Til e Op en Co urc S eries of Classics of S cien ce a nd

’ 7 fiilo o /z s . p y, M 3

T H E G EO M ET R I C A L L EC T UR E S

O F I S A A C BA R RO W

T R SL T ED T NO T ES AND PRO O FS AND AN A , WI H , A DISCUSSIO N O N T H E ADVAN CE MADE T HE REIN O N T H E WO RK O F H I S PREDEC ESSO RS IN T H E INFINIT ESIMAL CALCULU S

I D M . CH L J.

E . L D . B A . S C . . ( ON

C H I CAG O AN D L O ND O N T H E O P E N CO UR T P UBL I S H I NG CO MP A NY

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P R E F A C E

I S A A C B A R R O W wa s t/ie firs t in ven to r o tb c [nfi n ites ima t f . Ca lc u lu s N ewto n go t t/ze main idea of it fro m B a r ro w by pers o n a l commu nication a n d L eibn iz als o wa s in s o me ’ de te to B a rro w s wo rk bt c r m t meas u re in b d , o aining onfi a ion o bis o wn r l ideas a n d s u es t s o r tbeir u r t/cer f o igina , gg ion f f ’ ’ develo men t ro m t/ce c o B a r ro w s b b t/za t lze u rc b a s ea p , f opy f oo p

in 1 673 . The above is the ultimate conclu sion that I have arri ved ’ re s u lt n at , as the of six months close study of a si gle book , B my first essay in historical research . y the I n fi n ite s imal ” “ c f Cal ulus , I intend a complete set of standard orms for f e i both the di fer ntial and integral sect ons of the subject ,

together with rules for their combination , such as for a u c product, a quotient , or a power of a f n tion ; and also a recognition and demonstration of the fac t that differen tiation ” e t and integration are inverse op ra ions . m B T he case of Newton is to y mind clear enough . arrow was m arab o lifo rms fa iliar with the p , and tangents and areas m 1 6 1 660 connected with the , in from 55 to at the very latest ; hen c e he cou ld at this time differentiate and inte grate by Izis o wn met/2 0d any rational positive power of a m an d s u . f variable , thus also a of such powers He urther 1 66 2 — — in developed it in the years 3 4, and the latter year

probably had it fairly complete . In this year b e com mu n ic at e d to Newton the great secret of his geometrical u c m constr tions , as far as it is humanly possible to judge fro a collec tion of tiny scraps of circumstantial evidence and it was probably this that set Newton to work on an attempt

to express everything as a sum of powers of the variable . “ D uring tbc n ex t y ea r Newton began to reflect on his ” flu x io n s A method of , and actually did produce his nalys is £ u a tio n es T 1 er . 666 p 9 his , though composed in , was not

published until 1 7 1 1 . ’ v iii BAR R O WS GE OM E TR I CA L LE C T UR E S

‘ The c ase o f Leibniz wants more argument that I am in a i o t o . p sition at present to g ve, nor is this the place give it I hope to be able to s u bmit this in another place at some fu ture m m time . T he striking points to y ind are that Leibniz ’ B 1 6 “ bought a copy of arrow s work in 73 , and was able to ” c ommunic ate a candid ac c ou nt of his c alculus to Newton 6 f ’ 1 . o f L z in 77 In this connection , in the ace eibni per sistent denial that he rec eived any assistance whatever from ’ ’ B mu we ll m L z f arrow s book, we st bear in ind eibni two old idea of the “ calculus ” m (i) the freeing of the matter fro geometry,

(ii) the adoption of a convenient notation . e c m c m H n e , be his denial a ere quibble or a andid state ent “ ” without any thought of the i dea of what the calc ulus e r ally is, it is perfectly certain that on these two points . at any rate he derived not the slightest assistance from ’ Barrow s work ; for the first of them wo u ld be dead against ’ ’ B tbe s eco u a B arro w arrow s practice and instinct, and of ’ b a a n o k n o wledge w/za t ener : T hese points have made the c c m fo r h al ulus the powerful instru ent that it is , and t is the world has to thank Leibniz ; but their in c eption does not m o f m c c u T the ean the invention the infinitesi al al ul s . his , i m o f w c c m e ep to e the ork of his prede essors, and its o pl tion by his own discoveries until it formed a perfected method o f d e aling with the problems o f tangents and areas for a n u r n ra l e m c ve in e e i . i n y g , . odern phraseology, the d ifferentiation and integration o f any function whatever B ’ m (such as were known in arrow s ti e), must be ascribed B to arrow . L m f ma c e m est the atter that ollows y be onsid red ra bling , m i and arred by repetitions and other defects , I g ve first some acc ount of the circumstan c es that gave rise to this

M . B . m F t o f r P . E u volu e . irs all, I was asked by Jo rdain to write a short acc ount o f Barrow for the M o n is t ; the request being ac companied by a first edition c o py o f ’ B L ectio n es O tion et Geo met rica m arrow s p . At this ti e, I m B ’ do not mind confessing , y only knowledge of arrow s claim to fame was that he had been “ Newton ’s tutor ” a ’ notoriety as unenviable as being known as Mrs S o - and - S o s ” F o r c if fo r husband . this arti le I read , as a review, the M book that had been sent to me . y attention was arrested F R E E / I CE

7 ’ e m B rectz iea b c c clo ia c by a theor in which arrow had j t y , whi h

' fi

u u y b e e n c t o S ir C . .

I happened to know has s all as ribed n Wren My interest th u s aroused impelled me to make a laborious (for I am no classic al scholar) translation o f t he whole o f Th m c u e ls e u . e the geo etri al lect res , to see what I co ld find ' conclusions I arrived at were s e nt to the M o ms t for pu blic a tion 5 but those who will read the artic le and this volume will find that in the article I had by no means reached t he L stage represented by this volume . ater, as I began to still c c m further appre iate what these le tures really eant, I con c e ive d the idea o f publishing a full translation o f the lectures together with a summary of the work of Barrow’s more m c m m im ediate prede essors , written in the sa e way fro a o f personal translation the originals , or at least of all those i P . O n U that I could obtain applying to the nivers ty ress ,

m m f d R v B . L c C e . a bridge , through y rien , the . J o k , I was referred by P rofessor Hobson to the re c ent work of P rofe ssor Z mm . O n c Mr u d euthen o unicating with Jo r ain , I was invited to elaborate my artic le fo r the M o n is t into a z o - r r o page volume fo the Open Co u t S er ies of Cla s s ics . I can lay no c laim to any great perspic acity in this dis c o ve r m ma y of ine , if I y call it so all that follows is due t o c c rather to the lack of it, and the lucky a ident that made me (when I c o u ld not follow the demonstration) turn one ’ B l c m h of arrow s theorems into a gebrai al geo etry . W at I found induc ed me to treat a number of the theorems in the m sa e way . As a result I came to the conclusion that ’ Barrow b a a g o t t/i e calc u lu s ; but I queried eve n then B z c o m whether arrow himself recogni ed the fact . O nly on le t in m o f e p g y annotation of the last chapter this volum , L c ect . XII , App . III , did I ome to the conclusion that is given as the Opening sentenc e of this P refa c e ; fo r I then found that a batc h o f theorems (whic h I had on first reading m noted as very interesting , but not of uch service), on careful the m revision , turned out to be few missing standard for s , n eces s a ry f o r co mpleting tb e s et f o r in teg r a tio n 5 and that one of his pro blems was a practical rule fo r finding the area as c under any curve, such would not yield to the theoreti al — rules he had given , under the guise of an inverse tangent problem . The reader will then understand that the conclusion is ’ x B A R R O WS GE OME TR I CA L L E CT UR E S the effect o f a gradu al acc u mu lation o f evid e n c e (mu c h as a detec tive picks up c lues) on a mind previo u sly blank as m f e T i . regards this atter, and there ore p rfectly unb ased his he will see reflected in the gradu al transformation from tentative and imaginative suggestions in the Introduc tion m c to direct state ents in the notes , whi h are inset in the o f text of the latter part the translation . I have purposely f m c re ra ed from altering the Introdu tion , which preserves the m m M o n is t m for of y article in the , to accord with y final i c u radu al m deas , be a se I feel that with the g develop ent thus indicated I shall have a greater chanc e of c arryi ng my m c c u readers with me to my own ulti ate on l sion . The order of writing has been (after the first fu ll trans — c n S V lation had been made) Introdu tio , ections I to III , excepting I II then the text with notes 5 then S ections III and I X of the Introduc tion ; and lastly some slight altera

tions in the whole and S ection X . S u In ection I , I have given a wholly inadeq ate account ’ o f the work o f Barrow s immediate predecessors 5 but I felt ’ b e e n lar e d that this could g at any reader s pleasure , by reference to the standard historic al authorities 3 and that it

was hardly any of my business , so long as I slightly expanded my Mo n is t artic le to a su fficiency for the purpose of showing m B that the ti e was now ripe for the work of arrow, Newton , T L z . and eibni his , and the next section , have both been taken from the pages o f the E n cy clope dia B rit a n n ica (Times

edition) . The m remainder of my argument simply expresses y own ,

i . as I have said, gradually formed opin on I have purposely refrained from consulting any authorities other than the B iblio t/zeca B rita n n ica in work cited above , the (for the dates S D ictio n a r o N tio n al io r a b fo r ection III), and the y f a B g p y ( ’ Canon O verton s life o f Barrow) but I mu st ac knowledge ’ the servic e rendered me by the dates and notes in S o t he ran s e u rr en t o L it er t u r Th P ric C f a e . e translation too is entirely — my own Without any help from the translation by S tone or other assistance— from a first edition of Barrow’s work

dated 1 670 .

As regards the text, with my translation beside me, I ’ have to all intents rewritten Barrow s book ; althou gh throughout I have adhered fairly closely to Barrow’s own P R E F A CE

c e m words . I have only retained those parts whi h se ed to

‘ ur o s e ln me to be absolutely essential for the p p hand . Thus the reader will find the first fe w chapte rs very much abbreviated , not only in the matter of abridgment, but also in respect of proofs omitted , explanations cut down , and e figures left out, when ver this was possible without breaking T the continuity . his was necessary in order that room l might be found for the critica notes on the theorems , the c B in lusion of proofs omitted by arrow, which when given B ’ i in arrow s style , and afterwards translated into analys s , had an important bearing on the point as to how he found out the more diffi c ult of his constructions ; and lastly for c m f dedu tions therefro that point steadily, one a ter the B c other, to the fact that arrow was writing a cal ulus and k n e t/za t lie was inven tin a rea t tit in w g g g . I can make no b u t t ran s la claim to any classical attainments, I hope the c m tion will be found corre t in al ost every particular . In the wording I have adhered to the order in which the

- original runs , because thereby the old time flavour is not lost ; the most I have done is to alter a passage from the vice vers a active to the passive or , and occasionally to change the punctuation . I have used three distinct kinds of type : the most widely ’ spaced type has been used for Barrow s own words ; only very occasionally have I inserted anything of my own in this , and then it will be found enclosed in heavy square brackets , that the reader will have no chance of confusing my explanations with the text ; the whole of the I n t ro du c B ’ P tion , including arrow s refaces , is in the closer type ; m this type is also used for y critical notes , which are m m generally given at the end of a lecture, but also so eti es occur at the end of other natural divisions of the work , when it was thought inadvisable to put off the explanation until the end of the lecture . It must be borne in mind B that arrow makes use of parentheses very frequently , so that the reader must understand that only remarks in lzeaoy s u a re q brackets are mine , those in ordinary round brackets ’ B The are arrow s . small type is used for footnotes only . In the notes I have not hesitated to use the L eibniz n notatio , because it will probably convey my meaning better ; but there was really no absolute necessity for this , ’ x ii B A R R O WS GE OM E TR Z CA L L E C T UR E S

’ B a e m n t b b arrow s and , or its oder equivalen , and , would ha ve done quite as well .

‘ I cannot c lose this P reface without an acknowledgment of my great indebtedness to Mr Jourdain for frequent advic e and help I have had an unlimited call on his wide ic f B reading and great histor al knowledge ; in act, as arrow C am him u c c him says of ollins , I hardly doing j sti e in alling m M All c fu y ersenne the same, I ac ept ll responsibility fo r any opinions that may seem to be heretical or otherwise o f M Mr o f out order . y thanks are also due to Abbott, e C C m n J sus ollege, a bridge, for his kind assistance in looki g m u p references that were inac c essible to e . M L D C . J. . HI D E R BY E G L A D , N N , X mas 1 1 , 9 5.

— . S c P . S in e this volume has been ready for press , I have c u v au e o f ons lted se eral thorities , and , through the kindn ss Mr S Walter tott, I have had the opportunity of reading ’ The S tone s translation . result I have set in an appendix T he at the end of the book . reader , will also find there a B ’ solution , by arrow s methods , of a test question suggested by Mr Jourdain after examining this I doubt whether any reader will have room fo r doubt concerning the correc tness of my main con c lus ion . I have also given two spec imen ’ pages of Barrow s text and a S pecimen of his folding plates ’ m o f B o f diagrams . Also , I have given an exa ple arrow s graphical integration of a func tion ; fo r this I have c hosen c c c a fun tion whi h he could not have integrated theoreti ally, 4 m I I —x 0 x na ely, /J( ), between the limits and when the u m m m 1 pper li it has its aximu value, , it is well known that the integral can be expressed in Gamma func tions ; this c u c o f m was used as a check on the ac ra y the ethod .

. M . J . c T A BL E O F C O NT E N T S

P AGE INTR OD UCTION ’ The wo rk o f B arro w s g re at p re de c e s s o rs

Life o f arro an d his c o n n e c io n it N e t o n B w, t w h w The wo rk s o f Barro w ’ E s timat e o f B arro w s ge n iu s ’ The s o u rc e s o f B arro w s id e as Mu tu al in flu e n c e o f N ewto n an d Barro w D e s c riptio n o f the bo o k fro m whic h the tran s l atio n mad e The p re fac e s H o w Barro w made his c o n s tru c tio n s ’ An alytic al e q u ivale n ts o f B arro w s c hie f t he o re ms

TR AN S LATION — ma i u e s Mo e s o f mo tio n an d E R E e n e ratio n o f n . L CTU I . G g t d d

th u an ti o f the mo iv e fo rc e ime as t he in e e n e n e q ty t . T d p d t

v aria e ime a s an a re a io n o f in s an t s c o m are bl . T , gg g t t , p d it a in e as an a re atio n o f o in s w h l , gg g p t

“ E R E — e n e ratio n o f ma n i u e s o c a mo v e L CTU II . G g t d by l l ” me n t s The s im e mo t io n s o f t ra n s a io n an d ro t atio n . pl l t

LE C R E —Co m o s i e a n d c o n c u rre n mo t io n s Co m TU III . p t t . p o s itio n o f re c tilin e ar an d paralle l mo t io n s

L E C R E I V — ro e r ie s o f c u rv e s aris in fro m c o m o s i io n TU . P p t g p t

ra ie n o f he t an n t n e ra iz io n o f mo io n s . The t e e a t g d t g . G l t o f a ro e m o f a i e o Cas e o f in fin it e v e lo c it p bl G l l . y

V - r r o i r u r L E C R E u e r e r e s o f c u v e s . an e n s C v e s TU . F th p p t T g t .

th o i M im min im m in e s ik e e c c N o rma s . ax u m an d u l y l d . l l — L E C TU R E V L Le mmas d e te rmin at io n o f c e rtain c u rv e s c o n s tru c te d ac c o rdin g t o giv e n c o n ditio n s mo s t ly hype rbo las

—~ L E C R E V I I S imi ar o r an a o o u s c u rv e s . E x o n e n t s TU . l l g p

r n i e s Ari hm t c a me ric a ro re s io n o c . e i an d e o I d t l G t l P g s s . The o re m an alo go u s t o t he a ppro x imatio n t o the Bin o mial e o re m fo r a rac tio n a n e x A s m t o e s Th F l I d . y p t ’ x iv B A R R O WS GE OME TR I CA L L E C T UR E S

— LE C R E V Co n s t ru c tio n o f t an e n t s me a s o f au x i iar TU III . g by n l y

c u rv e s o f ic t he t an e n s re k n o n Diff re n t iat i n wh h g t a w . e o o f a s u m o r a iffe re n c e A n a tic a e u iv a e n s d . ly l q l t

L E C R E I X — an e n t s t o c u rv e s fo rme a i m t a an d TU . T g d by r th e ic l

e o me t ric a me n ms u i a s . P arab o lifo r C rv e s o f e r o l c g l . hyp b an d e i tic fo rm D . iffe ren tiatio n o f a frac io n a o e r ll p t l p w , pro d u c ts an d qu o tie n ts

’ L E C R E X —R i o o min i n o s al n i . r u s e t e r a o f a x D iffe re t a TU g d t / .

t io n as t he in v x h e rs e o f in t e g ratio n . E plan atio n o f t e “ D iffe re n t ia rian e me t o it e x am e s D iffer l T gl h d ; w h pl . e n tiatio n o f a t rigo n o me t ric al fu n c tio n

L E C R E XI — C an e o f the in e e n e n t v aria e in in t e TU . h g d p d bl

rati io n t h v r iff r n i n D iffe r o n n e rat e in e s e o f e e atio . g . I t g d t e n tiatio n o f a u o tie n t A re a an d c e n t re o f rav i o f a q . g ty

ara o ifo rm Limit s fo r t he arc o f a c irc e an d a e r o la . p b l . l hyp b

E s t imatio n o f 7r

L E C R E X I — m r ifi at io n an ar I . e n e ra t e o re s o n e c c S TU G l h t . t d d fo rms fo r in t e g ratio n o f c irc u lar fu n c tio n s by re d u c tio n t o

t he u a ratu re o f t he e r o a Me t o o f c irc u ms c ri e q d hyp b l . h d b d

n d i i u rfac e s a n s c ri e fi u r Me as u re me n o f c o n c a s . b d g e s . t l

u a ra u re o f the e r o la Diffe re n t ia io n an d n e ra Q d t hyp b . t I t g

t io n o f m n n l u rt e r s t an ar a L o garith an d an E x p o e t ia . F h d d fo rms

L E C R E X — e s e t e o re ms av e n o t e e n in s ert e TU III . Th h h b d

POS TS C R I P T E x t rac ts fro m S tan d ard Au th o ritie s

A P P E N D I X ’ I o lu tio n o f a t e t s tio n arro s me tho . S s q u e by B w d ’ ra ic a in t r o arro s me t o II . G ph l e g ati n by B w h d ' R e u c e fac s imil f a rro s a e s an d fi u re s III . d d e s o B w p g g

I ND E X INT R O D UCT I O N

TH E WO R K O F BAR R O W’S GR E AT P R ED E CE S S O R S

T H E o f I n fi n it e s imal C beginnings the alculus, in its two main divisions , arose from determinations of areas and h n o f t an e n t s Th t e . e volumes , and findi g g to plane curves ancients attacked the problems in a strictly geometrical “ ”

m . manner, aking use of the method of exhaustions In “ ” modern phraseology, they found upper and lower limits , as closely eq ual as possible , between which the quantity m to be determined ust lie ; or, more strictly perhaps , they c d showed that , if the quantity oul be approached from two “ ” sides , on the one side it was always greater than a certain thing , and on the other it was always less hence it must be T finally equal to this thing . his was the method by means of which Archime des proved most of his discoveries . B u t m there seems to have been some distrust of the ethod , for we find in many cases that the dis c overies are proved by a ’ 7 ’ rea u c tio a o a bs u rau m c E , su h as one is familiar with in uclid . T o Apollonius we are indebted for a great many of the pro t m m c pe ties , and to Archi edes for the easurement , of the onic sections and the solids formed from them by their rotation about an axis . T he m first great advance, after the ancients, ca e in the c 6 —1 beginning of the seventeenth entury . Galileo (1 5 4 642 ) would appear to have led the way , by the introduction of the theory of composition of motions into me c hanics ; he in fi n it e s imals also was one of the first to use in geometry, “ and from the fact that he uses what is equivalent to virtual veloc ities it is to be inferred that the idea of time as the

him K 1 1 - 6 independent variable is due to . epler ( 57 1 3 0) was the first to introduce t he idea of infinity into geometry

’ S e M c S c ien c e o Mec ha n ics fo fu er e t ai s e a h s f r ll d l . ’ 2 B A R R O WS GE OME TR TCA L L E CT UR E S and to note that the increment of a variable was evanescent fo r values of the variable in the immediate neighbourhood o f m m m m mu 1 6 1 a axi u or ini m in 3 , an abundant vintage drew his attention to the defec tive methods in use for m c c s esti ating the cubi al ontents of ves els, and his essay on the subject (N o va S ter eo metria D o lio r u m) entitles him to rank amongst those who made the discovery of the in it e im l l 1 6 i fi n s a a . c lcu us possible In 3 5, Cavalieri publ shed “ ” o f a theory indivisibles, in which he considered a line as o f s u e rfi c ie s made up an infinite number of points , a p as c c composed of a su ession of lines , and a solid as a succession “ of s u pe rfic ie s thus laying the fo u ndation for the aggre

atio n s B . R g of arrow oberval seems to have been the first, m or at the least an independent, inventor of the ethod but l c he ost redit for it, because he did not publish it, preferring to keep the method to himself for his o wn use this seems to have been qu ite a u sual thing amongst learned me n of that time, due perhaps to a certain professional jealousy . The z b m method was severely critici ed y conte poraries , i G P 1 6 2 —1 662 espec ally by uldin , but ascal ( 3 ) showed that the method of indivisibles was as rigorous as the method of exhaustions , in fact that they were practically identical . In all probability the progress o f mathematical thought is much indebted to this defen c e by P as c al . S ince this method u is exactly analogo s to the ordinary method of integration , Cavalieri and R oberval have more than a little claim to be ‘ regarded as the inventors of at least the one branch of the c al c ulu s if it wer e n o t f o r tbc fa ct that they only applied it c c t o z to spe ial ases , and seem have been unable to generali e c it owing to umbrous algebraical notation, or to have failed to “ perceive the inner meaning of the method when concealed f P f under a geometrical orm . ascal himsel applied the m c a ethod with great su cess, but also to special c ses only ; The o f such as his work on the cycloid . next step was a more analytical nature ; by the method of indivisibles , Wallis (1 6 1 6- 1 703 ) reduced the determination of many areas and volumes to the calculation of the value of the m m m m o + 1 2 1 n i. e . series ( + + ) , the ratio of the m m n ean of all the terms to the last ter , for integral values of and later he extended his method , by a theory of interpola c o f n T th a f the I tion , to fra tio nal values . hus e ide o ntegral I N TR OD UC TI ON

Calculus was in a fairly advanced stage in the days imme di B ately antecedent to arrow . C R a What avalieri and oberval did for the integral c lculus, — D escartes (1 596 1 650) accomplished for the differential

' f al bran c h by his work on the application o ge b ra t o geometry . Cartesian coordinates made possible the extension of in ve s tigatio n s on the drawin g of tangents to special c u rves to o f T o the more general problem for curves any kind . this ’ ’ must be added the fact that he lia bit u ally n s ea tne in a ex no ta tio n ; for this had a very great deal to do with the possibility of Newton ’s discovery of the general binomial D expansion and of many other infinite series . escartes faileld m , however, to ake any very great progress on his own account in the matter of the drawing of tangents , owing to what I cannot help but call an u nfortunate choice of a ’ definition for a tangent to a curve in general . E uclid s circ le - tangent definition being more or less ho peless in the D general case, escartes had the choice of three

1 ( ) a secant, of which the points of intersection with the curve became coincident 2 m c ( ) a prolongation of an ele ent of the curve, whi h was to be considered as composed of an infinite succession of infinitesimal straight lines

(3 ) the direction of the resultant motion , by which the

curve might have been described .

D escartes chose the first ; I have called this choice u n fo r t u n at e , because I cannot see that it would have been possible D m f for a escartes to iss the dif erential triangle, and all its

consequences , if he had chosen the second definition . His choice leads him to the following method of drawing a

n . D c ta gent to a curve in general escribe a cir le , whose x centre is on the axis of , to cut the curve ; keeping the c m entre fixed , di inish the radius until the points of se c tion o f coincide ; thus , by the aid of the equation the curve, t he problem is reduced to finding the condition for equ al roots of an equation . 2 F o r = a oc , y 4 b a instance let e the equation to parabola , 2 2 2 x - + r and ( p) y the equation of the circle . T hen we h have (x If t is I S a perfect sq uare 3 , i. e 90 2 a . 2 a [ ; the subtangent is equal to . ’ 4 B A R R O WS GE OME TR TCA L L E CT UR E S

The method, however, is only applicable to a small m c ffi nu ber of simple cases, owing to algebrai al di culties . o f In the face this disability, it is hard to conjecture why D escartes did not mak e another c hoice of definition and use the sec ond one given above for in his rule for the tangents to roulettes , he considers a curve as the ultimate form of a

‘ Th e fi n i . e d it o n polygon third , if not originally due to G f alileo, was a direct consequence o his conception of the c omposition of motions this definition was used by R oberval (1 60 2 - 1 6 75) and applied succ essfully to a doz en or so of the well - known cu rves ; in it we have the germ o f “ ” x the method of fi u io n s . Thus it is seen that R oberval m u occupies an al ost nique position , in that he took a great part in the work preparatory to the invention o f bo t/z bra n c/zes of the infinitesimal calculus ; a - fact that seems to have F m 1 0—1 66 ’ escaped remark . er at ( 59 3 ) adopted K epler s notion of the increment of the variable becoming evanes x u m m cent near a ma im m or inimu value, and upon it m F ’ based his ethod of drawing tangents . ermat s method of finding the maximum or minimum value of a function in f volved the dif erentiation of any explicit algebraic function , ’ m tna t a ears in a n - in the fo r pp y begin n er s tex t bo ob of to day (though Fermat does not seem to have the “ function ” m idea) that is , the maximum or minimu values of f (x ) are ’ the roots of where f (x ) is the limiting value of [f (x b) only Fermat uses the lettere or E instead li u v c o n of . Now, if YYY is any c r e, wholly — o r t o i TZ YZ cave ( convex) a stra ght line A D , a tangent to it at the point Ywhose ordinate T A is N Y, and the tangent meets A D in ; NYZ N also, if ordinates are drawn on either Y N side of NY , cutting the curve in and the tangen t in Z ; then it is plain that the N ratio YN NT is a maximum (or a mini mum) when Y is the point of contact of D the tangent . Here then we have all the essentials for the calculus ; fo r c but only expli it integral algebraic functions , needing n o f e wt o n the binomial expa sion , or a general method of rationaliz ation which did not impose too great algebraic ffi c i n di ult es , for their further development ; also, o the

6 B A R R O W’S GE OM E TR I CA L L E C T UR E S

L F E O F B R R O AND H I S E T I A W, CO NN C I ON WIT H N EWTO N

B 1 6 0 s o n - Isaac arrow was born in 3 , the of a linen draper L C S in ondon . He was first sent to the harterhouse chool, where inattention and a predilection for fighting created a e ra bad impression ; his fath r was overheard to say (p y , “ according to one account) that if it pleased G o d to take c L one of his children , he ould best spare Isaac . ater, he 1 6 seems to have turned over a new leaf, and in 43 we find him S t P ’ C C m entered at eter s ollege , a bridge, and afterwards w m i T . n o e at rinity Having beco e exc ed ngly studious , he c i made onsiderable progress in literature, natural ph losophy, — anatomy, botany, and chemistry the three last with a v m f — iew to edicine as a pro ession , and later in chronology , m c geometry, and astrono y . He then pro eeded on a sort “ ” o f G T F S o n rand our through rance, Italy , to myrna, C s t an tin O le G m p , back to Venice , and then home through er any and Holland . His stay in Constantinople had a great influence on his after life ; for he here studied the works of

u . Chrysostom , and thus had his thoughts t rned to divinity Bu t for this his great advance on the work of his pre de c e s s o rs in - the matter of the infinitesimal calculus might have been developed to s u ch an extent that the name of Barrow would have been inscribed on the roll of fame as at least the equal of his mighty pupil Newton . E was d Immediately on his return to ngland he ordaine , and a year later, at the age of thirty, he was appointed to t he Greek professorship at Cambridge his inaugural lectures R beto ric were on the subject of the of Aristotle, and this choice had also a distinct effect on his later mathematic al 1 662 P work . In , two years later, he was appointed rofessor of Geometry in Gresham College ; and in the following year c L u c as ian C o f M c u he was ele ted to the hair athemati s , j st T founded at Cambridge . his professorship he held for five f c L ectio n es years , and his o fice created the oc asion for his M a t/zema ticaz 1 66 — —6 , which were delivered in the years 4 5 ta t abrz ce T w H abi Ca n i . ( g ) hese lectures ere published ,

m E n rit . P . m c B according to rof Benja in Willia son ( cy . I N TR OD UCTI ON

me e i 1 6 0 Ti s . I n fi n t e s imal C ( dition), Art on alculus) in 7 ; : this , however, is wrong they were not published until 1 8 t n t w s 6 L ec io es M a /i ema tica . a 3 , under the title of What published in 1 6 70 was the L ectio n es Optica et Geo met rica ; the L ec tio n es I VI a tnema tica were philosophical lectures on the fundamentals of mathematics and did not have much

m u . T bearing on the infinitesi al, calc lus hey were followed by the L ectio n es Opticoe an d lectures on the works of T Archimedes , Apollonius , and heodosius ; in what order these were delivered in the schools of the U niversity I have been unable to find out b u t the former were p u blished in ” 1 66 m u M 1 668 9, I primat r having been granted in arch , so that it was probable that they were the professorial lect u res for 1 66 7 thus the latter would hav e been delivered 1 668 1 6 in , though they were not published until 75, and then Th L ectio n es eo metrica C . e G probably by ollins great work , , did not appear as a separate publication at first : as stat ed o f above , it was issued bound up with the second edition L ectio n es O t icce u the p ; and , j dging from the fact that there n o t does , according to the above dates , appear to have been L u c as ian L any time for their public delivery as ectures , since Imprimatur was granted for the combined edition in ’ 1 669 ; also from the fact that Barrow s P reface speaks of “ six out of the thirteen le c tures as matters left over from c m the O ptics , which he was induced to o plete to form a all separate work ; also from the most conclusive fact of , that on the title - page of the L ect io n es Geo met ricce there 1 5 no t he H abitae Can t ab ri iae mention at all of usual notice g , “ t/za t tb c L ectio n es judging from these facts , I do not believe ” r del r d a u a a n L e t u r Geo metrica we e ive e s L c s i c es . S hould this m f be so , it would clear up a good any di ficulties it would corroborate my suggestion that they were for a great part e volved d u ring his professorship at Gresham College ; also it wou ld make it almost certain that they would have been giv en as internal college lectures , and that Newton would 1 66 —1 66 have heard them in 3 4. 1 66 B fi rs t Now, it was in 4 that arrow ~ came into close N personal contact with ewton ; for in that year, he E examined Newton in uclid , as one of the subjects for m T C a mathe atical scholarship at rinity ollege, of which Newton had been a subsi z ar for three years and it was due 8 B A R R O W’S GE OME TR I CA L L E CT UR E S

’ to Barrow s report that Newton was led to st u dy the E lemen ts more carefully and to form a better estimate o f The c m their value . conne tion once started ust have n o t O B c developed at a great pace, for nly does arrow se ure o f e Lu c as ian the succession N wton to the chair, when he it 1 66 b u t m c relinquished in 9, he com its the publi ation of his L ectio n es Opt ica to the foster care of Newton and m e m n Collins . He hi s lf had now deter i ed to devote the rest of his life to divinity entirely in 1 670 he was created D o f D 1 6 2 D r P a octor ivinity , in 7 he succeeded earson as

M T 1 6 c - C c aster of rinity, in 75 he was chosen Vi e han ellor o f U 1 6 the niversity ; and in 77 he died, and was buried m in West inster Abbey, where a monument, surmounted by f e his bust , was soon a terwards er cted to his memory by his friends and admirers .

TH E WO R K S O F BAR R OW

v m Barrow was a very olu ino u s writer . O n inqu iring o f the Librarian of the Cambridge University Library wheth e r he c ould supply me with a complete list of the works of B arrow in order of publication , I was informed that the complete list oc c upied f o u r co lu mn s in tbe B ritis li M u s eu m Cata logu e ! T his of course would include his theological

f - works , the several dif erent editions , and the translations o f

h w . his Latin works . T e follo ing list o f his mathematical as fo r m works, such are important the atter in hand only, m B iblio tb eca B r ita n n ica b R is taken fro the ( y obert Watt , E 1 8 2 dinburgh , 4) ’ 1 E u clid s E lemen ts . C . 1 6 . , amb 55 ’ E u clid s D a ta 2 . 6 . C . 1 , amb 57

L ectio n es O tico ru m P /zen o in en dn L . 6 . . 1 6 3 p , ond 9

. L ectio n es O ticce et Geo metrica 1 L . 6 0 4 p , ond 7 6 2 . 1 s E S (in vols , 74 ; tran , dmond tone, e ti n M a tnema tica . L c o es L 68 1 . 5 , ond . 3 This list makes it absol u tely certain that Williamson is wrong in stating that the le c tures in geometry were published “ ” M m c L . T under the title of athe ati al ectures his , how ’ m ever, is not of uch consequence ; the important point in I N TR OD UC TI ON

the list, assuming it to be perfectly correct as it stands, is that the lec t u re s on O ptics were first published s eparately 1 66 in 9. In the following year they were reissued in a c m revised form with the addition of the le tures on geo etry . The above books were all in Latin and have been translated by different people at one time or an other .

E S T I MAT E O F BAR R OW S G E NIUS

The i B c in wr ter of the article on arrow, Isaa , the ninth ’ Times E n /c lo ceaia B rita n n ic a f m ( ) edition of the g p , ro which m S n m ost of the details in ectio II have been taken , re arks “ By his E nglish contemporaries Barrow was c onsidered a mathematician second only to Newton . Continental him t/zeir u d men t is ro b writers do not place so high , and j g p ably tb e mo re co rrect o n e . F L ectio n es Geo met rica ounding my Opinion on the alone, I f z fail to see the reasonableness o the remark I have italici ed . f O course, it was only natural that contemporary continental m B m mathe aticians should belittle arrow , since they clai ed for F e rmat and Leibniz the invention o f the infinitesimal w c o n calculus before Newton , and did not ish to have to B sider in arrow an even prior claimant . We see that his own countrymen placed him on a v ery high level ; and surely the only way to obtain a really adequate opinion of a ’ scientist s worth is to accept the u nbiased opinion that has w been expressed by his contemporaries , who were a are of all the facts and conditions of the case ; or, failing that, to u try to form an nbiased opinion for ourselves , in the position of his contemporaries . An obvious deduction may be drawn from the controversy between Newton and Hooke the Opinion of Barrow ’s own countrymen would not be likely to

- err on the side of over appreciation , unless his genius was great enough to outweigh the more or less natural jealousy that ever did and ever will ex ist amongst great men occupied m on the same investigations . Most odern criticism of f ancient writers is apt to fail, because it is in the hands O r e t the experts ; perhaps to some deg ee this must be so , y

w . m n K . a you ould hardly allow a C to be a fitting for a jury . ’ 1 0 BAR R O WS GE OME TR I CA L L E CT UR E S

C m riticism by experts , unless they are the selves giants like z unto the men whose works they critici e , compares , perhaps u c c n onsciously, their dis overies with facts that are now m c com on knowledge, instead of onsidering only and solely tbe adva n ce ma de u o n w/za t was t/zen co mmo n k n owle e p dg . T hus the skilled designers of the wonderful electric engines of to day are but as pigmies compared with such giants as a F araday . F B urther, in the case of arrow , there are several other u c things to be taken into account . We m st onsider his dis o f position, his training, his changes intention with regard S man to a career, the accident of his connection with uch a c m c u as Newton , the ircu stan es brought abo t by the work of m his im ediate predecessors , and the ripeness of the time for his discoveries .

His disposition was pugnacious , though not without a touch o f humour there are many indic ations in the L ectio n es Geo metricce n ma c alone of an i clination to what I y all, for m z lack of a better ter , a certain contributory la iness ; in this way he was somewhat like F ermat with his usual I have o u got a very beautiful proof of if y wish , I will send it to yo u ; but I dare say you will be able to find it for your ” B ’ self ; many of arrow s most ingenious theorems , one or — two of his most far reaching ones , are left without proof, though he states that they are easily deduced from what has v m gone before. He e idently knows the i portance of his discoveries ; in one place he remarks that a certain set of m “ m i theore s are a mine of infor ation , in wh ch should any one investigate and explore, he will find very many things ” L t him m him f . e o . this kind do so who ust , or if it pleases f o f i He omits the proo a certa n theorem , which he states has been very useful to him repea tedly ; and n o wonder it u v f has , for it turns out to be the eq i alent to the di ferentia “ o f u o tien t u f me tion a q and yet he says , It is s ficient for ”

m 0 r fo r . to ention this , and indeed I intend to st p he e a while It is not at all strange that the work of such a man should

come to be underrated . His pugnacity is shown in the main Object pervading the whole of the L ec tio n es Geo met ricce he sets out with the o n e express intention of simplifying and generaliz ing the existing methods of drawing tangents to curves of all kinds and of I N TR OD UCTI ON

finding areas and volumes ; there is distinct humour in his “ ” r : glee at wiping the eye of some other geomete , ancient or modern , whose solution of some particular problem he has not only generaliz ed but simplified . “ G S t regory Vincent gave this , but (if I remember rightly) proved with wearisome prolixity . Hence it follows immediately that all curves of this kind are touched at any one point by one straight line only . E uclid proved this as a special case for the circle, Apollonius for the conic sections , and other persons ” in the case of other curves . n His early traini g was promiscuous , and could have had no other effect than to have fostered an inclination to leave G others to finish what he had begun . His reek professor ship and his study of Aristotle would tend to make him a “ ” c confirmed geometri ian , revelling in the elegant solution and more or less despising Cartesian analysis because of its then (freq uently) cumbersome work, and using it only with certain qualms of doubt as to its absolute rigour . F o r z instance , he almost apologi es for inserting , at the L X very, end of ecture , which ends the part of the work devoted to the equivalent of the differential calculus , his ” — ” a and e method the prototype of the lz and b method

- - of the ordinary text books of to day . Another light is thrown o n the matter of Cartesian L geometry, or rather its applications , by ecture VI in this , fo r b the purpose of esta lishing lemmas to be used later , Barrow gives fairly lengthy proofs that

2 2 1 m x = mx b 2 x x —m = mx r ( ) y i y / , ( ) i y +g y /

n represent hyperbolas , instead of merely stati g the fact on m 2 z b x nzx r x . The account of the factori ing of / i y , / i y lengthiness of these proofs is to a great extent due to the fact that, although the appearance of the work is algebraical , the reasoning is almost purely geometrical . It is also to be is noted that the index notation rarely used , at least not till n very late in the book in places where he could do nothi g else , although Wallis had used e ven fractional indices a doz en years before . In a later lecture we have the truly terrifying equation (rrbé rrff + 2 fmpa )/bb (rrmm m B ’ Again we ust note the fact that all arrow s work, ’ 1 2 BA R R O WS GE OM E TR I CA L L E CT UR E S

u c was m u f witho t ex eption , geo etry, altho gh it is airly evident t hat he used algebra for his own purposes . F rom the above, it is quite easy to see a reason why Barrow should not have turned his work to greater account b u t in estimating his genius one must make allowance for c m this disability in , or dislike for, algebrai geo etry , read into his work what co u ld have been got out of it (what I w L iz o u t am certain both Ne ton and eibn got of it), and c u not stop short at just what was a tually p blished . It must chiefly be remembered that these old geometers could u s e their ge ome tric al fa c ts far more re adily and s u rely than many mathematicians o f the present day can use their u m m analysis . As a j stification of the extre ely high esti ate v m f m L ectio n es Geo metr icce I ha e for ed , ro the alone, of ’ B c t arrow s genius , I all the a tention of the reader to the list of analytical equivalents of Barrow ’s theorems given on 0 if c u page 3 , he has not the patien e to wade thro gh the ru nning commentary which stands instead of a fu ll trans lation of this book .

T H E S O UR CE S O F BAR R O W ’S I D E AS There is too strong a resemblance between the methods to leave room for do u bt that Barrow owed much of his idea in te ra tio n G C R o u of g to alileo and avalieri (or oberval , if y w O n u t he u c ill) . the q estion as to so rces from whi h he derived his notions on differentiation there is c onsiderably more difficulty in deciding ; and the c omparatively narrow range o f my reading makes me diffide n t in writing anything that may be considered dogmatic on this point and yet if.

I do not do so, I shall be in danger of not getting a fair T h m u f hearing . e following re arks m st there ore be con “ side re d in the nature o f the ple a o f a c o u nsel fo r the ” ’ f c u de en e, who believes absol tely in his client s case or as if c m suggestions that possibly, even not probably, o e very near t o the truth . The general Opinion would seem to be that Barrow was a mere improver on F e rmat . B u t Barrow was c ons c ientious to a fault ; and if we are to believe in his honesty, the source F ermat F o r B re o f his ideas could not have been . arrow

’ 1 4 B A R R O WS GE OME TR I CA L L E C T UR E S

f x ing the appendices , with the possible exception o Appendi ' P f h t B rr w 3 to L ect . XII) ; for we read in the re ace t a a o falling in with his wishes (I will not say ve rywillin gly) added ” “ ” c u . The e L ibra riu s the first five le t res word his r fers to , — c e o f edit o r which for la k of a b tter word, or , which I do — not like I have translated as t/ze p u blis /zer ; but I think it ’ fo r B refers to Collins , , in arrow s words , John Collins looked ” after the publication . T he opinion I have formed is that the idea of the differ ial m e n t triangle, upon which all attention see s (quite c wrongly) to be focussed , when onsidering the work of w o ri in a l co n ce t c Barro , was altogether his own g p ; and to all m F m ’ it a mere i provement on er at s method , in that he uses T h . e two increments instead of one, is absurd discovery m B ’ h was the outco e of arrow s definition of a tangent, w olly F “ and solely and the method of ermat did not consider this . The m B o f mental picture that I for of arrow , and the m m events that led to this discovery , a ongst others far ore P f o f G G e m important, is that of the ro essor eometry at r sha o n College, who has to deliver lectures his subject ; all o n d he reads up that he can lay his hands , ecides o f u that it is all very decent stuff a sort, yet pugnacio sly ” determines that he can and will go one better . In the

I

i A Fi . . F . g . g B

m i c o u rse of his researches , he is led fro one th ng to ario the r until he comes to the paraboliform construction o f L c fu m e t . IX, 4, perceives its use lness and inner eaning, and immediately conc eives the idea o f the differential e m t wo u e triangle . I think if any read r co pares the fig r s Fi fo r u arab o lifo rms above, g . A used his constr ction of the p , B diff re n t ial trian le n o u F . ig for the e ‘ g , he will longer inq ire I N TR OD UCTI ON

B ’ for the source of arrow s idea , unless perhaps he may

0 L . 1 1 . prefer { refer it to ect X, P do il b t Ofi ersonally, I have no that it was a flash nspira ’ u B tion , suggested by the first fig re , and that it was arrow s c c lu k to have first of all had o casion to draw that figure, and sec ondly to have had the genius t o note its significan c e and be able to follow up the clue thus afforded . As further corroborative evidence that Barrow’s ideas were in the main his own creations , we have the facts that he was alone in using b abitu ally the idea o f a curve being a succession of t an infini e number of infinitely short straight lines , the pro lo n gatio n of any one representing the tangent at the point the fo r c on curve whi h the straight line, or either end of f of it, stood ; also that he could not see any dif erence between indefinitely narrow re c tangles and straight lines as

“

. e the constituents of an area If his m thods had req uired it, which they did not , he would no doubt have proved rigor o u s ly that the error could be made as small as he pleased c by making the number of parts , into whi h he had divided his area, large enough ; this was indeed the substance ’ ’ ” P o f C i m of ascal s defence aval eri s ethod of indivisibles ,

6 . is L . . and the idea used in ect XII , App I I , m B I have re arked that , in considering the work of arrow, all attention seems to be quite wrongly focussed on the differ ial i o f m e n t triangle . I hope to conv nce readers this volu e that the differential triangle was only an important side issue in the L ectio n es Geo metrica ; certainly Barrow only

. B d considered it as such arrow really had , concealed un er m a dis a co m lete trea tis e the geometrical for th t was method , p t me t o tb o n /ze ele n s f c ca lcu lu s . T he ma question y then be asked why, if all this is true , did Barrow not finish the work he had begun ; and the answer , I take it, is inseparably bound up with the peculiar B disposition of arrow , his growing desire to forsake m mathe atics for divinity, and the accident of having first

c o - as his pupil and afterwards as his worker, and one in c him man e close personal conta t with , a lik Newton , whose analytical mind was so peculiarly adapted to the task of carrying to a successful conclusion those matters which Barrow saw could not be developed to anything lik e the x t O n e e tent by his own geome rical method . writer has ’ 1 6 B A R R O WS GE OME TR I CA L L E C T UR E S

B ‘ i stated that the great genius of arrow must be adm tted, if only for the fact that he re c ogniz ed in the early days of ’ Newton s career the genius of the man , his pupil , that was m hi . afterwards to overshadow Also, if I fail to make ’ out my contention that Barrow s ideas were in the main m u original, the same re ark can with j stice be applied to him that William Wallace in 51 m ilar c ircumstances applied D c to es artes, that if it were true that he borrowed his ideas f “ on algebra from others , this act, would only illustrate the genius o f the man who c ould pi c k out from other works all that was productive , and state it with a lucidity that makes ” it look his own discove ry ; for the lucidity 1 S there all right B an al in this work of arrow, only it wants translating into y tical language before it can be readily grasped by anyone but a geometer .

MUTUAL INFLUE NCE O F NE WT O N AND BAR R OW B ’ m I can image that arrow s interest, as a confir ed ’ e v g ometer, would ha e been first aroused by j oung Newton s poor show in his scholarsh ip paper on E uclid . This was in ’ 1 66 e O f B w e April 4, the y ar of the delivery arro s first l ctures ’ Lu c as ian P as rofessor, and , according to Newton s own u e c words , j st about the time that he, N wton , dis overed his m his ethod of infinite series , led thereto by reading of the w n o works of D e scartes and Wallis . Ne ton doubt attended o f B a he these lectures rrow, and the probability is that would have shown Barrow his work on infinite series ; for this would seem to ha v e been the etiquette or c ustom of the time ; for we know that in 1 669 Newton communicated to Collins through Barrow a compendium of his work on flu x io n s (note that this is the year of the pre paration fo r press of the L ectio n es Optica et Geo metrica ) . Barrow cou ld not help being struck by the in c ongrui ty (to him) of a man ’ of Newton s calibre not appre c iating E uclid to the full ; at t he same time the o n e great mind would be drawn to the c c other, and the onne tion thus started would have ripened a B inevit bly . I suggest as a consequence that arrow would w o wn sho Newton his geometry, Newton would naturally I N TR OD UCTI ON ask Barrow to explain how he had got the idea for some of m f i B w his ore di ficult construct ons , and arrow ould let him “ into the secret I find out the constructi ons by this little ” B t Of . u list rules , and methods for combining them , my fi n d dear sir, the rules are far more valuable than the mere ” “ ing of the tangents or the areas . All right , my boy, if you m think so, you are welco e to them , to make what you like of, ’ o t or what you can only do not say you g them from me, I ll ” stick to my geometry . This was probably the o ccasionwhen Newton persu aded Barrow that the differential triangle was more general than all his other theorems put t ogether also n G o t fo r later whe the eometry was being g ready press , Newton probably asked Barrow to produce from his stock B ’ of theorems others necessary to complete his, arrow s,

C L . . alculus , the result being the appendices to ect XII The B w rest of the argument is a matter of dates . arro P G 1 660 1 662 P was rofessor of reek from to , then rofessor G G C 1 662 1 66 of eometry at resham ollege from to 4, and Lu c asian P rofessor from 1 664 to 1 669 Newton was a T C 1 66 1 member of rinity ollege from , and was in residence until he was forced from Cambridge by the plague in the ’ summer of 1 665 from manuscript notes in Newton s hand it h writing , was probably during , and owing to, t is enforced C o absence from ambridge (and , I suggest, away fr m the geometrical influence of Barrow) that he began to develop the method Of flu x io n s (probably in accordance with some such permission from Barrow as that suggested in the purely imaginative i nterview above) . The similarity of the two methods of Barrow and Newton is far too close to admit of them being anything else but the outcome of o n e single idea ; and I argue from the dates given above that Barrow had developed most o f his geometry from the researches begun for the necessities of lectures at ’ Gresham College . We know that Barrow s wo rk o n the d ifficult theorems and pro blems of Archimedes was largely a suggestion of a kind of analysis by which they were reduced to their simple component problems . What is then m o re likely than that this is an intentional or unintentional crypto grammatic key to Barrow ’s own method ? I suggest that it

O f c o u rs e t his is imagin at ive re t ro s pe c t iv e pro phe c y ; I b e g that n o o n e wi t a e t he in ve rt e d c o mmas t o s i n if u o t at io n s ll k g y q . ” 1 8 B A R R O N/ S GE OME TR I CA L L E CT UR E S

— 1T I S . is more than likely, As I said, the similarity of the B two methods of Newton and arrow is very striking . F o r the flu x io n al method the procedure is as follows

’ (1 ) S ubstitu te x + oco for x and y +yo for y in the giy e n n in in ' th n t x equation c o t a g e flu e s and y .

2 S u 0. ) ubtract the original eq ation , and divide through by R o t o 3) egard as an evanescent quan ity, and neglect and its powers . B ’ arrow s rules , in altered order to correspond, are ’ (2 ) After the equation has been formed (Newton s rule reject all terms consisting of letters denoting constant or m c h a e deter ined quantities , or terms whi do not contain or ’ ' (which are equivalent to Newton s yo and oco respec tively) for these terms brought over to one side of the equation will ’

z 2 . always be equal to ero (Newton s rule , first part) 1 m ( ) In the calculation , o it all terms containing a power a e fo r of or , or products of these letters ; these are of no ’ 2 value (Newton s rule , second part, and rule m a t (3 ) Now substitute (the ordinate) for , and (the sub ’ e T x tangent) for . ( his corresponds with Newton s ne t step , at x the obtaining of the ratio y, which is e actly the same as ’ Barrow s e a . ) ’ T he only difference is that Barrow s way is the more suited “ t o his geometrical purpose of finding the quantity of the ” ’ c subtangent , and Newton s method is pe uliarly adapted to

a m m . B an lytical work, especially in proble s on otion arrow left his method as it stood, though probably using it freely u s ita tu m 1 1 re u en ta tive (mark the word on page 9, which is af q u to r derivative of , I use) to obtain hints for his tangent n problems , but not thinki g much of it as a method compared with a strictly geometrical method ; yet admitting it into his work, on the advice of a friend , on account of its en eralit O n g y . the other hand , Newton perceived at once the immense possibilities of the analytical methods intro du c e d D e by escartes, and developed the id a on his own lines , to suit his own purposes . T P here is still another possibility . In the reface to the ' “ O c de lic at e mo t he rs c o m pti s , we read that as are wont , I mitt e d m dis to the foster care of friends, not unwillingly, y ” ca rded c/zild T B m . hese two friends arrow entions by “ name : I s a ac N ewto n (a man of ex ceptional ability I N TR OD UCTI ON

m C o f and re arkable skill) has revised the opy, warning me many things to be corrected and adding some things from ” wn 5 his o work . Newton additions were probably con fined to a great extent to the Optics only but the geometrical lectures (seven of them at least) were originally designed as m O supple entary to the ptics , and would be also looked over by Newton when the c ombined publication was being pre “ ” C . pared . John ollins has attended to the publication B Hence , it is just possible that Newton showed arrow his m flu x io n s B ethod of first, and arrow inserted it in his own way ; this supposition would provide an easy explanation of the treatment accorded to the batch of theorems that m t h x L for e third appendi to ect . XII ; they seem to be m hastily scrambled together, co pared with the orderly treat ment of the rest of the book , and are without demonstration f c and this , although they orm a ne essary complement for u the completion of the standard forms and rules of proced re .

I say that this is possible, but I do not think it is at all ’ probable ; for it is to be noted that Barrow s description rs t erso n s in u lar of the method is in the fi p g (although , when “ u f giving the reason for its introd ction , he says requently u sed by and remembering the authentic accounts of B ’ arrow s conscientious honesty , and also judging by the o f e later work Newton , I think that the only alt rnative to be considered is that first given . Also , if that is accepted, we have a natural explanation of the lack of what I call the B ’ true appreciation of arrow s genius . Barrow could see the limitations imposed by his own geometrical methods (none so well as he , naturally, being probably helped to this con e lusion by his disc ussions with Newton) ; he felt that the c orrect development of his idea was on purely analytical lines , he recognised his own disability in that direction and ’ the peculiar aptness of Newton s genius for the task, and , v lastly, the growing desire to forsake mathematics for di inity made him only too willing to hand over to the foster care “ of Newton and Collins his discarded child t o be led out ” ” mt /it s eem o o d to taeui blan c he e and set forth as g g . Carte of such a sweeping character very often has exactly the opposite effect to that which is intended ; and so probably Newton and Collins forbore to make any serious alterations B f or additions , out of respect for arrow ; o r altho ugh the ’ 2 0 BA R R O WS GE OME TR I CA L L E CT UR E S

n allusion to the revision properly applies o ly to the Optics, it may fairly be assumed that it would be ex tended to the G eometry as well , and if not then , at any rate later, for, quoting a quotation by Canon Overton in the D i ctio na ry o N a tio n al B io ra /z f g p y (source of the quotation not stated) , which refers to Barrow ’s pique at the poor reception that — was accorded to the geometric al lectures and does not this w B sho the high opinion that arrow had of them himself, and lend colour to my suggestion that they were never delivered as L u c as ian L ? P f ectures also note his remark in the re ace, given “ e The I ex o s e mo re reel to o u r lat r, other seven , as I said , p f y y view lzo in tb a t t/zere is n o t/Zin in t/zem tuat t will d s leas e , p g g i i p tae eru dite to s ee m m , When they had been so e ti e in the o n s idered world, having heard of a very few who had read and c t/zem tb o ro u bl i me t g y , the little relish that such th ngs with helped to loose him more from those speculations an d heighten D his attention to the studies of morality and divinity . oes not this read like the disgust at people forsaking the le giti “ mate methods o f geometry for such unsatisfactory stuff (as I have suggested that Barrow would consider it) as analysis ” ? Who can say the form these lectures might have taken if there had been no Newton or if Barrow had taken kindly to Cartesian geometry ; or what a second edition, ” B revised and enlarged , might have contained , if arrow on his ret u rn to Cambridge as Master of Trinity and Vice Chancellor had had the energy or the inclination to have made one ; or if Newton had made a treatise of it, instead L B of a reprint of S cholastic ectures , as arrow warns his 1 6 readers that it is , and such as the edition of 74 in two vol m ? B u t B u es probably was arrow died only a few years later , i C Newton was far too occup ed with other matters , and ollins seems to have passed out of the picture , even if he had been the equal of the other two .

D E S CR IP TIO N O F T H E BO O K FR O M WHI CH T H E TR ANS LAT IO N H AS BE E N MAD E

The running commentary which follows is a précis of L a full translation of a book in the Cambridge ibrary . In t wo s one volume , bound in strong yellow calf, are the work ,

’ 2 2 B AR R O WS GE OME TR I CA L L E CT UR E S the ease with which the argument may be followed is also not by any means increased by Barrow ’s plan of running his work ‘ c on in one continuous stream (paper was lear in those days), with intermediate steps in brackets and this is made still worse by the use of the “ full stop as a sign of a ratio (division) instead of as a sign of a rec tangle (multiplication) thus — — : : DL DH. LN HO LH. HK stands , in modern symbols , for the extended statement — DHz HO DLz LN (DL HO) LHz HB DH: H0 LHz LB LHz HK; x —LHx HK= KO LH HKx LH whe reas DL LK , on the DL HK LH HK contrary, means , as is usual at present,

K0 LH HK LH, the minus sign thus being a weaker bond than that of multiplication , but a stronger bond than that ’ B I . S of ratio or division arrow s list of symbols , in full , “ F o r the sake of brevity certain signs are used, the meaning of which is here subjoined .

A B that is , A and B taken together .

e . A B A , B b ing taken away The f A B dif erence of A and B .

A X B A multiplied by, or led into , B .

B. e A divided by B , or applied to

A B A is equal to B . — A c B A is greater than B .

‘ A 3 B A is less than B . m B 0 . A. B A bears to the sa e ratio as to D

0 . A B C, D A , B , , D are in continued proportion , , — C . A . Bc—O. D A to B is greater than to D i t han C t o . A. B _ C. D A to B is less D T he ratios equal to

t t o . M. N A o B, C D greater than M to N compounded less than

Aq The square on A .

The . JA side, or square root of, A At T he cube of A . f JAq + Bq The side o the square made up of the

square o f A and the square o f B . i re O ther abbreviations , if there are any, the reader w ll e v cognise, by easy conjectur , especially as I have used ery ” little analysis . I N TR OD UCTI ON

The o n e B style of the text , as would expect from a arrow, “ ” v s is classical ; that is , full of long in olved entences , “ ” phrases such as through all of a straight line points ,

general inversion of order to enable the sense to run on ,

use of the relative instead of the demonstrative, and so on all agreeing with what is but an indistinct memory (thank 1 C goodness ) of my trials and troubles as a boy over icero ,

D e S en ect u te D e Amicitia - P , , and such like, studied ( ), by the ’

G L . way, in Newton s old school at rantham in incolnshire In this way there is a striking difference between the style of Barrow and the straightforward L atin of Newton ’s P rin ci ia L 1 8 2 2 L e p , as it stands in my atin edition of , by M m S c . ur and Jacquier y classical attain ents are, however,

so slight that , in looking for possible additions by Newton , I have preferred to rely on my proof- reading experience in ’ the matter of punctuation . T he strong point in Barrow s

somewhat awful punctuation is the use of the semicolon ,

combined with the long involved sentence, and the frequent z interpolation of arguments , sometimes running to a do en i n lines, parentheses ; Newton makes use of the short con m cise sentence , and rarely uses the se icolon , nor indeed

does he use the c olon to any great ex tent . Of course I do not know how much the printer had to do with the pu n c t u a

tion in those days , but imagine this distinction was a very O f n great matter the author . Compari g two analogous 2 00—2 0 passages , from each author, of about 5 words , we get the following table

arro N e t o n . B w. w

T his contrast is striking enough for all practical purposes B in addition , arrow starts three of his five s entences with a

relative, whilst Newton does not do this once in his ten . U sing this idea, I failed to find anything that could, with any probability, be ascribed to Newton . 2 4 B A R R O WS GE OME TR I CA L L E CT UR E S

“ L u astly, one strong feat re in the book is the continued use o f the parab o lifo rms as auxiliary curves ; this corroborates my contention that Barrow fully appreciated the importanc e o f m and inner meaning his theore , or rather construc tion L (see note to ect . IX, that is , he uses it in precisely the same way as the analytical mathematician uses its equi x im ff valent, the appro ation to the binomial and the di erentia i t on of a fractional power of a variable, as a foundation of

all his work . t Al hough there are two fairly long lists of errata, most w i u m probably due to Ne ton , there are st ll a great n ber of m misprints the diagrams are , however, unifor ly good , there being no omissions of important letters and only one or two 2 00 i slips in the whole set of , one of these ev dently being the fault of the en graver ; nevertheless they might have been much clearer i f Barrow had not been in the habit m s of using one diagra for a whole batch of allied theorem , thereby having to make the diagram rather complicated in order to get all the curves and lines necessary fo r the

- whole batch of theorems on the one figure, whilst only

using some of them for each separate theorem . In the

text which follows this introduction , only those figures have

been retained that were absolutely essential . T here is a book - plate bearing a medallion of George I “ ” and the words MU N I F I C E N TI A R EGIA 1 71 5 whic h points out that the book I have was o n e of the volu mes of books and manus c ripts c omprised in the library of Bishop m Moore of Norwich , which was presented to the Ca bridge L 1 1 G ac k n o wle d University ibrary in 7 5 by eorge I , as an g ment of a loyal address sent up by the University to the king

on his accession . It may have c ome into his possession as e f B in a p rsonal gift rom arrow ; at any rate, there is an “ ” fl - f u . scription on the first y leaf, A gi t from the a thor I am unabl e to ascertain whether Moore was a student at u i e u C ambridge at the date of the p bl cation of thes lect res , but the date o f his birth (1 646) would have made him twenty o f m four years age at the ti e , and this supposition would ex plain the presen c e o f a four- line Latin verse (Barrow had a weakness fo r turning things into Latin or Greek verse) on — the back of the title page of the geometrical section , which reads I N TR OD UCTI ON

* To a y o u ng ma n a t t k e Un ivers ity .

u mble wo rk o tk bro tk er ro n o u n ced o r to be H f y , p , N o w rz k tl a ea rs devo ted to tit ee g y pp , ; ’ k o u ld s t lear n ro m it a u k t bo tk k a a n d s u re S f g , ppy I n tk a tron l avo u r ermit it en du re y p y f p , and is in the same handwriting as the inscription .

T H E P R E FACE S

P In the following translation of the refaces, ordinary type B ’ is used instead of arrow s italics , in order that I may call o attention to p ints already made , or points that will be x possibly referred to later, in the notes on the te t, by means of italics . The P O first reface, which precedes the ptics

Communication to the reader .

Worthy reader, T hat this , of whatever humble service it may be, was not designed for you , you will soon understand m from any indications , if you will only deign to examine it nor, that you might yet demand it as your due, were other

. T o authorities absent these at least, truly quaking in mind and after great hesitation , I yielded chiefly because thereby I should set as an example to my successors the production e of a literary work as a duty, such as I mys lf was the first

to discharge ; if less by the execution thereof, at any rate

by the endeavour at advance, not unseemly , nor did it seem

f . T to be an ostentation foreign , to my o fice here was in addition some slight hO pe that there might be therein some o f u thing the nature of good fruit , s ch as in some measure

might profit you , and not altogether be displeasing to you . m m Also , re e ber, I warn those of you , who are more ad an c e v d in the subj ect of my book , what manner of writing you are handling ; not elaborated in any way for you alone

Wit a o o ie s fo r do e re b u t the g gg t ran s atio n is fair y c o s e in e fo r h p l l l l l , l ’ 2 6 B AR R O WS GE OME TR I CA L L E CT UR E S not produced on my own initiative ; no r by long medita o f tion , exhibiting the ordered concepts leisurely thought ; ‘ but S c holastic Lec tures ; fi rs t extracted from me by the n eces s ities of my ofi ice ; then from time to time ex panded over - hastily to complete my task within the allotted time ; lastly, prepared for the instruction of a promiscuous literary public , for whom it was important not to leave out many m lighter atters (as they will appear to you) . In this way you will not be looking in vain (and it is nec essary to warn c o u you of this , lest by expecting too mu h y may harm both l yourself and me) for anything elaborated , skilfu ly arranged ,

. F o r or neatly set in order indeed I know that, to make x the matter satisfactory to you , it would be e pedient to cut m out any things , to substitute many things , to transpose ‘ ’ F o r many things , and to recall all to the anvil and file . m this , however, I had neither the sto ach nor the leisure to take the pains nor indeed had I the capability to carry the matter through . And so I chose rather to send . them forth ‘ ’ ’ in Nature s garb , as they say , and just as they were born m rather than , by laboriously licking the into another shape, to fashion them to please . However, after that I had z entered on the intention of publication, either sei ed with e u disgust, or avoiding the troubl to be ndergone in making n c the e essary alterations , in order that I should not indeed n a s put off the rewriting of the greater part of these thi gs , delica te mo t/zers a re wo n t I co mmitted to tii e o s ter ca re o , f f rien ds n o t u n willin l m dis carded ck ild to be led o u t a n d f , g y , y , o o t tk e f o s et o rtk as it k t s eem d o m. O f r f n g which , I think it right that you should know them by name, Isaac c Newton , a fellow of our ollege (a man of exceptional ability m C ‘ and re arkable skill) has revised the opy, warning me of many things to be corrected , and adding some things from se e x his own work , which you will anne ed with praise here The m and there . other (who not undeservedly I will call M e the ersenn of our race, born to carry through such essays as this , both of his own work and that of others) John

C ollins has attended to the publication , at much trouble to himself. n o w o I could place other obstacles to y ur expectation , or show further causes for your indulgence (such as my meagre c ability, a la k of ex periments , other cares intervening) if I N TR OD UCTI ON

I were not afraid that that bit of wit of the elder Cato would be hurled at me Truly you publish abroad these things as if bound by ’ Am hic t o n e s a decree of the p y . “ m u At least fairness de anded a prolog e of this kind, and ’ in some degree a c ert ain parental affec tion for one s own ff m o spring enticed it forth, in order that it ight stand forth

the more excusable, and more defended from censure . “ Bu t if you are severe, and will not admit these excuses

into a propitious ear, according to your inclination (I do not mind) you may reprove as much and as vigorously as

you please .

The P G o second reface , which refers to the e metry M y dear reader, O f these lectures (wk ic/zy o u will n o w receive in a cert a in meas u re la te- bo rn ), seven (one being ex c e pt e d) I intended as the final accompaniments and as it were l O wk ic/z s t a n d o rt/z the things eft over from the ptical lectures , f la tel u blis /zed u y p ; otherwise, I imagine, I shall be tho ght

. little of for bringing out sweepings of this kind However, — — L ibrarius PC when the publisher [or editor ollins] thought, m for reasons of his own , that these atters should be prepared, separately removed from the others ; and moreover b e de sired something else to be furnished that should give the work a distinct quality of its own (so that indeed it might surp ass the siz e of a supplementary pamphlet) ; falling in with his wishes (I will n o t s ay very willingly) I a dded tb c rs t ve lectu res fi fi , cognate in matter with those following and as it were coherent ; which indeed I b ad devis ed so me y ears a o c g , but, as with no idea of publishing , so without that are F r which such an intentio n calls for . o they are clumsily and confusedly written ; nor do they contain anything

firmly, or anything lying beyond the use or the compre he n sio n of the beginners for which they are adapted where fore I warn those experienced in this subject to keep their awa ro m tk es e s ec tio n s to eyes turned y f , or at least give m the indulgence a little liberally . The I ex o s e mo re ree t o other seven , that I spoke of, p f ly o u r view k o in tk at tk ere is n o ticin in tk em tlzat it will y , p g g di l sp eas e tlze mo re eru dite t o s ee . ’ 2 8 B AR R O WS GE OME TR I CA L L E CT UR E S

' The man last lecture of all a friend (truly an excellent , v in one of the ery best, but in a business of this sort an m me m c satiable dun extorted fro ; or, ore orrectly,

claimed its insertion as a right that was deserved . F o r wk a t tk es e lec tu res brin o rt/z o r to wk a t the rest , g f , t/ze ma lead y y , you may easily learn by tasting the beginnings

of each . “ S ince there is now no reason why I should longer

detain or delay you ,

F AR EWELL .

IX H O W BAR R OW MAD E H I S CO NS TR UCTIO NS

In haz arding a guess as to how Barrow came by his con s tru c tio n s , one has , to a great extent , to be guided by his ma be other works , together with any hint that y obtained ' T from the o r der of his theorems in the text . aking the ff a x latter first, I will state the e ect the re ding of the te t had The on me . only thing noticeable, to begin with, was the r pairing of the p opositions , rectangular and polar ; the rest m z seemed ore or less a hapha ard grouping, in which one proposition did occasionally lead to another ; but c ertain of the more difficult constructions were apparently without

m . O any hint fro the preceding propositions nce, however, it began to dawn on me that Barrow was trying to write a m lete elemen ta r treatis e o n tb c calcu lu s co p y , the matter was F set in a new light . irst , the preparation for the idea of a s mall part of the tangent being s u bstituted fo r a small part L vice vers a c . u e of the arc , and ( e t V, this , of co rs , havm g been added later, probably, I suggest, to put the differential triangle on a sound basis ; then the lemmas on

hyperbolas , for the equivalent of a first approximation in the form ofy (ax d) fo r any equation in the form giving y as an explicit fun c tion o f x ; this first gave the clue pointing to his constr u c tions having been found out analytic ally then the work on arithmetical and geometric al means leading to the approximation to the binomial raised

F la itato r im ro bu s a s e c imen o f Barro vian u mo ur g p ; p h .

’ 30 B A R R O WS GE OM E TR I CA L L E CT UR E S

ANALYT ICAL EQ UIVALE N TS T O BAR R O W ’S C HI E F T H E O R E MS

F u n dame n t al T he o re m

n c If is any positive rational number, integral or fra tional , ” 1 x 2 1 n . x n ? 1 then ( ) , according as and, this inequality te nds to become an equality when x tends to z ero . P c L 1 —1 6 [roved without onvergence in ect . VII , 3 ]

S t an dard F o rms fo r Diffe re n t iat io n = 1 . I f c x z A y is any fun tion of , and /y 2 dx z d L dz A d x . then / ( /y ) y / ect VIII , 9 2 I f x z = . y is a function of , and y + 0,

dz dx z d dx L 1 1 then / y/ ect . VIII , 2 2 2 I f o f x 2 = a 3 . y is any function , and y , = z dz dx . d dx r then . / y y/ or, in anothe form , z ’ ’ if z dtr then aZ /ax [y /Jtfi fi ller/fix

L . 1 ect VIII , 3 2 2 2 = dz x == d . Z a z d d x 4 If y , then / y y / , 2 dz dx a d dx L . 1 or / )] y / ect VIII , 4 2 2 2 — d dx . I f z a z . z 5 y , then / 2 dz dx a L 1 or / [y /J( ect . VIII , 5 6 I f x z a b . y is any function of , and y , = d L . 1 dz dx b d x . then / . y/ ect IX, "

. I f x z 7 y is any function of , and X L . I then ect , 3 ”" 1 8 . : n x n S pecial case or . n L I X where is a positive rational ect . , 4

n 9 . The case when is negative is to be deduc ed from 1 the combination of F orms and 8 . 2 0 f = ta n x d dx z s ec x E x 1 . I . y , then y / , proved as 5 on the differential triangle at the end of L ect . X . m It is to be noted that the sa e two figures , as used t a n x f ffi for , can be used to obtain the dif erential coe cients of the other c ircular functions . I N TR 0D06 TI ON

L aws fo r Diffe re n t iat io n

— - ~ I f w -z LAW 1 . S u m o f Two F u n c t io n s y l ,

dw dx z d dx dz dx L c . then / y / / e t VIII , 5

— f w= z W 2 . I L A P ro du c t o f TWO F u n c t io n s y ,

L . 1 2 then ect IX,

— w= z LAW . I f 3 Q u o t ie n t o f T wo F u n c t io n s y / , v = I z then , if / , as has already

L . been obtained in ect VIII , 9 ; hence by the above

— B . I V. Note the logarithmic form of these two results , B corresponding with the subtangents used by arrow .

The remaining standard forms Barrow is unable ap pare n tly to obtain directly ; and the same remark applies S o to the rest of the laws . he proceeds to show that

Differentiation and Integration are inverse Operations . = = i L . 1 1 . . z d R dx h R. z dx ( ) If Jy , t en / y ect IX, = = . d dx L 1 z dx . R. z R. (ii ) If / y , then Jy ect XI , 9 Hence the standard forms for integration are to be obtained immediately from those already found for differ e n tiatio n B . arrow, however, proves the integration formula for an integral power independently, in the course of certain L theorems in ect . XI . He also gives a separate proof of the

L XI 2 . quotient law in the form of an integration , in ect . , 7

F u rthe r S t an dard F o rms fo r I n t e g rat io n

' A a x x = lo x . JI / g

.L . x ect . XII , App 3 , = x B . a dx k a 1 P . j ( ) rob 3 , 4

C . tu n eno l I 2 o i i A . co s 9 L . x J g ( ect , pp . o ) D — , s in 5

2 2 E . d /(a x ) {log (a (see F orm D ) F co s 0d ta n 9 db ta n 9 d co s 6 db ta n 9 co s 0 J ( ) J ( ) , s ec b dd both being equal to J , the only example of ” 8 s L . integration by part I have noticed ect XII , App . I , 2 2 2 G x a lo x x . d /J( ) g [{ J( 9 ’ 3 2 B AR R O WS GE OME TR I CA L LE CT UR E S

G raphic al I n t e g rat io n o f an y F u n c t io n

Fo r x any function , f( ), that cannot be integrated by the f B m d oregoing rules , arrow gives a graphical etho for Jf (x )dx as a logarithm of the quotient of two radii ve c tores of the cu rve and for d /f (x ) as ’ a differen c e of — L . 8 their reciprocals ect . XII , App III , 5

F u n dame n t al T he o re m in R e c t ifi c at io n

2 2 ds dx dx L c 1 d . He proves that ( / ) ( y / ) e t X , 5 He re c tifi e s the cyc loid (thus apparently anticipating

S a - Wren), the logarithmic pir l , and the three cusped hypo c c a m an d ycloid (as special ses of one of his general theore s), red u ces the rec tification of t he parabola to the quadrature O f u h b e rb o la c c the rectang lar y , from whi h the re tification follows at once .

'

X . 2 6 XI I 2 0 x I I E x 2 E . ( , App III , I , . ; XI , ; , ,

In addition to the foregoi n g theorems in the I n fi n it e s imal Calculus (for if it is not a treatise on the elements of the C P B n i alculus , what is it ), arrow gives the followi g nteresting m theore s in the appendix to Lec t . XI

Max ima an d Min ima

’ s m m m l x c —x He Obtains the axi u va ue of ( ) , giving the condition that x /r = (c x )/s ; also he shows that this is the — s c ondition fo r the minimum value of c) .

T rig o n o me t ric al Appro x imat io n s

Barrow proves that the c ircular measure of an angle a s in a 2 co s a s in a 2 co s 2 co s a lies between 3 /( ) and ( ) , m m the for er being a lower limit, and equivalent to the for ula O f S nellius ; each O f these approximations has an error o f the order o f T H E G E O M E T R I C A L L E C T U R E S

ABR I DGE D TR ANS LAT IO N

N E S D E D UC N S R S M E D WITH OT , TIO , P OOF O ITT BY AR R AND UR E R E XAM L E S B OW, F TH P OF H I S ME THO D

L E CT UR E I

’ Gen era tio n o ma n it u des Mo des o mo tio n a n d tk e f g . . f u a n ti o ti n mo ti e o r e Time a s tk e in de en den t a r q ty f v f c . p v i a ble T me a s a n . i a r e a te o in s t a n ts co m ared witk a , gg g f , p lin e a s t/ze a re a te o o in ts D edu ctio n s , gg g f p . .

B [In this lecture, arrow starts his subject with what he calls the generation of magnitude s ]

E very magnitude can be either supposed to be produced ,

n . T he or in reality can be produced , in in umerable ways most important method is that of local movements . In motion , the matters chiefly to be co nsidered are the mode of motion and the quantity of the motive force . S ince T q uantity of motion cannot be discerned without ime, it

I S T T m necessary first to discuss ime . i e denotes not an a x ctual e is tence, but a certain capacity or possibility for a continuity of ex istence ; just as S pace den o tes a capacity

i n T for nterveni g length . ime does not imply motion , as far as its absolute and intrinsic nature is concerned ; not any more than it implies rest ; whether things move or are T still, whether we sleep or wake , ime pursues the even

T n tenor of its way . ime implies motio to be measurable ; T without motion we could not perceive the passage of ime . ’ 36 BA R R O WS GE OME TR I CA L L E CT UR E S

’ O n Time a s Time tis et co n es s ed , , y f

F ro m mo vin tk in s dis tin ct o r t ra n u il res t g g , q , ” N o t/zo u /zt can be g ,

L is not a bad saying of ucre tius . Also Aristotle says

Wk en we o o u rs elves in n o wa alter t/ze t rain o o u r , f , y f

' t/zo u k t o r in deed i we ail to n o tice tk in s tlzat a re a ectin g , f f g fi g

t m do e n o t s eem to u s to k a e as e it i e s v s d. , p And indeed it does not appear that any, nor is it apparent how much , time

Bu t has elapsed, when we awake from sleep. from this, it is not right to conclude that I t is plain tk at Time do es n o t ex is t wit/co n t mo tio n an d ck a n e o o s itio n We do g f p . n o t erceive it tk er e o re it do es n o t ex is t p , f , is a fallacious inference ; and sleep is deceptive, in that it made us connect two widely separated instants of time . However, it is very

Wk atever tbc amo u n t o tk e mo tio n was s o true that f ,

u ck time s eems to k ave ass ed m p nor, when we speak of so

e much tim , do we mean anything else than that s o much

o e e motion could have g n on in betw en , and we imagine the continuity of things to have coex tended with its con

u x tin u o s ly succe ssive e tension . We evidently must regard Time as passing with a steady flow ; therefore it must be compared with some handy

e as st ady motion , such the motion of the stars , and especially of t he S u n and the Moon ; such a comparison

an d the is generally accepted , was born adapted for pur

D o f Go d G Bu t pose by the ivine design ( enesis i, how,

w S u n e you say, do we kno that the is carried by an qual

x o n e motion, and that one day , for e ample , or year, is

l o f a io ? I ex act y eq ual to another , or eq ual dur t n reply

’ 38 BA R R O WS GE OME TR I CA L L E CT UR E S

c o f as the vel o city . Q uantity of velo ity cann t be found rom

t the q uantity of the space traversed only, nor from the ime

. o ni taken only, but from both of these brought into reck ng togeth e r ; and quantity of time elapsed is not determined without known quantities of space and velocity ; nor is q uantity of space (so far as it may be found by this method) dependent on a definite quantity - o f velocity alone, nor on so much given time alone , but on the joint ratio of both .

T o every instant of time, or indefinitely small particle of m time , (I say instant or indefinite particle, for it akes no difference whether we suppose a line to be composed of points or of indefinitely small lin e le t s and so in the same m manner, whether we suppose time to be ade up of instants

t imele t s or indefinitely minute ) ; to every instant of time, I say, there corresponds some degree of velocity, which the moving body is considered to possess at the instant to this degree of velocity there corresponds some length of space described (for here the moving body is a point , and so we

Bu t consider the space as merely long). since , as far as

‘ this matter is concerned, instants of time in nowise depend on one another, it is possible to suppose that the moving body in the nex t instant admits of another degree of velo city (either equal to the first or differing from it in some proportion), to which therefore will correspond another m length of space , bearing the sa e ratio to the former as the latter velocity bears to the preceding ; for we cannot b u t

n suppose that our insta ts are exactly equal to one another . if Hence , to every instant of time the re is assigned a suit L E CT UR E I

t able degree of velocity, there will be aggregated out of hese

ch ‘ e a certain quantity, to any parts of whi resp ctive parts o f space traversed will be truly proportion ate ; and t h us a magnitude representing a quantity compos e d of these

e degrees can also represent the space described . H nce, if through all points of a line representing time are drawn straight lines so disposed that no one coincides with another

i. e a re ( . parallel lines), the plane surface that results as the gg

o f gate the parallel straight lines , when each represents the degree of velocity corresponding to t he point through which it is drawn , exactly corresponds to the aggregate of the

r deg ees of velocity, and thus most conveniently can be n adapted to represent the space traversed also . I deed this surface , for the sake of brevity, will in future be called the aggregate of the velocity or the representative of the space . It may be contended that rightly to represent each

t ime le t separate degree of velocity retained during any , a very narrow rectangle ought to be substituted for the right

u ite s o line and applied to the given interval of time . Q , bu t it co mes t o tk e s a me tk ing wk ick ever way y o u tak e it ; but as our method seems to be simpler and clearer, we will in future adhere to it .

B [ arrow then ends the lecture with examples , from which he obtains the properties of uniform and uniformly accel e rat e d motion ]

1 m ( ) If the velocity is always the sa e , it is quite evident from what has been said that the aggregate of the velocity attained in any definite time is correctly represented b y a m w parallelogra , such as AZZE, here the side AE stands for ’ 40 B A R R O WS GE OME TR I CA L L E C T UR E S

'

a definite time, the other AZ, and all the parallels to it, 82 OZ EZ , , DZ, , separate degrees of B C o s: veloci ty corresponding to the separate

instants of time, and in this case

plainly equal to one another . Also O the parallelograms AZZB , AZZ , AZZD,

e AZZE , conveniently repres nt , as has Z 2 Z Z Z been said , the spaces described in the F ’ 1 g ' °

AC AE. respective times, A B , , A D ,

2 I f ( ) the velocity increase uniformly from rest, then the aggregate o f the velocities is represented by the triangle f AEY. Also if the velocity increases uniformly rom some definite velocity to another definite velo city represented CY EY respectively by , , then the space is represented by a

z V . trape ium , such as GY E

(3 ) If the velocity increase according to a progression of

a re square numbers, the space described to represent the gg

f - gate o the velocity is the complement of a semi parabola .

[Fo r which Barrow gives a figure ]

and (2 ) all the properties of uniform motion m and of unifor ly accelerated motion are simply deduced, and the lecture concludes with the remark

T hese things , being necessary for the understanding of things to be said later, and theories of motion that are, I think , not on the whole q uite useless , it has seemed to be advantageous to explain clearly as a preliminary . Having

e . finished this task, I direct my st ps forward L E CT UR E I

NOTE m T here is not uch in this lecture calling for remark . The B P matter, as arrow says in his reface, is intended for T . e beginners her is , however, the point, to which attention B is called by the italics on page 3 9, that arrow fails to see any difference between the use of lines and narro w rect T ’ angles as constituent parts of an area . his is Cavalieri s “ ” o f P method indivisibles, which ascal showed incontro “ ” ve rtibl y was the same as the method of exhaustions , T as used by the ancients . here is evidence in later lectures that Barrow recogniz ed this ; for he alludes to the possi bility of an alternative indirect argument (dis cu rs u s apo o ic u s g g ) to one of his theorems , and later still shows his meaning to be the method of obtaining an upper and a T lower limit . here is also a suggestion that he personally

x - o f used the general modern method of the te t books , that proving that the error is less than a rectangle of which one side represented an instant and the other the difference between the initial and final velocities ; and that it could b t ak in be made evanescent y g the number of parts , into o i which the whole time was d vided , large enough . Also the attention o f those who still fight shy of graphical f proofs for the laws of uni ormly accelerated motion , if any

- f such there be to day, is called to the fact that these proo s B were given by such a stickler for rigour as arrow , with the remark that they are evident, at a glance, from the diagrams he draws . L E CT UR E I I

“ n Tbe Gen eratio n of mag n it u des by lo cal mo veme ts . s imple mo tio n s of t ra n s la tio n a n d ro ta tio n

Mathematic ians are not limited to the actual manner in which a magnitude has been produced ; they assume any method of generation that m ay be best s u ited to their * purpose . Magnitudes may be generated either by simple m c c otions , or by composition of motions , or by oncurren e of motions .

[E xamples of the difference of meaning that Barrow attac hes to the two latter phrases are given by him in a Th later lecture . e simple motions are considered in this lecture ] T m here are two kinds of si ple motions , translation and

i e m . rotation , . . progressive otion and motion in a circle

F o r m 1 a these otions , mathematicians assume that ( ) point c an progress straightforwardly from any fixed

n termi us , and describe a straight line of any length ;

(2 ) a straight line can proceed with one extremity moving

A s an e x am e t a e the c as e o f n din the v o u me O f a ri t c irc u ar pl , k fi g l gh l c o n e b in t e rat io n e re de n it io n the me t o d o f e n e rat io n is t he y g ; h , by fi , h g ro t atio n o f a right - an gle d t rian gle abo u t o n e o f t he re c t an gu lar s ide s ; b u t it is s u o s e d t o b e e n e rat e d fo r t he u r o s e o o f mo de rn in t e rat io n pp g , p p g , t he mo t io n o f a c irc e t at c o n s t an t in c re as e s in s iz e an d mo ve s by l , h ly , e t i f it i c n re n t h x f t h c n parall l o t s el w h t s e t o e a is o e o e . LE C T UR E I I

along any other line, keeping parallel to itself ; the former

en etrix b e a lied To the is called the g , and is said to pp latter which is called the direc trix by these are described paral le lo rammatic x x g surfaces, when the genetri and the directri

t he are bo th in same plane, and prismatic and cylindrical

x n e c e s surfaces otherwise . In general, the genetri may , if

sary, be taken as a curve , which is intended to include

x polygons , and the genetri and the directrix may usually

The be interchanged . same kinds of assumptions are made for simple motions of rotation ; and by these are described circles and rings and sectors or parts of these, when a straight line rotates in its own plane about a point in itself or in t he line produced ; if the directrix is a curve

(in the wider sense given above), and does not lie in the plane of the genetrix , of which one point is supposed to

x be fi ed , the surfaces generated are pyramidal or conical . From this kind of generation is deduced the similarity of parallel sections of such surfaces ; and thus it is evident that the surfaces can also be generated by taking the

e x genetrix of the first m thod as the directri , and the former directrix as the genetrix so long as it is supposed to shrink proportionately as it proceeds parallel to itself towards what

x was the verte or fixed point in the first meth o d .

F o r producing solids the chief meth o d is a simple

o o o rotation , ab ut s me fixed line as axis , of an ther line

lying in the same plane with it . In addition, there is the

“ method of indivisibles , which in most cases is perhaps

x the the most e peditious of all , and not least certain and infallible of the whole set . ’ 44 BA R R O WS GE OME TR I CA L L E CT UR E S

* n T ac u e t u s o The lear ed A . q more than nce objects to this method in his clever little book on “ Cylindrical and S i Annular olids , and therein thinks that he has fals fied it, because the things found by means of it concerning the surfaces of cones and spheres (I mean quantities of these) do not agree in measurement with the truths discovered and handed down by Archimedes .

F i 1 0 g , . T

T x DVY ake, for e ample, a right cone , whose axis is V K through every point of this suppose that there pass straight D KD m Z B Z O Z . lines ZA, , , (or ), etc ; fro these indeed according to the Atomic theory the right - angled triangle V D K is made up and from the circles described with these

m . Tk ere o re as radii the cone itself is ade f , he argues ,

fro m tk e perzpk eries of tk es e circles is co mpo s ed t/i e co n ic a l s u rfac e n o w tbis is fo u n d to be co n tra ry to tk e tr u tk ; k en ce tk e metk o d is fallacio u s .

I reply that the c alculation is wrongly made in this manner ; and in the computation of the peripheries of which

A n dre as ac u e t a e s u it o f An t e r : u is e d a o o o n the T q , J w p p bl h b k c ylinde r E le me n ta Geo met r iw (1 654) an d a bo o k o n Arit hme t ic m n t i n e d Wa (1 664) e o by lli s . ’ ~ ~ T he n u m e rin o f the dia rams is B arro w s an d is i n c o n s e u e n c e o f l b g g , q a rid me n t n o t c o n s e c u t ive . b g ,

’ 46 B A R R O WS GE OM E TR I CA L L E C T UR E S

' plane surfaces are triangles and the are a o f the curved surface is thus easily found .

There are other methods which may be used convenie nt ly in certain cases ; but enough has been said for the present concerning the constru c tion o f magnitudes by simple motions .

NOTE It would be interesting to see how Barrow would get ’ f u T T m over the di fic lty raised by acque t , if acquet s exa ple had been the case of the oblique circular cone . It seems B to me to be fortunate for arrow that this was not so . B 1 5 arrow also states , be it noted , that the method general u for any solid of revol tion , if the generating line is supposed to be straightened before the peripheries are applied ; in t he which case, area can be found for the curved surface only when the plane surface aggregated from the appli e d peripheries turns out to be one whose dimensions can be found . T hus , if the ordinate varies as the square of the arc m m easured fro the vertex, the plane equivalent is a semi 2 7rs r s i s parabola, and the area is /3 , where length of the

i r x . rotat ng arc , and is the ma imum or end ordinate L E CT UR E I I I

m u rr t n Co po s ite a n d c o n c en mo tio n s . Co mpo s itio of rectilin ear a n d a rallel mo tio n s p .

In generation by composite motions, if the remaining motions are unaltered , then , according as the velocity of one, or more, is altered , we usually obtain magnitudes differing not only in kind but also in q uantity, or at least differing in position every time . T hus , suppose the straight line A B is carried along the straight line A0 by a uniform parallel motion , and at the same time a point M descends uniformly in A B ; or suppose that, while AO descends with a uniform

mo parallel tion , it cuts A B also m moving unifor ly and to the right .

F m co m rom otions of this kind , F i 1 g . 3 . o s ite co n p in the former case, and c u rr en t in the latter, the straight line A M may be produced. h Again , if in the previous example, whilst t e motion of the straight line A B remains the same as before with respect

e n t to its v locity , but the uniform motio of the poin M , or ’ 48 B AR R O WS GE OME TR I CA L L E CT UR E S

of the straight line AC, is altered in velocity, so that indeed

t A the point M now comes to the point i , or O cuts A B in a t , , there is described by this motion another straigh line

f . An, in a dif erent position from the first F m urther, if once more, while the motion of A B re ains

' o the same, instead of the uniform moti n of the point M,

f t o or o the straigh line AO, we substitute a m tion that is called uniformly accelerated ; from such c o mposite or con

e current motion is produc d the parabolic line AMX, or in A Y another case the line n , according as the accelerated motion is supposed to be one thing or an o ther in degree .

x c o n In these e amples , it is seen that composite and current motio ns come to the same thing in the en d ; but frequently the generatio n of magnitudes is not so e asily

‘ e to be ex hibited by one of these methods as by the oth r .

Thus suppo se that a straight lin e A B is uniformly rotated

’ o round A , and at the same time the p int M , starting from A , is carried alo ng A B by a continuo us and uniform motion ;

co m o s ite o i fro m this p motion is pr duced a certa n line, namely

ral o Arc/zimedes i n t b e x the Spi f , wh ch can o e plained satis

l co n cu rren ce o o . O n the fac t o ri y by any of m ti ns other hand , if a straight line BA is rotate d with uniform mo tion about a

e t A0 centre B , and at the sam time a straigh line is moved in a parallelwise manner uniformly along A B , the continuous AG intersections of BA , , so moving, form a certain line, usually called the Qu adra trix and the generation o f this line is not so clearly shown or explained by any strictly s o - called composition of motion .

Magnitudes c an be compounde d and also decomposed in L E C T UR E I I I innumerable ways ; but it is impossible to take account of

me i r all of these, so we shall only discuss so mpo tant cases , such as are considered to be of most service and the more S easily explained . uch especially are those that are com

o f m n pounded rectilinear and parallel otions , or rectili ear and rotary motions , or of several rotary motions ; preference being given to those in which the constituent simple motions M o r . t are all, at least some of them , uniform oreover, here is not any magnitude that cannot be considered to have been

F o r a generated by rectilinear motions alone . every line th t lies in a plane can be generated by the motion of a straight line parallel to itself, and the motion of a point along it ; every surface by the motion of a plane parallel to itself and

an the motion of a line in it (that is , y line on a curved surface can be generated by rectilinear motions) in the same way solids , which are generated by surfaces , can be made to depend on rectilinear motions . I will only consider the generation of lines lying in one plane by rectilinear and parallel motions ; for indeed there is not one that cannot be produced by the parallel motion of a straight line together with that of a point carried alo n g * it ; but the motions must be combined together as the special nature of the line demands .

F o r n insta ce , suppose that a straight line ZA is always m AY oved along the straight line parallelwise, by any

n motio , uniform or variable , increasing or decreasing

I n o t e r o rds Barro s t at e s t at e v e r an e c u rve h as a Cart i h w , w h y pl e s an e u at io n re fe rre d t o e t e r o i u e o r re c t an u ar c o o rdin at e s e t it i q , i h bl q g l ; y s do u btfu l whe t he r he fu lly re c o gn iz e s t hat a ll t h e p ro pe rtie s o f t he c u rve c an f t he e u atio b e o bt ain e d ro m q n . ’ 50 B A R R O WS GE OME TR I CA L L E C T UR E S

. or alternating in velocity, according to any imaginable ratio ; and that any po mt M in it is moved in suc h a way that the motion o f the point is proportional to l t he m motion of the straight line , throughout any the sa e intervals of time ; then there will be certainly a straight line generated by these motions . e

Fo r , since we always have

AB z AC BM: Cu AB z MX AM: Xu , , or , , AC (M X being drawn parallel to ), it follows that the points

A , M , p. are in one straight line .

Bu t if, instead , these motions are comparable with one another in such fashion that, given any line D , then the

d if rectangle , contained by the ference between the line D and B M the distance traversed and B M itself, always bears the same ratio to the square on A B (the distance traversed by the line AZ in the same time) then a circle or an ellipse

c is des ribed a circle, if the supposed ratio is one of equality w and the angle ZAY is a right angle , and an ellipse other ise ; and in these there will be one diameter, situated in the line

w m c AZ in its first position , and dra n fro A in the dire tion

m . D. of Z, and this dia eter will be equal to L E CT UR E I I I

t he m If, however , otions are such that the rectangle con

” t ain e d by the sum of the lines D and B M an d B M itself bears always the same ratio to the square on the line A B , a hyper l bola will be produced ; a rectangu ar or equilateral hyperbola , if the assigned ratio is one of equality and the angle ZAY is

a right angle ; if otherwise, of another kind , according to the quality O f the assigned ratio ; of these the transverse diameter will be eq ual to D , being situated in ZA when occupying its first position , and being measured in a direction Opposite to Z ; and the parameter is given by the given ratio .

B u t if the rectangle contained by D and B M bears always m the sa e ratio to the square on A B , it is evident that a

c parabolic line is produced , of whi h the parameter is easily found from the given straight line D and the quantity of the assigned ratio .

Also in the first case of these, if the transverse motion

AY i along is supposed to be un form , the descending motion along AZ will also be uniform ; in the second and third AY cases, if the motion along is uniform , the descending

‘ motion along AZ will be c ontinually increasing and t he

t e i same thing being supposed in h last case, in wh ch the parabola is produced , the point M has its velocity increased uniformly .

c an In a similar manner, any other line be conceived to

Bu t be produced by such a composition of motion . we shall come across these some time or other as we go along ; let us see whether anything useful in mathematics can be

Obtained from a supposed generation of lines in this way . R ’ ME TR I E T R E S 52 B A R O WS GE O CA I . L C U

B u t for the sake of simplicity and c learnes s let us suppose

a o f o f the th t one the two motions , say that line preserving m m m parallelis , is always unifor ; and let us strive to ake out what general properties of the generated lines ari se from the general differences with regard to the velocity of the

x . other ; let us try, I say, but in the ne t lecture

NOTE We here see the reason for Barrow considering time as in de e d t hat the independent variable ; he states , , , the con s t ru c tio n s can be effected , no matter what is the motion of the line preserving parallelism but for the sake of simplic ity and clearness he decides t o take this motion as uniform for o f m m this the consideration ti e is necessary . At the sa e B fo r time it is to be noted that arrow, except the first case of the straight line , is unable to explicitly describe the m velocity of the point M , and uses a geo etrical condition as the law of the locus ; in other words , he gives the pure m geo etry equivalent O f the Cartesian equation . In later s lectures, we hall find that he still further neglects the use T of time as the independent variable . his , as has been

. du e explained already , is to the fact that the first five ’ c t er r B le tures were added af wa ds . In arrow s original de a sign, the independent variable is length along one of his T axes . his length is, it is true; divided into equal parts ; m but that is the only way , a subsidiary one , in which ti e e enters his investigations ; and even so , the modern id a of ” steps along a line , used in teaching beginners , is a better ’ analogue to Barrow s metho d that that whi c h is given by a ’ comparison with flu x io n s .

’ 54 B A R R O WS GE OME TR I CA L L E C T UR E S

arc r within the , and if produced , falls enti ely without the curve . This property was separately proved for the circle by

E fo r uclid , for the conic sections by Apollonius , cylinders

r n by S e e u s .

4. Curves of this sort are convex or c oncave to the same parts throughout This is the same as saying that a straight line only cuts the curve in t wo points nor does it differ from the definition

o f of hollow, as given by Archimedes at the beginning

c his book on the sphere and the ylinder .

5. All straight lines parallel to the genetrix cut the curve

and any one cuts the curve in one point only .

T o his was pr ved , separately, for the parabola and the

fo r hyperbola by Apollonius , and the sections of the

' c o n c o ids by Archimedes .

6 S x . imilarly all parallels to the directri cut the curve ,

and in one point only .

Apollonius proved this for the sections of the cone .

v m 7. All cho rds of the cur e eet the genetrix and all parallels to it, produced if necessary . Apollonius thought it worth while to prove this property f separately o r the parabola and the hyperbola .

8 S m c . i ilarly, all straight lines tou hing the curve, with o n e exception (see meet the same parallels .

T his also Apollonius showed for the conic sec tions in separate theorems L E CT UR E I V

x 9 . Also any straight lines cutting the genetri will also

cut the curve . Apollonius went to a very great deal of trouble to prove

a property of this kind in the case of the conic sections.

1 0 . i. e S . traight lines applied to the directrix, parallels

‘ to the genetrix, have a ratio to one another (when the less is the antecedent) which is less than the ratio of the corre

e . s po n din g spaces traversed by the moving straight line ; i.

S the ratio of the versed ines of the curve , the less to the

S . greater , is less than the ratio of the ines

T it his property of circles and other curves , will be found , is everywhere proved separately for each kind .

NOTE All the preceding properties are deduced in a very simple ’ manner from one diagram ; and Barrow s contin u al claim that his method not only simplifies but generaliz es the work of the early geometers is substantiated . T he full proof of the next property is given , to illustrate Barrow ’s way o f using one of the variants of the method of ex haustions .

1 1 Le t u . us suppose that a straight line TMS to ches a

i. e given curve at a point M ( . it does not cut the curve) ; and A let the tangent meet AZ in T, and through M let PMG be dra wn p parallel to AY. I may say that the velocity of the descending

c point, describing the urve by its

Fi 2 0, motion , which it has at the point g . 56 BA R R O WS GE OME TR I CA L L E CT UR E S

o f c e ic ontact M , is equal to the v locity by wh h the

n m straight line TP would be described u ifor ly, in the m m sa e ti e as the straight line 1 AZ AC is carried along or PM (or, A what comes to the same thing, I say t hat the velocity o f the de scending point at M has the same ratio to the velocity with which the straight line AZ is mo v ing as the straight line TP has to the straight line PM).

F o r , let us take anywhere in the tangent TM any point

K , and through it draw the straight line KG , meeting the

O AY . T curve in and the parallels , PG in D , G hen , since the tangent TM is supposed to be described by a twofold

m e uniform otion , partly of the straight lin TZ carried AC T parallelwise along or PM , and partly of the point T TZ descending from along and since, of these motions , AC the one along or P M is common with , or the same as ,

i s an d TZ that by which the curve described ; since, when is in the position KG , AZ will be in the same position as

r well ; therefore, when the point descending f om T is at K , 0 the point descending from A will be at , the intersection of the curve with KG (for the straight lin e KG cannot Cu t the curve in any other point, as has already been shown) .

0 e n Also the point is below K , because the tangent lies

t ti ely outside the curve . Now, if the point 0 is supposed t to be above the point of contact, owards T, since in that c 0G a ase is less than G K , it is clear th t the velocity of the e d scending point , by which the curve is described , at the L E CT UR E I V point 0 is less than the velocity of the uniformly descend d S i ing motion , by which the tangent is pro uced ; nce the

former, always increasing, in the same time (namely that represented by G M) traverses a smaller space than the latter which does not increase at all and as this goes on continu [

he 0G ally, the former describes t straight line whilst the [

. O n latter describes the straight line KG the other hand , if the point K is below the point of contact towards the end 0 8, since G is then greater than KG , it is clear that the velocity of the descending point, by which the curve is pro du c e d 0 as , at the point , in the same way before, is greater than the velocity of the uniformly descending motion , by which the tangent is described ; for the former motion , continually decreasing during the time represented by G M , traverses a greater space than the latter, which does not decrease at all, but keeping constant, describes indeed the

d ri space KG . Hence , since the velocity of the point e s c b ing the curve , at any point of the curve above the p o int of f contact towards A , is less than the velocity o the motion for TP and at any point of t he curve below the point of contact is greater than it it follows that it is ex actly equal to it at the point M .

1 2 Th i . e converse of the preceding theorem s als o true .

1 . F 3 rom these two theorems , it follows at once that curves of this kind are touched by any one straight line in one point only .

T E his , separately, uclid proved for the circle, Apollonius

for the conic sections, and others for other curves . 58 B A RR O WS GE OME TR I CA L LECTURES

F o ou t a rom this meth d , then , there comes an advant ge not to be despised , that by the one piece of work proposi

tions are proved concerning tangents in several cases .

1 The c 4. velo ities of the descending point at any two assigned points of a curve have to one another the ratio rec iprocally compounded from the ratios of the lines applied

i e to the straight line AZ from these points ( . . parallels to

AY) and the interc epts by the tan gents a t these points f measured rom the said applied lines . In other words , the ratio of the velocities is equal to the ratio of the intercepts divided by the ratio of the applied lines .

1 . e n 5 Incidentally , I here give a g neral solutio by my

method , and one quite easy to follow, of that problem

u which Galileo made much of, and on which he spent m ch T trouble, about which orricelli said that he found it most T skilful and ingenious . orricelli thus enunciates it (for the enunciation of Galileo is not at hand) : G i iven any parabola with vertex A , it is req u red to find some point above it , from which if a heavy body falls to A , and from A , with the velocity thus attained , is turned hori

z o n t all . y, then the body will describe the parabola

NO TE

B fo r arrow gives a very easy construction the point, and a short simple proof ; further his construction is perfec tly ” fo an m x i general r y curve of the for y , where n is a pos tive integer . 6 I T m d 1 , 7 . hese are two ingenious ethods for etermining the ratio of the abscissa to the s u btangent . Barrow remarks that the theorems will be proved more

m c v . geo etri ally later, so that they need not be gi en here LE CTURE [V

1 8 . A circle, an ellipse, or any returning curve of this - b kind , being supposed to be generated y this method , then the point describing any one of them must have an infinite velocity at the point of return .

F o r instance, let a quadrant APMbe so generated ; then

M _ since the tangent, T , is parallel to the diameter AZ, and only meets it at an infinite distance , therefore the velocity at M is to the velocity o f the uniform motion of AZ parallel

to itself as an infinite straight line is to P M ; hence, the velocity at M must certainly be infinite . And indeed it will be so for all curves of this kind ; but for others which are gradually continued to infinity (such as the parabola or the hyperbola) the velocity of the descending point at any point on the curve is finite .

L eaving this, let us go on to those other properties of the given curves which have to be expounded .

NOT E

B It is to be observed that, although arrow usually draws his figures with the applied lines at right angles to his directrix AZ, his proofs equally serve if the applied lines are

. T oblique, in all cases when not otherwise stated hat is , analytically, his axes may be oblique or rectangular . Having mentioned this point, since my purpose is largely with B ’ arrow s work on the gradient of the tangent, I shall always B draw the applied lines at right angles , as arrow does ex c ept in the few isolated cases where Barrowhas intentionally drawn them oblique . L E CT UR E V

Ta n e t e t/ze u o u r s . u e l k F ri/zer proper ties f c ve g n s . C m s i ’ ’ mu a n d ls . a x i m u C do za . N o r ma M m m m Zin es y ini .

n 1 . The angles made with the applied lines by the ta gents at different points o f a curve are unequal and those are less which are nearer to the point A , the vertex .

2 . Hence it may be taken as a general theorem that tan gents cut one another between the applied lines drawn t at right angles o AZ through the points of contact .

“ The PTM X N. 3 . angle is greater than the angle Q

4. Applied lines nearer to the vertex (and therefore also any straight lines parallel to other directions) c u t the curve at a greater angle than those more remote .

’ 6 2 B A RR O WS GE OM E TR TCA L LE CTURES

6 AY t o . A straight line , moving parallel itself, traverses

u v any c rve, either concave or con ex to the same parts ,

e u with uniform motion (that is to say, it passes over q al m parts of the curves in equal times), and si ultaneously any point is carried , also uniformly, along AY from A ; by the point moving in this manner there is generated

' a curve AMZ , of which it is required to find the tangent at any point M .

F i 2 g . 7.

T o Y do this , draw M P parallel to A to cut the curve A PX in P ; through P draw the straight line P E touching the curve A PX through M draw MH parallel to P E 3 take any

o mt M R o ff p R in H, and draw S parallel to P M ; mark RS

rc . as so that M R : R8 a A P P M (i e . the one uniform T M motion is to the other) ; j oin MS . hen S will touch the curve AMZ .

F o r an d , if any point Z be taken in this curve, through K it Z be drawn parallel to M P, cutting the curve APX in X , MH the tangent at P in E, the parallel to it in H , and MS in S 3 then , if (i), the point Z is above M towards A , P E arc PX

: arc PA P E arc PA arc PX . LECTURE V

— — B u t are PA : arc PX P M : PM XZ = PM E H XZ

fi — arc PA z arc PX P M : Z H EX PM: Z H;

arc : : : : Z H. hence, PA P E PM Z H or arc PA PM P E

Bu t arc PA z PM MR : RS MH z KH PE z KH;

: KH PE z Z H KH Z H. P E , and

EZ Z o r Now, since X P M E H , the point H is outside the curve AZM hence K is outside the curve AZM . S m B i ilarly [ arrow gives it in full] , (ii), if the point Z is below the point M , K will be outside the curve ; therefore the whole straight line KMKS lies outside the curve, and thus touches it at M .

After this digression we will return to other properties of the curve .

7. Any parallel to the tangent TM, through a point E i T . F . directly below , will meet the curve [ g

8 . x If E lies between the point T and the verte A , the

parallel to the tangent will cut the curve twice .

Apolloniu s was hard put to it to prove these two theorems

n for the conic sectio s .

9 . If any two lines are equally inclined to the curve,

v i. e. these straight lines di erge outwardly , they will meet one another when produced towards the parts to which the curve is concave .

I O . If a straight line is perpendicular to a curve, and along it a definite length H M is taken , then H M is the shortest of all straight lines that can be drawn to the curve from the point H . 64 B A R R O WS GE OME TR Z CA L L E CT UR E S

1 1 c C . It follows that the cir le, with entre H , drawn through M , touches the curve .

’ 1 2 s t rai ht i in e s . Conversely , if H M is the shortest of all g that can be drawn from H to the curve, then H M will be perpendicular to the curve .

h 1 3 . If H M is the shortest of all straight lines t at can be

u drawn from H , and if the straight line TMis perpendic lar

TM . to it, then touches the curve

1 . F h 4 urther, a line which is nearer to H M is shorter t an one which is more remote .

1 5. Hence it follows that any circle described with centre H meets the curve in one point only on either side of M that is, it does not cut the curve in more than two points altogether .

1 6 . If two straight lines are parallel to a perpendicular, the nearer of these will fall more nearly at right angles to

o n e the curve than the more remote .

1 7. If from any point in the perpendicular H M , two straight lines are drawn to the curve, the nearer will fall more nearly at right angles to the curve than the one more remote .

8 m 1 . Hence it is evident that by o ving away from the perpendicular, the obliquity of the incident lines with the curve increases, until that which touches the

e curv is reached ; this , the tangent, is the most oblique of all .

1 . n 9 If the point H is taken withi the curve, and if, of LE CT URE I7

1 t all lines drawn from to meet the curve, H M is the least then H M will be perpendicular to the curve or th e tangent

MT.

2 0 if . Also, H M is the greatest of all straight lines drawn to meet the curve, then H M will be perpendicular to the curve .

2 1 . MT Hence, if is perpendicular to H M , whether the

u latter is a maximum or a minim m , it will to uch the curve .

2 2 . It follows that, if a straight line is not perpendicular

to the curve, no greatest or least can be taken in it .

2 3 . If H M is the least of the lines drawn to the curve , and I any point I is taken in it ; then Mwill be a minimum .

2 t 4. If H M is the grea est of the lines drawn to meet the

I MH ‘ IM curve , and any point is taken on produced ; then will be a maximum .

F r o the rest, the more detailed determination of the greatest and least lines to a curve depends on the special

. nature of . the curve in question

[Barrow concludes these preliminary five lectures with the remark

B u t I must say that it seems to me to be wrong , and

c not in complete ac ord with the rules of logic , to ascribe

things which are applicable to a whole class, and which

u come from a common origin , to certain partic lar cases , or m to derive the from a more limited source .

NOTE

The x s ne t lecture is the first of the even, as originally 66 BARR O W’S GE OME TRJC A L LE CTURE S

m n O c . designed , that were to for a suppleme t to the pti s

' Barrow begins thu s

I have previously proved a number o f general propert ies

c c u c m m of urves of ontinuous curvat re, dedu ing the fro a certain mode of construction common to all ; and especially m those properties , as I entioned , that had been proved by the Ancient Geometers fo r the spec ial curves which they m investigated . Now it see s that I shall not be displeasing,

v m if I shall add to them se eral others ( ore abstruse indeed ,

u but not altogether uninteresting or seless) ; these will be,

c as usual, demonstrated as oncisely as possible, yet by the same reasoning as before ; this method seems to be in the highest degree scientific , for it not only brings out the truth

c of the con lusions , but opens the springs from which they

The matters we are going to consider are chiefly con cerned with

reed ro m Me lo a Ms o me (i) An investigation of tangents , f f l u rden o ca lcu latio n f , adapted alike for investigation and proof (by deducing the more complex and less easily seen from the more simple and well known)

(ii) The ready determination of the dimensions of many magnitudes by the help of tangents which have been

drawn . T hese matters seem not only to be somewhat difficult m of G m c o pared with other parts eo etry , but also they have

not been as yet wholly taken up and ex haustively treated

(as the other parts have) at the least Mey li ar/e n o t as y et

' a c rdin lo Mzs meMo d been co n s idered c o g Ma t 1 k n ow. S o We LE CTURE V ]

c will straightway ta kle the subject, proving as a preliminary

m l Ee c certain lem as , which we shall see wil of onsiderable

m e use in demonstrating or clearly and briefly what follows .

“ The original opening paragraph to the seven lectures t he The probably started with words , matters we are going a to consider, the first p ragraph being afterwards added to connect up the first five lectures . Barrow indicates that his s u bject is going to be the con sideration of tangents in distinction to the other parts of c geometry , whi h had been already fairly thoroughly treated ; he probably alludes to the work on areas and volumes by m the method of exhaustions and the ethod of indivisibles , of which some account has been given in the I ntroduction o f m when he treats areas and volumes hi self, he intends to use the work which , by that time , he has done on F the properties of tangents . rom this we see the reason why the necessity arose for his two theorems on the inverse nature of differentiation and integration . T hat Barrow himself knew the importance of what he t o e .was about do is perf ctly evident from the next para T graph . He distinctly says that angents had been investi gated neither thoro u ghly nor in general ; also he claims dis t in ctl Ma t to Me k es t o it is k n o wled e a n d belie lzis y , f g f, meMo d is q u ite o rigin al. He further suggests that it will be found a distinct improvement on anything that had been done before . In other words , he himself claims that he is inventing a new thing, and prepares to write a short text fi i im l o n I n n t e s a C . book the alculus And he succeeds , no matter whether the style is not one that commended itself D to his contemporaries, or whether the work of escartes had re volutioniz ed mathematical thought ; he succeeds in

man - his task . In exactly the same way as the who put the

- m eye of a needle in its point invented the sewing achine . Barro w sets out with being able to draw a tangent to a circle and to a hyperbola whose asymptotes are either given or can be easily found, and the fact that a straight line is n everywhere its own tangent . Whenever a constructio is not immediately forthcoming from the method of description 68 B AR R O W’S GE OME TR TCA L LECTURE S

u m of the curve in hand , he us ally has some eans of drawing a hyperbola to touch the curve at any given point ; he finds m o f b the asy ptotes the hyper ola, and thus draws the tangent An ti to it this is also a tangent to the c urve r e qu ired . ary c as a cally, for any urve whose equation is y f ( ), he uses as a x d first approximation the hyperbola y ( + ) . He then gives a construction fo r the tangent to the f m ‘ c general paraboli orm , and akes use of these urves as f u u c . a xiliary urves As will be o nd later, he proves that 1 n x is an approximation to (1 leading to the ” m i f » x d dx n x T f theore that ) , then y / H y / hus he ounds the whole o f his work on exac tly the same principles as o n the c u d m those which cal ulus always is fo nde , na ely, on the approximation to the binomial theorem ; and he does it in a way that do es n o t call f o r a ny dis cu s s io n of Me co n en e o Me k zn o mial o r a n o Mer s eries ver c . g f y _ F o r the benefit o f those who are beginners in mathe matic al o u t history , it may not be of place if I here reiterate the warning of the P reface (for P refaces are so often left unread) that Barrow k n ew n o t/Zing of Me Calcu lu s n o t a tio n of ’ B eo etr a l L eik n iz . m ic u arrow s work is g , as far as his p blished lectures go ; the nearest approac h to the calculus of to - day “ a e m L X is given in the and ethod at the end of ecture , o f Again , with regard to the differentiation the com f m plicated unction , given as a speci en at the end of this u B vol me, I do not say that arrow ever tackled such a thing . B What I do urge , however, is that arrow could have done m c u c so , if he had co e a ross s ch a fun tion in his own work . M u y argument, absol tely conclusive I think , is that I have B ’ been able to do so, using nothing but arrow s theorems

and methods .

’ 70 BA RR O WS GE OME TR I CA L LECTURES

f a that the part o the straight line so dr wn , intercepted

m o f B between the ar s a given angle A C, may be equal to a

” given straight line T.

Fo r , if the hyperbola (of the preceding article) is first described , and if with centre D, and a radius equal to the

c PO n given straight line T, a ircle Q is described , cutti g the

DC c m hyperbola in O, and is produ ed to cut the ar s of t he angle in M and N ; then it follows that M N D0 T.

L e t v 4. ABC be a given angle and D a gi en point ; and let the line 08 0 be such that , if through D any straight w line DN is drawn, the length M N intercepted bet een the arms of the angle bears always the same ratio (s ay X: Y) to the length MO intercepted between the arm BC and the c urve OBO then OBO will be a hyperbola .

5. If M0 is taken on the other side of the straight line

BC m . , the ethod of proof is the same

6 — . F E R C I f B IN EN E . a straight line Q divides the angle

ABC u , and thro gh the point D are drawn , in any manner, XY B two straight lines M N , , cutting the straight line Q in

0 c 0 the points , P , of whi h is the nearer to B5 then

MNz MO XY: XP M . B 7 oreover, if several straight lines Q, BG ABC if m divide the angle , and fro the point D the straight

DN . BC B BG O lines , DY are drawn, cutting , Q, , BA in M , ,

DN B ’ E, N and X , V , F, Y , being the nearer to , then

N E : M 0 YF : VX. m 8 . F ro what has gone before, it is also evident that

o f t wo through B (in one . directions) a straight line can be LECTURE v V l so drawn that the segments intercepted on lines drawn through D between the constr u cted line and BC shall have to the segments intercepted between BA and BC a ratio that is less than a given ratio .

. ABC 9 Again , suppose a given angle and a given point 000 u D ; also let the line be s ch that , if through D any D0 m straight line is drawn , cutting the ar s of the angle DM NO Y in M , N , then always bears to a given ratio (X say); then the line 000will be a hyperbola .

1 0 . A straight line I D being given in position , and a

D N c point D fixed in it , let N be a urve such that , if any N d point G is taken in I D , and a straight line G is rawn

IK v parallel to a straight line gi en in position , and if two

m k n an d straight lines whose lengths are and are take , if

DG x GN we put , and y , there is the constant relation z m x mx b t D o y y / hen NNwill be a hyperb la .

2 1 1 . m x mx o If the equation is y y / , the same hyper DM bola is obtained , only G must be taken in instead of

z x —m mx k DO. If, however , the equation is y y / , then G must be beyond M and the hyperbola conjugate to the former is obtained .

1 2 BDF DNN . If is a given triangle and the line is such BD that, if any straight line R N is drawn parallel to , cutting DF O DN the lines B F, , NN in the points R , G , N , and is j oined ; and if DN is then always a mean proportional e between R N and NG then the line DNNis a hyp rbola . DNN 1 3 . If I D is a straight line given in position ; and is

n t he a curve suc h that, if any point G is take in I D , and ’ 72 B A RR O WS GE OME TR Z CA L LECTURES

e t h i straight line GN is drawn parall l to IK, a s raig t line g ven

m r in position , and if straight lines whose lengths are g , , ,

‘ x N t he n er are taken and if we put DG , and G y , th e is 2 a c onstant relation x y + g x my mx /r then the line

ONN will be a hyperbola .

2 — x + x + m mx b If the equation is y g y / , then the same hyperbola is obtained , but the points G must then be taken between B and M (B being the point where the curve cuts the straight line I D) ; and if the points G are assigned to

. Bu t other positions , the signs of the equation vary it is not opportune to go into them at present .

1 Two 4. straight lines D B , DA , are given in position , and along the line D B a straight line CX is carried parallel to

BA also, by turning round the point D as a centre, a straight m line DY oves so that , if it cuts BA in X , there is always the same ratio between the lines B E and CD (eq ual to the ratio of some assigned length R to D B , say) ; then , if D E

DNN . cuts CXin N , the line is a parabola

S t bu t demo n s t ra ted wiM Gregory Vincent gave this ,

u s ro lix it . la bo rio p y , if I remember rightly We add the following

1 CX 5. If, other things remaining the same, and DY are moved in such a way that n o w B E and BC are always

m BD z R w in the sa e ratio ( , say) ; their intersections ill give a parabola also .

1 6 . If, with other things remaining the same, the straight

n o w line CX is not carried parallel to BA , but to some

- n an d the a o ther straight line D H , given in positio if r tio LECTURE V ] of B E to DC is always equal to the ratio of D B to R ; then

’ the intersections N will lie on a hyperbola .

1 M 7. oreover, other things remaining the same as in e CX the prec ding, if now moves in such a way that B E BD always bears the same ratio to BC ( R , say) ; the inter sections in this case will also lie on a hyperbola .

1 8 Le t i . two straight lines D B , DA be g ven in position , and a point D fixed in D B ; and let the line ONN be such

t tha , if any straight line GN is drawn parallel to BA , and

r DC two straight lines whose lengths are g , are taken , and ,

N x if r x x DNN G are called , y ; and y y g ; then the line will be a hyperbola .

If x —r x m a , however, the equation is y y g , we ust t ke

r m D E , and B0 g ( easured below the line D B) ; the proof is the same as before .

1 . Le t 9 two straight lines D B , BA be given in position ; and let the straight line FXmove parallel to D B , and let DY pass through the fixed point D , so that the ratio of B E to

B F is always equal to an assigned ratio, say D B to R ; then N the intersections of the straight lines DY, G lie on a

straight line .

2 0 Bu t . if, other things remaining the same, some other

0 we point is taken in A B , which take as the origin of reckon ing , so that the ratio B E to CF is always eq ual to the ratio

D B to R thenthe intersections will lie on a hyperbola .

i 2 1 M le t t he . oreover, other things remaining the same,

e straight line FXnow move not parallel to D B , but to anoth r

i i H t x in stra ght l ne D , so tha , a fi ed point 0 being taken 74 B A R R O W’ S GE OME TR JCA L LECTURES

BA , the ratio B E to CF is always equal to an assigned ratio (say DB to m) ; then the intersections will again lie on a

‘ hyperbola .

2 2 . L e t DYY A D B be a triangle and a line such that, if m any straight line PM is drawn parallel to D B , eeting A B PY w P 2 2 in M , is al ays equal to J( M DB ) ; then the line DYY is a hyperbola .

C O R . YS If is the tangent to the hyperbola DYY, then 2‘ 2 PM : PY PA : PS .

2 . I f o m 3 , ther things re aining the same, we have now PY J(PM2 DB2 ) then the line DYYis again a hyperbola

C O R . YS e If is the tang nt to the hyperbola, then

2 2 PM : PY PAz PS .

2 is a 4. If A D B triangle, having the angle A D B a right

u u angle, and the c rve CCD is s ch that, if any straight line

c u F EG is drawn parallel to D B , tting the sides of the triangle

t e in F, E , and the curve in G , h rectangle contained by EF and EG is equal to the squ are on DB ; then the curve CGD

- o f m . is an ellipse, which the se i axes are A D , AC

R L e t CO . GT be a tangent to the ellipse, then

2 2 : E z EF EG A AT.

2 i 5. If DTH is any rect lineal angle, and A is a fixed point

D m V GG u in T , one of its ar s ; if also the curve is s ch that,

u TD when any straight line EFG is drawn perpendic lar to ,

' D TH V E cutting the lines T , , GG in the points , F, G , and

E t o n V AF is joined , G is always equal A F ; the the line GG i wll be a hyperbola . LECTURE V ]

n ar NOTE . If a straight li e FQ is drawn perpendicul TH R E D to , and Q is taken eq ual to A (along T ), and G R

is joined ; then G R will be perpendicular to the hyperbola

VGG. T ake this on trust from me , if you will, or work it out

for yourself ; I will waste no words over it .

6 2 . Le t t wo AC BD straight lines , , , intersecting in X , be

in K given position ; then if, when any straight line P L is

AC BD a n d drawn parallel to BA , cutting , , in the points P PL K , is always equal to B K ; then the line ALL will be a

straight line .

L 2 7. e t a straight line AX be given in position and a fixed DNN point D ; also let the line be such that, if through D

any straight line M N is drawn , cutting AX in M , and the line D N DM DN N in N , the rectangle contained by and is eq ual to a given square, say the square on Z ; then the line DNN

will be circular . T hus you will see that not only a straight line and a

c hyperbola, but also a straight line and a ircle, each in its

own way, are reciprocal lines the one of the other .

Bu t here, although we have not yet finished our preliminary

m . theore s , we will pause for a while

NOTE

It has already been noted in the Introduction that the B fo r proofs which arrow gives these theorems, even in the c case where he uses an algebrai al equation , are more or less

“ “ l u lu m e x i e — I ard t in t a a o in n Ad C a c g . h ly h k h t B rr w t e ds by ma an a s is b u t he . ly , y 76 B A RR O W’ S GE OM E TR TCA L L E CT UR E S of a strictly geometrical c harac ter ; the terms of his equa tions are kept in the second degree, and translated into c re tangles to finish the proofs . In this connection , note the remark on page 1 97 to the effect that I cannot imagine B e eo metrica l rela ti o n arrow ev r using a g , in which the ex o f The m pressions are the fourth degree . asy ptotes of the hyperbolae are in every c ase found ; and this points to his intention of using these c urves as auxiliary curves for draw ing tangents ; c ases of this use will be noted as we c ome across them but the fact that the number of c ases is small arab o lifo rms m suggests that the p , which he uses ore

' e u m m re fr q ently, were to so e extent the outco e of his c a n sear hes rather than first intentio . The great point to notice, however, in this the first of the n i e T origi ally des gned sev n lectures , is that the idea of ime t n m he i. e c as independent variable, . the ki e ati al nature of u m his hypotheses , is neglected in favo r of either a geo etrical c The or an algebrai al relation , as the law of the locus . dis

- i tinction is also fa rly sharply defined . B u arrow , thro ghout the theorems of this lecture , gives u t o fig res for the particular cases that correspond rectangular, but his proofs apply to obliq u e axes as well (in that he does B not make any use of the right angle) . oth the figures and the proofs have been omitted i n order to save s pac e ; all m the ore so, as this lecture has hardly any direct bearing t he on infinitesimal calculus . The proofs tend to show that Barrow had not advanced very far in Cartesian analysis at least he had not reached the point of diagnosing a hyperbola by the fact that the terms of the second degree in its equation have real factors or perhaps he does not think his readers will be acquainted t with this method of obtaining the as ympo e s .

78 BARR O W’ S GE OM E TR TCA L L E CT UR E S

l — An in dex or ex po n en t is also defined thu s z I f the number f m f m m m O . o ter s ro the first ter , A say, t any other ter , F i t he say, is N (exclud ng first term in the count), then N

o f e . L n is the index or exponent the t rm F ater, a other m ex o n en t eaning is attached to the word p ; thus , if A , B , C are the general ordinates (or the radu vectores) of u three c rves , so related that B is always a mean of the m m a sa e order , say the third out of six e ns , between A and C so that the indic es or exponents of B and C are 3 and 7 respectively ; then is calle d the exponent of the c urve B B B . T here is no difficulty in rec ogniz ing which

‘ e B n m fo r meaning is int nded , as arrow uses / the latter

o f N M. T he c c case , instead / onne tion with the ordinary idea of indi c es will appear in the note to 1 6 of this lecture L and that to ect . IX , 4.

I L t . e t le t A , B be two quan ities of which A is the greater some third quantity X be taken then A X B X A B .

' F r : o X A : X A: A X B : he n c e e t c . , since X B , + + B ; ,

2 . Le t e L thre points , , M , N be taken in a straight line

N G z

F i . 6 1 g .

L YZ and between the points , M let any point E be taken , and another point C outside LN (towards Z) ; let EG be cut in F so that G E EF NL LM then F will fall between

M and Z .

F o r : NL : ML GE z EF : N E M E ) N E FE ,

F E M E .

L e t DC i 3 . BA , be parallel stra ght lines, and also BD, G P 5

through the point B draw two straight lines BT, BS, L E CT UR E V T] cutting GP in L and K ; then I assert that D8 DT KG : LG

F o r the ratio KG LG LG is compounded of KG G B and G B ,

: PS PT : PL that is, of P K and ,

i 62 : DS DT : F . . that is , of D B and D B , g and hence is equal to the ratio DT : DS .

Le t BDT 4. be a triangle, and let any two straight lines

BS , B R , drawn through B , meet any straight line G P drawn BD L parallel to the base in the points , K then I say that = KG . TD RD : S D.

[Barrow ’s proof by drawing parallels through L is rather c long and omplicated ; the following short proof, using a

d. different subsidiary construction , is therefore substitute Co n stru ctio n —D GXYZ raw , as in the figures , parallel T D to . S LG 8 2 GX x z T S s o ince , and

L T R . T hence R D S D G . KG S

LG . T R + R D . KG KG . TS + KG . S D

K T D . G . ]

. Bu t S 5 , if the points R , are not situated on the same D * m side of the point , then by a si ilar argument, — . RD . . LG TD KL . KG TD RD : S D

is is a mis t a e : D s o u d b e P an d LG T D —KL RD Th k h l , s ho u ld b e m i u o u in s i n a b g s g . ’ 8O B A R R O WS GE O/WE TR TCA L L E CT UR E S

6. L e t there be four eq u in u me rable series of quantities

c in ontinued proportion (such as you see written below),

of which both the first antecedents and the last c o n s e

i. e u quents are proportional to one another ( . A M , ,

: ( ) n c m and F I S the , the four in any the same olu n

being taken, they will also be proportional to one another 8 = (say , for instance, D P

“ V O 7T I , : : : P:

Fo r A BV Co D71“ E F0" n, , , , P:

are in continued proportion . 0 SP s R ( 68 and 7 , , , 1

T A . Ma. F0 68 herefore, since n and 9 , it is plain that

7: S : 8 P : 7r D P, and hence that D .

The conclusion applies equally to either Arithmetical or * Geometrical proportionality .

L t . e CD 7 A B , be parallel lines ; BD and let a straight line , given

o u t . in position , these Now let h FBF 5 t e curves E B E, be so related that, if any straight line PG is draw n parallel to BD, P F is always E a mean proportional of the same

I f t he s e rie s are arit me t ic a ro o rtio n a s t e n h l p p l , h

' - - ' ' A ' - B V C o D 1r E F 0 I pt , + , I , + , l p, + are arit me tic a ro o rtio n a s M N P 6 R S h l p p l o. + B+ 6 + , + , ¢ + } , , ‘ n d t he c o n dit io n fo r the rs t an t e c e de n t s an d the as t c o n s e u e n t s mu s t b e a fi , l q

- ' u M an d F - I a + S in t is c as e D rr p o r 0 - 5 + ¢ ; h + 3 + , is is n o t o f v e r re at im o rt an c e as in t he fo o in t e o re ms Th y g p , , ll w g h , B rro a are n t o n c o n s iders e o me t ric a ro o rtio n a s F o r it a w pp ly ly g l p p l . ar h me t ic a ro o rt lo n alit t he in e s A GB HEL mu s t b e ara e c u rve s i e l p p y , l , p ll l , . . E mu s t b e c o n s t a n t an d t e n the c u rve s KEK an d FBF are r G , h , pa alle l

i e RK is c o n s t an t . c u rve s , . . L E CT UR E V I ]

e order b tween PG and P E ; then , through any point E of a CD the given line E B E, let H E be drawn par llel to A B and and let another curve K EK be such that, if any straight line BD L QL is drawn also parallel to , cutting EB E in I , H E in , D S K FB F in R, and C in , then Q is always a mean propor t io n al L I of the same order between Q and Q (of that order , m I say, of which P F was a ean proportional between PG and FBF P E) then I assert that the lines , K E K are similar that is , the ordinates such as QR, QK bear a constant ratio to

t he w . one another, ratio hich P F bears to P E T his follows from the lemma j ust proved , as will be clear by considering the argument below .

R I Q8, Q , Q are proportionals L K I Q , SQ , Q such that S L PG , P F, P E Q Q PG P E ,

I : I : P E , P E , P E Q Q P E P E ;

QR z QK PF : PE

N . H OTE Instead of the straight lines A B , E, CD, we can substitute any parallels we please , even curved lines .

8 . A PB Again , let Q , ASG D be two straight lines meeting in

A , and let BD be a straight line given in position ; also let F E B E , FB be two curves so related that, if any straight line

PG is drawn parallel to BD, P F is always a mean propor t io n al m of the sa e order between PG and P E then , having AE n joined , let a other curve K E K be such that, if any

LI t o BD in straight line Q is drawn parallel , cutting AE L,

in FBF EB I I , and in R , QK is always a mean proportional 6 ’ 8 2 B A R R O WS GE OME TR JCA L LE CT UR E S

between QL and QI of the same order as‘ P F was between

PG and P E ; then the line FBF is similar to the line K EK ; R K that is , Q Q P F P E , in every position .

F o r NOTE . the three straight lines A B, A H , A D we can

u s bstitute any three analogous lines .

‘

. c is an d EBE 9 Also, if AG B is a circle whose entre D ; ,

c F B F are two other curves su h that, if any straight line DG D f is drawn through D, F is always a mean proportional o m DG the sa e order between and D E then , through E , let a

c cir le H E , with centre D , be drawn ; and let another curve

u c K EK be drawn s h that, if any straight line DL is drawn t L hrough D to meet the circle H E in , and E B E in I , DKis DL always a mean proportional between and D I , of the same order as DF was between DC and D E ; then the curves i i. e : DK DF F . FB , K EK will be s milar, D R D E, in every position . [DL meets FBF in R. ]

E NOT . In this case also, instead of the circles , we may substitute any two parallel or two analogous lines .

1 0 L . B FBF astly , let AG G, EB E be any two lines ; and let m be another line so related to the that, if any straight line m DG be drawn in any anner through a fixed point D , DF is always a mean proportional of the same order between DC

le t i e . c and D E then HEL be a line analogous to AG B ( . su h D L that, if through D a straight line S is drawn in any DS DL w manner, and are al ays in the same proportion) ;

t he c DL lastly, let line K EK be su h that, if be drawn in any

c EBE DK manner, utting in I , is always a mean proportional L m F between D and D I , of the sa e order as D was between LE CT UR E V T]

D F F G and D E then , in this case also, B is analogous to the line K EK .

1 1 . L e t C A B , , D , E , F be a series of quantities in Arith P metical rogression ; and , two terms D , F being taken in it, m let the nu ber of terms from A to D (excluding A) be N , and the number of terms from A to F (excluding A) be M

~ : ~ then A D A F N M .

Fo r , suppose the common difference to be X then = = . X A t X and F A i M. ; therefore

1 2 . Hence , if there are two series of this kind , and in

each a pair of terms , corresponding to one another in order, are taken (say D, F in the first, and P , R in the second) then where the series are

C O . A , B , , D , E , F and M , N , , P , Q, R

F o r each of these ratios is equal to that which the

numbers , N , M , as found in the preceding article , bear to one another .

T hese numbers N , M , in any series of proportionals , we shall usually call the ex po n en ts or in dices of the terms to which they apply ; and where we use these letters in what follows , we shall always understand them to have this meaning .

1 . L e t 3 any quantities A , B , C, D , E , F be a series in

P an d Arithmetical rogression ; let there be another set, m G P equal in nu ber, in eometrical rogression, starting ’ 84 BA R R O WS GE OME TR TCA L L E CT UR E S

m m e with the sa e ter A ; thus, [suppos the two series

A B o D E . , , , , , F

0 P . A , M , N , , Q

Also let the second term B of the Arithmetical P ro gre s s io h be not greater than M the second term of the Geometrical P rogression then any term in the Geometrical P rogression is greater than the term in the Arithmetical P rogression * that corresponds to it .

hence,

i. e . O > D; hence,

e . i. P > E; and so on , as far as we please .

1 . C P ro 4 Hence, if A , B , , D , E, F are in Arithmetical

re s s io n 0 Ge P g , and A , M , N , , P, Q are in ometrical rogression , and the last term F is not less than the last term Q (the number of terms in the two series being equal) then B is greater than M .

F o r , if we say that B is not greater than M , then F must be less than Q whic h is contrary to the hypothesis .

1 . 5. Also , with the same data , the penultimate E is greater than the penultimate P .

Al e b raic all z—I f t he s erie s are a a d a ad a d an d g y , + , + , + 3 , e a a r a r a r3 we ave W et e r r I t he fac t t at 1 , , , h , h h 2 , h ( I —r 1 —r3 are all o s it ive e n c e it fo o s t at (1 ( )( ), p ; h ll w h 2 2 a ——a r 2 a r a a r3 a r a r a a r4 a r a r3 I , + + , + + , e n c e s in c e a r is n o t e s s t han a + d it fo o s t at H , l , ll w h z 2 a 2 d a a r z a d o r a r . + ( + ) + , o r a r3 a d > + 3 , ’ an d s o o n in e x ac t e uiv a e n c e wit Barro s ro o f. , q l h w p

86 B A R R O W’S GE OM E TR I CA L L E CT UR E S

d e if form (the hyperbola of Apollonius in e d, P F is the

m c m simple geo etri al mean between PG and P E , but so e

c m if m curve of hyperboli for of a different kind, P F is a ean of some other kind) ; and it is plain from the last theorem S b that the line q g is an asymptote to the line V FF, corre i m s po n d n g to the points of the sa e kind of mean .

n o t c I do know whether this is of very mu h use, but indeed it was an incidental corollary for us to have obtained it here .

1 L e t i u . BC B 9 three stra ght lines BA , , Q be drawn thro gh i a given fixed point B to a straight line AC, fixed in posit on C let then , in Q produced some point D be taken as a fixed it point . T hen is possible to draw through B a straight B if line (B R say), on either side of Q, such that, any straight

h h DN c line is drawn t roug D,as , the part inter epted between BQ and B R is less than the part intercepted between BA and BC.

2 0 L e t . D , E , F be three points in a straight line DZ, and

o f an le c let F be the vertex a rectilineal g BFC, of whi h the arms are c ut by a straight line DBC let a straight line EG be drawn t hro u gh E ; then it is p ossible to draw through

' u c E a straight line EH s h that, on any straight line DK

drawn through the point D, the intercept between the lines EG, E H is less than the intercept between the lines

F0, F8 .

2 1 L e t BO c BA B . a straight line tou h a curve in ; and

let the length B0 of the straight line be equal to the arc

i f in arc BA of the curve ; then , any point K is taken the L E C T UR E ‘ V / f

I S o me d BA and K0 j , the straight line KO is greater than the arc KA.

2 2 . L o f Hence, if any two points K , , on the same side

n n the poi t of contact , , are take , one in the curve and the other in the tangent, and KL is joined ; then KL+ LO

are KA.

NOTE

From 1 6 we have the followi n g geometrical theorem. S u AB d1 vide d t wo ppose a line is into parts at C, and CB e that the part is divided at D , E , F, G , H in the figur on ' ' ' ' ' D E F G H u the left, and at , , , , in the fig re on the right, AC AE m c P ro so that , A D , , A F, AG , A H , A B are in Arith eti al ' ' ' ' re s s io n AC AD AE AE AG are Ge o g , and , , , , , A H , A B in P AD' ' metrical rogression ; then A D , A H A H

— -4 —— 0 4 l A C D E F G H B A B

E if xpressing this theorem algebraically , we see that, a a x t he m o f AC and CB , and nu ber of points section

‘ n —I r t h ic between C and B is , and F is the arithmet al , ' “ and F the r t h geometri c al me an point between C an d ' F c m B , then the relation A F A be o es a + r ax /n a

i . e . 1 r n x 1 r n + ( / ) ( where . ' c m m l m ff Also, as CB be o es s a ler and s aller, the di erence FF m m m s in c e it c a beco es s aller and s aller, is learly less th n CB FF' AC ; that is , the ratio / can be made less than any u assigned number by taking the ratio CB/AC small eno gh . c e Hen e the alg braical inequality tends to an equality, when x is taken smaller and s maller . rx n x Again , if we put / y , we have and then 1 ” e < w or i n , where n r ; and again the inequality tends to be c ome an i if eq ual ty y is taken small enough . 8 8 B A R R O W’S GE OME TR TCA L L E CT UR E S

2 man n x x x Naturally, a who uses the notatio for does Bu not state such a theorem about fraction al indices . t the approx imation to the binomial expansion is there just the m — W u m . e sa e, tho gh concealed under a geo etrical form may as well say that the ancient geometers did not know s in the expansion for (A B), when they used it in the form ’ P m B u of tole y s theorem , as say that arrow was naware of M e if n this . or over, further corroborative evidence is eeded , ) we have it in 1 8 . Here Barrow states that a line 96954 is m c c an asy ptote to a urve V FF, the distan e between the curve an d m m u its asy ptote, eas red along a line parallel to a fixed '

o f . direction , being the equivalent our FF in the work above Now the condition for an asymptote is that this distan c e should continually dec rease and finally become evanescent “ as we proceed to infinity . L e t us try to reason out the manner in which Barrow came to the conclusion that his m line was an asy ptote to his curve .

The figure O n the left is the o n e u sed by Barrow fo r 1 8 ; v f m PE as P mo es away ro V , and PG both increase withou t im o f l it , but it can readily be seen that the ratio EC to P E c T c an steadily de reases . his is all that be gathered from u far c an m m the fig re ; and , as as I see, it ust have been fro this that Barrow argued that the distance Fgh dec reased im m t m without l it and ulti a ely beca e evanescent . In other

‘ c faCt words , he appre iated the that the inequality tended to so m become an equality when was taken s all enough . Assum m c h ing that y suggestion is correct, the very fa t t at he has

‘ rec ogniz ed this impo rt an t truth leads him into a trap ; fo r b is n o t a n as m to te c i l n e c b e . i V F . Me bg q y p to the urve F , as we n T understa d an asymptote at the present day . aking the

L E CT U R E V I I I

Co n s tr u c tio n of ta ngen ts ky mea n s of a u x ilia ry cu rves of ' n n er n ti tio o a u o r a wizick Me ta ng e ts a re k n o w . D ifi e a n f s m d t a l u i a len ts . ifiier en ce. A n a ly ic eq v

Tr u ly I seem to myself (and perhaps also to you) to have S * done what that wise man , the coffer, ridiculed, namely, to have built very large gates to a very small c ity . F o r up

b u t to the present, we have done nothing else struggle

u e . towards the real thing , j st a littl nearer Now let us get

1 : . We assume the following

two OMO TMT u If lines , to ch one another, the angles between them (OMT) are less than any rec tilineal angle ;

v and con ersely, if two lines contain angles which are less

‘ h u c than any rectilineal angle, t ey to h one another (or at

u c least, they will be eq ivalent to lines that tou h) .

The m reason for this state ent has already been discussed , unless I a m mistaken .

2 if i PMP c . OMO Hence, any third l ne tou h two unes ,

i o OMO TMT u n e . TMT, the lines , w ll also to ch another

rat e s t he At e n ian i o s o e r Z e n o c a e d him S c u rra At t ic u s S o c . , h ph l ph ll the At tic S c o ffer, L E CT UR E V1 ] !

Le t 3 . a straight line FA touch a curve FX in F and let

FE be a straight line given in position ; also let EY, EZ be I two curves such that , if any straight line L is drawn parallel EF FX EY EZ L to , cutting FA in G and the curves , , in I , K , KL respectively , the intercept is always equal to the intercept

EY EZ . IG then the curves , touch one another

4. Again , let a straight line A F touch a curve AX , and let

EY EZ t wo , be curves such that , if through a fixed point D DL any straight line is drawn , cutting the given lines as in t he e m KL I preceding th ore , is always equal to G; then the

EY EZ u . curves , will to ch one another

T he c two foregoing con lusions are also true , and can be shown to be true by a like reasoning , if it is given I that G, KL always bear to one another any the same

Le t TEI i YFN 5. be a stra ght line , and let two curves , Z GO

e be so related that , if any straight line E FG is drawn parall l

r v to A B , a st aight line gi en in position , E EF the intercepts G, always bear to one another the same ratio ; also let

T Z O o f the straight line G touch G , one the curves in G and meet IE in T ;

TF u then , being joined , will to ch the F i 8 0 . curve YFN g .

F o r IL , let a straight line , parallel to A B , be drawn , cutting t he given line s as shown in the figure . T hen \ I L : I N > I0 : IN EG z EF > IN IK; ’ 9 2 B A R R O WS GE OME TR TCA L LE CT UR E S

* hence the line TF falls altogether without the curve YFN.

erwis e IL: IK CL NK OM . It can be shown that ; hence , h L GO FN . by 4 above , since C , touch , , FK also touc

6 M XEM YFN Z CD . oreover, if three curves , , are so related that , if any straight line EFG is drawn parallel to a line given E EF in position , G and are always in the same ratio ; also E T let the tangents ET, GT to the curves X M, Z GO meet in ;

‘

YFN. then TF, being j oined , will touch the curve I

L e t Y 7. D be a given point, and let XEM, FN be two curves so related that , if through D any straight line D EF is DF b “ drawn , the straight lines D E , always ear to one another m ' the sa e ratio and let the straight line F8 touch YFN , one F of the curves, and let ER be parallel to 8 ; then E R will touch the curve XEM. 1

8 . Let YFN Z O XEM, , G be three curves such that, if any DEFG straight line is drawn through a given point D , the

T he re aso n in g fo r t he s e t he o re ms give n by Barro w is n o t c o n c lu s iv e ; i de e n ds t o o mu c o n t he ac c ide n t o f t he ure drawn A t o u he t p h fig . l h gh s t at e s in a n o t e aft e r 6 t at h e a wa s c o o s e s t he s im e s t c as e s it is § h l y h pl , de s irable that t he s e S imple c as e s s ho u ld b e c apable o f be in g ge n erali z e d m I i it o u t a t e rin t he ar u e n t . n a ddit o n his ro o f o f is o n an d w h l g g , p 4 l g c o m ic at e d an d n e c e s s it at e s as a re imin ar e mma t he t e o re m o f pl , p l y l h 2 t is is a s o ro ve d in a far fro m s im m n r u L e c t . 0 e an e a t o VII , ; h l p pl , l h gh im e ro o f o f it S t i arr t e re is a v e r s . o mu s t ave had s o me o o d h y pl p _ ll B w h g re as o n fo r pro vin g t he s e t wo the o re ms by the me t ho d o f the van is hin g an e o f I fo r he s tat e s t at t e s e t e o re ms are s e t fo rt s o t at n o n e gl , h h h h , h f t he fo o in t e o re ms ma b e am ere d it do u t s H e s e e ms t o o ll w g h y h p w h b . do u t t he ri o u r o f the me t o d u s e d in o f w ic ave iv e n th e fu b g h 5, h h I h g ll pro o f fo r th e s ake o f e x emplifi c at io n ; t o ge t her wit h t he alt ern at ive pro o f — T h e ro o f o f t he att e r fo o s e as i t u s — S in c e FA ie s o by 4. p l ll w ly h l wh lly i f t he c u rv e FX EZ ie s o o n o n e s de o o n o n e s ide o f EY. , l wh lly

I f x F x are t re e c u rve s s u c t at t ere is a c o n y f ( ) , y ( ) , y h h h h s t an t re at io n + B F + C o ere A B C o t he t an e n t s at l A f , wh + + , g avin e u a a s c is s ae are c o n c u rre n t po in t s h g q l b . I H o mo t he tic c u rve s hav e paralle l t an ge n ts ; t his t hee re m an d the n e x t i n t f o f 6 re t he o ar e u va e s o t o s e . a p l q l h 5,

’ 94 BA R R O I/V S GE OME TR TCA L L E CTUR E S

ET E t ‘ T touches the curve X Mat E , and cu s V D in then , if D8 F8 is taken eq ual to DT and F8 is joined , will touch the * curve YFN at F.

Le t an i EF u y stra ght line I N be drawn parallel to , c tting the given lines as shown then

TP z PM TP z PI > TD : DE; also 8 P : PK D8 : DF ;

TP . 8 P : PM. PK > TD . 8 D : DE . DF

B t m u TS TD. , since D is the iddle point of , S D TP S P

m : D . TD S D . T S D PN hence all the ore, PM P K P M ,

. P . P PM K PM N or P K PN. T herefore the whole line FS lies outside the curve YFN .

E c in i . XEM o c NOT If the line is a straight line, and so TEI YFN dent with , the curve is the ordinary hyperbola, of which the centre is T and the asymptotes are T8 and a line through T that is parallel to EF.

1 0 . XEM YFN v Again , let D be a point , and , two cur es so EF related that, if any straight line is drawn through D, the rectangle contained by D E , DF is always equal to a certain

T h e an alyt ic al e qu ivale n t o f t his is I f i a fu n c ti n o f x an d z A t e n dz dx d dx s o . y , /y , h / y / z v x I t is t u s t at arro Als o t he s pe c ial c as e gi e s I / . h h B w s t art s ff u his re al wo rk o n the di e ren t ial c alc lu s . L E CT UR E V T] ! square (say the square o n Z) ; and let a straight line ER

: touch one curve XEM; then the tangent to the other is found thus

D hav 1n raw D P perpendicular to ER and , g made D P Z

: C CF F8 i Z D B , bisect D B at ; j oin and draw at r ght angles to CF; then F8 will touch the curve YFN .

1 1 . Le t Y e XEM and FN be two curv s such that, if any straight line F E is drawn parallel to a straight line given in

i v posit on , it is always equal to a gi en length ; also let a

u c straight line F8 to ch the urve YFN then R E , being drawn * F8 XEM. parallel to , will touch the curve

1 2 Le t u c . XEM be any c rve, whi h a straight line ER tou c hes at E ; also let YFN be another curve so related to

if a the former that , a straight line D EF is drawn in any m nner

c EF through a given point D , the inter ept is always equal to some given length Z ; then the tangent to this c urve is drawn thus T ake D H Z (along D EF), and through H draw A H

u a FG perpendicular to D H , meeting ER in B ; thro gh F dr w L LFS parallel to A B ; take G G B ; then , being drawn , will touch the curve YFN. I N XEM OTE . If is supposed to be a straight line, and so

c C coin ide with ER , then YFN is the ordinary true onchoid , or the Conchoid of Nic o me de s hence the tangent to this m c urve has been deter ined by a certain general reasoning .

— — T he an a tic a e u iv a e n t is I f is a fu n c t io n o f x an d z - c ly l q l y , y i , e re c i z dx x s a c o n s t an t t e n d d d . wh , h / y / F o r t he ro o f o f t is t e o re m arro a ain u s e s an au x i iar c u rve T p h h , B w g l y ,

me t . n a he e r o a de termin e d in L e c t . ly hyp b l VI , 9 96 B A R R O WS GE OME TR JCA L L E C T UR E S

1 . Le t an 3 VA be a straight line, and B E I y curve and let

DFG be another line such that, if any straight line PFE is

ar drawn parallel to a line given in position, the sq u e on P E is eq ual to the square on P F with the square on a given straight line Z also let the straight line TE to uch the curve 2 2 PE : PF PT : P8 n F D E I ; let ; the 8, being joined, will * touch the curve D FG .

T v L 2 2 [ his is pro ed by the use of ect . VI , , and corollary . ]

1 L e t n o w 4. other things be supposed the same, but let the square on P E together with the square on Z be equal

I 2 2 ‘ to the square on P F ; also let PE PF PT z PS ; then F8

L will touch the curve GFG. ]

F r B 2 o L . [ this, arrow uses ect VI, 5 3 , and its corollary . ]

1 Le t o 5. AFB , CGD be two curves having a comm n axis

- A D, so related to one another that , if any straight line E EG is drawn perpendicular to AD , cutting the lines drawn as

o sh wn, the sum of the squares on EF and EG is equal to the square on a given straight line Z ; also let the straight z 2 u o n e EF z EG line FR to ch A FB, of the curves ; and let ET ER then GT, being joined , will also touch the curve

CGD . I

F r B L 2 . o . [ this , arrow uses ect VI , 4, and its corollary ]

2 2 2 The an a t ic a e u iva e n t is —If is an fu n c t io n o f x an d 2 a ly l q l y y , y , = i iffe re n t fo rm i m h e n z dz dx . d dx o r n a d e re a s s o e c o n s t an t t . wh , / y y / ; , 2 2 2 2 : / il The art ic u ar c as e e n if z a a t e n dz dx . a . ( ) , h / y y ) p l , wh = x is t he e u iva e n t o f L e c t . 2 2 . y , q l VI , 2 2 1 A s imilar re s u lt fo r the c as e o f V (v + a ) o r 2 2 2 — 2 = / - / b a e n The c as e o f z a o r 4 a x . S in c e T R are t o e t I 4 ( y ) ( ) , k it e i f F o n o ppo s s de s o G .

96 B A R R O WS GE OME TR Z CA L L E C T UR E S

L 1 . e t c 3 VA be a straight line , and B E I any urve and let

DFG be another line such that, if any straight line PFE is drawn parallel to a line given in position , the square on P E is equal to the sq uare on P F with the square on a given straight line Z also let the straight line TE to uch the curve

2 2 PE : PF PT : PS F8 D E I ; let ; then , being joined, will * touch the curve DFG .

T L 2 2 [ his is proved by the use of ect . VI , , and corollary . ]

1 L e t m 4. other things be supposed the sa e, but now let

” the square on P E together with the square on Z be equal

2 2 to the square on P F ; also let PE PF PT : P8 then F8

’ ‘ will touch the curve GFG. I

F o r B 2 L . [ this, arrow uses ect VI, 5 3 , and its corollary . ]

1 Le t o 5. A FB, CGD be two curves having a comm n axis

A D, so related to one another that, if any straight line E EG is drawn perpendicular to A D , cutting the lines drawn as

o sh wn , the sum of the squares on EF and EG is equal to the square on a given straight line Z ; also let the straight z 2 u EF z EG line FR to ch A FB, one of the curves ; and let ET E R then GT, being j oined , will also touch the curve

CGD . 1

B L 2 . Fo r . [ this , arrow uses ect VI , 4, and its corollary]

2 2 2 The an a t ic a e u iva e n t is —I f is an fu n c t io n o f x an d 2 a ly l q l y y , y , = f rm ere a is s o me c o n s t an t t h e n z . dz dx . d dx o r in a diffe re n t o wh , / y y / ; , = 2 = 2 2 T h e art ic u ar c as e e n if z a t e n dz dx . a . ) , h / y y ) p l , wh = x is t he e uiv a e n t o f L e c t . 2 2 . y , q l VI , 2 3 A s imilar re s u lt fo r the c as e o f V (v + a ) o r = — 2 z — z The c as e o f z o r a/ a x . S in c e R are t o b e t a e n I y ) ( ) T , k it e i f F o n o ppo s s de s o G . LE C T UR E V T] !

1 L a x 6 . e t o f A FB be any curve, which A D is the is and

D B is applied to A D ; also let VCO be another curve so

n t related that, if any straight line Z F is draw hrough some

fixed point Z in the axis A D, and through F a straight line DBC EC Z F EFG is drawn parallel to , is equal to also let

FQ be at right angles to the curve A FB ; along A D, in the

o Z E R Z E RG i o me d directi n , take Q then , be ng j , will be

perpendicular to the curve VGC.

Fo r Ba u s e L . [ this, rrow makes of the hyperbola of ect VI , x 2 . 5, as the au iliary curve he did not give a proof of the “ theorem of that article, but left it to the

1 Le t 7. D P be a straight line, and DRS , DYXtwo curves

so related that, if any straight line R EY is drawn parallel to

o a straight line D B, given in p sition , cutting D P in E and

D DY : the curves RS, X in R , Y , the ratio RY DY is always

eq ual to the ratio DY : EY; also let the straight line R F

touch the curve DRS at R . It is req uired to draw the

tangent to the curve DYXat Y . S DY uppose the line O is such that, if any straight line G0 FP D is drawn parallel to D B , cutting the lines FR , , YO 0 D0 CC DD in the points G , P , , and is joined , DO PO then the curve DYO touches the curve DYXat Y .

B i 1 2 u t n L . , ect VI , , it has already been shown that the curve DYO is a hyperbola ; let Y8 touch the hyperbola ; then YS also touches the c u rve DYX.

NOTE . If the curve DRS is a circle, and the angle GDB DYX is a right angle , the curve is the ordinary Cissoid and thus the tangent to it (together with many other c urves

e m similarly produced) is d ter ined . ’ 9 8 B A R R O WS GE OME TR TCA L - LE C T UR E S

1 8 . Le t V D B , K be two lines given in position , and let

DYX h tra the curve be suc that, if from the point D any s ight DYH i line is drawn , cutt ng the straight line B K in H and the c DYX urve in Y , the chord DY is always equal to the straight line BH; it is required to draw the straight line touching DY the curve Xin Y . With centre D and radius D B describe the circle BRS ;

c let Y ER , drawn parallel to K B , meet the cir le in R ; join

. T : D R hen RY: YD Y D D E ; hence, the straight line touc hing the curve DYX can be found by the prec eding * proposition .

1 . L e t 9 DB , B K be two straight lines given in position ; also let BXX be a curve such that, if from a point D any BXX straight line is drawn , cutting B K in H and the curve H in X , HX is always equal to B it is required to draw the

t o B tangent the curve XX at X . S uppose that DYYis a curve such that DY is always e q u al to BH (such as we considered in the previous proposition) ,

le t Y o and T t uch this curve in Y, and cut B K in R then let NXN c m RT the hyperbola be des ribed , with asy ptotes R B , ,

. to pass through X ; then the hyperbola NXN touches t he

e BXX . T l curv at X hus , if the tangent to the hyperbo a,

X8 is drawn X8 will also to uch the curve BXX.

t rifle d However, we seem to have with this succession of th e orems quite long enough for one time ; we wfll leave off for a while .

I n I it is n o t e s s e n t ia t at t he c u rv e RS s o u d as s t ro u 7 , l h h l p h gh D ; t i s t at e me n t is u s t i a e he n c e h s j fi bl .

’ 1 00 BA R R O WS GE OME TR I CA L LE CT UR E S

common tangent KF. Le t now the third curve be aGo ; LO IM Lo IM 0 then , since K N , and K N , therefore, 0 2 KN is always equal to ; hence, by 5 3 above, the curves z Go Z GO LG , also touch , and is the common tangent .

Therefore the construction holds in this case also . I believe the omission of the theorem was intenti onal ; and I argue from it that Barrow himself was not completely satisfied with the theorems of 3 , 4, thus corroborating T my footnote . his theore m is of course equivalent to the B ff . di erential of a sum arrow may have thought it evident, or he may have considered it t o be an immediate con sequence o t his differential triangle ; but I prefer to think o f that he considered it as a corollary the theorem of 5. F o r this may be given analytically as

- w d r - n w r + n r z u d x . d dx n r If y ( ) , then / y / + ( ) dz dx e / . If we tak one curve a straight line, and this straight k the x d dx k . d dx line as a is, we have ( y )/ y / , or the sub “ ” tangents o f all multiple curves have the same subta ngent t h as the o riginal curve . Hence e co nstructions for the ” “ ” tangents to s u m and difference curves follow immediately

L et be an two c u rves o wk ick E is tak en AAA , B B B y , f F as a co mmo n a x is ; let N A B be any s tra zgk t lin e applied er en dic u la r to EF let Me t a n en ts AS c u t in p p ; g , B R , EF tak e a b e u al to res ectivel a n d als o let 8, R ; A , B q NA , N B p y , a NC NA+ NB, n d N D N A N B . o in a b in ters ectin in T an d draw TV er en dic u lar j 8 , R g , p p to EF.

“ “ T/zen will to uM t s u m cu rve 000 an d will TC , V D ‘ to u clz Me difi é ren ce cu rve DD D . B ’ It seems rather strange, considering arrow s usual custom , 1 2 that he fails to point out that, in , if the curve XEMis a c r YFN cir le passing th ough D , the curve is the Cardioid or o one of the other Limag n s . T he final words of t he lecture seem to indicate that Barrow now int ends to proceed to what he considers to an d be the really important part of his work ; , in truth , u this is what the next lecture will be fo nd to be . L E CT U R E I X

Ta ngen ts to cu rves fo rmed by a riMmetica l a n d geo met rica l l t mea n s . a ra b l r u r s o k erbo lic an d el i ic P o if o ms . C ve f yp p o rm D feren t atio n o a a t o n a l o wer ro du cts a n d . i c i f if f fi p , p u o tien ts q .

We will now proceed along the path upon which we

started .

1 Le t . the straight lines A B, V D be parallel to one another and let D B cut them in a given position also let the lines FBF E B E , pass through B , being so related that, if any

is straight line PG drawn parallel to D B , P F is always an arithmetical mean of the same order between PG an d P E ;

i F . g . 94

c v R and let the straight line B8 tou h the cur e . eq uired to

‘ draw the tangent at B to the curve FBF. ’ 1 02 B AR R O WS GE OME TR I CA L' LE CT UR E S

Le t Le c . 1 2 the numbers N , M (as explained in t VII , )

T n t ak E DT be the exponents of the proportio als P F, P E ; ,

: such that N M D8 DT, and join TB; then TB touches the line FBF

F o r i c u , in whatever posit on the line PG is drawn, tting the given lines as shown in the figure, we have = FG z EG N: M D8 : DT LG z KG. h EG LG Hence , since by ypothesis KG , KG ; and thus it has been shown that the straight line TB falls wholly * without the curve F BF.

2 . m All other things remaining the sa e, let now P F be m a geometrical ean between PG and P E (namely, the mean of the same order as in the former case of the arithmetical

“ mean) then the same straight line touches the curve FBF. F o r the lines constructed in this way fromarithmetical and geometrical means touch one another ; hence, since BT

' s h o the r. touche the one curve , it will also touc the j — E x a mple Suppose P F is the third of six means between

D8 : DT PG and P E , then M 7, and N 3 ; and

N o t e t at in t is c as e FG EG LG : KG an d t u s t is is a ar h h , ; h h p t ic u lar c as e o f t he c u rve s in L e c t . t he an a tic a e u iva e n t is VIII , 5 ; ly l q l —— d dx d(a l by ) b y / . ’ A n a t ic a e u iva e n t —I f is a n fu n c t io n o f x an d a’n = a n - T r t e n 1 ly l q l y y , y , h d dx e n 2 a dz dx r n . / ( / ) y / , wh y

’ 1 04 BA R R O WS GE OME TR TCA L L E C T UR E S

4. Note that, if the line EB E is supposed to be straight , then the line FBF is one of the parabolas or curves o f” t he

m ara bo li o rms T for of a parabola p herefore, that which

“ is usually held to b e known concerning these curves

n (deduced by calculation and verified by a sort of inductio ,

a but not, as far as I am aware, proved geometric lly) flows

m c fro an immensely more fruitful source, one whi h covers * innumerable curves of other kinds .

e 5. Henc the following deductions are evident I f l TD is a straight ine and E EE, FFF are two curves s o PEF related that, when straight lines are drawn parallel to

in d BD, a st raight line given position , the or inates P E vary E8 FT as the squares on the ordinates P F and if , , straight

e lines drawn from the nds of the same common ordinate, 2 TP . Bu t t he n touch these curves ; then 8P , if ordi ate TP S S P P E varies as the cube of P F, then if P E varies

o w TP 4S P as the fourth p er of P F, then ; and so on in the same manner to infinity “?

6 . Again , let AG B be a circle, with centre D and radius

FBF o D B , and let E B E , be two lines passing thr ugh B, so — e related to one another that, when any straight lin DC is D drawn through D, F is always an arithmetical mean of the same order between DC and D E ; also let the straight

S e e n o t e at t he e n d o f t his le c t u re ; Whe re it is s h o wn t hat t his t he o re m is e qu ivale n t t o a r igo ro u s de mo n s t rat io n o f t he me tho d fo r diffe re n t iat in g r f h v ria e a frac t io n al po we o t e a bl . This is a s pe c ial c as e o f t he p re c e din g th e o re m ; fo r P F is t he s imple ge o me t ric al me an be t we e n P E a n d a de fin it e le n gt h P G ; o r t he s e c o n d o f i m ric m n : t wo t he t rd o f t re e e t c . e o e t a e a s e t e e n P E an d P G t u , h h , , g l b w ; h s 2 3 2 P E P E . P G P F P E P G e t c . is e n a e s Barro w t o diffe re n t iat , , , Th bl e a n o e r o r ro o t O f x e n he c an diffe re n t iat e x it s e f y p w f ( ), wh f ( ) l . L E C T UR E TX line BO touch the curve E B E at B ; required to draw the tangent at B to the curve FBF. * T x his (demonstrated generally, to a certain e tent, in 8 Le . ct VII I , ) will here be specially shown to follow more

T : clearly and completely from the method ab o ve . hus

L e t t o BC 8 DQ be perpendicular D B , cutting in ; and

T es let N : M D8 : DT j oin BT. hen BT touch the curve

FBF. t

7. Hence, other things remaining the same as before , if the straight line DF is always taken as a geometrical mean (of the same order as before) between DG and D E , the same straight line BT will to uch the curve FBF also .

Fo r the lines formed from arithmetical means and from

o ne geometrical means of the same order touch another, and have a common tangent .

F n 8 . urther, other things remaini g the same as in the

a n preceding proposition , let y point P be taken in the curve

FBF; then the straight line that touches the curve at this point can be determined by a similar plan .

Le t the straight line DF be drawn , cutting the curve E B E in E ; also draw DQ perpendicular to DC cutting E0 the

t o EY tangent E B E in X ; make DX DY N M ; j oin , and T draw FZ parallel to EY. hen FZ touches the curve FBF.

Hence , not only the tangents to innumerable spirals , but also those to a bo undless number of others of different kinds, can be determined q uite readily

T he ac t u al c o n s t ru c t io n fo r the as ympt o t e s o r t an ge n t t o t he au x iliary r o a is n o t iv e n hype b l g . ’ 1 Barro w pro ve s his c o n s t ru c tio n by t he u s e o f an au x iliary hype rbo la u s in L e c t . an d . g VI , § 4, VIII , 9 ’ 1 06 B A R R O WS GE OME TR I CA L L E CT UR E S

. n 9 He ce, it is clear that , if two lines E EE, FFF are so

e o m related that, wh n any straight line D EF is drawn fr a E8 fixed point D , D E varies as the square on DF and if ,

FT e s are the tangents to the curv s at the end E , F, meeting

D 2 08 . the line perpendicular to D EF in S , T; then T * Bu t B DT 3 0 o n . , if D E varies as the cube of F, 8 and so

. Le t TB T I O V D, be two straight lines meeting in , and BD i let a straight line , g ven in position , fall across them

Fi . 1 00 g .

also let the lines E B E, FBF pass through B and be such

li P BD that, if any straight ne C is drawn parallel to , PF is always an arithmetical mean of the same order between PG

“ R i and P E ; also let B R touch the curve EBE. equ red to

t o draw the tangent at B the c u rve FBF.

T e x o f aking numbers N , M to repr sent the e ponents P F,

. TD+ M . : . TD RD : S D P E, make N ( N) R D M , and join

‘ BS ; then BS touches the curve FBF. 1

As the t e o re ms o f 6 8 are o n th e o ar e u iva e n ts o f 1 h , 7 , , 9 ly p l q l , 2 t he u re s an d ro o fs are n o t ive n t e ir in c u s io n arro , 3 , 5, fig p g ; h l by B w s u e s t s t a t he was a are o f the fac t t at it the u s u a mo de rn n o t at io n gg h w h , w h l , = t a n r d6 dr . ¢ . / 4. T he fo rm o f t h e e qu at io n s u gge s t s lo garithmic diffe re n t iat io n s e e n o t e a t e n d o f t is e c t u re h l .

1 08 BA R R O WS GE OME TR TCA L L E C T UR E S

E x a m le — If p P F is a third of six means , or M N 3 3 then

3 4 i e . 7 3 4 . 7 : . D D TD : D D . D TD R TD R RO TD R S , S /( )

1 2 a n o . It is evident that , if y p int F whatever is taken on

n the line FBF, the ta gent at F can be drawn in a similar

. T P manner hus , through F draw the straight line Gpar allel to D B , cutting the curve E B E at E , and through E let E R be drawn touching the curve EB E at E then make

. : P N TP + (M RP S , T and join S F. hen S F will touch the curve FBF.

1 . i. e . 3 Note that, if E B E is a straight line ( coinciding

B ihfi n it e with the straight line B R), the line F F is one of an number of hyperbolas or curv es of hyperbolic form ; and we have therefore included in the o n e theorem a method of drawing tangents to these, together with innumerable others of different kinds .

1 . do 4 If, however, the points T, R not fall on the same B E F side of D (or P) , the tangent S to the curve B is drawn — — by making N . RD (M N) . TD M. TD R D 8 D.

1 5. Hence also the tangents to not only all elliptic curves (in the case when E B E is supposed to be a straight line t o coinciding with B R), but an innumerable number of

ff e other curves of di erent kinds , can be determin d by the one method .

m l — I f f E x a e f i. o f e . p PF is the ourth our means, M 5, 5 — 4 . and N 4 ; then S D TD . RD/( RD TD)

E it ha . e n s , . TD NOT If pp that N R D (M N) , then D8 L E CT UR E [X

o Fi t is infinite or BS is parallel to V D . O ther p i s may be

bu t . noticed , I leave them

1 6 C . Amongst innumerable other curves , the issoid and the whole class of cissoidal curves may be grouped together

- . F o r DS B by this method , let be a semi right angle ; and i let 8GB, S EE be two curves so related that, if any stra ght n line G E is draw parallel to BD, cutting the given lines

in F, P , PG , P F, P E are in continued proportion ; also let the straight line GT touch the c urve 8GB at G then the line touching the curve S EE is foun d by making — 2 TP S P z TP S P : R P ; i S EE. and , n every case, if R E is joined, R E touches h T e proof is easy from what has gone before .

8 B a li Now, if the curve G is a circle , and the angle of pp P cation , S G, is a right angle, then the curve S EE is the ordinary Cissoid or the Cissoid of D iocles ; otherwise it

h Bu t will be a cissoidal curve of some ot er kind . I only

n mention this in passing, and will not now detain you lo ger over it .

NOTE

T his le cture is remarkable for the important note of 4. B ’ In it, arrow calls his readers attention to the fact that he has given a method for drawing tangents to any o f the parabolas or parab o lifo rms ; and apparently he refers in a more or less depreciative words to the work of W llis , whilst claiming that his own work is a geometrical demon i s t rat o n . , and therefore rigorous If we take a line parallel i to PG , and DV , as the coord nate axes , and suppose them

M M ‘ N N e PF PG . r ctangular or oblique, then PE gives

' M W'“ N x a N N k x W . y , or y . , as the general e , quation to c the urve FBF. ’ I I O B AR R O WS GE OM E TR TCA L L E CT UR E S

d dx PT PF PT P x 8 PS PF M N . Also, y / / ( / ) ( / ) ( / ) y / ; e u — or, if the ax s are interchanged , the eq ation to the_ _ curve N/M d d c . x x is y , and then y / PF/PT Note particularly that the form suggests logarithmic dif

fe re n t iatio n .

The 6 theorem of is a particular case of this, in which m 1 i. e a N , . P F is the first of any number of e ns between 2 o f u k x PG and P E , and the equations the c rves are y . , 3 4 “ k x k x . . , . , etc (the parabolas as distinguished from “ the parab o lifo rms h c It seems strange , unless per aps it is to be as ribed to ’ B c arrow s dislike for even positive integral indi es , that he does n o t make a second note to the effect that if the curve E B E is a hyperbola whose asymptotes are V D and a line FBF h e rb o lifo rms parallel to PC, then the curves are the yp . F o r in m , from this particular case, a manner si ilar to the " a x r foregoing , it follows that if y , where is any positive rational , either greater or less than unity, then dy /dx Bu t Barrow probably intends the recip L V —I f r c l . o a theorem of ect I II , 9, to be used thus " 7' k . x L c . x 2 1 c . y , let /y ; then from e t VIII , 9, 1 z dz dx 1 d dx we have ( / ) / ( /y ) y / ; also from the above , = z x d dx dz /dx r . / hence y / I suggest that c u Barrow found out these constru tions by analysis , sing a e d doc letters such as and instead of y and , and that the form of the results suggests very stro n gly that he first lz s e u a tio n lo a riMmica ll ex pres s ed i q g y . B rs t ri o ro u s Anyway, arrow was the fi to give a g demon " f f c o f x s tratio n of the form of the dif erential coe fi ient , f a n ra tio n a l wlza tever . ar where r is y As as I am aware, it is the only proof that has ever been given , that does not c o f t involve the onsideration of convergence infini e series , m f m M or of limiting values , in so e or or other . oreover, c m he gives it in a form , whi h yields, as a converse theore ,

d dx r . dz dx the solution of the differential equation y / / , B w although , of course, this is not noted by arro , simply because he had not the notation . n 8 Agai , considering , which is only 3 with the w c constant distance bet een the parallels , PG , repla ed by DG DB the c onstant radius , , we see that, if is the initial BDG 9 line , and the angle is , and the angles between the

’ 1 1 2 B A RR O WS GE OM E TR TCA L LE CT UR E S

T 1 2 e hus , putting N , and M , we have for the g neral theorem of 1 2 the remarkably simple results

f GGG a re two cu rves a n d is a s tr a z k t lin e f , EEE , P EG g a lied er en dicu la r to a n a x is PRT a n d T are Me pp p p , G , ER ta n en ts to a n d Men g G GG EE E,

i I is a n o Mer cu rve s o relate d t o Me o Mer two ( ) f H H H ,

Ma t . Men i H is Me t a n en t t o . W Z P H P E PG ; , f g H H H , me etin Me ax is in l W i. e is a W P . W g , / P o u rM ro o r io n a l t o a n d t P + P P . f p p R T, P R , T (ii) If K K K is a n o Mer cu r ve s o rela ted to GGG a n d EE E Ma t PK: Z : Men i KV is Me t a n en t t o P E PG , , f g K K K , meetin Me ax is in I l I o r is a PV PR P P.V g V , / / / T; o u r M r o o r o n a l to a ti P n d . f p p T P R , P R , PT T he elegance of the geo metrical results probably accounts B for the fact that arrow adheres to the subtangent, as used D c F e rmat by es artes , , and others ; and this would tend to keep from him the further discoveries and development that awaited the man who considered , instead of the subtangent,

. r the much more fe tile idea of the gradient , as represented ’ L z m d dx by eibni later develop ent, y / the germ of the idea “ ” of the gradient is of course contained in the a and e

c . method , but it is negle ted

Note the disappearance of the constant Z ; hence the n c curves may be drawn to any conve ient scale, whi h need m not be the sa e, for all or any, in the direction parallel to The P EG . analytical eq uivalents are

i w = z ( ) If y , then v z 1 d dx (ii) if y / , then ( /y ) y /

The first of these results is generally given in modern text s m book on the calculus , but I do not re ember seeing the

o . T c second in any b ok hus , for produ ts and quotients we may state the one rule

u v d u v du 1 dv 1 dw i de _ y ’ + ' wz dx wz dx v dx w dx z dx

u v w z a n d f c x . where , , , , y are all un tions of L E CT UR E X

‘ R i o ro u s deter min atio n o ds dx ~ D z é ren tiatio n a s Me g f / . fi f in vers e o in te ra tio n E x la a tio n o Me D i eren tia l f g . p n f f ’ ’ ’ Tria n le meM d wiM ex a m les D z eren tza tzo n o a g o p . fi f tr i o n o metric al u n ctio n g f . 0

1 Le t e . A EG be any curv whatever , and A F I another EF curve so related to it that, if any straight line is drawn parallel to a straight line given in position (which cuts A EG E EF AE in . and A F I in F) , is always equal to the arc of the ET curve A EG , measured from A ; also let the straight line E touch the curve A EG at E , and let T be eq ual to the arc AE; j oin TF; then TF touches the curve A F I .

2 M . oreover, if the straight line EF always bears any the AE FT same ratio to the arc , in just the same way can be * shown to touch the curve A F I .

Le t x an d 3 . AG E be any curve, D a fi ed point , A I F be another curve such that , if any straight line D EF is drawn

EF the AE through D , the intercept is always eq ual to arc ; and let the straight line ET touch the curve AG E ; make

S in c e the arc is a fu n c t io n o f t he o rdin at e t is is a s e c ia c as e o f , h _ p l i i f a s u m L e c I X 1 2 it is e u ivale n t t o h diffe re n t at o n o t . t e , , q dx d dx s e n o t e t o a ds / + y / e 5. ’ 1 1 4 BAR R O WS CE OME TR TCA L L E CT UR E S

TE AE TKF equal to the arc let be a curve such that, if

K' c any straight line D H is drawn through D, cutting the urve

TKF in K and the straight line TE in H , HK HT ; then let

” FS be drawn ; to touch TKF at F ; FS will touch the curve l A F also .

. Mo E 4 reover, if the straight line F is given to bear any the o the AE t he same ratio t arc , tangent to it can easily be L 8 o v . found from the ab e and ect . VII I ,

Le t 5. a straight line A P and two curves A EG , A F I be so

related that, if any straight line D EF is drawn (parallel to

K l H G

Fi 1 06 g . .

A B , a straight line given in position), c utting A P, A EG , A FI , DF in the points D , E, F respectively, is always equal to the

‘ AE o ET u AEG E TE arc als let touch the c rve at _ take t E TR equal o the arc A , and draw parallel to A B to cut

in . AP R then, if R F is joined, R F touches the curve AF I

F r LFL c e o , assume that is a urv such that, if any straight TE line PL is drawn parallel to A B , cutting A EG in G , in H , PL and LFL in L, the straight line is always equal to TH

T L are an d HG taken together . hen P A EG P I ; and

n i h me n s e T E A E are draw n t e s a s e . ,

’ ‘ L c t I 1 . B e . I y VII , _ 9

’ 1 1 6 B AR R O WS GE OME TR I CA L LE C T UR E S

in f whic h x there is a fi ed point D , and D H a straight line given

an o in t in position also let AG B be a curve such that, if yp

G is taken in it , and through G and D a straight line is

in drawn to cut the curve AEH E , and CF is drawn parallel

AE n to D H to cut A D in F, the arc bears to A F a give ratio, X to Y say ; also let ET touch the curve AEH along ET take

e AE O EV q ual to the arc ; let GO be a curve such that, if any DOL O straight line is drawn , cutting the curve OGO in and ET L O in , and if Q is drawn parallel to CF, meeting A D in

LV A . T Q, Q X Y hen the curve OGO is a hyperbola (as

T ' has been hen , if GS touches this curve, GS will touch the curve AG B also .

AEH If the curve is a q uadrant of a circle , whose centre is D , the curve AG B will be the ordinary Quadratrix . Hence the tangent to this curve (together with tangents to all curves produced in a similar way) can be drawn by this method .

I meant to insert here several instances of this kind ;

re all I f but y . think these are su ficient to indicate the

o a method , by which , without the lab ur of , calcul tion , one can find tangents to curves and at the same time prove the

Or . add constructions Nevertheless, I one two theorems, which it will be seen are of great generality, and not lightly to be passed over.

1 1 Le t u x . Z GE b e any c rve of which the a is is A D ; and l let ordinates applied to this axis , AZ , PG , D E, continua ly

h me t O n ro ve d fo r a s e c ia c as e in L e c t . 1 b u t t e o d c an ly p p l VI , 7 h z it u t if c u t b e ge n e rali e d w ho d fi l y . L E CT UR E X

N in c rease from the initial ordinate AZ ; also le t AIF be a line EDE such that, if any straight line is drawn perpendicular to A D , cutting the curves in the points E , F, and A D in D , the re c tangle contained by DF and a given length R is equal

: DF D to the intercepted space ADEZ also let D E R T, T and join DT. hen TF will touch the curve A I F.

Fi 1 0 g . 9.

Fo r o n , if any point I is taken in the line A I F (first the IG side of F towards A), and if through it is drawn parallel to AZ, and KL is parallel to A D, cutting the given lines as

LF LK DF DT : shown in the figure ; then D E R , or

R . LF LK D E .

B u t DF v , from the stated nature of the lines , P K , we ha e = = are a LK. DE are a P < R . LF P D EG ; therefore D EG D P D E hence LK D P LI. A gain , if the point I is taken on the other side of F, and l the same construction is made as before , p ainly it can be

‘ e as ily s ho wn that LK D P LI. From which it is quite clear that the whole of the line

TKFKlies within or below the curve AIFI. A Other things remaining the same , if the ordinates, Z ,

e PG , D E, continually decr ase , the same conclusion is ’ 1 1 8 B A R R O WS GE OME TR TCA L L E CT UR E S

mi m c attained by si lar argu ent ; only one distinction oc urs ,

me c a c u rve na ly, in this se; contrary to the other, the fiAIFI is c on c ave to the axis A D .

D . e C O R . It should be noted that D E . T R DF ar a * ADEZ .

2 F c an f 1 . rom the preceding we deduce the ollowing theorem .

Le t KF if Z GE, A be any two lines so related that, any m straight line EDF is applied to a co mon axis A D , the sq u are on DF is always equal to twice the s pac e ~ ADEZ D also take Q, along A D produced, equal to D E , and join

FQ then FQ is perpendicular to the curve AKF.

I will also add the following kindred theorems .

1 . Le t Z 3 AG E be any curve, and D a certain fixed point

r ii DG such that the ad , DA , , D E , drawn from D , decrease continually from the initial radius DA ; then let DKE be

' t he fi rs t another curve intersecting in E and such that ,

n i if a y stra ght line D KG is drawn through D, cutting the DKE curve AEZ in G and the curve in K , the rectangle contained by DK and a given length R is equal to the area

’ D DT be - A G also let drawn perpendicular to DE , so that

- 2 . T t he DT R; join TE hen TE touches curve DKE.

M c D oreover, if any point, K say , is taken in the urve KE,

DG DK 2 and through it D KG is drawn , and R P then, if DT 2 P TG is taken equal to and is joined, and also KS is K drawn parallel to GT; S will touch the curve BKE.

S e e n o t e at e n d o f t is e c tu re h l .

’ 1 2 0 BA R R O WS GE OME TR TCA L L E CT UR E S

L e t A P, PM be two straight lines given in position , of

’ c c e MT s u bs e d whi h PM cuts a given urv in M , and let be pp

c T. to tou h the curve at M , and to cut the straight line at T * In order to find the quantity of the straight line P ,

Se t o ff I an indefinitely small arc, M N , of h t e curve ; then I draw NQ, N R parallel to

MP = m PT = t = a M P, A P ; I call , , M R ,

e N R , and other straight lines , determined f by the special nature of the curve, use ul A 1 G p

F i . 1 1 for the matter in hand , I also designate g 5,

u m by name ; also I compare M R , N R (and thro gh the P M P , T) with one another by means of an equation obtained

s by calculation meantime ob erving the following rules .

R L 1 . U E In the calculation , I omit all terms containing

a e r m a power of or , or p oducts of these (for these ter s h ave no value).

R L 2 . U E After the eq uation has been formed , I reject all terms consisting of letters denoting known or deter

m do a e mined quantities , or ter s which not contain or

“ o f u (for these terms, brought over to one side the eq ation ,

z will always be equal to ero) .

R L E . m a t U 3 I substitute (or M P) for , and (or PT) for

e t PT . . Hence at length the quan ity of is found

M m e oreover, if any indefinitely s all arc of the curve ent rs

e S the calculation , an ind finitely mall part of the tangent, or of any straight line equi valent to it (on ac count of the

S e e n o t e at the e n d o f t is e c t u re h l . L E CT UR E X

indefinite ly small siz e of the arc) is substituted for the arc .

B u t these points will be made clearer by the following m exa ples .

NOTE

“ B o f f a arrow gives five examples this , the dif erenti l m triangle ethod . As might be expected , two of these are

- F m D e well known curves , namely the oliu of escart s , called B L a G ala n de x by arrow , and the Q uadratri ; a third is the

” ‘ general case of the quasi - circular curves x the f r tan 6 ourth an d fifth are the allied curves a . and

a t a n x . y It is noteworthy, in connection with my sug gestion that Barrow used calcul u s methods to obtain his c geometrical constru tions , that he has already given a purely m r a t a geo etrical construction for the curve n 0in Lect . 1 8 if VIII, , the given lines are supposed to be at right angles . B w I believe that arro , by including this example , intends to give a hint as to how he made out his geometrical con struction : thus : 4 2 2 2 2 The equation of the curve is x + x y a y the 2 2 x 2 x —x gradient, as he shows is ( ) ; using the D general letters x and y instead of his p and m. escartes has shown that a hyperbola is a curve having an eq uation B of the second degree , hence arrow knows that its gradient

is the quotient of two linear expressions , and finds by equating coefficients) the hyperbola whose gradient is x 2 x x —x x 0( O 0) the feasibility of this is greatly enhanced by the fact that Barrow would have written the t wo gradients as m t x (2 x x +yy ) : y (aa x x ) and = x 2 x x x x . 0( 0 0) T i x h hese two grad ents are the same at the point 0, y o ; ence

if he can find such a hyperbola, it _ will touch the curve he

can draw its tangent, and this will also be a tangent to The : his curve . curve does turn out to be a hyperbola 2 2 fi 2 z - - x x + x . x O r x for its equation is 0 0y0 y ayy 0 l y (d Where d This latterfo rm is ’ 1 2 2 B A R R O WS GE OME TR TCA L L E C T UR E S

L . easily seen to be equivalent to the construction , in ect VIII , v DYO re c t an ulaiz fo r for the cur e , when the axes are g ; Z t he u e eq uation gives DY Y E Y R . It also s gg sts that the construction for the original cu rve is transformable into 1 B 1 8 that of 7, as is proved by arrow in , and in order B 1 0 1 1 1 2 L that arrow may draw the tangent, , , of ect . VI are necessary to prove that the auxiliary is a hyperbola of which the asymptotes can be determined by a fairly easy B z his geometrical construction . arrow then generali es theorems for oblique axe s . I contend that this suggestion is a very probable one for three reasons : (i) it is quite

e - feasible, ev n if it is considered to be far fetched , (ii) we know that mathematicians o f this time were jealous of their c methods , and gave cryptogrammati hints only in their work ’ c N he (f . ewton s anagram) , and (iii) it is to my mind t only reason why this particular theorem should have been select e d

’ (especially as Barrow makes it his E xample fo rthe re is no great intrinsic worth in it . The x c as e ~ o f r = a t a n 9 fifth e ample, the the cu ve y , I have selected for giving in full, for several reasons . It is x the clearest and least tedious e ample of the method , it t wo e is illustrated by _ diagrams , one being deriv d from the other, and therefore the demonstration is less confused , it is connected with the one discussed above and suggests that Barrow was aware of the analogy of the differe ntial o C form of the p lar subtangent with the artesian subtangent , and that in this is to be found the reason why Barrow C gives , as a rule, the polar forms of all his artesian theorems i ic and lastly, and more part cularly, for its own intrins ’ B a . n d merits, as stated below arrow s enunciation proof are as follows

E X MP L L e t A E 5. D EB be a quadrant of a circle, to which BX is a tangent ; then let the line AMO be such

i ~AV an t hat, if in the straight l ne y part A P is taken equal n a to the arc B E , and PM is erected perpe dicul r to AV, then

rc PMis equal to BC the tangent of the a B E .

1 2 4 B A R R O WS GE OME TR LCA L L E CT UR E S

c m As regards this le ture , it only re ains to remark on the fac t that the theorem of 1 1 is a rigorous proof—« that f dif erentiation and integration are inverse operations , where mm B integration is defined as a su ation . arrow not only, h as is well known , was the first to recognise t is ; but also, judging from the fact that he gives a very careful and full proof (he also gave a second figure for the case in which the ordinates continually decrease), and in addition , as will L 1 be seen in ect . XI , 9, he takes the trouble to prove the m — c — he theore conversely, judging from these fa ts , I say, als o must have recognised the importance of the theorem . m m h It does not see , however, to have been re arked t at he e ever made any use of this theorem . He, how ver, does use it to prove formulae for the c entre of gravity and the ae o e area of a paraboliform , which formul he only qu t s with “

m - the re ark , of which the proofs may be deduced in t various ways from what has already been shown, withou

ffi L . much di culty (see note to ect XI , g “ The differential triangle method has already bee n referred to in the Introduction ; it only remain s to point

i . B out the s gnificance of certain words and phrases arrow , whilst he acknowledges that the method “ seems to be more profitable and mo re gen er al than those Which I have ” m discussed , yet is in so e doubt as to the advantage of z including it , and almost apologi es for its insertion ; has probably, as I suggested , because, although he found it a most useful tool for hinting at possible geometrical m m constructions , yet he co pares it unfavourably as a ethod with the methods of pure geometry . It is also to be x observed that his a es are not necessarily rectangular, although in the case of oblique axes , PT can hardly be accepted as the subtangent hence he finds it convenien t T h to tacitly assume that his axes are at right angles . e last point is that Barrow distinctly states that his method is expressly “ in order to find the quantity of the sub ” an d tangent, I consider that this is almost tantamount to a direct assertion that he has u s ed it frequently to get his n The first hi t for a construction in one of his problems . final significance of the method is that by it he can readily handle implicit functions . L E CTU R E XI

t n t e ra Ck a nge of Me in depen den t va ria ble in in teg ra io n . I g ' ’ t tio n Me en D eren tia tio n o a u o tien . in vers e of dzfi er tia tio n . ifi f q A rea d c t e r b li o r m L imits o r Me a n en r of gra vity cf a pa a o f . f a rc o a c r l n a s tima tio n o f i c e a d kyperbo la . E f

NOTE B In the following theorems , arrow uses his variation of the usual metho d of summation for the determination of an area . If ABKJ is the area under the curve AJ, he divides B K into an infinite number of equal parts and erects ordinates . In his figures b e generally makes four parts do duty for the infinite number . He then uses the notation already ABKJ mentioned , namely, that the area is equal to the sum of the ordinates A B ,

CD, EF, GH, JK. The same idea is involved when he speaks of the sum of the rectangles

CD . F . H E H FH K . G G . J D B , FD , , ; for this sum , where commas are used between the quantities instead of a plus B ‘ D F H K ' '

. ABKJ sign , does not stand for the area ABKJ, but for R , w ' ' F here an ordinate HG is such that R HG H G H, and R is some given length ; in other words , ordinates proportion al to each of the rectangles are applied to points

' of the line B K , and their aggregate or sum is found ; e h n d . O henc t is sum is of three imensions the contrary, d mm he uses the same phrase, with plus signs instea of co as , m m to stand for a si ple su mation . ’ 1 2 6 B A R R O WS GE OM E TR TCA L L E C T UR E S

2 2 T the AE BZ BF hus, in 3 of this lecture, sum of AZ , , 2 CZ . G . f in O , etc , is the area aggregated rom ord ates 2 2 2 B CZ . G . . F . BZ C proportional to AZ A E , , , etc , applied to the line V D ; and it is of the fourth dimension . 2 2 2 s u m LK Y KY + HL . L Whereas , in 3 , the HO + + K I 2 2 2 . H LY KY etc , is aggregated from ordinates equal to 0 , , , m etc . , applied to the line HD; and it is the sa e as the 2 2 2 LV KY . sum of HO , , , etc

1 V H s HD . If is a curve whose axi is V D , and is an

< ) ordinate perpendicular to V D , and 1t is a line such that,

‘ c m e if from any point hosen at rando on the curv , E say, a s t rai ht EP g line is drawn normal to the curve, and a Z straight line EA perpendicular to the axis, AZ is equal to the intercept A P ; then the area V D¢¢ will be equal to half the square on the line D H .

F o r HDO if the angle is half a right angle, and the straight line V D is divided into an infin ite number of C equal parts at A , B , , and if through these points straight i FAZ FBZ GCZ ar l nes , , , e drawn

HD i u parallel to , meet ng the c rve in E , F , G ; and if from these points are drawn straight lines E I Y , F Y LY HO K , G , parallel to V D or

s EP FP and if al o , , G P , H P are

c the normals to the urve, lines intersecting as in the figure ; then

‘ F i 1 2 2 the triangle HLG is similar to the g . .

PDH fo r o n u t he triangle ( , acco nt of the infinite section , small arc H G c an be considered as a straight line) .

’ 1 2 8 B A R R O WS GE OME TR TCA L L E C T UR E S

' NOTE

O n in 2 . the assumptions the proofs of , 3

T he summation used in 2 has already been given by B L arrow in ect . IV ; he states that it has been established ” i n another place by Wallis or others), and that it at “ eo er least is suffic iently known among g met s . v n It is easy, howe er, to give a demonstration accordi g ’ B in to arrow s methods of the general case ; and , since several cases Barrow is content with saying that the proof ma m y easily be obtained by his method , and someti es he “ f ” adds in several dif erent ways , I feel sure that he had m made out a proof for these su mations in the general case , T he L method given below follows the idea of ect . IV, n din a cu rve co n ven en t o r by fi g i f the summation , without proving that this curve is the only one that will do . O ther ’

i . B methods w ll be given later, thus substantiating arrow s statement that the matter may be proved in several ways ;

L . 2 L . 2 see notes following ect XI , 7, and ect . XI , App , . L t wo e t A H , KHO be straight lines H0 at right angles , and let A H R Le t P an d KH 8 . AEO be a curve such that (in the figure) UE is the first of A — I m W n geo etrical means between U s V and U , " ” ‘ 1 E UW . UV i e . U . ,

” ” ” “ 1 or DY A D R . D E . ' L e t A FK be a curve such that P F is the first of n geometrical means between ” +1 ” +1 ”+1 i. e 8 AD R DF. PL PF PL . PQ and , or , T hen the curve A FK is a curve that is fitted for the AED determination of the area under the curve , providing a suitable value of S/R is chosen . = 2 Fo r FD/DT 1 ) 1 ) D E 8/R ; and if S is taken equal to R/(n + FD : DT D E : R ; and L 1 1 therefore by ect . X , " ' DD' the sum DY . DD the sum D E R” ’ I area AD E " "+ 1 . R R DF S . AD / LE C TUR E X]

Hence we may deduce the following important theorems

L t c 4. e V Dtq be any space of whi h the axis V D is m equally divided (as in fig . then if we i agine

- o f c VAZ c V BZ S V CZ b . that each the spa es fi, q , q , etc , is

BZ CZ e t c . e multiplied by its own ordinate AZ , , , , resp ctively, the sum whic h is produced Will be equ al to half the squ are of the space V DtM) .

. u 5 If, however, each of the sq are roots of the spaces is i multiplied by its own ord nate, an aggregate is produced equal to two - thirds o f the square root of the c u be of

E x a m l — o f 6 . e Le t V Dt f p / be a quadrant of a circle, which the radius is R and the perimeter is P ; then the

V BZ VOZ mu segments VAZ, , , each ltiplied by its BZ CZ own sine , AZ, , , respectively , will together

2 2 make R P /8 .

s u m , e t c . Also the AZ J(VAZ ) + BZ . J< R3 P3 /8>

V Dt is n o f s u m m e Again , if b a segme t a parabola, the ad from the produ c ts into the ordinates will be equal to

2 2 V D D 1 r m t he c o f of 4 , and that f o produ ts the square roots o f the segments i nto the ordi n ates will be equal to 2 /3 of

3 3 3 3 8 2 1 J( / 7 of V D 011 ) or J(V D . D41

dx d 2 T he e u iva e n t s o f are re s e c t ive . x dx q l 4, 5, p lyf y f y ) fy ) a n d / dx dx dx o r i o re re o n iz r y « . y n a m c a e fo m f f y ) f ) g bl , i dx t e e z dz dx dx a n d o o n u t t n a fo r ar s . p g fy , h y ( / 1 30 BA R R O W’S GE OME TR TCA L LE CT UR E S

O ther similar things c oncerning the su ms mad e from the products of other powers and roots of the segments inTO the

n ordi ates or sines can be obtained .

. F f m f h 7 urther, it follows ro what has gone be ore t at in

if c t every case, the lines V P inter epted be ween the vertex and the perpendiculars are supposed to be applied throu gh B C AY CY the respective points A , , , say that , BY , , are e qu al to the respec tive lines V P ; then will the spac e

O H c f V S , onstituted by these applied lines , be equal to hal the square on the subtense V H.

8 . M if m a RXXS oreover, with the sa e dat , is a curve

u c | = KX= BP LX= CP s h that X A P, , , then the solid formed by t he rotation o f the space V DtM) about V D as an ax is is half the solid formed by the rotation o f the s pace DRS H m about the sa e axis V D .

. f m m 9 All the oregoing theore s are true, and for si ilar

‘ the lin e reasons, even if the curve V E H is convex to V D .

F m m i u rom these theore s , the di ens ons of a tr ly bound less number o f magnitudes (pro c eeding direc tly from their c ma onstruction) y be observed , and

i t l easily ver fied by ria .

I O f e . i V H c u ve Again , is a r , whos DZ Z axis is V D and base D H , and is

c u u c h u c a rve s h t at , if any point s h as E is taken o n the c urve V H and

~ c c ET is drawn to tou h the urve , and

e . 1 2 a straight line EIZ is drawn parall l to Fig 5.

’ 1 3 2 BA R R O WS GE OME TR TCA L L E CT UR E S

1 B f 3 . y similar reasoning , it ollows that

m 2 2 2 ’ s u I DK . L e t c Z KZ D LZ . tO the of D I , , , (applied HD)

three times the s u m o f the c u bes on all the ordi l AE BF CG . nates , , , app ied to V D

m if 1 . e DXH c u 4 With the sa data, is a urve s ch that any m ordinate to D H , as I X , is a ean proportional between the

IE IZ n e ordinates , co gruent to it ; then the solid form d by the spac e V DH rotated abou t the axis D H is dou ble the solid formed from the space DXH rotated about the same

1 I f h we v . c u D H . X 5 , o er (in fig the rve is supposed * u c C bime dian to be s h that any ordinate, X say, is a

the u IZ s u m o f between congr ent ordinates I E, ; then the — u o f IX KX LX e tc . o f s u m the c bes , , , , is one third the of the * c o f IE KF e t c . B u t if trime dian ubes DV , , , I X is a ; then

4 4 4 - s u m o f KX LX e t c . u o f the I X , , , , is eq al to one fourth the

4 4 4 u m IE KF e t c . o n fo r s of DV , , , and so all the other powers .

E t he me bimedia n f NOT . I call by na the first o two m t rimedia n t he o f ean proportionals , by first three, and so on .

The se results are ded u c ed and proved by similar re ason ing to that o f the previou s propositio n s but repetition is

’ n o in an y g. I

1 6 VY t he . Again , if Q is a line such that ordinate AYis

Q 2 AT BT s u m o f IZ KZ equal to , BY to , and so on then the , ,

2 e t c h t he s u m o f u o f i L2 , , t at is , the sq ares the ord nates

L it ra it ir s me t o c r c u c e o o . , lly, k y k L E C T UR E X ]

f u e u o the c rv DZ O applied to the line D H , is eq al to the

u V OH . BF . s u m VA AE AY+ AB BY + etc , that is , the fig re * multiplied by the figu re V DQ.

3 3 3 u m o f . 1 . s IZ KZ LZ 7 Also the , , , etc

2 2 t c . m . . BY e s u AE . + the VA . AY + A B B F ; “ V D that is, the figure V DH multiplied by the figure Q

squared .

T hese you c an easily prove by the pattern o f the proofs

given above .

1 8 T h . e same things are true and are proved in an

m if u V H t o exactly si ilar manner, even the c rve is convex

the straight line V D .

NOTE

T he 1 1 B l equality , which in is said by arrow to be we l 2 2 m t c s u CD . DH BC CG + e . known , namely , the

s u m DI . DK KF e t c . twice the of I E , , , is really an eq u ality between two expressions fo r the volume of the solid formed by the rotation o f V DH abo u t V D ; and ? c e dx z x d e the analyti al quivalent is (y j y y , with the int r

mediate step 2 ”y dy dx . T hus these theorems of Barrow are equivalent to the equality of the results obtained from t wo i a double integral , when the first ntegrals are obtained

by integrating with regard to eac h o f . the two variables in ih 1 1 turn . He says indeed that the first result, that § , is m mm d m a atter of co on knowle ge , but he re arks that the others that he uses in the following sections c an be obtained F m by similar reasoning . rom this , and fro indications in 6 1 i 1 L 1 L c . X ect . IV , , e t , , and 9 of this lecture , I feel

The e qu ivale n t s o f t he s e t he o re ms are ” 1 6 d = 03x - y f y . d = 1 . dx 7 y f y . ’ 1 34 B A R R O WS GE OME TR TCA L L E C T UR E S certain that Barrow had obtain e d th e se theorems i n the c o f c b u t in m n c e s ourse his resear hes , , as a y other as , he m f v m h o its the proo s and lea es the to the reader . All t e m u f f his m ore, beca se the proo s ollow very easily by ethods . T fo r o f m the c u o f ake , the sake exa ple, ase of the eq ivalent

e’ ’ d dx h c m B m o jjy y , for w i h I i agine arrow s ethod w uld have m w f been so e hat as ollows . In order to find the aggregat e o f all the points of a given V DH c mu c m space , ea h ltiplied by the ube of its distance fro m a c ff . the axis V D, we y pro eed in two di erent ways Mo 1 — M e d . B e y applying lin s PX , proportional to the o f BP o f i cube , to every point P the l ne B F , find the

aggregate o f t he produ c ts fo r t he line BF; then find the s u m o f e e t he e th s aggregates for all lines applied to V D . Her , 3 e BP c u v BXX u c sinc PX is proportional to , the r e is a c bi al — t he c BFX f u o f parabola, and spa e is one o rth the fourth power of B F ; and Barrow wo u ld write the resu lt of t he 4 4 4 mm m f 4 4 4 e t c u s u o AE BF CG . s ation as the / , / , / , (applied to V D) . — M et/zo d 2 . Find the aggregates along all the parallels s u m o f e D to V D , and then the these aggr gates applied to H Here the first aggre gates are re presented by rec tangles whose 3 IE KF . u DI bases are , , etc , and whose heights are eq al to , 3 DK e t c . c i B u , , respe t vely ; and arrow wo ld write this as the 3 3 s m o f . . e t e u DK KF c . . D I I E, , (appli d to D H) 3 3 L . . c 1 2 . I . Z AE astly, in fig 5, I D D I VA D I , etc ; hen e s . dx d all the res u lts follow immediately ; i e. ”y y is equal 4 3 t he dx x d B to either of integrals jy /4 , j y y ; and arrow

- f d proves that th e se are eq u al to one fourth o y .

’ 1 36 B A R R O WS GE OME TR TCA L L E C T UR E S

in fo r of EY, applied to BD and so on similar fashion the other powers .

2 2 L t c u . e DOK be any rve , D a fixed point in it, and DKE

c h a chord also let A FE be a curve su h t at, when any straight

u c rirve s D line D M F is drawn c tting the in M and F, S is

c DM M8 t he v e drawn perpendi ular to , is tangent to the cur

K c u D S R DO , tting 8 in , and is any given straight line, then 2 2 2 T DS R DM DF . hen the space A D E will be equal to * c e c the re tangl ontained by R and DK.

2 T he c 3 . data and the constru tion being otherwise the

m MI d c u t sa e, let KH and be rawn perpendi lar to the angents KT MS m DT D8 and , eeting , in H and I respectively ; and

E c u u c DK. let A be a rve s h that D E J( DH ), and also

T he c DE and so on . n the spa e A D E is

- equ al to one fou rth of the squ are on DK.

i i 2 . DO c u 4 If K is any rve, D a g ven po nt on it,and DK

DZ I v u any chord also if is a cur e s ch that, when any point

DOK DM e d D M is taken in the curve , is j oin , S is drawn

c u DM MS u perpendi lar to , is a tangent to the c rve , DP is

DK u DM PZ taken along eq al to , and is drawn perpendicular

PZ u D8 h c c DZ I to DK, then is eq al to ; in t is ase the spa e

u c c D is eq al to twi e the spa e DKO . T

2 T he c o n s t ru c tio n ~ b e in c 5. data and the g in other respe ts t he m PZ n o w u u sa e , let the ordinates be s p posed to be eq al

T he an alyt ic al e q u iv ale n t s are z d R dd dr dd 2 R r r d 2 . f l . / ) 2 = 2 2 r 2 dd r 2 dr dd dd r . 3 . f 1 / . / / ( / ) /4 z 2 r dd dd dr dr 4. f . ( / ) .

‘ S e e n o t e o n a e 1 8 . 1 p g ‘ 3 L E C T UR E X ]

MS x é to the respective tangents ; take any straight line , DOK DOM DON and distances along it equal to the arcs , , ,

é d md 72 d t he etc . , and draw the ordinates , , , etc . , equal to é d c KD . x hords , M D , N D , etc 5 then the space will be equal to the space D K I .

2 6 M . oreover if, other things remaining the same, any

k x k fi straight line g is taken , the rectangle g is completed , and the c urve DZI is supposed to be such that M D D8 kg : PZ ; then the rec tangle x ég/z will be equal to the space DK I .

c Hen e , if the space D K I is known , the quantity of the

curve DOKmay be found .

S m hould anyone explore and investigate this ine, he will

Le t find very many things of this kind . him do so who m ust, or if it pleases him .

P m m m erhaps at so e ti e or other the following theore ,

u m too , ded ced fro what has gone before, will be of service it has been so to me rep e atedly .

2 . L e t V EH 7 be any curve, whose axis is V D and base

D H , and let any straight line ET touch it ; draw EA parallel

. u to HD Also let GZ Z be another curve s ch that, when any

i EZ m c u i stra ght line is drawn fro E parallel to V D , tt ng the

HD I GZ Z o f base in and the curve in Z, and a straight line 2 2 = R I S R DT IZ . given length taken , then at all times DA T AE= R2 D| Z R hen DA space G 5 (or, if DA is made equal

: P to R D P , and Q is drawn parallel to D H , then the rect

DP I u Fi 1 1 . D Z I . angle Q is eq al to the space G ) [ g 3 , p

Th e following theorem is also added for future use .

2 8 . Le t AMB be any curve who s e axis is A D 5 also let t he 1 3 8 B A R R O WS GE OM E TR TCA L L E CT UR E S

e KZ L u c lin be s h that , if any point M is taken in AM B , and — from it are drawn a straight line M P perpendic u lar t?) AB c u u tting A D in P, and a straight line MG perpendic lar to A D

u c KZ L m c tting the urve in Z, at all ti es G M P M is equal to arc AM GZ ; then the spac e ADKL will be equal to half the square on the arc AM .

T e m ma m h se theore s , I say, y be obtained fro what has

u c f c d u f c gone before witho t mu h di fi ulty ; indee , it is s fi ient t o m m an d I ention the ; , in fact, intend here to stop for a while .

NO T E

The 2 —2 8 c theorems of 4 deserve a little spe ial notice . The first of these was probably devised by Barrow fo r the u S o f c m c u q adrature of the piral Ar hi edes ; it in l ded , as was “ ” u u him u m l s al with , inn erable spira s of other kinds , thus B m representing both , as arrow would consider it , an i prove ’ ment and a generaliz ation o f Wallis th e orems on this spiral “ ” m c o f I n fi n it s in the Arith eti e . if I c u e It is readily seen that DZ is a straight line, the rv AOK is the first b raric h or tu rn o f the Logarithmic or

‘ E u S if Ol t he c u v DOK is q iangular piral ; l is a parabola , r e the Circ u lar S piral or S piral o f Arc hime des and if the u f m c u DOK c rve is any paraboli or , the rve is a spiral whose m " ma r d B e A . equation y be In short, arrow has giv n a general theorem to find the polar area o f any cu rv e who s e u d dr fo r c in c eq ation is , all ases whi h he can

find the area u nder the cu rve y f (x ) . T he t 2 6 e heorem o f is indeed r markable , in that it is a ' * r eciz fca fzo n general theorem on j . It is stated that Wallis 1 6 c o f had shown , in 59, that certain urves were capable c c m 1 66 0 c re tifi ation , that Willia Neil , in , had re tified the ' ' ’ s emi- cu é zc a l a r a é o la u Wa /Zzs meMo d c p , sing , that the se ond c u c clo id t rve to be rectified was the y , and hat this was

' ‘ B r it m s it io n Art o n [n m zes zwza l Ca lc u lu s Ti e e d . . E n c . . y ( ) , fi n h e dat e s are wro n o we v e r ac c o rdin t o o t r Wi iams o . e s e ( ll ) T g , h , g h it s s u as R o u s e B au t o r c a . h ie , h ll

’ 1 40 B A R R O WS GE OME TR TCA L L E C T UR E S

L e t t he c u v b e if c m r e VXY such that, EA produ ed eets it : : D i e arc i n in Y , then always EA A D AY R . vid the Ev to u m o f L an d FEX an infinite n ber parts at F, , draw , LOX HD m O , parallel to , eeting V D in B , , and the c u V Yin F Z l rve X the points X ; also draw J , para lel to m e i HD d c GZ Z in V D , e t ng in , and the urve the points Z .

- h n . BD . BD . E . e AD R EA. R A T AY. ( AD and R . (EA. AD c XW c u n AY W hen e , if is drawn parallel to V D , tti g in , then 2

. A . BD E . . WY AD WY. D A AB) B u t or 2 — WY. D . T R . I . A I J A ) J DT. 2 2 DA : R DT l 2 z D : WY. AD lJ . T R

c s u m o f c Id IZ f Hen e, since the the re tangles only dif ers e e f m c e DGZ l s u m in the least d gr e ro the spa , and the of t he lengths WYis AY it follows immediately that 2 z s ac e z z R p DOZI R AY DA AE.

No w if DT are c o - e and D H taken as the ordinate axes , th n — W t he ffe e o f AY R x DT x dx a z Yis di r ntial or y / , and y /j ; 2 r t a u v f he c i e o . t h e refo e nalyti al eq al nt WY AD R DT . Id x d x is R . ( or (y / ) (an dy y B e e m i e i b u t if I arrow stat s it as a theor in nt grat on ; , c c u e his m o f f ai have orre tly s ggest d ethod proo , he obt ns hI S m i f i o f x 1 theore by the d f erentiat on _y/ (see pages 94, A P P E ND I X

' ‘ 1 ma n ea rs a o I m C clo mezrzea . When y y g exa ined the y of * man C H u e n iu s that illustrious , hristianus g , and studied it closely,I observed that two methods of atta c k were more

c him. espe ially used by In one of these , he showed that the segment of a c ircle was a mean between two parabolic

m m c seg ents , one inscribed and the other circu s ribed , and in this way he found limits to the magnitude of the former .

In the other, he showed that the centre of gravity of a cir c u lar segment was situated between the c entres o f gravity o f m u a parabolic segment and a parallelogra of eq al altitude ,

c u m and hence found limits fo r this point . It oc rred to e

c o f m that in pla e the parabola in the first ethod , and of

m m a r a é o lz o rm eu r ue the parallelogra in the second , so e p f

c c c c circums ribed to the ir ular segment ould be substituted , so that the matter might be c onsidered somewhat more t I c . n i losely O examining , soon found that this was

— The wo rk o f C hris t iaan H u yge n s (1 62 9 1 69 t he gre at D u t c h mat he mat ic ian as t ro n o me r me c han ic 1 a n an d s ic is t t at is re fe rre d t o ma , , , phy , h y ' ’ b e t he e s s a E x e las zs u a dra t u re e ir e u lz L e de n b u t mo re y q ( y , ' ' ' ‘ ‘ ro a is t he c o m e t e t re at is e D e c irc u le ma u z zu dzu e z rwem a t a t p b bly pl g , h ’ P u t t in t h e da t e o f B rr was pu blis he d t hre e ye ars lat e r. g a o w s s t u dy o f ’ H u yge n s wo rk a t n o t lat e r t han 1 656 (n o t e t he wo rds in t he firs t lin e abo v e — t at av e s e t in it a ic s — ma n ea rs a o a n d re me m e rin t at t is was h I h l y y g , b g h h ’ prin t e d in it fo llo ws fro m Barro w s me n tio n o f t he parabo lifo rm c u rve a s s o me t in e n o n t o him a n d fro m a re mar t at t he ro o fs h g w ll k w , k h p o f t he t he o re ms o f § 2 may b e de du c e d in v ario u s ways fro m what has a re ad ee n s o n wit o u t mu c dif c u t t at Barro w was in l y b h w , h h fi l y , h p o s s e s s io n o f his kn o wle dge o f the pro pe rt ie s o f his be lo v e d parab o lifo rms v I it n o t t e re fo re ro a e n a a mo t e e n e fo re t is da t e . s s c e rt ain b h h p b bl , y l , ‘ “ t at Barro in 1 6 a t zlze v er la t es t h a d n o e d e o f his t e o re m h w , 55 y , k wl g h e qu iv ale n t t o t h e of a f rae t zo u a l po wer ? 1 42 B A R R O WS GE OME TR I CA L LE CT UR E S

m correct ; oreover, I easily found that like methods could

—” be used for the magnitude of a hyperbolic segmentf As — the proofs fo r these theorems better perhaps than others t n — c c e hat might be i vented are short, and lear (be ause th y m follow fro or depend on what has been shown above) ,

m . I thought good to set the forth in this place I think ,

t o o u . , that they are in other respects not witho t interest

2 t u . L e s assume the following as known theorems ; of which the proofs may be deduc ed in various ways from

m c f . what has already been shown , without u h di ficulty m If BA E is a parabolifor curve , whose

axis is A D and base or ordinate is B D E , BT K a tangent to it, and the centre

u m of gravity 5 then , if its exponent is / , we have

o f . Area of BA E A D B E ,

TD F i 1 of A D, g . 3 3 .

KD m a 2 772 and /( ) of A D .

T he de n it io n o f L e c t . 1 2 u s e s N M t he v a u e o f T D A D - i fi VII , , / ; l / s fo u n d in L e c I X t . e re a s o t he de n it io n o f t h , § 4 ; wh l fi e parab o lifo rms is ve n gi . N o w it is c le ar fro m t he adjo in in g figu re t hat if HLE is a ara o ifo rm w o s e e x o n e n t is A p b l , h p r s I a s a t e n LK HK = a LM M / ( / y ), h / /A an d v c o n e rs e ly . L e t A IFB b e a c u rve s u c h t hat F M R LK HK L M / / a . /A M ;

t e n b L e c t . X I 1 are a A FB D R D E h , y , 9 , . Bu t in t is c a s e we a ve , h , h IG F M LM/A M HN/A N LM/A M A M LK — M H . L K : LM . . A N a HK : LM A ( . N ; FG GI 1 a I o f M F M / /( ) A / . e n c e A IFB is a ara o fo rm o s e v e rt e x is a x is AD e x o n e n t H p b li , wh A , , p

a 1 .

1 44 BA R R O WS GE OME TR JCA L LE C T UR E S

6 L t w . e t o BT ES c the straight lines , now tou h a hyper AEB is O 55 bola , whose centre and let other things the same as in the theorem just before ; then TD A D S P A P

L e t c mm th 7. the axis A D and the base BD be o on to e c c O ir le AEB whose centre is , and the paraboliform A FB 3

o f m u m also let the exponent the parabolifor be / , where

m a o f CA AD ( ) ,

m —u : 7 2 — 2 72 z 7 CA AD.

M h BT c oreover, let the straig t line touch the ircle then BT f will to u ch the paraboli orm also .

in 8 . u c c v It sho ld be noted this conne tion that, on ersely, if I S lv e n f m the ratio of A D to CA g , the paraboli or which

e touches the c irc l AEB at B is thereby determined . — F o r c if O s i l s instan e , A D/ A / , then ( ) will be f the exponent o f the requ ired paraboli orm.

m b f m 9 . With the sa e hypothesis as in § 7, the para oli or

c AFB will fall altogether outside the cir le AEB.

1 0 . m if Again with the sa e hypothesis, with a base G E (any parallel to BD) and axis A D another paraboliform is

o f me supposed to be drawn , the sa kind as AFB (or having t he m u m i u v e fo r sa e exponent / ) then th s c r also , the part

e c c AE abov G E , will fall altogether outside the ir le .

ma f m f 1 1 . Also it y be shown that the said paraboli or (o

c c like kind to AFB and onstru ted on the base G E), when

u n prod ced below G E to D B , will fall altogether withi the circle as regards this part .

1 2 F c m . urther, let A D be the axis and DB the base om on LE CT UR E X T— AP P E ND TX

AEB c O a to the hyperbola whose entre is , and the p ra b o lifo rm u r u A FB , whose exponent is / ; also let A D

- (2 72 u ) of CA and let BT touc h the hyperbola .

u m Then BT to c hes the parabolifor also .

1 . o v 3 Hence again, if the rati of A D to CA is gi en , the paraboliform tou ching the hyperbola at the point B

‘ . F o r O s z is thereby determined instance, if A D/ A / , t ( + s ) .

m 1 . 1 2 4 With the same hypothesis as in , the parabolifor h AEB A FB, will lie altogether wit in the hyperbola

1 . o m 5 Also , with the same hyp thesis , if you i agine a paraboliform of the same k ind to be c o n stru c ted with the base G E and axis AG ; it will fall within the hyperbola on

the upper side of G E .

1 6 M c m . oreover, if this se ond like parabolifor , constructed E on the base G, is supposed to be produced to D B ; then t he part of it intercepted by EG and BD will fall altogether

‘ e outside the hyp rbola .

1 Le t c AEB AFB 7. the ircle and the parabola have a c ommon axis A D and base BD then the parabola will fall w BD ithin the circle on the side above , and without the c ircle below BD.

c I f an ellipse is substituted for the cir le , the same result m holds and is proved in like anner .

t 1 8 . L e the hyperbola whose axis is AZ and parameter

h m an d A H , and the parabola AF B ave the sa e axis A D base

8 0; then the parabola will fall altogether outside the ’ 1 46 B A R R O WS GE OM E TR TCA L L E CT UR E S

BD b u t i it u c hyperbola above , w thin when prod ed w belo BD.

1 . F m w l for 9 ro hat has been said , the fol owing rules the m u ma ens ration of the c irc le y be obtain e d .

L e t BAE c c t he i be a part of a ir le, of which ax s is

o f c AD , and the base B E ; let G be the centre the ircle , and

EH equal to the right sine of the arc BA E ; also let I A D 0A s : .

T h 1 2 t 2 5 o f m BAE en ( ) ( ) A D . BE seg ent ;

(2 ) EH+ (4l as ) o f BH are BAE; m (3 ) of AD . BE seg ent BA E ;

(4) EH of BH arc BAE.

2 0 S im f o f . m ilarly, the ollowing ru les for the ensuration

ma u c the hyperbola y be ded ed .

’ L e t A D B be a segment of a hyperbola [Barrows figure is

c really half a segment] , whose entre is 0, axis A D , and base

z i D B ; and let AD CA s z .

T 1 m hen ( ) of A D . D B seg ent AD B ;

2 f m ( ) o A D D B seg ent A D B .

NOTE

The u 1 2 0 fo r c B m i res lts of 9, , whi h arrow o its any h nt

f e . as to proo , are thus obtain d 1 9 (1 ) A paraboliform whose exponent is (t s ) c an b e drawn t o u c hin c c BAE , , g the ir le at B , A , and E, and lying c ompletely outside it ; the area o f it cut o ff by t he ‘ c is 1 1 u 2 2 2 5 o f BE. hord B E , by , eq al to ( ) A D 3 ) A parabola is a paraboliform whose expon e nt is and o f me 2 the area the seg nt is o f A D BE. ( ) and (4) follow from (1 ) and (3 ) by u sing obvio u s relations fo r the

b . T ns circle , and are not o tained independently his explai

1 48 BA R R O WS GE OME TR I CA L L E C T UR E S

2 Th . e c o f o f e 4 entre gravity the parabola , L say, li s — * v i. e . abo e K ; KD is less than two fi ft hs o f AD.

2 L m . c n 5 est the present ethod of resear h , owi g to the gre at n u mber of methods of this kind fo r measuring the c c ma m ac c u we o n e ir le, y see to be of little o nt, will add or

o f fe w m two riders (if only for the sake these, the theore s given would deserve employme nt) ; from which indeed

Maxima and Minima o f things o f a kind may be de termined i m c n a great nu ber of ases .

L e t A BZ be a semic irc le whos e c entre is 0; also let AD B be a segment ; and to this let a paraboliform AFB be ads c rib e d e u m : A , whose expon nt is / , where A D C

772 2 72 m u .

L I f the parameter o f the parab o lifo rm i (that is a straight line suc h that some power o f it m u ltiplied by a power o f

o f m the axis the seg ent, A D say , produces a power of the

DB m imu m o f ordinate, say) is p then p will be a ax its kind .

F o r i , if any stra ght line G E is drawn parallel to D B , and a paraboliform o f like kind to A FB is s u pposed to be

t he m c e applied to G E , of which para eter is alled 9 ; th n ,

c AFB c c c sin e the paraboliform tou hes the ir le externally,

m m" ” " E - GF > GE. GF > G AG , , or p ,

m ” ) m Dm‘ 2 ” It should be observed that p Z D . A

The s e limit s are n o t re markable fo r c lo s e a ppro x imat io n u n le s s t he

if t he arc i o n e - t ird o f t he c irc u mfe re n c e s e me n t is v e r s a o . u s s g y h ll w Th h , A O t he limlt s fo r t he c irc le are o n ly 2 AD/5 an d 3 /5. T I t is t o b e o bs e rv e d t hat Barro w he re in dic at e s t hat t he e qu at io n t o the m/n ara o ifo rm is in e n e ra a x . p b l , g l y L E C T UR E X T— A P P E N D TX

~ Q‘m - m m m m 2 n e c Z D . Ab A o Z G . h n e G

m m ‘ zn * m m m. that is Z D . AD is a axi u

4 3 - m le 1 L e t 1 m e B . E x a . p n , and 3 ; th n ? Z A D 2 2 2 Z D BD OA 2 Z D . BD , or p ; and A D /

“ 1 0 4 2 L t u z m 1 0 Z D AD E x a m le . e p 3 , and ; then? , 7 5 2 3 4 4 7 OA . AD Z D . BD or p Z D . , and A D /

u 2 6 . Again , let AEB be an eq ilateral hyperbola whose O Z A f m AFB centre is , and axis ; and to it let a paraboli or ,

u m ads c rib e d whose exponent is / and parameter p, be (with a base D B) ; also suppose that AD CA 2 a m m n ; then p will be a minimu m of its kind .

— ‘ lm ” l m Qn m Z D . AD It is to be noted that fi , and also — — - n ) m 2 n m 2 n m (as in g 2 5) Z G AG ; hen c e Z D AD IS a min imu mnl

As in the preceding I have to u ched u pon the mensura

o f t he if c fe w tion circle, what I add in identally a theorems b e aring u pon i t whi c h I have by me ? The foll o win g g e neral

m m h e m . theore ust, ow ver, be given as a preli inary

2 L e t v 7. AG B be any cur e whose axis is A D , and let the T BD . straight lines , G E be ordinates to it hen the arc A B will bear a greater ratio to the arc AG than the straight line

BD to the straight line G E .

is 1 s e u iva e n t t o the a e raic a t e o re m t at if x a c o n s t an t Th q l lg b l h h , +y ” s is a m ax imu m w e n x r s t he n so y h / y / . is 1 5 e u iv a e n t t o th e a e raic a t e o re m t at If x a c o n s t an t Th q l lg b l h h , y , s min imu m e n x r s t he n fl /y is a wh / y / . T he ro o f o f t is t e o re m is e n e ra a s c ri e d t o R ic c i wh o ro v e d it p h h g lly b , p ‘ a e raic a in 1 666 an d u s e d it t o dra t he t a n e n t t o t he e n e ra lg b lly , w g g l ’ para b o lifo rm ; t hu s we s e e that Barro w s pro o f was in depe n de n t o f R ic c i n if B rro had n o t dis c o v e it e fo re Ri i o n r e ve er c c . ma e a s a o a w d b f y y g . 1 50 B A RR O WS GE OME TR JCA L - L E C T UR E S

2 8 . L e t c c o f c u A AM B be a ir le , whi h the radi s is O , and let OBE be a straight line perpendi c u lar to CA also le t ANE

a u c h MN d be c rve su that, when any straight line P is rawn

c the parallel to D E , cutting the cir le in M and curve in N , T the straight line PNis equal to the arc AM . hen the para bola described with axis A D and base D E will fall altogether outside the curve A N E .

2 F c m m 9 . rom the pre eding , and fro what is com only * m i o f known about the di ens ons the spaces A D B , A D E, the following formula may b e easily obtained S OA. arc A B .

F arc AB t o o f urther, if the is supposed be one 3 0

2 0A 1 1 C i m degrees , and 3 , then the whole rcu ference , calculated by this formula, will prove to be greater than

55 less a frac tion of unity .

0 arc AB 3 . Hence also , being given the , let arc A B p,

CA r DB e e t he f l u ma b e , and ; th n ol owing eq ation y used to find the right sine DB :

2 2 2 2 arid /(97 + 2 ) 1 2 flee/(97 + e > 2 2 2 i k fo r 7 7’ + we e or; substitut ng 3 p/(9 p ), hav — 2 — 2 — kp 4k e e ; or z é J(4E Ep)

1 . L e t c c u CA 3 A M B be a ir le whose radi s is , and let the straight line OBE be perpendic u lar to CA let also the curve

A N E be a part of t he c ycloid pertaining to the circle AMB and lastly let a parabola AOE be drawn with axis A D and T base D E . hen the parabola will fall altogether within the c yc loid .

e n o te at the e n d o f t is e c t u re S e h l .

’ 1 52 B A R R O WS GE OME TR TCA L LE C T UR E S

YZ c c DF V the major axis , and let the ir le O , whose centre is K FT o a point on the minor axis , pass through the p ints

O DOFP c c D , F, then I say that the part O of the ir le will lie within the part DMFNO o f the ellipse .

6 L e t 3 . DEG be a segment of a circle whose centre is L; D F and , any point F being taken in its axis G E , let M O be a

u c curve s h that, when any straight line RMS is drawn

R : RM= parallel to G E , S GE GF; then DMFO is an ellipse

u m z— F E F H th s deter ined ind H , such that G G GL G ; HZ D through H draw Y parallel to C, and let H Y equal LE;

i- then H Y , H F are the sem axes of the ellipse .

T G S t his is held to have been proved by regory Vincent ,

B I V P r0 1 ook , p . 54.

R L L — m C o O AR v . c D Hen e, seg ent EC segment DMFO

EG FG.

L e t c . D O 3 7 EO, D FO be portions of two ircles having a common c hord DO and axis GFE; then the greater portion

DEC will bear to the portion DOF0a greate r ratio than that which the axis G E bears to the axis GF

NOTE

2 B In 9 , arrow gives no indication of the source of the m ” m known di ensions , and there is also probably a isprint ” “ “ c for the spaces A D B , A D E , we should read the spa es ” NED AOED u B w A , , nless arro intended A D E to stand for c I f f 2 both the latter spa es . so, we have rom , area h o f . t e ANED AOED A D D E , and area of can thus be found by Barrow ’s methods m c ED F RS and C o plete the re tangle O , and draw Q parallel t o raw VZ e to A indefinitely near PMN d S parall l C, cutting — LE C T UR E _ X T A P P E N D TX

N F RT AC i PN CF P , C in V , Z, and parallel to , cutt ng , in T OP OM MT : MT : , Y ; then we have M R N V ,

° V M. MT. CP . N O H e n c e area ANED+ CD D E

the s u m o f V N . OP +

the s u m o f OM. MT + . s u m MT - i OM. (the of OM BD;

Area ANED OA BD OD D E . — BD CD . AOED CA. D E area o f A D . D E , F Y S 2 OA . BD D E , F l 1 1 2 arc g 5 ' or ( 0A+ OD) A B . It should be observed that the equ1 valent of the ex pres sion fo r the area A N E D is

“ 1 ” 1 l“ co s x /a dx x co s x /a

T he mu B 2 ) for la finally obtained by arrow , if we put 4 for the angle subt e nded at the c entre o f the circ le by the arc s in A B , reduces to 3 which is the f o f ' S u v ormula nelli s this , as I ha e already noted , has an 5 error o f t he order 95 ; a handier res u lt is obtained by

a 2 m a a 2 a 5 + . taking 4, when it beco es 3 sin /( cos )

F o r 2 c o f m arc PN § 3 , sin e M N ( this theore ) A M (in

fig . hen c e

1 area o f c yc loid area o f AMBD+ are a o f A N E O (fig . 51 ) 2 (CA arc A B OD BD)/ OA BD OD arc A B . T he remark made by Barrow in 3 3 indicates with almost c ertainty that the above was his method for t he cycloid .

N o w o f c , since the area the ycloid is less than the area t he c of corres ponding parabola, whi h is of A D . D E or of A D (D B + arc A B) ; hen c e we obtain

arc 2 A . D D A B ( 0 B+ O .

' T u v a s l a a 2 co s 2 c o s a his is eq i alent to ( ), a T limit obtained be fore in g 1 9 . hu s Barrow has here two v r c m in f e y lose li its , one in excess and the other de ect, each having an error of t he order ’ 1 54 B A R R O WS GE OME TR TCA L L E C T UR E S

The u o f m e res lts obtained, by the use these approxi at m ae 0 a re for ul , with the convenient angle of 3 degrees in “ f 6 8 The act 3 55 and 354 . formula obtained by Adrian , o f o f M z f o f the son Anthony , a native et and ather the better known Adrian Metius of Alkmaar ” is one of the “ ” m m c m c B ost re arkable lu ky shots in mathe ati s . y con s ide 1 in o f 6 M m g polygons 9 sides , etius obtained the li its 1 5 $3 5 and and then added n u merators and de 3 1 i u — 6 nom nators to obtain his res lt 3 2 2 11 or 3 1 1 3 B e u arrow seems to be cont nt, as us al , with giving the geometrical proof of th e formu la obtained by Metius which must have appeared atro c io u s to him as regards the method by means o f which the final result was obtained from the m I f B u fo r two li its . only arrow had not had s ch a distaste c u c B i long cal ulations , s ch as that by whi h r ggs found the logarithm of 2 (he extrac ted the square root o f 1 70 2 4 forty s e ven times s u cc essively and worked wit h over thirty- fi ve c e o f m u m m pla s deci als), it wo ld see to be i possible that Barrow shou ld not have had his name mentioned with that of V ieta and Van C e u le n and others as one of the great c ompu ters of Fo r he here gives both an upper and a z lower limit, and therefore he is only barred by the si e of

n fo r c c an m c . the a gle whi h he deter ine the hord Now, he wou ld certainly know the work of Vieta ; and this wou ld suggest to him that a s u itable angle for his formu lae wou ld ” ’ u f F o r V b e 7r 2 u . / , where was taken s ficiently large ieta s work would at once lead him to the formulae ” 2 co s 7r 2 2 2 ”i 2 / J[+ J{ n /( H] ,

2 52 72 - JP R H] , where there are u 1 root extrac ti ons i n eac h cas e .

“ a 8 u If, then , he took to be 4 , his angle wo ld be less than 46 1 2 e u 1 , and the error in his valu s wo ld be less than 45 this is abou t 1 0 ; hen c e Barrow has pra c tic ally in his hands the calc u lation o f 71 to as many decimal plac es as the number o f squ are root e xtrac tions he has the patience t o perform and the number o f dec imal plac es that he is willing to use .

1 56 B A R R O WS GE OM E TR I CA L L E C T UR E S

KZ L a 8 u c FZ u S . T lines , ¢ s h that M P and , q MF hen the * c spa es ADLKare equal .

2 c if c u . Hen e, the rve A M B is rotated about the axis AD , the ratio of t he s u rfac e produced to the spac e ADLK is that o f c c m c o f c c m e if a ir u feren e a ir le to its dia eter ; wh nce , n i ADLK t he u c . the space is k own , sa d s rfa e is known S m ome ti e ago we assigned the reason why this was so .

i 6 F i 1 F 1 ; . . g . 5 g 57

c u f c o f h t he 3 . Hen e the s r a es the sphere, bot spheroids ,

m m t F o r and the conoids receive easure en ? , if A D is the

o f c c c m c axis the oni se tion fro whi h these figures arise,

m o n e c c KZ there always exists so e line of the oni s, L, that m c ffi c . t can be found without mu h di ulty I erely state his , for it is now c onsidere d as common knowle dge .

m . AYl c c 4 With the sa e hypothesis , let be a urve su h that the ordinate FY is a mean proportional between the T m c FZ . orresponding FM , hen the solid for ed by the

t he u a 8 rotation of space abo t the axis , will be equal

s T he e qu ivale n t i y ds y . F o r t he c irc e t he u re A D LK is a re c tan e an d t he are a o f a z o n e l , fig gl ; i de d c i e an d s o o n is imme d at e ly u bl . L E CT UR E X ] ] to the solid formed by the rotation o f the spac e AD I about the axis A D .

B im l e ma u c h if 5. y s i ar r asoning , it y be ded ed t at , FY

u b ime di n FZ s u m is s pposed to be a a between FM and , the

f c u e u a b f m o the bes of the applied lin s , s ch as , g , ro the

01 68 u s u m curve 9 , to the straight line is eq al to the of the c u bes on the lines applied to the straight line A D from

m e m c . S the urve AY I i ilarly , the th ore holds for other powers .

6 . F t m c VXO ur her, with the sa e hypothesis , let the urve be such that EX M P ; and let the curv e wit/x be such that

T a a S c DVOB. P F . hen the sp ce mpfi the pa e

c 7. O bserve also that, if the urve A B is a parabola, whose axis is A D and parame ter R ; then the c urve VXO

m - will be a hyperbola , whose centre is D , se i axis DV , and the parameter of this axis eq u al to R . Also the space

l c e c f i w b e . a fitp wil a re tangl Hen e it ollows that , be ng

b c B given the hyper oli space DVO , is to be given the curve * - All A M B , and vice versa . this is remarked incidentally .

v 8 . It should also be possible to obser e that all the

a squares on the lines applied to the straight line fi, taken

m w c together, fro the curve ay, are equal to all the re tangles

c such as P F EX , applied to the line D B (or cal ulated) ; the m 2 s u PF . . cubes on ,t are equal to the of EX , etc and

S O O I] .

Ye t it h as a n impo rt an t s ign ific a n c e fo r it is t he firs t in dic at io n that B arro w is s e e kin g t he c o n n e c t io n be twe e n th e pro ble m o f t he re c t ific at io n t he ara an f t h u r t u re o f t he e r o H e is n o t o f p bo la d t hat o e q ad a hyp b la .

u t e s a t i e d Wt t i re s u t b u t n a s u c c e e ds i n 2 0 E x . s s . q i fi i h h l , fi lly , 3 ’ 1 58 B A R R O WS GE OM E TR TCA L L E C T UR E S

ma PM n u if FZ 9 . Also it y be noted that, Q bei g prod ced , l is P A e supposed to be equal to O, and ,a to O th n the spac e 0 88 is equal to the s pac e ADLK.

1 0 F le t u c u . MT urther, the straight line to ch the rve A B ,

c u v DXO c MT and let the r es , be su h that EX and

T e c u c e DXOB. Mb MF. h n the spa e is eq al to the spa

i 8 F . g I 5 .

1 1 t he f e o f t he m . Hence again , sur ac solid for ed by the rotation of the space A B D about the axis A D bears to the s pac e ADOB the ratio of the circ umfere nce o f a c ircle to

u if w c its radi s ; therefore, one is kno n , the other be omes

the known at same time .

Hence again one may measure the surfaces of spheroids and c onoids .

2 1 2 I f the u . c e line DY I is s h that EY EX . M P ; th n the solid formed by the rotation o f the space 088 about the axis 0 8 is equal to the solid formed by the rotation of the space DBI abou t t he axis I8 .

1 B ma m r o f 3 . y similar reasoning, one y co pa e the sums the c ubes and other powers of the ordinates with spaces c m o puted to the straight line D B .

‘ 1 60 BA RR O WS GE OME TR JCA L LE CT UR E S

1 8 . M ve if min e R oreo r , any deter ate l ngth is taken , and

“ 8 8 e u R if u c h , is tak n eq al to ; and the c rve DXX is su that j MF MP = R t o FX h c 0 8 u , t en the re tangle 8 5 will be eq al if c ADOXX. s ac e is c u v to the spa e Also, this p found, the r e f is orthwith known .

Many other theorems like this c ou ld b e set down ; but

f e m u f c I ear that these may already app ar ore than s fi ient .

1 u v d 9 . It sho ld be obser e , however, that all these

m e u u c an v d in c theore s are q ally tr e , and be pro e exa tly the same way if t he c u rv e A M B is c on v ex to the straight line A D .

2 0 f m m . Also , ro what has been shown , an easy ethod of

c c m drawing urves (theoreti ally) is obtained , such as ad it o f m u m o u ma c eas rement of so e sort ; in fact, y y pro eed thus

Take as you may any right- angle d trapez ial area (o f

o u u f c which y have s fi ient knowledge), bounded by two D parallel straight lines AK , L, a straight line A D , and any line KL whatever ; to this let anoth e r such area A D EO be FH so related that , when any straight line is drawn parallel

6 F i 1 62 . F i . 1 . g . g 3

c u t he OE KL n F to DL, tting lines A D, , in the poi ts , G , H ,

i i n and some determ n ate stra ght line Z is take , then the L E CT UR E X 1 ]

M e sq uare on FHis eq ual to the squares on FG and Z . or

c r GFI over, let the curve A I B be su h that, if the st aight line Z is produced to meet it, the rectangle contained by and

FI is eq ual to the space AFGO then the rectangle contained

c DLK The by Z and the urve A B is equal to the space A .

method is just the same, even if the straight line A K is

supposed to be infinite .

E x am le L e t p I . KL be a straight line, then the curve

Fi . OGE is a hyperbola . ( g

E x a m le 2 L p . e t the line KL be the arc of a circle whose

centre is D , and let A K Z ; then the curve AG E will

arc be a circle ; and the arc A B KL. F ( ig.

E x a mple 3 . L e t the line KL be an

equilateral hyperbola , of which the

centre is A , and the axis A K Z ; OGE then will be a straight line, and F i 1 6 g . 4.

the curve A B a parabola .

E x a m le . L e t KL p 4 the line be a parabola, of which the E axis is A D ; then the line OG will also be a parabola , and the curve A B one of the parab o lifo rms .

E x a m le L e t p 5. the curve KL be an inverse or infinite

2 3 c c FH paraboliform (for instan e, su h that Z /AF) ; then

c c the curve A B will be a cy loid , pertaining to the cir le whose diameter is equal to Z . (S e e figure on page

P o erhaps, if you consider, you may think of s m e examples that are neater than these . ’ 1 62 B A R R O WS GE OME TR Z CA L L E C T UR E S

NOTE

The c hief i n terest in the foregoing theorems lies in the f last o all . The others are mainly theorems on the change of the variable in integra tion (or rath e r that the equality (82 /8x ) 8y (8y /8x ) 82 holds tru e in the limiting form for o f B the purposes integration , although of course arrow ’ L z m a li does not use eibni sy bols) ; and secondly, the pp c ation o f this princ iple to obtain gen era l theorems on the o f u rectification curves , by a conversion to a quadrat re . ’ m B e aim ex res s l It must be borne in ind that arrow s sol , p y s ta ted was en era l b e m , to obtain g theorems and that erely introduced the c ases of the well - known c urves as examples of his theorems ; and to obtain the gradient of the tangent of a c u rve in general is the foundation of the differential c alculus . 1 6 In 59, Wallis showed that certain curves were capable “ ” of rectification the fi rs t - fru its of this was the recti fication

e m - m 1 660 of the s i cubical parabola by Willia Neil in , by ’ u s e m m m the of Wallis ethod . Al ost si ultaneously this ’ curve was also rectified by Van H u rae t (see Williamson s n t a l 2 f m I . C t he o , p . 49) by use the geometrical theore P rod u c e eac h ordinate of the curve to be rectified until the whole length is in a constant ratio t o the corresponding v c normal di ided by the old ordinate, then the lo us of the ex tremity o f the ordinate so produced is a c urve whose ” area is in a c onstant ratio to the length of the given c urve . Now this theorem is identical with the theorem o f 1 8 ;

m - hence, reme bering that the semi cubical parabola, whose 2 3 x arab o lifo rms c equation is R . y , is one of those p of whi h B r f c a row is so ond , and for whi h , as we have seen , he could find both the tangent at any point and the area under the t wo am curve between any ordinates, noting also the ex ples m 2 0 u given to the theore of 5 , it is beyond all do bt that Barrow must have perceived that for this particular para b o lifo rm 1 6 0 his curve OXX (fig . ) was the parabola 2 4y R . Why then did not Barrow give the ? The m res ult answer, I think , is given in his own re ark “ P m I X 1 do n o t lik e to u t before roble in Appendix III, p ' ' ’ m s zcrt le zn to a n o tlzer ma n s lzaro es t y , where he refers to the

work of James Gregory on invo lute and evolute figures .

1 64 BA R R O WS GE OME TR TCA L L E C T UR E S

O f course Barrow knew nothing about the notation 2 d dx — 2 Y/ or IJ(y Z )dx his wo rk would have de alfi it h small finite arc s and lines but the perv ading idea is better ’ ’ represented fo r argume nt s sake by the use of L eibni z ’ notation . I s u ggest that Barrow s proof wo u ld have run in something like the following form A C K D raw JPQR parallel and v ery IF H c near to G , cutting the urves as shown in the adjoining diagram , and draw JT perpendicular to I H then = are a . 2 . IT PFGQ PF FG ;

2 2 2 2 2 2 Z . Id 2 . 11 . 4 . s PF .

. FH. z PF .

m . e m . ADLK H nce , su ing , Z arc A B area

T B 2 0 E x . the hat arrow had , in , 5, really rectified w cy c loid is easily seen from the adjoining di agram. Barro i o f if starts , I suppose , w th the property the cycloid that, IT IM t he m T is , are tangent and nor al at I , then M per L t e n dic u lar D. e p to B AD Z, * w v c . IT TN FH e then sin e Z / , ha e 2 2 Z 2 2 2 FH Z . IT /TN Z TM. TN/TN 2 3 Z . TM/TN Z /A F . The area u nder the c u rv e KHL is giv en as proportional to the ordi B M nate of what I may c all its integral

c u v L . XI r e (see note to ect , and is easily shown to 2 be AF FH . arc AFHK Z 2 AF EH Z 2 IT Hence A I area / / ; that is , u c o f th e c c I c eq al t o twice the hord ir le parallel to T , whi h u is also eq al to it .

is fo o s at o n c e fro m t he u re at t e t o o f t he a e fo r Th ll w fig h p p g ; , Z Id P F F H an d IT : T N in t he o e r u re is e u a t o Id : dT in t he , ( l w fig ) q l ( u e r u re an d t is is e u a t o Id P F o r F H : 2 e n c e in o wer ure pp fig ) h q l h , l fig IT : T N FH : Z . A P P E ND I X I

S ta n da rd fo rms f o r in teg ratio n of circ u la r fu n ctio n s éy r u t to fi e u adra t u r e o a n er oo la ed c io n t q f yp .

it is be o n d tiie o r i in a l in ten tio n to to u clz Here, although y g

o n pa rticu lar tfieo rems in tfiis wo r/e and indeed to build

up these general theorems with such corollaries wo u ld tend

m m to leas e a rien d to swell the volu e beyond easure yet, p f * mdo tnin é s tfiem wo rtn tne t ro u ble v , I add a few obser a

o n c m tions tangents and secants of a ircle , ost of which

follow from what has already been set forth .

F i 1 66 g . .

G E R L F R E R D Le t NE A O WO . AOB be a quadrant of a

c BG n cir le , and let A H , be tange ts to it ; in H A , AO pro du c e d le t take A K , OE each equal to the radius OA; the

v O bs e r e the wo rds o f the o p e n in g paragraph whic h I have it alic iz e d . 1 66 B A R R O WS GE OME TR TCA L L E CT UR E S

KZ Z hyperbola be described through K , with asymptotes

le t h b e rb o la thfo u h AO, OZ and the y LEO be described g

m BO BG. E , with asy ptotes ,

arc Also let an arbitrary point M be taken in the A B ,

M u S MT and through it draw O S c tting the tangent A H in , MFZ B0 MPL touching the circle , parallel to , and parallel L to AC. astly, let arc A B , arc AM ; let the

0 rm- straight lines 7 , a d be perpendicular to and let

a A0 AB 03 1 71 MP. , , as , 01 , and ,

1 T h 0 F2 . e straight line 8 is equal to thus the sum of AC the secants belonging to the arc A M , applied to the line ,

5 equal to the hyperbolic spac e AFZ K.

m 2 . T he s u space that is, the of the tangents to the

arc AM , applied to the line is equal to the hyperbolic

space A FZ K .

Le t c v AXX 3 . the ur e be such that PX is equal to the

P s u m o f secant OS or CT ; then the space AO X, that is the OB the secants belonging to the arc A M , applied to the line ,

is twice the s e ctor AOM.

’ 1 68 BA R R O WS GE OME TR TCA L L E CT UR E S ‘

l when any point R is taken at random on the hyperbo a ,

RV DC then’ S R and a straight line S is drawn parallel to , , E SV OK O , are in continued proportion ; join ; then the space CEVI will be double the hyperbolic sector DOE.

1 0 . R eturning now to the circular q uadrant AOB, let CE OA AE and with axis , and parameter also equal to E A , let the hyperbola EKK be described now let the curve MFY AYY be supposed to be such that, when any ordinate

u n is drawn , FY is eq al to the tangent AS draw Y I K , cutti g CZ K in I and the hyperbola in K , and join O , then the space

AOIYA is double the hyperbolic sector ECK.

1 1 R L L R Y - B . C . O O O A Hence , if with pole E , a chord , and

CA c c YFM a sagitta , a on hoid AVV is described ; and if produced meets it in V ; then M V FY ; and thus the space AMV is equal to the space A FY .

1 2 m . Whence the di ensions of conchoidal spaces of this m kind beco e known .

L e t 1 3 . AE be a straight line perpendicular to RS (cutting

CE CA EYV c it in O) ; and let let AZZ, be two con hoids , conjugate to one another, described with the same pole E and a common chord RS from E draw an y straight line EYY RS EYZ , cutting , AZZ, in the points Y , Z, I ; also let

c m EKK be an equilateral hyperbola, with entre 0 and se i axis OE; draw IKparallel to AE and join OK.

T u - AE YZ hen the fo r sided space, bounded by , , and the

c c EY m c on hoidal ar s , AZ is equal to four ti es the hyper b o lic sec tor ECK. — ' L E CT UR E X I I A P P E N D I X 1 1 09

1 ll - 4. We will also add to these the fo owing well known

measurement of cissoidal space .

Le t c A M B be a semicircle whose entre is O, and let the straight line A H tou c h it ; and let AZZ be the cissoid that is congruent to it , having this property , that, if any point M

m t he is taken in the circu ference A M B , and through it BMS S straight line is drawn (cutting A H in ), and also a

MFZ n MZ AS straight line , cutti g the cissoid AZZ in Z , ;

arc then in a straight line take equal to the AM , and to

i 0 let straight l nes , 5 be applied perpendicular , and equal

T S to the versines A F of the arc A M . hen the trilinear pace

M AZ is double the space Hence, since the dimensions

c of the spa e are generally known , and indeed can be easily deduced from the preceding theorems , therefore the

dimension of the cissoidal space M AZ is obtained . Anyone

may make the calculation who wishes to do so .

The n followi g rider will close this a ppendix .

1 . Let AOB c B 5 be a quadrant of a cir le , and let A H , G

c c Z tou h the ircle ; also let the curves KZ , LEO be hyper

u bolas , the same as those that have been sed above ; let the arc A M be taken , and let it be supposed to be divided into parts at an infinite number of po mt s N through these

' u ON N d draw rad , and let the straight lines X ( rawn parallel m T to A H) meet the in the points X . hen the sum of the straight lines NX (taken along the radii) will equal to the

' space and the s u mof the straight lines NX (taken along parallels t o AH) will be equal to the space

PLQO/(3 radius) . A P P E ND I X 1 1

’

M et/zo d o E x /za u s tio n s . M eas u remen t o co n i al s u r aces f f c f . 0

F o r d e a n e s s an d the sake of brevity combined with f , especially for the latter, the proofs of the preceding theorems have been given by the direct method ; by h which not only is the truth firmly establis ed , but also

Bu t its origin appears more clearly . for fear anyone, less m Of accustomed to argu ents this nature , should hesitate to use them , we will add a few examples by which such

ma arguments y be made sure, and by the help of which indirect proofs of the propo sitions may be worked out .

1 . L e t the ratios A to X , B to Y , O to Z, be any ratios , each greater than some given ratio R' to S then will the ratio of all the antecedents taken together to all the con sequents taken together be greater than the ratio R to S .

2 . v n Hence it is e ident that, if any umber of ratios are

each of them greater than any ratio that can be assigned, then the sum of the antecedents bears a greater ratio to the sum of all the conseq uen ts than any ratio that can be as signed .

’ 1 72 B A R R O WS GE OME TR / CA L L E CT UR E S

h n throu gh these points draw straig t li es parallel to A D ,

T in bE me t cuttin g the c u rve the points X , and let these i OG by straight l nes M E , N F, , P H , drawn through the points

K A X R

F i 1 6 g . 7 .

OXPXRA X parallel to BD; also let the figure ADBMXNX , circumsc ribed to the segment AD B (c ontained by the

a straight lines A D , D B and the curve A B) , be gre ter than any space S ; then I say that the segme nt A D B is n o t less than the space S .

n 7. Also if it is supposed that the i scribed figure

HXGXFXEXZ DH is less than any S pace 8 ; then I say that the segment A D B is not greater than S .

8 c c c . Hen e , if there is any space , S say, the figure ir um

ua e AD MNOPRA an d scribed to which is eq l to the figur B , also the figure inscribed to it is equal to the figure HGFEZ DH then the space S will be equal to the segment

F o r b e AD B . , as has just been shown, it cannot greater

be than it, nor can it less .

Also these things can be altered to suit other modes o f c ircumscription and ins c ri ption ; it sho u ld be s u fficient to

m 1 have just made ent on of this . — LE CT UR E X I I A P P E N D I X I ] 1 73

NO T E —I n 6 B - , arrow uses the usual present day method o f translating the error fo r eac h rectangle across the diagram to s u m them up on the last rec tangle ; another point of interest is the striking similarity between t h e fig u re used by Barrow and the figure u sed by Newton in Lemma II o f B P rin ci ia ook I of the p , especially as Newton uses the

- c B four part division of his base, whi h is usual with arrow , B whereas in this place arrow, strangely for him , uses a

fi v - e part division of the base .

METH OD OF MEAS UR IN G T H E S U R F AC E O F C o N E s

L e t e A M B be any curve, whose axis is A D , and G a giv n

in BD . point it, a straight line at right angles to it Any

c v point M in the ur e being taken , draw M E touching the w curve, and from O dra CG perpendicular to M E ; also let CV be a straight line of given length , perpendicular to the

T n plane A D B ; join VG . he VG will be perpendicular to

. RS M G Also let be a line such that, if a straight line M I X

c BD is drawn parallel to A D , utting the ordinate in I , and t he R8 n : line in X , the M P M E VG I X ; or , if the line zAL MPY BD is such that, when is drawn parallel to , cutting

, t he L PE z ME axis A D in P , and the line A in Y , then

VG PY; then will either of the spaces BRS D or ADL b e dou ble the surface o f the cone formed by straight lines t m hrough V that ove along the curve AM B .

E x am le — Le t p . the curve AM B be an equilateral hyper b CV OA r ola, of which the centre is O, and let , and

CP x (for it helps matters in most cases to use a calcula t ion of this kind) ; j oin MO then the rectangle BRS D is do uble the area A M BV of the cone .

This elegant ex ample was furnished by that most excellent ’ 1 74 B A R R O WS GE OME TR I CA L LE CT UR E S

Sir F man , of outstanding ability and knowledge, rancis m f K t . o Jessop, , an Honorable orna ent of our college , which he was once a Fellow- commoner I shall venture to

ewel adorn my book , as with a j , not indeed at his request , nor yet I hope against his wish , by means of his cleverly

m . written work on this matter, kindly com unicated to me

P R OP OS ITION 1 m ABO If from a point E in the axis A of a right cone p, a EC straight line of unlimited length , , passes through the sur

o f o - face the c ne , and if with the end E kept at rest, the line

EC is c arried round until it returns to the place from which . o f it it started , so that always some part cuts the surface o f CFO the cone (say , through the hyperbola and the straight lines DA , AO situated in the surface of the cone), the solid contained by the surface or surfaces generated by the straight line EO so moved and by the portion of the

f o f O sur ace the cone bounded by the line or lines OF , DA , E AC, which the straight line O describes in the surface as

o f it is carried round , will be equal to the pyramid which

En c a m the altitude is equal to , the perpendi ul r drawn fro the point E to the side of the cone, and base equal to that part of the conical surface bounded by the line or lines

OFO, AD , AC, generated by the motion of the line EC.

P R OP OS ITION 2

L e t ABOp be a right cone ; let it be cut by the plane Am t CFD parallel to its axis ; let the straigh lines AO, A D be drawn from the vertex of the cone to the hyperbolic line CFD and , upon the triangle AOD let the pyramid EAOD

A P P E ND I X I I I

u a ra u r la D eren t t te Q d t e of tlze lzyper oo . ifi ia io n a n d I n r a tio o a lo a r it/zm a n d ex o n en tia l E u r tfi er s ta n da rd g n f g p . o r ms f .

O n lo o /lin o o er tfie recedin g p g , there seems to me to be m m so e things left out which it ight be useful to add . Anyone can easily deduce the proofs from has already been given , and will obtain more profit from them thereby .

P R OBLEM I

Le t K EG be any curve of which the axis is A D, and let

LMB c A be a given point in A D find a curve , say, su h that, when any straight line P EM perpendicular to the axis A D

c v u LMB AE cuts the ur e K EG in E and the c rve in M , and LM TM is joined , and TM is a tangent to the curve B , then shall be parallel to AE.

— ‘ T he construction is made as fo llo ws z Thro u gh any

i RZ po nt R , taken in the axis A D, draw a straight line S perpendicular to A D ; let EA produced meet it in , and in the straight line EP take PY equal to R8 ; in this way the nature of the curve OYY is determined ; then let the rectangle c ontained by A R and P M be equal to the space T AYYP (or PM is equal to the space AYYP/AR) . hen the curve AYYP shall have the proposed property .

It should also be easily seen that , other things remaining L E C T UR E X I I —A P P E N D I X [I ] 1 77

c if it the same, if the urve QXX is such that, EP cuts in X , PX AS then the space AXXP is equal to the rectangle con t ain e d XXP AR LM. by A R and the arc LM, or space A / arc

E x — a mple 1 . Le t AOG be a q uadrant of a circle ; if EP

an a . is y straight line perpendicul r to A D , join D E It is E M required to draw the curve A M B such that, if P produced MT meets it in M , and MT touches the curve, then shall be parallel to D E . — The construction is as follows : D raw AZ parallel to OS , and let D E produced meet it in S let the curve AYY m Y AS be such that, if P E produced eets it in Y, P then take P M space AY P/A D and the construction is effected . — NOTE I f the curve QXX is such that PX DS (or if

AO A D and QXX is a hyperbola bounded by the angle

AOG . ), then arc A M A D space AQXP

E x am l — p e 2 . Le t A EG be any curve whose axis is A D it such that, when through any point E taken in a straight EP AE line is drawn perpendicular to A D , and is joined , then AEis a given mean proportional between A R and A P of the order whose ex ponent is n /m. It is required to find the

TM AE. curve A M B , of which the tangent is parallel to

O n m bserve about the curve A M that AE arc A M .

n m E Now, if / (or A is the simple geometrical mean between A R and AP), then A EG will be a circle, and A M B

H en ce tne meas u remen t o t /te la tter the ordinary cycloid . f co mes n ral ru le o u t fro m a ge e . These also follow from the more general theorem added below . ’ 1 78 B A R R O WS GE OM E TR TCA L L E CT UR E S

NOTE

At first sight the foregoing proposition , stated in the f o f m m orm a proble , but (by i plication in the note above) B referred to by arrow as a theorem , would appear to be an “ - attempt at an inverse tangent problem. Bu t the sting is m in the tail this , and ost of those which follow, are really f m c c urther atte pts to re tify the parabola and other urves , by obtaining a quadrature fo r the hyperbola . T hat this is m e o E 1 so is fairly evident fro the not t x . above ; and it c m m c m P m be o es a oral certainty when we o e to roble IV, where Barrow is at last su c c essfu l . T he first s e nten c e o f the opening remark to this c lic m c Appendix , whi h I have put in ita s , akes it ertain that ’ ' B r wn The e f c at these were ar ow s o work . r eren e to Wallis “ ” the e n d o f P roblem I V almost shouts the fact that it was ’ through reading Wallis work that Barrow began to a c c u mu c a c o f en er al late, as was his invariable pra tice, ollection g t/zeo rems connecting an arc with an area ; it is also probable that it was only j u st before [ p u blication that he was able to complete his collection with the proof that the area under m a hyperbola was a logarith . The f B v proo , as arrow states, for the construction gi en P m 1 e m o u t wi in roble is very asily ade , by dra ng another ordinate NFQYparallel and near to MEPYand MWparallel u t F o r v to POto c NOin W. we ha e then

RS AR MWNW MP PT AE MT. P E/PA / PY/A R / / , // E xample I is not tru ly an example o f the problem if we ’ allow for Barrow s inversion o f the figure (a bad habit o f u u his that probably ca sed tro ble to his readers), to render m o f m o f this a true exa ple the ethod the problem , A D PM shou ld be made equal t o the spac e DYP instead o f the AYP e u v m c v space this , how ver, wo ld ha e ade the ur e lie m o f u a n in n ite on the sa e side A D as the q adrant , at fi dis t a n c e B u ; so arrow s btracts the infinite constant , equal t he u c u to area QADY , and th s gets a rve lying on the other f c side of the line A D, ulfilling the required onditions . E m 2 m the m xa ple , however, is a true exa ple of proble ; and it is particularly noteworthy on account o f the fact re c t ifi e s c u that it the cy loid , a res lt previously attained in

’ 1 80 B A R R O WS GE OME TR TCA L L E CT UR E S

P M and at right angles to the axis AD) ; and through the

’ points Y obtained in this way let the c u rve YYK be dfdwn if m APY then, PM is ade equal to the space /R , the nature

u of the c rve AM B will be established .

E x a l I — f i m e . L e t u o p AOG be a q adrant of a circle , wh ch

‘ the radius is equal to t he assigned length R ; let it be desired that the ratio of TP to P M shall be equal to that of

AE c : E PY R to arc ; then , sin e as prescribed , R arc A R ;

ar PY c AE; and h e n c e PM APY/R.

E x m le 2 — L t a , e AO p G be a quadrant of a circle, and suppose that the ratio TP PM has to be equal to that of

P E : R ; then PY will be equal to the tangent o f the arc T APYY . e E G E ; and the space is equal to R ar A . hen

PM arc AE.

P R O BLEM 3

Being given any figu re AMBD whose axis is A D and base

c D B , it is required to find a urve KZ L such that , when any Z straight line PM is drawn parallel to DB, cutting A D in P , and it is supposed that ZT touches the curve KZ L, then

TP P M . T he construction is as follows

L e t OYY be a curve such that , any finite straight line R

PMY : : PY being taken , and produced , P M R R ; then ,

L BD c LE taking any point in produ ed, draw at right angles * DL DL : R : LE m to , so that R ; then , with asy ptotes * DL, DG, describe the hyperbola EXX passing through E ;

LE c let the space XH be equal to the spa e DOYP, and pro

AD is ro du c e d t o G an d LE i in t h m n - D p s e s a e s e s e as G . T R E X I I — A P P E N I X [T] 1 8 1 L E C U D .

’ T duce XH and YP to meet ih l . hen will Z be a po int in

ZT TP . the required curve , and if is a tangent to it, P M

“ t he a It is to be noted that, if given figure is a rect ngle

KZ L . A D BO, the curve has the following property D H is a geometric mean between DL an d DO of the same order as

o r z DP is an arithmetic mean between DA and O ( ero) .

No w KZ L , if any curve is described with this property,

c h b e rb o lic and the tangent ZT is found pra tically, then the y

u space LEXH will be found , and this in all cases is eq al to * the rectangle contained by TP and AP. I t can also be seen that

(i) the space ADLK R(DL DO) ; 2 2 z s u D 2 m Z P . R L DO s u m (ii) the of , etc ( )/ , and the 3 U 3 DO3 3 Z P . R L of , etc ( )/ , and so on ; T (iii) if it is supposed that 95 is the centre of gravity of the

ADLK bLJ 5 figure , and g / is drawn perpendicular to A D and 05 4 6 DL bi DL+ DO R . to , then g p ( )/ , and 9 5 A D DO/LO.

P R OBLEM 4

Le t BOH a be a right angle , and B F par llel to OH; with

D B , D H as asymptotes , let a hyperbola FXG be desc ribed to pass through F ; with centre D describe the circle KZ L; lastly let AM B be a curve such that, if any point M is taken

in it, and through M the straight line D MZ is drawn , and it is also assumed that D I DMand I X is drawn parallel

H e re Barro w s e eks tlze c u rv e wlzo s e s u t ta ng en t is c o n s t a n t an d o bt ain s it b e o e ve r do e s n o t at rs t s e e m t o e rc e iv e the e x o n e n tia c , h w , fi p p l hara c t e r o f it F o r a t o u he s t at e s the ro e rt o f t he me t r . e o ic an d a i , l h gh p p y g r t hme tic me an s it is n o t t i in c o n n e c t io n it t h e n e x t ro e m t at he s t t , ll w h p bl h a e s that i h in t o do it o ari ms t h s as an yth g w h l g t h . As u s u a t e s e u an t itie s ave t o b e a ie d t o AD l , h q h ppl . ’ 1 8 2 B AR R O WS GE OME TR TCA L L E CT UR E S

BF I to B F, then the hyperbolic space X is equal to twice the circ u lar sector ZDK. It is required to draw the tangent at M to the curve AM B . D 2 08 DM . raw perpendicular to , and let D B B F R ;

m DK : R R z P : z then ake , and then D K P DM DT ; j oin TM then TMwill touch the curve A M B . It is to be observed that the curve has the following

m c property . D I is a geo etri mean between D B and DO

(or DA) of the same order as the arc KZ is an arithme tic

w 0 z T KL. mean bet een (or ero) and the arc hat is , if D I

' is a number in the geometric series beginning witt B and O KL ending with DA , and , are the logarithms of D B , DA ,

KZ . O r then will be the logarithm of D I , working the other way (the way in which ordinary logarithms go), if D I is a number in the geometric series starting with DO and

O DO t he ending with D B , and is the logarithm of , and arc L LK that of D B , then the arc Z will be the logarithm of D I .

ir m Now, the curve is co pletely drawn and the tangent

c c to it determined practi ally, it is evident that the ircular

o f » equivalent the hyperbolic space is given , or the hyper b o lic equivalent of the circular sector .

T t * ha most eminent man , Wallis , worked out most clearly the nature and measurement of this S piral (as well as of the space B DA) in his book on the cycloid and so I will say no more about it .

’ Wallis c hie f wo rks c o n n e c t e d wit h th e pro ble ms o f I n fi n it e s imal a c u u s are in c o u rs e o f re arat io n an d wi b e is s u e d s o rt s o t at C l l p p , ll h ly ; h it h as n o t be e n t ho u ght n e c e s s ary t o give he re an ythin g fu rther t han t his refere n c e .

’ 1 84 B A R R O WS GE OME TR TCA L L E CT UR E S

B supposing my suggestion is true , however, arrow would at that time have been unable to have proved his c eo met rica ll o r alculation g y , indeed in any other theoretical m u m anner, and so wo ld not have entioned the matter ; as t he rac ticall we see, he leaves constant to be determined p y Mecfi a n ice ( ), this way being just as good in his eyes as any o t other that was n geometrical . Le t AFB be a paraboliform suc h that P F is the first of m 1 means between PG and V c e u c P E . Also let KDbe another urv s h that

s pace AV D P R . P F .

T A B c o n hen , taking O O , to avoid a A a u FB is st nt, the eq ation to A PF'" A P PG and the eq uation of V D K is

m“ 1 m m‘ l (m AP R . PG Now area LKDP R (PF HL) H /m l/m W LP PD R . PG . (AP AL ) .

l ‘ l m c if x dx x / Hen e, we put for AP we obtain j / l ‘ l’m l/m I /m m m . m . s u . P of LP PD/R . G (AP AL ) Bu t m if is indefinitely increased, and R is taken equal m V KD m c to , the curve tends to beco e a re tangular hyper P m B bola ; and in roble s 3 , 4, arrow has shown that the a lo c l area is proportion l to g AP/A L. Hen e og A P/A L is the l /m limiting value of m(AP when m is indefinitely " lo x m u x 1 n increased , or g is the li iting val e of ( )/ , when n is indefinitely small . Now re membering that Briggs in his Ar it/zmetic a L og ar itlz mica had given the value o f I O to the power of as

' I 000 0 000 000 0 0000 1 2 8 1 1 2 00 2 u 7 9 493 3 3 5, it wo ld not have taken five minutes to work out log 1 0 2 3 058 50 9 c c m 0 B hen e, alling this nu ber , , arrow has

’ 1 B f fo r arab o lifo rms Considering arrow s ondness the p , it wo u ld seem almost to be impossible that he should not i have c arried out th s investigation ; although , if only for “ his us u al disinclination t o put his sickle into an other — L E CT UR E X I J A P P E N D TX [T] 1 8 5

’ P m man s harvest, as he remarks at the head of roble 9 , ’ he does not publish it ; he in fact . refers to Wallis work on the Logarithmic S piral as a reason why he should say

m b . P no ore a out it It is to be noted that, in roblem 4, B E S arrow constructs the quiangular piral , and then proves L S . it to be identical with the ogarithmic piral Hence ,

9 " r O r a l Ka if gg , and converse y ; thus

z x a dx ma m and j , where K , have to be determined . If we do not allow that Barrow had found out a value lo x for the base of the logarithm , yet assuming g to stand m B for a logarith to an unknown base, arrow has rectified a ra bo la ff the p , e ected the integration ta n d of , and the areas of many other spaces that he has reduced to the quad F o r rature of the hyperbola . instance ,

L . 2 0 E x . in ect XII , , 3 , he shows that are Z . A B area ADLK. 2 2 Now ATLK Z log AT/Z )/2 + Z /4 z - lo ADLK lZ g (AO+ DL. DL D2 2 In modern notation , since Z A / ,

2 2 2 J(x a )dx log [{x J(x

2 x % x . J( where the base of the logarithm has to be determined .

m L 2 S . i ilarly, in ect . XII , App I , § , he states that the s u m of the tangents belonging to the arc AM applied to the line is equal to the hyperbolic space AFZ K that is ,

ddd log A F/AO log co s d.

The theo rem of 51 is the same thing in another form .

L . Again , in ect . XII , App I , 4, we have the eq uivalent ’ of the integral of s in d; since Barrow s integrals are all

2 s in d dd 2 co s d z we . definite, find it in the form /

2 F we s ec 9 d s in 0 dd co s d rom 5, obtain ( ) / or j0 ’ 1 86 B A R R O WS GE OME TR TCA-L LE C T UR E S

— s ec d ddgiven as log + s in s in which o f course can be reduced immediately to the more u su al form log t an (d/z + 7r/4) the same result is obtained from ' 2 2 6 dx a —x , 7 ; or they can be exhibited in the form J /( ) {log (a T he theorem of g 8 is a variant of the preceding and proves that j co s d d(t a n d) is equal to I t a n d d(eo s d) t a n

d . co s d s ec d dd. , both being equal to j 2 ? The theorem of 9 reduces immediately to jdx /J(x a ) 2 log {x J(x T hu s Barrow complet e s the usual standard forms for the u c integration of the circular f n tions .

’ There is one other point worth remarking in this c o n n e c t io n ma c u n fo r , as it y a co t the rushing into print of this i x l rather und gested Appendi ; I have a ready noted that, B ’ from arrow s own words , this Appendix was added only just before the publication of t he book . I imagine this ’ was du e to Barrow s inability to c omplete the quadrature o f h B u t the hyperbola to his rat er fastidious taste . , in 1 668 u K u m L a t in e M c u , Nikola s a f ann ( er ator) p blished his L o a ritlzmo teclzn ia c m g , in whi h he gave a ethod of finding true hyperbolic logarithms (not N apieria n logarithms) ; o f “ a o ri — S this publication P rof. C j says tarting with the grand property of the rectangular hyperbola he m c m obtained a logarith i series , which Wallis had atte pted ” R B but failed to obtain . ( ouse all attributes the series to

Gregory S t Vincent . ) This may have settled any qualms that Barrow had con c erning the unknown base o f his logarithms , and decided him to include this batch of the o f theorems , depending solely on quadrature the n it hyperbola, and merely requiri g a defin e solution of the latter problem to enable Barrow to complete his standard K fm fo rms . au ann obtained his series by shifting one axis o f e x his hyp rbola, so that the equation became y ), m divis io n the expanded by si ple , and integrated infinite e m u u f seri s term by ter , th s obtaining the area meas red rom u the ordinate whose length was nity, and avoiding the infinite area close to the asymptote .

’ 1 88 B AR R O WS GE OME TR TCA -L L E C T UR E S

* T R . proportionals in the ratio D B to , or D I to R) hen

‘ take the space BIXF equal to double the space Z DIf ; and let DM D I then M will be a point on the required curve ;

DT DN. and if MT touches it,

P R OBLEM 7

Le t A D B be any figure, of which the axis is A D and the

i base is D B , and , any straight line PM be ng drawn parallel

c APM to D B , let the spa e be given (or expressed in some way) ; it is required from this to draw the ordinate PM , or to give some expression fo r it .

T . PZ APM ake any straight line R , and let R space ; in this way let the line AZ Z K be produced ; find Z O the

' perpendic u lar to it ; then PZ : P0 R PM .

s T 2 u OtIzer zo i e . ake PZ J( APM) let Z O be perpendic lar to the curve AZ K; then PM PO.

P R O BL EM 8

Le t AD B be any figure, bounded by the straight lines

u DA , D B and the c rve AM B , and through D let any straight

DM S line be drawn given the pace A DM , it is required to

find the straight line DM. T 2 ake any straight line R , and let DZ ADM/R draw

Z O c AZ K e r e n perpendi ular to the curve let D H , the p p 2 l r dic u a DM DM . . to , meet it ; then R DO

t 0tlzerwis e. Le DZ J(4ADM) ; and draw Z O per

e n dic u lar AZ K p to the curve ; let D H , the perpendicular to 2 DM DO. DZ, meet it ; then DZ

Z o r : 3 B F R /DB . — L E CT UR E X I I A P R E N E TX [1 1 1 89

NOTE These four problems are generally referred to by the “ - authorities as inverse tangent problems . I do not think B ’ T this was arrow s intention . hey are simply the com le tio n n p of his work on integratio , giving as they do a method of integrating a ny fu n c tio n, which he is unable to t dr a win a n d c a lcu la io n . do by means of his rules , by g T : m G hus , the probl of 5 reduces to iven any function , x t h u f ( ) say, cons ruct the curve w ose polar eq ation is r d r an d f ( ) , perfo m the given construction , the value ” o f x dx R lo D DI lo B j:f ( ) is equal to g B/ or R g D /DM.

S P r 6 imilarly, in oblem , the value of is given as

I DM I T he / /DB. construction as given demands the x m c ne t two proble s, or one of those whi h follow, called “ ” B w v by arro evolute and in olute construc tions . As an ‘ B alternative , arrow gives an envelope method by means of re the sides of the polar tangent triangle . It is rather B markable that as arrow had gone so far, he did not give the mechanical construction of derivative and integral curves in the form u sual in u p- to - date text - books o n m practical athematics , which depend solely on the property t hat differentiation is the inverse of integration . With regard to the propositions that follow under the “ ” ame m n of proble s on evolutes and involutes , it must be B d noted that , although at first sight arrow has ma e a mistake, since the involute of a circle is a spiral and cannot under any circumstances be a semicircle ; yet this is not B ’ a mistake , for arrow s definition of an involute (whether ’ he got it from James Gregory s work or whether he has misunderstood Gregory) is not the usual one , but stands for a polar figure equivalent in area to a given figure in rectangular coor din at e s vice ers a . m w , and v In a sense so e hat similar to this Wallis proves that the circular spiral is the involute of a parabola .

“ Hence, these problems give alternative P 6 methods for use in the given constructions for roblems 5, . T h hus in the adjoining diagram , it is very easily shown t at

D0 0 B are a DBMP. area , , é ’ 1 90 B A R R O WS GE OME TR JCAL LE CT UR E S

T m G o f hat brilliant geo eter, regory Aberdeen , has set on foot a beautiful investigation concerning involu te and

u t m s iclele in t o a n o tlzer evolute figures . I do not like to p y

’ ma n s b ar ves t m , but it is permissible to interweave a ongst these propositions one or two little observations pertaining

lza ve o bt ru ded tbems elves in a way to such curves , which l m upo n my n o tice zv/zils t I b ave been wo r eing a t s o etbing els e .

P R O BL EM 9

Le t A D B be any given figure, of which the axis is AD and the base is DB; it is required to draw the evolute

corresponding to it .

c 0 L c With entre , and any radius C , let a cir le LXX be

KZ Z c described ; also let be a curve su h that , when any MPZ D line you please is drawn parallel to B , the rectangle c ontained by PM and PZ is equal to the square on CL

Q (or PZ is equ al t o CL /PM) . T hen let the arc LX space DKZ P/CL (or sector LOX half the space DKZ P) and in

CX B0 0 o f take P M then the line , , is the involute

BMA c . , or the spa e of the space AD B

F o r instance, if A D B is a quadrant of a circle, the line

c c is a semi ir le .

CO R . 1 . It is to be observed that if the two figures A D B ,

AOG are analogous ; and the involutes are v and if CV DG c D B then , re iprocally,

8 4 4 6 0 01 DG : . , l D B

R 2 . h O . T c C e onverse of this is also true .

’ 1 9 2 E A R R O WS CE OME TR TCAL L E CT UR E S

C o ro llarie s

Tb o rem e 1 . If on the figure ,8095 is erected a cylinder hav ing its altitude equal to the whole circumference of the circle whose radiu s is CL; then the c ylinder will be equal

u u to the solid prod ced by rotating the fig re 0,8 HKabout 0K.

Tb eo r em 2 L . e t AM B be any curve of which the axis is DB AD and the base is , and AZ L a curve such that , when Z PM PZ 2 APM any straight line is drawn , J( ) ; and let

OYY v c be another cur e su h that, when the straight line

2 2 Z PMY Z P : R L is produced to meet it, P M astly,

DL R : LE u LDG let R , and thro gh E , within the angle ,

e l describe the hyp rbo a EXX; let the straight line Z HX, m T UO drawn parallel to A D , eet it in X . hen the space P Y

u c c will be eq al to the hyperboli spa e LXHE.

2 s u m 2 L H R . Hence, the of all such as P M/A PM EX /

/z T eo r em 3 . L e t AM B be any c u r v e whose axis is A D

u c and base is D B and let the c rve KZ L be su h that, if any Z M straight line R is taken , and an arbitrary line P is drawn

BD APM: PM : P ADLK parallel to , J R Z ; then the space

c c 2 O ‘ is equal to the re tangle ontained by R and JA B , or 2 * ADLK/ R JA OB . — E x a mple L e t AD B be a quadrant o f a circ le ; then the s u m of all s u c h as PM/ JAPM J(2 DA arc AB) .

Tlzeo rem L e t c 4. AM B be any curve of whi h the axis is

the E K YL - b e A D and base is D B , and let X , C two lines so

T he t e o re ms e u iva e n t t o e o re ms 2 an d ar e c e ar e n o u e v e n h q l Th 3 l gh , wit ho u t th e fin a l lin e in t he firs t o f t he p air ; B arro w in t e n ds t he m as s t an dard fo rms in in t e rat io n g . L E C T UR E ANT— AP P E N D I X [TL 1 93

related that , any point M being taken in the curve , and the straight lines MPX, MOY being drawn respectively parallel BD MT to and A D , and it being supposed that touc hes the

TP Y . T curve AM B , then P M Q PX hen will the figures

DBL . A DK E , G be equal to one another

f t he NOTE . O all the propositions so far this theorem is most fruitful ; sinc e many of the preceding are either con t in d e F r a e . o in it or can be easily deduc d from it , suppose m the line A M B is by nature indeter inate , then if one or K Y e other of the curves EX , C L is d termined to be anything m m you please, there will result fro the supposition so e theorem of the kind of which we have given a considerable

m f m I f fo r o . YL nu ber exa ples already , instance , the line C is supposed to be a straight line maki n g with BD an an gle equal to half a right angle (in which case the points D , G are d taken to be coinci ent), then we get the theorem of

L . 1 ect XI , .

i L c . 1 1 . If CYL is a l ne parallel to D B , we have e t XI ,

EKX c Again , if P M PX (or the lines AM B , are exa tly

m L X I 1 0 the sa e), hence follows ect . , .

Fu rther it is plain from the theorem that for any given space an infinite number of eq ual spaces of a different kind D L can be easily drawn thus , if the space C B is supposed to

c be a quadrant of a ircle, centre D , and A M B is a parabola whose axis is AD , we get this property of the curve EKX (by

r x R em - putting D B , A P , PX y , and for the s i para

2 Q 2 2 e o f the a 2 r R z b x x m ter p rabola or D B / A B), that / + y . ’ 1 94 B A R R O WS GE OM E TR TCAL L E C T UR E S

If, however, AM B is supposed to be a hyperbola , there

c c will be produ ed a urve EXKof another kind .

M o ore ver, on consideration , I blame my lack of fore sight, in that I did not give this theorem in the first place

(it and those that follow, of which the reasoning is similar and the use almost equal) ; and then from it (and the others

c an n that are added directly below), as I see be do e, have

deduced the whole lot of the others . Nevertheless , I think that this sort of P hrygian wisdom is not unknown either to me m or to others who may read this volu e .

NOTE

When I first c onsidered the title - pages of the volu me m m u fro which I have ade the translation , I was str ck by the fac t that the L ec tio n es Optica had directly beneath the main title the words Ca n ta br igix in S cb o lis pu blicis b a bita v c S C m (deli ered in the publi chools of a bridge), whereas no such notific ation appears on the separate title ~ page of ectio n es G eo met r ica the L . When later I found that the title - page o f the L ec tio n es M a t/zema ticce also bore this n o ti fi c at io n c m u c u n o , I be a e s spi io s that at any rate there was direc t eviden ce that these lectures on Geometry had ever t been delivered as professorial lec ures , though they might have been given to his stu dents by Barrow in his c apac ity of f l P f e college ellow and ecturer . As I considered the re ac , I was c onfirmed in this opinion and the above note would F o r if seem to corroborate this suggestion . surely these U L u S matters had been given in niversity ect res in the chools , it would not have been n ecessary to wait till they were ready fo r press before Barrow should find o u t that his T heorem 4 was more fruitfu l and general than all the c c others . His own words ontradi t the supposition that he e m fo r he initially did not know this theor , blames his “ T t e want of foresight . his raises the point as to h exact date when Newton was shown these theorems ; this has been dis c u s s edi n the Introductio n .

L E CT UR E X I I I

[The subject of this lecture is a discussion of the roots o f T certain series of connected equations . hese are very e ingeniously treated and are xceedingly interesting, but c have no bearing on the matter in hand ; a cordingly , as m c m y spa e is li ited , I have omitted them altogether . ]

’ L a u s D E O Opt imo M a fi mo F I NI S

c n 1 6 In the se ond editio , published in 74, there were e m added the thr e proble s given below, together with a set M m P i o f theorems on Maxima and ini a . roblem II s very interesting on ac c ou nt o f the diffic ulty in seeing how Barrow a i a l ebra ica ll rrived at the construct on , unless he did so g y .

P R BL E M L e t o f O I . any line AM B be given ( which the

’ AD an d c axis is , the base is required to draw a urve

MNG e A N E , such that if any straight line is drawn parall l to BD, cutting A N E in N , then the curve A N shall be equal to G M .

The c MT u c c u urve AN E is such that if to hes the rve A M B , 2 — z GN T M . and NS the curve AN E , then 30 G J(G TG ) E X TR A CTS F R OM S E CON D E D I TI ON 1 97

P R BL M O E II . With the rest of the hypothesis and con

c stru tion remaining the same, let now the curve A N E be required to be such that the arc A N S hall be always equal to the intercept M N .

L e t the curve A N E be suc h that 80 : GN 2 TG . GM

2 2 M T . (G G ), then A N E will be the required curve

M L P R BL . e t e O E III any curve DXX be given , whos axis is DA ; it is required to find a c u rve AM B with the property

GXM u that , if any straight line is drawn perpendic lar to A D ,

S MT n and it is given that is the ta gent to the curve A M , then MS CX. l TM C early the ratio TO (that is , the ratio of GO to MS

u T i or CX) is given ; and th s the ratio G G M is also g ven .

B m The arrow does not give proofs of these probl e s . only geometrical proof of the second I can make out is as follows D P R GNM raw Q parallel to , cutting the curves A N E , A M B MW in Q, R respectively, and draw , N V parallel to A D to T h e N m R W . RW V meet PQ in , R hen we av O Q fro the supposed nature of the c u rve also from the several differen we RWGP MG GT V GP NG tial triangles , have / / , Q / /GS , and NQ/GP N S/S P ; and therefore — NS . GT M C . G S GT . NO. S quaring , Z z 2 2 2 S . T . N . MO GS MG . G G G + GT NG ; 2 2 GS GM GT 2 MC GT . NG hence, ( ) . 2 — 2 or GS z GN 2 MG . GT : (GM GT ) .

B u t I can hardly imagine Barrow performing the opera al e tion of squaring , unless he was working with g braic symbols ; in this c ase he would be u sing his the ore m that 1 L ( ect . X, P O S TS CR I P T

E x t ra c t s f ro m S t a n da rd A u tb o rit ies

S i c u m c o n n e this vol e has been ready for press , I have s u lt e d the following au thorities for verifi c ation or c ontra i i d ct on of my suggestions and statements .

‘ R OUS E BALL (A S b o rt A cco u n t of tbe H is to ry of M a tbema tics )

“ ’ m . m m (i) It see s probable , fro Newton s re arks in the ’ flu x io n al c e ontroversy, that Newton s additions w re confined to the parts (o f the L ec tio n es Optica c l G eo met ricx ) whic h ” i c dealt w th the O pti s .

“ The 1 66 (ii) . lectures for 7 suggest the analysis ” c m f by whi h Archi edes was led to his chie results .

. c o n c c (iii) Wallis , in a tra t the ycloid , in identally gave the rec tification o f the semi- cubic al parabola in 1 659 the e m 1 6 probl having been solved by Neil , his pupil, in 57 m c e T e . the logarith ic spiral had been re tifi d by orricelli (i. “ before The next curve to be rectified was the 6 cycloid this was done by Wren in 1 58 .

T i h ( his contrad cts Williamson entirely ; I suggest t at, of B m e c the two, all is probably the or orrect, if only for the fact that this would explain why Barrow did not remark on the fac t that he had rectified both the cycloid and the logarithmic spiral . )

’ 2 00 B A R R O WS GE OME TR TCA L L E C T UR E S

it t L V E r tlz . r P n c c . B . Xl c d. A rofessor O E ( y , . f n n ites imal a lc u lu s fi C .

“ S t c (i) . Gregory Vin ent was the first to S how the c onnection between the area under the hyperbola and he s it logarithms , though did not expres analytically . Mercator used the connection to calc u late natu ral ” logarithms .

“ F e rmat x , to differentiate irrational e pressions , first of all rationaliz ed them ; and altho u gh in other works he ”

u do . used the idea of substit tion , he did not so in this case

h L ectio n es O ticce et eo met r a (iii) . T e p G ic were apparently written in 1 663

“ B (iv) . arrow used a method of tangents in which he compounded two velocities in the direction of the ax es of ” x u and y to obtain a res ltant along the tangent to a curve . “ In an appendix to t/zis bo o k he giv e s another method which differs from F ermat ’s in the introduction of a second ” differential .

(Both these statements are rather misleading . )

“ v 1 66 B ( ) . Newton knew to start with in 4 all that arrow knew, and that was practically all that was known about the subjec t at that time .

“ L z f m (vi) . eibni was the first to di ferentiate a logarith and an exponential in L X P B . I I . ( arrow has them both in ect , App I II , rob .

“ R . C 1 2 2 f (vii) oger otes was the first , in 7 , to dif erentiate ” a trigonometrical func tion . (It has already been pointed out that Barrow explicitly ff di erentiates the tangent, and the figures used are applic able to the other ratios ; he also integrates those of them which are n o t thus obtainable by his inversion theorem from the differentiations . Also in one case he integrates

it . an inverse cosine, though he hardly sees as such With 1 2 2 P L v f m regard to the date 7 , as rofessor o e kindly in or e d P 05 TS CR I P T

m him o f e on my writing to , this is the date the posthumous ’ publication of Cotes work ; P rofessor L ove referred me to the passages in Cantor from which the information was o btained . ) “ The v (viii) . integrating cur e is sometimes referred to ” as the Quadratrix . ’ ’

T L z B . ( his is eibni use of the term, and not arrow s B 1 5 c With arrow, the Quadratrix the parti ular curve whose eq uation is

‘ r —x zem wx z r ( ) / .

T here are a host of other things both in agreement with and in c o ntradic tion of my statements to be found in this erudite article ; nobody who is at all interested in the subject shou ld miss reading it . B u t I have only room fo r the few things that I have here quoted .

’ ’ ’ D r G R H R D E di/io n s o L ez bmz za n eff E A T ( f . )

’ M d H o it al L z w (i) . In a letter to the arquis p , eibni rites z B I recogni e that arrow has gone very far, but I assure S ir h m you , , that I have not got any elp fro his methods . As I have recogni z ed pu blicly those things for which I am indebted to Huygens and , with regard to infinite series , to d Newton , I should have one the same with regard to ” Barrow . m m (In this connection it is to be re e bered , as stated in ’ P f c L z c the re a e, that eibni great idea of the alculus was the freeing of t he work from a geometric al figure and the con v n i hi u f e e n t notation of s calcul s o difference s . T hus he might truly have received no help from Barrow in his m B estimation , and yet might, as Ja es ernoulli stated in ' t he A d a E r u dzfo r u m 1 6 1 for January 9 , have got all his m f B L B funda ental ideas rom arrow . ater ernoulli (Aria June 1 69 1 ) admitted that Leibniz was far in v B v ad ance of arrow, though both iews were alike in some ways . )

’ L iz et 07 Ca k u lz (ii) . eibn n “ ” states that he obtained his c haracteristic trian gle from 2 02 BA R R O WS GE OJWE TR Z CA L L E C T UR E S

' m o f P a lzas D e tt o n ville m so e work ascal ( ), and not fro

B . Th ma if he arrow is y very probably be the case, has n fo r B c not given a wro g date his reading of arrow , whi h he states to have been 1 675 this wou ld not seem to be an altogether unprecedented pro c eeding on the part o f L z c f u m C . eibni , a cording to antor It is di fic lt to i agine L z f c c o f B that eibni , a ter pur hasing a opy arrow on the c o f O O u advi e ldenburg , especially as in a letter to ldenb rg 1 6 m c of April 73 he entions the fa t that he has done so , should have put it by for two whole years ; unless his geometric al powers were not at the ti me equal to the task ’ o f me B finding the hidden aning in arrow s work . i (ii ) . Gerhardt states that he has seen the c opy of Barrow e R L n m r ferred to in the oyal ibrary at Ha over . He entions the fac t that there are in the margins notes written in ’ L z n u n eibni own otation, incl ding the sign of integratio . He also lays stress on the fact that opposite the Appendix to “ ” L . f ect XI there are the Latin words for knew this be ore . T L z his tells against eibni , and not for him , for this Appendix w c L z refers to the work of Huygens , hi h of course eibni “ ” G tim e knew before, and erhardt does not state that wo r ds o ccu r in a ny o tlzer co n n ectio n ; hence we may argue that this partic ular section was the only one that L eibniz ‘ “ ” f The o f knew be ore . sign integration , though I cannot m o f f 1 6 me find any ention it be ore 75, ans nothing , for it e c L z might be added on a second r feren e, after eibni had ’ “ found out the value o f Barrow s book . A striking c oin ” c ide n c e exists in t he fact that the two examples that L eibni z gives o f the application ‘ o f his c alculu s to geometry h B . t e are both given in arrow In the first, figure (on the assumption that it was taken from Barrow) has been altered c m in every con eivable way ; for the second , a theore of ’ G u B L z u regory s q oted by arrow , eibni gives no fig re, and ’ it was o n ly af ter referen ce to B a rro n/ s fig u re tna t Z co u la ’ ’ co mplete L ezbn zz co n s tr u ctio n from the verbal directions T L z he gave . his looks as if eibni wrote with a figure him wa s a lr ea d dra wn beside that y , possibly in a copy of ’ ’ m B u G . regory s work , or, as I think, fro arrow s fig re I have been unable to ascertain the date of p u blic ation of m G c this theore by regory, or whether there was any hance of its getting into the hands of L eibniz in the original .

’ 2 04 B A R R O WS GE OME TR I CA L L E CT UR E S

’ E D MUND S TON E (Tra n s la tio n of B arro ws Geo metr ica l

L ect u res u b. 1 , p 73 5)

T his translation is more or less useless for my purpose . F m u irst of all , it is a ere translation , witho t commentary of

e S . any sort, and without even a prefac by tone “ T he title - page given states that the translation is from L the atin edition revised , corrected and amended by the ” f f f late S ir Isaac Newton . I this re ers to the edition o 1 6 0 S Bu t 7 , tone is in error . , since at the end of the book, c m there is an Addenda, in whi h are given several theore s m u that appeared in the second edition , it ust be concl ded that S tone used the 1 6 74 edition . It is to be remarked are m m that these theorems on axi a and minima , and , c ac ording to the set given by Whewell , only form a part of those that were in the second e dition of Barrow some two or three very interesting geometrical theorems being omitted ’ x m B one of these is e tre ely hard to prove by arrow s methods , and one wonders how Barrow got his theorem ou t t/ze pro of “ ” dro s o u t b too u s e o d dx p y f y / , which may account for

B . T arrow having it, but not for S tone omitting it his seems e m to give a clue as w ll to an altogether unjustifiable o ission , by either Newton or S tone (I do not see how it could have o f been Newton , however) at the end the Appendix to X I T w m m in L c . o e t . theore s have been o itted ; their e lusion was only nec essary to prove a third and final theorem of the Append ix as it stood in the first edition ; CED OFD c c namely, that if , are two ir ular segments having m c OGO FE a com on hord , and an axis G , then the ratio of the s e g . CED to the seg . CFD is greate r than the ratio of G E to GF. In S tone the two lemmas are omitted and the m o n tr T h theore is directly c a dicted. e proof given in S tone depends on unsound reasoning equivalent to

I f a o c a z d o > c z d > , then + + ,

o c d without any reference to the value of the rati of to , as F as compared with that of a to o. inally the theorem c originally given is corre t, as can be verified by drawing m m . and measure ent, analytically, or geo etrically S I S e In addition to this alteration , in tone there add d a passage that does not appear in the first edition , nor is _ it P OS TS CR JP T

’ Bu t given in Whewell s edition . I seem to hear you ‘ ‘ ’ A / 55 a k a w e T e a M o f . crying out p v B é , r at something else “ In a table o f errata the last fo u r words are altered to Give ” T h us something else . e Greek (there should be no “ aspirate on t he first word) literally means S hake acorns m ” m S fro another oak . If this alteration was ade by tone, he m B the addition of the passage, after t anner of arrow, is m T he an i pertinence . point is not , however, very important l o u t c o m in itse f, but taken with other things , points the ’ parat iv e uselessness of S tone s translation as a clue to i mportant matters . T he whole thing seems to have been done carelessly and hastily ; there hardly seems to have been any attempt to render the Latin of Barrow into the best contemporary E nglish and frequently I do not agree with S tone ’s render

c h ma u i . ing , a remark whi y unfort nately cut e ther way O f ma l course the passage y, though it is hardly ikely, have been added by Barrow ; such an unimportant state ment wo u ld hardly have been added in those days o f dear o b oks ; it is also to be note d that Whewell does not give it . T he point could only be settled on sight of the edition from B m c S m . whi h tone ade his translation arrow , however, akes m L 1 0 a somewhat similar istake with ratios in ect . IX, , T and S tone passes this and even renders it wrongly . his error has been noted on page 1 07 ; the wrong render — fo llo ws z B FG EF TD z RD ing is as arrow has , by c d m a whi h , accor ing to his list of abbreviations , he e ns an d S FG EF not , as tone renders it , / TO R c n m o f / D , without noti i g that this does not ake sense the proof. P erhaps one sample of the carelessness with which the book has been revised will suffice : he has A x B A dividend (s ic ) by B A A multiplied or drawn into B B in any case want of space forbids further examples .

It is this untrustworthiness that make it impossible to take S tone’s statement on the title—page as incontrovertible n o r another statement that these lectures on geometry were ’ 2 06 B A R R O WS GE OME TR I CA L L E C T UR E S delive red as Lu c as ian Lec tu res it is also to b e noted that ’ he giv es as Barrow s P refa c e the one already referred to in the Introduction as t he P reface to the O ptic s and omits the

P reface to the Geometry .

H E LL Tno M a tnenza tical Wo rk s o I s aa c B a rro w W EW ( f ,

i . P r es s 1 8 60 Ca nt o. Un v , )

o f c (i) . S tress is only laid on two points one ourse is the differen t ial triangle ; the other is the mode o f finding the

’ areas o f curves by comparing them with the s u m o f the in c rc u c am t he scribed and i ms ribed parallelogr s , leading way ’ h m to Newton s met od of doing the sa e , given in the first P i s e c tion o f the rin cip a .

“ It is a matter of diffic u lty fo r a reader in these days to follow o u t the complex constru ctions and reasonings o f ’ a mathematician o f Barrow s time ; and I do not pretend that I have in all c ases gone thro ugh them to my satis f c T f a tion . ( his is proo positive that Whewell did not m i B ’ grasp the inner ean ng of arrow s work that being done, if u there is , I think, no d fic lty at all . )

The - L ectio n es M a t/zenza tica (iii) . title page of the states that these lectu res were the lec tu res deli vered as the

1 1 L . XV i Lu s ian L 1 66 66 666 I . s c a ectures in 4, 5, ; and ect headed

MAT H E MAT I C I P R O FE S S O R I S L E CT I O NE S D MD LXV A . C . ( . I)

XX m L . (iv) . ect IV starts the work on the ethod of m u c u Arc hi edes , which wo ld thus appear to be the le t res for

1 66 u me B . 7, as g essed by , and as stated by all

(v). Whewell gives the additions that appeared in the 1 T c f u m s e cond edition o f 6 74. hese onsist of o r theore s and a grou p o f propositions on Maxima and Minima . O n e m f n a theore is noteworthy , as its proo depe ds on the ddition ru le for differe n tiation and the fac t that 2 2 (ds /dx ) 1 (dy/doc)

’ 2 08 B A R R O WS GE OM E TR I CA L L E C T UR E S A P P E N D I X

L t iv X . e N (ii) SC be the ordinate for the g en abscissa, 5 , {Z the given subtangents let XX be a c urve such that : N N NX R N6 5 find Q, a fourth proportional to NX S e e NZ NX, NZ, then QX will touch the curve XX [ 1 1 2 note on page , rule

L t (iii) . e pp be a curve whose ordinate varies as the square root of the o rdl n at e of 71 7 ; the n i t s subtan gent N R 2 NP (pas e I O4)

t KK (iv) . L e be a curve whose ordinate varies as the cube root of the ordinate of XX then its subtangent NC 3 NQ (page 1

L t 0 0 (v) . e be a curve such that its ordinate is the sum KK c r of the ordinates of the curves , pp ; take N , N double of K N Cc Rr N , p respectively ; then , are the tangents to the

. K K curves whose ordinates are double those of the curves , pp ; let these tangents meet in s ; then s o will touch the 0 0 S e e curve . ( note on page d d If s is drawn perpendicular to R0 to meet it in , then d8 88 ff will touch the curve , whose ordinate is the di erence

d KK . S e e between the or inates of the curves , pp ( note on page

z n jfgéf A n a ly t ic a l E q u iv a len ts

I I I I du 1 dd) 1 de I f u d) 9 + + ’ ‘ NP NF NT u dx 4, dx 6 dx

‘ I 1 dv I d1; 1 dz L __ _ i I f v = E/Z» NQ NX NL

I f U / Zt U NR 2 NP N ,

I f v fi/ o NC 3 NQ ,

He n c e [wax y

U du i? i l i an dx 3 v dx

1 dd) I dfi «WE/Z) —- — i - ‘ i 2 ia clic O dj c l 3 ’ 2 1 0 B AR R O WS GE OME TR I CA L L E C T UR E S

2 1 2 BA R R O WS GE OM E TR I CA L L E CT UR E S

me n P e s a l I I I . S pe c i ag n d P at e

’ Two specimen pages and a specimen of one of Barrow s

. The u s e d b , plates here follow pages show the signs, y Barrow and the difficulty introduced by inconvenient or “ ” an d m n unu sual notation, by the ethod of run ing on the argument in one long string , with interpolations . B ’ The second page shows arrow s algebraic symbolism . E specially note which stands for since A m

’ The specimen plate shows the quality of Barrow s dia The Fi 1 6 grams . most noticeable figure is g . 7 , to be considered in connection with Newton ’s method as given P r a in the in cipi . A P P E N D I X

L s c r . I X. 73 D current es unétis S T erit fem er D T 2 . 8 . ( aud D E F con p , p L Fi - B t ut c ubi i rum D F erit fem er D T D 3 ac li g 99 G E fun pfa , p 3 ;

mili deinc c ps mo do .

' Si efi x D T B co nc u entes in T uas demiler o fi iio X. mr V , rr , q p

’ Fi . 1 o ne data reéca D B t ranfeant et am er B linear E BE F B F t a es g 5 i p , l , t aa ac un u P G adD B a a e a fit er etub P F co demo r u du qu q c p r ll l , p p ‘ e me Arithmeticc inte P C P E t an at autem B c u vam din dia r , g R r in t E B E o po rtet linear F B F t angent emad 8 det erm a e. Sum tis N M o rdinum in u bus fun: P F P E c x o nentibus p ( q i , p )

fiar N 6: c o naeéla F t ur B S ; hz c curvam B F co ntinget .

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