conceal the      
object from which they emanate1 and many
 analogous phenomena4 ,sound  is more   
difficult2 it tends largely to disengage
 itself from any such  object4 ,for     
example1 we see the bell1 but we hear   
the sound which  comes from the bell2   
yet we also say that we hear the bell4  
,again1 a  toothache is largely by      
itself1 and is only indirectly a        
perception  of the nerve of the tooth4  
,illustrations to the same effect can be
  accumulated from every type of        
sensation4    ,another difficulty arises
 from the fact of change4 ,the          
thought-object  is conceived as one     
thing1 wholly actual at each instant4   

,but since  the meat has been bought it 
has been cooked1 the grass grows and    
then  withers1 the coal burns in the    
fire1 the pyramids of ,egypt remain     
unchanged for ages1 but even the        
pyramids are not wholly unchanged4 ,the 
difficulty of change is merely evaded by
 affixing a technical ,latin  name to a 
supposed logical fallacy4 ,a slight     
cooking leaves the meat  the same       
object1 but two days in the oven burns  
it to a cinder4 ,when  does the meat    
cease to be8 ,now the chief use of the  
thought-object  is the concept of it as 
one thing1 here and now1 which later can
 be  recognised1 there and then4 ,this  
concept applies sufficiently well to    
most things for short times1 and to many
 things for long times4 ,but            
sense-presentation as a whole entirely  
refuses to be patient of the  concept4  
,we have now come to the reflective     
region of explanation1 which is         
science4    ,a great part of the        
difficulty is at once removed by        

applying the  principle of convergence  
to simplicity4 ,we habitually make our  
thought-objects too large2 we should    
think in smaller parts4 ,for  example1  
the ,sphinx has changed by its nose     
becoming chipped1 but by  proper inquiry
 we could find the missing part in some 
private house  of ,western ,europe or   
,northern ,america4 ,thus1 either part1 
the rest  of the ,sphinx or the chip1   
regains its permanence4 ,furthermore1 we
  enlarge this explanation by conceiving
 parts so small that they can  only be  
observed under the most favourable      
circumstances4 ,this is a  wide         
extension of the principle of           
convergence in its application to       
nature2 but it is a principle amply     
supported by the history of exact       
observation4    ,thus1 change in        
thought-objects of perception is largely
 explained as  a disintegration into    
smaller parts1 themselves               
thought-objects of  perception4 ,the    
thought-objects of perception which are 

presupposed in  the common thought of   
civilised beings are almost wholly      
hypothetical4  ,the material universe is
 largely a concept of the imagination   
which  rests on a slender basis of      
direct sense-presentation4 ,but none the
  less it is a fact2 for it is a fact   
that actually we imagine it4 ,thus  it  
is actual in our consciousness just as  
sense-presentation also is  actual      
there4 ,the effort of reflective        
criticism is to make these two  factors 
in our consciousness agree where they   
are related1 namely1  to construe our   
sense-presentation as actual realisation
 of the  hypothetical thought-objects of
 perception4    ,the wholesale          
employment of purely hypothetical       
thought-objects of  perception enables  
science to explain some of the stray    
sense-objects  which cannot be construed
 as perceptions of a thought-object of  
perception3 for example1 sounds4 ,but   
the phenomena as a whole defy           
explanation on these lines until a      

further fundamental step is taken1      
which transforms the whole concept of   
the material universe4 ,namely1  the    
thought-object of perception is         
superseded by the thought-object of     
science4    ,the thought-objects of     
science are molecules1 atoms1 and       
electrons4 ,the  peculiarity of these   
objects is that they have shed all the  
qualities  which are capable of direct  
sense-representation in consciousness4  
,they are known to us only by their     
associated phenomena1 namely1  series of
 events in which they are implicated are
 represented in our  consciousness by   
sense-presentations4 ,in this way1 the  
thought-objects  of science are         
conceived as the causes of              
sense-representation4 ,the  transition  
from thought-objects of perception to   
thought-objects of  science is decently 
veiled by an elaborate theory concerning
 primary  and secondary qualities of    
bodies4    ,this device1 by which       
sense-presentations are represented in  

thought  as our perception of events in 
which thought-objects of science are    
implicated1 is the fundamental means by 
which a bridge is formed  between the   
fluid vagueness of sense and the exact  
definition of  thought4 ,in thought a   
proposition is either true or false1 an 
entity  is exactly what it is1 and      
relations between entities are          
expressible  7in idea7 by definite      
propositions about distinctly conceived 
entities4  ,sense-perception knows none 
of these things1 except by courtesy4    
,accuracy essentially collapses at some 
stage of inquiry4                       
,,iii4 ,time and ,space                 
,recapitulation4--,relations of time and
 relations of space  hold between       
sense-objects of perception4 ,these     
sense-objects are  distinguished as     
separate objects by the recognition of  
either 7#a7  differences of             
sense-content1 or 7#b7 time-relations   
between them  other than simultaneity1  
or 7#c7 space-relations between them    

other  than coincidence4 ,thus          
sense-objects arise from the recognition
 of  contrast within the complete stream
 of sense-presentation1 namely1 from    
the recognition of the objects as       
related terms1 by relations which       
contrast them4 ,differences of          
sense-content are infinitely complex in 
their variety4 ,their analysis under the
 heading of general ideas is the        
unending task of physical science4      
,time-relations and space-relations  are
 comparatively simple1 and the general  
ideas according to which  their analysis
 should proceed are obvious4    ,this   
simplicity of time and space is perhaps 
the reason why thought  chooses them as 
the permanent ground for objectival     
distinction1  throwing the various      
sense-objects thus obtainable into one  
heap1 as  a first crude thought-object  
of perception1 and thence1 as described 
above1 obtaining a thought-object of    
perception4 ,thus a thought-object  of  
perception conceived as in the present  

of a short duration is a  first crude   
thought-object of perception either     
actual or hypothetical4  ,such a        
thought-object of perception1 confined  
within a short duration1  takes on the  
space-relations of its component        
sense-objects within that  same         
duration4 ,accordingly thought-objects  
of perception1 conceived in  their whole
 extents1 have to each other the        
time-relationships of their  complete   
existences1 and within any small        
duration have to each other  the        
space-relationships of their component  
sense-objects which lie  within that    
duration4    ,relations bind together3  
thus thought-objects of perception are  
connected in time and in space4 ,the    
genesis of the objectival analysis  of  
sense-presentation is the recognition of
 sense-objects as distinct  terms in    
time-relations and space-relations3 thus
 thought-objects of  perception are     
separated by time and by space4         
,whole and ,part4--,a sense-object is   

part of the complete stream  of         
presentation4 ,this concept of being a  
part is merely the statement  of the    
relation of the sense-object to the     
complete sense-presentation  for that   
consciousness4 ,also a sense-object can 
be part of another  sense-object4 ,it   
can be a part in two ways1 namely1 a    
part in time  and a part in space4 ,it  
seems probable that both these concepts 
of  time-part and space-part are        
fundamental2 that is1 are concepts      
expressing relations which are directly 
presented to us1 and are not  concepts  
about concepts4 ,in that case no further
 definition of the  actual presentation 
is possible4 ,it may even then be       
possible to  define an adequate         
criterion of the occurrence of such a   
presentation4  ,for example1 adopting   
for the moment a realist metaphysic as  
to the  existence of the physical world 
of molecules and electrons1 the vision  
of a chair as occurring for some        
definite person at some definite  time  

is essentially indefinable4 ,it is his  
vision1 though each of us  guesses that 
it must be uncommonly like our vision   
under analogous  circumstances4 ,but the
 existence of the definable molecules   
and  waves of light in certain definable
 relations to his bodily organs  of     
sense1 his body also being in a certain 
definable state1 forms an  adequate     
criterion of the occurrence of the      
vision1 a criterion which  is accepted  
in ,courts of ,law and for physical     
science is tacitly  substituted for the 
vision4    ,the connection between the  
relations whole and part and all and    
some is intimate4 ,it can be explained  
thus so far as concerns directly        
presented sense-objects4 ,call two      
sense-objects separated if there  is no 
third sense-object which is a part of   
both of them4 ,then an  object ,a is    
composed of the two objects ,b and ,c1  
if 7#a7 ,b and ,c are  both parts of ,a1
 7#b7 ,b and ,c are separated1 and 7#c7 
there is no part of  ,a which is        

separated both from ,b and from ,c4 ,in 
such a case the class   which is        
composed of the two objects ,b and ,c is
 often substituted  in thought for the  
sense-object ,a4 ,but this process      
presupposes the  fundamental relation   
whole and part4 ,conversely the objects 
,b and ,c  may be actual sense-objects1 
but the sense-object ,a which           
corresponds  to the class  may remain   
hypothetical4 ,for example1 the round   
world  on which we live remains a       
conception corresponding to no single   
sense-object at any time presented in   
any human being's consciousness4    ,it 
is possible1 however1 that some mode of 
conceiving the  whole-and-part relation 
between extended objects as the         
all-and-some  relation of logical       
classes can be found4 ,but in this case 
the  extended objects as here conceived 
cannot be the true sense-objects  which 
are present to consciousness4 ,for as   
here conceived a part of a  sense-object
 is another sense-object of the same    

type2 and therefore  one sense-object   
cannot be a class of other              
sense-objects1 just as  a tea-spoon     
cannot be a class of other tea-spoons4  
,the ordinary way  in thought by which  
whole-and-part is reduced to            
all-and-some is by  the device of       
points1 namely1 the part of an object   
occupies some  of the points occupied by
 the whole object4 ,if any one holds    
that  in his consciousness the          
sense-presentation is a presentation of 
point-objects1 and that an extended     
object is merely a class of such        
point-objects collected together in     
thought1 then this ordinary method  is  
completely satisfactory4 ,we shall      
proceed on the assumption that  this    
conception of directly perceived        
point-objects has no relation to  the   
facts4    ,in the preceding address on  
,the ,organisation of ,thought1 another 
mode  is suggested4 ,but this method    
would apply only to the thought-object  
of perception1 and has no reference to  

the primary sense-objects here          
considered4 ,accordingly it must reckon 
as a subordinate device for a  later    
stage of thought4    ,thus the          
point-object in time and the            
point-object in space1 and the  double  
point-object both in time and space1    
must be conceived as  intellectual      
constructions4 ,the fundamental fact is 
the sense-object1  extended both in time
 and space1 with the fundamental        
relation of  whole-to-part to other such
 objects1 and subject to the law of     
convergence to simplicity as we proceed 
in thought through a series of          
successively contained parts4    ,the   
relation whole-to-part is a temporal or 
spatial relation1 and  is therefore     
primarily a relation holding between    
sense-objects of  perception1 and it is 
only derivatively ascribed to the       
thought-objects  of perception of which 
they are components4 ,more generally1   
space and  time relations hold primarily
 between sense-objects of perception and

  derivatively between thought-objects  
of perception4    ,definition of        
,points4--,the genesis of points of time
 and of  space can now be studied4 ,we  
must distinguish 7#a7 sense-time and    
sense-space1 and 7#b7 thought-time of   
perception and thought-space of         
perception4    ,sense-time and          
sense-space are the actually observed   
time-relations  and space-relations     
between sense-objects4 ,sense-time and  
sense-space  have no points except1     
perhaps1 a few sparse instances1        
sufficient  to suggest the logical idea2
 also1 sense-time and sense-space are   
discontinuous and fragmentary4          
,thought-time of perception and         
thought-space of perception are the time
  and space relations which hold between
 thought-objects of perception4         
,thought-time of perception and         
thought-space of perception are each    
continuous4 ,by continuous is here meant
 that all thought-objects of  perception
 have to each other a time 7or space7   

relation4    ,the origin of points is   
the effort to take full advantage of    
the principle of convergence to         
simplicity4 ,in so far as this          
principle does not apply1 a point is    
merely a cumbrous way of  directing     
attention to a set of relations between 
a certain set of  thought-objects of    
perception1 which set of relations1     
though actual so  far as a              
thought-object is actual1 is 7under this
 supposition7 of no  particular         
importance4 ,thus the proved importance 
in physical science  of the concepts of 
points in time and points in space is a 
tribute to  the wide applicability of   
this principle of convergence4          
,euclid defines a point as without parts
 and without magnitude4 ,in  modern     
language a point is often described as  
an ideal limit by  indefinitely         
continuing the process of diminishing a 
volume 7or  area74 ,points as thus      
conceived are often called convenient   
fictions4  ,this language is ambiguous4 

,what is meant by a fiction8 ,if it     
means  a conception which does not      
correspond to any fact1 there is some   
difficulty in understanding how it can  
be of any use in physical  science4 ,for
 example1 the fiction of a red man in a 
green coat  inhabiting the moon can     
never be of the slightest scientific    
service1  simply because--as we may     
presume--it corresponds to no fact4 ,by 
calling the concept of points a         
convenient fiction1 it must be meant    
that the concept does correspond to some
 important facts4 ,it is1 then1         
requisite1 in the place of such vague   
allusiveness1 to explain exactly  what  
are the facts to which the concept      
corresponds4    ,we are not much helped 
by explaining that a point is an ideal  
limit4  ,what is a limit8 ,the idea of a
 limit has a precise meaning in the     
theory of series1 and in the theory of  
the values of functions2 but  neither of
 these meanings apply here4 ,it may be  
observed that1 before  the ordinary     

mathematical meanings of limit had      
received a precise  explanation1 the    
idea of a point as a limit might be     
considered as  one among other examples 
of an idea only to be apprehended by    
direct  intuition4 ,this view is not now
 open to us4 ,thus1 again1 we are       
confronted with the question3 ,what are 
the precise properties meant  when a    
point is described as an ideal limit8   
,the discussion which now  follows is an
 attempt to express the concept of a    
point in terms of  thought-objects of   
perception related together by the      
whole-and-part  relation1 considered    
either as a time-relation or as a       
space-relation4  ,if it is so preferred1
 it may be considered that the          
discussion is  directed towards a       
precise elucidation of the term ideal   
limit as  often used in this connection4
    ,the subsequent explanations can be 
made easier to follow by a small  piece 
of symbolism3 ,let a,eb mean that b is  
part of a4 ,we need  not decide whether 

we are talking of time-parts or         
space-parts1 but  whichever choice is   
supposed to be made must be conceived as
 adhered to  throughout any connected   
discussion4 ,the symbol ,e may be       
considered  as the initial letter of    
encloses1 so we read a,eb as a  encloses
 b4 ,again the field of ,e is the set of
 things which  either enclose or are    
enclosed1 i4 e4 everything a1 which is  
such  that x can be found so that either
 a,ex or x,ea4 ,a member of the  field  
of ,e is called an enclosure-object4    
,now1 we assume that this relation of   
whole-to-part1 which in the future  we  
will call enclosure1 always satisfies   
the conditions in that the  relation ,e 
is 7#a7 transitive1 7#b7 asymmetrical1  
and 7#c7 with its  domain including its 
converse domain4    ,these four         
conditions deserve some slight          
consideration2 only the first  two of   
them embody hypotheses which enter      
vitally into the reasoning4             
,condition 7#a7 may be stated as the    

condition that a,eb and b,ec  always    
implies a,ec4 ,the fact that an entity b
 can be found such  that a,eb and b,ec  
may be conceived as a relation between a
 and  c4 ,it is natural to write ,e#b   
for this relation4 ,thus the condition  
is now written3 ,if a,e#b;c1 then a,ec4 
,this can be still otherwise  expressed 
by saying that the relation ,e#b        
implies1 whenever it holds1  that the   
relation ,e also holds4    ,condition   
7#b7 is partly a mere question of       
trivial definition1 and  partly a       
substantial assumption4 ,the            
asymmetrical relation 7,e7 is  such that
 a,eb and b,ea can never hold           
simultaneously4 ,this property  splits  
up into two parts3 7#a7 that no instance
 of a,eb and b,ea and  a diverse from b1
 can occur1 and 7#b7 that a,ea cannot   
occur4  ,the first part is a substantial
 assumption1 the second part 7so far as 
we are concerned7 reduces to the trivial
 convention that we shall not  consider 
an object as part of itself1 but will   

confine attention to  proper parts4     
,condition 7#c7 means that a,eb always  
implies that c can be  found such that  
b,ec4 ,this condition1 taken in         
conjunction with  the fact that we are  
only considering proper parts1 is the   
assertion of  the principle of the      
indefinite divisibility of extended     
objects1 both  in space and in time4    
,an indivisible part will lack duration 
in time1 and extension in  space1 and is
 thus an entity of essentially a        
different character  to a divisible     
part4 ,if we admit such indivisibles as 
the only true  sense-objects1 our       
subsequent procedure is an unnecessary  
elaboration4    ,it will be found that a
 fourth condition is necessary owing to 
logical difficulties connected with the 
theory of an infinite number  of        
choices4 ,it will not be necessary for  
us to enter further on  this question1  
which involves difficult considerations 
of abstract  logic4 ,the outcome is1    
that apart from hypothesis we cannot    

prove the  existence of the sets1 each  
containing an infinite number of        
objects1  which are here called points1 
as will be explained immediately4       
,now consider a set of enclosure objects
 which is such that 7#a7  of any two of 
its members one encloses the other1 and 
7#b7 there  is no member which is       
enclosed by all the others1 and 7#c7    
there  is no enclosure-object1 not a    
member of the set which is enclosed  by 
every member of the set4 ,call such a   
set a convergent set of                 
enclosure-objects4 ,as we pass along the
 series from larger to smaller  members1
 evidently we converge towards an ideal 
simplicity to any  degree of            
approximation to which we like to       
proceed1 and the series as  a whole     
embodies the complete ideal along that  
route of approximation4  ,in fact1 to   
repeat1 the series is a route of        
approximation4    ,we have now to       
inquire if the principle of convergence 
to simplicity  may be expected to yield 

the same type of simplicity for every   
such  convergent route4 ,the answer is1 
as we might expect1 namely1 that this   
depends upon the nature of the          
properties which are to be simplified4  
,for example1 consider the application  
to time4 ,now1 time is  one-dimensional2
 so when this property of               
one-dimensionality has been  expressed  
by the proper conditions1 not here      
stated1 a convergent set  of            
enclosure-objects must1 considered as a 
route of approximation1  exhibit the    
properties of one unique instant of     
time1 as ordinarily  conceived by the   
euclidean definition4 ,accordingly1     
whatever simplicity  is to be achieved  
by the application to time of the       
principle of  convergence to simplicity 
must be exhibited among the properties  
of  any such route of approximation4    
,for space1 different considerations    
arise4 ,owing to its multiple           
dimensions1 we can show that different  
convergent sets of  enclosure-objects1  

indicating different routes of          
approximation1 may  exhibit convergence 
to different types of simplicity1 some  
more complex  than others4    ,for      
example1 consider a rectangular box of  
height h ft41 breadth  b ft41 and       
thickness c ft4 ,now1 keep h and b      
constant1 and  let the central plane    
7height h1 breadth b7 perpendicular to  
the  thickness be fixed1 then make c    
diminish indefinitely4 ,we thus obtain  
a convergent series of an indefinitely  
large number of boxes1 and there  is no 
smallest box4 ,thus this convergent     
series exhibits the route of            
approximation towards the type of       
simplicity expressed as being a plane   
area of height h1 breadth b1 and no     
thickness4    ,again1 by keeping the    
central line of height h fixed1 and by  
making  b and c diminish indefinitely1  
the series converges to the segment  of 
a straight line of length h4            
,finally1 by keeping only the central   
point fixed1 and by making h1  b1 and c 

diminish indefinitely1 the series       
converges to a point4    ,furthermore1  
we have introduced as yet no concept    
which would prevent  an enclosure-object
 being formed of detached fragments in  
space4 ,thus  we can easily imagine a   
convergent set which converges to a     
number of  points in space4 ,for        
example1 each object of the set might be
 formed of  two not overlapping spheres 
of radius r1 with centres ,a and ,b4    
,then by diminishing r indefinitely1 and
 keeping ,a and ,b fixed1 we  have      
convergence to the pair of points ,a and
 ,b4    ,it remains now to consider how 
those convergent sets which converge to 
a single point can be discriminated from
 all the other types of such  sets1     
merely by utilising concepts founded on 
the relation of enclosure4    ,let us   
name convergent sets by ,greek letters2 
by proceeding forward  along any such   
set let us understand the process of    
continually passing  from the larger to 
the smaller enclosure-objects which form

 the set4    ,the convergent set  will  
be said to cover the convergent set 1 if
  every member of  encloses some members
 of 4 ,we notice that if an             
enclosure-object x encloses any member  
7y7 of 1 then every member  of the      
tail-end of 1 found by proceeding       
forward along  from y1  must be enclosed
 by x4 ,thus if  covers 1 every member  
of  encloses  every member of the       
tail-end of 1 starting from the largest 
member of   which is enclosed by that   
member of 4    ,it is possible for each 
of two convergent sets to cover the     
other4 ,for  example1 let one set 77 be 
a set of concentric spheres converging  
to  their centre ,a1 and the other set  
77 be a set of concentric  cubes1       
similarly situated1 converging to the   
same centre ,a4 ,then   and  will each  
cover the other4    ,let two convergent 
sets which are such that each covers the
 other be  called equal4    ,then it is 
a sufficient condition to secure that a 
convergent set   possesses the point    

type of convergence1 if every convergent
 set  covered by it is also equal to it1
 namely1  is a convergent set with  the 
punctual type of convergence1 if  covers
  always implies that   covers 4    ,it 
can easily be seen by simple examples   
that the other types of  convergence to 
surfaces or lines or sets of points     
cannot possess this  property4          
,consider1 for example1 the three       
convergent sets of boxes in  the        
preceding illustration1 which converge  
respectively to a central  plane1 a     
central line in the central plane1 and  
the central point in  the central line4 
,the first set covers the second and    
third sets1 and  the second set covers  
the third set1 but no two of the sets   
are equal4    ,it is a more difficult   
question to determine whether the       
condition here  indicated as sufficient 
to secure the punctual type of          
convergence  is also necessary4 ,the    
question turns on how far               
thought-objects  of perception possess  

exact boundaries prior to the           
elaboration of  exact mathematical      
concepts of space4 ,if they are to be   
conceived as  possessing such exact     
boundaries1 then convergent sets        
converging to  points on such boundaries
 must be allowed for4 ,the procedure    
necessary  for the specification of the 
complete punctual condition becomes then
  very elaborate1#c and will not be     
considered here4    ,but such exact     
determination as is involved in the     
conception of  an exact spatial boundary
 does not seem to belong to the true    
thought-object of perception4 ,the      
ascription of an exact boundary  really 
belongs to the transition stage of      
thought as it passes from  the          
thought-object of perception to the     
thought-object of science4  ,the        
transition from the sense-object        
immediately presented to the            
thought-object of perception is         
historically made in a wavering         
indeterminate line of thought4 ,the     

definite stages here marked out  simply 
serve to prove that a logically         
explicable transition is  possible4     
,we accordingly assume that the         
condition laid down above to secure the 
punctual convergence of a convergent set
 of enclosure-objects is not  only      
sufficient1 but necessary4    ,it can be
 proved that1 if two convergent sets of 
enclosure-objects are  both equal to a  
third convergent set1 they are equal to 
each other4  ,consider now any punctual 
convergent set 774 ,we want to define   
the  point to which  is a route of      
approximation in a way which is  neutral
 between  and all the convergent sets   
which are equal to 4  ,each of these    
sets is a route of approximation to the 
same point  as 4 ,this definition is    
secured if we define the point as the   
class  formed by all the                
enclosure-objects which belong either to
  or to  any convergent set which is    
equal to 4 ,let ,p be this class  of    
enclosure-objects4 ,then any convergent 

set 77 which consists of                
enclosure-objects entirely selected from
 members of the class ,p  must be a     
route of approximation to the same point
 as does the  original punctual set 2   
namely1 provided that we choose a small 
enough  enclosure-object in 1 we can    
always find a member of  which encloses 
it2 and provided that we choose a small 
enough enclosure-object in  1 we can    
always find a member of  which encloses 
it4 ,thus ,p  only includes convergent  
sets of the punctual type1 and the route
  of approximation indicated by any two 
convergent sets selected from  ,p       
converges to identical results4    ,the 
,uses of ,points4--,the sole use of     
points is to facilitate the  employment 
of the principle of ,convergence to     
,simplicity4 ,by this  principle some   
simple relations in appropriate         
circumstances become  true1 when objects
 are considered which are sufficiently  
restricted in  time or in space4 ,the   
introduction of points enables this     

principle to  be carried through to its 
ideal limit4 ,for example1 suppose g 7a1
  b1 c7 represents some statement       
concerning three enclosure-objects1  a1 
b1 c1 which may be true if the objects  
are sufficiently  restricted in extent4 
,let ,a1 ,b1 ,c be three given points1  
then we define g 7,a1 ,b1 ,c7 to mean   
that whatever three  enclosure-objects  
a1 b1 c are chosen1 such that a is a    
member  of ,a1 b of ,b1 and c of ,c1 it 
is always possible to find  three other 
members of ,a1 ,b1 ,c1 namely1 x a      
member of ,a1 y  of ,b1 and z of ,c1    
such that a,ex1 b,ey1 c,ez1 and g 7x1   
y1 z74 ,so by going far enough down in  
the tail-ends of ,a1 ,b1  ,c we can     
always secure three objects x1 y1 z for 
which g  7x1 y1 z7 is true4    ,for     
example1 let g 7,a1 ,b1 ,c7 mean ,a1 ,b1
 ,c are three  points in a linear row4  
,this must be construed to mean that    
whatever  three objects a1 b1 c we      
choose1 members of ,a1 ,b1 ,c           
respectively1 we can always find three  

objects x1 y1 z1 also  members of ,a1   
,b1 ,c respectively1 and such that a    
encloses x1  b encloses y1 c encloses z1
 and also such that x1 y1 z  are in a   
linear row4    ,sometimes a double      
convergence is necessary1 namely1 a     
convergence of  conditions as well as a 
convergence of objects4 ,for example1   
consider  the statement1 the points ,a  
and ,b are two feet apart4 ,now1 the    
exact statement two feet apart does not 
apply to objects4 ,for objects  x and y 
we must substitute the statement1 the   
distance between  x and y lies between  
the limits 7#b  e7 feet4 ,here e is some
  number1 less than two1 which we have  
chosen for this statement4 ,then  the   
points ,a and ,b are two feet apart2 if1
 however we choose the  number e1       
whatever enclosure-objects a and b1     
members of ,a and  ,b respectively1 we  
consider1 we can always find            
enclosure-objects x  and y1 members of  
,a and ,b respectively1 such that a     
encloses  x and b encloses y1 and also  

such that the distance between x  and y 
lies between the limits 7#b  e7 feet4   
,it is evident1 since  e can be chosen  
as small as we please1 that this        
statement exactly  expresses the        
condition that ,a and ,b are two feet   
apart4    ,straight ,lines and          
,planes4--,but the problem of the       
intellectual  construction of straight  
lines and planes is not yet sufficiently
  analysed4 ,we have interpreted the    
meaning of the statement that three  or 
more points are collinear1 and can      
similarly see how to interpret  the     
meaning of the statement that four or   
more points are coplanar1  in either    
case deriving the exact geometrical     
statements from vaguer  statements      
respecting extended objects4    ,this   
procedure only contemplates groups of   
finite numbers of points4  ,but straight
 lines and planes are conceived as      
containing infinite  numbers of points4 
,this completion of lines and planes is 
obtained  by a renewed application of   

the principle of aggregation1 just as  a
 set of first crude thought-objects of  
perception are aggregated  into one     
complete thought-object of perception4  
,in this way repeated  judgments of the 
collinearity of sets of points are      
finally1 when  certain conditions of    
interlacing are fulfilled1 aggregated in
 the  single judgment of all the points 
of the groups as forming one whole      
collinear group4 ,similarly for         
judgments of coplanarity4 ,this process 
of logical aggregation can be exhibited 
in its exact logical analysis4  ,but it 
is unnecessary here to proceed to such  
details4 ,thus we conceive  our points  
as sorted into planes and straight      
lines1 concerning which  the various    
axioms of geometry hold4 ,these axioms1 
in so far as they  essentially require  
the conception of points1 are capable of
 being  exhibited as the outcome of     
vaguer1 less exact judgments respecting 
the  relations of extended objects4     
,empty ,space4--,it must be observed    

that the points1 hitherto  defined1     
necessarily involve thought-objects of  
perception1 and lie  within the         
space-extension occupied by such        
objects4 ,it is true that  such objects 
are largely hypothetical1 and that we   
can bring into our  hypotheses          
sufficient objects to complete our lines
 and planes4 ,but  every such hypothesis
 weakens the connection between our     
scientific  concept of nature and the   
actual observed facts which are involved
 in  the actual sense-presentations4    
,occam's razor1 ,entia non multiplicanda
 prter necessitatem1  is not an         
arbitrary rule based on mere logical    
elegancy4 ,nor is its  application      
purely confined to metaphysical         
speculation4 ,i am ignorant  of the     
precise reason for its metaphysical     
validity1 but its scientific  validity  
is obvious1 namely1 every use of        
hypothetical entities  diminishes the   
claim of scientific reasoning to be the 
necessary  outcome of a harmony between 

thought and sense-presentation4 ,as     
hypothesis increases1 necessity         
diminishes4    ,commonsense thought also
 supports this refusal to conceive of   
all  space as essentially depending on  
hypothetical objects which fill it4  ,we
 think of material objects as filling   
space1 but we ask whether any  objects  
exist between the ,earth and the ,sun1  
between the stars1 or  beyond the stars4
 ,for us1 space is there2 the only      
question is whether  or not it be full4 
,but this form of question presupposes  
the meaning of  empty space1 namely1 of 
space not containing hypothetical       
objects4    ,this brings up a wider use 
of the concept of points1 necessitating 
a wider definition4 ,hitherto we have   
conceived points as indicating          
relations of enclosure between objects4 
,we thus arrive at what now  we will    
term material points4 ,but the idea of  
points can now be  transformed so as to 
indicate the possibilities of external  
relations  not those of enclosure4 ,this

 is effected by an enlargement of the   
concept of ideal points1 already known  
to geometers4    ,define material lines 
to be complete collinear classes of     
collinear  points4 ,consider now the set
 of material lines which contain a      
certain  material point4 ,call such a   
set of lines an ideal point4 ,this set  
of  lines indicates a possibility of    
position1 which is in fact occupied  by 
that material point common to all the   
material lines4 ,so this ideal  point is
 an occupied ideal point4 ,now consider 
a set of three material  lines1 such    
that any two are coplanar1 but not the  
whole three1 and  further consider the  
complete set of material lines such that
 each is  coplanar with each of the     
three material lines first chosen4 ,the 
axioms  which hold for the material     
lines will enable us to prove that any  
two  lines of this set are coplanar4    
,then the whole set of lines1 including 
the three original lines1 forms an ideal
 point1 according to the  definition in 

its full generality4 ,such an ideal     
point may be occupied4  ,in that case   
there is a material point common to all 
the lines of the  set1 but it may be    
unoccupied4 ,then the ideal point merely
 indicates  a possibility of spatial    
relations which has not been realised4  
,it is  the point of empty space4 ,thus 
the ideal points1 which may or may not  
be occupied1 are the points of geometry 
viewed as an applied science4  ,these   
points are distributed into straight    
lines and planes4 ,but any  further     
discussion of this question will lead us
 into the technical  subject of the     
axioms of geometry and their immediate  
consequences4  ,enough has been said to 
show how geometry arises according to   
the  relational theory of space4        
,space as thus conceived is the         
thought-space of the material world4    
,,iv4 ,fields of ,force    ,the         
thought-objects of science are conceived
 as directly related to  this           
thought-space4 ,their spatial relations 

are among those indicated  by the points
 of the thought-space4 ,their emergence 
in science has  been merely a further   
development of processes already        
inherent in  commonsense thought4       
,relations within the complete          
sense-presentation were represented  in 
thought by the concept of               
thought-objects of perception4 ,all     
sense-presentation could not be         
represented in this way2 also the       
change and disappearance of             
thought-objects occasioned confusion  of
 thought4 ,a reduction to order of this 
confusion was attempted  by the concepts
 of permanent matter with primary and   
secondary  qualities4 ,finally1 this has
 issued in the secondary qualities being
  traced as perception of events        
generated by the objects1 but--as       
perceived--entirely disconnected with   
them4 ,also the thought-objects  of     
perception have been replaced by        
molecules and electrons and             
ether-waves1 until at length it is never

 the thought-object of science  which is
 perceived1 but complicated series of   
events in which they are  implicated4   
,if science be right1 nobody ever       
perceived a thing1 but  only an event4  
,the result is1 that the older language 
of philosophy  which still survives in  
many quarters is now thoroughly         
confusing  when brought into connection 
with the modern concepts of science4    
,philosophy--that is1 the older         
philosophy--conceives the thing as      
directly perceived4 ,according to       
scientific thought1 the ultimate thing  
is never perceived1 perception          
essentially issuing from a series of    
events4 ,it is impossible to reconcile  
the two points of view4    ,the         
advantage of the modern scientific      
concept is that it is enabled  to       
explain the fluid vague outlines of     
sense-presentation4 ,the  thought-object
 of perception is now conceived as a    
fairly stable  state of motion of a huge
 group of molecules1 constantly         

changing1  but preserving a certain     
identity of characteristics4 ,also stray
  sense-objects1 not immediately given  
as part of a thought-object of          
perception1 are now explicable3 the     
dancing light-reflection1 the  vaguely  
heard sound1 the smell4 ,in fact1 the   
perceived events of  the scientific     
world have the same general definition  
and lack of  definition1 and the same   
general stability and lack of stability1
  as the sense-objects of the complete  
sense-presentation or as the            
thought-objects of perception4    ,the  
thought-objects of science1 namely1     
molecules1 atoms1 and  electrons1 have  
gained in permanence4 ,the events are   
reduced to changes  in                  
space-configuration4 ,the laws          
determining these changes are the       
ultimate laws of nature4    ,the laws of
 change in the physical universe proceed
 on the assumption  that the preceding  
states of the universe determine the    
character of  the change4 ,thus1 to know

 the configurations and events of the   
universe  up to and including any       
instant would involve sufficient data   
from  which to determine the succession 
of events throughout all time4    ,but  
in tracing the antecedents of events1   
commonsense thought1 dealing  with the  
world of thought-objects of perception1 
habitually assumes  that the greater    
number of antecedent events can be      
neglected as  irrelevant4 ,consideration
 of causes is restricted to a few events
  during a short preceding interval4    
,finally1 in scientific thought it  has 
been assumed that the events in an      
arbitrarily small preceding  duration   
are sufficient4 ,thus physical          
quantities and their successive         
differential co-efficients up to any    
order at the instant1 but with  their   
limiting values just before that        
instant1 are on this theory  sufficient 
to determine the state of the universe  
at all times after  the instant4 ,more  
particular laws are assumed4 ,but the   

search for  them is guided by this      
general principle4 ,also it is assumed  
that the  greater number of events in   
the physical universe are irrelevant to 
the production of any particular effect1
 which is assumed to issue  from        
relatively few antecedents4 ,these      
assumptions have grown out of  the      
experience of mankind4 ,the first lesson
 of life is to concentrate  attention on
 few factors of sense-presentations1 and
 on still fewer of  the universe of     
thought-objects of perception4    ,the  
principle by which--consciously or      
unconsciously--thought has been  guided 
is that in searching for particular     
causes1 remoteness in time  and         
remoteness in space are evidences of    
comparative disconnection  of influence4
 ,the extreme form of this principle is 
the denial of  any action at a distance 
either in time or space4 ,the difficulty
 in  accepting this principle in its    
crude form is1 that since there are no  
contiguous points1 only coincident      

bodies can act on each other4 ,i can    
see no answer to this                   
difficulty--namely1 either bodies have  
the same  location and are thus         
coincident1 or they have different      
locations and  are thus at a distance   
and do not act on each other4    ,this  
difficulty is not evaded by the         
hypothesis of an ether1  continuously   
distributed4 ,for two reasons3 in the   
first place1 the  continuity of the     
ether does not avoid the dilemma2 and   
secondly1 the  difficulty applies to    
time as well as to space1 and the       
dilemma would  prove that causation     
producing change is impossible1 namely1 
no changed  condition could be the      
result of antecedent circumstances4     
,on the other hand1 a direct interaction
 between two bodies separated  in space 
undoubtedly offends the conception of   
distance as implying  physical          
disconnection as well as spatial        
relation4 ,there is no logical          
difficulty in the assumption of action  

at a distance as in the case  of its    
denial1 but it is contradictory to      
persistent assumptions of  that         
apparatus of commonsense thought which  
it is the main business  of science to  
harmonise with sense-presentation1      
employing only the  minimum of          
modification4    ,modern science is     
really unconcerned with this debate4    
,its  7unacknowledged7 conceptions are  
really quite different1 though the      
verbal explanations retain the form of a
 previous epoch4 ,the point of  the     
change in conception is that the old    
thought-object of science was  conceived
 as possessing a simplicity not         
belonging to the material  universe as a
 whole4 ,it was secluded within a finite
 region of space1  and changes in its   
circumstances could only arise from     
forces which  formed no essential part  
of its nature4 ,an ether was called into
  existence to explain the active       
relations between these passive         
thought-objects4 ,the whole conception  

suffers from the logical  difficulties  
noted above4 ,also no clear conception  
can be formed of  the sense in which the
 ether is explanatory4 ,it is to possess
 a type  of activity denied to the      
original thought-object1 namely1 it     
carries  potential energy1 whereas the  
atom possessed only kinetic energy1  the
 so-called potential energy of an atom  
belonging really to the  surrounding    
ether4 ,the truth is1 that ether is     
really excepted from the  axiom no      
action at a distance1 and the axiom     
thereby is robbed of all  its force4    
,the modern thought-object of           
science--not yet explicitly             
acknowledged--has the complexity of the 
whole material universe4 ,in  physics1  
as elsewhere1 the hopeless endeavour to 
derive complexity  from simplicity has  
been tacitly abandoned4 ,what is aimed  
at is not  simplicity1 but persistence  
and regularity4 ,in a sense regularity  
is a sort of simplicity4 ,but it is the 
simplicity of stable mutual  relations1 

and not the simplicity of absence of    
types of internal  structure or of type 
of relation4 ,this thought-object fills 
all space4  ,it is a field2 that is to  
say1 it is a certain distribution of    
scalar and vector quantities throughout 
space1 these quantities having  each its
 value for each point of space at each  
point of time1 being  continuously      
distributed throughout space and        
throughout time1 possibly  with some    
exceptional discontinuities4 ,the       
various types of quantity  which form   
the field have fixed relations to each  
other at each point  of time and space4 
,these relations are the ultimate laws  
of nature4    ,for example1 consider an 
electron4 ,there is a scalar            
distribution of  electricity1 which is  
what is ordinarily called the electron4 
,this  scalar distribution has a        
volume-density  at the time t at any    
point 7x1 y1 z74 ,thus  is a function of
 7x1 y1 z1 t71  which is zero except    
within a restricted region4             

,furthermore1 at  any time t1 as an     
essential adjunct1 there is a continuous
 space  distribution at each point of   
the two vectors 7,x1 ,y1 ,z71 which is  
the electric force1 and 71 1 71 which is
 the magnetic force4 ,lastly1           
individuality is ascribed to the scalar 
electric distribution1 so that  in      
addition to its conservation of         
quantity--involved in the assumed       
laws--it is also possible to assign the 
velocities with which the  various      
individual parts of the distribution are
 moving4 ,let 7u1 v1  w7 be this        
velocity at 7x1 y1 z1 t74    ,this whole
 scheme of scalar and vector quantities1
 namely1 1 7,x1  ,y1 ,z71 71 1 71 7u1 v1
 w7 is interconnected by the            
electromagnetic laws4 ,it follows from  
these laws that the electron1 in  the   
sense of the scalar distribution 1 is to
 be conceived as at each  instant       
propagating from itself an emanation    
which travels outwards  with the        
velocity of light in vacuo1 and from    

which 7,x1 ,y1  ,z7 and 71 1 7 can be   
calculated1 so far as they are due to   
that  electron4 ,thus the field1 at any 
time1 due to the electron as a whole    
depends on the previous history of the  
electron1 the nearer to the  electron   
the more recent being the relevant      
history4 ,the whole scheme  of such a   
field is one single thought-object of   
science3 the electron  and its          
emanations form one essential whole1    
namely one thought-object  of science1  
essentially complex and essentially     
filling all space4 ,the  electron       
proper1 namely1 the scalar distribution 
1 is the focus of the  whole1 the       
essential focal property being that the 
field at any instant  is completely     
determined by the previous history of   
the focus and of  its space relations   
through all previous time4 ,but the     
field and the  focus are not independent
 concepts1 they are essentially         
correlated in  one organised unity1     
namely1 they are essentially correlated 

terms in  the field of one relation in  
virtue of which the entities enter into 
our thoughts4    ,the fields of a group 
of electrons are superposed according to
 the  linear law for aggregation1       
namely1 pure addition for analogous     
scalar  quantities and the parallelogram
 law for analogous vectors4 ,the changes
  in motion of each electron depend     
entirely on the resultant field in  the 
region it occupies4 ,thus a field can be
 viewed as a possibility of  action1 but
 a possibility which represents an      
actuality4    ,it is to be noted that   
the two alternative views of causation  
are  here both included4 ,the complete  
field within any region of space        
depends on the past histories of all the
 electrons1 histories extending         
backwards in proportion to their        
distances4 ,also this dependence can  be
 conceived as a transmission4 ,but      
viewing the cause which effects  changes
 on the electron within that region1 it 
is solely that field  within the region1

 which field is coincident with that    
electron both in  time and in space4    
,this process of conceiving the         
actuality underlying a possibility is   
the uniform process by which regularity 
and permanence is introduced  into      
scientific thought1 namely1 we proceed  
from the actuality of the  fact to the  
actuality of possibility4    ,in        
conformity with this principle1         
propositions are the outgrowth  from    
actual thought-expressions1             
thought-objects of perceptions from     
crude sense-objects1 hypothetical       
thought-objects of perception from      
actual thought-objects of perception1   
material points from hypothetical       
infinite suites of hypothetical         
thought-objects of perception1  ideal   
points from material points1            
thought-objects of science from         
thought-objects of perception1 fields of
 electrons from actual mutual  reactions
 of actual electrons4    ,the process is
 a research for permanence1 uniformity1 

and simplicity  of logical relation4    
,but it does not issue in simplicity of 
internal  structure4 ,each ultimate     
thought-object of science retains every 
quality attributed to the whole         
scientific universe1 but retains them   
in a form characterised by permanence   
and uniformity4                         
,v4 ,conclusion    ,we commenced by     
excluding judgments of worth and        
ontological judgments4  ,we conclude by 
recalling them4 ,judgments of worth are 
no part of  the texture of physical     
science1 but they are part of the motive
 of  its production4 ,mankind have      
raised the edifice of science1 because  
they have judged it worth while4 ,in    
other words1 the motives involve        
innumerable judgments of value4 ,again1 
there has been conscious  selection of  
the parts of the scientific field to be 
cultivated1 and  this conscious         
selection involves judgments of value4  
,these values may  be sthetic1 or moral1
 or utilitarian1 namely1 judgments as to

 the  beauty of the structure1 or as to 
the duty of exploring the truth1 or  as 
to utility in the satisfaction of       
physical wants4 ,but whatever the       
motive1 without judgments of value there
 would have been no science4    ,again1 
ontological judgments were not excluded 
by reason of any lack  of interest4     
,they are in fact presupposed in every  
act of life3 in our  affections1 in our 
self-restraints1 and in our constructive
 efforts4  ,they are presupposed in     
moral judgments4 ,the difficulty about  
them is  the absence of agreement as to 
the method of harmonising the crude     
judgments of commonsense4    ,science   
does not diminish the need of a         
metaphysic4 ,where this need is  most   
insistent is in connection with what    
above has been termed the  actuality    
underlying a possibility4 ,a few words  
of explanation may  render the argument 
clearer1 although they involve a rash   
approach  to metaphysical heights which 
it is not the purpose of this paper to  

explore4    ,the conception of subject  
and object in careless discussion covers
  two distinct relations4 ,there is the 
relation of the whole perceiving        
consciousness to part of its own        
content1 for example1 the relation  of a
 perceiving consciousness to an object  
of redness apparent to it4  ,there is   
also the relation of a perceiving       
consciousness to an entity  which does  
not exist in virtue of being part of the
 content of that  consciousness4 ,such a
 relation1 so far as known to the       
perceiving  consciousness1 must be an   
inferred relation1 the inference being  
derived from an analysis of the content 
of the perceiving consciousness4    ,the
 bases for such inferences must be      
elements in consciousness  directly     
known as transcending their immediate   
presentation in  consciousness4 ,such   
elements are universal logical truths1  
moral and  sthetic truths1 and truths   
embodied in hypothetical propositions4  
,these are the immediate objects of     

perception which are other than the     
mere affections of the perceiving       
subject4 ,they have the property of     
being parts of the immediate            
presentations for individual subjects   
and  yet more than such parts4 ,all     
other existence is inferred existence4  
,in this chapter we are more directly   
concerned with truths embodied  in      
hypothetical propositions4 ,such truths 
must not be confused with  any          
doubtfulness which attaches to our      
judgments of the future course  of      
natural phenomena4 ,a hypothetical      
proposition1 like a categorical         
judgment1 may or may not be doubtful4   
,also like a categorial judgment1  it   
expresses a fact4 ,this fact is twofold3
 as a presentation in  consciousness1 it
 is just this hypothetical judgment2 as 
expressing a  categorical fact1 it      
states a relation which lies beyond     
consciousness1  holding between entities
 thereby inferred4    ,but this         
metaphysical analysis1 short though it  

be1 is probably wrong1  and at the best 
will only command very partial assent4  
,certainly2  and this admission brings  
out the very point which ,i wished to   
make4  ,physical science is based on    
elements of thought1 such as judgments  
registering actual perceptions1 and     
judgments registering hypothetical      
perceptions which under certain         
circumstances would be realised4  ,these
 elements form the agreed content of the
 apparatus of commonsense  thought4     
,they require metaphysical analysis2 but
 they are among the  data from which    
metaphysics starts4 ,a metaphysic which 
rejects them  has failed1 in the same   
way as physical science has failed when 
it is  unable to harmonise them into its
 theory4    ,science only renders the   
metaphysical need more urgent4 ,in      
itself  it contributes little directly  
to the solution of the metaphysical     
problem4 ,but it does contribute        
something1 namely1 the exposition of    
the fact that our experience of sensible

 apparent things is capable of  being   
analysed into a scientific theory1 a    
theory not indeed complete1  but giving 
every promise of indefinite expansion4  
,this achievement  emphasises the       
intimate relation between our logical   
thought and the  facts of sensible      
apprehension4 ,also the special form of 
scientific  theory is bound to have some
 influence4 ,in the past false science  
has been the parent of bad metaphysics4 
,after all1 science embodies  a rigorous
 scrutiny of one part of the whole      
evidence from which  metaphysicians     
deduce their conclusions4               
,,footnotes3    ,footnote #c3 ,cf4 ,rvue
 de ,mtaphysique et de ,morale1 ,may    
#aiaf1  where this question is dealt    
with by the author at the end of an     
article1 ,la thorie relationniste de    
l'espace4                               
,,chapter ,,viii                        
,,space1 ,,time1 ,,and ,,relativity     
7,paper read to ,section ,a at the      
,manchester ,meeting of the ,british    

,association1 #aiae1 and later before   
the ,aristotelian ,society7             
,,fundamental ,problems concerning space
 and time have been considered  from the
 standpoints created by many different  
sciences4 ,the object of  this paper is 
the humble one of bringing some of these
 standpoints into  relation with each   
other4 ,this necessitates a very cursory
 treatment of  each point of view4      
,mathematical physicists have evolved   
their theory of relativity to  explain  
the negative results of the             
,morley-,michelson experiment and of    
the ,trouton experiment4 ,experimental  
psychologists have considered the       
evolution of spatial ideas from the     
crude sense-data of experience4         
,metaphysicians have considered the     
majestic uniformity of space and  time1 
without beginning and without end1      
without boundaries1 and  without        
exception in the truths concerning them2
 all these qualities  the more arresting
 to our attention from the confused     

accidental  nature of the empirical     
universe which is conditioned by them4  
,mathematicians have studied the axioms 
of geometry1 and can now deduce  all    
that is believed to be universally true 
of space and of time by the  strictest  
logic from a limited number of          
assumptions4    ,these various lines of 
thought have been evolved with          
surprisingly  little interconnection4   
,perhaps it is as well4 ,the results of 
science  are never quite true4 ,by a    
healthy independence of thought perhaps 
we  sometimes avoid adding other        
people's errors to our own4 ,but there  
can  be no doubt that the normal method 
of cross-fertilising thought is by      
considering the same1 or allied problems
 to our own1 in the form which  they    
assume in other sciences4    ,here ,i do
 not propose to enter into a systematic 
study of these various  chapters of     
science4 ,i have neither the knowledge  
nor the time4    ,first1 let us take the
 ultimate basis of any theory of        

relativity4  ,all space measurement is  
from stuff in space to stuff in space4  
,the  geometrical entities of empty     
space never appear4 ,the only           
geometrical  properties of which we have
 any direct knowledge are properties of 
those shifting1 changeable appearances  
which we call things in space4  ,it is  
the sun which is distant1 and the ball  
which is round1 and the  lamp-posts     
which are in linear order4 ,wherever    
mankind may have got its  idea of an    
infinite unchangeable space from1 it is 
safe to say that it  is not an immediate
 deliverance of direct observation4     
,there are two antagonistic             
philosophical ways of recognising this  
conclusion4    ,one is to affirm that   
space and time are conditions for       
sensible  experience1 that without      
projection into space and time sensible 
experience would not exist4 ,thus1      
although it may be true to say that  our
 knowledge of space and time is given in
 experience1 it is not true  to say that

 it is deduced from experience in the   
same sense that the  ,law of            
,gravitation is so deduced4 ,it is not  
deduced1 because in the act  of         
experiencing we are necessarily made    
aware of space as an infinite  given    
whole1 and of time as an unending       
uniform succession4 ,this  philosophical
 position is expressed by saying that   
space and time are  a priori forms of   
sensibility4    ,the opposed            
philosophical method of dealing with the
 question is  to affirm that our        
concepts of time and space are          
deductions from  experience1 in exactly 
the same way as the ,law of ,gravitation
 is such  a deduction4 ,if we form exact
 concepts of points1 lines and surfaces1
  and of successive instants of time1   
and assume them to be related as        
expressed by the axioms of geometry and 
the axioms for time1 then we  find that 
we have framed a concept which1 with all
 the exactness of  which our            
observations are capable1 expresses the 

facts of experience4    ,these two      
philosophic positions are each designed 
to explain a  certain difficulty4 ,the a
 priori theory explains the absolute    
universality ascribed to the laws of    
space and time1 a universality  not     
ascribed to any deduction from          
experience4 ,the experiential theory    
explains the derivation of the          
space-time concepts without introducing 
any other factors beyond those which are
 admittedly present in framing  the     
other concepts of physical science4     
,but we have not yet done with the      
distinctions which in any discussion  of
 space or time must essentially be kept 
in mind4 ,put aside the  above question 
as to how these space-time concepts are 
related to  experience--,what are they  
when they are formed8    ,we may        
conceive of the points of space as      
self-subsistent entities  which have the
 indefinable relation of being occupied 
by the ultimate  stuff 7matter1 ,i will 
call it7 which is there4 ,thus1 to say  

that the  sun is there 7wherever it is7 
is to affirm the relation of  occupation
 between the set of positive and        
negative electrons which  we call the   
sun and a certain set of points1 the    
points having an  existence essentially 
independent of the sun4 ,this is the    
absolute  theory of space4 ,the absolute
 theory is not popular just now1 but it 
has very respectable authority on its   
side--,newton1 for one--so treat  it    
tenderly4    ,the other theory is       
associated with ,leibniz4 ,our space    
concepts are  concepts of relations     
between things in space4 ,thus there is 
no such  entity as a self-subsistent    
point4 ,a point is merely the name for  
some  peculiarity of the relations      
between the matter which is1 in common  
language1 said to be in space4    ,it   
follows from the relative theory that a 
point should be definable in  terms of  
the relations between material things4  
,so far as ,i am aware1  this outcome of
 the theory has escaped the notice of   

mathematicians1  who have invariably    
assumed the point as the ultimate       
starting ground  of their reasoning4    
,many years ago ,i explained some types 
of ways in  which we might achieve such 
a definition1 and more recently have    
added  some others4 ,similar            
explanations apply to time4 ,before the 
theories  of space and time have been   
carried to a satisfactory conclusion on 
the relational basis1 a long and careful
 scrutiny of the definitions  of points 
of space and instants of time will have 
to be undertaken1  and many ways of     
effecting these definitions will have to
 be tried and  compared4 ,this is an    
unwritten chapter of mathematics1 in    
much the same  state as was the theory  
of parallels in the eighteenth century4 
,in this connection ,i should like to   
draw attention to the analogy  between  
time and space4 ,in analysing our       
experience we distinguish  events1 and  
we also distinguish things whose        
changing relations form  the events4 ,if

 ,i had time it would be interesting to 
consider more  closely these concepts of
 events and of things4 ,it must suffice 
now to  point out that things have      
certain relations to each other which we
  consider as relations between the     
space extensions of the things2 for     
example1 one space can contain the      
other1 or exclude it1 or overlap  it4 ,a
 point in space is nothing else than a  
certain set of relations  between       
spatial extensions4    ,analogously1    
there are certain relations between     
events which we  express by saying that 
they are relations between the temporal 
durations of these events1 that is1     
between the temporal extensions of  the 
events4 ,the durations of two events ,a 
and ,b may one precede the  other1 or   
may partially overlap1 or may one       
contain the other1 giving  in all six   
possibilities4 ,the properties of the   
extension of an event  in time are      
largely analogous to the extension of an
 object in space4  ,spatial extensions  

are expressed by relations between      
objects1 temporal  extensions by        
relations between events4    ,the point 
in time is a set of relations between   
temporal extensions4  ,it needs very    
little reflection to convince us that a 
point in time  is no direct deliverance 
of experience4 ,we live in durations1   
and not  in points4 ,but what community1
 beyond the mere name1 is there between 
extension in time and extension in      
space8 ,in view of the intimate         
connection between time and space       
revealed by the modern theory of        
relativity1 this question has taken on a
 new importance4    ,i have not thought 
out an answer to this question4 ,i      
suggest1 however1  that time and space  
embody those relations between objects  
on which  depends our judgment of their 
externality to ourselves4 ,namely1      
location in space and location in time  
both embody and perhaps  necessitate a  
judgment of externality4 ,this          
suggestion is very vague1  and ,i must  

leave it in this crude form4            
,diverse ,euclidean ,measure ,systems   
,turning now to the mathematical        
investigations on the axioms of         
geometry1 the outcome1 which is most    
important for us to remember1  is the   
great separation which it discloses     
between non-metrical  projective        
geometry1 and metrical geometry4        
,non-metrical projective  geometry is by
 far the more fundamental4 ,starting    
with the concepts  of points1 straight  
lines1 and planes 7of which not all     
three need  be taken as indefinable71   
and with certain very simple            
non-metrical  properties of these       
entities--such as1 for instance1 that   
two points  uniquely determine a        
straight line--nearly the whole of      
geometry can  be constructed4 ,even     
quantitative coordinates can be         
introduced1 to  facilitate the          
reasoning4 ,but no mention of distance1 
area1 or volume1  need have been        
introduced4 ,points will have an order  

on the line1 but  order does not imply  
any settled distance4    ,when we now   
inquire what measurements of distance   
are possible1 we find  that there are   
different systems of measurement all    
equally possible4  ,there are three main
 types of system3 any system of one type
 gives  ,euclidean geometry1 any system 
of another type gives ,hyperbolic 7or   
,lobatchewskian7 geometry1 any system of
 the third type gives ,elliptic         
geometry4 ,also different beings1 or the
 same being if he chooses1  may reckon  
in different systems of the same type1  
or in systems of  different types4      
,consider the example which will        
interest us later4  ,two beings1 ,a and 
,b1 agree to use the same three         
intersecting lines  as axes of x1 y1 z4 
,they both employ a system of           
measurement of  the ,euclidean type1 and
 7what is not necessarily the case7     
agree as to  the plane at infinity4     
,that is1 they agree as to the lines    
which are  parallel4 ,then with the     

usual method of rectangular ,cartesian  
axes1  they agree that the coordinates  
of ,p are the lengths ,,on1 ,,nm1 ,,mp4 
,so  far all is harmony4 ,a fixes on the
 segment ,,ou#a1 on ,ox1 as being  the  
unit length1 and ,b on the segment      
,,ov#a1 on ,ox4 ,a calls his            
coordinates 7x1 y1 z71 and ,b calls them
 7,x1 ,y1 ,z74    ,then it is found     
since both systems are ,euclidean that1 
whatever  point ,p be taken1    ,x  x1  
,y  y1 ,z  z4     4    ,they proceed to 
adjust their differences1 and first take
 the  x-coordinates4 ,obviously they    
have taken different units of  length   
along ,ox4 ,the length ,,ou#a1 which ,a 
calls one unit1 ,b  calls  units4 ,b    
changes his unit length to ,,ou#a1 from 
its original  length ,,ov#a1 and obtains
 ,x  x4 ,but now1 as he must use the    
same unit for all his measurements1 his 
other coordinates are altered  in the   
same ratio4 ,thus we now have    ,x  x1 
,y  y_/1 ,z  z_/4    ,the fundamental   
divergence is now evident4 ,a and ,b    

agree as to their  units along ,ox4     
,they settled that by taking along that 
axis a  given segment ,,ou#a as having  
the unit length4 ,but they cannot agree 
as to what segment along ,oy is equal to
 ,,ou#a4 ,a says it is  ,,ou#b1 and ,b  
that it is ,,ou#b4 ,similarly for       
lengths along ,,oz4    ,the result is   
that ,a's spheres    x#b  y#b  z#b  r#b1
    are ,b's ellipsoids1                
,x#b  #b,y#b_/b  #b,z#b_/b  r#b1  i4 e4 
,x#b_/b  ,y#b_/b  ,z#b_/b  r#b_/b4      
,thus the measurement of angles by the  
two is hopelessly at variance4    ,if   
1 there is one1 and only one1 set of    
common rectangular  axes at ,o1 namely  
that from which they started4 ,if   1   
but   1  then there are a singly        
infinite number of common rectangular   
axes  found by rotating the axes round  
,ox4 ,this is1 for us1 the  interesting 
case4 ,the same phenomena are reproduced
 by transferring to  any parallel axes4 
,the root of the difficulty is1 that    
,a's measuring rod1 which for him is  a 

rigid invariable body1 appears to ,b as 
changing in length when turned  in      
different directions4 ,similarly all    
measuring rods1 satisfactory  to ,a1    
violate ,b's immediate judgment of      
invariability1 and change  according to 
the same law4 ,there is no way out of   
the difficulty4  ,two rods  and         
coincide whenever laid one on the other2
  is held  still1 and both men agree    
that it does not change4 ,but  is turned
  round4 ,a says it is invariable1 ,b   
says it changes4 ,to test the matter1   
is turned round to measure it1 and      
exactly fits it4 ,but while ,a is       
satisfied1 ,b declares that  has changed
 in exactly the same way as did  4      
,meanwhile ,b has procured two material 
rods satisfactory to him as  invariable1
 and ,a makes exactly the same          
objections4    ,we shall say that ,a and
 ,b employ diverse ,euclidean metrical  
systems4    ,the most extraordinary fact
 of human life is that all beings seem  
to  form their judgments of spatial     

quantity according to the same metrical 
system4                                 
,relativity in ,modern ,physics         
,owing to the fact that points of space 
are incapable of direct  recognition1   
there is a difficulty--apart from any   
abstract question  of the nature of     
space--in deciding on the motion to be  
ascribed to  any body4 ,even if there be
 such a thing as absolute position1 it  
is  impossible in practice to decide    
directly whether a body's absolute      
position has changed4 ,all spatial      
measurement is relative to matter4      
,newton's laws of motion in their modern
 dress evade this difficulty  by        
asserting that a framework of axes of   
coordinates can be defined  by their    
relations to matter such that1 assuming 
these axes to be at  rest1 and all      
velocities to be measured relatively to 
them1 the laws  hold4 ,the same         
expedient has to be employed for time1  
namely1 the laws  hold when the         
measurement of the flow of time is made 

by the proper  reference to periodic    
events4 ,thus the laws assert that the  
framework  and the natural clock adapted
 for their use have been successfully   
found4    ,but1 if one framework will   
do1 an infinity of others serve equally 
well2 namely1 not only--as is of course 
the case--all those at rest  relatively 
to the first framework1 but also all    
those which move  without relative      
rotation with uniform velocity          
relatively to the  first4 ,this whole   
set of frameworks is on a level in      
respect to  ,newton's laws4 ,we will    
call them ,dynamical frameworks4        
,now1 suppose there are two observers1  
,a and ,b4 ,they agree in their         
non-metrical projective geometry1 e4g41 
what ,a calls a straight  line so does  
,b4 ,they also both apply a ,euclidean  
metrical system of  measurement to this 
space4 ,their two metrical systems also 
agree in  having the same plane at      
infinity1 that is1 lines which are      
parallel  for ,a are also parallel for  

,b4 ,furthermore1 they have both        
successfully  applied ,newton's laws to 
the movement of matter1 and agree in    
having  the same sets of dynamical axes4
 ,but the framework 7among these sets7  
which ,a chooses to regard as at rest is
 different from the frame 7among  the   
same sets7 which ,b so regards4         
,without alteration of their respective 
judgment of rest1 they choose  their    
co-ordinate axes so that the origins 7,o
 for ,a1 and ,o for ,b7 are  in relative
 motion along ,,oo1 which is the axis of
 x for both4    ,further1 since ,,oo is 
the line of symmetry of their diverse   
,euclidean  systems1 we assume that the 
two measure-systems agree for planes    
perpendicular to ,,oo1 i4e41 we assume a
 symmetry round ,,oo4  ,then if1 for ,a 
at ,o1 the distance ,,oo be 1 the       
relations at any  instant between ,a's  
coordinates 7x1 y1 z7 and ,b's          
coordinates 7x1 y1 z7 for the same point
 ,p are  given by    x  7x - 71 y  y1 z 
z4    ,also1 according to ,a's clock1 ,o

 is moving forward with a uniform       
velocity v1 and we measure ,a's time    
from the instant of the  coincidence of 
,o and ,o4    ,thus      vt1    and    x
  7x - vt71 y  y1 z  z4    ,we now      
consider ,b's clock1 and ask for the    
most general supposition  which is      
consistent with the fact that their     
judgments as to the fact  of uniform    
motion are in agreement4    ,we do not  
assume that events in various parts of  
space which ,a  considers to be         
simultaneous are so considered by ,b4   
,but we assume that  at any point ,p1   
with coordinates 7x1 y1 z7 for ,a1 there
 is a  determinate relation between ,b's
 time ,t and x1 y1 z1 t4    ,put    ,t  
7x1 y1 z1 t74    ,write    ,p  ,t_/x1 ,q
  ,t_/y1 ,r  ,t_/z1 ,s  ,t_/t4    ,now  
suppose that the point ,p is moving1 and
 that 7u#a1  u#b1 u#c7 is its set of    
component velocities along  the axes    
according to ,a's space and clock       
system1 and 7,u#a1 ,u#b1  ,u#c7 is its  
set of component velocities according to

 ,b's space and  clock system4 ,then by 
mere differentiation it follows after a 
short  mathematical deduction that      
,u#a  7d_/dt77x - vt7  7u#a - v7_/,pu#a 
,qu#b  ,ru#c  ,s1  ,u#b  u#b_/,pu#a     
,qu#b  ,ru#c  ,s1  ,u#c  u#c_/,pu#a     
,qu#b  ,ru#c  ,s4    ,but we have       
assumed that1 whatever the direction of 
the resultant  velocity 7u#a1 u#b1 u#c71
 the velocities 7,u#a1 ,u#b1 ,u#c7  and 
7u#a1 u#b1 u#c7 are both uniform when   
either is uniform4    ,hence it is      
easily proved that 1 ,p1 ,q1 ,r1 ,s are 
independent of the  coordinates 7x1 y1  
z7 and of the time t4 ,in other words1  
they  are constant4    ,hence we obtain 
,u#a  7u#a - v7_/,pu#a  ,qu#b  ,ru#c    
,s1    and    ,t  ,px  ,qy  ,rz  ,st4   
,but we assumed that ,,oo1 i4e41 ,ox1 is
 an axis of  symmetry4 ,it follows from 
this assumption that    ,q  ,r  #j4     
,we thus obtain the simplified results  
,t  ,px  ,st1                           
,u#a  7u#a - v7_/7,pu#a  ,s71    7,i7   
,u#b  u#b_/7,pu#a  ,s71            ,u#c 

u#c_/7,pu#a  ,s74              ,here we 
remember that 7u#a1 u#b1 u#c7 are  the  
velocities of any particle according to 
,a's space and clock  system1 and that  
7,u#a1 ,u#b1 ,u#c7 are the velocities of
 the same  point according to ,b's space
 and clock system4 ,we have obtained the
  most general relations consistent with
 the facts that 7#a7 they both  employ  
,euclidean systems1 related as described
 above1 and 7#b7 they  agree in their   
judgments on the uniformity of velocity4
    ,we now compare their judgments on  
the magnitudes of velocities4    ,let   
the magnitude of the velocity of ,p be  
,v according to ,a's judgment1  and ,v  
according to ,b's' judgment4    ,then   
,v#b  u#ab  u#bb  u#cb1  ,v#b  ,u#ab    
,u#bb  ,u#cb4    ,also we can put    u#a
  l,v1 u#b  m,v1 u#c  n,v1    where 7l1 
m1 n7 have nothing to do with the       
magnitude ,v1 but  simply depend on the 
direction of motion4 ,in fact 7l1 m1 n7 
are  the direction cosines of the       
velocity according to ,a's judgment4 ,by

  substituting in the above equation for
 ,v#b we see that    l#b  m#b  n#b  #a4 
,now1 substituting for 7u#a1 u#b1 u#c7  
in  the equations 7,i7 above1 and       
squaring and adding1 and eliminating    
m#b  n#b by the relation just found1 we 
at once find    ,v#b  77#b - #a7,v#bl#b 
- #bb,vvl  #bv#b  ,v#b7_/7,,pvl  ,s7#b4 
,it is thus seen that in general the    
relation of ,v to ,v depends on the     
direction cosine l4 ,now l is the cosine
 of the angle  which the direction of   
the velocity ,v makes with ,ox1         
according  to ,a's judgment4    ,the    
meaning of this relation is1 that if ,a 
discharges1 from guns at  the point ,p1 
shells with a given muzzle velocity ,v  
according to his  judgment1 ,b will     
consider that their muzzle velocities   
are different  from each other1 except  
in the case of pairs of guns equally    
inclined  to the axis ,,oo4 ,instances  
of this type of diversity of judgment   
can  be noted any day by any one who    
looks out of the window of a railway    

carriage1 and forgets that he is        
travelling4    ,now1 suppose the        
velocity ,v bears a relation to the     
velocity ,v1 which  is independent of l4
 ,then l must disappear from the above  
formula4 ,there are two conditions to be
 satisfied    ,one condition is    ,v#b 
#bv#b_/7b - #a71    or in a more        
convenient form    #b  #a_/7a -         
v#b_/,v#b74    ,the meaning of this     
condition is1 that there is one1 and    
only one1  muzzle velocity ,v 7according
 to ,a's judgment71 namely1 the muzzle  
velocity given by the above formula1    
which can have the property that  ,b    
will judge that all the guns are firing 
in their diverse directions  with one   
common muzzle velocity4    ,let us now  
suppose that ,v has this peculiar value3
 that is1 if we look  on this value ,v  
as known1 we must suppose that  is given
 by the second  of the above formul4    
,the other condition allows ,p and ,s to
 be put in the forms    ,p  -v_/,v#b1 ,s
  _/1    where    ,v  ,v4    ,thus we   

have the bundle of formul    #b  #a_/7a 
- v#b_/,v#b71  ,t  t - vx_/,v#b_/1  ,v  
,v4    ,the value which we give to  is  
purely a matter for the adjustment of   
units4 ,if we want ,a and ,b to agree in
 their judgments of the magnitude  of   
this peculiar muzzle-velocity1 we put   
#a4    ,we then get the formul usually  
adopted1 namely    #b  #a_/7a -         
v#b_/,v#b71    ,t  t - vx_/,v#b1        
7,,ii7  ,v  ,v4                         
,but if we prefer that ,a and ,b should 
reckon 7according to ,a's judgment7  in 
the same units of time1 we put   1 and  
obtain    #b  #a_/7a - v#b_/,v#b71  ,t  
7t - vx7_/,v#b1  ,v  ,v4    ,but ,a and 
,b are in any case in such hopeless     
difficulties over their  comparisons of 
time-judgments that the detail of using 
the same units  does not help them much4
 ,accordingly the formul marked 7,,ii7  
are those  used4 ,thus ,a and ,b agree  
in their judgments as to the magnitude  
of one  special velocity ,v1 whatever   
may be the direction in which the entity

  possessing it is moving4    ,in order 
to reach this measure of agreement1 they
 have to disagree as  to their space    
judgments and their time judgments4 ,the
 root cause of  their disagreement is   
their diverse judgment as to which axis 
system is  to be taken at rest for the  
purpose of measuring velocities4        
,before discussing the nature of the    
disagreement disclosed in formula       
7,,ii71 let us ask why we should bring  
these difficulties on our heads by      
supposing that two people in relative   
motion1 who both 7for the purpose  of   
measuring velocities7 assume that they  
are at rest1 should agree in  their     
judgments in respect to this special    
velocity ,v4    ,such an agreement has  
no counterpart in any of our obvious    
judgments  made from railway carriages4 
,surely we can wait till the contingency
  occurs before discussing the confusion
 which it creates4    ,but the          
contingency has occurred4 ,it occurs    
when we consider the  velocity of light4

 ,perhaps ,i may venture to remind a    
philosophical  society that light moves 
so very quickly that it is difficult to 
consider its velocity at all4 ,so we    
need not be surprised that this         
peculiar fact concerning its velocity is
 not more obvious4    ,now ,v being the 
velocity of light1 unless v is large1   
v_/,v  7and still more v#b_/,v#b7 will  
be quite inappreciable4 ,the only       
velocity ready to hand which is big     
enough to give v_/,v an  appreciable    
value is the velocity of the earth in   
its orbit4    ,many diverse experiments 
have been made1 and they all agree in   
concluding that a man who assumes the   
earth to be at rest will find by        
measurement that the velocity of light  
is the same in all directions4          
,furthermore1 when the same man turns   
his attention to interstellar or        
interplanetary phenomena1 and assumes   
the sun to be at rest1 he will  again   
find the velocity of light to be the    
same in all directions4  ,these are     

well-attested experiments made at long  
intervals of time4    ,this is the exact
 contingency contemplated above4        
,again the velocity of light in vacuo   
has recently taken on a  new dignity4   
,it used to be one among other wave     
velocities such as  the velocity of     
sound in air1 or in water1 or the       
velocity of surface  waves in water4    
,but ,clerk ,maxwell discovered that all
 electromagnetic  influences are        
propagated with the velocity of light1  
and now modern  physical science half   
suspects that electromagnetic influences
 are  the only physical influences which
 relate the changes in the physical     
world4 ,accordingly the velocity of     
light becomes the fundamental  natural  
velocity1 and experiment shows that our 
judgment of its  magnitude is not       
affected by our choice of the framework 
at rest1 so  long as we keep to a set of
 dynamical axes4 ,these experiments on  
light  have been confirmed by other     
electromagnetic experiments not         

involving  light4    ,thus we are driven
 to equations 7,,ii71 where ,v is the   
velocity of light4    ,the first        
conclusion to be drawn from equations   
7,,ii7 is that two people  who make     
different choices of bodies at rest will
 disagree as to  their measuring rods in
 the way described above4 ,there is no  
peculiar  difficulty about that4 ,the   
only wonder is that all people agree so 
well  in their judgments as to metrical 
systems4 ,a mathematical angel would    
naturally expect incarnate men to be in 
violent disagreement on this  subject4  
,but the case of time is different4 ,for
 simplicity of statement we speak  of ,a
 as at ,o1 and ,b as at ,o4 ,we remember
 that ,o is moving relatively  to ,o    
with velocity v in direction ,,oo4      
,suppose ,a and ,b are  looking in this 
direction2 and they both measure their  
time from the  instant when they met1 as
 ,o passed over ,o4 ,then we have    ,t 
77t - vx7_/,v#b7_/77a - v#b7_/,v#b74    
,now1 suppose we consider all the events

 all over space which ,a  considers to  
have happened simultaneously at the time
 t4 ,the  events of this set which      
occurred anywhere on a plane            
perpendicular to  ,,oo at a distance x  
in front of ,o 7according to ,a's       
reckoning71  will have occurred         
according to ,b's reckoning at the time 
,t as given  above4 ,let us fix our     
attention on the fact that ,b does not  
consider  all these events to be        
simultaneous4 ,for let ,t#a1 and ,t#b be
 ,b's  times for such events on planes  
x#a1 and x#b4 ,then    ,t#a - ,t#b      
v7x#b - x#a7_/7,v#b - v#b74    ,thus if 
x#b be greater than x#a1 ,t#b is less   
than  ,t#a4 ,thus ,b judges the more    
distant events in front of him to have  
happened earlier than the nearer events 
in front of him1 and vice  versa for the
 events behind4 ,this disturbance of the
 judgment of  simultaneity is the       
fundamental fact4 ,obviously the        
measurement of time  intervals is a     
detail compared to simultaneity4 ,a may 

think a sermon  long1 and ,b may think  
it short1 but at least they should both 
agree  that it stopped when the clock   
hand pointed at the hour4 ,the worst of 
the matter is that so far as any test   
can be applied there is no method  of   
discriminating between the validities of
 their judgments4    ,thus we are       
confronted with two distinct concepts of
 the common world1  ,a's space-time     
concept1 and ,b's space-time concept4   
,who is right8 ,it is  no use staying   
for an answer4 ,we must follow the      
example of the wise  old ,roman1 and    
pass on to other things4    ,thus       
estimates of quantity in space and time1
 and1 to some extent1 even  estimates of
 order1 depend on the individual        
observer4 ,but what are  the crude      
deliverances of sensible experience1    
apart from that world  of imaginative   
reconstruction which for each of us has 
the best claim  to be called our real   
world8 ,here the experimental           
psychologist steps  in4 ,we cannot get  

away from him4 ,i wish we could1 for he 
is frightfully  difficult to understand4
 ,also1 sometimes his knowledge of the  
principles of mathematics is rather     
weak1 and ,i sometimes suspect----  ,no1
 ,i will not say what ,i sometimes      
think3 probably he1 with equal  reason1 
is thinking the same sort of thing of   
us4    ,i will1 however1 venture to     
summarise conclusions1 which are1 ,i    
believe1 in harmony with the            
experimental evidence1 both physical and
  psychological1 and which are certainly
 suggested by the materials for  that   
unwritten chapter in mathematical logic 
which ,i have already  commended to your
 notice4 ,the concepts of space and time
 and of  quantity are capable of        
analysis into bundles of simpler        
concepts4 ,in  any given sensible       
experience it is not necessary1 or even 
usual1 that  the whole complete bundle  
of such concepts apply4 ,for example1   
the  concept of externality may apply   
without that of linear order1 and the   

concept of linear order may apply       
without that of linear distance4        
,again1 the abstract mathematical       
concept of a space-relation  may confuse
 together distinct concepts which apply 
to the given  perceptions4 ,for example1
 linear order in the sense of a linear  
projection from the observer is distinct
 from linear order in the sense  of a   
row of objects stretching across the    
line of sight4    ,mathematical physics 
assumes a given world of definitely     
related  objects1 and the various       
space-time systems are alternative ways 
of  expressing those relations as       
concepts in a form which also applies to
  the immediate experience of observers4
    ,yet there must be one way of       
expressing the relations between objects
  in a common external world4           
,alternative methods can only arise as  
the  result of alternative standpoints2 
that is to say1 as the result of        
leaving something added by the observer 
sticking 7as it were7 in the  universe4 

,but this way of conceiving the world of
 physical science1 as composed  of      
hypothetical objects1 leaves it as a    
mere fairy tale4 ,what is really  actual
 are the immediate experiences4 ,the    
task of deductive science  is to        
consider the concepts which apply to    
these data of experience1  and then to  
consider the concepts relating to these 
concepts1 and  so on to any necessary   
degree of refinement4 ,as our concepts  
become  more abstract1 their logical    
relations become more general1 and less 
liable to exception4 ,by this logical   
construction we finally arrive  at      
conceptions1 7i7 which have determinate 
exemplifications in the  experience of  
the individuals1 and 7ii7 whose logical 
relations have  a peculiar smoothness4  
,for example1 conceptions of            
mathematical time1  of mathematical     
space1 are such smooth conceptions4 ,no 
one lives in  an infinite given whole1  
but in a set of fragmentary experiences4
  ,the problem is to exhibit the        

concepts of mathematical space and time 
as the necessary outcome of these       
fragments by a process of logical       
building up4 ,similarly for the other   
physical concepts4 ,this process  builds
 a common world of conceptions out of   
fragmentary worlds of  experience4 ,the 
material pyramids of ,egypt are a       
conception1 what is  actual are the     
fragmentary experiences of the races who
 have gazed on  them4    ,so far as     
science seeks to rid itself of          
hypothesis1 it cannot go  beyond these  
general logical constructions4 ,for     
science1 as thus  conceived1 the        
divergent time orders considered above  
present no  difficulty4 ,the different  
time systems simply register the        
different  relations of the mathematical
 construct to those individual          
experiences 7actual or hypothetical7    
which could exist as the crude  material
 from which the construct is elaborated4
    ,but after all it should be possible
 so to elaborate the mathematical       

construct so as to eliminate specific   
reference to particular  experiences4   
,whatever be the data of experience1    
there must be  something which can be   
said of them as a whole1 and that       
something is  a statement of the general
 properties of the common world4 ,it is 
hard  to believe that with proper       
generalisation time and space will not  
be  found among such properties4        
,commentary added on reading the ,paper 
before the ,aristotelian                
,society    ,the first six pages of the 
paper consist of a summary of ideas     
which  ought to be in our minds while   
considering problems of time and space4 
,the ideas are mostly philosophical1 and
 the summary has been made by  an       
amateur in that science2 so there is no 
reason to ascribe to it any  importance 
except that of a modest reminder4 ,there
 are only two points  in this summary to
 which ,i would draw attention4    ,on  
pp4 #aib and #aic there occurs--        
,wherever mankind 444 unending uniform  

succession4    ,if ,i understand ,kant  
rightly--which ,i admit to be very      
problematical--he holds that in the act 
of experience we are aware  of space and
 time as ingredients necessary for the  
occurrence of  experience4 ,i would     
suggest--rather timidly--that this      
doctrine should  be given a different   
twist1 which in fact turns it in the    
opposite  direction--namely1 that in the
 act of experience we perceive a whole  
formed of related differentiated parts4 
,the relations between these  parts     
possess certain characteristics1 and    
time and space are the  expressions of  
some of the characteristics of these    
relations4 ,then the  generality and    
uniformity which are ascribed to time   
and space express  what may be termed   
the uniformity of the texture of        
experience4    ,the success of          
mankind--modest though it is--in        
deducing uniform laws  of nature is1 so 
far as it goes1 a testimony that this   
uniformity of  texture goes beyond those

 characteristics of the data of         
experience  which are expressed as time 
and space4 ,time and space are necessary
  to experience in the sense that they  
are characteristics of our  experience2 
and1 of course1 no one can have our     
experience without  running into them4  
,i cannot see that ,kant's deduction    
amounts to much  more than saying that  
what is1 is--true enough1 but not very  
helpful4    ,but ,i admit that what ,i  
have termed the uniformity of the       
texture of  experience is a most curious
 and arresting fact4 ,i am quite ready  
to  believe that it is a mere illusion2 
and later on in the paper ,i suggest    
that this uniformity does not belong to 
the immediate relations of the  crude   
data of experience1 but is the result of
 substituting for them  more refined    
logical entities1 such as relations     
between relations1 or  classes of       
relations1 or classes of classes of     
relations4 ,by this means  it can be    
demonstrated--,i think--that the        

uniformity which must be  ascribed to   
experience is of a much more abstract   
attenuated character  than is usually   
allowed4 ,this process of lifting the   
uniform time and  space of the physical 
world into the status of logical        
abstractions has  also the advantage of 
recognising another fact1 namely1 the   
extremely  fragmentary nature of all    
direct individual experience4    ,my    
point in this respect is that           
fragmentary individual experiences  are 
all that we know1 and that all          
speculation must start from these       
disjecta membra as its sole datum4 ,it  
is not true that we are  directly aware 
of a smooth running world1 which in our 
speculations we  are to conceive as     
given4 ,in my view the creation of the  
world is the  first unconscious act of  
speculative thought2 and the first task 
of a  self-conscious philosophy is to   
explain how it has been done4    ,there 
are roughly two rival explanations4 ,one
 is to assert the world  as a postulate4

 ,the other way is to obtain it as a    
deduction1 not a  deduction through a   
chain of reasoning1 but a deduction     
through a chain  of definitions which1  
in fact1 lifts thought on to a more     
abstract  level in which the logical    
ideas are more complex1 and their       
relations  are more universal4 ,in this 
way the broken limited experiences      
sustain  that connected infinite world  
in which in our thoughts we live4 ,there
  are three more remarks while on this  
point ,i wish to make--    7i7 ,the fact
 that immediate experience is capable of
 this deductive  superstructive must    
mean that it itself has a certain       
uniformity of  texture4 ,so this great  
fact still remains4    7ii7 ,i do not   
wish to deny the world as a postulate4  
,speaking without  prejudice1 ,i do not 
see how in our present elementary state 
of  philosophical advance we can get on 
without middle axioms1 which1 in  fact1 
we habitually assume4    ,my position   
is1 that by careful scrutiny we should  

extrude such  postulates from every part
 of our organised knowledge in which it 
is  possible to do without them4        
,now1 physical science organises our    
knowledge of the relations between  the 
deliverances of our various senses4 ,i  
hold that in this department  of        
knowledge such postulates1 though not   
entirely to be extruded1 can  be reduced
 to a minimum in the way which ,i have  
described4    ,i have not the slightest 
knowledge of theories respecting our    
emotions1  affections1 and moral        
sentiments1 and ,i can well believe that
 in  dealing with them further          
postulates are required4 ,and in        
practice ,i  recognise that we all make 
such postulates1 uncritically4    7iii7 
,the next paragraphs on pp4 #aic and    
#aid are as follows--    ,the opposed   
philosophical method 444 physical       
science4    ,it will be noted that1 in  
the light of what has just been stated1 
the  first of these paragraphs 7which1  
,i hope1 faithfully expresses the       

experimental way of approaching the     
problem7 really obscures the point      
which ,i have been endeavouring to make4
 ,the phrase1 ,if we form the  exact    
concepts of points1 etc41 is fatally    
ambiguous as between the  method of     
postulating entities with assigned      
relations1 and the method  of forming   
logical constructions1 and thus reaching
 points1 etc41 as  the result of a chain
 of definitions4    ,turning now to pp4 
#aid-aie1 we come to the following      
paragraphs--    ,the other theory 444   
eighteenth century4    ,we note again   
that the relational theory of space from
 another point  of view brings us back  
to the idea of the fundamental          
space-entities  as being logical        
constructs from the relations between   
things4 ,the  difference is1 that this  
paragraph is written from a more        
developed  point of view1 as it         
implicitly assumes the things in space1 
and  conceives space as an expression of
 certain of their relations4  ,combining

 this paragraph with what has gone      
before1 we see that the  suggested      
procedure is first to define things in  
terms of the data of  experience1 and   
then to define space in terms of the    
relations between  things4    ,this     
procedure is explicitly assumed in the  
next short paragraph3 ,in  this         
connection 444 from the events4    ,the 
gist of the remaining paragraphs of this
 section is contained in  the paragraph 
at the bottom of p4 #aif3 ,the point in 
time 444 new  importance4    ,the       
sentence1 ,we live in durations1 and not
 in points1 can be  amplified by the    
addition1 ,we live in space-extensions  
and not in  space-points4    ,it must be
 noted that whole and part as applied to
 extensions in  space or time must be   
different from the all and some of      
logic1  unless we admit points to be the
 fundamental entities4 ,for spatial     
whole and spatial part can only mean all
 and some if they really  mean all the  
points and some of the points4 ,but if  

extensions and  their relations are more
 fundamental than points1 this          
interpretation  is precluded4 ,i suggest
 that spatial whole and spatial part is 
intimately connected with the           
fundamental relation between things from
  which our space ideas spring4    ,the 
relation of space whole to space part   
has many formal properties  which are   
identical with the properties of all and
 some4 ,also when  points have been     
defined1 we can replace it by the       
conception of all  the points and some  
of the points4 ,but the confusion       
between the two  relations is fatal to  
sound views on the subject4             
,diverse ,euclidean ,measure ,systems   
,the next section deals with the measure
 systems applicable to space4    ,a     
measure system is a group of congruent  
transformations of space  into itself4  
,consider a rigid body occupying all    
space4 ,let this body  be moved in any  
way so that the particles of the body   
which occupied  points ,p#a1 ,p#b1 ,p#c1

 etc41 now occupy points ,q#a1 ,q#b1    
,q#c1 etc4 ,then any point ,p#a in space
 is uniquely related to the             
corresponding point ,q#a in space by a  
one-to-one transformation with  certain 
characteristics4 ,by the aid of these   
transformations we can  achieve the     
definition of distance in a way which   
definitely determines  the distance     
between any two points1 provided that we
 can define what  we mean by a congruent
 transformation without introducing the 
idea of  distance4 ,if we introduced the
 idea of distance1 we should simply say 
that a congruent transformation is one  
which leaves all distances  unchanged1  
i4 e41 if ,p#a1 ,p#b are transformed    
into ,q#a1  ,q#b then the distance      
,p#a,p#b is equal to the distance       
,q#a,q#b4    ,but mathematicians have   
succeeded in defining congruent         
transformations  without any reference  
to distance4    ,there are alternative  
groups of such congruent                
transformations1 and  each group gives a

 different measure system for space4    
,the distance  ,p#a,p#b may equal the   
distance ,q#a,q#b for one measure       
system1  and will not equal it for      
another measure system4 ,all these      
different  measure systems are on the   
same level1 equally applicable4 ,a being
  with a strong enough head could think 
of them all at once as applying  to     
space4 ,the result so far as it         
interests us in respect to the  theory  
of relativity is explained on pp4       
#aig-bjj1 ending with ,the  most        
extraordinary fact 444 same metrical    
system4 ,this final sentence  bears on  
,poincar's assertion that the measure   
system adopted is  purely conventional4 
,i presume that by conventional a       
certain  arbitrariness of choice is     
meant2 and in that case1 ,i must express
  entire dissent4 ,it is true that      
within the circle of geometrical ideas  
there is no means of giving any         
preference to any one measure system1   
and any one is as good as any other4    

,but it is not true that if we  look at 
a normal carriage wheel1 and at an oval 
curve one foot broad  and ten feet long1
 we experience any arbitrariness of     
judgment in  deciding which has         
approximately the form of a circle4     
,accordingly to  ,poincar the choice    
between them1 as representing a circle1 
is entirely  conventional4    ,again1 we
 equally form immediate judgments as to 
whether a body is  approximately rigid4 
,we know that a paving stone is rigid1  
and that a  concertina is not rigid4    
,this again necessitates a determinate  
measure  system1 selected from among the
 others4    ,accordingly we conclude    
that 7i7 each being does1 in fact1      
employ a  determinate measure system1   
which remains the same1 except possibly 
for very small variations1 and 7ii7 the 
measure systems of different  human     
beings agree1 to within the limits of   
our observations4 ,these  conclusions   
are not the less extraordinary because  
no plain man has  ever doubted them4    

,it is an interesting subject to        
investigate exactly what are the        
fundamental uniformities of experience  
which necessitate this  conclusion4 ,it 
is not so easy as it looks1 since we    
have to divest  ourselves of all aid of 
scientific hypothesis if our conclusions
 are to  be demonstrative4              
,relativity in ,modern ,physics    ,pp4 
#bja-bjb1 ,owing to the fact 444 which  
,b so regards4    ,the fundamental      
formul for the theory of relativity are 
the relations  between diverse          
co-ordinate systems given on p4 #bjc1   
and formul ,,ii  at the bottom of p4    
#bjg4 ,the general explanation of one   
method in  which these formul           
arise--namely1 ,einstein's method--is   
given on  pp4 #bja-baa4 ,namely1 we seek
 the condition that for all dynamical   
axes the velocity of light should be the
 same1 and the same in all  directions4 
,it should be noted that the experiments
 which1 so far as  they go1 confirm     
these formul1 can also be explained in  

another way  which makes the theory of  
relativity unnecessary4 ,we need only   
ascribe  to the ether a certain property
 of contraction in the direction of     
motion1 and the thing is done4 ,so no   
one need be bludgeoned into  accepting  
the rather bizarre doctrine of          
relativity1 nor indeed any  other       
scientific generalisation4 ,the good old
 homely ether1 which we  all know1 can  
in this case serve the purpose4 ,just as
 an author of  genius1 if he lives long 
enough1 survives the inevitable         
accusation  of immorality1 so the ether 
by dint of persistence has outlived all 
reputation of extravagance4 ,but if we  
detach ourselves from the glib          
phraseology concerning it1 the          
scientific ether is uncommonly like the 
primitive explanation of the soul1 as a 
little man inside us1 which can         
sometimes be caught escaping in the form
 of a butterfly4 ,as soon as the  ether 
has to be patched up with special       
properties to explain special           

experiments1 its scientific use is      
problematical1 and its philosophic  use 
is nil4    ,philosophically the ether   
seems to me to be an ambitious attempt  
to give a complete explanation of the   
physical universe by making  an elephant
 stand on a tortoise4 ,scientifically it
 has a perfectly  adequate use by       
veiling the extremely abstract character
 of scientific  generalisations under a 
myth1 which enables our imaginations to 
work  more freely4 ,i am not advocating 
the extrusion of ether from our         
scientific phraseology1 even though at  
special points we have to  abandon it4  
,but the key to the reasons why it is   
worth while to consider seriously  the  
doctrine of relativity is to be found on
 pp4 #bji1 #baj3 ,again the  velocity of
 light 444 not involving light4 ,namely1
 we have begun to  suspect that all     
physical influences require time for    
their propagation  in space4 ,this      
generalisation is a long way from being 
proved4  ,gravitation stands like a lion

 in the path4 ,but if it be the case1   
then  all idea of an immediate          
presentation to us of an aspect of the  
world  as it in fact is1 must           
necessarily be abandoned4 ,what we      
perceive at  any instant is already     
ancient history1 with the dates of the  
various  parts hopelessly mixed4    ,we 
must add to this the difficulty of      
determining what is at rest  and what is
 in motion1 and the further difficulty  
of determining a  definite uniform flow 
of time4 ,it is no use discussing this  
matter  as though1 but for the silly    
extravagant doctrine of relativity1     
everything would be plain sailing4 ,it  
isn't4 ,you may be quite sure  that     
when1 after prolonged study1 you        
endeavour to give the simplest          
explanation of a grave difficulty1 you  
will be accused of extravagance4  ,i    
have no responsibility for the doctrine 
of relativity1 and hold no  brief for   
it1 but it has some claim to be         
considered as a comparatively  simple   

way out of a scientific maze4    ,in the
 first place1 we use the ,newtonian     
dynamical sets of axes1  and the        
,newtonian clock to extricate ourselves 
partially from the  difficulties of     
rest1 motion1 and time4 ,these have     
proved capable  of scientific           
determination within the limits of our  
experimental  accuracy4 ,thus the only  
thing left over is the choice of the    
axes  at rest1 which is a completely    
indeterminate problem on ,newtonian     
principles4    ,again1 so far as we can 
at present guess by adopting the theory 
that all metrical influence is          
electromagnetic1 all influences  are    
propagated with the velocity of light in
 vacuo4 ,this  electromagnetic          
hypothesis is by no means established1  
but it gives  the simplest of all       
possible results in respect to the      
propagation of  influence1 which we     
therefore adopt4    ,but what dynamical 
axes are we taking as at rest8 ,now our 
practical  choice gives a range of      

relative velocities small compared to   
that of  light4 ,so except for certain  
refined experiments it does not matter4 
,there are two possibilities--    7i7   
,we may assume that one set of axes are 
at rest1 and that the others  will show 
traces of motion in respect to the      
velocity of light2 or--    7ii7 ,that   
the velocity of light is the same in all
 directions whichever  be the dynamical 
axes assumed4    ,the first supposition 
is negatived by experiment1 and hence we
 are  driven to the second supposition2 
which immediately lands us in the  whole
 theory of relativity4    ,but if we    
will not have this theory we must reject
 the earlier  supposition that the      
velocity of light in vacuo is the same  
in  all directions4 ,this we do1 in     
fact1 by assuming an ether1 and assuming
  a certain law for its modification4   
,then we1 in fact1 adopt the first      
supposition so far as to hold that there
 are dynamical axes specially  at rest1 
namely1 at rest relatively to the       

undisturbed ether4 ,then an  assumed law
 for the modification of the ether so   
alters the velocity of  light that we   
explain why no dynamical axes show      
traces of motion4    ,i wish now to go  
back to the point which ,i made a few   
minutes ago1  that what we perceive at  
any instant is ancient history with its 
dates  hopelessly mixed4 ,in the earlier
 part of my comments ,i emphasised the  
point that our only data as to the      
physical world are our sensible         
perceptions4 ,we must not slip into the 
fallacy of assuming that we are         
comparing a given world with given      
perceptions of it4 ,the physical  world 
is in some general sense of the term a  
deduced concept4    ,our problem is1 in 
fact1 to fit the world to our           
perceptions1 and not  our perceptions to
 the world4             ,,printed ,,in  
,,great ,,britain ,,by ,,richard ,,clay 
,,sons1 ,,limited1          ,,brunswick 
,,st41 ,,stamford ,,st41 ,s4,e41 ,,and  
,,bungay1 ,,suffolk4          ,from     

,,williams  ,,norgate',s ,,list       ,a
 ,,spiritual ,,pilgrimage4 ,by the ,rev4
 ,r4 ,j4 ,,campbell4   7,the ,rev4 ,r4  
,j4 ,campbell's own ,story of his       
,religious ,life47 #bnd   ,impression4  
,demy #hvo4 #gs4 #f;d4 net4     ,this is
 eminently a book for which to be       
thankful--simple1   straightforward1    
kindly4 ,neither does it lack the saving
 grace of   humour4--,guardian4     ,mr4
 ,campbell devotes a chapter to his     
re-ordination1 and it must be   admitted
 that his defence of this act is        
effective4--,methodist   ,times4        
,,edinburgh4 ,by the ,right ,hon4 ,sir  
,,herbert ,,maxwell1   ,bart41          
,d4,c4,l41 ,author of ,life of the ,duke
 of ,wellington1 ,scottish   ,gardens1  
etc4 ,president of the ,society of      
,antiquaries of ,scotland1   #aiaj-ac1  
and ,chairman of the ,royal ,commission 
on ,scottish ,historical   ,monuments4  
,with a ,coloured ,frontispiece and #fd 
pages of ,illustrations   of the past   
and present city4 ,medium #hvo4 #ajs4   

#f;d4 net4     ,a ,history of the       
,scottish ,capital from the earliest    
times to the   nineteenth century4      
,,further ,,pages ,,of ,,my ,,life4 ,by 
the ,right ,rev4 ,w4 ,,boyd             
,,carpenter1 ,k4,c4,v4,o41 ,d4,d41      
,d4,c4,l41 ,formerly ,bishop of ,ripon1 
,author of ,some ,pages of ,my ,life1   
,the ,witness of ,religious             
,experience1 etc4 ,with ,illustrations4 
,medium #hvo4 #ajs4 #f;d4 net4          
,reflections mingled with intimate      
reminiscences and recollections of      
,royal personages and of men eminent in 
many spheres4     ,,some ,,pages ,,of   
,,my ,,life4 ,by the ,right ,rev4 ,w4   
,,boyd   ,,carpenter4 ,new and ,cheaper 
,edition4 ,large post #hvo4 #es4   net4 
,all who have listened to his eloquent  
preaching will read with delight   his  
musing on what life has brought the     
author4     ,,raphael ,,meldola1 ,hon4  
,d4,sc4 7,oxon471 ,hon4 ,,ll4,d4 7,st4  
,and471   ,f4,r4,s41 ,professor of      
,chemistry in the ,city and ,guilds of  

,london   ,technical ,college4          
,reminiscences of his worth and work1 by
 those who   knew him1 together with a  
,chronological ,list of his             
,publications   7#ahfh-aiae74 ,edited by
 ,,james ,,marchant4 ,with a ,preface by
 the   ,right ,hon4 ,,lord ,,moulton1   
,k4,c4,b41 ,f4,r4,s4 ,with a ,portrait4 
,crown #hvo4 #es4 net4     ,the book is 
one of the most delightful biographies  
of scientific   men that have been      
published during recent years4--,chemist
 and   ,druggist4     ,,the             
,,manufacture ,,of ,,historical         
,,material4 ,an ,elementary ,study in   
the ,sources of ,story4 ,by ,j4 ,w4     
,,jeudwine1 ,,ll4,b4 7,camb471 of       
,lincoln's ,inn1 ,barrister-at-,law1    
,author of ,the ,first ,twelve          
,centuries   of ,british ,story1 and    
other works4 ,crown #hvo4 #fs4 net4     
,a consideration of the successive      
phases of ,historical ,research2 it     
explains necessary processes through    
which all the material has to   pass    

before it is placed before us as        
history4                                
,,london3 ,,williams ,,and ,,norgate    
,,some ,,books ,,on ,,russia ,,and      
,,poland      ,with a ,coloured         
,frontispiece1 #ab ,photogravure        
,plates1 #bh ,illustrations         in  
the text1 and #h ,maps4 ,demy #hvo4 #gs4
 #f;d4 net 7postage                     
#f;d4 extra74                       ,a  
,thousand ,years of ,russian ,history   
,by ,,sonia ,e4 ,,howe4     ,we can     
recommend the volume as an excellent1   
careful1 and   well-written history of a
 great nation4--,daily ,telegraph4      
,demy #hvo4 ,cloth1 #fs4 net4           
,the ,false ,dmitri      ,a ,russian    
,romance and ,tragedy4 ,described by    
,british ,eye-witnesses1                
#afjd-afab4                    ,edited  
with a ,preface by ,,sonia ,e4 ,,howe4  
,illustrated by reproduction of         
contemporary portraits4     ,with more  
than a whiff of the atmosphere ,hakluyt 
got into his famous   volume4--,northern

 ,whig4     ,of special interest at the 
present time4--,oxford ,chronicle4      
,with #af ,plates and #bh ,illustrations
 in the text4 ,demy #hvo4 #gs4  #f;d4   
net4                   ,some ,russian   
,heroes1 ,saints and ,sinners     ,a    
portrait gallery of outstanding figures 
in ,russian history who were            
typical of their times and their        
country4                                
,by ,,sonia ,e4 ,,howe4     ,it gives a 
vivid study of ,russia 444 and          
interesting book4--,land   and ,water4  
,fcap4 #hvo4 ,cloth1 #as4 #c;d4 net4 ,in
 full leather presentation              
binding1 #bs4 #f;d4 net4                
,an ,outline of ,russian ,literature    
,by the ,hon4 ,,maurice ,,baring1       
,author of ,with the ,russians in       
,manchuria1 ,a ,year in ,russia1 ,the   
,russian ,people1 etc4      ,gives a    
clear and interesting account of the    
growth of ,russian   literature4        
,fcap4 #hvo4 ,cloth1 #as4 #c;d4 net4 ,in
 full leather binding1 #bs4  #f;d4 net4 

,poland                         ,by ,w4 
,,alison ,,phillips1 ,m4,a41            
,lecky ,professor of ,modern ,history in
 the ,university of ,dublin4      ,a    
concise survey of the ,history of       
,poland up to #aiae4                    
,,london3 ,,williams ,,and ,,norgate    
#ad ,,henrietta ,,street1 ,,covent      
,,garden1 ,w4,c4                        
,,transcriber',s ,,notes    ,simple     
typographical errors have been silently 
corrected2 unbalanced  quotation marks  
were remedied when the change was       
obvious1 and  otherwise left unbalanced4
    ,punctuation1 hyphenation1 and      
spelling were made consistent when a    
predominant preference was found in the 
original book2 otherwise they  were not 
changed4