^%^ BERKELEY ^librar"* Af ^LJ) ■^.^ s^imI^ V2 ^ SSiMklW^W ^^ /T- ^ PS A HISTORY OF THE CONCEPTIONS OF LIMITS AND FLUXIONS IN GREAT BRITAIN FROM NEWTON IO WOODHOUSE Copyright in Great Britain under the Ad of i<)ii TABLE OF CONTENTS INTRODUCTION PAGE I CHAPTER I Newton Newton's Publications printed before 1734 . . . , . 2 Principia .......... 3 Wallis's De Algebra Tractatus ...... 14 Quadratura Curvaruin{\']0\) 17 An Account of the Conwiercitim Epistoliciim ... 26 Newton's Correspondence and Manuscripts not in print in 1734 . 29 Remarks ........... 32 CHAPTER II Printed Books and Articles on Fluxions BEFORE 1734 John Craig, De Moivre, David Gregory, Patio de Duillier, Cotes Ditton. Cheyne . John Harris, 1702, 1705, 1710 Charles Hayts, 1704 . William Jones, 1706 . Humphry Ditton, 1706 Commerciiim Epistolicum D. Johannis Collins, 17 12 Joseph Raphson, 17 15 Brook Taylor, 17 15 . James Stirling, 17 17, 1730 . Edmund Stone, 1730 Remarks .... 37 40 41 43 43 47 49 50 50 50 55 VI LIMITS AND FLUXIONS CHAPTER III BERKELEY'S ANALYST (1734); CONTROVERSY VVITH JURIN AND WALTON The Analyst .... Jurin's first reply to Berkeley Walton's first reply to Berkeley . Berkeley's reply to Jurin and Walton Walton's second reply to Berkeley Jurin's second reply to Berkeley . Berkeley's second reply to Walton The second edition of Walton's second reply Remai- Ics ....... 57 64 69 72 78 80 85 87 89 CHAPTER IV Jurin's Controversy with Robins and Pemberton Robin s's Discourse on Fhixioiis ....... 96 Jurin's review ofhis own letters to Berkeley .... loi Robins's rejoinder . . . . . . . . .106 The debate continued ........ 109 Pemberton enters the debate . . . . . . .129 Debate over Robins's review of treatises by Leonhard Euler, Robert Smith, and James Jurin . . . . . .139 Remar ks . . . . . . . . . . .146 CHAPTER V TEXT-BOOKS IMMEDIATELY FOLLOWING Berkeley's Attack John Colson, 1736 149 James llodgson, 1736 155 Thomas Bayes, 1736 . . 157 John Muller, 1736 162 Anonymous translation of Newton's Methcd oj Fluxiovs, 1737 . 164 James Smith, 1737 165 Thomas Simpson, 1737 ........ 169 l')enjamin Martin, 1739, 1759 171 An anonymous text, 1741 ........ 172 John Rowe, 1741, 1757, 1767 • 175 Berkeley ten years after . . . . . . . .178 Remarks 179 TABLE OF CONTENTS VII CHAPTER VI Maclaurin's "Treatise OF Fluxions, 1742" Remarks CHAPTER VII TEXT-BOOKS OF THE MIDDLE OF THE CENTURY John Stewart, 1745 .... William Emerson, 1743 (?), 1757, 1768 Thomas Simpson, 1750 Nicholas Saunderson, 1756 John Rowning, 1756 Israel Lyons, 1758 William West, 1762 James Wilson, 1761 Remarks . 190 192 194 197 198 201 202 202 206 CHAPTER Vili Robert Heath and Friends of Emerson in Controversy wiTH John Turner and Friends of Simpson Robert Heath Main articles in the controversy . Ladies' Diary^ 1751, 1752. Popular impression of the nature of fluxions Remarks ....... 207 209 219 222 223 CHAPTER IX Abortive Attempts at Arithmetisation John Kirkby, 1748 • 225 John Petvin, 1750 . 230 John Landen, 1758 . 231 James Glenie, 1793 • 235 Remarks ......... . . 238 vili LIMirS AND FLUXIONS CHAPTER X Later Books and Articles on Fluxions Encyclopa:dia Britannica^ iJTi, i779, 1797 Robert Thorp, 1777 . F. Holliday, 1777 Charles Hutton, 1796, 1798 S. Vince, 1795, 1805 . Agnesi — Colson — Hellins, 1801 . T. Newton, 1805 William Dealtry, 1810, 18 16 New editions .... Keviarks ..... PAGE 240 241 243 244 245 247 250 252 253 253 CHAPTER XI Criticisms of Fluxions by British Writers under the Influence of D'Alembert, Lagrange, and Lacroix Review of Lagrange's Fonctions analyiigues, Review of a memoir of Stockler, 1799 Review of Lacroix's Calcul différentiel^ 1800 Review of Carnot's Kéjlexions, 1801 Robert Woodhouse, 1803 . William Hales, 1804 . Encyclopcedia Britannica, 18 io . Lacroix's Elenientary Treatise, 1816 Remar ks ..... 1799 255 259 260 262 263 267 269 271 274 CHAPTER XH Merits and Defects of the Eighteenth-Century British Fluxional Conceptions Merits 277 Defects . . 279 Addenda 289 Indkx 294 LIMITS AND FLUXIONS INTRODUCTION I. EVERV great epoch in the progress of science is preceded by a period of preparation and prevision. The invention of the differential and integrai calculus is said to mark a " crisis " in the history of mathe- matics. The conceptions brought into action at that great time had been long in preparation. The fluxional idea occurs among the schoolmen — among Galileo, Roberval, Napier, Barrow, and others. The differences or differentials of Leibniz are found in crude form among Cavalieri, Barrow, and others. The undeveloped notion of limits is contained in the ancient method of exhaustion ; limits are found in the writings of Gregory St. Vincent and many others. The history of the conceptions which led up to the invention of the calculus is so extensive that a good-sized volume could be written thereon. We shall not yield to the temptation of lingering on this pre-history at this time, but shall proceed at once to the subject-matter of the present monograph. CHAPTER I NEWTON 2. It was in the year 1734 that Bishop Berkeley made his famous attack upon the doctrine of fluxions, which was the starting-point of ali philo- sophical discussion of the new mathematics in England during the eighteenth century. In what foUows we quote from the writings of Newton that were printed before 1734 sudi parts as bear on his conceptions of fluxions, so that the reader may judge for himself what grounds there were for Berkeley's great assault. To assist us in the inter- pretation of some of these printed passages, we quote also from manuscripts and letters of Newton which at that time were stili unprinted. In the next chapter we give an account of the foundations of fluxions as displayed by other writers in books and articles printed in Great Britain before 1734. It is hoped that the material contained in these first two chapters will enable the student to foUow closely and critically the debates on fluxions. Front Neivton's Publications printed before 1734 I. Principia 3. Three editions of the Principia were brought cut in Newton's lifetime ; the first in 1687, the NEWTON 3 second in 1713, the third in 1726. We give extracts which bear on the theory of limits and fluxions and indicate the changes in phraseology introduced in the second and third editions. We give also trans- lations into EngUsh based on the text of the 1726, or third, edition. Principia, Book /, Section /, Lemma I First edition : 4. " Quantitates, ut & quantitatum rationes, quae ad aequaUtatem dato tempore constanter tendunt & eo pacto propius ad invicem accedere possunt quam prò data quavis differentia ; fiunt ultimo aequales. 5. "Si negas, sit earum ultima differentia D. Ergo nequeunt propius ad a^qualitatem accedere quam prò data differentia D : contra hypothesin." Second and third editions : 6. " Ouantitates, ut & quantitatum rationes, quae ad aequalitatem tempore quovis finito constanter tendunt, & ante finem temporis illius propius ad invicem accedunt quam prò data quavis differentia, fiunt ultimo xquales. 7. "Si negas, fiant ultimò ina^quales, & sit earum, etc." [As in the first edition.] Translation by Robert Thorp:^ 8. "Quantities, and the ratios of quantities, which, in any finite time, tend continually to equality; and ^ Matheniatical Principles of Naturai Philosophy, by Sir Isaac Newton, Knight. Trnnslated into Engiish, and illustrated with a Comtncntary, by Robert Thorp, M.A., voi. i, London, 1777. LIMITS AND FLUXIONS LF before the end of that time, approach nearer to each other than by any given difference, become ulti- mately equal. " If you deny it, let them be ultimately unequal ; and let their ultimate difference be D. Therefore, they cannot approach nearer to equality than by that given difference D. Which is against the supposition." Principia, Book /, Sectiott /, Lemma II Translation by Motte :^ 9. ''If in any figure AacK, terminated by the right lines Aa, A E, and the curve acY., there be inscribed any number of paral- lelograms Ab, B<:, Cd, etc, comprehended under equal bases AB, BC, CD, etc, and the sides ^b. Ce, Dd, etc, parallel to one side Aa of the figure ; and the parallelograms aKb/, bhejn, eMdn, etc, are completed. Then if the breadth of those parallelograms be supposed to be diminished, and their number be augmented in infinitum ; 1 say, that the ultimate ratios which the inscribed figure AK<^L<:M<^D, the circumscribed figure Aalbmcndo^., and curvilinear figure AabedK, will bave to one another, are ratios of equality. ^ T/ie Mathematical Principks of Naturai Philosophy, by Sir Isaac Newton; transiated into English ^j/ Andrew Motte, London, 1729. (Two volumes.) II m b pN L a 1 c M \ d \ BF C NE WTON 5 **For the difference of the inscribed and circum- scribed figures is the sum of the parallelograms K/, \^rn, M«, D(?, that is (from the equality of ali their bases), the rectangle under one of their bases K^ and the sum of their altitudes ha, that is, the rectangle AB/ + /^A. Therefore with the whole increments a and b of the sides, the increment aV> + ^A of the rectangle is generated. Q. E. D. " IL Wallis's De Algebra Tractatus 20. The Latin edition of John Wallis's Algebra, which appeared in 1693, contains on pages 390-396 a treatise on the "Quadrature of Curves " which Newton had prepared many years before, and from which he cited many things in his letter of October 24, 1676. In revised phraseology and with a new Introduction, the "Quadrature of Curves" was republished in 1704, as we shall see presently. Through the researches of Rigaud ^ we know now that what is given in Wallis's Algebra, p. 390, line 18, to p. 396, line 19, are Newton's own words, except, no doubt, the word " clarissimus," as applied to himself From this part we quote as foUows : 2 — 21. Page 391: ''V^x fluentes qua^ititates intelli- git indeterminatas, id est quae in generatione Cuvarum per motum localem perpetuo augentur vel diminuuntur, & per earum fluxionem intelligit celeri- tatem incrementi vel decrementi. Nam quamvis fluentes quantitates & earum fluxiones prima fronte conceptu difficiles videantur, (solent enim nova difficilius concipi), earundem tamen notionem cito faciliorem evasuram putat, quam sit notio momen- ' S. P. Rigaud, Historical Essay on Sir Isaac Newton^ s Principia^ Oxford, 1838, p. 22. 2 Johannis Wallis, S.T.D., De Algebra Tractatus; Historicus &f Practicus. Oxoniae, MDCXCIII. NEWTON 15 torum aut paì'tium ininimarum vel differentiaruin infinite par-varum : propterea quod figurarum & quan- titatum generatio per motum continuum magis naturalis est & facilius concipitur, & Schemata in hac methodo solent esse simpliciora, quam in illa partium. ..." 22. Page 392: " Sint v, x, y, ^ fluentes quanti- tates, & earum fluxiones his notis i), x, j>, i desig- nabuntur respective. Et quoniam hae fluxiones sunt etiam indeterminata^ quantitates, & perpetua muta- tione redduntur majores vel minores, considerat velocitates quibus augentur vel diminuuntur tanquam earum fluxiones, & punctis binis notat in hunc modum i», ir, j, z, & perpetuum incrementum vel decrementum harum fluxionum considerat ut ipsarum fluxiones, ..." 23. Page 392: " Sit enim 0 quantitas infinite parva, & sint 02, oj, ox Synchrona momenta seu incrementa momentanea quantitatum fluentium z, y, 8i X : & hae quantitates proximo temporis momento per accessum incrementorum momentaneorum eva- dent z-\-oz, y-\-oy, x-{-ox\ ..." After substitut- ing these in x'^ — xyy-^aaz = o, then subtracting the originai expression and dividing the remainder by 0, he remarks (page 393) : '' Terminos multiplicatos per 0 tanquam infinite parvos dele, & manebit aequat i o 3i'.r2 _ ^^ _ 2xyy + aaè = o. " Translation : 24. Page 391 : " By flowing quantities he under- stands indeterminates, that is, those which, in the i6 LIMITS AND FLUXIONS generation of curves by locai motion are always increased or diminished, and by their fluxions he understands the velocity of increase or decrease. For, however difficult of comprehension flowing quaìttities and their fluxions appear at first sight (for new things are usually perceived with diffi- culty), yet he thinks a notion of them will be obtained more easily than the notion of moments either oi least pai'ts or of infinitely small diffei^ences ; because the generation of figures and quantities is more naturally and easily conceived, and the draw- ings in this method are usually more simple than in that of parts." 25. Page 392: " Let the flowing quantities be designated v, x^ j', 2, and their fluxions by the marks v, x, j>, i, respectively. And since these fluxions are likewise indeterminate quantities, and by perpetuai motion become greater or lesser, he considers the velocities by which they are increased or diminished as their fluxions, and marks them with doublé dots in this way i), x, y, i, and he con- siders the perpetuai increase or decrease of these fluxions as fluxions of themselves. ..." 26. Page 392: ** Let 0 be an infinitely small quantity, and 02^ oy, ox the synchronous moments or momentaneous increments of the flowing quanti- ties z, y, X \ and these quantities at the next moment of time, by the accession of the momen- taneous increments become z-\-02, y-\-oy, x + ox: . . ." After substituting these in -r3—;i7j/-j-^(3:<3' = o, then subtracting the originai expression and divid- NEWTON 17 ing the remainder by 0, he remarks (page 393) : ''Destroy the terms multipUed by 0 as infinitely small, and there will remain the equation 2)^x^—^yy — 2X}'y + aaà = o. " III. Quadratura Curvarum,^ 1704 '' Introductio 27. " Quantitates Mathematicas non ut ex parti- bus quam minimis constantes, sed ut motu con- tinuo descriptas hic considero. Lineae describuntur ac describendo generantur non per appositionem partium sed per motum continuum punctorum, superficies per motum Hnearum, soHda per motum superficierum, anguli per rotationem laterum, tem- pora per fluxum continuum, et sic in ca^teris. Hse Geneses in rerum natura locum vere habent et in motu corporum quotidie cernuntur. Et ad hunc modum Veteres ducendo rectas mobiles in longi- tudinem rectarum immobilium genesin docuerunt rectangulorum. 28. "Considerando igitur quod quantitates aequa- libus temporibus crescentes et crescendo genita^, prò velocitate majori vel minori qua crescunt ac ^ Tractatus de Quadratura Curvarum, published in 1704 in London, as an appendix to Newton's Opiicks. It was reprinted under the editorship of William Jones in London in ihe year 171 1, in a volume containing also three other papers of Newton, viz., the De analysi per ceguationes infinifas, Enumcratio Hnearum tertii ordinis, and Methodus differentialis. An English translation of the Qìiadratura Curvarum, made by John Stewart, was brought out in 1745 at London, in a volume containing also Newton's Analysis hy Equations of an Infinite Ntiinber of Ternis. A German translation of the Quadratura Curvarum by Gerhard Kowalewski appeared at Leipzig in 1908 in OsiwalcP s Klassiker der exakten Wissenschaften, Nr. 164. 2 i8 LIMITS AND FLUXIONS generantur, evadunt majores vel minores ; metho- dum quaerebam determinandi quantitates ex veloci- tatibus motuum vel incrementorum quibus gener- antur; et has motuum vel incrementorum velocitates nominando Fluxiones et quantitates genitas nomin- ando FluenteSy incidi paulatim Annis 1665 et 1666 in Methodum Fluxionum qua hic usus sum in Quadratura Curvarum. 29. '' Fluxiones sunt quam proxime ut Fluentìum ^ f rc__.£ V A B 'A : ( FiG. : 5 a 2. augmenta aequalibus temporis particulis quam mini- mis genita, et ut accurate loquar, sunt in prima ratione augmentorum nascentium ; exponi autem possunt per lineas quascunque quae sunt ipsis pro- portionales. Ut si areae ABC, ABDG ordinatis BC, BD super basi AB uniformi cum motu pro- gredientibus describantur, harum arearum fluxiones erunt inter se ut ordinata^ describentes BC et BD, et per ordinatas illas exponi possunt, propterea quod ordinatae illae sunt ut arearum augmenta nascentia. NEWTON 19 Progrediatur ordinata BC de loco suo BC in locum quemvis novum bc. Compleatur parallelogrammum BCE<^, ac ducatur recta VTH qua^ curvam tangat in C ibsisque bc et BA productis occurrat in T et V : et abscissai AB, ordinata^ BC, et lineae curvae AC^ augmenta modo genita erunt B^, E^, et Qc ; et in horum augmentorum nascentium ratione prima sunt latera trianguli CET, ideoque fluxiones ipsarum AB, BC et AC sunt ut trianguli illius CET latera CE, ET et CT et per eadem latera exponi possunt, vel quod perinde est per latera trianguli consimilis VBC. 30. ''Eodem recidit si sumantur fluxiones in ultima ratione partium evanescentium. Agatur recta Qc et producatur eadem ad K. Redeat ordinata bc in locum suum priorem BC, et cceuntibus punctis C et e, recta CK coincidet cum tangente CH, et triangulum evanescens CE^ in ultima sua forma evadet simile triangulo CET, et ejus latera evanes- centia CE, E<; et C^ erunt ultimo inter se ut sunt trianguli alterius CET latera CE, ET et CT, et propterea in hac ratione sunt fluxiones linearum AB, BC et AC. Si puncta C et e parvo quovis intervallo ab invicem distant recta CK parvo inter- vallo a tangente CH distabit. Ut recta CK cum tangente CH coincidat et rationes ultimai linearum CE, Y.C et C^ inveniantur, debent puncta C et ^ coire et omnino coincidere. Errores quam minimi in rebus mathematicis non sunt contemnendi. 31. " Simili argumento si circulus centro B radio BC descriptus in longitudinem abscissae AB ad 20 LIMITS AND FLUXIONS angulos rectos uniformi cum motu ducatur, fluxio solidi geniti ABC erit ut circulus ille generans, et fluxio superficiei ejus erit ut perimeter circuii illius et fluxio lineae curvae AC conjunctim. Nam quo tempore solidum ABC generatur ducendo circulum illum in longitudinem abscissa:^ AB, eodem super- ficies ejus generatur ducendo perimetrum circuii illius in longitudinem curvai AC. . . . 32. '' Fluat quantitas X uniforiniier et inv emenda sit fluxio quafititatis x'\ Quo tempore quantitas x fluendo evadit x-\-o, quantitas x'"- evadet x-\-o\**, id est per methodum serierum infinitarum, x"-\-nox"~'^ -\-{nn — n)l 2 oox''~^ -\-etc. Et augmenta 0 et nox'"-'^ -\-{nn — n) / 2 <9 and nox*'-^-{-{n'^ — n)l2 oox''-^-{-etc. are to one another as i and nx''-'^-j-(n^ — n)l2 ox^-'^ + etc. Now let these augments vanish, and their ultimate ratio will be i to nx""-^. 42. " By like ways of reasoning, the fluxions of lines, whether right or curve in ali cases, as likewise the fluxions of superficies's angles and other quan- tities, may be collected by the method of prime and ultimate ratios. Now to institute an analysis after this manner in finite quantities and investigate the prime or ulti^nate ratios of these finite quantities when in their nascent or evanescent state, is con- sonant to the geometry of the ancients : and I was willing to show that, in the Method of Fluxions, there is no necessity of introducing figures infinitely small into geometry. Yet the analysis may be performed in any kind of figures, whether finite or infinitely small, which are imagin'd similar to the evanescent figures ; as likewise in these figures, which, by the Method of Indivisibles, used to be reckoned as infinitely small, provided you proceed with due caution." 43. In the Quadrature of Curves proper, under '* Proposition I" the proof of the rule for finding the fluxion of expressions like x^ — Xf^ -^ a'^z — l?^ = o contains the following passages which indicate the use made of the symbol " (? " and of the term NEWTON 25 '* moment," and the mode of passing to the limit. We quote : — * * Demonstratio 44. **Nam sit o quantitas admodum parva et sunto oz, oy, ox, quantitatum z, y, x, momenta id est incrementa momentanea synchrona. Et si quantitates fluentes jam sunt z, y et x, hae post momentum temporis incrementis suis oz, oy, ox auctae, evadent z-\-oz^ y-\-oy, x-\-ox, quae in aequatione prima prò z^ y Qt x scriptae dant aequationem ... '^xx'^ + ^xxox -\- i'^oo — xyy — 2xyy — 2xoyy — xoyy — xooyy + aaz = o. Minuatur quantitas 0 in infinitum, et neglectis terminis evanescentibus restabit ^xx"^ — xyy — 2xyy -^aaò=:0. O.E.U." Translation by John Stewart : '' Demonstration 45. " For let 6» be a very small quantity, and let oz, oy, oxhe the moments, that is the momentaneous synchronal increments of the quantities z, y, x. And if the flowing quantities are just now z, y, x, then after a moment of time, being increased by their increments oz, oy, ox : these quantities shall become z-\-oz, y-\-oy, x-\-ox: which being wrote in the first equation for z, y and x, give this equation . . . ^xx^ + ^xxox + ,i^oo — xyy — 2xyy — 2xoyy — xoyy — xooyy + aaz — o. 26 LIMITS AND FLUXIONS Let the quantity o be diminished infinitely, and neglecting the terms which vanish, there will remain ^xx^ — xyy — 2xyy + aaz = o. Q. E. D. " IV. An Account of the '*Commercium Epistolicum " 46. It is now generally accepted that the account ^ of the Commeixium Epistolicum^ published in the Philosophical Transactions, London, 17 17, was written by Newton. The reasons for attributing it to him are stated by De Morgan^ and by Brewster.^ In abstract the account is as follows : — 47. (Pp. 177-178.) In a letter of October 24, 1676, to Oldenburgh, Newton explained that in deducing areas he considered the area as growing "by continuai Flux"; ''from the Moments of Time he gave the Name of Moments to the momentaneous Increases, or infinitely small Parts of the Abscissa and Area generated in Moments of Time. The Moment of a Line he called a Point, in the Sense of Cavalerius, tho' it be not a geometrical Point, but a Line infinitely short, and the Moment of an Area or Superficies he called a Line, in the sense of Cavalerius, tho' it be not a geometrical Line, ^ Philosophical Transactions, voi. xxix, for the years 17 14, 17 15, 1716. Li)ndon, 1717. " An Account of the Hook entituled Commer- ciimi Epistolicum Collimi et aliorum, De Analysi promoia . . .," pp. 173-224. This account was translated into Latin and inserted in the edilion of the Commerciiim Epistolicum of 1725. 2 See De Morgan's articles in the Philosophical Magazine, S. 4, voi. iii, June, 1852, pp. 440-444; v )1. iv, November 1652, p. 323. ^ Sir David V>tcvi^\.ttx, Memoirs of the Life, Writings,and Discoveries of Sir Isaac Newton, 2nd ed., voi. ii, Edinburgh, 1860, pp. 35, 36. NE WTON 27 but a Superficies infinitely narrow. And vvhen he consider'd the Ordinate as the Moment of the Area, he understood by it the Rectangles under the geo- metrica! Ordinate and a Moment of the Abscissa, tho' that Moment be not always expressed." Again, p. 179: "And this is the Foundation of the Method of Fluxions and Moments, which Mr. Newton in his Letter dated Octob. 24, 1676, comprehended in this Sentence. Data cequatione quotcunque fluentes quantitates involvente, invenire Fluxiones ; et vice versa. In this Compendium Mr, Newton represents the uniform Fluxion of Time, or of any Exponent of Time by an Unit ; the Moment of Time or its Exponent by the Letter o ; the Fluxions of other Quantities by any other Symbols ; the Moments of those Quantities by the Rectangles under those Symbols and the Letter 0 ; and the Area of the Curve by the Ordinate inclosed in a Square, the Area being put for a Fluent and the Ordinate for its Fluxion. When he is demon- strating a Proposition he uses the Letter 0 for a finite Moment of Time, or of its Exponent, or of any Quantity flowing uniformly, and performs the whole Calculation by the Geometry of the Ancients in finite Figures or Schemes without any Approxi- mation : and so soon as the Calculation is at an End, and the Equation is reduced, he supposes that the moment 0 decreases in infiìiitum and vanishes. But when he is not demonstrating but only investi- gating a Proposition, for making Dispatch he supposes the Moment 0 to be infinitely little, and 28 LIMirS AND FLUXIONS forbears to write it down, and uses ali manner of Approximations which he conceives will produce no Error in the Conclusion. " In ^^yN\.ovC^ Principia " he frequently considers Lines as Fluents described by Points, whose Velocities increase or decrease, the Velocities are the first Fluxions, and their Increase the second. " The Compendium of his Analysis was written ''in or before the year 1669" (p. 180). ''And the same Way of working he used in his Book of Quadratures, and stili uses to this day"(p. 182). On p. 204 we read : '*Mr. Newton used the letter 0 in his Analysis written in or before the Years 1669, and in his Book of Quadratures, and in his Principia PhilosophicB, and stili uses it in the very same Sense as at first. . . . These Symbols 0 and i' are put for things of a different kind. The one is a Moment, the other a Fluxion or Velocity as has been explained above. . . . Prickt Letters never signify Moments, unless when they are multiplied by the Moment 0 either exprest or understood to make them infinitely little, and then the Rectangles are put for the Moments " (p. 204). Further on we read : " It [the method of fluxions] is more elegant [than the Differential Method of Leibniz], because in his Calculus there is but one infinitely little Quantity represented by a symbol, the symbol 0. We bave no Ideas of infinitely little Ouantities, and therefore Mr. Newton introduced Fluxions into his Method, that it might proceed by finite Ouantities 'as much as possible. It is more Naturai and Geometrical, because founded upon the NE WTON 29 primce quantitatum nascentiuin rationes, which have a Being in Geometry, whilst Indivisibles, upon which the Differential Method is founded, have no Being either in Geometry or in Nature. There are rationes primce quantitatum nascentium, but not quantitates prinicB nascentes. Nature generates Quantities by continuai Flux or Increase ; and the ancient Geometers admitted such a Generation of Areas and Solids " (p. 205). Front Newton' s Correspondence and Manuscripts not in print in 1734 48. Manuscripts of Newton, some of them stili unpublished, show that he first thought of fluents and fluxions in 1665 and 1666, when he was in his twenty-third and twenty-fourth years.^ The notation by dots occurs as early as 1665. As pointed out by De Morgan,- these early papers are infinitesimal in character. They were first published in 1838.^ A manuscript, dated Nov. 13, 1665, gives rules for finding the velocities p, q, ;-, etc. , of two or more lines x, y, z, etc, described by bodies A, B, C, etc, the lines being related to each other ^ See a list of Newton's manuscripts and publications on fiuxional calculus prepared by Philip E. B. Jouidain, in his edition of Augtistus De Morgati's Essays on the Life and Work of Nezvton, The Open Court Publishing Co., 1914, pp. 107- 1 12. " Augustus De Morgan, "On the Early History of Infinitesimals in England," 7'he London, Edinburgh, and Duhlin Philosophical Magazine, 4th S., voi. iv, 1852, pp. 321-330. This article is an important historical contribution, of which extensive use is made in the present history. ^ See S. P. Rigaud, Historical Essay on the first Public ation of Sir Isaac Newton's Principia, Oxford, 1838, Appendix, pp. 20-24. 30 LIMITS AND FLUXIONS by an equation, such ^i?, x^—2a^y-\- zzx—yyx-\-zyy — z'^—o. ''If the body A, with the velocity /, describe. the infinitely Httle line o in one moment, in the same moment B, with the velocity q, will describe the line oq / /," and the body C, with the velocity r, will describe the line or / /. So that, if the described lines be x, y and z "in one moment," they will be x-\-o, y-\-oq / /, z + or j p '*in the next." He finds that the relation of the velocities /, q, r, in the above example, is '^pxx + pzz —pyy — 2aaq — 2yxq + 2zyq + 2zxr -\-yyr •— ^zzr = o. In proving his rules for differentiation, Newton divides by o^ and in the resulting expression observes that ''those terms in which o is, are infinitely less than those in which it is not. Therefore, blotting them out, there rests " the relation sought. The notation by dots, " pricked letters," occurs on a leaf, dated May 20, 1665, which has never been printed. ^ 49. It is evident that Newton permitted twenty- eight years to pass between the time of his first researches on fluxions and 1693, the date when the earliest printed account of his notation of fluxions appeared from his pen in the Latin edition of Wallis's Algebra. Moments and fluxions are mentioned in his Principia^ as has been shown by our extracts. 50. Of importance in the interpretation of the meanings of "moment" in the second edition of ^ S. P. Kigaud, op. cif.f Appendix, p. 23. Consult also the remarks on this passage macie by CJ, Ene^tròm in Bibliotheca maihe>natica, 3. F., Bd. II, Leipzig, 1910-191 1, p. 276, and Bd. 12, 1911-1912, p. 268, and by A. Wiiiing in Bd, 12, pp. 56-60. See also A catalogne of the l'ortsinouth collection of books aìid papers^ written by or belonging to Isaac Newton, Cambridge, 1889. NEWTON 31 the Principia (17 13) is a letter of May 15, 17 14, from Newton to KeilV from which we quote the foUowing : — 51. ". . . altho I use prickt Letters in the first Proposition of the hook of Quadratures, yet I do not there make them necessary to the method. For in the Introduction to that hook I describe the method at large & illustrate it w"' various examples without making any use of such letters. And it cannot be said that when I wrote that Preface I did not understand the method of fluxions because I did not there make use of prickt letters in solving of Problems.^ The book of Quadratures is ancient, many things being cited out of it by me in my Letter of 24 Octob. 1676. . . . 52. ''ffluxions & moments are quantities of a different kind. ffluxions are finite motions, moments are infinitely little parts. 1 put letters with pricks for fìiuxions, & multiply fluxions by the letter 0 to make them become infinitely little and the rectangles I put for moments. And wherever prickt letters represent moments & are without the letter o this letter is always understood. Wherever •^> j) y^ 7) ^tc, are put for moments they are put for xo^ yo^ j/00, yo^. In demonstrating Propositions I always write down the letter 0 & proceed by the Geometry of Euclide and ApoUonius without any ^ J. Edleston, Correspondence of Sir Isaac A/ewlon and Professor CoUs, London, 1850, pp. 176, 177. yji^ John Bernoulii, in tlie Ada Ertiditorum for February and March, 17 13, had critici-sed a passage in the Principia^ and claimed ihal Newton did not understand the second fluxions when writinij that passagc. 32 LIMITS AND FLUXIONS approximation. In resolving Ouestions or investi- gating truths I use ali sorts of approximations w^'' 1 think will create no error in the conclusion and neglect to write down the letter o, and this do for making dispatch. But vvhere x, j>, j/, y are put for fluxions without the letter o understood to make them infinitely little quantities they never signify differences. The great Mathematician ^ therefore acts unskilfuUy in comparing prickt letters with the marks dx and dy^ those being quantities of a different kind. " Remarks 53. The extracts from Newton's writings demon- strate the following : — (i) At first Newton used infinitesimals (infinitely small quantities), as did Leibniz and other mathe- maticians of that age. As early as 1665, when Newton was a young man of twenty-three, he used them and speaks of " blotting them out."^ He uses infinitesimals in the Principia of 1687^ and in his account of the quadrature of curves in Wallis's Algebra of 1693, where Newton speaks of himself in the third person.^ It is worthy of emphasis, in contrast to Leibniz, that Newton uses only infinitesi- mals of the first order. Moreover, as De Morgan remarked long ago,^ ''the early distinction between the systems of the two is this, that Newton, holding to the conception of the velocity or fluxion, ' John Bernoulli. See Edleston, op. cit., p. 171. ~ See our § 48. 3 See our §§ io, 13, 16, 18. ^ See our §§ 21, 26. ^ De Morgan, Philosophtcal Alagaztne, 4 S., voi. iv, 1852, p. 324. NEWTON 33 used the infinitely small increment as a means of determining it ; while, with Leibnitz, the relation of the infinitely small increments is itself the object of determination. " (2) As early as 1665, Newton speaks of describing an ''infinitely little line" in " one moment," and then uses the expression "in the next " moment.^ Here ** moment" cannot mean a point of time, destitute of duration ; it means an infinitely small duration, an infinitesimal of time. Doubtless this use of "moment" with reference to time suggested the more extended and general use of the term " momentum " or " momenta " as found in the Principia ^ and later publications. (3) The use of dots, " prickt letters," to indicate velocities or fluxions goes back to 1665,^ but they are not used by Newton in print until 1693 ^" Wallis's Algebra ; they are used extensively in Newton's Quadrature of Curves of 1704.* (4) Newton first used the word " fluxion " in print in 1687 in the Principiai (5) The first refinement of the doctrine of fluxions is found in Newton's Principia, where he speaks of "prime and ultimate ratios"^ and of "limits. "^ (6) The high-water mark of Newton's efforts to place the doctrine of fluxions upon a thoroughly logicai basis is found in his Quadrature of Curves, 1704. It indicates the almost complete exclusion ^ See our § 48. - See our §§ 16, 18, 21, 24. ^ See our § 48. * See our §§ 22, 25, 44, 45. •"' See our §§ 16, 18. « See our §§ io, 13. ' See our §§ 4, 6, 8, io, 13. 34 LTMITS AND FLUXIONS of quantities infinitely little. " 1 consider mathe- matical quantities in this place not as consistiiig of very small parts, " says Newton.^ Also " the very smallest errors in mathematica! matters are not to be neglected, "- and "in the method of fluxions there is no necessity of introducing figures infinitely small into geometry. "^ In view of these statements the symbol o used in the Quadrature of Curves, a ''quantitas ad modum parva,"* must be interpreted as a small finite quantity. In this connection De Morgan's remarks are of interest : ^ "In 1704, Newton in the Quadratura Cuì'vai'um renounced and abjured the infinitely small quantity ; but he did it in a manner which would lead any one to suppose that he had never held it. . . . And yet, there is something like a recognition of some one having used infinitely small quantities in Fluxions, contained in the following words : volui ostendere quod in Methodo Fluxionum non opus sit figuras infinite parvas in Geometriam introducere : nothing is wanted except an avowal that the some one vvas Newton himself The want of this avowal was afterwards a rock of offence. Berkeley, in the Analyst, could not or would not see that Newton of 1687 ^^""d Newton of 1704 were of two different modes of thought. " We do not interpret Newton's expressions of 1704 as declarations that a logicai exposition of 1 See our §§ 27, 34. 2 See our §§ 30, 39. » See our §§ 33, 41. " See our §§ 44, 45. •■* De Morgan, Philosophical Magazine, 4 S., voi. iv. 1852, p. 328. NEWTON 35 fluxions cannot be given on the basis of infini- tesimals or that infinitely small quantities are impossible ; for he says/ ''the analysis may be performed in any kind of figures whether finite or infinitely small, which are imagined similar to the evanescent figures." In fact, not even in 1704 did Newton succeed in completely banishing from his doctrine of fluxions the infinitely little. If what he used in 1704 is not the infinitely little, it is so closely related thereto, that it cannot be called either a finite magnitude or an absolute zero. In 1704, fluxions are "in the fif-st ratio of the nascent augments, " or ' ' in the ultimate ratio of the evanescent parts. " ^ Unless the fully developed theory of limits is read into these phrases, they will involve either infinitely little parts or other quantities no less mysterious. At any rate, the history of fluxions shows that these expressions did not meet the demands for clearness and freedom from mysticism. Newton himself knew full well the logicai difiiculty involved in the words "prime and ultimate ratios " ; for in 1687 he said,^ "it is objected, that there is no ultimate proportion of evanescent quantities ; because the proportion, before the quantities have vanished, is not ultimate ; and, when they have vanished, is none." How does Newton meet this, his own unanswerable argument ? He does so simply by stating the difificulty in another 1 See our §§ 33, 42. '^ See our §§ 29, 30, 33, 36, i%, 39, 42. ' See our §§ u, 14. 36 LIMTTS AND FLUXIONS form : " But, by the same argument, it might as well be maintained, that there is no ultimate velocity of a body arriving at a certain place, when its motion is ended : because the velocity, before the body arrives at the place, is not its ultimate velocity ; when it has arrived, is none. But the answer is easy : for by the ultimate velocity is meant that . . . at the very instant when it arrives." If ''instant," as used here, is not an infinitesimal, the passage would seem to be difficult or impossible of interpretation. (7) A return to the open use of the infìnitely small quantities is seen in writings of Newton after the year 1704. It might be argued that such a return was necessary in the second edition of the Principia, 17 13, unless the work were largely re- written. Newton's Anafysis per cequationes numero terniinorum infinitas was first printed in 171 1, and might bave been rewritten so as to exclude infini- tesimals as fully as was done in the Quadrature of Curves of 1704. But the infìnitely little is per- mitted to remain.^ There is no disavowal of such quantities either in the Commercium Epistolicum, with the editors of which Newton was in touch, or in Newton's own account of this publication, contributed to the Pìiilosophical Transactions.^ (8) The theory of limits is involved in the first lemma of the Principia,^ and in the explanation of prime and ultimate ratios as given in that work. 1 See our § 66. - See our § 47. 3 See our §§ 4, 6, 8, 9, io, 12, 13, 15. CHAPTER li PRINTED BOOKS AND ARTICLES ON FLUXIONS BEFORE 1734 54. The earliest printed publication in Great Britain on the new calculus was from the pen of John Craig, a Scotsman by birth, who settled in Cambridge and became a friend of Newton. Later he was rector of Gillingham in Dorsetshire. He was " an inoffensive, virtuous man," fond of mathe- matics. In 1685 he published at London a book entitled, Methodus figuraruin . . . quadraiuras determinandi. At that time nothing could be known about fluxions except through private com- munication. In 1684 Leibniz published his first ideas of Differential Calculus in the Leipzig Acts. Craig used in 1685 the calculus of Leibniz and also the notation of Leibniz. Continental writers cali Craig the introducer of the theory of Leibniz into England. On p. 28 of his book, Craig derives dp = 4Snr*jy^d_y from p= lónr^jy^, and arrives at a differential equation (aiquationem differentialem). The meanings of <^, dy, dx, etc. , are not explained but taken for granted, reference being made to Leibniz. In 1693 Craig published another book in which the notation of Leibniz is used. He con- 37 38 LIMITS AND FLUXIONS tributed also several papers to the Philosophical Trans act io ns {l^ondon), but never, before 17 18, did he use fluxional symbols. In preparing the book of 1685 he had received from Newton the binomial theorem which he used before it had appeared in print, but he had no communication about fluxions. ''We have here the singular indifference, " says De Morgan, "which Newton at that time, and long afterwards, showed toward his own calculus."^ Craig wrote a tract in 1693, and articles for the Philosophical Transactzons \niy 01, 1703, 1704, 1708, using the differential calculus ali this time. In the issue No. 284, 1703, he employs the Leibnizian sign of integration /. Craig submitted to Newton one of his early manuscripts (probably the one printed in 1693). With regard to this event De Morgan wrote to Hamilton, the inventor of quater- nions : *' Few of us know that Leibniz was perfectly well known in England before the dispute, and that Newton's first provocative to an imperfect publica- tion was ds and infinitely small quantities paraded under his own eyes by an English writer (Craig), who lent him his MSS. to read."^ Craig's publica- tion of 17 18 foUowed the great controversy on the invention of the calculus ; now he uses fluxions exclusively and says not a word on the differential calculus. The book does not discuss fundamentals, and no explanation of x is given. As conjectured ^ De Morgan, Philosophùaì Magazine, 4 S., voi. iv, 1S52, p. 326. '^ Life of Str IVilliam Rowan Hamilton, by Robert F. Graves, voi. iii, 1889, p. 415. PRINTED BOOKS, ETC, BEFORE 1734 39 by De Morgan, it may have been Craig's manu- script that suggested to Newton the need of making bis own fluxions accessible to the public. At any rate, in 1693 there appeared the account of fluxions in Wallis's Algebra. [See Addenda, p. 289.] 55. Abraham De Moivre, a French mathemati- cian who in 1688, after the revocation of the Edict of Nantes, carne to London, contributed in 1695 to No. 216 of the Philosophical Transactions (London) an article in which he uses x, j>, x, y, and lets both ''fluxion" and "moment" stand for things infin- itely small. In the same number of the Transac- tions, the astronomer Edmund Halley has an article on logarithms in which he uses infinitely small ratiunculce and differentiolce, but neither the nota- tion of Leibniz nor that of Newton. In 1697, David Gregory used in No. 231 of the Transactions x ^nd speaks of " fluxio fluxionis " without, however, ex- plaining bis terms. 56. Patio de Duillier, a Swiss by birth, who had settled in London and become member of the Royal Society, wrote in 1699 a treatise, Lincee brevissimi dcscensus investigatio geometrica, uses fluxions as infinitely small quantities. This publi- cation is noted as containing a statement which started the Newton-Leibniz controversy on the invention of the Calculus. 57. It is remarkable that Roger Cotes, in 1701, when an undergraduate at Trinity College, Cam- bridge [Newton's own College], wrote a letter on mathematica! subjccts, in which x is used as 40 LIMITS AND FLUXIONS ''infinitely little."' In 1702-3 Humphry Ditton, in voi. xxiii of the Transactions ^ used the fluxional notation, without explanation. 58. Other vvritings that do not define their terms are the Fluxionum rnethodus inversa, 1704, by the London physician, George Cheyne, and De Moivre's Animadvei'siones in D. Georgii Cheynai Tractaium, London, 1704. However, Cheyne lets x— i, from which we infer that, with him, x was finite. [See Addenda, p. 289.] 59. The next writer on fluxions was John Harris, a voluminous author of books on various subjects. He was at one time Secretary of the Royal Society. In 1702 he published at London A New Short Treatise of Algebra, which devotes the last 22 pages, out of a total of 136 pages, to fluxions. It is the first hook in the English language in which this subject is treated. The doctrine of fluxions is the " Arithmetick of the I n finii e ly s mail Increments or Decrements of Indeterminate or Variable Qua?i- tities, or as some cali them the Moments or hifin- itely small Differences of such Variable Ouantities. These Infinitely small Increments or Decrements, our incomparable Mr. Isaac Newton calls very pro- perly by this name of Fluxions " (p. 115). A few lines further on it says that Newton ''calls the celerity or Velocity of the Augmentation of Diminu- tion of these Flowing Quantities, by the name of Fluxions.'' A second edition of this hook appeared in 1705. As authors on fluxions, Harris in 1705 ^ J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes, London, 1850, p. 196. PRTNTED BOOKS, ETC, BEFORE 1/34 4i mentions Newton, Wallis, Nieuwentiit, Carré, Leib- niz, l'Hospital, de Moivre, and Hayes. 60. John Harris also published a Lexicon Tech- nìcuin, of which the second volume, London, 17 io, contains an article, "Fluxions." " This general Method of finding the Fluxions of ali Powers and Roots, I had from the Hon. Fr. Robartes, Esq. If a Ouantity gradually increases or decreases, its immediate Increment or Decre- ment is called its Fliixion. Or the Fluxion of a Quantity is its Increase or Decrease indefinitely small. . . . Since xx . . . is infinitely smaller than 2xx^ whereby it can make no sensible Change in that Quantity, it may be laid aside as of no Value. . . . Authors' Nameswho have written of Fluxions: D. Bernoulli Ti-actatus de Principiis Calculi Exponen- tialis\ Nieuwentiifs Analysis Infinitoi-um^ Amster. , 1695 ; Dr. Cheyne's Fluxions^ with Moivre' s Anim- adversiotis on them, and the Doctor's reply ; Hays's Fluxions, Lond., 1704; Analyse des Infiniinent Petits. Par l'Hospital, Fr., Paris, 1696; Le Calcule Integrale, par M. Carré, Paris, 1700; Mr. Abraham de Moivre's Use of Fluxions, in the Solution of Geometrick Problems. See Philos. Trans., N. 216; Mr. Humphry Ditton's Institution of Fluxions. " 61. In the above list of writers are Charles Hayes and Humphry Ditton, authors of English texts now demanding our attention. Hayes starts his elucidation of fundamentals (p. i) as follows :^ ^ A Treafisc of F/uxions : or, An Introduction to Afathematical Philosophy, Charles Hayes, London, 1704. 42 LIMITS AND FLUXIONS " Magnitude is divisible in infinitum, and the Parts after this infinite Division, being infinitely little, are what Analysts cali Moments or Differ- ences ; And if we consider Magnitude as Indeter- minate and perpetually Increasing or Decreasing, then the infinitely little Increment or Decrement is call'd the Fluxion of that Magnitude or Quantity : And whether they be called Moments, Differences or Fluxions, they are stili suppos'd to have the same Proportion to their Whole's, as a Finite Number has to an Infinite ; or as a finite Space has to an infinite Space. Now those infinitely little Parts being extended, are again infinitely Divisible ; and these infinitely little Parts of an infinitelylittle Part of a given Ouantity, are tlG. 3. O ^ J 1 by Geometers call'd Injìnite- siince Infinitesiììiaruiìi or Fluxions of Fluxìons. Again, one of those infinitely little Parts may be conceiv'd to be Divided into an infinite Number of Parts which are call'd Third Fluxions, etc. " He endeavours to justify this doctrine by illus- trations. The angle of contact FAG formed by the line AE and the ordinary parabola AG, is less than any rectilineal angle ; the angle P'AD, formed by AE with the cubical parabola AD, is infinitely less than the angle FAG, and so on. Hayes defines the doctrine of Fluxions as the " Arith- metick of infinitely small Increments or Decrements PRINTED BOOKS, ETC, BEFORE 1734 43 of Indeterminate or variable Quantities." He cautions the reader : " But we must take great heed, not to consider the Fluxions, or Increments, or Decrements as finite Quantities " (p. 4). He rejects xèy and xèy ''as being incomparably less " than xzy. The same year in which Hayes wrote this first English hook on fluxions which could make any claim to attention, saw the appearance of Newton's Quadratuì-a Curvarum. The contrast in the defini- tion of " fluxion " was sharp. Hayes called it " an infinitely small increment " ; Newton called it a "velocity, " a finite quantity. 62. William Jones, in his Synopsis P almariot-uni Matheseos, London, 1706, devotes a few pages to fluxions and fluents, using the Newtonian notation. On p. 225 he gives, in substance, Newton's lemma, in these words : "Quantities, as also their Ratio's, that continually tend to an Equality, and therefore that approach nearer the one to the other, than any Difìference that can possibly be assign'd, do at last become equal." Then he says : "Hence ali Curved Lines may be considered as composed of an Infinite Number of Infinitely little right Lines." He uses "infinitely small" quantities, but defines a fluxion as "the Celerity of the Motion," fluxions being "in the first Ratio of their Nascent Aug- ments." Jones represents bere the Newton of the Principia, and of the Quadrature of Curves as given in 1793. 63. The earliest hook exhibiting a careful study 44 LI MIT S AND FLUXIONS of Newton's tract of 1704 was Humphry Ditton's Instiiution of Fluxions, 1706.^ Ditton was pro- minent as a divine as well as a mathematician. Like so many other Eiiglish writers on fluxions during the eighteenth century, he had not been at either of the great universities. He states in his preface that he has also consulted and drawn from the writings of John Bernoulli and some other Continental writers. 64. The reader of Ditton's hook is impressed by the fact that he labours strenuously to make every- thing plain. He takes the reader fully into his confìdence. This is evident in the extracts which follow (pp. 12-21) : — "Suppose any flowing Quantities, . . . as also their Increments . . . which Increment imagine to be generated in equal very small Tarticles of Time. I conceive we may say without Scruple, that the Fluxions are ihe velocities of those Increments, con- sideraci not as actually genei'ated, but quatenus Nascentia, as arising and beginning to be genei^ated. As there is a vast difference between the Increments consider'd as Finite, or really and actually generated ; and the same considered only as Nasce?ttia or in the first Moment of their Generation : So there is as great a difference also between the Velocities of the Increments, consider'd in this two fold respect. . . . ^ An Institution of Fluxions : Containmg the First Pnnciples, The Operatiovs, with some of the Uses and Applications of that Admirable Method ; According to the Schet/te prefixd to his Iract of Qiiadraturcs, by {its First Inventor) the hicoviparable Sir Isaac Newton. By Humphry Ditton, London, 1706. PRINTED BOOKS, ETC, BEFORE 1734 45 The Reason of that [difference], is this. Because there is (speaking strictly and accurately) an Infinity of Velocities to be consider'd, in the Generation and Production of a Real hicrernent ; ... So that if we conceiv'd the Fluxion, to be the Velocity of the Increment, as actually Generateci ; we must conceive it to be an Infinite Variety or Series of Velocities. Whereas the Velocity, with which any sort of In- crement arises, or begins to be generateli ; is a thing that one may form a very clear and distinct Idea of, and leaves the Mind in no Ambiguity or Confusion at ali. . . . However, if we take those Particles of time exceeding small indeed, and Neglect the Acceleration of the Velocity as inconsiderable, we may say the Fluxions are proportional to those In- crements ; remembering at the same time, that they are but nearly, and not accurately so. . . . If in the Differential Calculus, some Terms are reìected and thrown out of an Equation, because they are nothing Comparatively, or with respect to other Terms in the same Equation ; that is, because they are infinitely small in proportion to those other Terms, and so may be neglected upon that Score: On the other band, in the Method of Fluxions, those same Terms go out of the- Equation, because they are multiplied into a Ouantity, which . . . does at last really vanish. . . . N.B. Speaking bere of Infinitely small Quantities, or Infinitesimals as some Authors (and particularly Mr. Neiwentiit) chuse to term them, I cannot but take notice of a notion, which that Excellent and In- genious Ferson advances in bis Analysis htjìnitorum. 46 LIMITS AND FLUXIONS It is this ; That a Quantity Infinitely Great, a Finite or any given Quantity, an Infìnitesimal, a7id Nihilum Geornetricum, are in Geometrical Troportion. I confess I cannot discover the truth of this. . . . Let in denote an infinite Quantity, d any finite one ; then is d / ;;/ the Infinitesimal of d, according to Mr. Neiwentiit. Now his Assertion is, that m : d : : d I m : o ; therefore since from the nature of Geo- metrical Proportion, 'tis aiso m : d : : d / m : dd / mm ; it foUows that dd I mm is = o . . . then d / m = o. Now Mr. Neiwentiit will hardly allow his Infinitesimal to be nothing ; and yet ... I think it must follow, that d=o.'' Proceeding geometrically, Ditton ex- plains the fluxions of lines, areas, solids, and surfaces. Next he takes up algebraical expressions. To find the fluxion of x'% he lets x flow uniformly and re- presents the augment of ;r in a given particle of time by the symbol o. While x becomes x-\-o, x" becomes (x-\-oy. Expanding the binomial, he finds that the two augments are as i to nx''~'^-{-(n'^ — n)ox'''^ / 2 + etc. " And the Ratio of them (making o to vanish) will be that of i to nx""'^." According to his nota- tion i" is a fluxion of x, and .ir is a fluxion of a:. Taking ^ as a very small quantity, he lets the ex- pressions oà, oy represent the moinents, or increments of the flowing quantity z, y generated in a very small part of time. " If therefore now, at the present Moment, the flowing Quantities are z, y, x ;. the next Moment (when augmented by these Increments) they will become x+oò; y-\-oy, x-\-ox.'' He ex- presses the general mode of procedure for finding PRINTED BOOKS, ETC, BEFORE i734 47 the fluxion, vvhich coincides with the modem mode of finding a derivative. Ditton considers the in- crements as finite (p. 53). " These Momenta are in proportion to one another as the Fluxions of the flowing Quantities respectively, for 02, oy, ox, are as 2, j>, i' ; and Mr. Newton had before expresly told US ; that the Increments generated in a very small Farticle of time were very nearly, as the Fluxions." Evidently Ditton does not here overlook that 02, oy, ox represent the increments only *'very nearly." He observes (p. 98) that \ve may " go on with ease to the second, third, and any other Fluxions ; neither are there any new Difficulties to be met with." A second edition of Ditton's book was brought out in 1726 by John Clarke. 65. Ditton's first edition appeared at a time when the Newton-Leibniz controversy was under way. Leibniz had appealed to the Royal Society for justice. That Society appointed a committee which published a report containing letters and other material hearing on the case, in a book called the Commercium Epistolicum,'^ which figures prominently in the lamentable controversy. From this book the early use of infinitely small quantities on the part of Newton is conspicuously evident. The book makes it clear also that some of Newton's warmest supporters were guilty of gross inaccuracy in the use of the word "fluxion." ' Commercium Epistolicuìn D. /ohannis Collins, et aliortttn de analysi promota : jussu societatis regia in lucevi editum. Lendini, MDCCXII. 48 LIMITS AND FLUXIONS 66. Newton's Analysis per cequationes numero terminonim infinitas, which was sent on July 31, 1669, through Barrow to Collins, and which was first published at London in 171 1, was reprinted in the Commercium Epistolicuni. In this Analysis in- finitely small quantities are used repeatedly, but the word ' ' fluxion " and the fluxional notation do not occur. In a letter to H. Sloane, who was then Secretary of the Royal Society of London, written in answer to a letter of Leibniz dated March 4, 17 1 1, John Keill, professor of astronomy at Oxford, re- counts the achievements of Isaac Barrow and James Gregory, and says : " If in place of the letter 0, which represents an infinitely small quantity in James Gregory's Geometrice pars uìiiversaìis (1667), or in place of the letters a ox e which Barrow em- ploys for the same thing, we take the x or y of Newton or the dx or dy of Leibniz, we arrive at the formulas of fluxions or of the differential calculus."^ Thus Keill, the would-be great champion of Newton, instead of warning the reader against confusing differentials and fluxions, himself comes dangerously dose to conveying the erroneous idea that x and y are infinitely small, the same as dx and dy. He Comes so near to this as to be guilty of lack of caution, if not of inaccuracy. More serious is a statement further on. The en- ^ " Nam si prò Litera 0, quae in Jacobi Gregorii Parte Matheseos Uni- versali quantitaleni infinite parvam repraesentat ; aut prò Literis a vel e quas ad eandem designandam adhibet Barrovius ; ponamus x vel y Newtoni, vel dx seu dy Leibnitii, in P'ormulas Fliixionum vel Calculi Differentialis incidemus " (p. 112). PRINTED BOOKS, ETC, BEFORE 1734 49 listment of the services of a clever lawyer would be needed to acquit the editors of the Conimercium Epistolicum of gross error when, in the final summary of their case against Leibniz, they declare (p. 121), "that the Differential Method is one and the same with the Method of Fluxions, excepting the name and the notation ; Mr. Leibniz calling those Quantities Uifferences, which Mr. Newton calls Moments or Fluxions ; and marking them with the letter d^ a mark not used by Newton." òj . Joseph Raphson, in bis History of Fluxions (which appeared as a posthumous work at London, in 171 5, printed in English, and in the same year also in Latin, the Latin edition containing new corre- spondence hearing on the Newton-Leibniz contro- versy), says on p. 5 that Newton ' ' makes use of Points, and denotes those first Differences (which by a Name congruous to their Generation, being con- sider'd as the first Increments or Decrements of a continued Motion, he calls Fluxions) thus, viz. x, j>, -c. " This misrepresentation of Newton is the more astonishing when we recollect that Raphson was very partial to Newton, and also meant his History **to open a plain and easy way for Beginners to understand these Matters. " Newton never looked upon a fluxion as anything different from velocity ; with him it was always a finite quantity. To make matters worse, Raphson continues : "To these Quantities he adds others of another Gender, and which in relation to Finite ones may be conceiv'd as infinitely great, and denotes them thus 'x, 'y, 'z, 4 50 LIMITS AND FLUXIONS whereof the first or finite Ouantities themselves, viz. X, V, z, may be conceiv'd as Fluxions.'' And again, "a Point . . . may be consider'd as the Fluxion of a Line, a Line as the Fluxion of a Piane, and a Piane as the Fluxion of a Solid, and a finite Solid as the Fluxion of a (parfially) infinite one, and that again as the Fluxion of one of an higher Gender of Infinity, and so on ad inf. which we shall further illustrate in some Dissertations at the end of this Treatise." 68. Brook Taylor brought out at London in 17 15 his Methodus incrementonim dire età et inversa, in which he looks upon fluxions strictly from the stand- point of the Newtonian exposition in the Quadrature of Curves, 1704. 69. James Stirling uses x and y as infinitesimals in his LinecE tertii ordinis, Oxford, 17 17. He draws the inftnitely small right triangle at the contact of a curve with its asymptote, the horizontal side being " quam minima" and equal to i-, the vertical side being y. In the appendix to this booklet of 17 17, X and y are again infinitely small. In his Methodus differentialis, London, 1730, there is no direct attempt to explain fundamentals, any more than there was in 17 17, but on p. 80 he puts the fluxion of an independent variable equal to unity, from which we infer that a fluxion is with him now a finite velocity. 70. For twenty-four years after Ditton no new text appeared. In 1730 Edmund Stone, a self- taught mathematician who had studied De l'Hospital, PRINTED BOOKS, ETC. BEFORE 1734 51 sent forth a new book, the first part of which was a translation.^ The foUowing extract is from Stone's translation of De l'Hospital's Preface, the words in the square brackets [ ] being interpolateci by Stone : — " By means of this Analysis we compare the infinitely small (DiiTerences or) Parts of finite Magnitudes, and find their Ratio's to each other ; and hereby Hkewise learn the Ratio's of finite Magnitudes, those being in reality so many infinitely great Magnitudes, in respect of the other infinitely small ones. This Analysis may ever be said to go beyond the Bounds of Infinity itself ; as not being confined to infinitely small (Differences or) Parts, but discovering the Ratio's of Differences of Differ- ences, or of infinitely small Parts of infinitely small Parts, and even the Ratio's of infinitely small Parts of these again, without End. So that it not only contains the Doctrine of Infinites, but that of an Infinity of Infinites. It is an Analysis of this kind that can alone lead us to the Knowledge of the true Nature and Principles of Curves : For Curves being no other than Polygons, having an Infinite Number of Sides, and their Differences arising altogether from the different Angles which their infinitely small Sides make with each other, it is the Doctrine of Infinites alone that must enable us to determine the Position of these Sides, in order to get the ^ l^he Method of Fluxions^ both Direct and Inverse. The former bein^ a Translation from the Celebrated Marqiiis De PHospiiars Analyse des Injinenunts Petits : And the Latter Supply'd by the Trans lator, E. Stone, F.R.S. London, MDCCXXX. 52 LIMITS AND FLUXIONS Curvature formed by them ; and thence the Tangents, Perpendiculars, Points of Inflexion and Retrogres- sion, reflected and refracted Rays, etc. , of Curves. '*Polygons circumscribed about or inscribed in Curves, whose Number of Sides infinitely augmented till at last they coincide with the Curves, bave always been taken for Curves themselves. . . . It was the Discovery of the Ànalysis of Infinites that first pointed out the vast Extent and Fecundity of this Principle. . . . Yet this itself is not so simple as Dr. Barrow afterwards made it, from a dose Consideration of the Nature of Pohgons, which naturally represent to the Mind a Httle Triangle consisting of a Particle of a Curve (contained between two infinitely near Ordinates), the Differ- ence of the correspondent Absciss's ; and this Triangle is similar to that formed by the Ordinate, Tangent, and Subtangent. . . . Dr. Barrow . . . also invented a kind of Calculus suitable to the Method {Lect. Geoin., p. 80), tho' deficient. . . . The Defect of this Method was supplied by that of Mr. Leibnitz'z,^ [or rather the great Sir Isaac Newton].'^ He began where Dr. Barrow and others left off: His Calculus has carried him into Countries hitherto unknown. ... I must bere in justice own (as Mr Leibnitz himself has done, in Journal dcs S^avans for August 1694) that the learned Sir Isaac Newton likewise discover'd something like the Calculus Differentialis, as appears by his excellent A Ada Eruàit. Lips., ann. 1684, p. 467. 2 See Commcrcùim Epistoluuni. PRINTED BOOKS, ETC, BEFORE 1734 53 Principia^ published first in the Year 1687, which almost vvhoUy depends upon the Use of the said Calculus. But the Method of Mr. Leibnitz'z is much more easy and expeditious, on account of the Notation he uses. . . ." In the preface of " The Translator to the Reader " Stone points out that the work he is bringing out '*becomes the more necessary, because there are but two EngHsh Treatises on the Subject . . . the one being Hay's Introduction to Mathematical Philosophy, and the other, Ditton's Institutio7i of Fluxions'' \ the former " too prolix," the latter "much too sparing in Examples " and " too re- dundant " in the explanation of fluxions, so that " it is next to impossible for one who has not been conversant about Infinites to apprehend it. That of our Author is much easier, tho less Geo- metrica!, who calls a Differentia] (or Fluxion) the infinitely small Part of a Magnitude. " " But," con- tinues Stone, ' ' I would not bere be thought in any wise to lessen the Value of Sir Isaac Newton's Definition : When the Learner has made some Progress, I would bave him then make himself Master of it." Stone then proceeds to explain the nature of fluxions, foUowing closely Newton's language in his Quadrature of Curves. 71. In De l'Hospital's treatise, as translated by Stone, we read : "The infinitely small Part whereby a variable Quantity is continually increased or decreas'd, is called the Fluxion of that Ouantity." 54 LIMITS AND FLUXIONS Here Stone simply writes " fluxìon " where De l'Hospital writes " difìerence," which is a mischievous procedure, seeing that the two words stand for things totally difìferent. De rHospital's wordiiig is *'La portion infiniment petite dont une quantité variable augmente ou diminuè continuelle- ment, en est appellée la Difìférence. " Stone also changes from the Leibnizian to the Newtonian notation, by writing x instead of dx. Then foUow two postulates : ''Grant that two Quantities, whose Difìférence is an infinitely small Ouantity, may be taken (or used) indifferently for each other : or (which is the same thing) that a Ouantity, which is increased or decreas'd only by an infinitely small Quantity, may be consider'd as remaining the same. " Grant that a Curve Line may be consider'd as the Assemblale of an infinite Number of in- finitely small right Lines : or (which is the same thing) as a Polygon of an infinite Number of Sides, each of an infinitely small Length, which determine the Curvature of the Line by the Angles they make with each other." De l'Hospital's " prendre la difìférence" is rendered by Stone " to find the fluxions." The fluxion oi xy is found by taking the product of x-\-x and J+j, and neglecting xy, " because ij is a Quantity infinitely small, in respect of the other Terms yx and xj/." 72. Further on in Stone's translation (p. 73) we read : PRINTED BOOKS, ETC, BEFORE 1734 55 *'The infinitely small Part generateci by the con- tinuai increasing or decreasing of the Fluxion of a variable Quantity, is called the Fluxion of the Fluxion of that Quantity, or second Fluxion.'' In like manner he defines third Fluxion ; ' ' fluxion of the second fluxion" taking the place of " différence de la différence seconde." In the appendix, containing Stone's Inverse Method of Fluxions, a fluent is defined thus : ' ' The fluent or flowing Quantity of a given fluxionary Expression, is that Quantity whereof the given fluxionary Expression is the Fluxion." Remarks 73. The earliest treatment of the new analysis which became current in England was that of Leibniz. The Scotsman Craig used it for over a quarter of a century before rejecting it in favour of fluxions. Harris, Hayes, and Stone drew their inspiration from French writers who followed Leibniz. A hopeless confusion arose in the use of the term ''fluxion." Newton always took it to be a velocity, but many writers, including Newton's friends who prepared the Corniate rcium Episiolicum, simply said "fluxion" instead of " differential, " thus putting a home label upon goods of foreign manufacture. A strict foUower of the Newton of 1704 was Ditton ; fluxions are taken as infinitesimals by Fatio de Duillier, Cotes (in 1701), Harris, Hayes, Raphson, Stirlìng (in 17 17), and Stone. Stone Comes out strongly with the view that a 56 LIMITS AND FLUXIONS circle is a polygon of an infinite number of sides. He also uses the infinitesimal triangle. Hayes and Stone have no hesitation in speaking of '' fluxions of fluxions," and " infinitely little parts of an in- finitely little part." No writers, unless we except Newton (1704) and Ditton, dispense with the use of infinitely small quantities. The dropping of such quantities from an equation was usually permitted without scruple. What an opportunity did this medley of untenable philosophical doctrine present to a dose reasoner and skilful debater like Berkeley ! [See Addenda, p. 289.] CHAPTER III BERKELEY'S ANALYST '(1734) ; CONTROVERSY WITH JURIN AND WALTON 74. BiSHOP BERKELEY'S publication of the Analyst'^ is the most spectacular event of the century in the history of British mathematics. The arguments in the Analyst were so many bombs thrown into the mathematica! camp. The views expressed in the Analyst are fore- shadowed in Berkeley's Principles of Human Know- ledge (§§ 123-134), published nearly a quarter of a century earUer. The " Infidel mathematician," it is generally supposed, was Dr. Halley. Mathe- maticians complain of the incomprehensibiUty of rehgion, argues Berkeley, ,but they do so unreason- ably, since their own science is incomprehensible. " Our Sense is strained and puzzled with the perception of objects extremely minute, even so the Imagination, . . . is very much strained and puzzled to frame clear ideas of the least particles of time, or the least increments generated therein : ^ The Analyst : or, a Discourse addresscd to an Infide l Mathe- matician. Wherein it is exainined whether the Oò/'ect, Principles, and Inferences of the Modem Analysis are more dislinctly concewed, or more evidently deduced, than relii^otis Mystenes and Points of Faith. London, 1734. 57 58 LIMITS AND FLUXIONS and much more so to comprehend the moments, or those increments of the flowing quantities in staiu nascenti, in their very first origin or beginning to exist, before they become finite particles. And it seems stili more difficult to conceive the abstracted velocities of such nascent imperfect entities. But the velocities of the velocities — the second, third, fourth, and fifth velocities, etc. — exceed, if 1 mistake not, ali human understanding " {Analyst, § 4). . . . 75. '*In the calculus differentialis . . . our modem analysts are not content to consider only the differences of finite quantities : they also consider the differences of those differences, and the differences of the differences of the first differ- ences : and so on ad infiniturn. That is, they consider quantities infinitely less than the least discernible quantity ; and others infinitely less than those infinitely small ones ; and stili others infinitely less than the preceding infinitesimals, and so on without end or limit " (§ 6). 'j^. "■ I proceed to consider the principles of this new analysis. . . . Suppose the product or rectangle AB increased by continuai motion : and that the momentaneous increments of the sides A and B are a and b. When the sides A and B are deficient, or lesser by one-half of their moments, the rectangle was À-4^xB-i ^, i.e. h.^-\ a^-l bK^-\ ab. And as soon as the sides A and B are increased by the other two halves of their moments, the rectangle becomes A + -| ^ x B + J ^ or AB + | aV> + \ bA + | ab. From the latter rectangle subduct the former, and BERKELEY' S ANALYST (1734) 59 the remaining difìference will be à^-\-bh. There- fore the increment of the rectangle generateci by the entire increments « and <^ is <3:B + ^A. Q.E.D. But it is plain that the direct and true method to obtain the moment or increment of the rectangle AB, is to take the sides as increased by their whole increments, and so multiply them together, A+ — \h). '' . . . Rigorously speaking, the moment of the rectangle AB is not, as you suppose, the increment of the rectangle AB ; but it is the increment of the rect- angle A — J <2 X B — J /^. " A moment may be either an increment or a decrement ; you obtain the increment aB-\-òA+aò, the decrement of AB is aB + òA —ab. Which of those two will you cali the moment of AB ? "I apprehend the case will stand thus : àB^bA-\-ab-\-aB-\-bA — ab making twice the moment of the rectangle AB ; it foUows that oB-^bA will make the single moment of the same rectangle";^ the velocity which the flowing rectangle has, is its velocity ''neither before nor after it becomes AB, but at the very instant of time that it is AB." In like manner with the moment of the rectangle. Let me advise you hereafter to "first examine and weigh every word he [Newton] uses. " Lastly, 1 must observe that the moment of AB, namely oB-^bA, and the increment of the same rectangle, aB-\-bA-\-ab, "are perfectly and exactly equal, supposing a and b to be diminished ad infiniium.'' 88. As to your second instance of false reason- ing, in Newton's hook on Quadraiurcs, apparently that is "so truly Boeotian a blunder" that I know not how "a Newton could be guilty of it. " You ^ Jurin, op. cit., p. 46. 68 LIMITS AND FLUXIONS interpret '' Evanescant jani augìiienta illa'^ (our § 32), as "let novv the increments vanish, i.c, let the increments be nothing, or let there be no incre- ments. " But ' ' do not the words ratio ultima stare US in the face, and plainly teli us that though there is a last proportion of evanescent increments, yet there can be no proportion of increments which are nothing, of increments which do not exist ? " You grossly misinterpreted Newton. 89. As to the third head of your objections, since New^ton did not reason falsely, "he had no occasion to make use of arts and fallacies to impose upon his foUowers." " Having now . . . driven you entirely out of your intrenchments ... I should Sally out and attack you in your own. " "But as they seem rather designed for shew, than use, . . . to dazzle the imagination . . . [they] will likewise immediately disappear like the Ghost of a departed quantity,'' if you exorcise them " with a few words out of the first section of the Principia.''' You say that the paradox, "that Mathematicians should deduce true Propositions from false Principles " is accounted for by the fact that one error " is compensated by another con- trary and equal error." But the two are no errors at ali, as is evident from the fact that true results foUow when only the first operation is carried out, so that no compensation is possible. Jurin argues that the first supposed fallacy, without the second, gives as the subtangent of y'^=^ax, the value 2x{2y -\- dy) -h- {2y) \ the second supposed fallacy, BERKELETS ANALYST (1734) 69 without the first, gives 2x{2y)-^{2y + (fy). Both these expressions are equal to 2.r, " which is the result either of two errors, or of none at ali." li you claim that 2x{2y + dy)-^{2y)> 2x, how much greater is it, supposing 2,r= 1000 miles ? Not as much as the thousand-millionth part of an inch. Jurin ends vvith a discussion of Lock on abstract ideas. Wa/ton's First Reply to Berkeley 90. Little is known about John Walton. He was Professor of Mathematics in Dublin, and partici- pated in this controversy. Otherwise, practically nothing about him has been handed down. His reply to Berkeley was published in 1735 at Uublin.i Berkeley attacked the method of fluxions more particularly as given in Newton's earlier exposition ; Walton defended the theory on the basis of the later treatment as given by Newton in his Quadratura Curvaruni (1704), and in the Prmcipia, Book II. 91. Walton begins by stating that inasmuch as the credulous may " become infected" by Berkeley's attack on fluxions, it seems necessary to give a short account of the nature of fluxions. "The momentaneous Increments or Decrements of flow- ^ A Vindiiatioìi of Sii- Isaac Neiuton x l'timiples of Fluxions, agaiusi the Oift'idotis lontaiiied in fhe Analyst. By J. Walton. — Siquid novisti reclius isiis, candidus imperti : bi non, his utere inecum. Hor. In the fuhiess of his Suftìciency he shall he in Straits : Every Hand of the IVickcd shall come upon him. y<^/^— Duhlin, Printed ; and reprinted at London, and sold by J. Robcrts in Warwick-Lane. 1735- [Price Six Pence.] 70 LIMITS AND FLUXIONS ing Quantities, he [Newton] elsewhere calls by the name of Moments, . . : By Moments we may understand the nascent or evanescent Elements or Principles of finite Magnitudes, but not Particles of any determinate Size, or Increments actually generated ; for ali such are Quantities, themselves generated of Moments." 92. "The magnitudes of the momentaneous Increments or Decrements of Quantities are not regarded in the Method of Fluxions, but their first or last Proportions only ; that is, the Proportions with which they begin or cease to exist." . . . "The ultimate Ratios with which synchronal Increments of Quantities vanish, are not the Ratios of finite Increments, but Limits which the Ratios of the Increments attain, by having their magni- tudes infinitely diminish'd. . . . There are certain determinate Limits to which ali such Proportions perpetually tend, and approach nearer than by any assignable Difference, but never attain before the Quantities themselves are infinitely diminish'd ; or 'till the Instant they evanesce and become nothing." "The Pluxions of Quantities are very nearly as the Increments of their Fluents generated in the least equal Particles of Time," and they "are accurately in the first or last Proportions of their nascent or evanescent Increments." "The Fluxions of Quantities are only velocities. ..." Again, ". . . to obtain the Ratios of Fluxions, the corresponding synchronal or isochronal Incre- ments must be lessened in infiniiuni. For the BERKELEY'S ANALYST (1734) 71 Magnitudes of synchronal or isochronal Increments must be infinitely diminished and become evan- escent, in order to obtain their first or last Ratios, to which Ratios the Ratios of their corresponding Fluxions are equal. " The moment of the rectangle AB is Ab+Ba, for consider Ab-\-Ba-\-ab and Ab + Ba, "under a Constant Diminution of the Incre- ments a and b . . . [they] constantly tend to an Equality . . . [and] they become equal, and their Ratio becomes a Ratio of Equality. ..." Hence Ab-\-Ba-\-ab " is not the Moment or Fluxion of the Rectangle AB, except in the very Instant vvhen it begins or ceases to exist." Here fluxions appear to be no longer velocities (finite magnitudes) but moments. Walton next quotes a Latin passage from the Quadratura Curvai um. He says that Berkeley seems ''to have been deceived by an Opinion that there can be no first or last Ratios of mathematical Ouantities," but Walton insists that if quantities are generated together, or if they vanish together, they will do so "under certain Ratios, which are their first or last Ratios." Walton claims that Berkeley's lemma " is in no Way pertinent to the Case for which it was in- tended " ; he explains the Newtonian process of finding the fluxion of x*^, supposing x to increase uniformly, and points out that this is done without rejecting quantities " on account of their exceeding smallness." Commenting on Berkeley's contention that "no geometrie Ouantity, by being infinitely diminished, can ever be exhausted or become 72 LIMITS AND FLUXIONS nothing, " Walton states that the fluxional calculus assumes that '^Ouantities can be generateci by Motion . . . and consequently they may also by Motion be destroy'd." 93. Walton's Vindication foUows Newton's ex- position closely ; Berkeley's claim that Walton followed in Jurin's track and borrowed from him, is, I believe, incorrect. Take the vital question of rejecting infinitesimals : Jurin claims that, being so very small, they do not appreciably affect the result ; Walton takes the stand that there is no rejection whatever of infinitesimals. The main criticism to be passed on Walton's first essay con- sists, in our judgment, in a failure to meet Berkeley's objections squarely and convincingly. Beì'kelefs Reply to Jurin and Walton 94. Jurin's and Walton's articles were answered by Berkeley in a publication entitled, A Dcfence of Free- TJiinking in Mathematics. ^ Berkeley restates the purpose he liad in writing the Analyst '. " Novv, if it be shewn that fluxions are really most incomprehensible mysteries, and that those who believe them to be clear and scien- tific do entertain an implicit faith in the aiithor of that method : will not this furnish a fair argumen- tum ad hominem against men who reject that very thing in religion which they admit in human learn- ^ A Defence of Frec-'J'hitikmi^ in A/athet/iaiics. In Ansivo- io a Pamphlet of Philalethes Cantabrigiensis. . . . Also an Appendix concernine; Mr. Walton^ s Vindication. . . . By the Author of " 7'he Minute Philosopher,'' Dublin, 1735. BERKELEY'S ANALYST (1734) n ing?i (§ 3) . . . I say that an infidel, who believes the doctrine of fluxions, acts a very inconsistent part in pretending to reject the Christian religion — because he cannot believe what he doth not comprehend " (§ 7). . . . ^ Berkeley is not the only one who invoked the aid of the Doctrine of Fluxions in theological discussion. In a criticism {A Keviexv of the Fiery Eniption^ etc, London, 1752, p. 128) of Bishop William Warburton's Julian^ concerning earthquakes and fiery eruptions, which, Warburton argued, defeated Julian's attempi to rebuild the tempie at Jerusalem, it is stated that a connection (needed in the argument) was established between the preservation of Christianity and tìie destruction of Judaism by the following clever procedure: — "The great modem Father of the mathematics had invented a new and curious way of improving that science by a fiction ; according to which quantities are supposed to be generated by the continuai flux or motion of others. In the application of this method it became neces- sary to consider these quantities, sometinìes in a nascent, and at other times in an evanescent state, by which ingenious contrivance they could be made either continually to tend to and at last absolutely to become nothing, or vice versa, according to the intention and occasions of the Artist. Now by extending this noble invention to the two religions, it evidently appeared, that, from the time of the first coming of Christ, Judaism entered into its evanescent state, as on the other band Christianity did into a nascent state, by which means both being put into a proper flux, one was seen continually decaying, and the other continually improving, till at last by the destruction of the Tempie Judaism actually vanished and became nothing, and the Christian religion then bursted out a perfectly generated Entity. . . . As the great author of the mathematical method of fluxions had for very good reasons studiously avoided giving any definition of the precise magni- tude of those moments, by whose help he discovers the exact magnitude of the generated quantities, so our Author [Warburton] by the same rule of application, and under the influence of the same authority, was fairly excused from defining that precise degree of perfection and imperfection in which the two religions subsisted, during the respective evanescent and nascent state of each, by the help of which he discovered the precise time when Judaism was perfectly abolished, and Christianity perfectly established. But we may well suppose, that the most alluring charm in this extraordinary piece of ingenuity, was the creating of a new character by it : For questionless he may now be justly stiled the great founder and inventor of the fluxionary method of theology. . . . This fancy of a necessary connexion between the Temple-edifice, and the being of Christianity, . . . this pretended Christianity which is of such an unsubstantial nature, that it must necessarily vanish at the restoration of the Tempie, can be nothing else but a mere Ghost^ . . . evidently the Ghost of departed Judaism." 74 LIMTTS AND FLUXIONS 95. " I bave said (and I venture stili to say) that a fluxion is incomprehensible : that second, third, and fourth fluxions are yet more incomprehensible : that it is not possible to conceive a simple infini- tesimal : that it is yet less possible to conceive an infinitesimal of an infinitesimal, and so onward. What bave you to say in answer to this ? Do you attempt to clear up the notion of a fluxion or a difference ? Nothing like it " (§ 17). 96. Berkeley quotes from Newton's Principia and QuadratUì^e of Curves, and then asks, " Is it not plain that if a fluxion be a velocity, then the fluxion of a fluxion may, agreeably thereunto, be called the velocity of a velocity? In like manner, if by a fluxion is meant a nascent augment, will it not then foUow that the fluxion of a fluxion or second fluxion is the nascent augment of a nascent augment?" (§ 23). 97. "I had observed that the great author had proceeded illegitimately, in obtaining the fluxion or moment of the rectangle of tvvo flowing quan- tities. ... In answer to this you allege that the error arising from the omission . . . is so small that it is insignificant (§ 24). . . . If you mean to defend the reasonableness and use of approxi- mations ... I bave nothing to say. . . . That the method of fluxions is supposed accurate in geometrica! rigour is manifest to whoever considers what the great author writes about it . . . In rebus mathernaticis errores guam minimi 7ion sunt contemnendi'' (§ 25 ; our § 30). BERKELEY'S ANALYST (1734) 75 98. Berkeley justifies his use of the expression '' increment of a rectangle " by quoting from Newton (our § 1 7), " rectanguli incrementum aB -\-òA." '* You say ' you do not consider AB as lying at either extremity of the moment, but as extended to the middle of it ; as having acquired the one half of the moment, and as being about to acquire the other ; or, as having lost one half of it, and being about to lose the other.' Now, in the name of truth, I entreat you to teli what this moment is, . . . Is it a finite quantity, or an infinitesimal, or a mere limit, or nothing at ali ? . . . If you take it in either of the two former senses, you con- tradict Sir Isaac Newton. And, if you take it in either of the latter, you contradict common sense ; it being plain that what hath no magnitude, or is no quantity, cannot be divided " (§ 30). "... You observe that the moment of the rectangle determined by Sir Isaac Newton, and the increment of the rectangle determined by me are perfectly and exactly equal, supposing a and ò to be diminished ad infiniiurn : and, for proof of this, you refer to the first lemma of the first section of the first hook of Sir Isaac's Principles. I answer that if a and /; are real quantities, then ab is some- thing, and consequently makes a real difference : but if they are nothing, then the rectangles whereof they are coefficients become nothing like- wise : and consequently the monientuvi or ina-e- mentuni, whether Sir Isaac's or mine, are in that case nothing at ali. As for the above-mentioned 76 LIMITS AND FLUXIONS lemma, . . . however that way of reasoning may do in the method of cxhaustions, where quanti ties less than assignable are regarded as nothiiig ; yet, for a fluxionist writing about momentums, to argue that quantities must be equal because they have no assignable difference, seems the most injudicious step that could be taken ; . . . for, it will thence foUow that ali homogeneous momentums are equal, and consequently the velocities, mutations, or fluxions, proportional thereto, are ali likewise equal" (§ 32). 99. As regards Newton's evanescant jam augvienta illa{p\xx § 32), Berkeley argues that it means either 'Met the increments vanish," or else " let them become infinitely small," but the latter " is not Sir Isaac's sense, " since on the very same page in the Introduction to the Quadrature of Curves he says that there is no need of considering infinitely small figures. Taking advantage of the fact that the Newton of the Principia (1687) differed from the Newton of the Quadratura Curvaruvi (1704), Berke- ley broke out into the following philippic : " Vou Sir, with the bright eyes, be pleased to teli me, whether Sir Isaac's momentum be a finite quantity, or an infinitesimal, or a mere limit ? If you say a finite quantity ; be pleased to reconcile this with what he saith in the scholium of the second lemma of the first section of the first hook of his Principles (our § 12): Cave intellii^as quarititatcs lìuigniliidiìic deierminatas, sed cogita seìuper diniinucìidas sinc limite. If you say, an infinitesimal ; reconcile this BERKELEY'S ANALVST (1734) 77 with what is said in his Introduction to the Ouadra- tures (our § òZ)'- Volui os tendere quod in niethodo fluxionuni non opus sii figuras infinite pai'vas in geovietriaìH introducere. If you should say, it is a mere limit ; be pleased to reconcile this with what we find in the first case of the second lemma in the second hook of his Principles (our § 17): Ubi de lateribus A et B deerant monientorum dimidia, etc. — where the moments are supposed to be divided. I should bc very glad a person of sudi a luminous intellect would be so good as to explain whether by fluxions we are to understand the nascent or evanescent quantities themselves, or their motions, or their velocities, orsimply their proportions . . . that you would then condescend to explain the doctrine of the second, third, and fourth fluxions, and show it to be consistent with common sense if you can" (§ 36). 100. In an appendi x to the Defence of Free-Think- ing in Matheìnatics^ Berkeley replies to Walton, stating that the issues raised by him had been previously raised by "the other," that he delivered a technical discourse -without elucidating anything, that his sclìolars have a right to be informed as to the meaning of fluxions and should therefore ask him "the foUowing questions. " Then foUow many questions, of which we give a few : " Let them ask him — Whether he can conceive velocity without motion, or motion without ex- tension, or extension without magnitude ? . . . Whether nothing be not the product of notliing 78 LIMITS AND FLUXIONS multiplied by something ; and, if so, . . . when ab is nothing, whether Ab-\-V>a be not also nothing ? ì.e. whether the momentum of AB be not nothing? Let him then be asked, what his momentums are good for, when they are thus brought to nothing ? . . . I wish he were asked to explain the differ- ence between a magnitude infinitely small and a magnitude infinitely diminished. . . . Let him be farther asked, how he dares to explain the method of Fluxions, by the Ratio of magni tudes infinitely diminished, when Sir Issac Newton hath expressly excluded ali consideration of quantities infinitely small? If this able vindicator should say that quantities infinitely diminished are nothing at ali, and consequently that, according to him, the first and last Ratio's are proportions between nothings, let him be desired to make sense of this. . . . If he should say the ultimate proportions are the Ratio's of mere limits, then let him be asked how the limits of lines can be proportioned or divided ? " Walton's Second Reply to Berkeley lOi. In a second reply ^ to Berkeley, Walton States that in the Appendix to the Defence^ Berkeley "has composed a Catechism which he recommends to my Scholars " and which Walton quotes. I am first to be asked, " Whether I can conceive Velocity without Motion, or Motion without Extension. . . . ^ J. Walton, Catechism of the Author of the Minute Philosopher FuUy answer'd. Printed at Dublin. Reprinted at London, and sold by J. Roberts, 1735. It is a pamphlet of 30 pages. BERKELEY'S ANALYST (i734) 79 I answer, I can conceive Velocity and Motion in a Point of Space ; that is, without any assignable Length or Extension described by it . . . for . . . if a cause acts continually upon a given Thing . . . there must be a continuai Increase of its Velocity : the Velocity cannot be the same in any two difìferent Points," as in the case of falling bodies, Referring to A^ + B^, Walton continues : ''I agree with him that nothing is the Product of nothing multipl'd by something ; but must know what he means by the vanishing of the Gnomon ^ and Sum of the two Rectangles . . . before I give him a direct Answer. If by vanishing he means that they vanish and become nothmg as Areas, I grant they do ; but absolutely deny, upon such an Evan- escence of the Gnomon and Sum of the two Rectangles by the moving back of the Sides of the Gnomon till they come to coincìde with those of the Rectangle, that nothing remains. For there stili remain the moving Sides, which are now become the Sides of the Rectangle, . . . the Motion of the Gnomon is the same with the Sum of the Motions of the Two Rectangles, when they evanesce, and are converted into the two Sides of the Rectangle AB. If a point moves forward to generate a Line, and afterwards the same Point moves back again to destroy the Line with the very same Degrees of Velocity, in ali Parts of the Line ^ If a parallelogram is extended in length and breadth and if the originai parallelogram be removed, the remaining figure is called the gnomon. 8o LIMITS AND FLUXIONS which it had in those Parts when moving forward to generate it ; in the Instant the Line vanishes as a Length . . . the generating point will remain, together with the Velocity it had at the very Beginning of its Motion, And the Case is the very same with respect to the Rectangle increas- ing by the Motion of its Sides." This point is elaborated with great fullness. After some illustra- tions, Walton exclaims : *'This is a full and clear Answer to this part of the catechism, and shows that its Author has been greatly mistaken in supposing that I explained the Doctrine of Fluxions by the Ratio of Magnitudes infinitdy diminish' d, or by Proportions between nothings. ... I do not wonder that this Author should bave no clear Ideas or Conceptions of second, third or fourth Fluxions, when he has no clear Conceptions of the common Principles of Motion, nor of the first and last Ratios of the isochronal Increments of Quantities generated and destroyed by Motion. ... In order to prevent my being Catechised any more by this Author," Walton makes a confession *'of some Part of my Faith in Religion." Jurin's Second Reply to Berkeley 102. Jurin brought out a second publication,^ of 1 1 2 pages, which was in reply to Berkeley's Defence of Free-Thinking. Passing by unimportant pre- liminaries, we come to Jurin's definitions of '*flow- ' Tlìi' Minute Matheinatìclaii : or. The FrecThiither no Jiist- Thinkei. By Philaleihes Cantabiigiensis. London, 1735. BERKELEY'S ANALYST (1734) 81 ing quantity, " " fluxion " ("the velocity with which a flowing quantity increases or decreases "), " incre- ment," " nascent increment " ("an increment just beginning to exist from nothing . . . but not yet arrived at any assignable magnitude how small so- ever"), "evanescent increment" (similarly defined). He then endeavours to prove the proposition : "The Fluxions, or Velocities of flowing quantities . . . are exactly in the first proportion of the nascent increments, or in the last proportion of the evanescent increments." He insists that "the first ratio of the nascent increments must be the same, whether the velocities be uniform or variable " ; hence, "the nascent increments must be exactly as the velocities with which they begin to be generated." In further explanation, Jurin says that, according to Newton, nascent increments are " less than any finite magnitude," " their magnitude cannot be assigned or determined," "the proportion between them . . . being ali that is requisite in his Method." In further explanation of the pro- portion of evanescent increments he says, it " is not their proportion before they vanish, " " nor is it their proportion after they bave vanished," "but it is their proportion at the instant that they vanish." Jurin then states that Berkeley has " taken as mudi pains as . . . any man living, except a late Philosopher of our University, to make nonsense of Sir Isaac Newton's principles; " There is no occurrence in Newton's writings of "velocity with- out motion," " motion without cxtcnsion," which 6 82 LIMITS AND FLUXIONS Berkeley pretends to derive from them. Jurin succeeds, we think, in establishing the contention that there is no greater difficulty in explaining the second or third fluxion, than there is in explaining the first. " The second fluxion is the velocity with which the first fluxion increases." Jurin confesses that his statement in his first reply to Berkeley, to the effect that certain errors were of "no significance in practice," was intended for popular consumption, for men such as one meets in London. 103. "One of them, indeed, could make nothing of what I had said about the length of a subtangent, or the magnitude of the orb of the fixed stars ; but was fully satisfied by the information given him by one of his acquaintance to the following effect. The Author of the Minute Philosopher has found out that, if Sir Isaac Newton were to measure the height of St. Paul's Church by Fluxions, he would be out about three quarters of a hair's breadth : But yonder is one Philalethes at Cambridge, who pretends that Sir Isaac would not be out above the tenth part of hair's breadth. Hearing this, and that two books had been written in this controversy, the honest gentle- man flew into a great passion, and after muttering something to himself about some body's being over- paid, he went on making reflections, which I don't care to repeat, as not being much for your honour or mine." 104. Jurin thereupon takes up the rectangle AB. The terms "moment" and " increment " are involved in the discussion of it. Jurin de- BERKELEY' S ANALYST (1734) 83 clares : "I absolutely and fully agree with you that the incremenium in the conclusion is the momentum in the Lemma," that "the momenium in the Lemma" is "the momentU7n of the rectangle AB." Further, Jurin says, "the incremenium in the conclusion is manifesti)^ the excess of the rectangle A+|<2xB4-|^, above the rectangle K — \ay.^ — \b, i. e. the increment of the rectangle A — J<2xB — 1^. Therefore we are agreed that the moment of the rectangle AB is the increment of the rectangle A — Ja only, and not the whole increment Aò-}-Ba-\-aòf is called the momentum of the rectangle under A,B. 120. Of this Discourse, a long account of twenty- six pages, written by Robins himself, although his name does not appear,^ was given in The Present State of the Republick of Letters, London, October, 1735, in which it is stated that Robins wrote his Discourse with the view of removing the doubts which had lately arisen concerning fluxions and ^ This account is republished in the Maihematical Tracis of the late Benjamin Robins, edited by James Wilson, London, 1761, voi. ii, p. 78. loo LIMITS AND FLUXIONS prime and ultimate ratios ; that Robins carefully distinguished both these methods from the method of indivisibles and also from each other. After an historical excursion viewing the works of the ancients, of Cavalieri and Wallis, the introduction by Newton of the concept of motion is taken up. " If the proportion between the celerìty of increase of two magnitudes produced together is in ali parts known, " then '' the relation between the magnitudes themselves must from thence be discoverable." This is the basis for fluxions. The "method of prime and ultimate ratios proceeds entirely upon the consideration of the increments produced." By it Newton avoids "the length of the ancient demon- strations by exhaustions, " on which, according to Robins, the method of fluctions rests. "Newton did not mean, that any point of time was assign- able, wherein these varying magnitudes would become actually equal, or the ratios really the same ; but only that no difìference whatever could be named, which they should not pass." Newton's term momentum is used simply for greater brevity, hence need not be considered. Newton's descrip- tion is capable of an interpretation too much resembling the language of indivisibles — in fact, he sometimes did use indivisibles at first ; Robins has freed the doctrine from this imputation in a manner that "shall agree to the general sense of his [Newton's] description. " JURIN V. ROBINS AND PEMBERTON loi Jurìn^s Review of his own Lettei's to Berkeley 121. In the November, 1735, number of the Republick of Letters, Philalethes Cantabrigiensis (Jurin) appears with an article, Considerations upon some passages contained in two Letters to the Author of the Analyst. The two letters in question are the two repHes Jurin himself had made to Berkeley. The article is really a reply to Robins, though Robins's name is not mentioned. Jurin claims to have adhered strictly to Newton's language ; some other defenders of Newton, says he, are guilty of departing from it. Their objections to his own defence are threefold : " I. My explication of Lemma i, Lib. I, Princip." See our §§ 4, 6, 8. '* IL The sense of the Scholium ad Sect. i, Libr. I, Princ, particularly as to, *' I. The doctrine of Limits, 2. The meaning of the term evanescent, or vanishing. " See our §§ 10-15. "III. The sense of Z67;2w<2: 2, Lib. \\, Princip." See our §§ 16-19. 122. As to the first objection, Jurin insists that Newton's v^oràsfitcnt ultimo cequales mean that the quantities 'Mo at last become actually, perfectly, and absolutely equal." He takes the hands of a clock between 11 and 12. The arcs traced by the hands '' i. Constantly tend to equality, 2. During an hour, 3. And will come nearer to one another than to have any given difference, 4. Before the I02 LIMITS AND FLUXIONS end of the hour ; . . . at the end of the hour, the two quantities must become equal. " Further, " by taking the consideration of a finite time, Sir Isaac Newton is able to assign a point of time, at which he can demonstrate the quantities to be actually equal." Consider, says Jurin, the ordinate to a point of a hyperbola and that ordinate continued to the asymptote : they do not become equal in a finite time; Newton's Lemma is, **with great judgment, so worded on purpose, as necessarily to exclude this and such like cases. " Thus Newton's inscribed and circumscribed rectangles of Principia^ Lib. I, Sec. I, Lemma 2 (fig. i in our § 9), were thought by Nieuwentiit and others never to be capable of coincidence with the curve (say the quadrant of a circle) ; but Jurin assujues the varia- tion to be of such a nature that the limit is actually reached, as demanded by Newton's Lemma. For, suppose a point to move on the horizontal radius from the circumference to the centre A in one hour ; suppose also that, when this moving point is at B on that radius, there be two rectangles described upon AB (one in- scribed, the other circumscribed), and that upon every other part of the horizontal radius that is equal to AB, namely the parts BC, CD, DE, taken in order, rectangles be similarly erected ''at the same point of time," then as the moving point nears the centre, the rectangles diminish in size and increase in number, and they must together become equal to the quadrant at the end of the JURIN V. ROBINS AND PEMBERTON 103 hour. Jurin points out that he has introduced bere ali the suppositions of Newton's first Lemma^ namely that, (i) the two figures tend constantly to equality, (2) during one hour, i.e. a finite time, (3) and come nearer to one another than to any gìven difìference, (4) before the end of the hour, i.e, before the end of a finite time. Jurin continues : " If any man shall say, that a right-angled figure, inscribed in a curvilineal one, can never be equal to that curvilineal figure ; much less to another right-lined figure, circumscribed about the curve ; I agree with him. I am ready to own that, during the hour, these figures are one inscribed, and the other circumscribed ; that neither of them is equal to the curvilineal figure, much less one to another. But then, on the other band, it must be granted me, that, at the instant the hour expires, there is no longer any inscribed or circumscribed figure ; but each of them coincides with the curvi- lineal figure, which is the limit, the limes curvi- lìneus, at which they then arrive. " 123. Jurin thereupon proceeds to Lemma 7 of Book I, Section i in Newton's Principia, which, he says, requires additional consideration. It relates to fig. zj, where ACB is any are and "the points A and B approach one another and meet." Lemma 7, in Andrew Motte's translation, reads as follows : — ''The same things being supposed ; I say, that the ultimate ratio of the are, chord, and tangent, any one to any other, is the ratio of equality." Jurin says that bere the chord AB, the arch ACB, 104 LIMITS AND FLUXIONS FlG. and the tangent AD come to vanish when B reaches A, and their last ratio is unity. Newton " directs Olir imagination, not to these vanishing quantities themselves, but to others which are proportional to them, and always preserve a finite magnitude," such as A<^, the arch Kcb^ Ad. Since at the instant when A and B coincide, ''the angle BAD, or bAd, will vanish ; it is easy to conceive that, . . . the chord Ab must coincide with the tangent Ady . . . consequently, AB, AD must likewise, at the same instant of time, arrive at the same proportion of a perfect equality. " 124. Proceeding to the last Scholium in Book I, Section I of the Principia, Jurin starts by defining the word liniit. " I apprehend therefore that, by the limit of a variable quantity, is meant some determinate quantity, to which the variable quantity is supposed continually to approach, and to come nearer to it than to have any given difiference, but never to go beyond it. And by the limit of a variable ratio, is meant some determinate ratio, to which the variable ratio is supposed continually to approach, and to come nearer to it than to have any given difference, but never to go beyond it. By arriving at a limit I understand Sir Isaac Newton to mean, that the variable quantity, or ratio, becomes absolutely equal to the determinate quan- tity, or ratio, to which it is supposed to tend." fURIN V. ROBINS AND PEMBERTON 105 VVith unusual lucidity, for that period, Jurin says Oli the subject of limits : * ' Now whether a quantity, or ratio, shall arrive at its limit, or shall not arrive at it, depends entirely upon the supposition we make of the time, during which the quantity, or ratio, is conceived constantly to tend or approach towards its limit." If we assume the approach to be made in a finite time, the limit is reached, other- wise it is not reached. Of a variable which "can never attingere limitem " Newton gìves one illustra- tion at the end of the Scholium : that of two quantities having at first a common difference and increasing together by equal additions, ad infinitum. Since they can never be really and in fact increased ad infinitum, says Jurin, their ratio cannot arrive at its limit. What Newton wanted to meet was the objection, "that if the last ratio's of evanescent quantities could be assigned, the last magnitudes of those quantities might likewise be assigned." Newton says No, "for those last ratio's, wìth which the quantities vanish, strictly speaking, are not the ratio's of the last quantities ... but they are the limits" which those ratios can never "arrive at," "before the quantities are dÀvcì\m^\i^à. ad infinitum.'' As to the sense in which Newton uses the word evanescent or vanishing, in the Scholium under consideration, Jurin inclines to the view that '*both imply one single instant, or point of time." 125. In the Principia, Book II, Section 2, Lemma 2 (our §§ 16-19), Newton defines moment as " a momentaneous increment, or decrement, of a io6 LIMITS AND FLUXIONS flowing quantity, proportional to the velocity of the flowing quantities." According to Jurin, Newton puts a, b, e to signify either the moments, or the velocities, of the flowing quantities A, B, C. Leibniz looks upon them as differences. Newton, says Jurin, never used indivisibles, and his method to find the differences of variable quantities is not *'rigorously geometrical," but is more rigorous than the treatment given by Leibniz. Robins's Rejoinder 126. Robins replied in the Republick of Letters for December, 1735, in a Rcview of some of the Principal Objections that have been made to the Doctrine of Fluxions and Ultimate Propoj'tions ; with some Remarks on the different Methods that have been taken to obviate them. Robins does not here men- tion Philalethes any more than the latter referred directly to Robins. The objections to fluxions, says Robins, are levelled at Newton's expression, fluxiones sunt in ultima ratione decrementoi-um evan- escentium vel prima nascentium. " Which being usually thus translated, that fluxions are in the ultimate ratio of the evanescent decrements, or in the first ratio of the nascent augments, it has from hence been ask'd, what these nascent or evanescent augments are ? " There are difficulties of interpre- tation, whether the augments have quantity or have not. One way out of this difiiculty which has been pointed out, is to say : "the limit of the proportion that the decrements bear to each other JURIN V. ROBINS AND PEMBERTON 107 as they diminish, is the true proportion of the fluxions"(p. 438). Here a new difficulty arises : Does the varying ratio reach its limit **actually, perfectly, and absolutely," or does it not ? On the understanding that it does not reach the Hmit, '' ali that has at any time been demonstrated by the ancient method of exhaustions may be most easily and elegantly deduced." Rigour of demonstration does not require ultimate coincidence. *'Coinci- dence of the variable quantity and its limit, could it be always prov'd, would yet bring no addition to the accuracy of these demonstrations " (p. 441). Hence, *'why to the naturai difficulty of these subjects should the obscurity of so strained a con- ception be added ? " Is this view a correct inter- pretation of Newton ? A literal translation of bis Lemma i, Section i, Book I, Principia (see our §§ 4, 6, 8), is : " Ouantities, and the ratios of quantities. that during any finite time constantly approach each other, and before the end of that time approach nearer than any given difference, are ultimately equal. " What is the meaning of *' given difference"? If it be a "difference first assigned" according to which the degree of approach of these quantities may be afterwards regulated ; then . . . ratios, and their limits, tho' they do never actually coincide, will come within the de- scription of this Leinuia ; since the difference being once assign'd, the approach of these quantities may be so accelerated, that in less than any given time the variable quantity, and its limit, shall differ by io8 LIMirS AND FLUXIONS less than the assign'd difìference. " Mere Robins expresses the idea that the rapidity of approach toward the limit can be arbitrarily altered, yet he does not apparently perceive — certainly he does not admit — that this rapidity may be altered in such a way that the variable actually does reach its limit. On the contrary, he maintains that *'where the approach is determin'd by a subdivision into parts," **it is obvious, that no coincidence can ever be obtain'd." A coincidence such as Philalethes explains in the case of rectangles circumscribed and inscribed in a curve, if it could take place, is not a coincidence such as Newton intended, for Newton did not in this instance use motion, but continuai subdivision. Robins tries to establish his view of the matter by giving an instance of erroneous results being deduced by letting the variable reach its limit. He takes a hyperbola and revolves its principal axis in the piane of the curve, around the point of intersection of this axis and an asymptote, until the two lines coincide. At the end '*the hyperbola coincides with the asymptote," which is "absurd." As a matter of fact there is no absurdity. In lAx^ — a?'y'^ — a^U^ ^ the slope of the asymptote is m = b j a. Robins's process amounts to making in = o, which gives a real locus when b = o, namely the \oc\xs y^ = o. The only objection lies in stili calling the final curve a " hyperbola." JURIN V. ROBINS AND PEMBERTON 109 The Debate Continued 1 27. Robins's article was foUowed in the January, 1736, number of the Republick of Letters by Philalethes's Considerations occasioned by a Paper in the last Republick of Letters^ concerning some late Objections against the Do et r ine of Fluxioìis^ and the different Methods that have beeii taken to obviate them, Jurin denies having said that there was an ** intermediate state" between augments being **any real quantity" and being "actually vanished"; he says he gave Newton's declaration that " their magnitude cannot be assigned or determined. " Such intermediate magnitudes, in Jurin's opinion, cannot be " represented to the mind," but their ratio can be represented to the mind, by contemplating the ratio, *'not in the vanishing quantities themselves, but in other quantities permanent and stable, which are always proportional to them " (p. 'jG). As to Newton's Lemma i in Section i, Book I of the Principia^ if the great author meant to conclude^ that the quantities '*approach nearer than any given difference," then he first supposed what he would prove, and proved only what he had before supposed. Of this he could not be guilty. Besides, Newton's words,^ " fiunt eequales," do absolutely subvert such an interpretation. Jurin says that he does not claim that coincidence is necessary for rigorous proof; he admits that in Robins's treat- ment of prime and ultimate ratios, coincidence is 1 Newton's words are " fiunt ultimo aequales." See our § 4. no LIMITS AND FLUXIONS not necessary ; only, Robins's method is not that of Newton. To establish this last point, Philalethes quotes Newton's lemma in Latin, then prints Robins's and his own translation of it. In case of variation, the upper line is Robins's translation, the lower is Jurin's : — Quantities ^ and \ j \ ratio's of quantities, that , . ^ -. .- . .7 {approach eacÌLOthevA ciurins!; any finite time constantlyX , ,. J- ^ -"-^ -^ \ tend to equahty , J and before the end of that time approach nearer i than any given difference, are ultimately equal. \ \ to one another than to have any given difference^ do \ at last become equal. ) It is not clear to Jurin what Robins means by **are ultimately equal," nor can Jurin conceive "how quantities, which do never become actually equal, . . . can come within the description of a Lemma, which Lemma expressly affirms, that they become equal. " Fiunt ultimo cequales means * * become at last equal." Jurin quotes different places in the Principia which support his point. He denies that Newton proceeds, in the case of inscribed and circumscribed rectangles, by continuai divisions of the base of the figure, and gives references in support of his contention. Of interest are the foUowing admissions made by Jurin (p. 87): ''This equality therefore we are obliged to acknowledge, although we should not be able, by stretch of imagination, to pursue these figures, and, as it were, to keep them in sight ali the way, till the JURIN V. ROBIN S AND PEMBERTON ni very point of time that they arrive at this equality. For a demonstrated truth must be owned, though we do not perfectly see every step by which the thing is brought about." *'We have therefore no occasion for the delinea- tion of a line less than any line that can be assigned. We acknowledge such delineation to be utterly impossible ; as likewise the conception of such a line, as an actually existing, fixed, invariable, determinate quantity," Jurin here begins to dis- avow infinitesimals. 'M am very free to own that Sir Isaac Newton does not always consider this coincidence, or rather equality, of the variable quantity, or ratio and its ultimate, as necessary in his method." 128. The debate between Jurin and Robins had reduced itself by this time, not so much to the discussion of the logicai foundations of fluxions, as to the discussion of what Newton's own views on the subject had been. Robins prepared a long paper on the subject for the Aprii, 1736, issue of the Republick of Letters^ under the title : A Dis- sertation shewing, that the Account of the doctrine of Fluxions, and of prime and ultimate ratios, delivered in a treatise entitledy ' A discourse concerning the nature and certainty of Sir Isaac Newton's methods of fluxions, and of prime and ultimate ratios, ' is agreeable to the real sense and meaning of their great inventor. The paper covers 45 pages. Robins repeats the fundamental definitions and historical statements given in his earlier papers, and directs 112 LIMITS AND FLUXIONS some attacks against Berkeley, To set forth the views of Newton, quotations are made from his Works, He quotes from the Introduction to the Quadratura Cuì^ai-um (see our §§ 27-42). From the Quadratura Cuì-varum itself he quotes: *' Quantitates indeterminatas, ut motu perpetuo crescentes vel decrescentes, id est, ut fluentes vel defluentes, in sequentibus considero, designoque literis z, j'y X, v, et earum fluxiones, seu celeritates crescendi noto iisdem Hteris punctatis. Sunt et harum fluxionum fluxiones, sive mutationes magis aut minus celeres, quas ipsarum ^, j, ;r, v fluxiones secundas nominare licet," etc. Robins quotes also from the anonymous account of John ColHns's Commerdum Epistolicum, which figures so prominently in the controversy between the foUowers of Newton and of Leibniz. This account was published in the Philosophical Tì'ans- actions, voi. xxix, for the years 17 14, 1715, 17 16, of the Royal Society of London, of which Robins was a member.' Robins goes on the assumption that the anonymous article was written by Newton himself, an assumption denied by no one at that time or since, though Jurin in a reply wants to know on what authority Newton's authorship is asserted. Robins quotes as follows (see our § 47): *'When he [Newton] considers lines as fluents described by points, whose velocities increase or decrease, the velocities are the first fluxions, and their increase the second. " 129. Robins says that Berkeley, '*for the support JURIN V. ROBINS AND PEMBERTON 113 of his objections against this doctrine [of fluxions], found it necessary to represent the idea of fluxions as inseparably connected with the doctrine of prime and ultimate ratios, intermixing this plain and simple description of fluxions with the terms used in that other doctrine, to which the idea of fluxions has no relation : and at the same time by confounding this latter doctrine with the method of Leibniz and the foreigners, has proved himself totally unskill'd in both. These two methods of Sir Isaac Newton are so absolutely distinct, that their author had formed his idea of fluxions before his other method was invented, and that method is no otherwise made use of in the doctrine of fluxions, than for demonstrating the proportion between different fluxions. For, in Sir Isaac Newton's words [see our §§ 29, 36], as the fluxions of quantities are nearly proportional to the contemporaneous increments generated in very small portions of time, so they are exactly in the first ratio of the augmenta nascentia of their fluents. With regard to this passage the writer of the Analyst has made a two-fold mistake. First, he charges Sir Isaac Newton, as saying these fluxions are very nearly as the increments of the flowing quantity generated in the least equal particles of time. Again, he always represents these augmenta nascentia, not as finite indeterminate quantities, the nearest limit of whose continually varying pro- portions are here called their first ratio, but as quantities just starting out from non-existence, and 8 114 LIMITS AND FLUXIONS yet not arrived at any magnitude, like the infini- tesimals of differential calculus. But this is con- trary to the express words of Sir Isaac Newton, who after he had shewn how to assign by his method of prime and ultimate ratios the proportion, that difìerent fluxions have to one another, he thus concludes. In finitis autem quantitatilms Analysin sic institueì'e et finitarum nascentium vel evafiescentiufn rationes p7'imas vel ultimas investigare consonum est geo7netrice veteì-uni: et volui estendere^ quod in methodo fluxionum non opus sit figuras infinite pai-vas in geometriam introducei-e.'' (See our §§ 33, 41.) 130. Robins proceeds to explain that the method of prime and ultimate ratios is ''no other.than the abbreviation and improvement of the form of demonstrating used by the ancients on the like occasions. " It has nothing to do with infinitely small quantities, which have led into error not only Leibniz in studying the resistance of fluids and the motion of heavenly bodies, but also Bernoulli like- wise in the resistance of fluids and in the study of isoperimetrical curves. Such infinitely small quanti- ties led Parent to make wrong deductions. It was argued that because a heavy body descends through the chord of a circle terminating at its lowest point in the same time as along a vertical diameter, " the time of the fall through the smallest arches must be equal to the time of the fall through the diameter." To relieve Newton of the suspicion of not being free from the obscure methods of indivisibles, Robins says he [Robins] defined an ** ultimate JURIN V. ROBIN S AND PEMBERTON 115 magnitude " and ** ultimate ratio" as limits. This exposition Robins had given in full in his Discourse. The difference of interpretation of Newton's Lemma I in the Principia (Book I, Section i), given by him- self and by Jurin, arises from Jurin's misinterpre- tation of Newton's word given. He **supposes it to stand for assignable\ whereas it properly signi- fies only what is actually assigned." Jurin claims that by our interpretation, ''Newton is rendred obnoxious to the charge of first supposing what he would prove " (p. 307). Robins says in reply that the statement, quantities which **are perpetu- ally approaching each other in such a manner, that any difference how minute soever being given, a finite time may be assigned, before the end of which the difference of those quantities or ratios shall become less than that given difference," is an obvious but not an identical proposition. Robins argues, "that Sir Isaac Newton had neither demonstrated the actual equality of ali quantities capable of being brought under this lemma, nor that he intended so to do " (p. 309) ; when quanti- ties "are incapable of such equality, the phrase of ultimately equal must of necessity be interpreted in a somewhat laxer sense," as in Principia^ Book I, Prop. 71, "prò ajqualibus habeantur, are to be esteemed equal." When Newton says that the number of inscribed parallelograms should be augmented in infinituDi^ he does not mean that the number bccomes infinitely great, but that they are augmented endlessly. The nature of the motion ii6 LIMITS AND FLUXIONS assumed by Jurin to explain how the limit may be reached is excessively complex. Moreover, ''to assert that any collection of these inscribed and circumscribed parallelograms can ever become actually equal to the curve, is certainly an impro- priety of speech, . . . the essence of indivisibles consists in endeavouring to represent to the mind such inscribed or circumscribed figure, as actually subsisting, equal to the curve" (p. 312). Our interpretation " thus removes this doctrine quite beyond the reach of every objection " (p. 315). Robins argues that Newton's ultimce rationes^ quibuscuni quantitates evanescunt are not rationes quantitatum ultìmaruin ; but only limits, to which the ratios of these quantities, which themselves decrease without limit, continually approach ; and to which these ratios can come within any differ- ence, that may be given, but never pass, nor even reach those limits" (p. 316). ''Newton has expressly told us, that the quantities, he calls nascentes and evanescentes, are by him always con- sidered as finite quantities" (p. 321). 131. The 7nome?ita of quantities occur in Newton's De analysi per cBquationes numero terminoìmm infinitas, drawn up in 1666. Newton says "that he there called the moment of a line a point in the sense of Cavalerius, and the moment of an area a line in the same sense " (see our § 47), that " from the moments of time he gave the name of moments to the momen- taneous increases, or infinitely small parts of the abscissa and area generated in moments of time . . . JURIN V. ROBINS AND PEMBERTON 117 because we have no ideas of infinitely little quantities, he introduced fluxions into his method, that it might proceed by finite quantities as much as possible." Prime and ultimate ratios he introduced later. Newton says in that place that in his proofs he uses 0 for a finite moment of time, though some- times, for brevity, he supposes 0 infinitely little. Thus Newton used 0 in two senses ; in the fluxions published in 1693 in Wallis's algebra, 0 is used in the sense of indivisibles ; in 1704 he gave it a second signi fication in the Quadratura Curvarum. Robins sums up his dissertation thus: ** Hence it is very manifest, that as Sir Isaac Newton used at first indivisibles, so he soon corrected those faulty notions by his doctrine of fluxions, and afterwards by that of prime and ultimate ratios. And ali the absurdity of expression, and ali the inconsistency with himself charged on him by the author of the Analyst, arises whoUy from mis-representation." This paper was badly arranged and below the level of Robins's earlier contributions. 132. Robins's long paper in the Republick of Letters was followed in the July and August (voi. xxviii, 1736) numbers by Considerations upon some passages of a Dissertation conce rning the Doctrine of Fluxions^ published by Mr. Robins in the Republick of Letters for Aprii last, by Philalethes Cantabrigiensis. The paper extends over 136 pages, and could not be easily accommodated in a single number. From now on the disputants, particularly Jurin, are no longer in a calm frame of mind. . The ii8 LIMITS AND FLUXIONS debate is one over words and ceases to be illumi- nating. Their judgments were perverted by the heat of controversy. Even theological or politicai controversies could not easily surpass the verbosity and haze exhibited here. Jurin's first objection to Robins's last analysìs is the statement that the method of fluxions has no relation to the method of first and last ratios ; Jurin quotes from Newton in support of his contention. The charge that he (Jurin) represents augmentia nascentia not as finite, but as just starting out of non-existence, 'Mike infinitesimals of the differential calculus," Jurin denies, saying : Leibniz's differ- entials **are fixed, determinate, invariable " ; he himself has represented the nascent augments as "quantities just starting out from non-existence, and yet not arrived at any magnitude, and not as finite quantities" (p. 52), and quotes Newton in support of this view. According to the article in the Pìiilosophical Transactions, No. 342, attributed to Newton, moments are represented " by the rectangles under the fluxions and the moment 0 " ; "in his calculus there is but one infinitely little quantity represented by a symbol, the symbol 0 : it is also said, Prick'd letters never signify moments, unless when they are multiplied by the moment 0 either exprest or understood to make them infinitely little, and then the rectangles are put for moments." Jurin charges that Robins has now published four different interpretations of Newton's much-discussed lemma. Newton's phrase, jfìunt ultimo cequales, the JURIN V. ROBINS AND PEMBERTON 119 use of the words ** perpetually " and ** endlessly," ''the last difference," are again discussed at length. Jurin quotes from Robins a passage which appears to show that " Mr. Robins is now of opinion, that Sir Isaac's demonstration is appHcable to such quantities, as at last become actually equal, as well as to quantities, which only approach without Hmit to the ratio of equaHty " (p. 6"]) ; therefore, the lemma, ''by Mr. Robins's own confession, may be taken in the sense 1 have always understood it in " (p. 68). However, this is in direct conflict with Robins's earlier assertions. In the discussion about the inscribed rectangles, both Robins and Jurin agree that if the * ' base of the curve " (our abscissa) be continually subdivided as in Euclid I io or V IO, it is manifest ''that such subdivision can never be actually finished " (p. 78) ; but Newton proceeds difìferently — he supposes a line to be described by a moving point. Jurin thereupon repeats exactly the argument in Zeno's " Dichotomy," though he does not mention Zeno, to show that a point moving across the page in, say, one hour passes over i / 2 of the distance, then over i / 4 of it, then over 1/8, i / 16, etc. , and insists that "ali the possible subdivisions of the line " will be " actually finished " and " brought to a period at the end of the hour." This is given in support of bis previous argument that the rectangles inscribed in a curve may reach the limit. " If Mr. Robins will teli me, that the imagination cannot pursue these parallelo- grams to the very end of the hour, I may ask him, I20 LIMITS AND FLUXIONS whether the imagination can any better pursue the subdivision of the line, or even of the hour itself, to the end of the hour, vvhìch subdivisions he must own to be brought to a period by the end of the hour. But there is no need to strain our imagination^ to labour in every case, or indeed in any case, after some idea of motion however intricate ; no subtle inquiry is at ali necessary, since we are obliged to own the conclusion to be true and certain. ..." " However, since Mr. Robins is pleased to talk so much about straining our imagination, . . . let us see, if we cannot find some plain and easy way of represent- ing to the imagination, that actual equality, at which the inscribed and circumscribed figures will arrive with each other, and with the curvi- linea! figure, at the expiration of the finite time " (p. III). Let the curvilineal figure ABE equal in area the rectangle with sides EA and AF. When the moving point describing the base EA in a finite time is at C, let the rectangle with the base EA and height Cd be equal to the sum of the parallelograms inscribed in ABE (not drawn in the figure) which stand on CA and upon as many other adjoining parts of EA as can be taken equal to CA. Let lidd be the curve traced by the moving point d. eE JURIN V. ROBINS AND PEM BERTO N 121 Let the area of the rectangle with EA as base and CD as height = sum of the circumscribed parallelograms (not drawn in the figure) standing on CA and upon as many other parts of EA as can be taken equal to CA and adjoining to it ; also let GDD be the curve traced by the movable point D. Then as the curvilineal, the inscribed, and the circumscribed figures are respectively equal to EA x AF, EA X C<^, EA x CD, these figures must be pro- portional to AF, C<:/, and CD. These three lines will ' ' be equal to one another at the end of the finite time." Now since C<:/ and CD approach each other, during a finite time, within less than any given distance before the end of that time, these three lines will, by that Leniina, be equal to one another at the end of the finite time. The limit is reached (p. 1 14). 133. As a further illustration, Jurin takes a rectilinear figure, the right triangle ABE, where EA = AB = ^, AF=J^, EC=;r, the point C mov- ing from E to A as before. Upon AC as a base, imagine an inscribed rectangle (height CH), and a circumscribed rectangle (height CK). As in the previous figure, imagine other inscribed and circumscribed rectangles, standing upon as many other parts of EA as can be taken equal to CA, and adjoining to it in order. When CA is an aliquot part of AE, then a x Cd is the sum of the inscribed rectangles and a x CD is the sum of the circumscribed rectangles, where Cd—x / 2, and C'D = a-x I 2. Let K^/=CD. The ordinate Kd, 122 LIMITS AND FLUXIONS B K K N < ^ ^ D c e FiG. 6. e E drawn to the base BG, will be terminate by EF. When CA is not an aliquot part of AE, if we divide the base into as many parts as may be, there will be left a portion Eé?, which, let us cali r. Then Qd=x-\-rv.a — r^{2à) and ali these ordinates will be bounded by Y.ddY. In the same way, Y^d =a—x-j-rxa-~r-^ (2«), and the ordinate will be bounded by EddF. When x=a, r vanishes, Cd= | a and Kd= la. Hence the inscribed and circumscribed figures do then become equal to each other, and to the triangle ABE ; again, the limit is reached. Jurin takes Robins to task for asserting that ** equality can properly subsist only between figures distinct from each other." To Robins's query, *'Does Philalethes here suppose the truth of Sir Isaac Newton's demonstrations to depend on this actual equality of the variable quantity and its limit?" Jurin answers, " I do . . . In the manner Mr. Robins defines, and treats of prime and ultimate ratios, I allow his demonstrations to be just without this actual equality. But Sir Isaac Newton does not define and treat of prime and ultimate ratios, in the same manner with Mr. Robins ; nor are Mr. Robins's demonstrations at ali like Sir Isaac Newton's demonstration " (p. 128). The inability of our imagination to pursue the rectangles in reach- JURIN V. ROBINS AND PEMBERTON 123 ing the limit is no valid argument against the con- tention that the limit is reached ; even in the ancient geometry there are demonstrated truths that He beyond the reach of the imagination, as for instance, that three cones may equal a cylinder, ali of the same base and height (p. 130). The meaning of moment^ a truly diffìcult concept, is dis- cussed again, Jurin holding that Newton took it as *'a mcmentaneous increment, . . . less than any finite quantity whatsoever, and proportional to the velocity of the flowing quantity," while Robins seem- ingly claimed that Newton meant them to be finite quantities (p. 151). With respect to Newton's early use of the infinitely little, Jurin and Robins were in disagreement, and Robins was in our opinion nearer the truth. Robins claimed that Newton at first used infinitely little quantities ; that afterwards he improved his method by discarding them ; Jurin claimed that Newton's alleged absurdity of expression and inconsistency with himself, as charged by Berkeley and others, **arises wholly from misinterpretation, or misunderstanding him " (P. 179). 134. Jurin's article appeared in the July and August numbers, 1736, of the Republick of Letters. Robins could not wait in patience until the entire article of Jurin had been printed. In the August number he replies to the part of Jurin's article that had appeared in the July number. The August number was given up to Jurin and Robins, to the entire exclusion of ali othcr articles and of the usuai 124 LIMITS AND FLUXIONS hook reviews. On the last page of the August number, the editor apologises to the readers and assures them * * they shall hereafter have no occa- sion to complain upon this head." In Robins's reply, ^ both ''Robins" and '* Fhilalethes " appear in the third person, as if the writer were some out- sider. Robins says : ' * Newton does not intermix his simple and plain description of fluxions with the terms used in the doctrine of prime and ultimate ratios ; for his description of fluxions is contained in the two first paragraphs of his Introduction to the Quadratures, in which no terms of the other doctrine occur " (p. 89). The Lemma is, of course, taken up again, Robins claiming his interpretation legitimate, " for two quantities may constantly tend to equality during some finite space of time, and before the end of that time come nearer together than to have any difference, which shall be given ; and yet at the end of that time have stili a real difference," while Jurin's interpretation was not "any difference that shall be given," but '*any assignable difference," which would mean that the limit must be reached. Mr. Robins says (p. 97) : '* It is not difificult to assign a very probable reason, which led Sir Isaac Newton to the use of this expression [fiunt ultimo aequales], for before him it had not been unusual for geometers to speak of the last sums of infinite progressions, which is an ex- ^ " Remarks on the Considerations relating to Fluxiojis, eie, that were published by Philalethes Cantabrigiensis in the Republick of Letters for the last month," Republick of Letters, August, 1736, pp. 87-110. fURIN V. ROBINS AND PEMBERTON 125 pression something similar to this. Surely here no one vvill pretend, that an infinite number of terms can in strict propriety of speech, and without a figure, be said to be capable of being actually summed up and added together. " Robins makes the only direct reference that was niade in this debate to Zeno's paradoxes. He mentions Achilles and the Tortoise, but in a manner devoid of interest. Referring to the line which Jurin supposes traced in one hour, Robins says : " Perhaps it may be easiest understood by comparing the present point with the old argument against motion from Achilles and the Tortoise. It is impossible to pursue in the imagination their motion by the means proposed in that argument to the point of their meeting, because the motion of each is described by the terms of an infinite progression." Robins does not seem fully to realise that Achilles and the Tortoise present a case in which a variable reaches its limit. 135. The editor of the Republick of Letters permitted the two disputants to continue their wranglings in an Appendix to the September issue.^ Philalethes's attempt to represent to the imagina- tion the actual equality at which the inscribed and circumscribed figures will arrive with each other, and with the curvilinear figure, is criticised by Robins ^ Ali Appendix to the Present State of the Republick of Letters for the Moìith of September y 1736. Being Remarks on the Remainder of the Considerations relating to Fluxions, etc.^ that was piiblished by Philalcthes Cantabrigiensis in the Republick of Letters for the last Month. To which is added by Dr. Pemberton a Postscript occasioned by a passage in the said Considerations. London, 1736. 126 LIMITS AND FLUXIONS on the ground that the continued curve *'is not to be described, but by an endless number of para- bola's " (of which the curve is the envelope) ; thus, Philalethes gave *'as an equation expressing the nature of a single curve, one which in reality includes an infinite series." " Philalethes supposed a last form of the inscribed figures, that was equal to the curve." Robins observed "that equality implies the things, which bave that property, to be distinct from each other. For to say a thing is equal to itself is certainly no proper expression." But ''there is no such last form distinct from the curve," as Philalethes admits ; hence Philalethes **gives up the point. " 136. In the Principia^ Newton does not deliver the doctrine of fluxions, but the doctrine of prime and ultimate ratios. ''The understanding of this book is attended with difificulty. " The expression ultima sumina is defective : **Can any sum of a set of quantities, whose number is supposed infinite, in strict propriety of speech be called their last sum ? " Later, Robins says : *' Let Philalethes reconcile the actual arrivai of these quantities to the ratio supposed, and at the same instant vanishing away. Is not this saying, that the two quantities become nothing, and bear proportion at the same instant of time ? " (p. (14)). Philalethes " has thought himself unjustly accused by Mr. Robins of supposing a nascent increment to be some intermediate state of that increment between its finite magnitude, and its being absolutely nothing. To bave proved this JURIN V. ROBINS AND PEMBERTON 127 assertion groundless he ought to have shown, that this definition does not attempt at describing sudi an intermediate state" (p. (15)). Robins asserts : **Whoever has read Sir Isaac Newton's Lectiones OpticcB^ and will deny, that he has at any time made use of indivisibles, must be very much a stranger to that doctrine, and to the style of those writers who foUow it " (p. (19)). '' What reflexion is it upon Sir Isaac Newton to suppose, that he made use of the methods he had learned from others before he had invented better of his own : or that in an analysis of a problem for dispatch he stili continued to make use of such methods, when he conceived it would create no error in the conclusion ? Has not Sir Isaac Newton said this of himself, and has Mr. Robins said anything more?" (p. (15)). "Does Philalethes here mean, that a quantity can become less than any finite quantity whatsoever, before it vanishes into nothing? If not, then the point is given up to Mr. Robins, who only contends, that vanishing quantities can never by their diminu- tion be brought at last into any state or condition, wherein to bear the proportion called their ultimate : if otherwise, since Philalethes supposes . . . that it is nonsense, that it implies a contradiction to imagine a quantity actually existing fixed, deter- minate, invariable, indivisible, less than any finite quantity whatsoever ; because this imports as much as the conception of a quantity less than any quantity, that can be conceived : how can a quantity supposed to be less than any finite quantity whatso- 128 LIMITS AND FLUXIONS ever be rendered more the object of the conception by being understood to be brought into this con- dition by a Constant diminution from a variable divisible quantity?" (p. (20)). ''Sir Isaac Newton has introduced into use the term moment throughout the whole second book of the Principia^ and for no other purpose than for the sake of brevity ; for his doctrine of prime and ultimate ratios had been before fully explained, and every proposition of the second book might bave been treated on without the use of this term, though perhaps with a somewhat greater compass of words " (p. (23)). '* Mr. Robins has endeavored to defend Sir Isaac Newton both against the accusation of the author of the Analyst^ and the misrepresentation of Philalethes. He has shown, that Sir Isaac Newton's doctrine of prime and ultimate ratios has no connexion with indi- visibles, and that, if he ever allowed himself in the use of indivisibles, he knew that he did so, and did not confound both the methods together, as the author of the Analyst accuses him, and Philalethes without knowing it has owned " (p. (27)). ' ' Had Philalethes been versed in the ancients, and in the later writers who bave imitated them, he could have been at no loss about the true sense of data quavis differentia used by Sir Isaac Newton in his first Lemma. For this expression is borrowed from the writers, that made use of exhaustions " (p. (29)). " What separates the doctrine of prime and ultimate ratios from indivisibles is the declaration made in the Scholium to the first Section of the PrÌ7icipia, JURIN V. ROBINS AND PEMBERTON 129 that Sir Isaac Newton understood by the ultimate sums and ratios of magnitudes no more than the limits of varying magnitudes and ratio's ; and he puts the defence of his method upon this, that the determining any of these limits is the subject of a problem truly geometrical. To insist, that the variable magnitudes and ratio's do actually attain, and exist under these limits, is the very essence of indivisibles " (p. (34)). Robins's reply in the August and September, 1736, numbers of the Republick of Letters is con- densed in form, yet covers 61 pages, It is im- possible for us to convey an adequate idea of the amount of detail entering in the discussion. Altogether Robins shows greater willingness to admit that Newton's views were different at differ- ent periods in his career, and that even Newton may be guilty of modes of expression that are not free from obscurity. Moreover, Robins speaks in general with greater sincerity than his opponent. But Jurin proves himself the superior of Robins in adhering to a broader and more comprehensive conception of variables and limits. Pemberton enters the Debate 137. At this stage a new party enters the debate — Henry Pemberton, who had studied medicine and mathematics at Leyden and Paris, had been a friend of Newton, and had edited the third edition of the Principia. In an article follo wing the one of Robins in the '' Appendix " (August and September 1736), 9 I30 LIMITS AND FLUXIONS Pemberton says : ''I . . . am fuUy satisfied, that Mr. Robins has expressed Sir Isaac Newton's real meaning. " Pemberton quotes from Newton's Intro- duction to the Quadrature of Curves about prime and ultimate ratios (see.our §§ 33, 42), and then remarks; ** Here Sir Isaac Newton expressly calls the quan- titates nascentes and evanescentes, whose prime and ultimate ratios he investigates, by the appellation of finite. Now I desire Philalethes to reconcile this passage with his notion of a * nascent quantity being a quantity not yet arrived at any assignable magnitude how small soever. ' And I must farther ask Philalethes, whether he has not here attempted to define a non-entity. " 138. Robins's last article and Pemberton's rash challenge led to another flow of words, covering 77 pages in the ' ' Appendix " to the Republick of Letters for November, 1736, in an article by Jurin, entitled Obseiuations upon some Remarks relating to the Method of Fluxions, published in the Republick of Letters for August last, and in the Appendix to that for September. Jurin insists that *'the method of fluxions, as it is drawn up by Sir Isaac Newton, could not possibly h^ formed befoi^e the method of first and last ratio's was invented'' (p. (6)). Robins " takes no notice of the letter 0 being used in the hook of Quadratures^ in the very same sense as in the Analysis'' (p. (8)). "That symbol never denotes any quantity, but what, by a con- tinuai decrease, becomes infinitely little, i.e. less JURIN V. ROBJNS AND PEMBERTON 131 than any quantity, and at last vanìshes into nothing" (p. (8)). "He is grossly mistaken in thinking, that quantities, which, before the end of a finite time, come nearer together than to have any assignable difference, will therefore become equal before the end of that time " (p. ( 1 2)). ' ' I have clearly proved in November and January last, that Sir Isaac Newton designed no quantities or ratio's to be com- prehended within the sense of this lemma, which do not become actually equal " (p. (13)). " Has then Mr. Robins, . . . offered to shew, that any quantities or ratio's incapable of an actual equality are compared in this lemma? I think not" (p. (22)). In January, '' I use the following words, ' This determinate proportion of the finite quantities a and e, is what I understand by the proportion of the evanescent augments.' This, l say, ought to have been attended to, before this charge against me was renewed " (p. (24)). As regards the ratio between the inscribed and circumscribed figures, **have not I truly expressed it ? If my expression be too complex, let these great Geometers shew me a simpler, if they can, and I will make use of that " (p. (34)). Robins's argument about the last form of parallelograms differing from the limiting curve is defective in the minor of the syllogism : " Things which are equal are distinct from each other. " *' Is it," says Jurin, " the part of a candid and ingenious adversary, to insist always upon the word equal, when a more proper expression, as that of co incidi ng, has 132 LTMITS AND FLUXIONS been used by bis antagonist ? " If bis argument is sound, " it will bold against my expression, tbat tbe figures inscribed and circumscribed do at last coincide witb tbe curviHnear figure." Jurin claims '*tbat if Mr. Robins's interpretation of tbe first Lemma be admitted, Sir Isaac's demonstra- tions, as tbey now stand, will not be accurate, nor geometrically rigorous," for, " as tbey now stand, tbe examples be bas given in tbe several Lemmata of tbe first Section, are of sucb quantities and ratio's only, as do actually arrive at tbeir respective limits" (pp. (42) and (43)). '' Mr. Robins and I bave been disputing some time, wbetber Sir Isaac Newton used indivisibles. Tbat Gentleman main- tains tbat be used tbem ; and grounds bis ebarge upon tbe term infinitely little^ wbicb is sometimes to be met witb in Sir Isaac Newton's writings : but be does not explain tbe meaning of tbat term, wben used eitber by Sir Isaac, or by tbe writers of indi- visibles. I, on tbe contrary, distinctly explain wbat I apprebend to be meant by it, botb wben used by Sir Isaac Newton, and wben used by tbe writers of indivisibles. ... I supposed tbe writers upon indivisibles, by an infinitely little quantity, to mean a quantity actually existing, fixed, deter- minate, invariable, indivisible, less tban any finite quantity wbatsoever " (p. {JZ)). Robins quotes Pascal and Barrow as using tbe term indefinite in place of infinite^ but tbe writers I quoted use infinite and infinitely little. Tbere is difference of usage among followers of Cavalieri. '' It is not denied, JURIN V. ROBINS AND PEMBERTON 133 but that Sir Isaac Newton, by the term infinitely little^ meant a quantity variable, divisible, that, by a Constant diminution, is conceived to become less than any finite quantity whatsoever, and at last to vanish into nothing. By which meaning ali that is faulty in the method of indivisibles, is entirely avoided ; and that being allowed, the rest is only a dispute about a word " (p. (74)). Jurin declares in a ''Postscript" that " to carry on two controversies at once is more than I have leisure for " ; later "I intend to accept of Dr. Pemberton's invitation " ; meanwhile Jurin inserts an attestation of " his learned friend Phileleutherus Oxoniensis'' to the effect that this friend is " fuUy satisfied, Mr. Philalethes has expressed Sir Isaac Newton's real meaning." The language of this attestation foUows exactly the language of Pember- ton, except that Philalethes, and not Robins, is now declared the correct interpreter of Newton. 139. In the December issue, 1736, of the Republick of Lctters^ Robins says in an " Advertisement " that " since Philalethes has given loose to passion," he " cannot think anything farther necessary for the satisfaction of impartial readers " (p. 492), and now takes *Meave of Philalethes," but cannot resist a few parting shots. Nor could Philalethes resist making reply to this " Advertisement " in an " Appendix " to the December number, 1736, of the Republick of Letters, in which he expresses regret "that so long a correspondence should end in dis- content or ili humour." Jurin justifies the practice 134 LIMITS AND FLUXIONS he exercised in this controversy of offering poetry (usually in Latin) for the sake of readers who are under necessity /'of exercising their faith, rather than their reason in this dispute," for ** A verse may catch him, who a sermon flies, " and for the sake of enlivening the subject for others, ''who are judges of the dispute." 140. In this December " Appendix " Jurin then contributes A Reply to Dr. Pemberton's Postscript^ which takes up 31 pages. Referring to Newton's Lemma i, Jurin says that in his former expres- sion, the quantities "come nearer to equaUty than to bave any assignable difference between them," it never was his intention to assert "that during the time of the approach, the difference between the quantities is not always assignable " ; he meant "that, though they shall always bave a difference during the finite time, yet, before the end of that time, their difference shall become less than any quantity that can be assigned. And if my words are taken in this sense, the Dr. 's objection immediately falls to the ground " (p. (24)). Mr. Jurin then gives a " demonstration " of the foUowing proposition : " If two lines (i) tend con- stantly to equality with each other, (2) during any finite time, as, for instance, an hour; (3) and thereby, their difference become less than any quantity that can be assigned, (4) before the end of the hour ; then, at the end of that finite time, or at the end of the hour, the lines will be equal." As to Dr. Pemberton's charge that Jurin misinterprets Newton's JURIN V, ROBINS AND PEMBERTON 135 nascent and evanescent increments, Jurin says that he discusseci this question with Robins. Newton's words in the Quadratura Cujvarum, viz. finitarum nascentìuin vel evanescentìum^ may mean ''(i) finite nascent or evanescent quantities, or (2) finite quan- tities when they begin to be, or when they vanish. But the former sense contradicts the second Lemma of the second Rook of the Principia, where Sir Isaac Newton says, cave intellexeris particulas finitas . . . and indeed it is contrary to the whole tenor of his doctrine." The second interpretation is *'perfectly conformable to ali the rest of Sir Isaac Newton's Works" (p. (32)). Jurin repeats that a nascent in- crement is '*an increment not yet arrived at any assignable magnitude, how small soever. " To Dr. Pemberton's query, whether Jurin " has not here attempted to define a non-entity," Jurin replies that it *'ought not to be called simply a non-entity, nor simply an entity. It is a non-entity passing into entity, or entity arising from non-entity, a begin- ning entity, something arising out of nothing " (P- (37)). 141. The discussion is carried on from this time in a journal called The Works of the Learned, into which the Republick of Letters and another journal had merged. In the February, 1737, issue Dr. Pemberton appears with Some Obsei-vations on the Appendix to the Present State of the Republick of Letters for December, 1736, which enjoys the merit of brevity, being limited to only two pages. Pem- berton declares that in Newton's passage in the 136 LIMITS AND FLUXIONS Quadratura Curvarum^ " Philalethes cannot remove my objection by straining one or two of the words to fit his sense " ; Newton meant there that vanish- ing quantities should not he ''otherwise than finite quantities" (p. 157). Moreover, " what kind of nothings they must he, which with any propriety can be said to pass into somethings, and for this reason can be capable of bearing proportions, before they are become anything, certainly requires explanation." A reply by Jurin in The Works of the Learned for March, 1637, is kept within the very moderate compass of io pages. The title of his contribution is The Contents of Dr. Pemberton's Observations pub- lished the last month. Nothing bere is of interest in the interpretation of Newton. Dr. Pemberton's reply in the Aprii issue refers to Jurin's phrase, *'they come nearer to equality than to bave any assignable difference between them " : *' My objection to the interpretation of Philalethes [in the Minute Mathematician, p. 88] is, that these words, which compose the third article of that inter- pretation, in conjunction with the fourth article can have no other signification, than that the quantities come nearer to equality than to have any difference between them before that point of time, wherein they are supposed by the second article to become equal ; ali which amounts to this inconsistency, that there is a time, when the quantities have no difference, and yet are not equal " (p. 306). Dr. Pemberton again gives his endorsement of Robins's interpretation of Newton. JURIN V. ROBINS AND PEMBERTON 137 Jurin appears with a 12-page article in the May, 1737, number of The Works of the Learned, say- ing : "He stili ascribes to my words a meaning, which I have again and again utterly disavowed ; not only so, but he changes the words themselves, putting any difference instead of any assignable differ- ence'' {^. 388). As to the Introduction to Newton's Quadratura Curvarum^ " in that very Introduction Sir Isaac Newton has made use of infinitely little quantities, in the sense I understand them, that is, quantities which being at first finite, do by a graduai diminution at last vanish into nothing and conse- quently must, during their diminution, become less than any quantity that can be assigned " (p. 389). As to evanescent quantities being entities or non- entities, "If this page were divided from top to bottom into two equal parts, one black, and the other white,^ and Dr. Pemberton were to ask me, whether the middle line, which divides the two parts, were black or white, I apprehend it would be a direct answer to say, it is neither ; it cannot properly be called either a black line, or a white line ; it is the end of the white and beginning of the black, or the end of the black and beginning of the white" (p. 389). '*I was apprised that Mr. Robins had ali along expressed the sentiments of Dr. Pemberton " (p. 393). Dr. Pemberton stili refuses to give his interpretation of Newton's ^ As far as I know, Jurin is the first to use colour devices to illustrate subtle points in evanescent quantity or in number. Jules Tannery, in his Le^ons cf Algebre et cTAnalyse, Paris, 1906, p. 14, uscs colour imagery to illustrate the discussion of irralional numbers. 138 LIMITS AND FLUXIONS Lemma, •' Every body will be satisfied that the true reason of bis backwardness, is the fear he is under, that I shall make good my promise, in shew- ing, that bis explanation is either a false one, or, in case it be true, is to ali intents and purposes the very same with mine " (p. 396). In June, 1737, Dr. Pemberton replies again, by re- peating bis previous assertion against Philalethes's explanation of Newton's Le^nma^ given in the Minute Mathematician, but does not permit him- self to be drawn into giving an explanation of his own of Newton's Lemma. In Jurin's article in the July issue, 1737, we read : **I did indeed take notice of the prudence Dr. Pemberton used, in passing by my second inter- pretation, which was so clear and plain, and was so fully illustrated by examples, that there was no possibility of perverting the sense of it " (p. 70). *'But since this dispute, which began upon matters of science, . . . unless Dr. Pemberton shall see fit to revive it by giving his so long demanded explica- tion, I shall not judge it worth while to take notice of what he may hereafter write. " Dr. Pemberton followed with some Observations in the August, 1737, number, while in the September number there appears " the last reply of Philalethes," and in the October number the final answer by Pemberton. Thus ended a dispute which had for some time ceased to contain much of scientific and historic value. JURIN V. ROBINS AND PEMBERTON 139 Debate over Robins's Review of Treatises written by Leonhard Euler, Robert Smith, and Jurin 142. Being in a somevvhat combative mood, Robins made attacks upon Euler's treatise on motion, Dr. Robert Smith's optics,and Jurin's essay on vision.^ Robins's criticisms of Euler concern mainly the philosophy of the Calculus. Robins quotes Euler's third proposition, " That in any unequal motion the least element of the space described may be conceived to be passed over with an uniform motion," and then says, this ''is not universally true," as, for instance, ''when those spaces are compared together, which a body acceleratcd by any force described in the beginning of its motion ; for the ultimate proportion of the first of two contiguous spaces, thus described in equal times to the second, is not that of equality, but the ratio of i to 3, as is well known to every one acquainted with the common theory of falling bodies " (p. 2). In another place (p. 4) Robins argues that the path assigned by Euler to a certain body " is false even on the confused principles of indivisibles." Some passages in Robins involve the Leibnizian notation in the calculus, and look quite odd in an eighteenth- century publication prepared by a Briton in Great Britain. Robins concludes that most of Euler's errors " are owing to so strong an attachment to the principles, he had imbibed under that inelegant ^ Kemarks on Mr. Euler's IVeatise of Motion^ Dr. Smith's Covipleat System of Opttcks, a7id Dr. Jnrins Essay upon Distinct and IndistÌ7ict Vision. By Benjamin Robins, London, 1739. HO LIMITS AND FLUXIONS computist, who was his instructor, that he was afraid to trust his own understanding even in cases, where the maxims, he had learnt, seemed to him contradictory to common sense " (p. 30). This master was John Bernoulli. 143. Never losing an opportunity to engagé in controversy, Jurin wrote a treatise in reply. ^ We refer only to such parts of this pamphlet, and the ones which followed it, as bear on fluxions or the parties engaged in the discussions on fluxions. In the preface Jurin says : " I, it seems, am the Reputed Author of the late dissertations under the name of Philalethes Cantabrigiensis, and the other Gentleman [Dr. Robert Smith] is . . . suspected of being my associate. . . . If Dr. Smith were to teli Mr. Robins, what he has often professed to other persons, that he had no band in those papers ; if to confirm this he were to remind him, that Philalethes has declared more than once, he wrote alone and unassisted ; if I — But what signifies pleading, when the execution is over ? Mr. Robins has already vented his Resentment to the utmost. ..." 144. Not without interest is the following refer- ence to young Euler in St. Petersburg, whose scientific achievements bave been so very extra- ordinary. Jurin says that to make no reply to Robins's criticisms " might be such a discouragement to the hopeful young writer, whose name is prefixed ^ A Reply to Air. Robins^ s Remarks on the Essay upon Distinct and Indistinct Vision Published at the End of Di-, Smith^s Compleat System of Opticks. By James Jurin, M. D. , London, MDCCXXXIX. JllRIN V. ROBIN S AND PEMBERTON 141 to theìr common labours, and who possibly, when he Comes to study suo Marte, and to see with his own eyes, or to meet with abler instructors, may make some figure in the Learned World, that pure humanity induces me to oblige them with this one Reply"(p. 54). 145. Of course, Robins wrote a tract in reply,^ but only the preface of this tract demands our attention. In answer to the charge made by Jurin, that he (Robins) had conducted the controversy "with passion and abuse," Robins proceeds to explain their past relations to each other. '* About six years since a pamphlet was publish'd under the title of the Analyst ; in which the author endeavors to shew, that the doctrine of fluxions in- vented by Sir Isaac Newton is founded on fallacious suppositions. As that writer had a false idea of this doctrine, ... I thought the most effectual method of obviating his objections would be to explain . . . what Sir Isaac Newton himself had delivered with his usuai brevity. . . . And with this view I pub- lished a Discourse on Sir Isaac Newton's method of fluxions, and of prime and ultimate ratios, But in the mean time a controversy was carrying on between the author of the Analyst and another, who under the name of Philalethes Cantabrigiensis had undertaken the defence of Sir Isaac Newton : and as I at last perceived, both by the concessions ^ A Full Confutation of Dr. Jurin* s Reply io the Rcmarks on his Essay upon Distinct and Indistiiut Vision. By Benjamin Robins, London, 1740. 142 LIMITS AND FLUXIONS of Philalethes, and the avowed opinions of others, that the erroneous conceptions of the writer of the Analyst on thìs head were more prevalent even amongst those, who approved of the method of fluxions, than I had at first beHeved ; I thought, it might be no unacceptable task more particularly to shew those, who were thus misled, how irreconcile- able their opinions were with the tenets of Sir Isaac Newton, and how impossible it would be to defend the accuracy of his doctrine on these their mistaken suppositions ; and it was with this intention, that in an account of my hook inserted in the Present State of the Republick of Letters, some of the errors con- tained in the writings of Philalethes Cantabrigiensis were endeavoured to be obviated. ''But tho' this discourse was written with great caution, and only mentioned the principles objected to without so much as naming or even insinuating the treatises, from whence they were taken ; yet, as Dr. Jurin, who was generally reputed the author of them, was one, that I often conversed with ; at my request, before this paper was printed, a common friend carried to him the manuscript, and, without pretending to suppose, whether he was, or was not Philalethes, desired him to read it, and asked him if he thought, Philalethes could be displeased with any thing contained in it ; he was also told at the same time, that if he believed any part of it could give offence to that gentleman, whoever he were, it should be struck out, or that 1 would even let the whole design fall, if he desired it. JURIN V. ROBINS AND PEMBERTON 143 *'My friend brought me the Doctor's answer importing, that he could not believe, my paper would displease any one, since, if the tenets, I excepted to, were really erroneous, it was reason- able, they should be exposed ; and if otherwise, it was the business of Philalethes to defend them . . . it was however added, that I had in two places censured doctrines, whìch, if 1 supposed them to be the opinions of Philalethes, I must bave mis- apprehended him. Now ... 1 immediately ex- punged them, and published the remaining part in the Republick of Letters for October 1735, as an account of my hook on Sir Isaac Newton's method of fluxions, and of prime and ultimate ratios. " To this Philalethes answered in the foUowing month, and I again replied, till five papers were successively written in this controversy, that is, three by me, and two by him. And ali this time so very desirous was I on my part of avoiding irritating circumstances . . . that I thought even the most intimate friend . . . could not be offended with it. . . . But alas . . . Philalethes in his reply, part of which was published in the July foUowing, and the rest in the succeeding month, runs out into the most extravagant heats of passion . . . charg- ing me with dishonestly writing against the con- victions of my own judgment. . . . After so gross and unprovoked an abuse, ... I should surely bave been acquitted of any breach of decency, if . . . I had sharply exposed his ignorance in the subjects, 144 LIMITS AND FLUXIONS he had attempted. But I chose, ìf possìble, to avoid the ridicule of quarreling on a matter of mere speculation ... I again requested my friend to speak to Dr. Jurin, and to represent to him the inconveniencies, that would arise from the persever- ance of Philalethes in his rash and groundless calumny. My friend accordingly went to Dr. Jurin, and carried with him an answer to so much of Philalethes's paper, as was then pubHshed, and told the Doctor, that he carne to propose to him a method, that might prevent the controversy betwixt m.e and Philalethes from degenerating into a passionate personal altercation . . . that therefore, ifDr. Jurin thought it expedient, my paper should be given to a certain gentleman, to whose impartiality and knowledge of the subject in debate no exception could be taken on either side ; and that if, when that gentleman had perused it, he should believe, I had in any instance changed my opinion from my first entering into this dispute, I did then promise to submit patiently and without reply to any censures of unfairness and dishonesty, that Philalethes . . . should hereafter think proper, . . . [otherwise] it would then be but common justice, that Philalethes should moderate the remaining part of his performance. . . . But this proposai was rejected. . . . It was immediately given out, that my friends had confessed me to have been foiled in the argument ; and were now only soUicitous to support me from the charge of unfairness. . . . The reader will not wonder, if I resolved for the future JURIN V. ROBIN S AND PEMBERTON 145 to treat him with that freedom, which his unskilful- ness authorised. ..." 146. The above preface constitutes what we may cali Robins's apologia prò vita sua. It seems only fitting that Jurin should appear with a similar docu- ment. This he did in a long Letter.^ We make the following quotations from Jurin (p. 8) : ** About five years ago some passages in a paper of Mr. Robins, were shown to me . . . and a question was put to me, whether I should take it ili, if those passages were printed, it being intimated, that Philalethes, against whom they were designed, might possibly be some friend of mine : and indeed, several persons were then guessed at, ali of which happened to be my friends. To this ... I gave answer, that I should not at ali take it ili. But I added, that as I had read the controversy between Philalethes and the Author of the Analyst, with some attention, it seemed to me that in one or two passages Mr. Robins imputed opinions to Philalethes, which . . . that gentleman did not hold. . . . Also, I took notice, that Mr. Robins did not rightly explain Sir Isaac Newton's first Lemma. . . . But when 1 desired to talk with Mr. Robins about the Lemma, before the papers went to the press, as imagining I could convince him that he was in the wrong, answer was made, that the question was ^ A Letter to . . . Esquire^ In Answer to Mr. Robins^s Full Confu- tation of the Reply to his Remarks on the Essay upon distinct and indistinct Vision. By James Jurin, M.D., London, 1741. IO 146 LIMITS AND FLUXIONS not whether I thought him in the right or in the wrong, but only whether I should take anything amiss ; to which I replied as before. Upon talking with another friend of Mr. Robins a day or two after, I repeated my desire to talk with Mr. Robins about his explanation of the Lemma, before his papers went to the press : but was told that could not be, for that the part of the papers where the Lemma was spoke of, was to go to the press that afternoon. ... I do not remember, that any offer was made to me of ' letting the whole design fall, if I desired it. ' Had any such offer been made, I had at that time so much regard for Mr. Robins, that I think I should at least have desired him to stop the design, till he and I had examined the Lemma together, in order to prevent his exposing himself in the manner he has since done. As to the second application made to me near a year after, it may easily be judged, that I, who gave these gentlemen no reason to think I had any in- fluence over Philalethes, or so much as knew who he was, could neither comply with nor reject their proposai " (p. 9). Remarks 147. The debate between Jurin and Robins is the most thorough discussion of the theory of limits carried on in England during the eighteenth century. It constitutes a refinement of previous conceptions. Jurin possessed the more general conception of JURIN V. ROBINS AND PEMBERTON 147 a limit in insisting that there are variables which reach their limits. His interpretation of Newton on this point appears to us more nearly correct than that of Robins ; Jurin's geometrie illustrations of limit-reaching variable, intended to aid the imagina- tion, though as he admits incapable of exhibiting the process "ali the way," are nevertheless interesting (see our §§ 124, 132, 133). The imagination is subject to limitations where the reason is stili free to act. Robins, and after him Pemberton, deserve credit in clearly, openly, and completely breaking away from infinitely little quantities, and from prime and ultimate ratios. Robins's conception of a limit was narrow, but this narrowness had certain peda- gogical advantages, since it did not involve a mode of advance to the limit which altogether tran- scended the power of the imagination to follow ali the way (see our §§ 117, 118, 129, 130). It is interesting to observe that both Jurin and Robins disavow belief in the possibility of a sub- division of a line into parts so as to reach a point — they assert "that such subdivision can never be actually finished " (see our §§ 126, 132). Robins discarded the use of Newton's moinents in developing the theory of fluxions (see our §§ 119, 120). Toward the end of his long debate with Robins, Jurin begins to disavow infinitely small quantities. He brings out the difference between infinitesimals as variables, and infinitesimals as constants. He 148 LIMITS AND FLUXIONS rejects ali quantity *'fixed, determinate, invariable, indivisible, less than any finite quantity whatsoever," but he usually admits somewhat hazily a quantity "variable, divisible, that, by a Constant diminution, is conceived to become less than any finite quantity whatever, and at last to vanish into nothing. " (See our §§ 132, 138, 141.) VVhile Berkeley's Analyst and Berkeley's replies to Jurin and Walton involved purely destructive criticism, the present controversy between Jurin and Robins brought forth valuable constructive results. Jurin's papers against Robins are decidedly superior to those he wrote against Berkeley, though here too they contained much that was not pertinent to the subject and was intended merely to amuse the general reader. CHAPTER V TEXT-BOOKS IMMEDIATELY FOLLOWING BERKELEY'S ATTACK 148. The Analyst was published in 1734 ; two years later appeared four books on fluxions. Thus, more British text-books on this subject were published in 1736 than in ali the thirty years preceding. That the Analyst controversy was largely the cause of this increased productivity there can be no doubt. We proceed to give an account of the books which preceded the publication of Maclaurin's Treatise of Fluxions^ 1742. John Colson^ 1736 149. Newton's Method of Fluxions,'^ S3.\d to have been written in 1671, was translated and first published in 1736 by John Colson. Colson had been a student at Christ Church, Oxford, which he left without taking a degree. He was appointed ^ The Method of Fhixions and Infinite Stries ; with its Application to the (Jeometry of Curz'e- Lines. By the Inventar^ Sir Isaac Newton, Kt., Late Presi dent of the Koyal Society. Translated from the Author^s Latin Originai not yet made puhlick. To which is subjoined, A Perpetuai Commenttipon the Whole Work, . . . By John Colson, M. A. and F. R.S., Master of Sir Joseph Williamson's free Mathematical- School at Rochester. London, M.DCC.XXXVI. This hook was reissued in 1758. 149 ISO LIMITS AND FLUXIONS master of a new mathematica! school founded at Rochester, and, in 1739, Lucasian professor of mathematics at Cambridge, in succession to Nicholas Saunderson. Colson was a man of great industry but only ordinary ability. In his preface, Colson refers to the controversies on fluxions, and says that the defenders as well as their opponents were little acquainted with Newton's own exposition, that this hook now published for the first time is "the only genuine and originai Fountain of this kind of knowledge. For what has been elsewhere deliver'd by our Author, concerning this Method, was only accidental and occasionai" (p. x). Colson accompanies Newton's hook "with an ampie Commentary " and " particularly with an Eye to the fore-mention'd Controversy " (p. x). Colson in this preface represents Newton as hold- ing the principle "that Quantity is infinitely divisible, or that it may (mentally at least) so far continually diminish, as at last, before it is totally extinguish'd, to arrive at Quantities that may be call'd vanishing Quantities, or which are infinitely little, and less than any assignable Quantity. Or it supposes that we may form a Notion, not indeed of absolute, but of relative and comparative infinity " (p. xi). Colson opposes " indivisibles," as also the " infinitesimal method" and "infinitely little Quantities and infinite orders and gradations of these, not relatively but absolutely such " (p. xii). He argues against " imaginary Systems of infinitely great and infinitely little Quantities, and their TEXT-BOOKS, 1736-1741 151 several orders and properties, which, to ali sober Inquirers into mathematica! Truths, must certainly appear very notional and visionary " (p. xii), for "Absolute Infinity, as such, can hardly be the object either of our Conceptions or Calculations, but relative Infinity may, under a proper regula- tion " (p. xii). Newton " observes this distinction very strictly, and introduces none but infinitely little Quantities that are relatively so." Colson answers Berkeley's criticism in the Analyst of Lemma 2, Book li, in the Principia in the follow- ing manner : — "Let X and Y be two variable Lines. . . . Let there be three periods of time, at which X becomes K — \a, AjA-f-J^; and Y becomes V> — \b, B, and B + i^ . . . Then ... the Rectangle XY will become . . . KV>-\a?>-\bK^\ab, AB, and AB + i^B + i<^A + i^^. Now in the interval from the first period of time to the second . . . its whole Increment during that interval is \aY>-\-\bK — \ab. And in the interval from the second period of time to the third, ... its whole Increment during that interval is \à^-\-\bh.-\-\ab. Add these two Incre- ments together, and we shall bave a^ + b\ for the compleat Increment of the Product XY " (p. xiii), called the " Moment of the Rectangle" when a and b are infinitely little. Another mode of procedure is this: ''the Fluxions or Velocities of increase, are always proportional to the contemporary Moments." " When the Incre- ments become Moments, that is, when a and b are 152 LIMITS AND FLUXIONS so far diminish'd, as to become infinitely less than A and B ; at the same time ab will become infinitely less than either a^ or ^A (for ^ aV) . ab -. -.Vi .b^ and ^A . ab \ : A . a)^ and therefore it will vanish in respect of them. In which case the Moment of the Product or Rectangle will be a^-{-bA, as before " (p. xv). Newton, however, prefers the more direct way previously explained. Proceeding to Newton himself, we find (on p, 24) the following definition : " The Moments of flowing Quantities (that is, their indefinitely small Parts, by the accession of which, in infinitely small por- tions of Time, they are continually increased) are as the Velocities of their Flowing or Increasing. Wherefore if the Moment of any one, as ;r, be repre- sented by the Product of its Celerity x into an indefinitely small Quantity 0 (that is, by x 0), the Moments of the others v, j/, z, will be represented by vo, fo, éo ; because vo, xo, yo, and èo^ are to each other as v^ x, y, and i." On p. 25 terms contain- ing (? as a factor " will be nothing in respect of the rest. Therefore I reject them." 1 50. Colson appended extensive annotations to Newton's treatise. In these annotations, p. 250, Colson speaks of '' smallest particles," but the term '' smallest " does not occur in Newton's definition. However, Colson says that he does not mean **atoms" nor "definite and determinate magni- tude, as in the Method of Indivisibles," but things "indefinitely small; or continually decreasing, till ^ Here a^ . ab : : V> . h means a^ : ab : -. ^ : b. TEXr-BOOKS, 1736-1741 153 they are less than any assignable quantities, and yet may then retain ali possible varieties of pro- portion to one another. Becoming stili more deeply involved in the metaphysics of the subject, Colson adds " that these Moments are not chimerical, visionary, or merely imaginary things, but have an existence sui generis^ at least Mathematically and in the Understanding, is a necessary consequence from the infinite Divisibility of Quantity, which I think hardly anybody now contests " (p. 251). This he qualifies, ''perhaps the ingenious Author of . . . The Analyst must be excepted, who is pleased to ask, in his fifth Ouery, whether it be not unnecessary, as well as absurd, to suppose that finite Extension is infinitely divisible " (p. 251). By ultimate ratio Colson means the ratio when the arguments ''become Moments " (p. 255). Fearing that moments, infinitely little quantities, and the like, **may furnish most matter of objection," he says (p. 336) that the symbol 0 at first represents a finite quantity, which then diminishes continually till " it is quite exhausted, and terminates in mere nothing. " But *'it cannot pass from being an assignable quantity to nothing at once ; that were to proceed per saltum, and not continually " ; hence " it must be less than any assignable quantity what- soever, that is, it must be a vanishing quantity. Therefore the conception of a Moment, or vanishing quantity, must be admitted as a rational Notion " (P- 336). Again : "The Impossibility of Concep- tion may arise from the narrowness and imperfection 154 LIMITS AND FLUXIONS of our Faculties, and not from any inconsistency in the nature of the thing" ; these quantities "escape our imagination. " Referring to imaginaries, a J — i in the solution of cubie equations, Colson says (PP- 338-9)* "These impossible quantities . . . are so far from infecting or destroying the truth of these Conclusions, that they are the necessary means and helps of discovering it. And why may we not conclude the same of that other species of impossible quantities, if they must needs be thought and call'd so ? . . . Therefore the admitting and retaining these Quantities . . . 'tis enlarging the number of general Principles and Methods, which will always greatly contribute to the Advancement of true Science. In short, it will enable us to make a much greater progress and proflcience, than we otherwise can do, in cultivating and improving what I have elsewhere call'd The Philosophy of Quantity, " 151. A review 1 of this book contains the follow- ing historical exposition. Sir Isaac Newton, 1665, " found the Proportions of the Increments of inde- terminate Quantities. These Increments or Aug- menta Momentanea he called Moments, which others called Particles, infinitely small Parts, and Indi- visibles ; and the Velocities by which the Quantities increased he called Motions, Velocities of Increase, and Fluxions. He considered Quantities not as composed of Indivisibles, but as generated by locai Motion, after the manner of the Ancients . . . and represented such Moments [of Time] by the Letter 0, ^ Republick of Letters, Art. XI, pp. 223-235, 1736. TEXT-BOOKS, 1736-1741 I55 or by any other Mark drawn into an Unit " (p. 228). ' ' Fluxions are not Moments, but finite Quantities of another kind." *'When Mr. Newton is demon- strating any Proposition, he considers the Moments of Time in the Sense of the Vulgar, as indefinitely small, but not infinitely so ; and by that means performs the whole work, in finite Figures, by the Geometry of Euclid and Apollonius, exactly without any Approximation : and when he has brought the work to an Equation, and reduced the Equation to the simplest Form, he supposes the Moments to decrease and vanish ; and from the terms which remain he deduces the Demonstration. But when he is only investigating any Truth, or the Solution of any Problem, he supposes the Moment of Time to be infinitely little, in the Sense of Philosophers, and Works in Figures infinitely small." James Hodgson, 1736 152. James Hodgson, a mathematical teacher and writer, and a fellow of the Royal Society of London, is the author of a book, The Doctrine of Fluxions.^ Hodgson says in his Introduction that *' it is now some years since the greatest Part of this Book was prepared for the Press." There is no direct refer- ence in the book to the Analyst controversy, but the declaration is made that the principles upon which fluxions rest need " fear no Opposition." ^ The Doctrine of Fluxions, fotinded on Sir Isaac Newton s Afethod, Piiblished by Himself in his Tract upon the Quadrature of Curves^ By James Hodgson, London, MDCCXXXVI. 156 LIMITS AND FLUXIONS Hodgson also says in his Introduction that most books on fluxions that bave hitherto appeared proceeded on the same principles as the Difìferential Calculus, so that " by calling a Differential a Fluxion^ and a second Differential a second Fluxion, etc. , they bave . . . confusedly jumbled the Methods together," although the principles are really "very difìferent. " ' ' The Differential Method teaches us to consider Magnitudes as made up of an infinite Number of very small constituent Parts put together ; whereas the Fluxionary Method teaches US to consider Magnitudes as generated by Motion ... ; so that to cali a Diffei'ential a Fluxion, or a Fluxion a Differential is an Abuse of Terms. " In the method of fluxions, " Quantities are rejected, because they really vanish " ; in the differential method they are rejected '* because they are in- finitely small." Hodgson adds that he always used the differential method '''till I became acquainted with the Fluxionary Method." He considers fluxions of quantities (p. 50) '*in the first Ratio of their nascent Auginents, or in the last ratio of their evanescent Decrements,'" and gives an able and faithful exposition of Newton's ideas as found in his Quadrature of Curves. He cannot think ''there is any more difificulty in conceiving or forming an adequate Notion of a nascent or evanescent Quantity, than there is of a Mathematica! Point " (p. xi). In explaining the derivation of the fluxion of the product xy^z he apparently permits (p. xv) the small quantity ^ to " vanish," and thereupon divides TEXT-BOOKS, ly 7,6-1741 157 both sides of the equation Xfo-]-fXo = èo by 0. However, in the exposition given on p. 50 he is more careful and divides by 0 while 0 is an incre- ment, and obtd.ìns f±-\-±y-\-J^:i;o = v. Then he says : " Imagine the Quantity 0 to be infinitely dimin- ished, or, which is the same thing, the Quantity xy to return back again into its arising State ; then the Quantity ±yo, in this Case, into which 0 is multi- pHed, will vanish ; whence we shall have xy-\-yji'=v for the Fluxion of the Quantity proposed. " Hodgson foUows Newton closely and permits the variable to reach its limit. Thomas Bayes, 1736 153. An anonymous pamphlet of 50 pages, on the Doctrine of Fluxions,^ has been ascribed to Rev. Thomas Bayes. This author contributed in 1763 to the Philosophical Transactions a meritorious article on the doctrine of chances. The pamphlet of 1736 represents a careful effort to present an unobjectionable foundation of fluxions. *'The fluxion of a flowing quantity is its rate or swiftness of increase or decrease. " Let «, b,x^ and y be flowing quantities, let A and B be permanent quantities \\{ a\ b = h.'^x : B=Fjj/, during any time T, and at the end of that Time, a^ b, x^ y ali vanish ; then . . . the ratio of A to B is the last ratio of the vanishing quantities a and b (p. 13). This definition is *' in efifect the same" as that given by Newton. ^ Introduction to Doctrine of Fluxions and Defence of the Mathe- maticiaus agaitist . . . t/ie Ana/j'st, 1736. 158 LIMITS AND FLUXIONS The author speaks of *'that most accurate defìnì- tion of the ultimate ratio's of vanìshing quantities ; which we have at the latter end of Sch, Lemma XI Princip. [see our §§ 10-15], ^^^ which is so plain, that I wonder how our author [Berkeley] could help understanding it ; which had he done, I am apt to think that ali his Analyst says concerning the pro- portion of quantities vanishing with the quantities themselves, had never been heard : For according to this definition, we are not obliged to consider the last ratio as ever subsisting between the vanishing quantities themselves. But between other quantities it may subsist, not only after the vanishing quantities are quite destroyed, but before when they are as large as you please. And the reason why we consider quantities as decreasing continually till they vanish, is not in order to make, but to find out, this last ratio. Sir Isaac Newton does not indeed say that this last ratio is the ratio with which the quantities themselves vanish ; but whether he herein speaks with the utmost propriety or not, is a mere nicety on which nothing at ali depends " (p. 16, note). Velocity " signifies the degree of quickness with which a body changes its situation in respect to space"; the fluxion of a quantity ''signifies the degree of quickness with which the quantity changes its magnitude. " "And when our author asserts, that in order to conceive of a second fluxion, we must conceive of a velocity of velocity, and that this is nonsense ; he plainly appeals to the sound and TEXT-BOOKS, 17 36-1 741 159 not the sense of words . . . if . . . you make it synonymous to the word Fluxion, then the velocity of velocity . . . is nothing but plain common sense" (p. 19). Moments are not used by the author. The author says that, were he to write a treatise on fluxions, **in order to understand equations where Fluxions of different orders are jumbled together ; it would be convenient to re- present ali Fluxions not as before, but as quantities of the same kind with their Fluents. . . . The Fluxion of a quantity anyhow flowing at any given instant is a quantity found out by taking it to the Fluxion of an uniformly flowing quantity in the ultimate proportion of those synchronal changes which then vanish " (pp. 34, 35). The variables X and x"^ have the synchronal augments 0 and nox''~^-\-^ — —^o^x"~^-\-, etc. , which are to one 2 another as i : nx''~'^ -\-- ^-ox""'^-]-, etc. *'Let 2 now these arguments vanish, and their last ratio will be I : nx^'K" *'This our author says is no fair and conclusive reasoning, because when we suppose the * increments to vanish, we must suppose their proportions, their expressions, and everything else derived from the supposition of their existence to vanish with them. ' To this I answer, that our author himself must needs know thus much, viz. That the lesser the increment 0 is taken, the nearer the proportion of the increments of x and x"" will arrive to that of i to nx"'"^, and that by supposing i6o LIMITS AND FLUXIONS the increment o continually to decrease, the ratio of these synchronal increments may be made to approach to it nearer than by any assignable difìference, and can never come up with. it before the time when the increments themselves vanish. . . . For tho', strictly speaking, it should be allowed that there is no last proportion of vanishing quantities, yet on this account no fair and candid reader would find fault with Sir Isaac Newton, for he has so plainly described the proportion he calls by this name, as sufficiently to distinguish it from any other whatsoever : So that the amount of ali objections against the justice of this method in finding out the last proportion of vanishing quantities can arise to little more than this, that he has no right to cali the proportions he finds out according to this method by that name, which sure must be egregious trifling. However, as on this head our author seems to talk with more than usuai confidence of the advantage he has over his oppo- nents, and gives us what he says is the amount of Sir Isaac's reasoning, in a truly ridiculous light, it will be proper to see on whom the laugh ought to fall, for I am sure somebody must bere appear strangely ridiculous, ... I readily allow whatever consequence he is pleased to draw from it, if it appears that Sir Isaac, in order to find the last ratios proposed was obliged to make two incon- sistent suppositions. To confute which nothing more need be said than barely to relate the sup- positions he did make. TEXT-BOOKS, 1736-1741 161 *' I. Then he supposes that x by increasing bè- comes;ir + ^, and from hence he deduces the relation of the increment of x and ;r". '*2. Again, in order to find the last ratio of the increments vanishing, he supposes 0 to decrease till it vanishes, or becomes equal to nothing. . . . These are evidently no more inconsistent and con- tradictory, than to suppose a man should first go up stairs, and then come down again. To suppose the increment to be something and nothing at the same time, is contradictory ; but to suppose them first to exist, and then to vanish, is perfectly con- sistent ; nor will the consequences drawn from the supposition of their prior existence, if just, be any ways affected by the supposition of their subsequent vanishing, because the truth of the latter supposition no ways would have been an inconsistency ; but to suppose them first unequal, and afterwards to become equal, has not the shadow of difficulty in it. . . . must confess there seems to be some objection against considering quantities as generated from moments. What moments, what \)i\^ principia janijam nascentia finitaruiìi qziantitatum, are in themselves, I own, I don't understand. I can't, I am sure, easily con- ceive what a quantity is before it comes to be of some bigness or other ; and therefore moments considered as parts of the quantities whose moments they are, or as really fixed and determinate quanti- ties of any kind, are beyond my comprehension, nor do I indeed think that Sir Isaac Newton himself did thus consider them " (pp. 35-41). Il i62 LIMITS AND FLUXIONS John Muller, 1736 154. John Muller, a German by birth, dates bis Mathematical Treatise^^ 173^, from the Tower of London, and dedicates it to the master-general of the ordnance. He was appointed in 1741 head- master of the Royal Military Academy, Woolwich. He was * ' the scholastic father of ali the great engìneers this country employed for forty years. " The author's method of explaining fluxions is somewhat unique. *' I make no use of infinitely small quantities nor of nascent or evanescent velocities ; and yet I think to bave explained those Principles, so that any Person of a moderate capacity . . . may be fully convinced of the Truth thereof" (Preface). He begins his conic sections with the postulate: ''Grant that two infinite quantities, differing from each other by a finite quantity, may be esteemed equal. " He then explains that this postulate '*is bere of use only to shew the connection of the Conic-Sections," and hastens to assure the reader that ''whenever we make use of it in the demonstration of any Proposi- tion, we shall give always another Demonstration independent on it." In the Republick of Lctters, June, 1736, occurs the foUowing comment : " He introduces this [Conic-Sections] by a Postu- latum that sounds very absurdly to those that are ^ A Mathematical Treatise : Containing a System of Conic-Sections ; with the Doctrine of Fluxions and Fluents, Applied to various Subjects. By John Muller. London, 1736. TEXT-BOOKS, 1736-1741 163 not vers'd in mathematica! Speculations. 'Grant,' says he, ' that two infinite Quantities, differing from one another by a finite Quantity, may be esteemed equal. ' Such would imagine that there could not be two infinite Quantities ; or that if there could, they must necessarily be absolutely and not only reputedly equal. But however Hobbes or Berkeley may talk of geometrica! Fallacies, or these unex- perienced People think, the Adepts in this Science very well know, that more infinite Quantities than two are possible, and that one Quantity may be in- finitely greater than an infinite one, and yet be itself infinitely less than a third. But enough of these Ludibria Scientiae, that I may inform the Publick of the more useful Theorems ..." (pp. 422, 423). Muller considers in his text a curve generated by a point " urged by two powers acting in two dififerent directions, the one parallel to the Abscisses and the other parallel to the Ordinates. I prove from thence, that if this point (when arrived at a given place) did continue to move with the velocity it has there, it would proceed in a right line touch- ing the Curve in that place ... So that the three Directions being known in each place, the propor- tion between the velocities of the urging powers is likewise known." Fluxions are defined as velocities. To find the fluxion of j^^^ he puts y'^=x\ the sub- tangent of the parabola is 2y'^ Since the subtangent is to the ordinate as the velocities along the abscissa and ordinate, he has 2y^ -. y \ \x \ y, or ,r=2yy, and 2yy is the required fluxion. Similarly, to find the 64 LIMITS AND FLUXIONS fluxion of j|/^, let x =y^. Take u~z^^ the u—x=z —y y.zz-\-2y-\-yy^ or z^y : u—x=i : zz-\-zy-\-yy. li now j/ and z approach continually until they coincide with an intermediate ordinate, then z=y and the chord through the extremities of the ordinatesi and z will likewise coincide with the tangent. Therefore, the ordinate is to the subtangent as i is to 3j/^. Hence the proportion i : 3XF=>' : i', or i'=3>x^, the fluxion required. The same argument is applied to y^". In these demonstrations appeal is made to a geometrie figure, and no attention is directed to the ratio z—y \u—x for the difficult case when j = 5. The author remarks that " though we commonly say that . . . mjy'*''^ is the Fluxion ofy ; yet that expression is not sufficiently accurate : Therefore, the sense in which we desire to be understood is, that I : 7ny"^~^ : : y : 7nyy'"'^, that is, unity is to uiy"'''^, or j> is to myy'"~^^ as the fluxion or velocity with which y is generated, is to the fluxion, or contemporary velocity with which j/"' is generated, and so for the rest " (p. 79). Thus, the emphasis. is placed upon the ratios of velocities. Anonymous Trans lation ^ of Neiuton's ' ' Method of Fluxio ns,'' 1737 i^^a. Colson's translation from the Latin ol' Newton's Method of Fluxions, published in 1736,, was foUowed in 1737 by a second translation, which ^ A Treatise of the Method of Fluxions and Infinite Series^ With its Application to the Geonictry of Curve Lines. By Sir Isaac Newton, Kt, Translatcdfrom the Latin Orioinal not yet published. Designed by the^ Author for the Use of Lear neri. London, MDCCXXXVII. TEXT-BOOKS, 1736-1741 165 was anonymous. In it no mention is macie of Colson's edition. The anonymous translator says in the preface : ' * We have reason to believe that what is here delivered, is wrought up to that Fer- fection in which Sir Isaac himself had once intended to give it to the Publick. The ingenious Dr. Pemberton has acquainted us that he had once pre- vailed upon him to complete his Design and let it come abroad. But as Sir Isaac's Death un- happily put a stop to that Undertaking, I shall esteem it none of the least Advantages of the present Publication, if it may prove a means of exciting that Honourable Gentleman, who is possessed of his Papers, to think of communicating them to some able Hand ; that so the Piece may at last come out perfect and entire." As remarked by G. J. Gray,^ the two translations were made *'from copies of the same manuscript," and differ from each other only "in the mode of expressing the work in English. " James Smith ^ ^717 155. In his New Treatise of Fluxions,^ Smith says (Preface) : '' What I cali here the New Method, and the Six Propositions immediately following, ^ A Bibliography of the Works of Sir Isaac Newton. By George J. Gray. Second edition, Cambridge, 1907, p. 47. 2 A New Treatise of Fhixions, containitig, I. The Elements of Fluxions, demonstrated in Two easy Propositions, without first or last Ratios, IL A Treatise of Nascent and Evanescent Quantities, first and last Ratios, III. Sir Isaac Newton s Detnonsiration of the Tluxions enlarged and illustrated : IV. Ansiucrs to the Principal Ohjections in the Analyst. By James Smith, A.M., London, 1737. i66 LIMITS AND FLUXIONS are entirely New . . . Our common Definition of Motion, trans latto corporis de loco in locum is certainly imperfect, and 1 am inclined to think, that Aristotle's old exploded Definition of Motion will, some time or other, come into Vogue again. Actus entis in potentia^ cpiatcnus in potcìitia est. Motion is an Efifect, and every Effect has a coin- staneous Existence with the Action by which it is produced." The definitions with which Smith starts out are not very reassuring. "The fluxion of a surface is the Velocity of the generating Line." "The velocity of a generating Line is the Sum of the Velocities of ali the Points of that Line, whether these Points move with equal, or unequal Veloci- ties. " The rectangle xy * * flows or increases by the flowing of both its contiguous Sides " together ; but it "flows into Length " by the velocity ja-, and **it flows into Breadth at the very same Instant of Time" by the velocity xy. " Therefore the Velocity with which it flows into Length and Breadth is the Sum of the synchronic Velocities," xy-\-yx. Nor is the second topic displayed with illumina- tion. "A nascent Ouantity is a Quantity in the Instant of its commencing to exist." Similar to this is the definition of " evanescent Quantity," as are also the definitions of first and last ratios. Interesting is the following proof that if " two Quantities begin and cease to exist in any finite Time T, . . . they have a first and a last Ratio," TEXT-BOOKS, 1736-1741 167 for, '*if they have not a first Ratio, they have not a second nor a third Ratio, etc. Therefore they have no Ratio in the Time T ; but in the Time T they are Ouantities, " and * * two quantities of the same kind, as soon or so long as they have any Quantity, Being or Existence {i.e. are not absolutely nothing], have a Ratio the one to the other," that is, *'they have a Ratio and they have not a Ratio in the Time T, which is absurd." Smith argues also that since two quantities ''cannot be in their first Ratio, neither before nor after the Beginning of the Time T, they must have been in their first Ratio at the very Beginning of the Time T, just as they began to exist." Near the dose of this part of his hook, Smith reveals some of the subtleties of his topic by stating an **Objection" and the ''Answer" to it. The Objection : *'Nascent and evanescent Quantities are Something or Nothing ; for, Inter ens et non-ens non daiur medium. If Something, then the Ratio of evanescent Ouantities is the same with the Ratio before they were evanescent, or when they had any finite Magnitude. . . . If they are mere Nothing, or Non-quanta ; then B^/E^ = 0/0 = 0; . . . which is absurd." In the *'Answer" Smith says : "Evanescent Quantities are really nothing, or Non-quanta ; for it is evident . . . that upon ^'s coinciding with B, and /s co- inciding with E, the Increments B/; and E^ are annihilated, and evanescent Quantities are never accurately evanescent, but upon this or the like Coincidence. And yet it does not follow that their i68 LIMITS AND FLUXIONS last Ratio, or the Ratio they nihilesce with, is Nothìng. For B^ / E^ is neither B/; nor E^, nor B^ and E^, but a Mark or Expression of their Ratio, which may be expressed as well by any other Character. . . . The Increments are indeed annihilated and gone, but their last Ratio remains, and is as real as any Ratio they ever had " ; . . . they have as real a Ratio at the last Instant of their Existence ; that is, when they are ceasing to be Something, and commencing to be Nothing, as they had at any instant preceding the last Instant of their Existence." . . . ''There is, sometimes, something very strange in the Nature of these evanescing Augments, and it is literally true of them, what Juvenal figuratively says of Man. — Mors sola fatetur^ Quantula sunt honiinum corpuscula — We know nothing of them till they be dead and gone." Of Part III, in which Smith '* demonstrates " Newton's Method of Fluxions, we quote only the last sentence : *' I have made use of infinitely little Quantities, and of a second Point as being next to a first Point ; but this was only for Illustration sake. There is not the least Occasion for any of these Notions in the Demonstration. " In the last part of Smith's hook, Berkeley's con- tention, '* No just Conclusion can be drawn from two contrary Suppositions," is answered by the statement, ''This is certainly true, Ì7i sensu com- posito^ but in sensu diviso is intirely false." TEXT-BOOKS, 1736-1741 169 We are tempted to make the remark that in 1737 Smith left the subject even more mysterious than he found it. Thomas Siinpson^ ^72)7 156. Thomas Simpson, the son of a vveaver, was a self-taught mathematician, and acquired a know- ledge of fluxions through Stone's translation of De L'Hospital's A^ialysc des infinimcnt pctits. Simpson was a mathematician of marked power, and influenced considerably the teaching of mathematics in England. In 1737 he broughtout his New Treatise of Fluxions ,'^ which contains some novel features. ** The Fluxions of variable Ouantities are always measured by their Relation to each other ; and are ever expressed by the finite Spaces that would be uniformly described in equal Times, with the Veloci- ties by which those Quantities are generated." He finds it easy to show that the fluxion of a rectangular area of Constant height and uniformly variable base is as the height drawn into the velocity with which the base changes ; also that the fluxion of a curvilinear area generated by an abscissa moving with uniform velocity is at a given point, as the ordinary y for this point, multiplied by that velocity. This last result is applied to finding the fluxion of ,tT. Avoiding infinitely small quantities, Simpson finds ^ A Ne7v Treatise of Fluxions: whercin the direct and ifiverse Method are demonstrated after a new, clear and concise Manner, with their Application to Physics and Astronomy, By Thomas Simpson, London, 1737. I70 LIMITS AND FLUXIONS the ratio of the fluxions of x and x^ thus : Let the points iìi and n move so that the distance h de- scribed by n shall always equal the square of the distance g described by ni in the same tinne. Then (AR)2 = CS, (AR-Rr)2=:Cj, and jS = 2AR x Rr- (Rr)=^. But jS is described with accelerated velocity when ;;/ moves uniformly, hence ^S will be 'Mess than that which would be uniformly described in the same time with the Velocity at the point S, and greater than that which would be described with the Velocity at the ,g point s ; and therefore must be equal to the Dis- tance that would be uni- formly described with the Velocity at another point e posited somewhere be- tween S and s, in the same Time that the other point m is moving over the Distance rR ; therefore rR : 2ARxRr-(Rr)2 : : ^:^(2AR — Rr), the Distance that* would be de- scribed with the Velocity of n^ at the point e, in the same Time that in is moving over the Distance g : Now therefore when the points r and s coincide with R and S, then will e coincide with S ; . . . and consequently (2AR — Rr)^^ will then . . . become 2ARx^, equal to h the required Distance." The criticai part of this proof is ' ' when the points r and s coincide with R and S, then will e coincide with S." A modification of this proof is applied to x"". Simpson's text marks a departure from Newton rt . r R H' m m m ' r , S e S o* n n n 9 -/? TEXT-BOOKS, 1736-1741 171 in the definition of fluxion. Newton makes it a velocity, Simpson makes it a finite distance. On the necessity and wisdom of this change there can readily be difference of opinion. But there can be no denying that Simpson developed his theory of fluxions in a manner almost, though not entirely, free from the objections against fluxions that had been advanced by Berkeley ; infinitely small quan- tities are nowhere used. A short but appreciative review of this text appeared in The Works of the Learned for July, 1737. Benjamin AI art in, 1739, 1759 157. Benjamin Martin was a mathematician, an optical instrument maker, and a general compiler. He was a self-educated man, and at one time taught reading, writing, and arithmetic. His exposition of fluxions, as found in his Eleinents of ali Geonietry ^ and in a later work, is below the standard usually reached by him in mathematica! writing. This book, intended as an introduction to modem mathematics, contains in an Appendix an epitome of the doctrine of fluxions. '* Since Fluxions are the very small Increnients and Decrenients of the Flowing Ouantities, or the Velocities of the Motions whereby they increase or decrease, 'tis plain that those Fluxions, or Velocities, themselves may be consider'd as Flowing Quantities, and their Fluxions are call'd Second Fluxions ..." It would seem 1 nANrEi2METPIA ; or ihe Elements of ali Ceometry. By B. Martin, London, M.DCC.XXXIX. 1/2 LIMITS AND FLUXIONS that in this statement a fluxion is " very small " and at the same time a *'velocity." A little later the author refers to fluxions as ''in the first Ratio of Augmenta nascentia, " Evidently, in this Appendix, covering twelve pages, the author has not succeeded in presenting a consistent theory of fluxions. A fuller exposition was given twenty years later in the System of Mathcmatical InstitutionSy agrecable to the Present State of the Newtonian Mathesis^ by Benjamin Martin, voi. i, London, MDCCLIX. The theory is stili confusing. " Indefinitely small Spaces " (p. 362) are represented by x and j>, which are called the fluxions oi x andjj^, and said to repre- sent the velocities of moving points. Newton is reported to have at first delivered the idea of what Martin calls a fluxion, under the name of momentum^ " a Term used in Mechanics to denote the Quantity of Motion generated by a given Quantity of Matter (A), and the Velocity {a) with which it moved con- jointly. This Momentum therefore was properly represented by (A^). . . . But instead of this mechanical Notation, we now use xx and yy for the Momenta^ or Fluxions. ..." It is seldom that one encounters a more grotesque conglomeration of unrelated ideas than is presented here. Martin gives John Rowe's mode of deriving the fluxions of xy and xyz. An Anonynious Text, 1741 I 58. An Explanation of Fluxions in a Short Essay on the Theo7y. London: Printed for W. Innys, at the TEXT-BOOKS, 1*736-1741 173 West-End of St. Paul's, MDCCXLI. This anony- mous publication of 16 pages was reprinted in 1809 in the fourth edition of John Rowe's Doctrine of Fluxions ; it constitutes a real contribution to the logie of fluxions. The pamphlet is offered " as an Explanation of the Doctrine itself, and not of Sir Isaac's Manner of delivering it." '*About that," he says, " I don't mean, nor pretend to take a Part in any Controversy. " He defines fluxions thus : **The word Fluxion properly apply'd always sup- poses the Generation of some Quantity (term'd Fluent or Flowing Quantity) with an equable, accelerated, or retarded Velocity, and is itself the Quantity which might be uniformly generated, in a Constant Portion of Time, with the Amount or Remainder of that Velocity, at the Instant of find- ing such Pluxion." " Hence, it will appear that the first Fluxions of Quantities are as the Velocities with which those Quantities are increas'd ; that second Fluxions are as the Increase or Decrease of such Velocities ; and that by second, third, fourth, etc. , Fluxions are meant Fluxions, whose Fluents are themselves Fluxions to other proposed Quan- tities ; and the manner of considering and determin- ing them is the very same as tho' they were first Fluxions, they being actually so to the Quantities from which they are immediately derived " (p. 7). Then follows the lemma : "The Fluxion of the Area ABC, whether tri- .angular or curvilinear, is the Rectangle ij." Suppose B to move along hV while the ordinate 174 LIMITS AND FLUXIONS G C ^ — 1 V / ^ f X oc y terminatcs in the curve AC ; " And, at any pro- peseci Position BC, conceive y to become Constant," while B " moves uniformly any Constant Time, ?//;/, with the Velocity at B, over the Distance x or BD ; for then vigili y in the Time mn uniformly generate the Rectangle ij, which Rectangle is plainly the Fluxion of ABC in this Position {per Definii.).'" Then follows the illuminating scholium : " It has been commonly objected to the Accuracy of Fluxions, that the Trapezium or curvi- linear Space BC<^^D, not the Rectangle xy^ is the Fluxion geometrically exact. But, ^ this Objection is built, I apprehend, upon a false Idea of the Thing. It supposes a Fluxion a complete Part of r a flowing Quantity, and an Infinity of Fluxions to con- stitute the flowing Quantity, which are Mistakes {per Definition and Lemma) . . . if i- be imagined ìnfinitely little, an Infinity of Increments may constitute the Area ABC. But, in Fluxions, our Reasoning is quite differenti a Pluxion can no more be called a Part of the Fluent, than an Efìfect a Part of the Cause. For Instance; from the Fluxion given we know the Fluent, and vice versa, just as when a Cause is known to produce a certain Effect, we can infer the one from a Knowledge of the other. " We shall find that later this reference to cause and B FiG. 8, TEXT-BOOKS, 1736 1741 175 efifect figured in a controversy carried on against Simpson. As regards the lemma given above, we shall see that the same idea is elaborated in detail by Maclaurin in his work and that a short and even more convincing statement than the one given here is found in the later, revised, text of John Rowe. From the above lemma, the derivation of the fluxion of xy becomes easy by considering the rect- angle ABCG as made up of two parts AHCB and AHCG, and applying the lemma to each part. John Rowe, 1741, 1757, 1767 159. The first edition (1741) of John Rowe's Doctrine of Fluxions ^ appeared anonymously. A copy in the British Museum has the following added by band after the preface : " This is the first edition of John Rowe's Fluxions. The second came out withhis name in 1757 with alterations and additions, and the third came out in 1767 much improved. " In the first edition Rowc begins by stating his pro- gramme: ''To render the Doctrine of Fluxions plain and easy" by explaining their nature *'as deliver'd both by Sir Isaac Newton and by Leibniz." Accord- ing to Newton, ** Pluxion is the same as velocity." ''Definition II [Foreigners Definition], Quantities are here suppos'd to be generated by a continuai In- crease, as before; and the indefinitely small Particles ^ An Introdudion to the Doctrine of Fluxions. Revised by several Gentlemen well skiird in the Mathematics. Felicibus inde Ingeniis aperitur IXar—Claiidian. London, M.DCC.XLI, 176 LIMITS AND FLUXIONS whereby they are continually increas'd, are call'd the Fluxions of these Quantities " (p. 3). '*This is the Notion of Fluxions as deliver'd by Leibnitz and his Followers. But these Fluxions, we shall, in the following Sheets, cali by the Names of Moments^ Increments and Dccreinents ; that is, Moments or Increments when the variable Quantities are increas- ing, and Decrements when they are decreasi ng " (p. 4). "As the Point b is continually nearer to a Coincidence with the Tangent TBG the nearer it approaches the Point of Contact B; so if we conceive the Ordi- nate cb to be moved on till it concides with CB ; the very first moment before its Coincidence, the Curve B(^, and Right line BG will be infinitely, or rather indefinitely near a Coincidence with each other ; and conse- quently, in that Case, the Increments B^, and eb will come indefinitely near to measure the Ratio of the Fluxions of the Absciss and Ordinate AC, and CB, or the Velocities with which they flow at the Point B . . . and therefore (because when any Ouantity is increas'd or decreas'd, but by only an infinitely or indefinitely small Particle, that Ouantity may be consider'd as remaining the same as it was before ;) these Increments may be taken as Proportional to, or for the Fluxions in ali Opera- TEXT-BOOKS, 1736-1741 77 tions ; and, on the contrary, the Fluxion for the Increment " (pp. 5, 6). Accordingly, he deduces the rules of operation by the use of increments, and in the result substitutes the fluxion for the increment. In finding the fluxion of xy he lets x' and y' be the increments, then the " increase in its nascent state" is such that x'y' ''bears no assignable Ratio to either x'y or xy (for as x'y' : x'y : :y' : y and y' by Supposition is infinitely less than y,'' and can be " expunged or rejected. " 160. The third edition (1767) was commented upon by J. Stubbs, Fellow of Queen's College, Oxford, as follows : *' I received your valuable present, and was much surprised to find it so prodigiously improved. Indeed, it so much resembles a New Work, when compared with the First Edition, that I almost wish you had made no mention of its being the Third ; but left the two former to be forgotten." The fluxion of xy is now deduced thus : "The fluxion of the curvilinear space AEI is less than the fluxion of the rectangle (of Constant altitude) AH before EH reaches BC, and greater after EH passes BC ; hence at BC the two fluxions are alike and equal to yi: Similarly, it follows that the rectangle AG (of Con- stant base) has the same fluxion xy at DB as has the curvilinear space AFI. Hence the rectangle 12 H 1 H B / / / / / 1 / G y E C FlG. IO. 1/8 LIMITS AND FLUXIONS of variable base and altitude AEIF with the vertex I moving along the curve through B has the fluxion In a footnote Rowe expressed the belief that this mode of deriving the rule is not open to criticism as was the method of using increments vvhich in 1 734was **smartly attacked by the late acute Dr. Berkeley." Rowe proves by a geometrical method similar to the above that the fluxion of a pyramid of fixed vertex and slant edges, whose variable base xy moves parallel to itself and whose variable altitude is z^ is xyz. Taking a parallelopipedon as equal to three pyramids, he finds the fluxion of xyz to be xyz-\-xyz-\-xyz. This new way of deriving the fluxion of xyz was copied by *'his friend" Benjamin Martin in the MatJwnatical Institutions. At the end of the third edition of Rowe's Fluxions is a bibliography of English works on this subject, and he **particularly refers to the Works of his two celebrated Friends, Mr. Emerson and the late Mr. Simpson." Berkeley Ten Years After i6i. Berkeley, in his Siris ^ of 1744, expressed himself as follows : " Concerning absolute space, that phantom of the mechanic and geometrical philosophers (§ 250), it may sufiìce to observe that it is neither perceived by any sense, nor proved by any reason, and was accordingly treated by the greatest of the ancients as a thing merely visionary. ^ George Bcrkelcy's Works. Edition by A. C. l'raser, voi. ii, Oxford, 1871, p. 468 and note. TEXT-BOOKS, 1736-1741 179 From the notion of absolute space springs that of absolute motion ..." He continues in a foot- note : " Our judgment in these matters is not to be overborne by a presumed evidence of mathematica! notions and reasonings, since it is plain the mathe- maticians of this age embrace obscure notions, and uncertain opinions, and are puzzled about them, contradicting each other and disputing like other men : witness their doctrine of Fluxions, about which, within these ten years, I bave seen published about twenty tracts and dissertations, whose authors being utterly at variance, and inconsistent with each other, instruct by-standers what to think of their pretensions to evidence." Remarks 162. In these publications no reference is made to the Jurin-Robins controversy, though Berkeley's Analyst is frequently discussed. Excepting only in Benjamin Martin, the definition of a fluxion as a '' differential " nowhere appears. Therein we see a step in advance. The influence of Newton's Quadrature of Curvcs (1704) is evident almost everywhere. An improve- ment in the mode of deriving the fluxion of a *' product " appears in the anonymous Explanation of Fluxions and in the revised text by John Rowe (our §§ 158, 160). Noteworthy is Thomas Simpson's new definition of fluxions ; this new definition plays an important ròle during the rest of the century. i8o LIMITS AND FLUXIONS 163. We quote Sir William Rowan Hamilton's remarks on the lemma of the anonymous Explana- tion of Fluxions (1741) and the derivation of the fluxion of xy^ based upon it. Hamilton knew this proof as it is given in a later edition of Simpson's fluxions. Says Hamilton :^ *' I notice that Thomas Simpson treats fluxions as finite . . . Thomas Simpson's conceptions appear to have been very clear and distinct, and I do not venture to say that the geometrical investigation which he gives of the fluxion of a rectangle, avowedly supplied to him by a young but unnamed friend, is insufficient in itself, but it fails to convince me, perhaps because I was not early accustomed to fluxions. Certainly there is no neglecting of ab^ or .1 j> ^'^ small ; for in fact that rectangle of the fluxions is not represented at ali in his Figure ... He conceives the varying rectangle xy to be the suin of two mixtilinear triangles, of which the two separate fluxions are yx and xy. This is very ingenious, but I do not feel sure to what degree I could rely on it and build upon it any superstructure, if I were now coming, for the first time, as a learner^ to the subject. However, I suppose that a pupil, if reasonably modest or even prudent, vvill take, for a while, his teacher's state- ments upon trust ; reserving to himself to return upon them, and to examine closely their truth and logie when he shall have acquired some degree of familiarity with the subject taught." ^ Life of Sir Williain Rowan Hamillon, by R. P. Graves, voi. iii, p. 571. CHAPTER VI MACLAURIN'S TREATISE OF FLUXIONS, 1742 164. Colin Maclaurin was educateci at the Uni- versity of Glasgow, and through the influence of Newton was elected professor at the University of Edinburgh. Maclaurin's hook on fluxions has been considered the ablest and most rigorous text of the eighteenth century. It was pronounced by Lagrange "le chef d'oeuvre de geometrie qu'on peut comparer à tout ce qu' Archimede nous a laissé de plus beau et de plus ingénieux."^ In the preface to his Trcatise of Fluxions^ Maclaurin says : "A Letter published in the Year 1734, under the Title of the Analyst, first gave Occasion to the ensuing Treatise. ... In the mean Time the Defence of the Method of Fluxions, and of the great Inventor, was not neglected. Besides an Answer to the Analyst that appcared very early under the Name of Philalethes Canta- brigiensis ... a second by the same Hand in Defence of the first, a Discourse by Mr. Robins, a ^ Mém. de tAcad. de Berlin, 1773 ; quoted in the art. " Maclaurin " in Sidney Lee's Dict . of National Biop-aphy. ^ A Treatise of Fluxions in 'fwo Books. By Colin MacLaurin, A.M., Professor of Mathematics in the University of Edinburgh, and Fellow of the Royal Society. Edinburgh, MDCCXLU. i82 LIMITS AND FLUXIONS Treatise of Sir Isaac Newton's with a Commentary by Mr. Colson, and several other Pieces were published on this Subject. After I saw that so much had been written upon it to so good Purpose ; I was the rather induced to delay the Publication of this Treatise, till I could finish my Design. . . . The greatest Part of the first Book was printed in 1737 ; But it could not bave been so useful to the Reader without the second. . . . In explaining the Notion of a Fluxion, 1 bave foUowed Sir Isaac Newton in the first Book ... ; nor do I think that I bave departed from bis Sense in the second Book ; and in both I bave endeavoured to avoid several Expressions, which, though convenient, might be liable to Exceptions, and, perhaps, occasion Disputes. I bave always represented Fluxions of ali Orders by finite Ouantities, the Supposition of an infinitely little Magnitude being too bold a Postulatum for such a Science as Geometry. But, because the Method of Infinitesimals is much in use, and is valued for its Conciseness, I thought it was requisite to account explicitly for the Truth, and perfect Accuracy, of the Conclusions that are derived from it . . . " 165. In the Introduction to bis Fluxions Maclaurin says : "... When the certainty of any part of geometry is brought into question, the most efìfectual way to set the truth in a full light, and to prevent disputes, is to deduce it from axioms or first prin- ciples of unexceptionable evidence, by demonstra- tions of the strictest kind, after the manner of the {To face page 182. H^^^__^ '-., ^\, ^ /'"'^^^^^H| H /^B '1 ^1 ||a| |T i ■1 ^^H ^^^^^^^^H 1 1 1 ""s/^^^B '"'-"■^^c::;:,:';. .r-r^'f r ■ r 1 e 0 1, 1 > >:iA CLAI^KCv, A 31. 1 MACLAURIN'S TREATISE, 1742 183 antient geometricians. This is our design in the following treatise ; wherein we do not propose to alter Sir Isaac Newton's notion of a fluxion, but to explain and demonstrate his method, by deducing it at length from a few self-evident truths, in that strict manner : and, in treating of it, to abstract from ali principles and postulates that may require the imagining any other quantities but sudi as may be easily conceived to have a real existence. We shall not consider any part of space or time as indivisible, or infinitely little ; but we shall consider a point as a term or limit of a line, and a moment as a term or limit of time ... [p. 41]. If we are able to join infinity to any supposed idea of a deter- minate quantity, and to reason concerning magni- tude actually infinite, it is not surely with that perspicuity that is required in geometry. In the same manner, no magnitude can be conceived so small, but a less than it may be supposed ; but we are not therefore able to conceive a quantity infinitely small ..." 166. In the posthumous work, An Account of Sir Isaac NewtoìCs Philosophical Discoveries^ by Colin Maclaurin, 2nd ed., London, 1750, there is printed a life of Maclaurin, from which we glean the follow- ing (pp. viii, ix, and xviii) relating to Berkeley 's attack in the Analyst : " Mr. Maclaurin found it necessary to vindicate his favourite study, and repel an accusation in which he was most unjustly included. He began an answer to the bishop's hook ; but as he proceeded, i84 LIMITS AND FLUXTONS so many discoveries, so many new theorìes and problems occurred to him, that, instead of a vindi- catory pamphlet, his work carne out a complete system of fluxions, with their application to the most considerable problems in geometry and naturai philosophy. This work was published at Edinburgh in 1742. . . . Piis demonstrations had been, several years before, communicated to Dr. Berkeley, and Mr. Maclaurin had treated him with the greatest personal respect and civility : notwithstanding which, in his pamphlet on tar- water, ^ he renews the charge, as if nothing had beeh done ; for this ex- cellent reason, that difìerent persons had conceived and expressed the same thing in different ways. . . . Mr. Maclaurin found it necessary, in demonstrating the principles of fluxions, to reject altogether those exceptionable terms {infinite and infinitesimal\ and to suppose no other than finite determinale quan- tities, such as Euclid treats of in his geometry." 167. In Chapter 1, p. 57, Maclaurin defines a fluxion : " The velocity with which a quantity flows, at any term of the time while it is supposed to be generated, is called its Fluxion which is therefore always measured by the increment or decrement that would be generated in a given time by this motion, if it was continued uniformly from that term without any acceleration or retardation : or it may be measured by the quantity that is gener- ated in a given time by an uniform motion which is equal to the generating motion at that term." 1 In the second cdition Berkeley gave the article the name oi Siiis, MACLAURIN'S TREATISE, 1742 185 The term velocity had been under dispute, par- ticularly in the controversy between Berkeley and Walton. Maclaurin evidently perceived the diffi- culty in arguing that variable velocity is a physical fact ; he says (p. 55), "the velocity of a variable motion at any given term of time is not to be measured by the space that is actually described after that term in a given time, but by the space that would have been described if the motion had continued uniformly from that term. If the action of a variable power, or the velocity of a variable motion, may not be measured in this manner, they must not be susceptible of any mensuration at ali " — an argument not liii- will be "infinitely less " than vx or xi). Here the fluxions x, v,i\ are looked upon as mfinitely small. In the account of the life of Nicholas Saunderson, printed in the first volume of his Elements of Algebra ^ Cambridge, 17^0, p. xv, we read: " Our Professor would not be induced by the Desires and Expecta- tions of any, to engagé in the war that was lately waged among Mathematicians, vvith no small Degree of Heat, concerning the Algorithm or Principles of Fluxiotis. Yet he wanted not the greatest Respect for the Memory of Sir Isaac Newton, and thought the whole Doctrine entirely defensible by the strictest Rules of geometry. He owned indeed that the great Inventor, never expecting to bave it canvassed with so much trifling Subtility and Cavil, had not thought it necessary to be guarded every where by Expressions so cautious as he might bave otherwise used." folm Rowning, 1756 1 74. A graduate of Magdalene College, Cambridge, and a Fellow there,Rowning interested himself chiefly in naturai philosophy, but wrote also A Prelinmiary Discoiirse^ on fluxions, with the intention of writing ^ A Pieliminary Discotirse to an ititended Treatise on the Fluxiouary Method. Hy John Rowning, M.A. London, 1756. TEXT-BOOKS OF MIDDLE OF CE N TUR Y 199 a full treatise. But the treatise in question never appeared. After a popular exposition of the ideas of fluxion and fluent, and of Leibniz's infinitely little quantities and their summation, showing how these methodsyield importantresults in naturai philosophy, he refers to Berkeley's attacks and the defence made by Philalethes Cantabrigiensis, Walton, and Robins, also Maclaurin, who ' * declined entering the Combat," but endeavoured to treat the subject *'in a Manner less exceptionable." *'But no Body, that I know of," continues Rowning, '*has explained it in so easy and familiar a VVay as I apprehend the Subject capable of." Moreover, Jurin and Walton " carry things ... no farther than Sir Isaac had done before. They leave them, as to the Objections made by the Analyst, exactly as they found them." The diffìculties do not He in the idea of a first fluxion — a velocity. *' In this there is Nothing either infinitely great or infinitely little : Nothing obscure. " As to higher fluxions, "these Things indeed elude our Senses ; but they do not surpass the Understanding " (p. 85). Berkeley's objection to 'Mnfinitely small Quantities" is not fatai, " because finite Measures might bave been made use of." His other objection, that " such Quanti- ties are in some Cases retained and made use of for a while, and afterwards, to use his own Expression, like Scaffolds to a Building, are rejected as of no Significancy," may be met by the proof that those quantities "are always such as ought by no means to be retained." In further explanation of his 200 LIMITS AND FLUXIONS position Rowning says (p. 88), '*that the Velocity of any Body is the same at any one Point, or at any one Time, whether the Body moves with an uniform, accelerated, or with a retarded Motion at that Point or Time." This is elucidated by reference to geo- metrie figures, and amounts, in the main, to the explanation given by Rowe in finding the fluxion of xy. One objection to such explanations, which had been raised by Berkeley, was that one could not speak of the velocity a body had at a point of space. That such a phraseology is admissible is tacitly assumed by Rowning. What the latter emphasises is that no use is made of the concept of the * ' infinitely little. " As to Berkeley's second objec- tion, that the supposition which is made at the beginning of the process is later displaced by its contrary, as when the symbol o is at first made an actual increment and later in the same process taken as no increment, Rowning argues that terms involv- ing factors oo^ ooo, etc. , "do arise in consequence of the Acceleration wherewith the Power of ;ir flows, when X itself flows uniformly ; and consequently that they arise from the second and higher Fluxions of that Power ; and that, therefore, when the first Fluxion of that power is only inquired after . . . they are to be left out and rejected, as appertain- ing to another Account." It can hardly be claimed that Rowning made a contribution to the theory of fluxions. However, he has a pleasant way of expressing himself. His book was favourably re- viewed in the Monthly Review (voi. xiv, p. 2 86). TEXr-BOOKS OF MIDDLE OFCENTUR Y 201 Israel Lyons, 1758 175. Lyons was a mathematician and botanist. His Trcatise of Fluxions, London, 1758, is dedi- cated to Robert Smith, Master of Trinity College, Cambridge, " being the first Essay of a young and unpractised Writer " which ' ' ovved its first rude Beginning to the early Encouragement " receìved from the Master, as the author modestly states. His treatment is geometrie. He says : "I reject no Ouantities as infinitely smaller than the rest, nor suppose difìferent Orders of Infinitesimals and infinitely great Ouantities. But consider the Ratio of the Fluxions as the same as that of the con- temporaneous Increments, and take Part of the Increment before and Part after the Fluent is arrived at the Term, where we want the Fluxion, since it is not the Increment after, or the Increment before that we want, but at the very instant, which can no otherwise be found but by considering Part of the Increment before and Part after" (Preface). Fluxions are defined as velocities. '*The moments of quantities are the indefinitely small parts, by the addition or subtraction of which, in equal particles of time,they are continually increased or diminished. " The author proves the proposition : **The indefi- nitely small spaces described in equal indefinitely small times are as the velocities," since, " when the time is diminished ad infinitum, the difference of the velocities at the beginning and ending of that time will vanish." If two flowing quantities 202 LIMITS AND FLUXIONS X and y are to each other in a given ratio, then in xy = z it is argued that 2y = ìucr. of ^-^incr. of x = è-^x; hence é=2yx. Whenj' = x, this becomes è=2x±; one has also, fluxion x-\-y =2x-\-vxx-\-y = 2 X xx-\-yx-\-xy -\-yy. From this is derived the fluxion of any rectangle 2 xy, thus : The fluxion of xy, or x'^-\-2ry-\-^y is also equal to 2;ri-H- 2/j> + fluxion of 2xy. Hence fluxion of 2xy = 2xy + 2;ir;/. ** In the same manner as the quantities ;r, j/, 2, are conceived to flow, and to have their fluxions, so may the quantities i-, j>, i-, be supposed to be variable, and therefore have their fluxions, which are thus represented x,y, z, and are called the second fluxions of x,y, z" (p. II). **The fluent of any quantity as x"'x is represented thus \x'"x\.'^ William West, 1762 176. William West's Mathematics'^ is a posthum- ous work ; the author died in 1760. Fluxions are treated from the earlier Newtonian standpoint, infinitely little quantities being used. Some novelty is claimed for this text in the treatment of maxima and minima. Janics Wilson^ 1761 177. In 1761 Wilson coUected some of Benjamin Robins's mathematical tracts in a two-volume hook, 1 Mathematics. By the late Rev. Mr. Wm. West of Exeter. Revised by John Rowe, London, 1762. There appeared a second, corrected, edition in 1763. TEXT-BOOKS OF MIDDLROF CRNTURY 201 entitled Matheniatical Tracts of the late Benjamin Robins. In an Appendix, Wilson inserts some matters of historical interest regarding certain manuscripts of Newton ; Wilson also defends Robins against criticisms passed by a French writer, and States his views of Maclaurin's indebtedness to Robins. Newton's Method of Fluxions (see our § 149) was brought out in Paris in 1740 by George Louis Le Clerk, Comte de Buffon, under the title, La méthode des Fluxions^ et des suites infiìiies. Buffon prepared a historical Preface, in which he criticised severely Berkeley and Robins for presUming to take excep- tion to anything Newton had written on fluxions or to modify Newton's mode of exposition. Buffon praises Jurin, and then speaks of Robins thus (pp. xxvii-xxix) : ''. . . il commence par le censurer & par dés- aprouver sa maniere trop brève de présenter les choses ; ensuite il donne des explications de sa facon, & ne craint pas de substituer ses notions incomplettes aux Démonstrations de ce grand homme. Il avoué que la Geometrie de l'Infini est une science certaine, fondée sur des principes d'une vérité sùre, mais enveloppée, & qui selon lui n'a jamais été bien connué ; Newton n'a pas bien lù les Anciens Géometres, son Lemme de la Méthode des Fluxions est obscur & mal exprimé ... : malheursement les Mathematiciens ont été plus incrédules que jamais, il n'y a pas eu moyen de leur faire croire un seul mot de tout cela, de sorte que 204 LIMITS AND FLUXIONS Philalethes comme défenseur de la vérité, s'est chargé de lui signifier qu'on n'en croyoit rien, qu'on entendoit fort bien Newton sans Robin, que les pensées & les expressìons de ce grand Philo- sophes sont justes & très-claires . . ., ce sont des piéces d'une mauvaise critique. ..." Buffon presents no argument against the views expressed by Robins, but abuses him for presum- ing to think independently. This doting attitude toward Newton is justly attacked by James Wilson, in bis Appendix to the Mathematica/ Tracts of the late Benjamin Robins^ voi. ii, London, 1761, pp. 325-327. Wilson rightly says that if it was a crime for Robins to make mention of the great brevity with which Sir Isaac Newton wrote, Robins was foUowed in it by Maclaurin and Saunderson. "The truth is," says Wilson, "Sir Isaac Newton at first made the same use of indivisibles, others had done : in his Analysis pei' cequationes numero terminorum infinitas, he expressly says, * Nec vereor loqui de unitate in punctis, sive lineis infinite parvis ^ ; ' and in his Lectiones Opticce he demon- strated by indivisibles." Wilson contends further- more that Buffon is wrong in claiming that the mathematicians paid no regard to what Robins had said, that in fact "the best writers soon after trod in Mr. Robins's steps." In fairness to Buffon it should be said, however, that he printed his Preface in 1740, and that Maclaurin, Saunderson, de Bougainville, and d'Alembert, whom 1 Cojnm. Epist, p. 85. TEXr-BOOKS OF MIDDLEOF CENTUR F205 Wilson mentions as foUowing Robins, wrote at a later date. 178. James Wilson claims ^ that Maclaurin in his Fluxions " conformed himself entirely to Mr. Robins's sentiments in regard to Sir Isaac Newton's doctrine," and ''has even expressly followed his pian in treating the subject. " Jurin had contended (says Wilson) "that Sir Isaac Newton's method, by proving the varying quantities carne up to their limits, was more perfect than that of the ancients. Whereas Sir Isaac Newton never claimed such superiority ; . . . The coincidence contended for, and thus highly praised by Philalethes, is the very essence of indivisibles. " Wilson rightly insists that Buffon 's criticisms of Robins are unfair. '*When he talks of the obscurity of Mr. Robins's ideas, the insignificancy of his phrases, and the unintelligible- ness of his style ; he gives the most certain proof, that he had never carefully read his writings, . . . for Mr. Robins is much admired bere for the con- trary excellencies, on whatever subjects he has employed his pen." 179. Wilson represents Philalethes (Jurin) as championing the use of the infinitely little and of indivisibles. This is putting the case too strongly. In his papers against Berkeley, Jurin uses quantities infinitely little. But toward the end of his debate with Robins he begins to disavow them. Never did Jurin use indivisibles. Few eighteenth-century * Mathematical Tracts of the late Benjamin Robins, voi. ii, London, 17Ó1, pp. 312, 315, 320. 2o6 LIMITS AND FLUXIONS writers have brought out as distinctly and clearly as has Jurin the difference between infinitesimals as variables, and indivisibles ; Jurin disavowed ali quantity "fixed, determinate, invariable, indivisible, less than any finite quantity whatsoever," but he usually did admit somewhat hazily a quantity ''variable, divisible, that, by a Constant diminu- tion, is conceived to become less than any finite quantity whatever, and at last to vanish into nothing." Remarks i8o. None of the works mentioned in this chapter are great works. Those of William Emerson and Thomas Simpson were the best and the ones most vvidely used. The first edition of Simpson is of earlier date (1737). CHAPTER Vili ROBERT HEATH AND FRIENDS OF EMERSON IN CONTROVERSY WITH JOHN TURNER AND FRIENDS OF SIMPSON i8i. The principals, Simpson and Emerson, do not themselves appear in this controversy. During the period of this debate, Robert Heath was editor of The Ladies' Diary, which appeared once every year as an ahiianac. We begin with one of his articles. Robe7-t Heath ^ 1746 182. In an article, Of the Idea, and Nature of Fliixions,^ Heath says : ' * The Distinction betwixt the Increments and Fluxions of Magnitudes, has been this ; that the former approach in Ratio infinitely near the latter, so that their Difference is unassignable. . . . What we cali the Fluxions, or Velocities of Magnitudes, are only the Fluxions in Chief, or in Part, with which they are born ; the Part neglected in the Ratio exactly corresponding with what is rejected in the finite Ratio of the infinitely small Increments, which is therefore the same as the Ratio of our ^ The Ladies Diary : or, the IVoman^s Almanack.for 1746. 207 2o8 LIMITS AND FLUXIONS Fluxions. And hence, whether we cali those finite Ratios, Pluxions, or Increments, their Idea, Nature, and Originai appear to be the very same thing. For ali Things are relative. ..." He argues that while we consider a line or piane, generating an area or solid, as of no thickness in the mind, in our notation we represent them as of unit thickness, **and consequently each Line or Piane should be express'd by ^ x L, and o x P, to denote them as they are in the Mind. But L x ^ to o, and P X P- 34- Six numbers of this journal appeared in London in 1750-1752. No. V bears the date 1752 ; No. VI has no date. Readers are invited " to send their Performances (whether new Problems, Paradoxes, Solutions, etc.) Post paid, to be left with Mr. James Morgan, at the Three-Cranes, in Thames- street . . ." In this connection a statement made by Charles Hutton, in his Memoirs of the Life and Writings of the Author [Thomas Simpson], printed in Thomas Simpson's Select Exercises in the MathematicSy new edition, London, 1792, p. xviii, is of interest : " It has also been commonly supposed that he [Thomas Simpson] was the real editor of, or had a principal share in, two other periodica! Works of a miscellaneous mathematical nature ; viz. the Mathematician , and Turner's Mathematical Exercises^ two volumes, in 8vo, which carne out in periodica! numbers, in the years 1750 and 1751, etc. The latter of these seems especially to bave been set on foot to afford a proper place for exposing the errors and absurdities of Mr. Robert Heath, the then conductor of the Ladies Diaryaxid the Palladium ; and which controversy between them endcd in the disgrace of Mr. Heath, and expulsiou trom his office nf editor to the Ladies^ Diary, and the substi- tution of Mr. Simpson in his stcad, in the year 1753." ROBERT HEATH v. JOHN TURNER 211 which, he tells us, himself, is not exactly the same as that of Sir Isaac Newton." Mr. Simpson is also charged with plagiarism from Cotes's Estimatio Errorum. John Turner says : " Here his Remarks on the Author's Definition of a Fluxion first demand our Consideration : Mr. Simpson makes it to be, * the Magnitude by which a flowing Quantity would be uniformly increased in a given Time.' This Definition the Critic represents as a very old one ; and with regard thereto advances the two following, extraordinary, Positions : '* I. That, in Ouantities uniformly generated, the Fluxion must (according to the said Definition) be the Fluent itself, or else a Part of it. **2. And that, in other Quantities generated by a variable Law, the Fluxion will not be a real, but an imaginary Thing. '*To the first of these Objections I answer, that the Fluxion is neither the Fluent itself nor a Part of it : it is a Quantity of the same Kind with the Fluent ; but the Fluent being the Quantity already produced by the generating Point, Line or Surface, supposed stili in Motion, and the Fluxion what will arise, hereafter, from the Continuation of that Motion ; the latter can no more be denominated a Part of the former than the ensuing Hour a Part of the Time past. "But his second Observation is a stili more glaring Instance of his Disingenuity, and Want of Judgment. Does it follow, because a Body, really, moves over a certain Distance, in a given Time, 212 LIMITS AND FLUXIONS with an accelerateci, or a retarded Velocity, that there is no Distance over which it might pass in the same Time, with its first Velocity uniformly continued. The Space over which a Body would uniformly move with such, or such, a proposed Velocity, is no less real because no Part of it is actually described with that Velocity " (pp. 36, 37). 185. Then foUows an article reprinted from the Daily Gazetteeriox December 4, last [1750], in which one who signs himself ''Honestus" (said to have been John Turner himself) charges that the compiler of the Ladies' Diary (Robert Heath) is also the compiler of the Palladium, and the best material designed by contributors for the Diary are reserved by him for the Palladium ; that the latter publica- tion is owned by the compiler, while the former is not. Robert Heath wrote a reply in the Daily Gazetteer of December 6 ; four letters follow on this subject. 186. John Turner's defence of Simpson led to the publication of what Turner called a ''scurrilous Pamphlet." This pamphlet is without doubt the TrutJi Triuiìiphant : or Fluxions for the Ladies,^ London, 1752, or else those parts in that pamphlet which appear over the pseudonyms *' X Primus " ^ The fuller title of the pamphlet is thus : Trttih Triumphant : or, Fluxions for the Ladies. Shewing the Cause to be before the Effect^ and different from it ; That Space is not Speed^ nor Magnitude Motion. With a Philosophic Vision, Most humbly dedicated to his Illustrious High and Serene Excellence, the Sutt. For the Information of the Public^ by X, F, and ^, who are not of the Family of jt, y, z, biit near Relations of x\ y\ and z . . . . London. Printed for W. Owen, M.DCC.LII. ROBERT HEATH v. JOHN TURNER 213 and **Y Secundus. " These documents evidently emanated from the pen of Robert Heath, assìsted possibly by some other adherents of William Emerson. At the risk, perhaps, of not observing strict chronological sequence, we proceed to the consideration of ali parts of TrutJi TriinupJiant. In the dedication ''to the Sun," it is stated that "the Family of the Wou'd-be's in this Island is become very numerous, by uniforvily continuing in their Errors." Thus, both the title-page and the dedication play on Simpson's definition of fluxions and its alleged defects. In the Preface one reads : "Fluxions, then, Ladies, that have so puzzled our wise Mathematicians to define, are the respective Degrees of Motion, at any Instant of Time, of any two things or Bodies that continually flow, or move on, over Space." Four pages are devoted to the explanation of fluxions. 187. Then follow the two criticisms of John Turner's defence of Simpson, signed "X Primus " and "Y Secundus," to which we have alluded above. In the former of these articles John Turner is treated with contempt. " Who this John Turner is, whether he is Mr. Simpson's Clerk, or his Pupil, or some Dependant on him ; or whether he be Mr. Simpson himself, is not very material to the Reader ..." Turner is continually referred to as "John." To Turner's reply to the first criticism on Simpson's text, "X Primus" makes rejoinder : ''''John says, the Pluent being the Ouantity already produced — Pray how was this Ouantity produced. 214 LIMITS AND FLUXIONS by some magic Art, without any Fluxion ? I believe not. . . . For my Part, l know of no Body that ever saìd, that the Parts of the Fluent that went before were generated by the Fluxion that is to come after, but every Part by its proper Fluxion. ..." To Turner's reply to the second criticism, ''X Primus " makes rejoinder : '* If there be no Magnitude by which the flowing Ouantity is really increased, such a Magnitude is not real, but an imaginary Thing only . . . But John thinks, that every Thing that exists in his Imagina- tion, really exists in Nature . . . Sir Isaac Newton defines PTuxions by the Velocities of the Motions. But Mr. Simpson declares against this, and likewise tells US, that by taking Fluxions for mere Velocities, the Imagination is confin'd, as it were, to a Point. How /lis Imagination is confin'd I don't know ; but Sir Isaac Newton chused to define it thus, as very well knowing, that this is the only soà'd Foundation upon which it could be defended against ali the impertinent Cavils of ignorant or weak Pretenders. " The parting shot by " X Primus" is — your Great Master will not ^'think you a fit Champion to engagé in his Cause for the future ; so, good Night, John." i88. The reply made by '* Y Secundus " is to the effect that the defender of Simpson is ''equally in the Dark" with Simpson himself, ''otherwise he would not have gone about to defend so defenceless a Cause, as to vindicate an Absurdity, by repre- senting a Pluxion to be of the same Kind with the ROBERT HEATH v. JOHN TURNER 215 Fluente uniformly generateci ; when the one is a Ouantity of actual Velocity, and the other a Ouantity of Space, described by that Velocity, which can be only proportional to it." 189. After some poetry **To the Family of the the Wou'd-be's," foUovv " Animadversions on Mr. Simpson's Fluxions," " By 2 Tertius," vvho quotes a criticism of Simpson from the pen of J. Landen. Where Landen's review first appeared we do not know. As quoted here, Landen objects to the definition of fluxions ^ ' diS faulty, by the Author's difìferent Idea given of them to that by the Inventor"; Landen disapproves of *'denoting ali Ouantities whatsoever by Lines, to bring them to one Denomination, and those Lines, to be described by Bodies in Motion." In criticism of fluxions in general, Landen says that the finding, from the velocities, the spaces passed over, and vice versa, " may be' managed by common Algebra, without the least Obscurity. The Business had always been better considered in that Light, without ever making Use of the Term Fluxions, as if a new Kind of Analysis, tho', in Faci, only the Do e trine of Motion improved, and applied to Purposes before unthought of. " 190. The next article in Truth Triumphant is a reprint of the first criticism of Simpson, contributed in 1750 by Cantabrigiensis to the MontJily Review. Eight more articles concerning motion, fluxions, and mechanics bring the pamphlet to a dose ; they make no reference to Simpson, '' Heliocentricus " 2i6 LIMITS AND FLUXIONS explains higher fluxions in a way that cannot be called illuminating. 191. Then " Amicus " speaks '* of the Use of the Algebraic Cypher, in finding the Fluxions of Alge- braic Ouantities," letting x increase or decrease, and become ;fzb^, where the *' Increments or Decre- ments are seen to be ±<^," and **dividing by algebraic 0" thereupon " algebraically considering 0 of insensible Value, as before it was consider'd of real sensible Value." Taking a reminiscent mental attitude, *' Amicus " says : ' ' This algebraic Ratio of the Fluxions of Quantities, to which the diminishing Value of the algebraic Increments or Decrements, from their limited State or Value, tend together, to their geometrical vanish- ing (by supposing the variable Value less and less) has been misconceived, as vanishing together with the ;t^/ geometrical Increments or Decrements they are the Value of\ whence o has been denominated a departed^ instead of an algebraic Quantity, by a famous B— /, tho' it's Reality and Presence stili existed before his Eyes ; but if 0^ the Cypher- Value, or algebraic Quantity, call'd also Nothing^ be made to signify Nothing, because it is so call'd, the Word Nothing with as much Propriety, may be called no Word, be allowed to bave no Signification among other Words, and be deem'd a mere Blank, as no Subject capable of Consideration." Further on in the pamphlet, the query whether there can be *' real Motion in no Time," for ''any one Point of inter- mediate Space gone over; especially since an infinite ROBERT HEATH v. JOHN TURNER 217 Number of Points can never actually constitute real Magnitude," and whether '' Motion, or Fluxion, can aciually exist, and be known, but by the next Increment of Space gone over, in some real and next Monnent of Time"? These are fundamental problems indeed. Zeno is not mentioned in the pamphlet, but the query involves Zeno's subtle paradox of the ' ' arrow. " Nor is the answer given devoid of interest. '* But Time, and Motion, flow- ing over Space, . . . {since no Quantity can be assign'd, or imagin'd so smalla but there will stili be smaller) the respective Degree of Velocity of Motion, or Fluxion (i.e. instantaneous velocities) of that Flowing at any Instant of Time, and Point of Space gone over, will be everywhere assignable by the immediate Increments, as Effects of those preceding Velocities, as has been shewn. Whence it will follow, that certain Degrees of Motion, Fluxion, or Velocity, exist at every instant of Time taken, and Point of Space respectively described ; contrary to the dijfferential Notion that P^oreigners have of this Matter. " The weak spot here resides in the words ''immediate Increments"; do immediate increments exist in view of the statement in the above paren- thesis ? The lack of a satisfactory arithmetical con- tinuum Comes to view more fuUy in the antagonism between geometrie increases and algebraic increases exhibited in the following passage taken from the pamphlet : "AH the Values of the geometrical Increases flow'd over, in finite Time, can never be algebraically 2i8 LIMITS AND FLUXIONS express'd in infinite Time ; in which Sense the algebraic Increases being again diminish'd, are said never to converge to the Limits of their geometrical Magnitudes in Motion, but will stili have sensible Value ; yet supposing the geometrie Increases, and their algebraic Values to flow and decrease alike, to something determinate, then o^ and —x)=x'^^xy-\-y^, which is equal to 3;^^ when j;/ = ;r. We proceed to give an application in Landen's own words : 203. (Page 5) " Fluxionists, in determining the limit of the ratio of the increments of x and x"" , commonly have recourse to the binomial theorem (which is mudi more diffìcult to investigate than the limit they are seeking) : But how easily may that limit be found, without the help of that theorem, by the equation exhibited in page 5 ! Thus, the incre- ment of x being denoted by x\ the increment of x"* is x-{-x'» —x'\ and the ratio of those incre- ments is 234 LIMITS AND FLUXIONS X-\-X *^ —X*^ X + X ** —X' x+x' —X =x-\-x' -1 1+ ^ X 1 ' X ^x+x' 3 (?«) V ' x+x' ' XJtx' ,+ ^ ^ X " ?f^ X 3^» r («) ' x+x' x-^x' x-\-x which, when x vanishes, is manìfestly equal to 7/^ ^f - 1 — ;r« , the limit of the said ratio." n The explanation of the method of drawing tan- gents is too long for quotation, and we shall limit ourselves to the following outline of it, as given by Landen : *' I consider the curve as already described, with- out any regard to its generation, and find the value of a certain line (terminated by the curve and its tangent), in algebraic terms involving {s) the subtangent with other quantities ; which algebraic expression I observe, from an obvious property of the line it is found to denote, must have a certain property with respect to being positive or negative in certain cases. I therefore assume that expres- sion equal to another which is known to have that very property ; and from thence, by means of the theorem mentioned in page 5, readily find the re- quired value o{ s" (p. io). 204. Landen's Discourse was attacked by. an anonymous writer in the Monthfy Review for June 1759, who claims that the Residuai Analysis " is no ATTEMPTS AT ARITHMETISATION 235 other than Sir Isaac Newton's method of differences ; and it is well known, that if the differences are diminished so as to vanish, their vanishing ratio becomes that of fluxions " (p. 560), ''that his pre- tended Residuai Analysis renders the investigations more tedious and obscure than any other. " Landen wrote a reply in the July number, from which we quote only the part relating to the word **function." Says Landen : ''He objects to prime nuinber\ function, etc, as terms never heard before. — Alas ! hovv egregiously does he betray his ignor- ance ! " James G lenze ^ 1793 205. James Glenie graduated from the University of St. Andrews, and became a military engineer. He was a prominent Fellow of the Royal Society of London. In his Antecedental Calculus,^ 1793, he begins with the statement, " Having, in a Paper, read before the Royal Society, the 6th of March, 1777, and pubhshed in the Philosophical Trans- actions of that Year, promised to deliver, without any consideration of Motion or Velocity, a Geo- metrica! Method of Reasoning applicable to every purpose, to which the much celebrated Doctrine of Fluxions of the illustrious Newton has been or can ^ 7'he Antecedental Calculus^ or a Geometrical Method of Reasoning, without any Consideratiojt of Motion or Velocity applicable to every Purpose^ to which Fluxions have been or can be applied. By James Glenie, Esq., M.A. and F.R.S. London, 1793. According to G. Vivant! (see M. Cantor's Vorlesungen uber Geschichte der Mathematik, voi. iv, Leipzig, 1908, p. 667), James Glenie (1750-1817) was an artillery officer in the war of the American Revolution, later professor of mathematics in the military school of the East India Company. 236 LIMTTS AND FLUXIONS be, applied ; and having taken notice of the same Method, in a small Performance, written in Latin, and printed the i6th of July, 1776, I now proceed to fulfil my promise with as much conciseness as perspicuity and precision will admit of." In his A?itecedental Calculus^ p. io, he says of Newton : * I am perfectly satisfied, that had this great Man, discovered the possibility of investigating a general Geometrica! Method of reasoning, without introduc- ing the ideas of Motion and Time, ... he would have greatly preferred it, since Time and Motion have no naturai or inseparable connection with pure Mathematics. The fluxionary and differential Caculi are only branches of general arithmetical proportion." Glenie speaks (p. 3) of '' the excess of the magni- tude, which has to B a ratio having to the ratio of A + N to B the ratio of R to Q (when R has to O any given ratio whatever), above the magnitude, which has to B a ratio having to the ratio of A to B the same ratio of R to 0, is geometrìcally expressed by " a complicated fraction vvhose denominator is B(R-Q)/Q^ and whose numerator is the result of ex- panding by the binomial theorem (A -f N)^^^ and then subtracting A^^^ therefrom. A similar expression is given for the case in which A — N takes the place of A + N: '*The excess of the magnitude, which has to B a ratio, having to the ratio of A to B the ratio of R to Q, above the magni- tude, which has to B a ratio, having to the ratio of A — N to B the ratio of R to Q, is geometrically ATTEMPTS AT ARITHMETISATION 237 expressed by " a fraction whose denominator is g(R-Q)/Q^ and whose numerator is obtained by ex- panding and simplifying Ai^/^-(A - N)R/Q. *'But if A + N and A — N stand to B in relations nearer to that of equality than by any given or assigned magnitude of the same Kind, these general expres- sions become R/0 . A^^-QVQ. N-fB^^-QVQ. This I cali the antecedental of the magnitude which has to B such a ratio as has to the ratio of A to B the ratio of R to Q. Now if N the antecedental of A be denoted by À or A . . . [and] if Q = i and "^AÀ R = 2, 3, 4, 5, etc. , this expression gives 3A^A B* B respectively. " For the ''antecedent A 2AÀ A— he finds the '* antecedental " or 2 A B B 2MÀ H-*^ T. (putting M for A — B). Glenie shows that at a point of a curve the antecedentals of the ab- scissa, ordinate and curve, are as the sub-tangent, the ordinate and the tangent, respectively. Glenie's calculus involves extremely complicated identities of ratios and examines the antecedents of ratios having given consequents. The style of ex- position is poor. In deriving the antecedentals, Glenie quietly drops out ali the terms in the numerator that involve powers of N higher than the first power. As this calculus plays no part in the later history of fluxions, we shall give only one more quotation ; it relates to the Binomial Theorem 238 LIMITS AND FLUXIONS (not used by him in the development of his funda- mental formulas). He says (p. ii) : '* It may not perhaps be improper to add, that, if to the ex- pressions delivered above for the excess of the magnitude, which has to B a ratio, having to the ratio of A + N to B, the ratio of R to O, above the magnitude, which has to B a ratio, having to the Ratio of A to B the same ratio of R to O ; and for the excess of the magnitude, which has to B a ratio, having to the ratio of A to B the ratio of R to Q, above the magnitude, which has to B a ratio, having to the ratio of A — N to B the ratio of R to O, be prefixed the magnitude, which has to B a ratio, having to the ratio of A to B the ratio of R to Q, we get a geometrical Binomial, of which, when it is supposed to become numerical, the famous Binomial Theorem of Sir Isaac Newton is only a particular case. " Reuiarks 206. The classic treatment of fluxions in Great Britain, during the eighteenth century, rests prim- arily on geometrical and mechanical conceptions. Attempts to found the calculus upon more purely arithmetical and algebraical processes are described in this chapter. Ali these attempts are either a com- plete failure or so complicated as to be prohibitive. Easily the ablest among these authors was John Landen. De Morgan says of his Analysis^: " It is the limit of D'Alembert supposed to be attaìned, ^ Penny CycloJ>iedia, Art. " Differential Calculus." ATTEMPTS AT ARITHMETISATION 239 instead of beìng a terminus which can be attained as near as we please. A little difference of algebraical suppositions makes a fallacious difference of form : and though the residuai analysis draws less upon the disputable part of algebra than the method of Lagrange, the sole reason of this is that the former does not go so far into the subject as the latter." In the same article De Morgan speaks of Kirkby's Ultimators thus : '* A something between Landen and D'Alembert, as to principle, published in 1748, was called the * Doctrine of Ultimators, containing a new Acquisi- tion, etc. , or a Discovery of the true and genuine Foundation of what has hitherto mistakenly pre- vailed under the improper names of Fluxions and the Differential Calculus. ' " CHAPTER X LATER BOOKS AND ARTICLES ON FLUXIONS Encyclopcedia Britannica^ ^77^, 1/79) ^797 207. The article ''Fluxions" in the first edition of the Encyclopcedia Britannica, Edinburgh, 1771, gives this definition : "The fluxion of any magni- tude at any point is the increment that it would receive in any given time, supposing it to increase uniformly from that point ; and as the measure will be the same, whatever the time be, we are at liberty to suppose it less than any assigned time." The fluxion of a rectangle is the increment, with the small rectangle at the corner omitted ; the latter *' is owing to the additional velocity wherewith the parallelo- gram flows during that time and therefore is no part of the measure of the fluxion." "The incre- ment a quantity receives by flowing for any given time, contains measures of ali the different orders of fluxions; for if it increases uniformly, the whole in- crement is the first fluxion ; and it has no second fluxion. If it increases with a motion uniformly accelerated, the part of the increment occasioned by the first motion measures the first fluxion, and the 240 LATER BOOKS AND ARTICLES 241 part occasioned by the acceleration measures the second fluxion. ..." The same article is reprinted in the second edition (1779) and the third edition (1797). Robert Thorp, lyyy 208. Thorp made a translation of part of New- ton's Principia.'^ In the ** advertisement " we read : '* The doctrine of prime and ultimate ratios . . . is established, so as to remove the various objections which have been raised against it, since it was first published. To the relations of finite quantities alone the reasoning on this subject is confined." The translation of quantitates quam mininice, evanescentes^ ultimce^ in- finite rnagnce^ and the like, has not been literal, yet they are " explained in that sense under which the author cautions his readers to understand them. This is the more necessary, as the terms infinite^ infinitesimal, least possible^ and the like, have been grossly misapplied and abused." 209. In the Commentary to Lemma i in Sect. i of Bk. I in the Principia, Thorp says : * * The prime and ultimate ratios of magnitudes . . . are investi- gated by observing their finite increments or decre- ments, and thence finding the limits of the ratios of those vai'iable magnitudes ; not the ratios to which the magnitudes ever actually arrive (for ^ Mathematical Principks of Naturai Philosophy. By Sir Isaac Newton, Knight. Translated into English, and illiistrated with a Commentary, by Robert Thorp, M.A., voi. i, London, 1777. 16 242 LIMITS AND FLUXIONS they are never, strictly speaking, either prime or ultimate in fact), but those limits to which the ratios • of magnitudes perpetually approach ; which they can never reach, nor pass beyond ; but to which they appear nearer than by any assignable difìference. " . . . " We now proceed to explain this Lemma more particularly than perhaps might seem necessary, had it not been much controverted, mis- represented, and misunderstood. " As one of the conditions of the proposition, Thorp states, is " that quantities and the ratios of quantities must con- tinually tend to equality. The one must never become equal to, nor pass beyond the other : their difference must never either vanish to nothing, or become negative." In this restriction Thorp goes even further than had Robins. The following passage from Thorp's commentary is thoroughly in the spirit of Robins :**... That we may not be led, from the expression ultiniately equal, to suppose, that there is an ultimate state, in which they are actually equal, we are cautioned in the scholium at the end of this Section [of Principia^ Bk. I, Sect. i] in these words, The ultimate ratioSy in which quantities vanishy are not in reality the ratios of ultimate quantities ; but the limits to which the ratios of quantities continually decreasing always approach ; which they never can pass beyond ^ nor arrive at^ unless the quantities are continually and indefinitely dimin- ished, According to Thorp, the inscribed or cir- cumscribed polygon can never arrive at the curve. He quotes from Saunderson's Fluxions. By the LATER BOOKS AND ARTICLES 343 doctrine of indivisibles there *'has been introduced into mathematica! reasoning ali that absurd jargon concerning quantities infinitely great, and infinitely little, which has been so much objected to by mathe- maticians. And, though it has often been elegantly applied by some able geometers to the demonstra- tion of many noble theorems ; yet in the hands of less accurate reasoners, it has often led to false conclusions" (p. 71). F. Holliday, 1777 210. In a somewhat lengthy preface to his Intro- duction to Fluxions^ the author tells that, when in 1745 he was in London, in company with W. Jones and De Moivre, they expressed great approbation of Emerson's Fluxions^ with regard to the method of treatment, but thought his book too high for beginners. The author tries to be more diffuse in the laying down of first principles. He derives the fundamental results in two ways : first, by the aid of nascent or evanescent quantities, as suggested by Newton's Principia; second, ''without using any infinitely small quantities, or vanescent Parallelo- grams, which perhaps will be more acceptable to many of my Readers. " Holliday explains at great length the Scholium (see our §§ 10-15) on prime and ultimate ratios, and gives a short account of the invention of fluxions as given in the review of ^ An hitroduction to Fluxions^ Designedfor the Use, and Adapted to the Capacities of Beginners. By the Reverend F. Holliday, Vicar of West Markham and Bothamsall, Nott's. London, 1777. 244 LIMITS AND FLUXIONS CoUins's Commercium Epistolicurn in the Philosophical TransactionSy 17 17. Though following Newton closely, variations were bound to arise. Thus, Holliday says (p. 73), *' Fluxions are not magnitudes but the velocities with which magnitudes, varying by a continuai motion, increase or decrease." It cannot be claimed that HoUiday made any contribution to the philosophy of fluxions, nor even that he profited as much as he might by the refinements in the logie which had been made by English writers since the time of Newton. Charles Hutton, 1796, 1798 211. In his M athematical Dictionary^ London, 1796, Charles Hutton makes reference to the advantage of Simpson's definition of a fluxion as a magnitude uniformly generated in a finite time, the imagination being now no longer confined to a single point and to the velocity at that point ; moreover, * ' higher orders of Fluxions are rendered much more easy and inteUigible. " 212. From the part on fluxions in Hutton's Course of Mathematics ^ we take the following : * * The rate or proportion according to which any flowing quantity increases, at any position or instant, is the Fluxion of the said quantity, at that position or instant : and it is proportional to the magnitude by which the flowing quantity would be uniformly increased, in a given time with the ^ A Coìirseof Mathematics. By Charles Hutton. London, 4th ed., 1803-1804, voi. ii, p. 279. [First ed., 1798-1801.] LATER BOOKS AND ARTICLES 245 generating celerity uniformly contìnued during that time." *'. . . If the motion of increase be accelerated, the increment so generated, in a given finite time, will exeeed the fluxion : . . . But if the time be indefinitely small, so that the motion be considered as uniform for that instant ; then these nascent increments will always be proportional, or equal, to the fluxions, and may be substituted instead of them in any calculation." The fluxion oi xy is derived in two ways : the first by the method of considering the rectangle composed of two parts, as previously expounded by Rowe. The second method finds algebraically the incre- ment xy +yx'' +xy, " of which the last term x'y is nothing, or indefinitely small, in respect of the other two terms, because x' and y are indefinitely small in respect of x and y. . . . Hence, by substitut- ing X and j> for x' and y, to which they are propor- tional, there arises xy-\-yx for the true value of the fluxion oi xy." S. Vince, 1795, 1805 213. Vince's Principles of Fluxions appeared in 1795 as the second volume of the Principles of Mathematics and Naturai Philosophy in Four VolumeSy^ which were brought out under the general editorship of James Wood. A second ^ The Principles of Mathematics and Naturai Philosophy in Four Volumes. Voi. II, The Principles of Fluxions : Designed for the Use of Students in the University. By the Rev. S. Vince, A.ÀI., F.K.S., Cambridge, 1795, 246 LIMITS AND FLUXTONS edition of Vince vvas printed in 1805. From thìs second edition we quote : P. I : *'The velocities with which flowing quan- tities increase or decrease at any point of time, are called ih.e. Jluxìons of those quantities at that instant. " As the velocities are in proportion to the increments or decrements uniformly generated in a given time, such increments or decrements will represent the fluxions. "^ Vince also quotes Newton on the generation of quantities by motion : '*Sir I. Newton, in the Introduction to his Quadrature of Curves^ observes that ' these geneses really take place in the nature of things, and are daily seen in the motion of bodies. And after this manner, the ancients, by drawing moveable right lines along immoveable right lines, taught the geneses of rectangles.'" Vince gives no formai definition of a //;;/// ; but his philosophy of this subject is disclosed by the two following quotations (pp. 4 and 5): " By keeping the ratio of the vanishing quantities thus expressed by finite quantities, it removes the obscurity which may arise when we consider the quantities themselves ; this is agreeable to the reasoning of Sir I. Newton in his Principia^ Lib. I, Sect. I, Lem. 7, 8, 9." " It has been said, that when the increments are ^ "This is agreeable to Sir I. Newton's ideas on the subject. He says : ' I sought a method of determining quantities from the velocities of the motions or increments with which they are generated ; and call- ing these velocities of the motions or increments, fluxions^ and the generated quantities fluents, I fell by degrees upon the method of fluxions.'— Introd. to Quad. Curves" LATER BOOKS AND ARTICLES 247 actually vanished, it is absurd to talk of any ratio between them. It is true ; but we speak not here of any ratio then existing between the quantities, but of that ratio to which they have approached as their liniit ; and that ratio stili remains. Thus, let the increments of two quantities be denoted by ax^ + inx and bx^ + nx ; then the limit of their ratio, when ;r = o, is m : n \ for in every state of these quantities, ax'^-\-mx : bx^-{-nx : : ax-{-7n : òx + n : : (when x=o) m : n. As the quantities therefore approach to nothing, the ratio approaches to that oi m \ n as it's limit. We must therefore be careful to distinguish between the ratio of two evanescent quantities, and the limit of their ratio ; the former ratio never arriving at the latter, as the quantities vanish at the instant that such a circumstance is about to take place." By aid of the binomial theorem, Vince finds the fluxion of ;ir", when the fluxion of ;ir is given ; he then finds the fluxion of xy by considering {x •\- yf = x'^ -\- 2xy+y'^y by which the fluxion of 2xy can be found in terms of the fluxions (x-^yY, x^ aindy^. Agnesi — Colson — Hellins^ 1801 214. The Analytical Institutions^ is the first cal- culus that was written by a woman. The authoress ^ Analytical Institutions , in four books : Originally written in Italian, by Donna Maria Catana Agnesi, Professor of the Mathe- maticks and Philosophy in the University of Bologna. Translated into English by the late Rev. fohn Colson, A/. A., F.P.S., and Lticasian Professor of the Mathematicks in the University of Cambridge. Now first printed,from th» Translators Manuscript , under the inspection of the Rev.fohn Hellins, B.D., F.R.S. Vols. i and il. London, rSoi. 248 LIMITS AND FLUXIONS is the noted Maria Gaetana Agnesi, of the University of Bologna. The Italian originai was first published at Milan in 1748. The two volumes of the translation were printed at the expense of Baron Maseres. In an introduction, Hellins points out that Colson hoped to interest the ladies of England in the study of fluxions by his translation of the work of the great Italian lady, " And, in order to render it more easy and useful to the Ladies of this country, ... he [Colson] had designed and begun a popular account of this work, under the title of The Pian of the Lady's System of Analyticks ; explaining, article by article, what was contained in it. But this he did not live long enough to finish." 215. Colson dealt wìth Agnesi's work somewhat as Stone had dealt with that of De L'Hospital, inasmuch as both translators substituted the nota- tion of Newton in place of that of Leibniz. The word fluxions (''flussioni") occurs in the originai Italian of Agnesi's masterly work. How Colson's conscience may have troubled him, when a fluxion stood out in his translation as something " infinitely little," may be judged when we consider that in 1736 he brought out an English translation, with an extensive comment, of Newton's Method of Fluxions. With Newton a fluxion always meant a velocity. We quote a few passages from Colson's Agnesi (voi. ii, pp. 1,2): "The Analysis of infinitely small Quantities, LA TER BOOKS AND ARTICLES 249 which is otherwise called the Differential Calculus^ or the Method of Fluxions^ is that which is con- versant about the differences of variable quantities, of whatever order those differences may be. " "Any infiiiitely little portion of a variable quantity is called it's Difference or Fluxion ; when it is so small, as that it has to the variable itself a less pro- portion than any that can be assigned ; and by which the same variable being either increased or diminished, it may stili be conceived the same as at first." On p. 3 we read that certain lines in a figure " will be quantities less than any that can be given, and therefore will be inassignable ^ or differentials^ or infinitesimals^ or, finally, fluxions. Thus, by the common Geometry alone, we are assured that not only these infinitely little quantities, but infinite others of inferior orders, really enter the composi- tion of geometrical extension." "These propositions," says a reviewer^ of the translation, '*may appear exceptionable, in point of language, to the rigorists in geometry ; but they are nevertheless founded on good principles, and furnish rules for the comparison of evanescent quantities, which will prove safe guides in investigation. The demonstrations appear to us to be perfectly sound (if the word infinite be taken in its true sense, as denoting merely the absence of any limit), with the exception, perhaps, of the first theorem, which (as is not a little curious to remark) is liable to the * Edinburgh Review^ voi. iii, 1805, p. 405, 2 50 L1MTTS AND FLUXIONS same objection that has been macie of ^the lemma of Newton's Principia. In both instances, also, the error is rather apparent than real. " The first theorem in question states that the two intersecting perpendì- culars to a curve drawn at the ends of *' an infìnitely little portion of it of the first order," *'may be assumed as equal to each other." We wonder what Robins and Maclaurin would bave thought, had they been alive in i8oi and 1805, and read these defini- tions and comments ! What horrible visions would these ghosts of departed quantities bave brought to Bishop Berkeley, had he been alive ! But the nine- teenth century was destined to bring back to British soil stili greater accentuations of infinitesimals. T. Newton, 1805 216. The Rev. T. Newton says in the preface of bis lllustrations of Sir Isaac Newton's Method : ^ * ' Every Mathematician now considers the whole doctrine of Prime and Ultimate Ratios in no other light, than as a Doctrine of Limits. " Young readers of Sir Isaac Newton's Principia encounter difficulties because commentators bave made ''use of the terms of Indivisibles, in their explana- tions ; . . . Newton expressly says, that by the ultimate ratios of quantities he means the ratios of their limits. 2 And when he wants to infer the ^ An Illitstration of Sir Isaac Newton^ s Method of Reasonitì}(. By Prime and Ultimate Ratios. By the Rev. T. Newton, Kector of Tewin, Herts ; late Fellow of Jesus College, Cambridge. Leeds, 1805. '^ Seeour§§ 12, 15. LATER BOOKS AND ARTICLES 251 equality of inequality of those limits from some relation of the variable quantities, which are never supposed absolutely to reach their limits, it cer- tainly requires something more than a definition to shew this. ... It is not my intention to detain the reader, with answering the objections of the Arialyst and his foUowers, because it has been already done by others in a satisfactory manner. . . . Notwithstanding the assertions of some modem writers, the method of ultimate ratios is extremely perspicuous, strictly logicai, and more concise than any other of modem invention ; . . . it neither involves the strange notion, that a straight line may be a part of a curve, and a piane superficies a part of a concave or convex one ; nor the unintelligible idea of adding and subtracting indivisibles, or inconceivably small magnitudes. Whatever magnitudes are compared, according to this method, they are always supposed to be finite." T. Newton begins with the following two defini- tions (p. i) : '*If a variable quantity, either increasing or de- creasing, approaches to a fixed quantity, the differ- ence between them being continually diminished, so as at length to become less than any assignable quantity ; the fixed quantity is called the Limit of the variable quantity." '' If the ratio of two variable quantities continu- ally approaches to a fixed ratio, so as to come nearer to it than by any assignable difference ; the 252 LIMITS AND FLUXIONS fixed ratio is called the Limiting Ratio of the variable quantities. " William Dealtry, 1810, 18 16 217. In the preface of Dealtry's Principles of Fluxions ^ ( 1 8 1 6) we read : "The method of Fluxions rests upon a principle purely analytical ; namely, the theory of Hmiting ratios ; and the subject may therefore be considered as one of pure mathematics, without any regard to ideas of time and velocity. But the usuai manner of treating it is to employ eonsiderations resulting from the theory of motion. This was the pian of Sir Isaac Newton in first delivering the principles of the method ; and it is adopted in the foUowing Work, from the belief, that it is well adapted for illustration." Dealtry defines a ''fluxion of a quantity at any point of time" as **its increment or decrement, taken proportional to the velocity with which the quantity flows at that time." . . . " When a quantity increases with a velocity which continually varies, the quantity, which measures the fluxion, is a limit between the preceding and succeeding increments, and is ultimately equal to either of them. " He explains that *'the word ultimately is intended to denote that particular instant, when the time is diminished sine limite^''' ^ The Principles of Fluxions : Designed for the Use of Students in the Universities. By William Dealtry, B.D., F.R.S., late Fellow of Trinity College, Cambridge. 2nd ed., Cambridge, 1816. LATER BOOKS AND ARTICLES 253 and quotes Newton's Scholium, Sect. i, in the Prin- cipia. He points out, also, that if x increases uni- formly, x'^ increases with accelerateci velocity, and the part of the increment x'^ is the efifect of the acceleration, and therefore, by his definition of fluxion, to be " omitted in taking the fluxions " (P. 8). New Editions, 1 801- 1809 218. William Davis, who was a bookseller in London and editor of the Companion to the Gentle- man's Diaryy appears also as the editor of new editions of three dififerent texts on fluxions. In 1801 he saw through the press the second edition of Maclaurin's Treatise of Fluxions \ in 1805 the third edition of Thomas Simpson's Doctrine and Application of Fluxions. In 1809 appeared the fourth edition of John Rowe's Doctrine of Fluxions^ revised **by the late William Davis." Remarks 219. Among some of the authors of this period there is less concern than among writers of former years about the attainment of the rigour of the ancients. Perhaps the effects of the revival of the ideals of Euclid and Archimedes which followed the publication of the Analyst were gradually subsiding. It would not be fair to this age to judge its mathe- matical status altogether by the authors which we bave selected. There was a movement under way 254 LIMITS AND FLUXIONS at this time which is reflected in the literature that will be under consideration in the next chapter. Both before the time of Berkeley's Analyst and after the time of Maclaurin's Fluxions there appeared in Great Britain texts which superposed British symbols and phraseology upon the older Continental concepts. The result was a system, destitute of scientific interest. Newton's notation was poor and Leibniz's philosophy of the calculus was poor. That result represents the temporary survival of the least flt of both systems. The more recent international course of events has been in a diametrically opposite direction, namely, not to superpose Newtonian symbols and phraseology upon Leibnizian concepts, but, on the contrary, to superpose the Leibnizian notation and phraseology upon the limit-concept, as developed by Newton, Jurin, Robins, Maclaurin, D'Alembert, and later writers. CHAPTER XI CRITICISMS OF FLUXIONS BY BRITISH WRITERS UNDER THE INFLUENCE OF D'ALEMBERT, LAGRANGE, AND LACROIX Review of Lagrange's '■^ Fonctions analytiques^'' 1799 220. Important is a review ^ of Lagraiige's Théorie des fonctions analytiques^ which, as is well known, is an attempt to deduce the principles of the calculus, diverted of ali reference to infinitely small or evan- escent quantities, limits or fluxions, and reduced to the algebra of finite quantities. The reviewer gives a general criticism of the methods of fluxions and the difìferential calculus. He discusses the principle of motion : '' It will not be denied that this principle is introduced purely for the purpose of illustration, . . . on th^ ground of convenience. . . . The mathe- matica! principle, on which the doctrine of fluxions depends, is a definition . . . and fluxions were defined to be velocities. . . . Novv velocity is nothing real, but is only the relation between the space described and the time of describing it ; — of which relation we have a clear idea when the motion is uniform." The reviewer continues : *' In variable ^ Monthly Review, London, voi. xxviii, 1799, Appendix. 255 256 LIMITS AND FLUXIONS motion, however, we inquire what velocity is ; and here it is defined to be the relation between the space which would be described were the motion continued uniform from any point, and the time. Stili difficulties remained ; this definition might con- vey to the mind a general idea of the nature of velocity, but was of no mathematical use, since the space which would be described could not be immedi- ately ascertained and determined. Another step was therefore to be made, and which was made by establishing this proportion ; if V be the velocity, S the space, which would be described, and T the time, S' the space really described, and T' the S corresponding time; then V = ^ = ultimate rjatio of S' , when S' and T' are indefinitely diminished. " Again he says : *' On the ground of perspicuity and evidence, the understanding is not much assisted by being directed to consider ali quantity as generated by motion ; . . . when such quantities as weight, density, force, resistance, etc. , become the object of inquiry . . . then the true end of the figurative mode of speech, illustration, is lost. . . . That which happened to Aristotle has happened to Newton ; his foUowers have bowed so implicitly to his authority, that they have not exercised their reason. The method of fluxions had never so acute, so learned, and so judicious a defender as Maclaurin : — yet who- ever consults it . . . finds the author speaking of CRiriCISMS BY BRITISH WRITERS 257 ' causes and effects,' of 'the springs and principles of things,' and proposing to deduce the 'relation of quantities by comparing the powers which are conceived to generate them ' ; — will be convinced that this could only happen from so able a mathe- matician having failed to seize the right principles." ** If English mathematicians first adopted Newton's method from veneration to him, . . . they have since persevered in it (we may almost say) against conviction. " The reviewer claims that the criticisms of D'Alembert, Torelli, and Landen have shown that the use of motion is unnecessary and unreal. We have given citations from Landen in an earlier chapter (see our §§ 202, 203). D'Alembert is quoted as saying fifty years previous : ** Introduire ici le mouvement, c'est y introduire une idée étrangère, et qui n'est point nécessaire à la démonstration : d'ailleurs on n'a pas d'idée bien nette de ce que c'est que la vitesse d'un corps à chaque instant, lorsque cette vitesse est variable. La vitesse n'est rien de réel ; . . . c'est le rapport de l'espace au tems, lorsque la vitesse est uniforme ; . . . Mais lorsque le mouvement est variable, ce n'est plus le rapport de l'espace au tems, c'est le rapport de la différentielle de l'espace à celle du tems ; rapport dont on ne peut donner d'idée nette, que par celle des limites. Ainsi, il faut nécessairement en revenir à cette dernicre idée, pour donner une idée nette à^s fiuxiofis,'''^ ^ Art. "Fluxion" in Encyclopédie^ oti Didionnaire raisonné des Sciences^ etc, t. 6, Paris, 1756. 17 258 LIMITS AND FLUXIONS 221. The reviewer states that foreign mathe- maticians have written treatises in which motion is entirely excluded, "and in some of these treatises, the principles of the doctrine in question have been laid down with a considerable degree of evidence and exactness." The Residuai Analysis of Landen rests on *'a process purely algebraical : but the want of simplicity . . . is a very great objection to it." The reviewer is of the opinion that Euler and D'Alembert give " the most clear and precise notions of the principles on which the differential calculus is established. " He refers to Euler's Institutiones calculi differentialis, 1755. D'Alembert, says the re- viewer, *' observes that the method is really founded on that of prime and ultimate ratios, or of limits, which latter method is only an algebraical transla- tion of the former ; that, in fact, there are no such things as inftnitely small quantities ; and that, when such quantities are mentioned, it is by the adoption of a concise mode of speech for the purpose of simplifying and abridging the reasoning ; — that the true object of consideration is the limit of the ratio of the finite differences of quantities." The reviewer continues : ''The explanations given by Euler and D'Alembert, beyond ali doubt, deserve much consideration, yet their method of consider- ing the doctrine of fluxions is not completely satis- factory, but is objectionable on two grounds : first, that we have no clear and precise notion of the ratio of quantities, when those quantities are in their vanishing state, or cease to be quantities ; CRITICISMS BY BRITISH WRITERS 259 secondly, the connection and naturai order of the Sciences are interrupted, if we give a di^tinct and independent origin to that which in fact, is a branch of analysis derived from the same common stock, whence ali the other branches are deduced. " Then follows a sympathetic account of the foundations for the calculus laid by Lagrange in his Thcorie des fonctions analytiques^ ^798- In passing, the reviewer remarks that ** Emerson, Stone, Simpson, Waring, etc. , have published treatises on fluxions ; in none of which, however, are the principles clearly laid down." Review of a Memoir of Stockler, 1799 222. In the same journal^ there is a review of a memoir on fluxions written by the Portuguese mathematician, Garcao Stockler, who modifies the explanation of fundamentals by the introduction of a '' hypothetical fluxion " (a uniform velocity that generates a quantity equal to the real increment generated during the actually variable motion), which is always contained between the proper fluxions at the first and second instant under consideration. By diminishing the interval of time, the hypothetical fluxion approaches the true fluxion more nearly than by any assignable quantity. Here also, the real object of consideration is the liinit. The reviewer argues that the fundamental principles are not new, and that the objections to Newton's fluxions apply equally to those of Stockler. In a reply to ^ Monthly Review ^ voi. xxviii, London, 1799, p. 571. 200 LIMirS AND FLUXIONS the Monthly Review, Stockler denies the reviewer's allegation that he [Stockler] supposed quantity to be generated by motion. *'The idea of motion, and the idea of velocity, are too particular to be admitted into a general theory of fluent quantities. " ^ Review of Lacroix's '^ Calcul dìfférentiel" 1800 223. A review of S. F. La Croix's Traìté dii calcul différentiel'^ served as the occasion of further comments and criticisms of fundamental concepts : '' Who would direct his ridicule against the refine- ments, subtleties, and trifling of the schoolmen, if he read what has been written by some men who were presumed to be the greatest masters of reason, and whose employment and peculiar privilege consisted in deducing truth by the justest inferences from the most evident principles ? The history of the differential calculus, indeed, shows that even mathe- maticians sometimes bend to authority and a name, are influenced by other motives than a love of truth, and occasionally use (like other men) false meta- physics and false logie. No one can doubt this, who reads the controversial writings to which the inven- tion of fluxions gave rise : he will there find most exquisite reasonings concerning quantities which survived their grave, and, when they ceased to exist, did not cease to operate ; concerning an in- finite derivation of velocities, — and a progeny of ^ Monthly Review, voi. xxxii, p. 497. ^ Monthly Review^ voi. xxxi, London, 1800, p. 493. CRI TI CTS MS BY BRITISH WRITERS 261 infinitesimals smaller than the * moonshine's wat'ry beams,' and more numerous than ' Autumnal leaves that strow the brooks In Vallombrosa.' (Milton, Par. Lost^ i, 302.) **The contemporaries and partizans of Newton were men infinitely inferior to him in genius, but they had zeal, and were resolved to defend his opinions and judgments. Hence they undertook the vindication of fluxions, according to the principles and method of its author ; although it may be fairly inferred, from the dififerent explanations given of that doctrine by Newton in dififerent parts of his Works, that Newton himself was not perfectly satis- fied of the stabiHty of the grounds on which he had established it." The reviewer quotes (p. 497) from Lacroix's preface : **These notions [velocities, motions], although rigorous, are foreign to geometry, and their appHca- tion is difficult. . . . Properly speaking, fluxions were to him [Newton] only a means of giving a sensible existence to the quantities on which he operated. The advantage of the method of fluxions over the differential calculus in , point of meta- physics, consists in this ; that, fluxions being finite quantities, their moments are only infinitely small quantities of the first order, and their fluxions are finite ; by these means, the consideration of in- finitely small quantities of superior orders is avoided. ... I can only mention a method which Landen gave in 1758, to avoid consideration of infinity of 262 LIMITS AND FLUXIONS motions, or of fluxions, since it rests on a very elegant algebraic theorem which cannot be given in a work of this nature. The freedom with which Landen divests himself of national prejudice stamps a remarkable character on his work ; he is perhaps the only English mathematician, who has acknow- ledged the inconvenience of the method of fluxions." . . . *'We can always descend from the function to the differential coefiìcient or from the primitive function to the derived function : but, generally speaking, the reverse step is attended with the greatest difficulty." **The rivals of Newton thought and invented for themselves ; had they been influenced by his authority, and devoted their talents to the perfec- tion of synthesis, science must have been con- siderably retarded. To the improvement of the algebraical analysis, is to be attributed the amazing advances of physical astronomy."^ Review of Carnofs ^ ^ Réflexiofts ,'' i8oi 224. In the Montili)' Review'^ (London) for 1801 there is a short and unimportant account of Lazare N. M. Carnot's new book, Réflexions sur la viéta- physique du calcul infinitésininl^ 1797- Carnot explains the correctness of results obtained by the infinitesimal calculus of Leibniz on the theory of compensation of errors — a theory which had been ^ Monthly Review, voi. xxxii, p. 491. 2 Monthly Review^ voi. xxxiv, 1801, p. 463. CRITICISMS BY BRITISH WRITERS 263 advanced much earlier by Berkeley in his Analyst. Mr Philip E. B. Jourdain has found clear indica- tions of this theory in Maclaurin's Fluxions and in Lagrange's Théorie des fonctions analytiques. The method of limits is explained by Carnet in the manner of D'Alembert. ''Of fluxions, indeed," says the reviewer, " as founded on the strange basis of velocity, there is no account." Robert Woodhouse^ 1803 225. In 1803, Robert Woodhouse published his Principles of A nalytical Calculation, ^ Woodhouse had graduated B. A. at Caius College, Cambridge, in 1795, as senior wrangler. He then held a scholar- ship and a fellowship at Caius College, devoting himself to mathematics. He has the distinction of being the first to strongly encourage the study in England of the mathematical analysis which had been created on the Continent by Swiss and French mathematicians. In his Principles of Analytical Calculation he discussed the methods of infinitesi- mais and limits, and Lagrange's theory of function, pointing out the merits and defects of each. '' By thus exposing the unsoundness of some of the Continental methods, he rendered his general support of the system far more weighty than if he had appeared to embrace it as a blind partisan. "^ 226. The ideas set forth in this book are, on the ^ The Princif^les of Analytical Calculation^ by Robert Woodhouse, A.M., F.R.S. Caniliridge, 1803. "^ Art. " Woodhouse, Robert," in Sidney Lee's Dictionary of National Biography. 264 LTMITS AND FLUXIONS whole, in such dose agreement with those advanced in the preceding reviews, that the query naturally arises, whether Woodhouse is not the author of those reviews. We have reached no final decision on this point. In the preface Woodhouse passes in review the difìferent methods of establishing the foundations of the calculus. He criticises the use of motion in the proof of the binomial and other related theorems. '' It required no great sagacity to perceive, that a principle of motion, introduced to regulate processes purely algebraical, was a foreign principle." If the binomial theorem and related theorems for the development of a function be established by algebra, independently of motion, then * * from the second term of this expansion, the fluxion or differential of a quantity may be immediately deduced, and in a particular application, it appears to represent the velocity of a body in a motion. The fluxionists pursue a method totally the reverse ; they lay down a principle of motion as the basis of their calculus, thence deduce some of the first processes, and establish the binomial theorem, by which it is said, the extraction of roots may be effected. . . . The project of extracting the square and cube roots of algebraical quantities by a principle of motion, is surely revolting to the common sense." ** Of his own method. Newton left no satisfactory explanation : those who attempted to explain it, according to what they thought the notions of its author, and . . . by reasoning which fairly may be CRITICISMS BY BRITISH WRITERS 265 called tedious and prolix. Of the commentators on the method of fluxions, Maclaurin is to be esteemed most acute and judicious, but his Introduction exhibits rather the exertions of a great genius struggling with difficulties, than a clear and distinct account of the subject he was discussing." To remove this prolixity, it was proposed, conformably to the notions of Newton, to cali in the doctrine of prime and ultimate ratios or of limits. Euler and D'Alembert, on the other hand, rejected motion, but retained limiting ratios, failing, however, in supply- ing a satisfactory explanation therefor. Wood- house is the earliest English mathematician who speaks in respectful and appreciative terms of the services to mathematics rendered by Bishop Berkeley. In fact, Woodhouse admits as valid some of Berkeley's objections which had been declared invalid. The methods of treating the calculus **all are equally Hable to the objection of Berkeley, concerning the fallacia suppositionis, or the shifting of the hypotJiesis.'' Thus, in fluxions and the method of limits, x is increased by /, and, in the case of x'"^ the increment of the function, divided by /, is inx"'~^-\ — ^^ ^;ir'"~VH-, etc. ; then, z putting, /=o, there results mx"'~^. But since the expansion of {x-^-i)'" was effected '*on the express supposition, that i is some quantity, if you take /=o, the hypothesis is, as Berkeley says, shifted, and there is a manifest sophism in the process " (p. xii). 266 LIMITS AND FLUXIONS 227. As another objection to limits, or prime and ultimate ratios, Woodhouse declares that * ' the method is not perspicuous, inasmuch as it considers quantities in the state, in which they cease to be quantities." Moreover, "the definition of a limit, is neither simple nor concise" (p. xvii). ' ' The name of Berkeley has occurred more than once in the preceding pages : and I cannot quit this part of my subject without commending the Analyst and the subsequent pieces, as formlng the most satisfactory controversial dis- cussion in pure science, that ever yet appeared : into what perfection of perspicuity and of logicai pre- cision, the doctrine of fluxions may be advanced, is no subject of consideration : But, view the doctrine as Berkeley found it, and its defects in metaphysics and logie are clearjy made out. If, for the purpose of habituating the mind to just reasoning ... I were to recommend a book, it should be the Analyst.'' "The most diffuse and celebrated antagonists of Berkeley, are Maclaurin and Robins, men of great knowledge and sagacity : but the prolixity of their reasonings confirms the notion, that the method they defend is an incommodious one. " " Landen, I believe, first considered and proposed to treat the fluxionary calculus merely as a branch of Algebra : After him, M. Lagrange, a name ever to be celebrated, in the Berlin Acts for 1772, laid down its analytical principles ; and subsequently in his Théoric des fo7ictio)is mialytiques^ 1796, he has resumed the subject : in this treatise, the author CRITICISMS BY BRTTISH WRITERS 267 expressly proposes, to lay down the principles of the differential calculus, independently of ali considera- tion of infinitely small, or vanishing quantities, of limits, or of fluxions " (p. xviii). While Wood- house considers Lagrange's discussion as very valu- able, he does not find it free from logicai faults. William Hales, 1804 228. As a protest against the new movement and a vindication of Newton from the attacks upon fluxions in the Monthly Review^ William Hales pre- pared a hook, the Analysis F/uxionum, which was published in Maseres' Scriptores Logarithniici^ voi. v, London, 1804. Hales endeavours to show that the doctrine of prime and ultimate ratios is really the same as the doctrine of the limits of the ratios. Hales's fundamental definitions are : '' Rationes ultimai sunt limites, ad quos quanti- tatum sine fine decrescentium rationes, i, semper appropinquant ; et, 2, quas propiùs assequi possunt quàm prò data quàvis differentià ; 3, nunquam vero transgredi ; 4, nec priùs attingere, quam quantitates ipsai diminuuntur in infinitum." "Momentum est fluentis augmentum aut decre- meniuiìi iìwinentaneuiii\ id est, tempore quam minimo genitum. Estque fluxioni proportionale." After Hales's work had gone to press, he became acquainted with Benjamin Robins's Discourse oi 1735, and published in appendices^ numerous extracts from ^ Maseres, Scriptores Logarif limici, voi. v, pp. 848, 854, 856. 268 LIMITS AND FLUXIONS it. Says Hales : " It is far superior indeed to the subsequent explanations of professed commentators ; and it is a high gratification to myself to find, that the mode of explanation, which I adopted of the Doctrine of Limits, is precisely the same as Robins's; long before 1 had seen his admirable treatise, which did not fall into my hands until lately, a considerable time after the publication of the Afialysis Fluxionum.'" Maseres calls the Discourse oi Robins ''the ablest tract that has ever been published on the subject." Hales's text and the appendices to it contain con- siderable historical material, consisting mainly of references to and quotations from earlier writings. In view of the testimony of Laplace, Legendre, and Lacroix on the superiority of the method of fluxions, " how was it possible," asks Hales, that the eyes of the Monthly Reviewers **could stili be so holden . . . as stili to assert, that Newton himself was not perfectly satisfied of the stability of the ground on which he had established his Method of Fluxions ! " Hales's motive in opposing Continental ideas was probably partly theological. D'Alembert, con- sidered by him a hostile critic of Newton, is called *' a philosophizing infidel," one " of the originai con- spirators against Christianity," **at once the glory and disgrace of the French Academy of Sciences," whose last words were *'a terrific contrast to the death of the Christian Philosopher," Colin Maclaurin.^ The publication of Hales's Fluxions in large ^ Maseres, Scriptores Logarithmici^ voi. v, pp. 176-182. CRITICI SMS BY BRITISH WRITERS 269 quarto form and in the Latin language, the in- clusion in the Appendix of matters foreign to the subject of the hook, together with the attempt to maintain a system of notation and mode of exposition that was beginning to be considered provincial, caused the hook to ''fall still-born from the press." Encyclopcedia Britannica, 18 io 229. In the fourth edition of the Encydopcsdia Britannica, Edinburgh, 18 io, the article "Fluxions" is wholly rewritten, and is much more extensive than the article in former editions. There is a lengthy historical introduction, and emphasis is placed upon work done on the Continent. It observes "that there is no work in the English language that ex- hibits a complete view of the theory of fluxions, with ali the improvements that bave been made upon it to the present time." Mention is made then of "several excellent works in the French language," mentioning Cousin, Bossut, La Croix, L'Huilier. Letting « be " any function " of x, the //;;/// of the ratio {u —u) \ h is defined as "a quantity to which the ratio may approach nearer than by any assignable difìference, but to which it cannot be con- sidered as becoming absolutely equal." The article asserts that the method of fluxions " rests upon a principle purely analytical, namely the theory of limiting ratios ; and this being the case, the subject may be treated as a branch of pure mathematics, 270 LIMITS AND FLUXIONS without having occasion to introduce any ideas foreign to geometry. Sir Isaac Newton, however, in first delivering the principles of the method, thought proper to employ considerations drawn from the theory of motion. But he appears to have done this chiefly for the purpose of illustration, for he immediately has recourse to the theory of limiting ratios, and it has been the opinion of several mathe- maticians of great eminence (such as Lagrange, Cousin, La Croix, etc. , abroad, and Landen in this country) that the consideration of motion was intro- duced into the method of fluxions at first without necessity, and that succeeding writers on the subject ought to have estabhshed the theory upon principles purely mathematica!, independent of the ideas of time and velocity, both of which seem foreign to investigations relating to abstract quantity." **By the fluxions then of two variable quantities having any assigned relation to each other, we are in the foUowing treatise always to be understood to mean any indefi^tite quantities which have to each other the limiting ratio of their simultaneous inci'cjuents (we . . . mean the ratio of the nuinerical values of the increments, which may always be compared with each other, whether the variable quantities be of the same kind, as both lines, or both surfaces, etc, or of different kinds, as the one a line, and the other a surface). The Newtonian notation is used in the article exclusively." CRITICI SMS BY BRITISH WRITERS 271 Lacroix's '■^ Elementary Treatise,'' 18 16 230. The translation of Lacroix's Elementary Trenti se on the Differential and Integrai Calculus ^ in 18 16 marks an important period of transition. From the '* Advertisement " we quote : This work of Lacroix '' may be considered as an abridgement of his great work on the Differential and Integrai Calculus, although in the demonstra- tion of the first principles, he has substituted the method of limits of D'Alembert, in the place of the more correct and naturai method of Lagrange, which was adopted in the former. The first part of this Treatise, which is devoted to the exposition of the principles of the Differential Calculus, was translated by Mr. Babbage. The translation of the second part, which treats of the Integrai Calculus, was executed by Mr. G. Peacock, of Trinity College, and by Mr. Herschel, of St. John's College, in nearly equal proportions." On p. 2 the process of differentiation of u^ax^ is explained, so that 2ax " is the Hmit " of the ratio {u' — u) I h, or it is **the value towards which this ratio tends in proportion as the quantity // diminishes, and to which it may approach as near as we choose to make it. " Thus Lacroix's definition, like D'Alembert's, does not prohibit the limit to be reached. In Note A, added by the translators, we read : ^ Alt Elementary Treatise on the Diffei ential and Integrai Calculus. By S. F. Lacroix. Translated from me French. Cambridge. 1816. 272 LIMirS AND FLUXIONS ** A limit, according to the notions of the ancients, is some fixed quantity, to which another of variable magnitude can never become equal, though in the course of its variation it may approach nearer to it than any difìference that can be assigned. " Thus, the method of limits is bere ascribed by the translators to the ancients, which is an act of reading into the ancient expositions a theory not actually there. The ancient * ' Method of Exhaustions " is merely a prelude to the theory of limits. Peacock gives in Note A a history of the theory of limits, in which researches on the Continent are dwelled upon and the contribution made by Newton is explained, but no reference is made to Jurin, Robins, and Maclaurin. In Note B Peacock states that the method used by Lacroix in this treatise " was first given by D'Alembert, in \}[i^ Encyclopédie'' z.xWq\q '* Différentiel. " Evidently Peacock was not altogether friendly toward this method, for in Note B he proceeds " directly to show in what manner this calculus may be estab- lished upon principles which are entirely indepen- dent of infinitesimals or limits," and then informs the reader **that we are indebted for the principal part of the contents of this note, to the Calcul des P'onctions of Lagrange and the large treatise by our author, on the Differential and Integrai Calculus." Peacock proceeds to give an account of Lagrange's calculus of functions and of the method of fluxions. Attention is called to ''the difìfìculty of denoting the operations of finding the difìferent orders of CRITICISMS BY BRITISH WRITERS 273 fluxions " according to the Newtonian notation, "when for u we put the function itself, which it represents. " 23 1. The attitude of some British mathematicians of the early part of the nineteenth century toward the discussions of the fundamental concepts of the calculus carried on during the eighteenth century is exhibited in the foUowmg passage from John Leslie's Dissertation on the progress of mathematica! and physical science : ^ *'The notion of flowing quantities, . . . appears on the whole, very clear and satisfactory ; nor should the metaphysical objection of introducing ideas of motion into Geometry have much weight. Maclaurin was induced, however, by such cavelling, to devote half a volume to an able but superfluous discussion of this question. As a refinement on the ancient process of Exhaustions, the noted method of Prime and Ultimate Ratios . . . deserves the highest praise for accuracy of conception. It has been justly commended by D'Alembert, who ex- pounded it copiously, and adapted it as the basis of the Higher Calculus. The same doctrine was like- wise elucidated by our acute countryman Robins ; . . . Landen, one of those men so frequent in England whose talents surmount their narrow education, produced in 1758, a new form of the Fluxionary Calculus, under the title of Residuai Analysis, which, though framed with little elegance, ^ Dissertation Fourth, in the Encyclopcedia Britannica^ 7th ed., voi. i, 1842, pp. 600, 601. 18 274 LIMITS AND FLUXIONS may be deemed, on the whole, an improvement on the method of ultimate ratios. " Remar ks 232. The first part of the nineteenth century marks a turning-point in the study and teaching of mathematics in Great Britain. i\ttention has been directed to the efforts of Woodhouse to introduce the higher analysis of the BernoulHs, Euler, Clairaut, and Lagrange. His efforts were strongly and ably seconded by three other young men at Cambridge, John Frederick William Herschel, Charles Babbage, and George Peacock, who used to breakfast together on Sunday mornings, and in 18 12 founded the " Analytical Society at Cambridge," for the promo- tion, as Babbage humorously expressed it, of *' the principles of pure D-ism in opposition to the Dot-digo. of the University." The translation into English of Lacroix's Elementary Treatise and the publication, in 1820, of Exaniples with their solutions, brought the more perfect notation of Leibniz and the re- fined analytical methods to the attention of young students of mathematics in England.^ ^ Before the nineteenth century, the use in England of the Leibnizian notation dz and Jydz is exceedingly rare. In our § 54 we saw that about the beginning of the eighteenth century these symbols were used by John Craig in articles published in the London Philosophical Transactions. When criticising Euler, Benjamin Robins once used the Leibnizian notation ; see our § 142. Mr. Philip E. B, Jourdain has brought to my attention the fact that the sign of integration f occurs also in a hook, entitied, Second Volume of ihe Instrtutions given in the Drawitt^ School established by the Dtiblin Society. . . . Under the Direction of [oseph Fenti, heretofore Professor of Philosophy in the University of Nantes. Bub/in, UDCCLXXll. De Morgan refers to this work in a letter to Hamilton. See Graves' Life of Sir William Rowan Hamilton^ voi. iii, p. 488. See also our Addenda^ p. 2S9. CRTTICISMS BY BRITISH WRITERS 275 As usually happens in reformations, so here, some meritorious features were discarded along with what was antiquated. William Hales, in 1804, referred to the much neglected Discourse of Benjamin Robins (i735)> with its full and complete disavowal of infinitesimals and clear-cut, though narrow, con- ception of a limit. By a curious turn in the process of events, Robins was quite forgotten in England, and D'Alembert's definition was recom- mended and widely used in England. Now Robins and D'Alembert had the same conception of a limit ; both held to the view that variables cannot reach their limits. However, there was one difìference between the two men: Robins embodied this restric- tion in his definition of a limit ; D'Alembert omitted it from his definition, but referred to it in his explanatory remarks. D'Alembert says : ^ ** On dit qu'une grandeur est la limite d'une autre grandeur, quand la seconde peut approcher de la première plus près que d'une grandeur donnée, si petite qu'on là puisse supposer, sans pourtant que la grandeur qui approche, puisse jamais surpasser la grandeur dont elle approche ; ensorte que la différ- ence d'une pareille quantité à la limite est absolu- ment inassignable." Further on in the same article we read : "A proprement parler, la limite ne co- incide jamais, ou ne devient jamais égale à la quantité dont elle est la limite ; mais celle-ci s'en ^ Art. "Limite" in the Encyclopédie, ou dictionnaìre rais sonni' des Sciences des arts et des tnétiers, puh li é par M. Diderot, et M. D Alembert, Paris, 1754. See also the later edition of Geneva, 1772. 276 LIMITS AND FLUXIONS approche toujours de plus en plus, & peut en différer aussi peu qu'on voudra. . . . On dit que la somme d'une progression géométrique décroissante dont le premier terme est <^ & le second b, est (a — ò) / (aa) ; cette valeur n'est poit proprement la somme de la progression, c'est la limite de cette somme, c'est-à- dire la quantité dont elle peut approcher si près qu'on voudra, sans jamais y arriver exactement." 233. That even the best expositions of limits and the calculus that the Continent had to offer at that time were recognised in England to be imperfect, is shown by a passage in a letter which William Rowan Hamilton wrote in 1828 to his friend John T. Graves : ^ **I have always been greatly dissatisfied with the phrases, if not the reasonings, of even very eminent analysts, on a variety of subjects. . . . An algebraist who should thus clear away the meta- physical stumbling-blocks that beset the entrance of analysis, without sacrificing those concise and powerful methods which constitute its essence and its value, would perform a useful work and deserve well of Science." ^ Life of Siy William Rowan Hamilton, by Robert P. Graves, voi. i, 1882, p. 304. CHAPTER XII MERITS AND DEFECTS OF THE EIGHTEENTH- CENTURY BRITISH FLUXIONAL CONCEPTIONS Merits 234. There are, perhaps, no intuitional concep- tions available in the study of the calculus which are clearer and sharper than motion and velocity. There is, therefore, a certain advantage in approach- ing the first study of the difìferential calculus or of fluxions by the consideration of motion and velocity. Even in modem teaching of the elements to beginners, we cannot afìford to ignore this advantage offered by the eighteenth-century British mode of treating the calculus. A second point of merit lies in the abandonment of the use of infinitely little quantities. Not ali English authors of the eighteenth century broke away from infinitesimals, but those who did vvere among the leaders : Robins, Maclaurin, Simpson, Vince, and a few others. The existence of infini- tesimals (defined as infinitely small constants) was looked upon by philosophers and by many mathe- maticians as doubtful. Their subjective existence was hardly more probable than their objective exist- ence. These mystic creations occupied a hypo- 277 278 LIMITS AND FLUXIONS thetical twilight zone between finite quantity and no quantity. Their abandonment added to the clearness and logicai rigour of mathematics. From the standpoint of rigour, the British treatment of the calculus was far in advance of the Continental. It is certainly remarkable that in Great Britain there was achieved in the eighteenth century, in the geometrical treatment of fluxions, that which was not achieved in the algebraical treatment until the nineteenth century ; it was not until after the time of Weierstrass that infinitesimals were cast aside by many mathematical writers on the Continent. 235. There is a perversity in historic events exhibited in the fact that after infinitesimals had been largely expelled in the eighteenth century from Great Britain as undesirable, unreal, and mischief-making, they should in the nineteenth century be permitted to return again and to flourish for a time as never before. About 18 16 the Leibnizian notation of the calculus and the vast treasures of mathematical analysis due to the Bernoullis, Euler, D'Alembert, Clairaut, Lagrange, Laplace, Legendre, and others, which were ali ex- pressed in that notation, found their way into England. This influx led to enrichment and advance- ment of mathematics in England, but also to a recrudescence — this return of the infinitely small. How thoroughly the infinitesimal invaded certain parts of British territory is seen in Frice's large work on the Infinitesiinal Calculus^ a work which in many ways is most admirablc and useful. MERITS AND DEFECTS 279 236. After the development of the theory of limits by the English mathematicians and by such Continental writers as D'Alembert and Lacroix, it would hardly seem necessary even for the sake of brevity to reintroduce the old-time infinitesimal which could be *'dropped" whenever it was very small, yet stood in the way. But at ali times, and particularly in the eighteenth and beginning of the nineteenth centuries, there have been mathematicians who cared little for the logicai foundations of their science. Fascinated by the ease with which the calculus enabled them to dispose of difficult prob- lems in the theory of curves, ordinary mechanics, and celestial motions, they felt more like poets, and held the sentiments toward logie that a distinguished hard entertained toward pure intellectualism when he contemplated the beauties of the rainbow : " Triumphal arch that fiU'st the sky, When storms prepare to part, I ask not proud philosophy To teach me what thou art." Defects 237. AH the eighteenth-century expositions of the foundations of the calculus — even the British — are defective. Without attempting an historical treatment or a logicai exposition of later develop- ments, we desire to point out briefly what some of these defects were. In the first place, the doctrine of fluxions was so closely associated with geometry, to the neglect of 28o LIMITS AND FLUXIONS analysis, that, apparently, certain British writers held the view that fluxions were a branch of geo- metry. In the preface to the GentlemarCs Diary of London, the new editor, Mr Wildbore, said at the commencement of his editorship in 1781, *'the doctrine of fluxions depends on principles purely geometrical, as is very satisfactorily demonstrated by that incomparable geometer, the late Dr Robert Simson of Glasgow in his Opera posthuma. " In the second place, as pointed out by Landen and Woodhouse, there was an unnaturalness in founding the calculus upon the notions of motion and velocity. In a real way, these notions seem to apply only to a limited field in the applications of the calculus, namely, to dynamics. In other fields, motion and velocity are wholly foreign concepts which, if applicable at ali, are so only in a figurative sense. 238. Newtonian writers lay great stress upon such conceptions as a line generated by the motion of a point, a surface generated by the motion of a line, and a solid generated by the motion of a surface. We have already referred to the pedago- gica! advantages of this view, in teaching beginners. But as a final logicai foundation this view is inade- quate. Not ali continuous curves can be conceived as traceable by the motion of a point. An example frequently quoted, in discussions of this sort, is the curve y — 0 for ;r = o, y = x sin for x =^ 0. MERITS AND DEFECTS 281 Let US try to trace this curve by the motion of a point starting from the origin of co-ordinates. In which direction must the point move from the origin ? To answer this question we differentiate, and find dy j dx=s\n {i I x) — {i / x) cos ( i / x). At the origin we have x = o and 7 = 0. No value can be assigned to dy / dx, because i / x has no meaning when^ = o ; moreover,the equationjr = ^' sin (i / x) ìs expressly stated above to apply only when x is not zero. There is, therefore, no way of ascertaining the direction in which the point must depart from the origin. Perhaps we can do better if the moving point is started at another part of the curve. An attempt to plot the curve reveals the fact that it lies between two right Hnes, of which one makes with the ;i'-axis an angle of 45°, the other an angle of —45°. As the point moves along the curve toward the origin, the curve is found to oscillate with ever-increasing rapidity. When we try to determine the direction by which it jumps into the origin, we encounter the same difficulty as before. As long as x is finite, the direction of motion is determinable. But as soon as we try x=o, the determination is impossible. This conclusion must be accepted, in spite of the fact that the curve is contiiiuous in ali its parts, including the origin. This example illustrates the inadequacy of motion as a fundamental concept. 239. Difficulties are encountered in the notion of velocity. Is variable velocity an objective reality ? Take a body falling from rest. We say that its 282 LIMITS AND FLUXIONS velocity is ds / dt=^gt. At the end of the first second, the velocity is ^^. If we ask ourselves, How far does the body move with the velocity gì we must admit that no distance can be assigned. We cannot say that the body moves from a certain point to the point immediately beneath ; there is no such point immediately beneath. For, as soon as we try to locate such a point, it occurs to us that we can imagine at least one point located between the two points under consideration. This intermediate point serves our purposes no better, for a fourth point located between it and the initial point is easily detected, and so on, without end. Thus it is seen that no distance, however small, can be assigned through which a body falls with a given velocity. We are thus compelled to reject variable velocity as a physical fact. What, then, is ds j dt=gtì Clearly it is merely a limit, a mathematical concept, useful in mathematical analysis, but without physical reality. To say that ds / dt represents the distance a body would fall in unit time after the instant indicated by /, is to assign it merely hypothetical meaning, destitute of concreteness. While these considerations in themselves may not debar the use of velocity as a mathematical concept upon which to build the calculus, they show that the concept is not as simple as it would seem to be at first approach. The reader will have observed that in ali discus- sion of limits during the eighteenth century the question of the existence of a limit of a convergent sequence was never raised ; no proof was ever given MERITS AND DEFECTS 283 that a limit actually exists. In this respect the treatment vvas purely intuitive. 240. Another defect in the British exposition of fluxions was in the use of the word ''quantity. " No definition of it was given, yet quantities were added, subtracted, multiplied, and divided. It is possible to treat quantities or magnitudes without the use of number. The fifth and tenth books of Euclid's Elenients contain such treatment. We may speak of the ratio of one magnitude to another magnitude, or we may speak of the ratio of one number to another number. Which was meant in the treatment of fluxions ? Straight Hnes were drawn and the ratios of parts of these lines were written down. What were these the ratios of? Were they the ratios of the Hne-segments themselves, or the ratios of the numbers measuring the lengths of these line- segments ? No explicit anewer to this was given. Our understanding of authors like Maclaurin, Rowe, and others is that in initial discussions such phrases as *' fluxions of curvih'neal figures," ''fluxion of a rectangle," are used in a non-arithmetical sense ; the idea is purely geometrical. When later the finding of the fluxions of terms in the equations of curves is taken up, the arithmetical or algebraical conception is predominant. Rarely does a writer speak of the difference between the two. Perhaps " His notions fitted things so well, That which was which he could not teli." 241. Analytical geometry practically identìfied geometry with arithmetic. It was tacitly assumed 284 LIMITS AND FLUXIONS that to every distance corresponds a number and to every number there corresponds a distance. Number was thus given a geometrical basis. This situation continued into the nineteenth century. This metrical view involved the entire theory of measurement, which assumed greater difficulties with the advent of the non-EucHdean geometries. The geometrical theory of number became less and less satisfactory as a logicai foundation. Hence the attempts to construct purely arithmetical theories.^ A good share of those difficulties arose from irrational numbers, which could not be avoided in analytical geometry. This occurrence is not merely occasionai ; irrational ratios are at least as frequent as rational ones. What is an irrational number ? How do we operate with irrational numbers ? What constitutes the sum, difìference, product or quotient, when irrational numbers«are involved? No explicit answer was given to these questions. It was tacitly assumed without fear, that it is safe to operate with irrational numbers as if they were rational. But such assumptions are dangerous. They might lead to absurdities. Even if they do not, this matter demands attention when mathematical rigour is the aim. 242. Perhaps it may be worth while to recali to the reader's mind illustrations of the danger result- ing from taking operations known to yield consistent ^ For a historical account of the number concept and the founding of the theory of transfinite numbers during the nineteenth century, read Philip E. B. Jourdain's " Introduction " to Cantor's Transjinite Numbers, The Open Court Publishing Co,, 191 5. MERITS AND DEFECTS 285 results when a certain limited class of numbers is involved, and applying them to numbers of a more general class. Suppose a and b to be rational, positive numbers, not zero ; we find, let us agree, consistent results in the operation a-\-by a — b when a>by and av^b, and a-^b. Let us now consider the class composed of rational numbers, both positive and negative ; suppose, moreover, that we introduce o in order to give interpretation to the operation a — a. If in this extended class of numbers we admit the four operations a-\-by a — by axb, a-^by trouble arises even after due considera- tion has been given to the negative numbers. There may arise the following well-known paradox. Let a — b=\y then a^ — b'^ = a — b. Divide both sides of the last equation by a — b^ and we have a-\-b=i, or 2=1. Where is the diffìculty ? The answer is known to every schoolboy : We have used a — b^ or o, as a divisor ; we have extended the operation of division to the larger class of numbers, and to zero, without first assuring ourselves that such an extension is possible in every case ; division by zero is inadmissible. 243. A less familiar example is the following. Let US suppose that, for real exponents, it is estab- lished that ( A^)^ = A'-^. When we apply this process to imaginary exponents, trouble arises. Take the equation e""*'"" — e'^**'"\ where in and n are distinct integers, z'= 7- I, 7r=3'i4i59 • • ., and ^=2 7 18 . . . That this equation holds is evident, for ^''"" = cos 2?«7rH-/sin 2;;/7r = cos 2;/7r + /sin 2;/x = ^^''". If 286 LIMITS AND FLUXIONS both sides of ^^""" = £* '"' are raised to the power i / 2, vve obtain ^-""^ = ^-«\ Here ali the letters stand for real numbers ; since ;;/ and n are not equal to each other, this last equation is an absurdity. The assumption that a rule of operation vab'd for real exponents was valid also for imaginary exponents, has led to papable error. Examples of this sort emphasise the need of caution when operations, known to be valid for a certain class of numbers, are applied to numbers belonging to a larger class. Special examination is necessary. These remarks are pertinent when operations applicable to rational numbers are ex- tended to a class which embraces both rational and irrational numbers. What are the numbers called irrational? It is hardly sufificient to say that an irrational number is one which cannot be expressed as the ratio of two rational numbers. A negative definition of this sort does not even establish the existence of irrational numbers. Considerale attention has been paid to the definition of irrational numbers as limits of certain sequences of rational numbers. Thus, ^2 may be looked upon as the limit of the sequence of rational fractions obtained by the ordinary process of root-extraction, namely, the sequence, i, 1*4, 1*41, I'4I4, I"4I42, . . . This attempt to establish a logicai foundation for irrational numbers was not successful. We endeavour, in what foUows, to make this matter plainer. 244. Let US agree that in building up an aiith- MERITS AND DEFECTS 287 mctical theory vve have reached a development of rational numbers (integers and rational fractions). We wish, next, to define //;//// and also irrntional number. An early nineteenth-century definition of limit was : '' VVhen the successive values attributed to a variable approach a fixed value indefinitely so as to end by difìfering from it as little as is wished, this fixed value is called the limit of ali the others. " Since, according to our supposition, we are stili in the field of rational numbers, this limit, unless it happens to involve only rational numbers and to be itself only a rational number, is, in our case, non- existent and fictitious. If now, as stated above, an irrational number is defined as the limit of certain sequences of rational fractions, trouble arises. The existence of such a Hmit is often far from evident. But aside from that general consideration, the difficulty of the situation in our case is apparent : Irrational numbers are limits, but limits themselves are non-existent or fictitious, unless they are rational numbers. To avoid this breakdown in the logicai development, it was found desirable to define irrational number without using limits. 245. Wìth the view of avoiding the use of limits in the definition of irrational number, and at the same time avoid inelegant and difficult assumptions, involving complicated considerations relating to the nature of space, ^ devices were invented by several ^ On this point consult the article " Geometry " in the Encyclopitdia Britannica^ iith edition, the part on Congruence and Measurement. 288 LIMITS AND FLUXIONS logicians independently, which freed the number con- cept from magnitude and established number theory on the concept of order. Chief among the workers in this field were Méray, Weierstrass, Dedekind, and Georg Cantor. It is to them that we owe re- presentations of number, both rational and irrational, which have yielded a much more satisfactory theory of limits, and in that way have vastly improved the logicai exposition of the differential calculus. These theories have brought about the last stages of what is called the aritJunetisation of mathematics. As now developed in books which aim at extreme rigour, the notion of a limit makes no reference to quantity and is a purely ordinai notion. Of this mode of treatment the eighteenth century had never dreamed. ADDENDA TO §§ 54, 58, 73 246. Additional data on the fundamental concep- tions relating to fluxions and on the use of the Newtonian and the Leibnizian notations in England during the Hfetime of Newton are contained in George Cheyne's Philosophical Principles of Relzgioti, Part li, London, 17 16. Fart I of this book appeared first in 1705. Like Berkeley's Analyst, which was written later, Cheyne's book, Fart I and Fart II, had for its primary purpose the refutation of atheism. Cheyne says in his preface to the third edition of Fart I, "that Atheism, may be eternally confounded, by the most distant Approaches to the true Causes of naturai Appearances. And that if the Modem Fhilosophy demonstrates nothing else, yet it infallibly proves Atheism to be the most gross Ignorance. " Fart 1 1 of Cheyne's book consists of three chapters and of seven pages of " Additions. " He says in his preface to this part that, excepting one short note, the third chapter and the "Additions" are " what the reverend and ingenious Mr. John Craig sent me about seven years ago, when 1 desired him . . . to vvrite me down his Thoughts on, correct or alter, 2^9 19 290 LIMITS AND FLUXIONS vvhat I had formerly published on this Head in the first Edition of this Work." Cheyne uses in Part II, p. 20, the notation ± io denote a distance B^ when he supposes " b infinitely near to B." In § 58 we pointed out that in 1704 Cheyne wrote once x= i, but nowhere in the present hook does x denote a finite quantity. He argues that I / i'= co I, that i / o=co ; hence thati-=^, or "relative nothing," which is ** the least Part of the Finite, to which it is related or compared. " On p. 21 he calls x " an infinitely little Part of x." On p. 12 he speaks of the " absolute infinite" as " ad- mitting of neither Increase, nor Diminution, or of any Operation that mathematica! Quantity is sub- jected to," while (p. 13) " absolute nothing" is ''neither capable of increasing nor diminishing, nor of any wise altering any Mathematica! Quantity to which it is apply'd, but stands in full opposition to absolute Infinite. " On the other band, "indefinite" or ' ' relative infinite " quantities (p. 29") ' ' are not properly either Finite or Infinite, but between both. " The " relative nothing " (p. 8) " is an infinitely little Quantity, as it stands related to a given Finite, by the perpetuai Subtraction of which from it self it is generated. Let 0 stand for relative nothing. Thus 190, 219, 241- 243, 253 ; quotations from, 2- •4, 43- Quadratura Curvarum (Newton's), 17-26, 28, 31-34, 36, 43, 50. 53, 67, 69, 71, 74, 76, 77, 112, 124, 130, 135-137, 156, 179, 190, 246. Quadrature of curves ; see Quadra- tura Curvarum. Raphson, Joseph, 49, 55 Ratios, first and last ; see Prime and ultimate ratios. Relative infinite ; see Infinity, relative. Relative motion, 86. Relative nothing ; see Nothing, relative. Republick of Letters ; see Present State o/the Republick of Letters. Residuai analysis (Landen's), 232- 235, 239, 258, 273. Rigaud, S, P., 14, 29, 30. Robartes, F., 41. Roberval, i. Robins, B., 96-148, 179, 181, 188, 189, 199, 242, 250, 254, 2Ó6, 267, 272-275 ; Robins'sil/a/A^- matical Tracts, 202-206 ; Robins's Discourse, 96-100, 115, 141, 142, 267, 268, 275; explains history of conlroversy with Jurin, 141-145. Rowe, John, 172, 173, 175, 178, 179, 195, 202, 253, 283. Rowning, John, 198-200. Saunderson, Nicholas, 150, 197, 198, 204, 242. Simpson, Thomas, 169-171, 175, 178-180, 194-196, 206, 209, 210-215, 218, 220, 221, 223, 224, 244, 253, 259, 277 ; his definition of fluxions. 244, Simson, Robert, 280. Sloane, H., 48. Smith, James, 165-169. Smith, Robert, 139, 140, 201. Solidus, use of, 12. Stevin, S. , 99. Stewart, John, 17, 21. 25, 190-192. Stirling, James, 50, 55. Stockler, G., 259. Stona, E., 50. 51, 53-56, 169, 248, 259. St. Vincent, Gregory, i, 99. Stubbs, J., 177. Tacquet, Andrew, 99. Tannery, Jules, 137. Taylor, Brook, 50. Thorp, R., 241-243; translator of Principia, 3, 8. Torelli, G., 257. Truth Triumphant, 212, 213, 215, 218, 222, 223. Turner, John, 210-214, 218, 219. INDEX 299 Ultimate ratios, 6, 7, 9, io, 19, 22- 24, 35, 68, 70, 78, 86, 98, 99, 103, 115, 153, 158-161, 168, 191, 219, 223, 242. Ultimate velocity, 7, io; see Velocity. Ultimato! s, doctrine of. 225. Valerius, Lucas, 99. Varignon, P. , 293. Velocity, II, 13-18, 21, 28-30, 32, 33, 36, 43-45, 49, 50, 58, 62, 63, 66, 70, 71, 74, 76-80, 83, 85. 87, 88, 93, 106, 112, 123, 151, 152, 158, 163, 164, 166, 169-171, 173, 175, 184, 185, 193-197, I99> 200, 201, 207, 209, 212, 214, 215, 217, 219, 220, 231, 235, 244, 246, 252, 261, 263, 270, 277, 280 ; velo- city, ultimate, 6, 9, 36 ; velo- city criticised as a fundamental concept, 255, 264, 280-282. Vince, S., 245-247, 277, Vivanti, G., 235. Wallis, John, 41, 100, 292 ; Wallis's Algebra, 14, 30, 32, 33, 39. Walton, John, 57, 69, 77, 85, 86, 91-96, 148, 185, 199; his first reply to Berkeley, 69-72 ; his second reply, 78-80 ; second edition of second reply, 87. Warburton, Bishop William, ']'},. Waring, E., 259. Weierstrass, K.. 288. Weissenborn, H., 92. West, William, 202. Whately, 90. Whiston, William, 197. Wildbore, 280. Wilson, James, 96, 99, 189, 202- 206, Witting, A., 30. Wood, James, 245, Woodhouse, R., 93, 263-267, 274, 280. Works of the Learned, 135-137, 171. 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