,,university ,,of ,,california          
,,publications                          
,,in                                    
,,mathematics                ,vol4 #a1  
,no4 #i1 pp4 #ahg-bji     ,february #ag1
 #aibj                            ,,on  
,,the ,,history ,,of ,,gunter',s ,,scale
 ,,and                          ,,the   
,,slide ,,rule ,,during ,,the           
,,seventeenth ,,century                 
,,by                                    
,,florian ,,cajori                      
,,university ,,of ,,california ,,press  
,,berkeley                              
,,table ,,of ,,contents                 
,,page    ,i4 ,introduction             
#ahg    ,,ii4 ,innovations in ,gunter's 
,scale                                  
#ahh      ,changes introduced by ,edmund
 ,wingate                               
#ahh      ,changes introduced by        
,milbourn                               
#ahi      ,changes introduced by ,thomas
 ,brown and ,john ,brown                
#aij      ,changes introduced by        

,william ,leybourn                      
#aib    ,,iii4 ,richard ,delamain's     
8,grammelogia0                          
#aib      ,different editions or        
impressions                             
#aid      ,description of ,delamain's   
instrument of #afcj                     
#aie      ,delamain's later designs1 and
 directions for using his               
instruments                             
#aig    ,,iv4 ,controversy between      
,oughtred and ,delamain on the invention
 of            the circular slide rule  
#aii    ,v4 ,independence and priority  
of invention                            
#bjc    ,,vi4 ,oughtred's 8,gauging     
,line10 #afcc                           
#bjf    ,,vii4 ,other seventeenth       
century slide rules                     
#bjg                                    
,i4 ,,introduction        ,in my history
 of the slide rule#a1 and my article on 
its invention#b it  is shewn that       
,william ,oughtred and not ,edmund      
,wingate is the inventor1 that          

,oughtred's circular rule was described 
in print in #afcb1 his rectilinear  rule
 in #afcc4 ,richard ,delamain is        
referred to as having tried to          
appropriate the invention to himself#c  
and as having written a scurrilous      
pamphlet against ,oughtred4 ,all our    
information about ,delamain was taken   
from ,de ,morgan1#d who1 however1 gives 
no evidence of having read any of       
,delamain's writings on the slide rule4 
,through ,dr4 ,arthur ,hutchinson of    
,pembroke ,college1 ,cambridge1 ,i      
learned that ,delamain's writings on the
  slide rule were available4 ,in this   
article will be given3 ,first1 some     
details of the changes introduced during
 the seventeenth century in the  design 
of ,gunter's scale by ,edmund ,wingate1 
,milbourn1 ,thomas ,brown1 ,john  ,brown
 and ,william ,leybourn2 second1 an     
account of ,delamain's book of #afcj  on
 the slide rule which antedates         
,oughtred's first publication 7though   
,oughtred's date of invention is earlier

 than the date of ,delamain's alleged   
invention7 and of ,delamain's later     
designs of slide rules2 third1 an       
account of the controversy between      
,delamain and ,oughtred2 fourth1 an     
account of a later book on the slide    
rule written by ,william ,oughtred1 and 
of other seventeenth century books on   
the slide rule4                         
,,ii4 ,,innovations ,,in ,,gunter',s    
,,scale                                 
,changes introduced by ,wingate      ,we
 begin with ,anthony ,wood's account of 
,wingate's introduction of  ,gunter's   
scale into ,france4#e      ,in #afbd he 
transported into ,france the rule of    
proportion1 having a    little before   
been invented by ,edm4 ,gunter of       
,gresham ,coll4 and    communicated it  
to most of the chiefest mathematicians  
then residing in    ,paris3 who         
apprehending the great benefit that     
might accrue thereby1    importun'd him 
to express the use thereof in the       
,french tongue4 ,which    being         

performed accordingly1 he was advised by
 monsieur ,alleawne the    ,king's chief
 engineer to dedicate his book to       
monsieur the ,king's only    brother1   
since duke of ,orleans4 ,nevertheless   
the said work coming forth    as an     
abortive 7the publishing thereof being  
somewhat hastened1 by reason    an      
advocate of ,dijon in ,burgundy began to
 print some uses thereof1 which         
,wingate had in a friendly way          
communicated to him7 especially in      
regard    ,gunter himself had learnedly 
explained its use in a far larger       
volume4#f    ,gunter's scale1 which     
,wingate calls the 8rule of proportion10
 contained1 as  described in the ,french
 edition of #afbd1 four lines3 7#a7 ,a  
single line of  numbers2 7#b7 a line of 
tangents2 7#c7 a line of sines2 7#d7 a  
line1 one foot  in length1 divided into 
#ab inches and tenths of inches1 also a 
line1 one  foot in length1 divided into 
tenths and hundredths4    ,the ,english 
editions of this book which appeared in 

#afbc and #afbh are  devoid of interest4
 ,the editions of #afde and #afeh       
contain an important  innovation4#g ,in 
the preface the reasons why this        
instrument has not been  used more are  
stated to be3 7#a7 the difficulty of    
drawing the lines with  exactness1 7#b7 
the trouble of working thereupon by     
reason 7sometimes7 of  too large an     
extent of the compasses1 7#c7 the fact  
that the instrument is  not readily     
portable4 ,the drawing of ,wingate's    
arrangement of the scale in  the        
editions of #afde and #afeh is about #ff
 cm4 7#bf4e in47 long4 ,it contains     
five parallel lines1 about #ff cm4 long1
 each having the divisions of one  line 
marked on one side and of another line  
on the other side4 ,thus each  line     
carries two graduations3 7#a7 ,a single 
logarithmic line of numbers2 7#b7  a    
logarithmic line of numbers thrice      
repeated2 7#c7 the first scale          
repeated1 but beginning with the        
graduations which are near the middle of

  the first scale1 so that its          
graduation reads #d1 #e1 #f1 #g1 #h1 #i1
 #a1 #b1 #c2  7#d7 a logarithmic line of
 numbers twice repeated2 7#e7 a         
logarithmic line of  tangents2 7#f7 a   
logarithmic line of sines2 7#g7 the rule
 divided into #ajjj  equal parts2 7#h7  
the scale of latitudes2 7#i7 a line of  
inches and tenths of  inches2 7#aj7 a   
scale consisting of three kinds1 viz41 a
 gauge line1 a line  of chords1 and a   
foot measure1 divided into #ajjj equal  
parts4    ,important are the first and  
second scales1 by which cube root       
extraction  was possible 8by inspection 
only1 without the aid of pen or         
compass20  similarly the third and      
fourth scales1 for square roots4 ,this  
innovation is  due to ,wingate4 ,the    
#afde edition announces that the        
instrument was made in  brass by ,elias 
,allen1 and in wood by ,john ,thompson  
and ,anthony ,thompson in  ,hosier      
,lane4                                  
,changes introduced by ,milbourn        

,william ,leybourn1 in his ,the ,line of
 ,proportion or ,numbers1 ,commonly     
called  ,gunter's ,line1 ,made ,easie1  
,london1 #afgc1 says in his preface 8,to
 the  ,reader30      ,the ,line of      
,proportion or ,numbers1 commonly called
 7by ,artificers7    ,gunter's ,line1   
hath been discoursed of by several      
persons1 and variously    applied to    
divers uses2 for when ,mr4 ,gunter had  
brought it from the    ,tables to a     
,line1 and written some ,uses thereof1  
,mr4 ,wingate added divers    ,lines of 
several lengths1 thereby to extract the 
,square or ,cube ,roots1    without     
doubling or trebling the distance of the
 ,compasses3 ,after him ,mr4            
,milbourn1 a ,yorkshire ,gentleman1     
disposed it in a ,serpentine or ,spiral 
,line1 thereby enlarging the divisions  
of the ,line4    ,on pages #abg and #abh
 ,leybourn adds3      ,again1 ,one ,t4  
,browne1 a ,maker of ,mathematical      
,instruments1 made it in a              
,serpentine or ,spiral ,line1 composed  

of divers ,concentrick ,circles1        
thereby to enlarg the divisions1 which  
was the contrivance of one ,mr4         
,milburn a ,yorkshire ,gentleman1 who   
writ thereof1 and communicated his      
,uses to the aforesaid ,brown1 who      
7since his death7 attributed it to      
himself3 ,but whoever was the contriver 
of it1 it is not without                
inconvenience2 for it can in no wise be 
made portable2 and besides    7instead  
of compasses7 an opening ,joynt with    
thirds threads must be    placed to move
 upon the ,centre of the ,instrument1   
without which no    proportion can be   
wrought4    ,this ,mr4 ,milburn is      
probably the person named in the diary  
of the  antiquarian1 ,elias ,ashmole1 on
 ,august #ac #afdf82 8,i bought of ,mr4 
,milbourn  all his ,books and           
,mathematical ,instruments40#h ,charles 
,hutton#i says that  ,milburne of       
,yorkshire designed the spiral form     
about #afej4 ,this date is  doubtless   
wrong1 for ,thomas ,browne who1         

according to ,leybourn1 got the  spiral 
form of line from ,milbourn1 is         
repeatedly mentioned by ,william        
,oughtred in his ,epistle#aj printed    
some time in #afcb or #afcc4 ,oughtred  
does not mention ,milbourn1 and says    
7page #d7 that the spiral form 8was     
first hit upon by one ,thomas ,browne a 
,joyner1 4 4 4 the serpentine           
revolution being but two true           
semicircles described on severall       
centers40#aa                  ,changes  
introduced by ,thomas ,brown and ,john  
,brown    ,thomas ,brown did not publish
 any description of his instrument1 but 
his  son1 ,john ,brown1 published in    
#affa a small book1#ab in which he says 
7preface7 that he had done 8as ,mr4     
,oughtred with ,gunter's ,rule1 to a    
sliding and circular form2 and as my    
father ,thomas ,brown into a ,serpentine
  form2 or as ,mr4 ,windgate in his     
,rule of ,proportion40 ,he says also    
that  8this brief touch of the          
,serpentine-line ,i made bold to assert1

 to see if ,i  could draw out a         
performance of that promise1 that hath  
been so long  unperformed by the        
promisers thereof40 ,accordingly in     
,chapter ,,xx he gives a  description of
 the serpentine line1 8contrived in five
 7or rather #ae7  turn40 ,whether this  
description1 printed in #affa1 exactly  
fits the  instrument as it was developed
 in #afcb1 we have no means of knowing4 
,john  ,brown says3      #a4 ,first next
 the center is two circles divided one  
into #fj1 the other    into #ajj parts1 
for the reducing of minutes to #ajj     
parts1 and the    contrary4      #b4    
,you have in seven turnes two inpricks1 
and five in divisions1 the    first     
,radius of the sines 7or ,tangents being
 neer the matter1 alike to    the first 
three degrees17 ending at #e degrees and
 #dd minutes4      #c4 ,thirdly1 you    
have in #e turns the lines of numbers1  
sines1 ,tangents1 in    three margents  
in divisions1 and the line of versed    
sines in pricks1    under the line of   

,tangents1 according to ,mr4 ,gunter's  
cross-staff3 the    sines and ,tangents 
beginning at #e degrees1 and #dd minutes
 where the other    ended1 and          
proceeding to #ij in the sines1 and #de 
in the ,tangents4 ,and the    line of   
numbers beginning at #aj1 and proceeding
 to #ajj1 being one entire    ,radius1  
and graduated into as many divisions as 
the largeness of the    instrument will 
admit1 being #aj to #aj #ej into #ej    
parts1 and from #ej to    #ajj into #bj 
parts in one unit of increase1 but the  
,tangents are divided    into single    
minutes from the beginning to the end1  
both in the first1    second and third  
,radiusses1 and the sines into minutes2 
also from #cj    minutes to #dj degrees1
 and from #dj to #fj1 into every two    
minutes1 and    from #fj to #hj in every
 #eth minute1 and from #hj to #he every 
#ajth1 and the    rest as many as can be
 well discovered4      ,the versed sines
 are set after the manner of ,mr4       
,gunter's ,cross-staff1    and divided  

into every #ajth minutes beginning at   
#j1 and proceeding to #aef    going     
backwards under the line of ,tangents4  
#d4 ,fourthly1 beyond the ,tangent of   
#de in one single line1 for one ,turn is
    the secants to #ea degrees1 being   
nothing else but the sines reitterated  
beyond #ij4      #e4 ,fifthly1 you have 
the line of ,tangents beyond #de1 in #e 
turnes to #he    degrees1 whereby all   
trouble of backward working is avoided4 
#f4 ,sixthly1 you have in one circle the
 #ahj degrees of a ,semicircle1 and     
also a line of natural sines1 for       
finding of differences in sines1 for    
finding hour and ,azimuth4      #g4     
,seventhly1 next the verge or outermost 
edge is a line of equal parts    to get 
the ,logarithm of any number1 or the    
,logarithm sine and ,tangent of    any  
ark or angle to four figures besides the
 carracteristick4      #h4 ,eightly and 
lastly1 in the space place between the  
ending of the    middle five turnes1 and
 one half of the circle are three prickt

 lines    fitted for reduction4 ,the    
uppermost being for shillings1 pence and
    farthings4 ,the next for pounds1 and
 ounces1 and quarters of small          
,averdupoies weight4 ,the last for      
pounds1 shillings and pence1 and to be  
used thus3 ,if you would reduce #afs4   
#c;d4 #bq4 to a decimal fraction1 lay   
the hair or edge of one of the legs of  
the index on #af4 #c in the line of     
#a4 s4 d4 and the hair shall cut on the 
equal parts #ha #af2 and the            
contrary1 if you have a decimal         
fraction1 and would reduce it to a      
proper    fraction1 the like may you do 
for shillings1 and pence1 and pounds1   
and    ounces4                          
,the uses of the lines follow4      ,as 
to the use of these lines1 ,i shall in  
this place say but little1 and    that  
for two reasons4 ,first1 because this   
instrument is so contrived1    that the 
use is sooner learned then any other1 ,i
 speak as to the manner1    and way of  
using it1 because by means of first     

second and third    radiusses1 in sines 
and ,tangents1 the work is always right 
on1 one way or    other1 according to   
the ,canon whatsoever it be1 in any book
 that treats    of the ,logarithms1 as  
,gunter1 ,wells1 ,oughtred1 ,norwood1 or
 others1 as in    ,oughtred from page   
#fd to #ajg4      ,secondly1 and more   
especially1 because the more accurate1  
and large    handling thereof is more   
then promised1 if not already performed 
by more    abler pens1 and a large      
manuscript thereof by my ,sires meanes1 
provided    many years ago1 though to   
this day not extant in print2 so for his
 sake ,i    claiming my interest        
therein1 make bold to present you with  
these few    lines1 in order to the use 
of them3 ,and first note1      #a4      
,which soever of the two legs is set to 
the first term in the question1    that 
,i call the first leg always1 and the   
other being set to the second    term1  
,i call the second leg 4 4 4    ,the    
exact nature of the contrivance with the

 8two legs0 is not described1  but it   
was probably a flat pair of compasses1  
attached to the metallic  surface on    
which the serpentine line was drawn4 ,in
 that case the instrument  was a slide  
rule1 rather than a form of ,gunter's   
line4 ,in his publication  of #affa1 as 
also in later publications1#ac ,john    
,brown devoted more space  to ,gunter's 
scales1 requiring the use of a separate 
pair of compasses1 than  to slide rules4
                        ,changes        
introduced by ,william ,leybourn    ,the
 same remark applies to ,william        
,leybourn who1 after speaking of ,seth  
,partridge's slide rule1 returns to     
forms of ,gunter's scale1 saying3#ad    
,there is yet another way of disposing  
of this ,line of ,proportion1 by        
having one ,line of the full length of  
the ,ruler1 and another ,line of the    
same ,radius broken in two parts between
 #c and #d2 so that in working your     
,compasses never go off of the ,line3   
,this is one of the best contrivances1  

but here ,compasses must be used4 ,these
 are all the ,contrivances that ,i      
have hitherto seen of these ,lines3     
,that which ,i here speak of1 and will  
shew how to use1 is only two ,lines of  
one and the same ,radius1 being set     
upon a plain ,ruler of any length 7the  
larger the better7 having the           
beginning of one ,line1 at the end of   
the other1 the divisions of each        
,line being set so close together1 that 
if you find any number upon one of      
the ,lines1 you may easily see what     
number stands against it on the other   
,line4 ,this is all the ,variation4 4 4 
4      ,example #a4 ,if a ,board be #a  
,foot #fd parts broad1 how much in      
length of    that ,board will make a    
,foot ,square8 ,look upon one of your   
,lines 7it    matters not which7 for #a 
,foot #fd parts1 and right against it on
 the other    ,line you shall find #fa2 
and so many parts of a ,foot will make a
 ,foot    square of that ,board4        
,this contrivance solves the equation   

#a4fdx#a1 yielding centesimal parts of  
a foot4    ,james ,atkinson#ae speaks of
 8,gunter's scale0 as 8usually of       
,boxwood 4 4 4  commonly #b ft4 long1 #a
 inch broad0 and 8of two kinds3 long    
,gunter or  single ,gunter1 and the     
sliding ,gunter4 ,it appears that during
 the  seventeenth century 7and long     
after7 the ,gunter's scale was a rival  
of the  slide rule4                     
,,iii4 ,,richard ,,delamain',s          
,,grammelogia      ,we begin with a     
brief statement of the relations between
 ,oughtred and  ,delamain4 ,at one time 
,delamain1 a teacher of mathematics in  
,london1 was  assisted by ,oughtred in  
his mathematical studies4 ,in #afcj     
,delamain  published the ,grammelogia1 a
 pamphlet describing a circular slide   
rule and  its use4 ,in #afca he         
published another tract1 on the         
,horizontall  ,quadrant4#af ,in #afcb   
appeared ,oughtred's ,circles of        
,proportion#ag  translated into ,english
 from ,oughtred's ,latin manuscript by  

another pupil1  ,william ,forster1 in   
the preface of which ,forster makes the 
charge 7without  naming ,delamain7 that 
8another 4 4 4 went about to            
pre-ocupate0 the new  invention4 ,this  
led to verbal disputes and to the       
publication by ,delamain  of several    
additions to the ,grammelogia1          
describing further designs of  circular 
slide rules and also stating his side of
 the bitter controversy1  but without   
giving the name of his antagonist4      
,oughtred's ,epistle was  published as a
 reply4 ,each combatant accuses the     
other of stealing the  invention of the 
circular slide rule and the horizontal  
quadrant4    ,the two title-pages of the
 edition of the ,grammelogia in the     
,british ,museum               in       
,london which we have called            
8,grammelogia ,,iv40                    
,different editions or impressions      
,there are at least five different      
editions1 or impressions1 of the        
,grammelogia which we designate1 for    

convenience1 as follows3                
,grammelogia ,i1 #afcj4 ,one copy in the
 ,cambridge ,university ,library4#ah    
,grammelogia ,,ii1 ,i have not seen a   
copy of this4      ,grammelogia ,,iii1  
,one copy in the ,cambridge ,university 
,library4#ai      ,grammelogia ,,iv1    
,one copy in the ,british ,museum1      
another in the ,bodleian    ,library1   
,oxford4#bj      ,grammelogia ,v1 ,one  
copy in the ,british ,museum4    ,in    
,grammelogia ,i the first three leaves  
and the last leaf are without           
pagination4 ,the first leaf contains the
 title-page2 the second leaf1 the       
dedication to the ,king and the preface 
8,to the ,reader20 the third leaf1 the  
description of the ,mathematical ,ring4 
,then follow #bb numbered pages4        
,counting the unnumbered pages1 there   
are altogether #cj pages in the         
pamphlet4 ,only the first three leaves  
of this pamphlet are omitted in         
,grammelogia ,,iv and ,v4    ,in        
,grammelogia ,,iii the ,appendix begins 

with a page numbered #eb and bears  the 
heading 8,conclusion20 it ends with page
 #fh1 which contains the same two  poems
 on the mathematical ring that are given
 on the last page of  ,grammelogia ,i   
but differs slightly in the spelling of 
some of the words4  ,the #ea pages which
 must originally have preceded page #eb1
 we have not seen4  ,the edition        
containing these we have designated     
,grammelogia ,,ii4 ,the reason  for the 
omission of these #ea pages can only be 
conjectured4 ,in ,oughtred's  ,epistle  
7p4 #bd71 it is stated that ,delamain   
had given a copy of the  ,grammelogia to
 ,thomas ,brown1 and that two days later
 ,delamain asked for the  return of the 
copy1 8because he had found some things 
to be altered  therein0 and 8rent out   
all the middle part40 ,delamain labored 
8to recall  all the bookes he had given 
forth1 7which were many7 before the     
sight of  ,brownes ,lines40 ,these      
spiral lines ,oughtred claimed that     
,delamain had  stolen from ,brown4 ,the 

title-page and page #eb are the only    
parts of the  ,appendix1 as given in    
,grammelogia ,,iii1 that are missing in 
the ,grammelogia  ,,iv and ,v4          
,grammelogia ,,iv answers fully to the  
description of ,delamain's pamphlet     
contained in ,oughtred's ,epistle4 ,it  
was brought out in #afcb or #afcc1 for  
what appears to be the latest part of it
 contains a reference 7page #ii7 to  the
 ,grammelogia ,i 7#afcj7 as 8being now  
more then two yeares past40  ,moreover1 
it refers to ,oughtred's ,circles of    
,proportion1 #afcb1 and  ,oughtred's    
reply in the ,epistle was bound in the  
,circles of ,proportion  having the     
,addition of #afcc4 ,for convenience of 
reference we number the two  title-pages
 of ,grammelogia ,,iv1 8page 7#a70 and  
8page 7#b710 as is done by  ,oughtred in
 his ,epistle4 ,grammelogia ,,iv        
contains1 then1 #aac pages4 ,the page   
numbers which we assign will be placed  
in parentheses1 to distinguish them     
from the page numbers which are printed 

in ,grammelogia ,,iv4 ,the pages        
7#dd7-7fe7 are the same as the pages    
#a-bb1 and the pages 7#fh7-7hc7 are the 
same as the pages #ec-fh4 ,thus only    
thirty-eight pages have page numbers    
printed on them4 ,the pages 7#fg7 and   
7#hc7 are identical in wording1 except  
for some printer's errors2 they contain 
verses in praise of the ,ring1 and  have
 near the bottom the word 8,finis40     
,also1 pages 7#bb7 and 7#bc7 are        
together identical in wording with page 
7#aac71 which is set up in finer  type1 
containing an advertisement of a part of
 ,grammelogia ,,iv explaining  the mode 
of graduating the circular rules4 ,there
 are altogether six parts  of           
,grammelogia ,,iv which begin or end by 
an address to the reader1 thus3 8,to    
the ,reader10 8,courteous ,reader10 or  
8,to the courteous and benevolent       
,reader  4 4 410 namely the pages 7#h71 
7#bb71 7#fh71 7#hi71 7#ij71 7#ajh74 ,in 
his ,epistle  7page #b71 ,oughtred      
characterizes the make up of the book in

 the following  terms3      ,in reading 
it 4 4 4 ,i met with such a patchery and
 confusion of    disjoynted stuffe1 that
 ,i was striken with a new wonder1 that 
any man    should be so simple1 as to   
shame himselfe to the world with such a 
hotch-potch4    ,grammelogia ,v differs 
from ,grammelogia ,,iv in having only   
the second  title-page4 ,the first      
title-page may have been torn off from  
the copy ,i  have seen4 ,a second       
difference is that the page with the    
printed numeral #bb  in ,grammelogia    
,,iv has after the word 8,finis0 the    
following notice3      ,this instrument 
is made in ,silver1 or ,brasse for the  
,pocket1 or at any    other bignesse1   
over against ,saint ,clements ,church   
without ,temple ,barre1    by ,elias    
,allen4    ,this notice occurs also on  
page #bb of ,grammelogia ,i and ,,iii1  
but is omitted  from page #bb of        
,grammelogia ,v4                        
,description of ,delamain's instrument  
of #afcj    ,in his address to ,king    

,charles ,i1 in his ,grammelogia ,i1    
,delamain emphasizes  the ease of       
operating with his slide rule by stating
 that it is 8fit for  use 4 4 4 as well 
on ,horse backe as on ,foot40 ,speaking 
8,to the ,reader10 he  states that he   
has 8for many yeares taught the         
,mathematicks in this ,towne10  and made
 efforts to improve ,gunter's scale 8by 
some ,motion1 so that the  whole body of
 ,logarithmes might move proportionally 
the one to the other1  as occasion      
required4 ,this conceit in ,february    
last #afbi ,i struke upon1  and so      
composed my ,grammelogia or             
,mathematicall ,ring2 by which only with
 an  ocular inspection1 there is had at 
one instant all proportionalls through  
the said body of ,numbers40 ,he dates   
his preface 8first of ,january1 #afcj40 
,the fifth and sixth pages contain his  
8,description of the ,grammelogia10 the 
term ,grammelogia being applied to the  
instrument1 as well as to the book4     
,his description is as follows3         

,the parts of the ,instrument are two   
,circles1 the one moveable1 and the     
other fixed2 ,the moveable is that unto 
which is fastened a small pin to    move
 it by2 the other ,circle may be        
conceived to be fixed2 ,the             
circumference of the moveable ,circle is
 divided into unequall parts1           
charactered with figures thus1 #a4 #b4  
#c4 #d4 #e4 #f4 #g4 #h4 #i4 these       
figures    doe represent themselves1 or 
such numbers unto which a ,cipher or    
,ciphers    are added1 and are varied as
 the occasion falls out in the speech of
    ,numbers1 so #a4 stands for #a4 or  
#aj4 or #ajj41 c4 the #b4 stands for #b4
 or    #bj4 or #bjj4 or #bjjj41 c4 the  
#c4 stands for #cj4 or #cjj4 or #cjjj41 
c4    ,after elaborating this last point
 and explaining the decimal subdivisions
  on the scales of the movable circle1  
he says that 8the numbers and  divisions
 on the fixed ,circle1 are the very same
 that the moveable are1 4  40 ,there is 
no drawing of the slide rule in this    

publication4 ,the  twenty-two numbered  
pages give explanations of the various  
uses to which  the instrument can be    
put3 8,how to performe the ,golden      
,rule0 7pp4 #a-c71  8,further uses of   
the ,golden ,rule0 7pp4 #d-f71 8,notions
 or ,principles  touching the disposing 
or ordering of the ,numbers in the      
,golden ,rule in  their true places upon
 the ,grammelogia0 7pp4 #g-aa71 8,how to
 divide one  number by another0 7pp4    
#ab1 #ac71 8to multiply one ,number by  
another0 7pp4  #ad1 #ae71 8,to find     
,numbers in continuall proportion0 7pp4 
#af1 #ag71 8,how to  extract the ,square
 ,root10 8,how to extract the ,cubicke  
,root0 7pp4 #ah-ba71  8,how to performe 
the ,golden ,rule0 7the rule of         
proportion7 is explained  thus3         
,seeke the first number in the moveable1
 and bring it to the second number    in
 the fixed1 so right against the third  
number in the moveable1 is the    answer
 in the fixed4      ,if the ,interest of
 #ajj4 li4 be #h4 li4 in the yeare1 what

 is the ,interest    of #fe4 li4 for the
 same time4      ,bring #ajj4 in the    
moveable to #h4 in the fixed1 so right  
against #fe4 in    the moveable is #e4b4
 in the fixed1 and so much is the       
,interest of #fe4 li4    for the yeare  
at #h4 li4 for #ajj4 li4 per annum4     
,the ,instrument not removed1 you may at
 one instant right against any    summe 
of money in the moveable1 see the       
,interest thereof in the fixed3    the  
reason of this is from the ,definition  
of ,logarithmes4    ,these are the      
earliest known printed instructions on  
the use of a slide  rule4 ,it will be   
noticed that the description of the     
instrument at the  opening makes no     
references to logarithmic lines for the 
trigonometric  functions2 only the line 
of numbers is given4 ,yet the title-page
 promised  the 8resolution of ,plaine   
and ,sphericall ,triangles40 ,page #bb  
throws light  upon this matter3      ,if
 there be composed three ,circles of    
equal thicknesse1 ,a4,b4,c4 so that     

the inner edge of ,d should be ,b and   
the outward edge of ,a bee    answerably
 graduated with ,logarithmall signes    
sines1 and the outward    edge of ,b and
 the inner edge of ,a with ,logarithmes2
 and then on the    backside be         
graduated the ,logarithmall ,tangents1  
and againe the    ,logarithmall signes  
oppositly to the former graduations1 it 
shall be    fitted for the resolution of
 ,plaine and ,sphericall ,triangles4    
,after twelve lines of further remarks  
on this point he adds3      ,hence from 
the forme1 ,i have called it a ,ring1   
and ,grammelogia by    annoligie of a   
,lineary speech2 which ,ring1 if it were
 projected in the    convex unto two    
yards ,diameter1 or thereabouts1 and the
 line ,decupled1 it    would worke      
,trigonometrie unto seconds1 and give   
proportionall numbers    unto six places
 only by an ocular inspection1 which    
would compendiate    ,astronomicall     
calculations1 and be sufficient for the 
,prosthaphaeresis of    the ,motions3   

,but of this as ,god shall give life and
 ability to health and    time4    ,the 
unnumbered page following page #bb      
contains the patent and copyright on    
the instrument and book3      ,whereas  
,richard ,delamain1 ,teacher of         
,mathematicks1 hath presented vnto ,vs  
an ,instrument called ,grammelogia1 or  
,the ,mathematicall ,ring1 together     
with a ,booke so intituled1 expressing  
the use thereof1 being his owne         
,invention2 we of our ,gracious and     
,princely favour have granted unto the  
said ,richard ,delamain and his         
,assignes1 ,privilege1 ,licence1 and    
,authority1 for the sole ,making1       
,printing and ,selling of the said      
,instrument and ,booke3 straightly      
forbidding any other to ,make1 ,imprint1
    or ,sell1 or cause to be ,made1 or  
,imprinted1 or ,sold1 the said          
,instrument    or ,booke within any our 
,dominions1 during the space of ten     
yeares next    ensuing the date hereof1 
upon paine of ,our high displeasure4    

,given under    our hand and ,signet at 
our ,palace of ,westminster1 the fourth 
day of    ,january1 in the sixth yeare  
of our ,raigne4          ,delamain's    
later designs1 and directions for using 
his instruments    ,in the ,appendix of 
,grammelogia ,,iii1 on page #eb is given
 a description of an  instrument        
promised near the end of ,grammelogia   
,i3      ,that which ,i have formerly   
delivered hath been onely upon one of   
the    ,circles of my ,ring1 simply     
concerning ,arithmeticall ,proportions1 
,i will    by way of ,conclusion touch  
upon some uses of the ,circles1 of      
,logarithmall    ,sines1 and ,tangents1 
which are placed on the edge of both the
 moveable    and fixed ,circles of the  
,ring in respect of ,geometricall       
,proportions1 but    first of the       
description of these ,circles4          
,first1 upon the side that the ,circle  
of ,numbers is one1 are graduated on    
the edge of the moveable1 and also on   
the edge of the fixed the               

,logarithmall ,sines1 for if you bring  
#a4 in the moveable amongst the         
,numbers to #a4 in the fixed1 you may on
 the other edge of the moveable and     
fixed see the sines noted thus #ij4 #ij4
 #hj4 #hj4 #gj4 #gj4 #fj4 #fj4 c4 unto  
#f4f4 and each degree subdivided1 and   
then over the former divisions and      
figures #ij4 #ij4 #hj4 #hj4 #gj4 #gj4 c4
 you have the other degrees1 viz4 #e4   
#d4 #c4 #b4 #a4 each of those divided by
 small points4      ,secondly1 7if the  
,ring is great7 neere the outward edge  
of this side of    the fixed against the
 ,numbers1 are the usuall divisions of a
 ,circle1 and    the points of the      
,compasse3 serving for observation in   
,astronomy1 or    ,geometry1 and the    
sights belonging to those divisions1 may
 be placed on    the moveable ,circle4  
,thirdly1 opposite to those ,sines on   
the other side are the ,logarithmall    
,tangents1 noted alike both in the      
moveable and fixed thus                 
#f4f4g4g4h4h4i4i4aj4aj4ae4ae4bj4bj4 c4  

unto #de4de4 which numbers or           
divisions serve also for their          
,complements to #ij4 so #dj gr4 stands  
for    #ej4 gr4 #cj4 gr4 for #fj gr4    
#bj4 gr4 for #gj4 gr4 c4 each degree    
here both    in the moveable and fixed  
is also divided into parts4 ,as for the 
degrees    which are under #f4 viz4     
#e4d4c4b4a4 they are noted with small   
figures over    this divided ,circle    
from #de4dj4ce4cj4be4 c4 and each of    
those degrees    divided into parts by  
small points both in the moveable and   
fixed4      ,fourthly1 on the other edge
 of the moveable on the same side is    
another    graduation of ,tangents1 like
 that formerly described4 ,and opposite 
unto    it1 in the fixed is a           
,graduation of ,logarithmall sines in   
every thing    answerable to the first  
descrition of ,sines on the other side4 
,fifthly1 on the edge of the ,ring is   
graduated a parte of the quator1        
numbered thus #aj #bj4 #cj4 unto #ajj4  
and there unto is adjoyned the degrees  

of the ,meridian inlarged1 and numbered 
thus #aj #bj4cj unto #gj4 each degree   
both of the quator1 and ,meridian are   
subdivided into parts2 these two        
graduated ,circles serve to resolve such
 ,questions which concerne    ,latitude1
 ,longitude1 ,rumb1 and ,distance1 in   
,nauticall operations4      ,sixthly1 to
 the concave of the ,ring may be added a
 ,circle to be elevated    or depressed 
for any ,latitude1 representing the     
quator1 and so divided    into houres   
and parts with an ,axis1 to shew both   
the houre1 and ,azimuth1    and within  
this ,circle may be hanged a ,box1 and  
,needle with a ,socket for    a staffe  
to slide into it1 and this accommodated 
with scrue pines to    fasten it to the 
,ring and staffe1 or to take it off at  
pleasure4    ,the pages bearing the     
printed numbers #ec-fh in the           
,grammelogia ,,iii1 ,,iv and  ,v make no
 reference to the dispute with ,oughtred
 and may1 therefore1 be  assumed to have
 been published before the appearance of

 ,oughtred's ,circles  of ,proportion4  
,on page #ec1 8,to the ,reader10 he     
says3      4 4 4 you may make use of the
 ,projection of the ,circles of the     
,ring upon    a ,plaine1 having the feet
 of a paire of compasses 7but so that   
they be    flat7 to move on the ,center 
of that ,plaine1 and those feet to open 
and    shut as a paire of ,compasses 4 4
 4 now if the feet bee opened to any two
    termes or numbers in that           
,projection1 then may you move the first
 foot to    the third number1 and the   
other foot shall give the ,answer2 4 4 4
 it hath    pleased some to make use of 
this way4 ,but in this there is a double
    labour in respect to that of the    
,ring1 the one in fitting those feet    
unto    the numbers assigned1 and the   
other by moving them about1 in which a  
man    can hardly accommodate the       
,instrument with one hand1 and expresse 
the    ,proportionals in writing with   
the other4 ,by the ,ring you need not   
but    bring one number to another1 and 

right against any other number is the   
,answer without any such motion4 4 4 4  
upon that the ,ring ,i write1    shewing
 some uses of those ,circles amongst    
themselves1 and conjoyned with    others
 4 4 4 in ,astronomy1 ,horolographie1 in
 plaine ,triangles applyed to           
,dimensions1 ,navigation1               
,fortification1 etc4 4 4 4 ,but before  
,i come to    ,construction1 ,i have    
thought it convenient by way            
introduction1 to    examine the truth of
 the graduation of those ,circles 4 4 4 
,these are the words of a practical man1
 interested in the mechanical           
development of his instrument4 ,he      
considers not only questions of         
convenience but also of accuracy4 ,the  
instrument has1 or may have now1 also   
lines of sines and tangents4 ,to test   
the accuracy of the circles of          
,numbers1 8bring any number in the      
moveable to halfe of that number in the 
fixed3 so any number or part in the     
fixed shall give his double in the      

moveable1 and so may you trie of the    
thirds1 fourths c4 of numbers1 vel      
contra10 7p4 #ed74 ,on page #ee are     
given two small drawings1 labelled1 8,a 
,type of the ,ringe and ,scheme of this 
,logarithmicall projection1 the use     
followeth4 ,these ,instruments are made 
in ,silver or ,brasse by ,john ,allen   
neare the ,sauoy in the ,strand40       
,,iv4 ,,controversy ,,between ,,oughtred
 ,,and ,,delamain ,,on ,,the ,,invention
 ,,of ,,the                             
,,circular ,,slide ,,rule    ,delamain's
 publication of #afcj on the            
#8,mathematicall ,ring' does not appear 
at that time to have caused a rupture   
between him and ,oughtred4 ,when in     
#afca ,delamain brought out his         
,horizontall ,quadrant1 the invention of
 which  ,delamain was afterwards charged
 to have stolen from ,oughtred1         
,delamain was  still in close touch with
 ,oughtred and was sending ,oughtred in 
the ,arundell  ,house1 ,london1 the     
sheets as they were printed4 ,oughtred's

 reference to  this in his ,epistle 7p4 
#bj7 written after the friendship was   
broken1 is as  follows3      ,while he  
was printing his tractate of the        
,horizontall quadrant1 although    he   
could not but know that it was injurious
 to me in respect of my free    gift to 
,master ,allen1 and of ,william         
,forster1 whose translation of my       
rules was then about to come forth3 yet 
such was my good nature1 and his        
shamelessnesse1 that every day1 as any  
sheet was printed1 hee sent1 or         
brought the same to mee at my chamber in
 ,arundell house to peruse which ,i     
lovingly and ingenuously did1 and gave  
him my judgment of it4    ,even after   
,forster's publication of ,oughtred's   
,circles of ,proportion1 #afcb1         
,oughtred had a book1 ,a canon of ,sines
 ,tangents and ,secants1 which he had   
borrowed from ,delamain and was then    
returning 7,epistle1 page 7#e774 ,the   
attacks which ,forster1 in the preface  
to the ,circles of ,proportion1 made    

upon ,delamain 7though not naming       
,delamain7 started the quarrel4 ,except 
for  ,forster and other pupils of       
,oughtred who urged him on to castigate 
,delamain1 the controversy might never  
have arisen4 ,forster expressed  himself
 in part as follows3      4 4 4 being in
 the time of the long vacation #afcj1 in
 the ,country1 at the    house of the   
,reverend1 and my most worthy friend1   
and ,teacher1 ,mr4    ,william ,oughtred
 7to whose instruction ,i owe both my   
initiation1 and    whole progresse in   
these ,sciences47 ,i vpon occasion of   
speech told him of    a ,ruler of       
,numbers1 ,sines1  ,tangents1 which one 
had be-spoken to be made    7such as it 
vsually called ,mr4 ,gunter's ,ruler7 #f
 feet long1 to be vsed    with a payre  
of beame-compasses4 8,he answered that  
was a poore invention1    and the       
performance very troublesome3 ,but1 said
 he1 seeing you are taken    with such  
mechanicall wayes of ,instruments1 ,i   
will shew you what deuises    ,i have   

had by mee these many yeares40 ,and     
first1 hee brought to mee two    ,rulers
 of that sort1 to be vsed by applying   
one to the other1 without any           
compasses3 and after that hee shewed mee
 those lines cast into a circle    or   
,ring1 with another moueable circle vpon
 it4 ,i seeing the great                
expeditenesse of both those wayes2 but  
especially1 of the latter1 wherein    it
 farre excelleth any other ,instrument  
which hath bin knowne2 told him1    ,i  
wondered that hee could so many yeares  
conceale such vseful inuentions1    not 
onely from the world1 but from my selfe1
 to whom in other parts and    mysteries
 of ,art1 he had bin so liberall4 ,he   
answered1 8,that the true way    of ,art
 is not by ,instruments1 but by         
,demonstration3 and that it is a        
preposterous course of vulgar ,teachers1
 to begin with ,instruments1 and    not 
with the ,sciences1 and so in-stead of  
,artists1 to make their    ,schollers   
only doers of tricks1 and as it were    

,iuglers3 to the despite of    ,art1    
losse of precious time1 and betraying of
 willing and industrious    wits1 vnto  
ignorance and idlenesse4 ,that the vse  
of ,instruments is indeed    excellent1 
if a man be an ,artist3 but             
contemptible1 being set and opposed     
to ,art4 ,and lastly1 that he meant to  
commend to me1 the skill of             
,instruments1 but first he would haue me
 well instructed in the ,sciences4      
,he also shewed me many notes1 and      
,rules for the vse of those circles1 and
    of his ,horizontall ,instrument1    
7which he had proiected about #cj yeares
    before7 the most part written in    
,latine4 ,all which ,i obtained of him  
leaue to translate into ,english1 and   
make publique1 for the vse1 and         
benefit of such as were studious1 and   
louers of these excellent ,sciences4    
,which thing while ,i with mature1 and  
diligent care 7as my occasions would    
give me leaue7 went about to doe3       
another to whom the ,author in a louing 

confidence discouered this intent1 using
 more hast then good speed1 went        
about to preocupate2 of which vntimely  
birth1 and preuenting 7if not           
circumuenting7 forwardnesse1 ,i say no  
more3 but aduise the studious           
,reader1 onely so farre to trust1 as he 
shal be sure doth agree to truth        
,art4    ,while in this dedication      
reference is made to a slide rule or    
8ring0 with a  8moveable circle10 the   
instrument actually described in the    
,circles of  ,proportion consists of    
fixed circles 8with an index to be      
opened after the  manner of a paire of  
,compasses40 ,delamain1 as we have seen1
 had decided  preference for the        
moveable circle4 ,to ,oughtred1 on the  
other hand1 one  design was about as    
good as the other2 he was more of a     
theorist and  repeatedly expressed his  
contempt for mathematical instruments4  
,in his  ,epistle 7page 7#be771 he says 
he had not 8the one halfe of my         
intentions  upon it0 7the rule in his   

book71 nor one with a 8moveable circle  
and a  thread1 but with an opening      
,index at the centre 7if so be that bee 
cause  enough to make it to bee not the 
same1 but another ,instrument7 for my   
part  ,i disclaime it3 it may go seeke  
another ,master3 which for ought ,i     
know1  will prove to be ,elias ,allen   
himselfe3 for at his request only ,i    
altered a  little my rules from the use 
of the moveable circle and the thread1  
to the  two armes of an ,index40    ,all
 parts of ,delamain's ,grammelogia ,,iv1
 except pages #a-bb and #ec-fh          
considered above1 were published after  
the ,circles of ,proportion1 for they   
contain references to the ill treatment 
that ,delamain felt or made believe     
that he felt1 that he had received in   
the book published by ,oughtred and     
,forster4 ,oughtred's reference to      
teachers whose scholars are 8doers of   
tricks10 8,iuglers10 and ,forster's     
allusion to 8another to whom the ,author
  in a loving confidence0 explained the 

instrument and who 8went about to       
preocupate0 it1 are repeatedly          
mentioned4 ,delamain says1 7page 7#hi77 
that  at first he did not intend to     
express himself in print1 8but sought   
peace  and my right by a private and    
friendly way40 ,oughtred's account of   
,delamain's course is that of an        
8ill-natured man0 with a 8virulent      
tongue10  8sardonical laughter0 and     
8malapert sawsiness40 ,contrasting      
,forster and  ,delamain1 he says that1  
of the former he 8had the very first    
moulding0 and  made him feel that 8the  
way of ,art0 is 8by demonstration40 ,but
 ,delamain was  8already corrupted with 
doing upon ,instruments1 and quite lost 
from ever  being made an ,artist40      
7,epistle page 7#bg774 ,repeatedly does 
,oughtred assert  ,delamain's ignorance 
of mathematics4 ,the two men were       
evidently of wholly  different          
intellectual predilections4 ,that       
,delamain loved instruments is  quite   
evident1 and we proceed to describe his 

efforts to improve the  circular slide  
rule4    ,the ,grammelogia ,,iv is      
dedicated to ,king ,charles ,i4         
,delamain says3      4 4 4 ,everything  
hath his beginning1 and curious ,arts   
seldome come to the    height at the    
first2 ,it was my promise then to       
enlarge the invention by a    way of    
decuplating the ,circles1 which ,i now  
present unto your sacred    ,majestie as
 the quintessence and excellencie there 
of 4 4 4    ,his enlarged circular rules
 are illustrated in the ,bodleian       
,library copy of  ,grammelogia ,,iv by  
four diagrams1 two of them being the two
 drawings on the  two title-pages at the
 beginning of the ,grammelogia ,,iv1 #d 
inches in  external diameter1 and       
exhibiting eleven concentric circular   
lines carrying  graduations of different
 sorts4 ,in the second of these designs 
all circles  are fixed4 ,the other two  
drawings are each #aj inches in external
 diameter  and exhibit #ah concentric   
circular lines2 the folded sheet of the 

first of  these drawings is inserted    
between pages 7#bc7 and 7#bd71 the      
second folded  sheet between pages 7#hc7
 and 7#hd74 ,all circles of this second 
instrument  are fixed4 ,counting in the 
two small drawings in ,grammelogia      
,,iii1 there are  in all six drawings of
 slide rules in the ,bodleian           
,grammelogia ,,iv4 ,on pages  7#bd7 to  
7#dc7 ,delamain explains the graduation 
of slide rules4 ,he takes  first a rule 
which has one circle of equal parts1    
divided into #ajjj equal  divisions4    
,from a table of logarithms he gets log 
#b  #j4cja2 from the  number #cja in the
 circle of equal parts he draws a line  
to the center of  the circle and marks  
the intersection with the circles of    
numbers by the  figure #b4 ,thus he     
proceeds with log #c1 log #d1 and so on2
 also with log sin  x and log tan x4    
,for log sin x he uses two circles1 the 
first 7see page  7#bg77 for angles from 
#cd #bd to #e #dd #bb1 the second circle
 from #e #dd  #bb to #ij4 ,the drawings 

do not show the seconds4 ,he suggests   
many  different designs of rules4 ,on   
page 7#bi7 he says3      ,for the single
 projection of the ,circles of my ,ring1
 and the dividing and    graduating of  
them3 which may bee so inserted upon the
 edges of ,circles    of mettle turned  
in the forme of a ,ring1 so that one    
,circle may moove    betweene two fixed1
 by helpe of two stayes1 then may there 
be graduated    on the face of the      
,ring1 upon the outer edge of the       
mooveable and inner    edge of the      
fixed1 the ,circle of ,numbers1 then    
upon the inner edge of    that mooveable
 ,circle1 and the outward edge of that  
inner fixed ,circle    may be inserted  
the ,circle of ,sines1 and so according 
to the description    of those that are 
usually made4    ,in addition to these  
lines he proceeds to mention the circle 
giving the  ordinary division into      
degrees and minutes1 and two circles of 
tangents on  the other side of the rule4
    ,next ,delamain explains an         

arrangement of all the graduation on one
 side of  the rule by means of 8a small 
channell in the innermost fixed ,circle1
 in  which may be placed a small single 
,index1 which may have sufficient length
  to reach from the innermost edge of   
the ,mooveable ,circle1 unto the outmost
  edge of the fixed ,circle1 which may  
be mooved to and fro at pleasure1 in    
the channell1 which ,index may serve to 
shew the opposition of ,numbers0 7p4    
7#ca774 ,from this it is clear that the 
invention of the 8runner0 goes back  to 
the very first writers on the slide     
rule4    ,after describing a            
modification of the above arrangement1  
he adds1 8many  other formes might be   
deliverd1 about this single projection0 
7p4 7#cb774    ,proceeding to the       
8enlarging0 of the circles in the ,ring1
 to1 say1 the  8,quadruple to that which
 is single1 that is1 foure times        
greater10 the  8equall parts0 are       
distributed over four circles instead of
 only one  circle1 but the general      

method of graduation is the same as     
before 7p4  7#cc772 there being now four
 circles carrying the logarithms of     
numbers1 and  so on4 ,next he points out
 8severall wayes how the ,circles of the
  ,mathematicall ,ring 7being inlarged7 
may be accommodated for practicall      
use30 7#a7 ,the ,circles are all fixed  
in a plain and movable flat compasses   
7or better1 a movable semicircle7 are   
used for fixing any two positions2  7#b7
 ,there is a 8double projection0 of each
 logarithmic line 8inlarged on a        
,plaine10 one fixed1 the other movable1 
as shown in his first figure on the     
title-page1 a single index only being   
used2 7#c7 use of 8my great ,cylinder   
which ,i have long proposed 7in which   
all the ,circles are of equall          
greatnesse17 and it may be made of any  
magnitude or capacity1 but for a  study 
7hee that will be at the charge7 it may 
be of a yard diameter and of  such an   
indifferent length that it may containe 
#ajj or more ,circles fixed  parallel   

one to the other on the ,cylinder1      
having a space betweene each of  them1  
so that there may bee as many mooveable 
,circles1 as there are fixed  ones1 and 
these of the mooveable linked1 or       
fastened together1 so that they  may all
 moove together by the fixed ones in    
these spaces1 whose edges both  of the  
fixed1 and mooveable being graduated by 
helpe of a single ,index will  shew the 
proportionalls by opposition in this    
double ,projection1 or by a  double     
,index in a single ,projection0 7p4     
7#cf774    ,next follows the detailed   
description of his ,ring 8on a ,plaine1 
according  to the diagramme that was    
given the ,king 7for a view of that     
projection7  and afterwards the ,ring it
 selve40 ,the diagram is the large one  
which we  mentioned as inserted between 
pages 7#bc7 and 7#bd74 ,the instrument  
has two  circles1 one moveable1 upon    
each of which are described #ac distinct
  circular graduations4 ,the lines on   
the fixed circle are3 8,the ,circle of  

degrees and calendar10 ,e4 8,circle of  
equall parts1 and part of the ,equator1 
and ,meridian10 ,,tt4 8,the ,circle of  
,tangents10 ,s4 8,the ,circle of        
,sines10 ,d4  8,the ,circle of          
,decimals10 ,n4 8,the ,circle of        
,numbers40 ,the lines on the  movable   
circle are3 ,n4 8,the ,circle of        
,numbers10 ,e4 8,the ,circle of equated 
figures1 and bodies10 ,s4 8,the ,circle 
of ,sines10 ,,tt4 8,the ,circle of      
,tangents10 ,y4 8,the ,circle of time1  
yeares1 and monethes40    ,on pages     
7#hd7-7hh7 ,delamain explains an        
enlargement of his ,ring for            
computations involving the sines of     
angles near to #ij4 ,on page 7#hf7 he   
says3      ,i have continued the ,sines 
of the ,projection unto two severall    
revolutions1 the one beginning at       
#gg4;gr4 #de4m4 #f4s4 and ends at       
#ij4;gr4    7being the last revolution  
of the decuplation of the former1 or the
    hundred part of that ,projection7   
the other beginning at #hf4;gr4 #f4m4   

#dh4s4    and ends at #ij4;gr4 7being   
the last of a ternary of decuplated     
revolutions1 or the thousand part of    
that ,projection7 and may bee thus      
used4    ,he explains the manner of     
using these extra graduations4 ,thus he 
claims to  have attained degrees of     
accuracy which enabled him to do what   
8some one0  had declared 8could not bee 
done40 ,it is hardly necessary to point 
out that  ,delamain's ,grammelogia ,,iv 
suggests designs of slide rules which   
inventors  two hundred or more years    
later were endeavouring to produce4     
,which of  ,delamain's designs of rules 
were actually made and used1 he does not
 state  explicitly4 ,he refers to a rule
 #ah inches in diameter as if it had    
been  actually constructed 7pages 7#hf71
 7#hh774 ,oughtred showed no            
appreciation of  such study in designing
 and ridiculed ,delamain's efforts1 in  
his ,epistle4    ,additional            
elucidations of his designs of rules1   
along with explanations of  the         

relations of his work to that of ,gunter
 and ,napier1 and sallies  directed     
against ,oughtred and ,forster1 are     
contained on pages 7#h7-7ba7 of  his    
,grammelogia ,,iv4                      
,v4 ,,independence ,,and ,,priority ,,of
 ,,invention      ,the question of      
independence and priority of invention  
is discussed by  ,delamain more         
specifically on pages 7#hi7-7aac72      
,oughtred devotes his entire  ,epistle  
to it4 ,it is difficult to determine    
definitely which publication is  the    
later1 ,delamain's ,grammelogia ,,iv or 
,oughtred's ,epistle4 ,each seems to    
quote from the other4 ,probably the     
explanation is that the two publications
  contain arguments which were          
previously passed from one antagonist to
 the  other by word of mouth or by      
private letter4 ,oughtred refers in his 
,epistle  7p4 7#ab77 to a letter from   
,delamain4 ,we believe that the ,epistle
 came after  ,delamain's ,grammelogia   
,,iv4 ,delamain claims for himself the  

invention of the  circular slide rule4  
,he says in his ,grammelogia ,,iv4 7p4  
7#ii771 8when ,i had  a sight of it1    
which was in ,february1 #afbi 7as ,i    
specified in my ,epistle7 ,i  could not 
conceale it longer1 envying my selfe1   
that others did not tast of  that which 
,i found to carry with it so delightfull
 and pleasant a goate  taste 4 4 40     
,delamain asserts 7without proof7 that  
,oughtred 8never saw it  as he now      
challengeth it to be his invention1     
untill it was so fitted to  his hand1   
and that he made all his practise on it 
after the publishing of  my ,booke upon 
my ,ring1 and not before2 so it was     
easie for him or some  other to write   
some uses of it in ,latin after         
,christmas1 #afcj and not the  ,sommer  
before1 as is falsely alledged by some  
one 4 4 40 7p4 7#ia774  ,delamain's     
accusation of theft on the part of      
,oughtred cannot be seriously           
considered4 ,oughtred's reputation as a 
mathematician and his standing in  his  

community go against such a supposition4
 ,moreover1 ,william ,forster is a      
witness for ,oughtred4 ,the fact that   
,oughtred had the mastery of the        
rectilinear slide rule as well1 while   
,delamain in #afcj speaks only of the   
circular rule1 weighs in ,oughtred's    
favour4    ,oughtred says he invented   
the slide rule 8above twelve yeares     
agoe10 that  is1 about #afba1 and 8,i   
with mine owne hand made me two such    
,circles1 which  ,i have used ever      
since1 as my occasions required10       
7,epistle p4 7#bb774 ,on  the same page1
 he describes his mode of discovery     
thus3      ,i found that it required    
many times too great a paire of         
,compasses in    using ,gunter's line1  
which would bee hard to open1 apt to    
slip1 and    troublesome for use4 ,i    
therefore first devised to have another 
,ruler with    the former3 and so by    
setting and applying one to the other1  
,i did not    onely take away the use of
 ,compasses1 but also make the worke    

much more    easy and expedite3 when ,i 
should not at all need the motion of my 
hand1    but onely the glancing at my   
sight3 and with one position of the     
,rulers1    and view of mine eye1 see   
not one onely1 but the manifold         
proportions    incident unto the        
question intended4 ,but yet this        
facility also wanted    not some        
difficulty especially in the line of    
tangents1 when one arch was    in the   
former mediety of the quadrant1 and the 
other in the latter3 for    in this case
 it was needful that either one ,ruler  
must bee as long againe    as the other2
 or else that ,i must use an inversion  
of the ,ruler1 and    regression4 ,by   
this consideration ,i first of all saw  
that if those lines    upon both ,rulers
 were inflected into two circles1 that  
of the tangents    being in both        
doubled1 and that those two ,circles    
should move one upon    another2 they   
with a small thread in the center to    
direct the sight1    would bee          

sufficient with incredible and          
wonderfull facility to worke all        
questions of ,trigonometry 4 4 4        
,oughtred said that he had no desire to 
publish his invention1 but in the       
vacation of #afcj finally promised      
,william ,forster to let him bring out a
  translation4 ,oughtred claims that    
,delamain got the invention from him at 
,alhallontide ,november #a1 #afcj1 when 
they met in ,london4 ,the accounts of   
that meeting we proceed to give in      
double column4                          
,delamain's ,statement                  
,grammelogia ,,iv1 page 7#ih7      84 4 
4 about ,alhalontide #afcj4 7as our     
,authors reporteth7 was the time he     
was circumvented1 and then his intent in
 a loving manner 7as before7 he         
opened unto me1 which particularly ,i   
will dismantle in the very naked        
truth3 for1 wee being walking together  
some few weekes before ,christmas1      
upon ,fishstreet hill1 we discoursed    
upon sundry things ,mathematicall1      

both ,theoreticall and ,practicall1 and 
of the excellent inventions and         
helpes that in these dayes were         
produced1 amongst which ,i was not a    
little taken with that of the           
,logarythmes1 commending greatly the    
ingenuitie of ,mr4 ,gunter in the       
,projection1 and inventing of his       
,ruler1    in the lines of proportion1  
extracted from these ,logarythmes for   
ordinary    ,practicall uses2 ,he       
replyed unto me 7in these very words7   
,what will yov    say to an ,invention  
that ,i have1 which in a lesse extent of
 the ,compasses    shall worke truer    
then that of ,mr4 ,gunters ,ruler1 ,i   
asked him then of    what forme it was1 
he answered with some pause 7which no   
doubt argued his    suspition of mee    
that ,i might conceive it7 that it was  
,arching-wise1 but    now hee sayes that
 hee told mee then1 it was ,circular    
7but were ,i put to    my oath to avoid 
the guilt of ,conscience ,i would       
conclude in the former47    ,at which   

immediately ,i answered1 ,i had the like
 my selfe1 and so we    discoursed not a
 word more touching that subject 4 4 4  
,then after my    coming home ,i sent   
him a sight of my ,projection drawne in 
,pastboard3 ,now    admit ,i had not the
 ,invention of my ,ring before ,i       
discoursed 4 4 4 it was    not so facil 
for mee 4 4 4 to raise and compose so   
complete1 and absolute    an ,instrument
 from so small a principle1 or glimpse  
of light 4 4 40                         
,oughtred's ,statement                  
,epistle1 page 7#bc7      8,shortly     
after my gift to ,elias ,allen1 ,i      
chanced to meet with ,richard           
,delamain in the street 7it was at      
,alhallontide7 and as we walked together
    ,i told him what an ,instrument ,i  
had given to ,master ,allen1 both of the
    ,logarithmes projected into circles1
 which being lesse then one foot        
diameter would performe as much as one  
of ,master ,gunters ,rulers of sixe     
feet long3 and also of the              

,prostaphaereses of the ,plannets and   
second    motions4 ,such an invention   
have ,i said he3 for now his intentions 
7that    is his ambition7 beganne to    
worke3 4 4 4 ,but he saith1 ,then after 
my    comming home ,i sent him a sight  
of my projection drawne in past-board4  
,see how notoriously he jugleth without 
an ,instrument4 ,then after3 how    long
 after8 a sight of my projection3 of how
 much8 ,more then seven weekes    after 
on ,december #bc1 he sent to mee the    
line of numbers onely set upon a        
circle3 4 4 4 and so much onely he      
presented to his ,majesty3 but as for   
,sine or tangent of his1 there was not  
the least shew of any4 ,neither    could
 he give to ,master ,allen any direction
 for the composure of the    circles of 
his ,ring1 or for the division of them3 
as upon his oath ,master    ,allen will 
testify how hee misled him1 and made him
 labour in vain above    three weeks    
together1 until ,master ,allen himselfe 
found out his ignorance    and          

mistaking1 which is more cleare then is 
possible with any impudence    to be    
outfaced40    ,oughtred makes a further 
statement 7,epistle1 p4 7#bd77 as       
follows3      ,delamain hearing that    
,brown with his ,serpentine had another 
line by which    he could worke to      
minutes in the #ij degree of sines 4 4 4
 gave the his    booke to ,browne3 who  
in thankfulnesse could not but gratify  
,delamain with    his ,lines also3 and  
teach him the use of them1 but          
especially of the    great ,line3 with  
this caution on both sides1 that one    
should not meddle    with the others    
invention4 ,two dayes after ,delamain 4 
4 4 because he had    found some things 
to be altered therin1 4 4 4 asked for   
the booke 4 4 4    but as soone as he   
had got it in his hands he rent out all 
the middle    part with the two ,schemes
  put them up in his pocket  went his   
way 4 4    4 and 4 4 4 laboureth to     
recall all the bookes he had given forth
 4 4 4    ,and shortly after this he got

 a new ,printer 7who was ignorant of his
    former ,schemes7 to print him new3  
giving him an especiall charge of the   
outermost line newly graven in the      
,plate1 which indeed is ,brownes very   
line3 and then altering his book 4 4 4  
,this and other statements made by      
,oughtred seem damaging to ,delamain's  
reputation4 ,but it is quite possible   
that ,oughtred's guesses as to          
,delamain's motives are wrong4          
,moreover1 some of ,oughtred's          
statements are  not first hand knowledge
 with him1 but mere hearsay4 ,one may   
accept his  first hand facts and still  
clear ,delamain of wrong doing4 ,there  
is always  danger that rival claimants  
of an invention or discovery will       
proceed on  the assumption that no one  
else could possibly have come           
independently upon  the same devices    
that they themselves did2 the history of
 science proves  the opposite4 ,seldom  
is an invention of any note made by only
 one man4 ,we  do not feel competent to 

judge ,delamain's case4 ,we know too    
little about  him as a man4 ,we incline 
to the opinion that the hypothesis of   
independent  invention is the most      
plausible4 ,at any rate1 ,delamain      
figures in the  history of the slide    
rule as the publisher of the earliest   
book thereon and  as an enthusiastic and
 skillful designer of slide rules4      
,the effect of this controversy upon    
interested friends was probably small4  
,doubtless few people read both sides4  
,oughtred says3#ba 8this scandall 4 4  4
 hath with them1 to whom ,i am not      
knowne1 wrought me much prejudice and   
disadvantage 4 40 ,aubrey1#bb a friend  
of ,oughtred1 refers to ,delamain 8who  
was so sawcy to write against him0 and  
remembers having seen 8many yeares      
since1 twenty or more good verses made0 
against ,delamain4 ,another friend of   
,oughtred1 ,william ,robinson1 who had  
seen some of ,delamain's publications1  
but not his ,grammelogia ,,iv1 wrote in 
a letter to ,oughtred1 shortly before   

the appearance of the latter's ,epistle3
      ,i cannot but wonder at the       
indiscretion of ,rich4 ,delamain1 who   
being    conscious to himself that he is
 but the pickpurse of another man's wit1
    would thus inconsiderately provoke  
and awake a sleeping lion 4 4 4 he      
hath so weakly 7though in my judgment1  
vaingloriously enough7 commended    his 
own labour 4 4 4#bc    ,delamain        
presented ,king ,charles ,i with one of 
his sun-dials1 also with a  manuscript  
and1 later1 with a printed copy of his  
book of #afcj4 ,a drawing  of his       
improved slide rule was sent to the     
,king and the ,grammelogia ,,iv is      
dedicated to him4 ,the ,king must have  
been favorably impressed1 for ,delamain 
was appointed tutor to the ,king in     
mathematics4 ,his widow petitioned the  
,house of ,lords in #afde for relief2 he
 had ten children4#bd    ,anthony ,wood 
states that ,charles ,i1 on the day of  
his execution1 commanded  his friend    
,thomas ,herbert 8to give his son the   

duke of ,york his large  ring-sundial of
 silver1 a jewel his maj4 much valued40 
,anthony ,wood adds1  8it was invented  
and made by ,rich4 ,delamaine a very    
able mathematician1 who  projected it1  
and in a little printed book did shew   
its excellent use in  resolving many    
questions in arithmetic and other rare  
operations to be  wrought by it in the  
mathematics40#be                        
,,vi4 ,,oughtred',s ,,gauging ,,line1   
#afcc      ,it has not been generally   
known1 hitherto1 that ,oughtred designed
 a  rectilinear slide rule for gauging  
and published a description thereof in  
#afcc4bf ,in his ,circles of            
,proportion1 chapter ,,ix1 ,oughtred had
 offered a  closer approximation than   
that of ,gunter for the capacity of     
casks4 ,the  ,gauger of ,london         
expostulated with ,oughtred for         
presuming to question  anything that    
,gunter had written4 ,the ensuing       
discussion led to an  invitation        
extended by the ,company of ,vintners to

 the instrument maker  ,elias ,allen to 
request ,oughtred to design a gauging   
rod4#bg ,this he did1  and ,allen       
received an order for 8threescore0      
instruments4 ,on page #ai  ,oughtred    
describes his #8,gauging ,rod3'      ,it
 consisteth of two rulers of brasse     
about #cb ynches of length1 which       
also are halfe an ynch broad1 and a     
quarter of an ynch thick 4 4 4 ,at one  
end of both those rulers are two little 
sockets of brasse fastened on           
strongly3 by which the rulers are held  
together1 and made to move one    upon  
another1 and to bee drawne out unto any 
length1 as occasion shall    require3   
and when you have them at the just      
length1 there is upon one of    the     
sockets a long ,scrue-pin to scrue them 
fast4    ,there are graduations on three
 sides of the rulers1 one graduation    
being  the logarithmic line of numbers4 
,he says 7p4 #ci71 8the maner of        
computing  the ,gauge-divisions ,i have 
concealed40 ,w4 ,robinson1 who was a    

friend of  ,oughtred1 wrote him as      
follows3#bh      ,i have light upon your
 little book of artificial gauging1     
wherewith ,i am    much taken1 but ,i   
want the rod1 neither could ,i get a    
sight of one of    them at the time1    
because ,mr4 ,allen had none left 4 4 4 
,i forgot to ask    ,mr4 ,allen the     
price of one of them1 which if not much 
,i would have one of    them40 ,oughtred
 annotated this passage thus3 8,or in   
wood1 if any be made    in wood by      
,thompson or any other40    ,another of 
,oughtred's admirers1 ,sir ,charles     
,cavendish1 wrote1 on ,february  #aa1   
#afce thus3#bi      ,i thank you for    
your little book1 but especially for the
 way of    calculating the divisions of 
your gauging rod4 ,i wish1 both for     
their own    sakes and yours1 that the  
citizens were as capable of the         
acuteness of    this invention1 as they 
are commonly greedy of gain1 and then ,i
 doubt not    but they would give you a 
better recompense than ,i doubt now they

 will4    ,on ,april #bj1 #afch1 we find
 ,oughtred giving ,elias ,allen         
directions#cj  8about the making of the 
two rulers40 ,as in #afcc1ca so now1    
,oughtred  takes one ruler longer than  
the other4 ,this #afcc instrument was   
used also  as 8a crosse-staffe to take  
the height of the ,sunne1 or any ,starre
 above  the ,horizon1 and also their    
distances40 ,the longer ruler was called
 staffe1  the shorter transversarie4    
,while in #afcc he took the lengths of  
the two in  the ratio 8almost #c to #b10
 in #afch1 he took 8the transversary    
three  quarters of the staff's length1 4
 4 4 that the divisions may be larger40 
,,vii4 ,,other ,,seventeenth ,,century  
,,slide ,,rules      ,in my ,history of 
the ,slide ,rule ,i treat of ,seth      
,partridge1 ,thomas ,everard1  ,henry   
,coggeshall1 ,w4 ,hunt and ,sir ,isaac  
,newton4#cb ,of ,partridge's ,double    
,scale of ,proportion1 ,london1 ,i have 
examined a copy dated #affa1 which is   
the earliest date for this book that ,i 

have seen4 ,as far as we know1 #affa  is
 the earliest date of publications on   
the slide rule1 since ,oughtred and     
,delamain4 ,but it would not be         
surprising if the intervening #bh years 
were  found not so barren as they seem  
at present4 ,the #affa and #affb        
impressions  of ,partridge are          
identical1 except for the date on the   
title-page4 ,william  ,leybourn1 who    
printed ,partridge's book1 speaks in    
high appreciation of it  in his own     
book4#cc    ,in #affa was published also
 ,john ,brown's first book1 ,description
 and ,use of  a ,joynt-,rule1 previously
 mentioned4 ,in ,chapter ,,xviii he     
describes the use  of 8,mr4 ,whites     
rule0 for the measuring of board and    
timber1 round and  square4 ,he calls    
this a 8sliding rule40 ,the existence1  
in #affa1 of a  8,whites rule0 indicates
 activities in designing of which we    
know as yet  very little4 ,in his book  
of #agfa1 previously quoted1 ,brown     
gives a drawing  of 8,white's sliding   

rule0 7p4 #aic72 also a special         
contrivance of his own1  as indicated by
 him in these words3      ,a further    
improvement of the ,triangular          
,quadrant1 as ,i have made it    several
 times1 with a sliding ,cover on the    
in-side1 when made hollow1 to    carry  
,ink1 ,pens1 and ,compasses2 then on the
 sliding ,cover1 and ,edges1 is    put  
the ,line of ,numbers1 according to ,mr4
 ,white's first ,contrivance for        
manner of operation2 but much augmented1
 and made easie1 by ,john ,brown4    ,he
 gives no drawing of his 8triangular    
quadrant10 hence his account of it  is  
unsatisfactory4 ,he explains the use of 
8gage-points40 ,his placing  logarithmic
 lines on the edges of instrument boxes 
was outdone in oddity  later by ,everard
 who placed them on tobacco-boxes4#cd   
,in ,brown's  publication of #agjd the  
,white slide rule is given again1 8being
 as neat and  ready a way as ever was   
used40 ,he tells also of a 8glasier's   
sliding rule40  ,william ,leybourn      

explains in #afgc how ,wingate's double 
and triple lines for  squaring and      
cubing1 or square and cube root1 can be 
used on slide  rules4#ce    ,beginning  
early in the history of the slide rule1 
when ,oughtred designed  his 8gauging   
rod10 we notice the designing of rules  
intended for very  special purposes4    
,another such contrivance1 which enjoyed
 long popularity1  was the ,timber      
,measure by a ,line1 by ,hen4           
,coggeshall1 ,gent41 ,london1 #afgg1  a 
booklet of #ce pages4 ,coggeshall says  
in his preface3      ,for what can be   
more ready and easie1 then having set   
twelve to the    length1 to see the     
,content exactly against the ,girt or   
,side of the    ,square4 ,whereas on    
,mr4 ,partridge's ,scale the ,content is
 the ,sixth ,number1    which is far    
more troublesome then even with         
,compasses4    ,one line on             
,coggeshall's rule begins with #d and   
extends to #dj1 these  numbers being the
 8,girt0 7a quarter of the              

circumference71 which in  ordinary      
practice of measuring round timber lies 
between #d inches and #dj  inches4 ,this
 8,girt line0 slides 8against the line  
of ,numbers in two  ,lengths1 to which  
it is exactly equal40 ,a second edition1
 #afhb1 shows some  changes in the rule1
 as well as an enlargement and change of
 title of the  book itself3 ,a ,treatise
 of ,measures1 by a ,two-foot ,rule1 by 
,h4 ,c4 ,gent1  ,london1 #afhb4 ,in     
this1 the description of the rule is    
given thus3      ,there are four ,lines 
on each flat of this ,rule2 two next the
 outward    edges1 which are ,lines of  
,measure2 and two next the inward edges1
 which    are ,lines of ,proportion4 ,on
 one flat1 next the inward edges1 is the
    ,square-line ,girt-line in round    
timber measurement with the ,line of    
,numbers his fellow4 ,next the outward1 
a ,line of ,inches divided into         
,halfs1 ,quarters1 and ,half-,quarters2 
from #a to #ab on one ,rule2 and from   
#ab    to #bd on the other4 ,on the     

other flat1 next the inward edges1 is   
the    double ,scale of ,numbers for    
solving proportions4 ,next the outward  
on    one ,rule a ,line of ,inches      
divided each into ten parts2 and this   
for    gauging1 etc4 ,on the other a    
foot divided into #ajj parts4    ,later 
further changes were introduced in      
,coggeshall's rule4#cf    ,it is worthy 
of note that ,coggeshall's slide rule   
book1 ,the ,art of  ,practical          
,measuring1 was reviewed in the ,acta   
eruditorum1 anno #afia1 p4  #dgc2 hence 
,leupold's description#cg of the        
rectilinear slide rule in his  ,theatrum
 arithmetico-geometricum1 ,leipzig1     
#agbg1 ,cap4 ,,xiii1 p4 #ga1 is not  the
 earliest reference to the rectilinear  
rule found in ,german  publications4    
,the above date is earlier even than    
,biler's reference to a  circular slide 
rule in his ,descriptio instrumenti     
mathematici universalis  of #afif4      
,two noted slide rules for gauging were 
described by ,tho4 ,everard1            

,philomath1 in his ,stereometry made    
easie1 ,london1 #afhd4 ,he designates   
his  lines by the capital letters ,a1   
,b1 ,c1 ,d1 ,e4 ,on the first           
instrument1 ,a on  the rule1 and ,b and 
,c on the slide1 have each two radiuses 
of numbers1 ,d  has only one1 while ,e  
has three4 ,the second rule is described
 in an  ,appendix2 it is one foot long1 
with two slides enabling the rule to be 
extended to #c feet4    ,everard's      
instruments were made in ,london by     
,isaac ,carver who1 soon after1  himself
 wrote a sixteen-page ,description and  
,use of a ,new ,sliding ,rule1          
projected from the ,tables in the       
,gauger's ,magazine1 ,london1 #afhg1    
which was  8printed for ,william ,hunt0 
and bound in one volume with a book by  
,hunt1  called ,the ,gauger's ,magazine1
 ,london1 #afhg4 ,this appears to be the
 same  ,william ,hunt who later brought 
out descriptions of his own of slide    
rules4  ,the instrument described by    
,carver 8consists of three pieces1 two  

whereof  are moveable to be drawn out   
till the whole be #cf inches long40 ,it 
has  several non-logarithmic            
graduations1 together with logarithmic  
lines marked  ,a1 ,b1 ,c1 ,d1 of which  
,a1 ,b1 ,c are 8double lines10 and ,d a 
8single line0 used  for squares and     
square roots4 ,it is designed for the   
determination of the  vacuity of a      
8spheroidal cask lying10 a 8spheroidal  
cask standing10 and a  8parabolical cask
 lying40    ,another seventeenth century
 writer on the slide rule is ,john      
,atkinson1 whom  we have mentioned      
earlier4 ,he says3#ch 8,the ,lines of   
,numbers1 ,sines and  ,tangents1 are set
 double1 that is1 one on each side1 as  
the middle piece  slides3 which middle  
piece is so contrived1 to slip to and   
fro easily1 to  slide out1 and to be put
 in any side uppermost1 in order to     
bring those  ,lines together 7or against
 one another7 most proper for solving   
the  ,question1 wrought by              
,sliding-,gunter40    ,the data         

presented in this article show that1    
while the earliest slide  rules were of 
the circular type1 the later slide rules
 of the seventeenth  century were of the
 rectilinear type4#ci                   
,january #ab1 #aiae4                    
,footnotes      #a;,f4 ,cajori1 ,history
 of the ,logarithmic ,slide ,rule and   
,allied ,instruments1     ,new ,york1   
#aiji1 pp4 #g-ad1 also ,addenda i-vi4   
#b;,f4 ,cajori1 8,on the ,invention of  
the ,slide ,rule10 in ,colorado ,college
     ,publication1 ,engineering ,series 
,vol4 #a1 #aiaj4 ,an abstract of this is
     given in ,nature 7,london71 ,vol4  
#hb1 #aiji1 p4 #bfg4    #c;,f4 ,cajori1 
,history etc41 p4 #ad4    #d;,art4      
8,slide ,rule0 in the ,penny            
,cyclopaedia and in the ,english        
,cyclopaedia ,arts and ,sciences4       
#e;,anthony ,wood1 ,athenae oxonienses  
7,ed4 ,p4 ,bliss71 ,london1 ,vol4 ,,iii1
 #ahag1     p4 #dbc4    #f,the full     
title of the book which ,wingate        
published on this subject in     ,paris 

is as follows3       ,l',vsage  de la   
,reigle de  ,proportion  en             
l',arithmetique       ,geometrie4  ,par 
,edmond ,vvingate1  ,gentil-homme       
,anglois4          1   4       ,in      
tenui1 sed n tenuis vsusve1 laborne4    
,a ,paris1  ,chez ,melchior ,mondiere1  
demeurant en l',isle du ,palais1        
la  ru de ,harlay aux deux ,viperes4    
,m4 ,,dc4 ,,xxiv4  ,auec ,priuilege du  
,roy4        ,back of the title page is 
the announcement3       ,notez que la   
,reigle de ,proportion en toutes faons  
se vend  ,paris chez     ,melchior      
,tauernier1 ,graueur  ,imprimeur du ,roy
 pour les ,tailles douces1     demeurant
 en l',isle du ,palais sur le ,quay qui 
regarde la ,megisserie      l',espic    
d'or4    #g,the title-page of the       
edition of #afeh is as follows3         
,the ,use of the ,rule of ,proportion in
 ,arithmetick  ,geometrie4 ,first       
published at ,paris in the ,french      
tongue1 and dedicated to ,monsieur1 the 
then king's onely ,brother 7now ,duke of

 ,orleance74 ,by ,edm4 ,wingate1 an     
,english ,gent4 ,and now translated into
 ,english by the ,author4 ,whereinto    
is now also inserted the ,construction  
of the same ,rule1  a farther use       
thereof 4 4 4 #bnd edition inlarged and 
amended4 ,london1 #afeh4    #h,memories 
of the ,life of that ,learned           
,antiquary1 ,elias ,ashmole1 ,esq42     
,drawn up by himself by way of ,diary4  
,with ,appendix of original ,letters4   
,publish'd by ,charles ,burman1 ,esq41  
,london1 #agag1 p4 #bc4                 
#i,mathematical ,tables1 #ahaa1 p4 #cf1 
and art4 8,gunter's ,line0 in his ,phil4
     and ,math4 ,dictionary1 ,london1   
#ahae4    #aj,to the ,english ,gentrie1 
and all others studious of the          
,mathematicks1     which shall bee      
readers hereof4 ,the just ,apologie of  
,wil3 ,ovghtred1     against the        
slaunderous insimulations of ,richard   
,delamain1 in a ,pamphlet     called    
,grammelogia1 or the ,mathematicall     
,ring1 or ,mirifica logarithmorum       

projectio circularis4 ,we shall refer to
 this document as ,epistle4 ,it was     
published without date in #cb unnumbered
 pages of fine print1 and was     bound 
in with ,oughtred's ,circles of         
,proportion1 in the editions of #afcc   
and #afci4 ,in the #afcc edition it is  
inserted at the end of the volume       
just after the ,addition vnto the ,vse  
of the ,instrument etc41 and in that    
of #afci immediately after the preface4 
,it was omitted from the ,oxford        
edition of #affj4 ,the ,epistle was also
 published separately4 ,there is a      
separate copy in the ,british ,museum1  
,london4 ,aubrey1 in his ,brief ,lives1 
edited by ,a4 ,clark1 ,vol4 ,,ii1       
,oxford1 #ahih1 p4 #aac1 says quaintly1 
8,he     writt a stitch't pamphlet about
 #afc78d7 against 4 4 4 ,delamaine40    
#aa,thomas ,browne is mentioned by      
,stone in his ,mathematical             
,instruments1     ,london #agbc1 p4 #af4
 ,see also ,cajori1 ,history of the     
,slide ,rule1 ,new     ,york1 #aiji1 p4 

#ae4    #ab,the ,description and ,use of
 a ,joynt-,rule3 4 4 4 also the use of  
,mr4     ,white's ,rule for measuring of
 ,board and ,timber1 round and square2  
,with     the manner of ,vsing the      
,serpentine-line of ,numbers1 ,sines1   
,tangents1 and     ,versed ,sines4 ,by  
,j4 ,brown1 ,philom41 ,london1 #affa4   
#ac;,a ,collection of ,centers and      
,useful ,proportions on the ,line of    
,numbers1     by ,john ,brown1 #affb7871
 #af pages2 ,description and ,use of the
 ,triangular     ,quadrant1 by ,john    
,brown1 ,london1 #afga2 ,wingate's ,rule
 of ,proportion in     ,arithmetick and 
,geometry3 or ,gunter's ,line4 ,newly   
rectified by ,mr4 ,brown     and ,mr4   
,atkinson1 ,teachers of the             
,mathematicks1 ,london1 #afhc2 ,the     
,description and ,use of the            
,carpenter's-,rule3 ,together with the  
,use of     the ,line of ,numbers       
commonly call'd ,gunter's-,line1 by     
,john ,brown1     ,london1 #agjd4       
#ad,william ,leybourn1 op4 cit41 pp4    

#abi1 #acj1 #acb1 #acc4    #ae;,james   
,atkinson's edition of ,andrew ,wakely's
 ,the ,mariners ,compass     ,rectified1
 ,london1 #afid ,wakely's preface dated 
#affd1 ,atkinson's     preface1 #afic4  
,atkinson adds ,an ,appendix containing 
,use of ,instruments     most useful in 
,navigation4 ,our quotation is from this
 ,appendix1 p4 #aii4    #af,r4          
,delamain1 ,the ,making1 ,description1  
and ,use of a small portable            
,instrument 4 4 4 called a ,horizontall 
,quadrant1 etc41 ,london1 #afca4        
#ag,oughtred's description of his       
circular slide rule of #afcb and his    
rectilinear slide rule of #afcc1 as well
 as a drawing of the circular     slide 
rule1 are reproduced in ,cajori's       
,history of the ,slide ,rule1           
,addenda1 pp4 ii-vi4    #ah,the full    
title of the ,grammelogia ,i is as      
follows3       ,gramelogia  or1  ,the   
,mathematicall ,ring4  ,shewing 7any    
reasonable ,capacity that hath  not     
,arithmeticke7 how to resolve and       

worke  all ordinary operations of       
,arithmeticke4  ,and those which are    
most difficult with greatest  facilitie3
 ,the extraction of ,roots1 the         
valuation of  ,leases1 c4 ,the measuring
 of ,plaines  and ,solids4  ,with       
the resolution of ,plaine and           
,sphericall  ,triangles4  ,and that     
onely by     an ,ocular ,inspection1    
and a ,circular ,motion4  ,naturae      
secreta tempus     aperit4  ,london     
printed by ,john ,haviland1 #afcj4      
#ai;,grammelogia ,,iii is the same as   
,grammelogia ,i1 except for the addition
     of an appendix1 entitled3       ,de
 la ,mains  ,appendix  ,vpon his        
,mathematicall  ,ring4 ,attribuit       
nullo 7praescripto tempore7 vitae       
vsuram nobis ingeniique ,deus4          
,london1        4 4 4 ,the next line or 
two of this title-page which probably   
contained     the date of publication1  
were cut off by the binder in trimming  
the     edges of this and several other 
pamphlets for binding into one volume4  

#bj;,grammelogia ,,iv has two title     
pages4 ,the first is ,mirifica          
,logarithmoru'     ,projectio           
,circularis4 ,there follows a diagram of
 a circular slide rule1     with the    
inscription within the innermost ring3  
,nil ,finis1 ,motvs1     ,circvlvs vllvs
 ,habet4 ,the second title page is as   
follows3       ,grammelogia  ,or1 the   
,mathematicall ,ring4  ,extracted from  
the     ,logarythmes1 and projected     
,circular3 ,now published in the        
inlargement     thereof unto any        
magnitude fit for use3 shewing any      
reason-  able     capacity that hath not
 ,arithmeticke how to resolve and worke1
  all     ordinary operations of        
,arithmeticke3  ,and those that are most
 difficult     with greatest facilitie1 
the extracti-  on of ,rootes1 the       
valuation of     ,leases1 c4 the        
measuring of ,plaines and ,solids1  with
 the resolution     of ,plaine and      
,sphericall ,triangles applied to the   
,practicall parts of     ,geometrie1    

,horologographie1 ,geographie           
,fortification1 ,navigation1            
,astronomie1 c4  ,and that onely by an  
ocular inspection1 and a ,circular      
motion1 ,invented and first published1  
by ,r4 ,delamain1 ,teacher1 and         
,student of the ,mathematicks4  ,naturae
 secreta tempus aperit4        ,there is
 no date4 ,there follows the diagram of 
a second circular slide     rule1 with  
the inscription within the innermost    
ring3 ,typus proiectionis     ,annuli   
adaucti vt in ,conslusione ,lybri praelo
 commissi1 ,anno #afcj     promisi4     
,there are numerous drawings in the     
,grammelogia1 all of which1             
excepting the drawings of slide rules on
 the engraved title-pages of            
,grammelogia ,,iv and ,v1 were printed  
upon separate pieces of paper and       
then inserted by hand into the vacant   
spaces on the printed pages     reserved
 for them4 ,some drawings are missing1  
so that the ,bodleian     ,grammelogia  
,,iv differs in this respect slightly   

from the copy in the     ,british       
,museum and from the ,british ,museum   
copy of ,grammelogia ,v4                
#ba;,epistle1 p4 7#h74    #bb;,aubrey1  
op4 cit41 ,vol4 ,,ii41 p4 #aaa4         
#bc,rigaud1 ,correspondence of          
,scientific ,men during the #agth       
,century1 ,vol4     ,i1 ,oxford1 #ahda1 
p4 #aa4    #bd;,dictionary of ,national 
,biography1 ,art4 8,delamain1 ,richard40
 ,see also     ,rev4 ,charles ,j4       
,robinson1 ,taylors' ,school1 from      
,a4,d4 #aefb to #ahgd1 ,vol4     ,i1    
#ahhb1 p4 #aea2 ,journal of the ,house  
of ,commons1 ,vol4 ,,iv41 p4 #aig;b2    
,sixth ,report of the ,royal ,commission
 on ,historical ,manuscripts1 ,part ,i1 
,report and ,appendix1 ,london1 #ahgg4  
,in this ,appendix1 p4 #hb1 we read the 
following3       ,oct4 #bb #afde        
,petition of ,sarah ,delamain1 relict of
 ,richard ,delamain4     ,petitioner's  
husband was servant to the ,king1 and   
one of ,his ,majesty's     engineers for
 the fortification of the kingdom1 and  

his tutor in     mathematical arts2 but 
upon the breaking out of the war he     
deserted the     ,court1 and was called 
by the ,state to several employments1 in
 fortifying     the towns of            
,northampton1 ,newport1 and ,abingdon2  
and was also abroad     with the armies 
as ,quartermaster-,general of the ,foot1
 and therein died4     ,petitioner is   
left a disconsolate widow with ten      
children1 the four     least of whom are
 now afflicted with sickness1 and       
petitioner has     nothing left to      
support them4 ,there are several        
considerable sums of     money due to   
the petitioner1 as well from the ,king  
as the ,state4 ,prays     that she may  
have some relief amongst other widows4  
,see ,l4 ,j41 ,,vii4 #f4     #feg4      
#be;,anthony ,wood1 ,athenae ,oxonienses
 7,edition ,bliss7 ,vol4 ,,iv41 ,london1
     #ahbj1 p4 #cd4    #bf,the ,new     
,artificial ,gauging ,line or ,rod3     
together with rules concerning     the  
use thereof3 ,invented and written by   

,,william ,,oughtred1 etc41 ,london1    
#afcc4 ,the copy we have seen is in the 
,bodleian ,library1 ,oxford4 ,the book  
is small sized and has #dj pages4       
#bg,oughtred1 op4 cit41 p4 #aa4         
#bh,s4 ,j4 ,rigaud1 ,correspondence of  
,scientific ,men of the #agth ,century1 
,oxford1 ,vol4 ,i1 #ahda1 p4 #ag4       
#bi,rigaud1 loc4 cit41 p4 #bb4          
#cj,rigaud1 loc4 cit41 pp4 #cj1 #ca4    
#ca,oughtred1 ,an ,addition vnto the    
,vse of the ,instrument called the      
,circles     of ,proportion1 ,london1   
#afcc1 p4 #fc4    #cb;,f4 ,cajori1      
,history of the ,slide ,rule1 ,new      
,york1 #aiji1 pp4 #af-bb1     ,addenda1 
pp4 vi-ix4    #cc,w4 ,leybourn1 op4     
cit41 #afgc1 ,preface1 and pp4 #abh-bi4 
#cd;,cajori op4 cit41 ,addenda1 p4 ix4  
#ce,william ,leybourn1 op4 cit41 #afgc1 
p4 #ce4    #cf,see ,cajori1 op4 cit41   
pp4 #bj1 #bh1 ,addenda1 p4 ix4          
#cg,see ,f4 ,cajori1 8,a ,note on the   
,history of the ,slide ,rule10          
,bibliotheca     mathematica1 #c ,f41   

,vol4 #aj1 pp4 #afa-afc4    #ch;,john   
,atkinson1 op4 cit41 #afid1 p4 #bjd4    
#ci,probably the oldest slide rule now  
in existence is owned by ,st4 ,john's   
,college1 ,oxford1 and is in the form of
 a brass disc1 #a ft4 #f in4 in         
diameter4 ,it was exhibited along with  
other instruments in ,may1 #aiai4       
,according to the ,catalogue of a ,loan 
,exhibition of ,early ,scientific       
,instruments in ,oxford1 opened ,may    
#af1 #aiai1 the instrument is inscribed 
with the name of the maker 78,elias     
,allen fecit07 and with the name of     
the donor1 ,georgius ,barkham4 ,it is   
dated #afce1 which is only three years  
after the first publication of          
,oughtred's description of his circular 
slide rule4 ,it is stated in the        
,catalogue3 8,unfortunately all the     
movable parts but the base-plate and a  
couple of thumb-screws are     missing4 
,the face of the instrument is engraved 
with ,oughtred's     ,horizontal        
,instrument4 ,the back is engraved with 

eleven ,circles of     ,proportion as   
described in ,arthur ,haughton's book1 a
 copy of which was     presented to ,st4
 ,john's ,college by ,george ,barkham1  
to explain the use of     the           
instrument40 ,as ,arthur ,haughton's    
,oxford edition of ,oughtred's          
,circles of ,proportion did not appear  
until #affj1 it would seem that the     
instrument was probably not presented to
 the ,college before #affj4 ,as far     
as is known1 the next oldest slide rule 
is of the year #afed1 kept in the       
,south ,kensington ,museum1 ,london1 and
 is described in ,nature of ,march #e1  
#aiad4 ,it is a rectilinear rule1 8of   
boxwood1 well made1 and bound           
together with brass at the two ends4 ,it
 is of the square type1 a little        
more than #b ft4 in length1 and bears   
the logarithmic lines first             
described by ,edmund ,gunter4 ,of these1
 the num1 sin and tan lines are         
arranged in pairs1 identical and        
contiguous1 one line in each pair being 

on the fixed part1 and the other on the 
slide40 ,the instrument is              
inscribed1 8,made by ,robert ,bissaker  
for ,t4 ,w41 #afed40 ,nowhere else have 
we seen reference to ,robert ,bissaker4 
,his slide rule seems to antedate       
the 8,whites rule0 mentioned above4     
,this foot-note was added on ,october   
#ae1 #aiai4                             
,,university ,,of ,,california          
,,publications      ,note4--,the        
,university of ,california ,publications
 are offered in exchange for  the       
publication of learned societies and    
institutions1 universities and          
libraries4 ,complete lists of all the   
publications of the ,university will be 
sent upon request4 ,for sample copies1  
lists of publications or other          
information1 address the ,manager of the
 ,university of ,california ,press1     
,berkeley1 ,california1 ,u4 ,s4 ,a4 ,all
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addressed to ,the ,exchange ,department1
 ,university ,library1 ,berkeley1       

,california1 ,u4 ,s4 ,a4                
,,mathematics4--,mellen ,w4 ,haskell1   
,editor4 ,price per volume1 #e4jj4      
,cited as ,univ4 ,calif4 ,publ4 ,math4  
,vol4 #a4 #a4 ,on ,numbers which        
,contain no ,factors of the ,form       
p7kp#a71 by ,henry ,w4 ,stager4 ,pp4    
#a-bf4 ,may1 #aiab                 #j4ej
        #b4 ,constructive ,theory of the
 ,unicursal ,plane ,quartic by          
,synthetic ,methods1 by ,annie ,dale    
,biddle4 ,pp4 #bg-ed1 #ca               
text-figures4 ,september1 #aiab         
4#ej        #c4 ,a ,discussion by       
,synthetic ,methods of ,two ,projective 
,pencils of          ,conics1 by        
,baldwin ,munger ,woods4 ,pp4 #ee-he4   
,february1 #aiac     4#ej        #d4 ,a 
,complete ,set of ,postulates for the   
,logic of ,classes          ,expressed  
in ,terms of the ,operation             
8,exception01 and a          ,proof of  
the ,independence of a ,set of          
,postulates due to ,del          ,r1 by 
,b4 ,a4 ,bernstein4 ,pp4 #hg-if4 ,may1  

#aiad                   4#aj        #e4 
,on a ,tabulation of ,reduced ,binary   
,quadratic ,forms of a                  
,negative ,determinant1 by ,harry ,n4   
,wright4 ,pp4 #ig-aad4 ,june1           
#aiad                                   
4#bj        #f4 ,the ,abelian ,equations
 of the ,tenth ,degree ,irreducible in a
          ,given ,domain of             
,rationality1 by ,charles ,g4 ,p4       
,kuschke4 ,pp4          #aae-afb4 ,june1
 #aiad                                  
4#ej        #g4 ,abridged ,tables of    
,hyperbolic ,functions1 by ,f4 ,e4      
,pernot4          ,pp4 #afc-afi4        
,february1 #aiae                        
4#aj        #h4 ,a ,list of ,oughtred's 
,mathematical ,symbols1 with ,historical
          ,notes1 by ,florian ,cajori4  
,pp4 #aga-ahf4 ,february1 #aibj         
4#be        #i4 ,on the ,history of     
,gunter's ,scale and the ,slide ,rule   
during the          ,seventeenth        
,century1 by ,florian ,cajori4 ,pp4     
#ahg-bji4          ,february1 #aibj     

4#ce        #aj4 ,on a ,birational      
,transformation ,connected with a       
,pencil of          ,cubics1 by ,arthur 
,robinson ,williams4 ,pp4 #baa-bbb4     
,february1          #aibj               
4#ae        #aa4 ,classification of     
,involutory ,cubic ,space               
,transformations1 by          ,frank    
,ray ,morris4 ,pp4 #bbc-bdj4 ,february1 
#aibj                  4#be             
,,agricultural ,,sciences4--,charles ,b4
 ,lipman1 ,ernest ,b4 ,babcock1 and     
,john ,w4 ,gilmore1 ,editors4 ,price per
 volume1 #e4jj4 ,volumes ,i and    ,,iii
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,,and ,,ethnology4--,alfred ,l4         
,kroeber1 ,editor4    ,volumes ,i-,,xii 
completed4 ,volumes ,,xiii to ,,xvi in  
progress4    ,,astronomy4--,william ,w4 
,campbell1 ,director of the ,lick       
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,publications of the ,lick ,observatory 
7,quarto74 ,volumes ,i-,,xiii           
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,,viii1 #e4jj2 ,volume ,,ix1 #b4be2     
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,t4 ,allen1 ,editors4 ,price per volume1
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,,iii 7#c4ej71 and ,,iv 7#b4ej7         
completed4 ,volumes ,,ii and ,v in      
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,price per volume1 #e4jj4 ,volume ,i    
completed4 ,volume ,,ii in progress4    
,,geology4--,andrew ,c4 ,lawson and     
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completed4 ,volume ,,viii in progress4  
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,editor4 ,volume ,i completed4 ,volumes 
,,ii and ,,iii in progress4    ,,modern 
,,philology4--,charles ,m4 ,gayley1     
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volumes #e4jj4 ,volumes ,,xviii    to   
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,,the ,,university ,,of ,,california    
7,quarto74 ,john ,c4 ,merriam1          
,editor4 ,volume ,i 7in progress74      
#a4 ,triassic ,ichthyosauria1 with      
,special ,reference to the              
,american ,forms1 by ,john ,c4 ,merriam4
 ,september1 #aijh          #c4jj       
#b4 ,the ,fauna of ,rancho ,la ,brea1 by
 ,john ,c4 ,merriam4          ,part ,i1 
,occurrence                             
4#cj          ,part ,,ii1 ,canidae      
4#hj      ,volume ,,ii4 ,the ,silva of  

,california1 by ,willis ,linn ,jepson4  
#aiji                                   
#e4jj2 cloth1 #g4ej      ,volume ,,iii4 
,business ,cycles1 by ,wesley ,c4       
,mitchell4 #aiac                        
#e4jj2 cloth1 #g4ej      ,volume ,,iv4  
#a4 ,the ,battle of the ,seven ,arts1 by
 ,louis ,j4 ,paetow4      ,july1 #aiad  
4#ge    ,,university ,,of ,,california  
,,chronicle4--,an official record of    
,university life1 issued quarterly4     
,george ,m4 ,calhoun1 ,editor4          
,price1 #a4jj per year4 ,current1       
,volume ,,xxii4                         
,transcriber's ,notes      --,inserted  
subchapter headings from the ,table of  
,contents into the    text4             
--,corrected two page numbers in the    
,table of ,contents    --,corrected a   
,greek quotation in footnote #f by      
reference to the    original sources4