**Author**: Sloman, H

**Title**: The claim of Leibnitz to the invention of the differential calculus. Translated from the German with considerable alterations and new addenda by the author.

**Publisher**: Cambridge [Eng.] Macmillan, 1860.

**Tag(s)**: leibniz, gottfried wilhelm, freiherr von, 1646-1716; differential calculus; leibnitz; newton; oldenburg; calculus; wallis; differential; commercium epistolicum; newton's letter; john bernoulli; series; integral calculus

**Contributor(s)**: Eric Lease Morgan (Infomotions, Inc.)

**Versions**: original; local mirror; HTML (this file); printable; PDF

**Services**: find in a library; evaluate using concordance

**Rights**: GNU General Public License

**Size**: 44,489 words (really short) **Grade range**: 13-17 (college) **Readability score**: 36 (difficult)

**Identifier**: claimofleibnitzt00slomrich

Tweet Bookmark this on Delicious

Discover what books you consider "great". Take the Great Books Survey.

*^E V F CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORR %8i = .iC« — —^7 e^ 3 = 6 F CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALI F CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFOR! BRARY OF THE UNIVERSITY OF CALIFORN llllSv; IR A R Y OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA RARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF T /@E%3 LIBRARY OF T RARY OF UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA V\\ LIBRARY OF Tl *%&£&& ^ & = m THE CLAIM OF LEIBNITZ INVENTION OF THE DIFFERENTIAL CALCULUS. ' . ' The present publication is a revised and enlarged edition of an Essay which appeared in German in 1858. (Leipsig and Kiel. Schwers'schc Buchhandlung.) « VMBRIDI VH1NI BD )IY n I l:ll > M 1.1 1 I . THE CLAIM OF LEIBNITZ TO THE INVENTION OF THE DIFFERENTIAL CALCULUS, BY DR. H. SLOMAN. TRANSLATED FROM THE GERMAN WITH CONSIDERABLE ALTERATIONS AND NEW ADDENDA BY THE AUTHOR. MACMILLAN AND CO. AND 23, HENRIETTA STREET. COVENT GARDEN, LONDON. 1860. SIR DAVID BR E W S T E R. THE BIOGRAPHER OF NEWTON, AND THE REV. J. EDLESTON, IS KESPECTEULLY DEDICATED BY THE AUTHOR. CHAPTER I. BARROW AND THE METHOD OF TANGENTS. From about the year 1650, the vigorous mathematical life, in which England had never been deficient, is seen to receive there an extraordinary impulse, and attain to such a degree of development, that that country became the centre of all the mathematical activity of the period, while in France, after the death of Descartes, there are no important men to name in mathematics.* * Perhaps even Descartes was much indebted to the English Harriot. For not only does the upright Wallis, who would never knowingly have uttered an untruth, affirm this with zealous warmth in many passages of his Tractatus Algebra historicus et practicus, but it was also believed by contemporaries, and at the same time countrymen of Descartes's, who are spoken of in Baillet's Vita Cartesii, and by Roberval, qui s ' entretenant un jour avec Milord Cavendish, lui temoignant etre inquiet, d'ou etait venu a Descartes I'idee, d'egaler tons les termes d'une equation a zero, Milord Cavendish lui dit, qiCil ri ignorait cela que parcequ'il etait Frangais, et lui offrit de lui montrer le livre auquel Descartes devait cette invention. En effet il le mena chez lui, et lui montra Vendroit de Harriot, ou Von voit la inane chose; sur quoi Roberval, transports de joie, s'ecria, "il Va vu, il Va vu!" et il le publia de toute part. We quote this out of Montuclat. II., p. 144. When Colbert in 1666 was looking about him for men, out of whom to form an Academie des Sciences, he found no geometers or astronomers in France, except the following: viz., Auzout, Buot, Carcavi, Couplet, Frenicle, Niquet, Picart, Richer, Roberval and De la Voye — none of them, with the exception of Roberval, who died soon after, persons of any great eminence. It was on them and their immediate successors that Leibnitz and Bernouilli, who were both their colleagues, pronounced the following judgment : (See Gerhardt's edition of the Math. Works of Leibnitz, p. 814 : the earlier editions B 2 BARBOW AND THE METHOD OF TANGENTS. Two problems occupied at that time the attention of geometers, namely the problem of Tangents and that of Quadratures, in which Barrow and "Wallis, in England, had achieved the most advanced positions. The two problems had as yet no mutual connexion ; for the object contemplated was measure in one of them, and direction in the other. It will be readily understood, that Barrow's method of tangents cannot be left unnoticed in an enquiry like ours; and so indeed a great deal is said respecting him, by the most modern writers in France and Germany — as Biot in the "Journal des Savants/' and Gerhardt in his various writings — who have aroused in the present day a lively interest in the question, was Leibnitz the discoverer of the Differential Calculus, and to what extent? AYe need not on this point speak at much length. Barrow says,* Nulla est magnitude*) qua- non innumeris modi's intelligi producta jiossit, of the Correspondence do not contain this passage) Verissimum est, quod de nonniillis Academicis notas — ct sane qua a se habent plerumque sunt mediocria, ne dicam ridicula — et si quid boni cdunt, dubitare non lied, quin ab aliisfurati sint. * Compare p. 15 of his principal work, and the one which made the greatest noise at that time, entitled, Lectiones Geometrical in quibus prcesertim generalia curvarum symptomata declarantur. Of this work the date is not without importance : it was puhlished in 1670, (and not in 1674, as Gerhardt says in his tract of 1848, p. 15 — nor yet in 1672, as he supposes in his tract of 1855, p. 45). That Leihnitz before his discovery of the Differential Calculus either in 1676 or in 1675 or in 1674, should not have read this work, (as Gerhardt affirms in the place quoted,) is inconceivable. Books were not so abundant in those times. Indeed evidences to the contrary are contained in the documents, which Gerhardt himself produces. In App. 1, to Gerhardt's tract of 1848, p. 32, Leibnitz says expressly, that he had seen from Barrow's Lectiones " cum prodirent" — what they contained. This proves that Leibnitz possessed Barrow's book not long after its first appearance, 1670. unit gives another document, (Tract of 1855, p. 129,) from which the same conclusion may be drawn. This document is, as Gerhardt affirms, dated in Leibnitz' hand-writing 1 Nov., 167.3, and therein we have again Leibnitz' own words: Plera- que theoremata Geometries indivisibilium, qua apud VavdUerium, J'inccntium, Gre- gorium, Barrovium, extant, etc. I BARROW AND THE METHOD OF TANGENTS. 3 per motus locales, per inter sectiones magnitudinum, per quantitate por- tioneque determinatas ab assignatis locis distantias, per ductus magni- tudinum in magnitudines, per applicationes magnitudinum ad magnitudines, per aggregationem magnitudinum ordine certo dispositarum, per appo- sitionem magnitudinum ad alias, vel subductionem ab aliis. Horum modus primarius, et quern alii omnes quodammodo supponant oportet est iste per motum localem. In spite of this idea, which involves his peculiar mode of contemplating the subject, Barrow is entirely devoted to the more important method of Cavalleri, who considers every figure as composed of parts infinitely minute and numerous, and every curve of an infinity of straight lines. So he says, for example, at p. 15 : curva aliquis, vel e rectis (angulos efficientibus) composita, quae curvae quoque nomen merito ferat; Archimedes enim e rectis compositas lineas [uti figurarum circuit's inscriptarum perimetros) KafMirvXwv <ypafjt/j,tov nomine complectitur, ut et vicissim curvae quaevis lineae censeri possunt e rectis innumeris quidem illis indefinite par vis adjacentibus et deinceps secum angxdos efficientibus, confectae. So likewise, in Lectio II. §. 21 : curva quaedam superficies, circularibus quasi peripheriis constans, [Atomistarum enim plirasin facilitatis, perspicuitatis, brevitatis, addere licet et veri- siinilitudinis, causa non illibenter usurpo). And modus — dimensiones investigandi juxta methodum indivisibilium, omnium expeditissimam, et modo rite adhibeatur Jiaud minus certain et infallibilem.*) We owe it to Gerhardt that attention has been again directed to this side of the Idea of Barrow. Now in the 4th Lectio, p. 40, Barrow proves a very important pro- position for the application and draw- ing of tangents to figures according to this method. He says: in figura" 26 tangant rcctae TM, XN, dico curvae arcum MN recta NH majorem esse; * Compare also p. 21 ad Jin. ; we quote the edition of 1670. B2 4 BARBOW AND THE METHOD OF TANGENTS. recta vero ME minorem. For draw the chord MN, and draw NR parallel to XT, then NHM shall be an obtuse angle, and therefore the chord MN and a fortiori its arc shall be greater than EN. On the other hand, because RNE is a right angle, HE shall be greater than RN, and therefore ME > MR + RN '; but according to Archimedes, Archirnedceis assumptis) MR + RN (the polygon circumscribed about a curve,) is already greater than the curve that is inscribed in it ; therefore the arc MN will be less than ME. Perutilis, says Barrow, est hcec propositio in tangentium demonstrationibus expediendis. Etenim hi nc consectatur, si arcus MN indefinite parvus ponatur, ejusce loco alterutram tangentis particulam ME vel NH tuto substitui. Lectio X. begins with these words : Institutum circa tangentes negotium < id hue urgco, and when the theorems which it remained necessary to supply on the subject of tangents, have been exhibited in proper geometric form, we read on page 80 :* Ita propositi nostri (ptriore, guam innuebamus parte) quodammodo defuncti sumus. Cui supplendae, appendiculae instar, svbnectemus a nobis usitatam metkodum ex Calculo tangentes reperiendi. Quamquam haud scio, post tot eiusmodi pervulgatas atque protritas mi tkodoSj an id ex usu sit facere. Facto saltern ex Amici\ consilio ; eoque lubentius, quod prae ceteris, quas tractavi, compendiosa videtur, ac generalis. In hum procedo modum. Sint AP, PM positione datae lineae et MT curvam tangere ponatur, recta* PT quantitatem exquiram; curvae arcum MN indefinite parvum statuo; nomine MP=y ; PT—t; MR = a; NR = e; ipsas MR, NR (et mediantibus illis ipsas MP, PT) per aequationem e Calculo depre- hensam inter se comparo ; regidas interim has observans 1, inter com- putandum omncs abjicio terminos, in quibus ipsarum a, vel e potestas ' AVe quote verbatim, except that we call the abscissa and its ordinata x and y, and not with Barrow/ and m. t Weissenborn in Ids " Beitrag zur Geschichte der Mathematik oder Principien der hoheren Analysis," Halle 1856, takes for granted that Newton is the friend here intimated. BARROW AND THE METHOD OP TANGENTS. habetur vel in quibus ipsae ducuntur in se (etenim isti termini nihil valebant). 2. Post aequationem constitutam omnes abjicio terminos, Uteris con- stantes quantitates notas, seu determinatas designantibus ; aut in quibus non kabentur a, vel e (etenim illi termini semper ad unam aequationis partem adducti, nihilum aequabunt). 3. Pro a ipsam y (vel IIP) pro e ipsam t (vel PT) substituo. Hinc demum ipsius PT quantitas dignoscetur. Quod si calculum ingrediatur curvae cujuspiam indefinita particula ; substituatur ejus loco tangentis particula rite sumta ; vel ei quaevis (ob indefinitam curvae parvitatem) aequipollens recta. Haec autem e subnexis Exemplis clarius elucescent. Exempt. Sit recta EA positione ac magnitudine data et curva EMO proprietate talis, ut ab ea utcunque ductd recta MP ad earn perpendiculari summa cuborum ex AP et MP aequetur Cubo rectae AE ; x 3 +y 3 =r 3 . (Fiant quae praescripta sunt) Nominatis AE—r; AP=x; (and as before MP = y; PT = t ; MR = a, NE = e); unde AQ = x-\-e,etAQcub = x 3 + Sx'e + 3xe 2 + e 8 (seu rejectis uti monitum est rejiciendis)=x 3 +3x'*e. Item NQ cub = cub (y—a)=y 3 — 3y*a 4 By a 3 — a 3 (hoc est) = y 3 — By z a. Quapropter est x 3 + Bx 2 e ■+ y 3 — By 2 a = (AQ cub -\-NQ cub = AE cub =) r 3 abjectisque datis est x l e — y l a = seu x 2 e = y* a subrogatisque loco a . . . , o y 3 et e tpsts y et t erit xt = y ; seu t = *% . That is, the equation of the curve y 3 + x 3 = r 3 gives by this mode of calculation — which at first assumes infinitely small increments MP and NR, a and e } which afterwards again vanish — the expression V* for the sub-tangent t — ^ . 6 BARKOW AND THE METHOD OF TANGENTS. This process Barrow illustrates by four further examples, and at page 84 he passes on with the words, Hcec sufficere videntur liuic methodo fflustirandce, to other geometrical investigations, that is to Lectio XI., which begins with the words; Reliquis utcunque paratis, apponemus jam quce ad magnitudinem e tangentibus seu e perpendicidaribus ad curvas dimensiones eUciendas pertinentia se objecerunt tkeoremata. Then follows Barrow's geometrical method of Quadratures, of which we shall not speak at present. We repeat therefore that Barrow's Method of Tangents of 1670, which Leibnitz had read, consisted in applying to the problem of Tan- gents the idea of neglecting the higher powers of infinitely small quantities. When Sluse, some time after Barrow, proposed a more convenient rule for the expression of tangents, Newton wrote, that he also had a rule for tangents, which was peculiarly suitable for quadratures; this rule Newton gives in his well-known letter of the 10th Dec, 1672, about which there has been so much controversy, some affirming that Leibnitz was acquainted with it, and others that he was not. Ex ammo gaudeo (writes Newton to Collins, 10th Dec. 1672) D. Barrovii amici nostri Rev. Lectiones matkematicas exteris adeo placuisse, neque parum me juvat intelligere eos in eandem mecum incidisse ducendi Tangentes Metho- (Jiuii. Qualem earn esse conjiciam ex hoc exemplo percipies. Pone CB applicatam ad AB, in quovis angulo dato, termi- nari ad quamvis curvam AC, et dicatur AB — x et BC=y, habitudoque inter x et y cxprimatur qualibet aequatione, puta x 3 —2xxy + bxx — bbx + byy - y 3 = 0, qua ipsa determinatur Curva. Regula ducendi Tangentem liaec est; multiplica aequationis ter- minos per quamlibet progrcssionem arithmeticam juxta dimensiones y, puta x 3 — 2xxy + bxx — bbx + byy — y s ; ut et juxta dimensiones x, puta 1 2 3 x 3 — 2xxy + bxx — bbx + buy — y 3 . Priiis productum erit Numerator, et 3 2 2 10 posterius division per x Denominator Fractionis, quae exp-imet longitu- BARROW AND THE METHOD OF TANGENTS. 7 dinem BD, ad cujus extremitatem D ducenda est Tangens CD: est ergo lonqitudo BB— ~ XX y ^r ~r . J Sxx — kxy + 2ox — bo Hoc est unum particulare, vel corollarium potius Methodi generalis, quae extendit se, citra molestum ullum calculum, non modo ad ducendum Tangentes ad quasvis Curvas, sive Geometricas, sive Mechanicas, vel quomodocunque rectas lineas aliasve Curvas respicientes ; veru?n etiam ad resolvendum alia abstrusiora Problematum genera de Curvitatibus, Areis, Longitudinibus, Centris Gravitatis Curvarum, etc. Neque [quem- admodum Suddenii metliodus de Maximis et Minimis) ad solas restrin- gitur aequationes illas, quae quantitations stirdis sunt immunes. Hanc methodum intertexui alteri isti, qua Aequationum Exegesin instituOj reducendo eas ad Series injinitas. Memini me ex occasione aliquando narrasse D. Barrovio, edendis Lectionibus suis occupato, in- structum me esse huiusmodi methodo Tangentes ducendi: Sed nescio quo diverticula ah ea ipsi describenda fuerim avocatus. Slusii Methodum Tangentes ducendi brevi publice prodituram con- fido : quamprimum advenerit exemplar ejus, ad me transmittere ne grave ducas. On reading this letter in the present century, we are impelled to ask for the demonstration, but in the year 1672 every one knew from the mention of the Huddenian Method de Maximis et Minimis, (which was based upon and proved by an infinitely small increment to the abscissa), and furthermore from the frequent mention of Barrow, that the foundation and proof of this Newtonian Method of Tangents lay also in that infinitely small increment. We see that Newton's letter commends this method as a universal one, while Barrow's rule, it is said, did not yet embrace all cases as bearing on one another ; at the same time the letter says, that Newton's rule is a corollary to his method of Quadratures. We shall hereafter return to this letter, but will previously obtain a clear insight into the method employed at that period for Quad- ratures. CHAPTER II. WALLIS AND THE PROBLEM OF QUADRATURES. With the invention of the Differential Calculus Wallis has a more direct connexion than even Barrow, and it is not bj mere accident that his two contemporaneously published Tractata de conicis sectionibus and Aritlimetica infinitorum exhibit the first steps in that direction, for he makes express reference to what still remained to be done, and has since actually been done by the Differential Calculus.* His labours occupy the very same period, in which the discovery took place. We take the course of his progress from the preface to the Aritlimetica infinitorum and to the treatise de sectionibus conicis, (published at the same time, and to which he refers in the aritlimetica infinitorum). Opus hoc, he says, plane novum. Exeunte anno 1650 incidi in ToriceUi scripita, * His principal work aritlimetica infinitorum is a well-known one; it appeared in the year 1G55, and its title in full was, ar. inf. sive nova mcthodus inquirendi in curvilineorum quadraturam. Barrow had handled the subject of quadratures and tangents in the main geometrically, and had made a mere adjunct of his analytical method of calculating, because he had a less warm attachment to arithmetic and algebra. Appendicula inslar, (compare p. 4), says he, when he has already handled the theory of tangents in a geometrical form, subnectemus a nobis nsitatam methodnni ex calculo tancjentes reperiendi, and immediately he goes back again to the forms of geometry, in order to investigate the Dimensiones curvarum ex tangentibus sen e perpendicularibtu ad curvas. "Wallis on the contrary had a greater partiality and respect for calculations, for arithmetic, and in consequence came nearer to the Differential Calculus than Barrow. WALLIS AND THE PROBLEM OF QUADRATURES. 9 ubi Cavalleri Oeotnetriam Indivisibilium exponit. Ipsius meihodus, Wallis continues, mihi quidem eo gratior erat, quod nescio quid eius- modi ex quo prirnum fere Mathesin salutaverim, ammo observabatur. Ubi huiusmodi jam obtinuisse methodum persenseram cogitare apud me coepi, num non hinc aliquid de circuit Quadratura, quam summos semper viros exercuisse notum est, luminis accedat. Quod spem facere videbatur, hoc erat. Infinitorum Coni circulorum, ad totidem Cylindri, ratio jam erat cognita, nempe ut 1 ad 3. Manifestum etiam erat rectas trianguli esse Arithmetics proportio- nales, sive ut I, 2, 3 etc. ergo circulos coni (in diametrorum ratione du- plicata) ut I, 4, 9 etc. Hoc autem si universali aliqud methodo invenire possem, de Circuit Quadraturd satis prospectum esset. Beducto ita problemate Geometrico ad pure Arithmeticum. Aggressus igitur sum primo (ut a simplicioribus inchoarem) series simplices. Adeoque hinc statim Geometriam auctam persensi ; cum enim antea ex figuris curvilineis sola fere Parabola quadraturam nacta erat, jam Paraboloeidium omnium infinita genera una quidem et generali me- thodo unica propositio quadranda doceat. Transii deinde ad series auctas (quas voco) et diminutas sive mu- tilatas ; quae ex duarum pluriumve serierum vel aggregatis aut diffe- rentiis constant. Atque hie etiam successum minime contemnendum reperi. Nempe eas omnes ad series aequalium redigere non erat difficile, adeoque Conoeidea et Spheroeidea, vel etiam Pyramidoeidea, non modo recta sed inclinata, ad Cylindros et Prismata redigere rem nullius esse negotii perspexi. De seriebus autem istis sive auctis sive diminutis non ipsis solum, sed et quae in earum ratione duplicatd, triplicatd, aut idterius multipli- cata procedunt, eandem inquisitionem eodem successu continuavi, uti ex it's quae deinceps sequuntur propositionibus videndum est. Ubi simul nu- merorum figuratorum, puta triangidarium, pyramidalium etc., (quorum nullus vel exiguus hactenus fuerat tisus et fere ludicrus), usus insignes ex insperato deteguntur. c 10 WALLIS AND THE PROBLEM OF QUADRATURES. Verum ubi de seriebus aliis quae sint in istarum auctarum vel di- minutarum ratione subduplicatd, et subtriplicatd etc., agendum erat, quod Circuli, Ellipseos, et Hyperbolae quadrat tiro m directe quidem et immediate spectabat, et quae sola jam superfuit difficultas: videbam illic aquam ha rere. He lias therefore, he intimates, proposed a problem on that subject to geometricians. It was this problem that afterwards gave a more direct occasion to the discovery of the Differential Calculus. In the treatise itself Wallis says: Siqtpono in limine (juxta Bona- venturae Oavallerii Geometriam Indivisibilium) Planum quodlibet quasi ex infinitis iineis parallelis constare: Vel potius (quod ego mallem) ex infinitis Paralhlogrammis aeque altis ; quorum singulorum altitudo sit tot ins altitudinis -- sive aliquota pars infinite parva, adeoque omnium simul altitudo aequalis altitudine figurae. Propositio II. Si triangidum rectis basi parallelis secetur, erunt ab- scissa triangula secto triangulo similia, et propterca latera kabebunt propor- tionalia (ut notum est). Adeoque — rectae cca, /3/3, 77 etc. — propter aequales excessus Fa, a/3, fiy etc. erunt arith- metice proportionales hoc est ut 1, 2, 3 etc. — Ideoque si rectae iliac supponan- tur numero infinitae, rrit — totum Tri- angulum aggregation rectarum numero injinitaruin, quarum minima est punc- tual, maxima est BS, Basis Trianguli. Prop. III. De area Trianguli. Itaque cum Triangulum constet ex infinitis sin- Iineis^ sire Parcdlelogrammis, aritJimetice proportionalibuSj a puncto inchoatis et ad basin continuatis : Brit Area Trianguli aequalis Basi in Altitudinis semissem ductae. Est enim notissima apud Aritlnucticos regula: Summeum* Arithme- * Here we have the idea of summation. WALLIS AND THE PEOBLEM OF QUADEATUEES, 11 ticae progressions, sive omnium quotcunque terminorum aggregatum, aequari aggregate) extremorum in semissem numeri terminorum ducto. Nam si terminus minimus supponatur (prout hie supponitur) idem erit extremorum aggregatum atque ipse terminus maximus. Altitudinem vero figurae pro numero terminorum substituo, quoniam cum numerus termi- norum supponatur oo erit omnium longitudinum aggregatum — Basis [quia Basis jam est extremorum aggregatum). Cum autem cujuslibet (lineae vel parallelogrammi inscrip>ti vel adscripti) crassities sive Alti- tudo supponatur - - Altitudinis figurae, in illam summa longitudinum ( — Basis ) ducenda est : adeoque — Alt. in — Bas. erit area. Est \ 2 / M co 2 autem — A x — B= \AB. Adeo ut A [figurae altitude-) non solum longitudinum numerum, sed eundem in communem omnium altitudinem ductum exhiheat, quae quidem communis altitudo tanto minor supponenda est, quanto termini seu longitudines sunt plures. The demonstration in Wallis is therefore as we see of the following sort: (1) In the progression of the natural numbers 1, 2, 3, 4, 5, etc., the sum is - — ] because, as we should now say, in all arithmetical series two terms at an equal distance from the beginning and the end respectively will always, when added together, produce an equal sum ; inasmuch as such a series can be annexed to itself in an inverted order, as 1, 2, 3, 4, 5 5, 4, 3, 2, 1 6, 6, 6, 6, 6 (1 + 5) 5 whence, it is evident, that the sum of the series must be - — — ^— , or in general terms - — — — . c2 12 WALLIS AND THE PROBLEM OF QUADRATURES. (2) In the triangle, if it be conceived as consisting of lines or small parallelograms, which we can suppose to be inscribed in or circum- scribed about the triangle, these series of parallelograms, when compared with one another according to their relative magnitude, must form an arithmetical progression, of which the first term will be (?, and the last term the base of the triangle. But the height of the triangle is the number of terms multiplied by the small alti- tude (or breadth), which is given to the lines or small parallelograms. (3) The triangle being the sum of all these small parallelograms, we require only the arithmetical summation of the numerical progression from zero up to that term, which measures the fundamental line (or Base), and we are enabled to satisfy the formula s = , even when the number of terms is infinite, and when we have therefore to all appearance, neither t nor n given, (neither the last term nor the number of terms) — because for this purpose we pass from the domain of arith- metic into that of geometry, and we find that (a = zero) and /, the longest of the parallel lines, or the longest of the many small parallelo- grams = the Base ; and that thus, since a = zero, a + 1, the first and last term together = Base ; which base we now therefore measure — and because the altitude of the triangle is equal to the thickness, which we assign to each line or parallelogram, multiplied into the supposed number of the said lines, therefore however thin and how many soever they may be, the altitude of the triangle is always = — altitude x m ; that is to say, the seemingly indefinite number of the parallelograms, (the altitude whereof is — of the altitude of the triangle) is the altitude of the WALLTS AND THE PROBLEM OF QUADRATURES. 13 triangle, which therefore takes the place of the factor n. The formula of summation s = — — - n is therefore satisfied, when the base is formally substituted for the t and the altitude for the n. Geometry and Arith- metic are thus reconciled with one another. Arithmetic says that the sum of every arithmetical progression = — — n ; and Geometry shows that this progression occurs in all triangles, when considered as com- posed of lines; besides which Geometry tells me, that the number of these terms (of which arithmetically I only know that it is infinite) comes to the same as the altitude or some other measurable part of the figure that lies before me, because each of the infinitesimal terms is an infinitesimal part of the altitude. It is equally manifest that the last term of this infinite series is the base of the figure, and the first term zero. We now give briefly the leading propositions of the application, which Wallis proceeds to make of this idea; we must remark however that he has another demonstration of the rule s = — — — n which he expresses as follows: Si sumatur series quantitation Arithmetice proportionalium, continue crescentium, a puncto vel inclwatarum et numero quidem vel finitarum vel infinitarum {nulla enim. discriminis causa est) erit ilia ad seriem totidem maximae aequalium ut 1 ad 2. Simplicissimus investigandi modus in hoc et sequentibus aliquot Pro- blematis est rem ipsam aliquousque praestare et rationes p>rodeuntes ob- servare^ ut inductione tandem universalis propiositio innotescat. Est igitur exempjli gratia: + 1 2 = t = i 1 + 1 0+1 + 2 2 + 2 + 2 0+1+2+3+4+5 5+5+5+5+5+5" 15—1 14 WALLIS AND THE PROBLEM OF QUADRATURES. Et pari modo, quantumlibet progrediamur, prodxbit semper ratio sub- dwpla [\ ad 2). It is therefore to the inductive method that Wallis gives the pre- ference ; and inasmuch as the first term is zero, the expression n reduces itself to — ; that is, the sum of the series hears to that of a A like number of terms, each of which is as great as the greatest term of the scries, [or to nt^\ the proportion of 1 to 2, [erit series ad seriem totidem maximal cequalium, ut 1 ad 2. In like manner Wallis proves the further arithmetical propositions, for instance: Prop. XIX. Si proponatur series Quant itatum in duplicata ratione Arithmetics proportwnalium continue crescentium a puncto vel inclioa- tarum {puta ut 0, 1, 4, 9 etc.) propositum sit inquirere, quam liabeat ilia rationem ad seriem totidem maximae aequalium? Fiat investigatio per modum inductionis [ut in prop. 1) eritque 0+1 = 1 =s=m 1 + 1 = 2 ° . a ' * 0+1+4= 5 t 4 + 4 + 4=12 * + T * 0+1+4 + 9 = 14 t 9 + 9 + 9 + 9 = 36 ~3 +T * 0+ 1+ 4+ 9 + 16=30 + A 16 + 16 + 16 + 16+16 = 80 s" 1 "**' Et sic deinceps. Ratio proveniens est ubique major quam subtripla seu ^. Excessus autcm perpetuo decrescit prout numerus terminorum augetur. Adeoque (Proqws. XXL) si proponatur series infinita, erit ilia ad seriem totidem maximae aequalium ut 1 ad 3. Wallis had in the meantime remarked, in general terms, how great must be the continually decreasing excess, viz., erit (2)osito multitudine terminorum m. ct ultimi latere 1) — .Z 2 + — - - — -.I 2 : the last part of which 3 6m — 6 ' WALLIS AND THE PROBLEM OF QUADRATURES. 15 expression Z 2 is the excessus, qui perpetub decrescit, prout numerus ter minor um augctur. Thus Wallis had obtained the two important propositions — first, that the arithmetical progression, taken with a finite or infinite number of terms, bears to the sum of as many terms, all of the magnitude of the last term, the proportion of 1 to 2. Secondly, that the infinite number of terms, if the series be com- posed of the second powers of an arithmetical progression, bears to the corresponding multiple of the last term the proportion of 1 to 3, which is here not exactly the case for a finite number of terms, but with an excess that is continually decreasing. From these Wallis draws a number of corollaries, all which we here pass over; then he proves Prop. XXXIX., XL. and XLL, that if the series be conceived in triplicatd ratione aritlimetice proportionalium continue crescentium, as for instance 0, 1, 8, 27, 64, etc., their sum will be somewhat greater than \ that of as many numbers, each of which is assumed to be as great as the last term, but exactly \ thereof if they are taken in an infinite number. Propositions XL1I. and XLIII. next prove, that if a series is com- posed of fourth powers, their sum comes out in a similar manner = ^, and in the fifth power = |, etc., all which Wallis comprehends in the proposition : erit totius seriei ratio ad seriem totidem maximae aequalium y ut in hac tabella Primanorum \ id est 1 ad 2 Secundanorum ^ . . 1 ad 3 Tertiaiiorum \ . . I«i4 Quartanorum -| . . 1 ad 5. Et sic deinceps. Wallis therefore calls the simple arithmetical pro- gressions primana, the second powers secundana, and so on. From the above follow then, in considering the law for these sums of series, the sums of the series of square roots, cube roots, etc., and thereafter Theorema LIV: Si intelligitur series infinita quantitation a 16 WALLIS AND THE PROBLEM OF QUADRATURES. puncto inchoatarum et continue crescentium pro ratione radicum quad- raticarum, cubicarum : biquadraticarum etc. numerorum Arithmetice pro- portionalium (quam appello seriem Subsecundanorum, Subtertianorum, Subquartanorum etc.) erit totius ratio ad seriem totidem maximae aequalium ea quae sequitur in hac tabella, nempe: Subsecundanorum § id est ut 1 ad 1^ Subtertianorum f .... 1 ad lj Subquartanorum £ .... \ ad 1\ Subquintanorum £ .... \ ad \\. Et sic deinceps. Without entering here also into the applications, which Wallis everywhere appends as corollaries to his theo- a t t t t rems, we will here annex the one directly d /-^\0; following Corollary. Prop. LV. Wallis says, Ergo planum semijKirabolae est vel etiam Para- bolae ad Parallelogrammum circumscriptum ut 2 ad 3. Est enim Planum Scnuparabolae (aid d t— etiam Parabolae) series infinita subsecunda- norum {per 8 prop. Con. Sect.) ; ParaUeio- grammum autem series totidem maximo aequa- lium. Ergo illud ad hoc ut 1 ad 1^ vel ut 2 d~ ad 3. The eighth proposition of the tract De canicis seciionibus, to which Wallis here refers us, as he does throughout as treatise, exhibited in a purely geometrical form the property of the parabola, that in it the Quad rata applicatarwm sunt interceptis Diametris proportionality and that therefore the abscissae AD are to one another as the squares of the applicatcs [ordinatcs] (DO)*; so that consequently if the applicates stand at an equal distance from one another, their squares, when imagined in a series, form an arithmetical progression, or the applicates them- selves a progression of the roots of such a scries. Therefore Wallis just says here briefly, that an infinite series subsecundanorum V#, (for instance the series Vo, VI, V2, V3, VI, V5 continued ad infinitum) WALLIS AND THE PROBLEM OF QUADRATURES. 17 bears to the rectangle (namely, to the infinite series of lines of equal magnitude, all as great as the last line of the parabola,) the proportion of 2 to 3. After the several series of squares, cubes, &c, and also the series of square-roots, cube-roots and so on, have been summed, (and when thus the quadrature has been effected of all figures whose equations are y = ax or g = a"x 1 ) the 58th proposition remarks concisely, that from the above follows the rule for the series composed of powers and roots; for since, e.g. series subtertianorum (puta Vo, Vl, V2, \/3, Vl, etc.) rationem habeant (ad seriem totidem maxima? cequalium) earn quce est 3 ad 4, eorum quadrata [quce eadem sunt et radices cubical seriei secundanorum, puta Vo, Vl } Vl, V9, Vl6, etc.) rationem habebunt ad totidem maximal ozqualia, earn quoz est 3 ad 5. Quia nempe § , f , § sunt arith- metice proportionalia. It is now obvious, that all curves, or other regular geometric functions, whose equations are y = ax n , are squared by this rule ; and this is all comprehended* in Prop. LXIV., to which Wallis refers us in his preface, in order to call attention to its im- portance. It will be remembered that in the Preface (after reference made to this most general proposition) we are told, Deinde transii ad series auctas [quas voco,) et mutilatas sive deminutas, quae ex duarum pluriumve serierum aggrcgatis constant. Of this we will give one example, which will make it readily in- telligible. Sit verbi gratia (in Prop. CVIII.) terminus aequalium (et prima- norum maximus) It ; ejusque pars infinite parva dicatur a = — ; numerus * Wallis treats also of the cases with negative exponents, in which occur the several series - , -, -, -, etc., — — , -— , — — , -— , etc., and consequently 12^4 Vl V2 V3 V4 the fractional equation y = — . 18 WALLIS AND THE PROBLEM OF QUADRATURES. terminorum omnium [vel jigurae altitudo) A. R-Oa R + Oa R-la R+la R-2a R + 2a R-3a R+3a etc. etc. etc. etc. si termini continuentur .... R-R R + R erit (Residuorum) summa.. AR-\AR. et (Aggregatorum) summa \AR + \AR. Nempe si termini This is repeated in a still more general fonn in Prop. CXI. ? E' + Oa* R 3 + 0a 3 E* + ld> R 3 +la 3 R 2 + 2a* R 3 + 2a 3 R 2 + Sa 2 R 3 + 3a 3 etc. etc. etc. etc. continuentur usque ad R? + R* R? + R 3 If + Oa* i^ + la 4 R 4 + 2a i R' + Sa* etc. etc. R i + R* Summae Hoc est residuorum summa et Aggregatorum summa AR' + ^AR 2 AR 3 + \AR 3 AR i + iAR^ It is evident that Wallis had discovered Integration in the easier cases. The geometrical idea of Cavalleri and Gregoire de St. Vincent, of conceiving a surface as composed of lines, was by Wallis converted into an arithmetical one, and the process of summation was thus brought into the foreground. Even now a learner, who never readily compre- hends what integration as a method of effecting the quadrature of geo- metrical surfaces really is, cannot have a more distinct idea of it given him than by the plan and teaching of Wallis, who, beginning from WALLIS AND THE PROBLEM OF QUADRATURES. 19 the simplest case in the triangle, shows that the infinite number of small parallelograms, (or if you will, of lines) added together give the height of the triangle, and that consequently the formula of summation s = - — —-^— , in the case in which the number of terms is infinite, is applicable to geometry, and that so are likewise other arithmetical formulas of summation in other figures. n'2 CHAPTER III. THE INVENTION OF THE DIFFERENTIAL CALCULUS. We have seen that Barrow appended an arithmetical method of tangents to his geometrical works, and that for geometrical quadratures Wallis taught in general the arithmetical process of summation. When Mercator had discovered the quadrature of the hyperbola, of which Wallis had said hie hceret aqua, and when this created great sensation, Barrow wrote back j* A friend of mine brought me the other day some papers, wherein he hath set down methods of calculating the Dimensions of Magnitudes like that of Mr. Mercator for the Hyperbola, but very general ; and a couple of days later, / am glad my Friend' ] s Paper gives you so much satisfaction ; his name is Newton, a Fellow of our College, and very young, but of an extraordinary Genius and Proficiency in these things. These papers were headed, De analysi per cequationes numero termi- norum infinitas, and begin with the words Methodum generalem, quam * Amicus quidam, according to the Latin version in the Commereium Ejristolicum, nudius fortius chartas quasdam mihi tradidit, in quibus magnitudinum dimensiones supputundi Methodos Mercatoris methodo pro Hyperbola similes, maxime vero gene- rales descripsit est Mi nomen Netotonits, juvenis, et qui cximio quo est aenmine magnos in hue re progressus fecit. This is the passage that Weissenborn did not cite, but Ave suppose had in view when he assumed (what in the meantime is however doubtful) that Newton was also that amicus who (just at this time, for Barrow's work was published 1670) pressed Barrow, as Barrow said, to append to his geometrical work the arithmetical process for calculating tangents (see above, page 4). THE INVENTION OF THE DIFFERENTIAL CALCULUS. 21 de Curvarum quantitate per infinitam terminorumi Seriem mensuranda olim excogitaveram^ in sequentibus breviter expUcatam potius quhm accurate demonstratam habes. The contents of this compendium are, in the brief words of Newton, (leaving out all the calculations and applications) as follows: Basi AB (Curvae aliouius AD) sit applicata BD perpendicularis : Et vocetur AB—x^ BD=y et sint a, &, c etc. Quantitates datae et m, n } Numeri integri. Deinde CURVARUM SIMPLICIUM QUADRATURA. Trt+n Regula I. Si ax n —y; erit an x 7n + n RES EXEMPLO PATEBIT (1) fit x* = \x l = y; erit ±x 3 = ABB. (4) Si -2 = x~ 2 = y; erit = — - x ' = — of 1 = — = aBD. — 1 x infinite versus a protensae ; quani calculus ponit negativam, propterea quod jacet ex altera parte lineae BD. = Area ABB. COMPOSITARUM CURVARUM QUADRATURA EX SIMPLICIBUS. ReGULA II. St valor ipsius y ex pluribus istiusmodi Terminis componitur, Area etiam componetur ex Areis quae a singulis Terminis emanant. exempla : Si x 2 + x* = y; erit |cc 8 + fa 1 = ABB. Et si 3x - 2x 2 + x 3 -5x* = y ; Erit §x* - f x 3 + \x* - x h = ABD. ■" THE INVENTION OP THE DIFFERENTIAL 0ALC1 LUB. ALtOBUM OMNIUM QUADBATUBA. I,'m;i i. a III. ,SV valor ipeius y ril aliquis <jns Terminus sit prae- oedentibus magis compoeitus. in Ternwnoe Simpliciores reducendus est* operondo in Uteris ad ev/ndem Modv/rn. quo Arithmetici in nunteris Decir malibus dividunt^ Radioes extrahunt, vel affectas aequationes eolvuntj 1 1 i.i- ml is 'I'ii' hi in IS i/iunsi/mu Ciicrar sii/n rfu ii Wl, /nr />r<u mh nli s Hi - aulas deinceps < Itou Si/ h I kxkmi'I.a DIVIDENDO. // ; ('iirni niuijii r.ristrnlr 1 1 i/pi rhola I )irisionrm in- a* a'x aV <>:.■' stitnu it sir rirr liujiis ij proilil // ' | ' - <— etc. si nr istn injinilr emit inn/tin J Ailroi/ur (j>rr linjiilam Scat in/mil) Area . , /1/l/r . ,r.r aV aV aV quaCMta AltlK. ertt . ... I ... - ,, etc. in/iiiifar ilium si in i. 1 1 2// 80 Ah • ' CUIUS fin/nil jiiiiui initiates sunt in nsiim i/iu inn's satis r.racti, si nioila ./• sit aliguoties minor guam h eto. (nix pages) • Et liai r ili arris currant in in r< ::! ii/a/ulis dicta snf/!riant. ImO i'iiiii I'rulilrinata omnia ilr ctirrartim Lom/itiulinc i/c i/ttantita/r it tuperfiote Solidorum^ deque Oentro Oravitatis. possunt eo tandem re~ duOt "t iiiiai ratnr i/iiau/i/as Snji, rjiciri ji/amu lima cttrra terminator, mm opus i si quicquam de lis adjumgere. In istts utitrm (/tin n/o o/u rar moila ilicttnt hrrrissintc. APPLIOATIO PRAEDIOTORUW AD RELIQUA ISTIUSMODl PBOBLEMATA. 8it ABD citrra i/inuris, cl Al/h'l! nrtaui/n/iim cni'is lattts All rrl IU\ est nnitas. Et ci'ijita net, im DBS. nnifar- u miter ah All motam ) arras A tin rt AK </,•- scriherc ; it i/tuul />7\'(1) sit munirii/nm i/iio A K {.r) ,t />'/>(_//) momentum quo ABD gra- tlattm aiu/,tur; rt ijiioil r.r monii nta />'/> jnr- prlim '/<//<>, possis, jn r />ru, ,/icl, is rc/n/as, arcam AlU> ipso desoriptcm investigate^ rive <-nm arm Ah {.r) momenta I ilcscrijita confrere. THE INVENTION OF THE DIFFERENTIAL CALCULUS. 23 Jam qua ratione Superficies A 111) ex momento suo perpetim dato. per praecedentes regulas elicitur. eadem quael/ibet aha quantitas ex momento buo sic (Into eUcietur. TSxemplo reejiet clarior. LONGUTUDINES CUEVA BOM I KVEN I Jti:. Hit A1)1jK circuluH en jus cvrcus A I) longitudo est wdoganda. Ducto tanijcMc l)lll\ ct com/plcjo indefinite pa/rvo recta/ngulo IM1I>K ) et posito AE = 1 = 2 AC. Writ ut BK §ive (Ull^ momentum Basis AB(x). ad III) momentum Arcus A I):: HT: 1)1':: HI) 1 2 Vx — XX est momentum, Arcus A J). Quod rcductwm fit i '/' A R II {1)11). Adeoque i 2 V./; - XX sire {Jx-xx) :DO(i)i: 1 {BK) : 2 V:/: - 2x/X '2x — 2xx \x '■ + \x + fox* I l,x + sU x ' + ,,' ! iV' ; \ <■'<>- Qua/re per regulam secundantj longitudo Arcus A I) est ■'■' I I.-- I fa> I- rw +Tffe fl r + 2^,-'- V etc. sive x i in 1 + \x + fi/x* -f , \ 2 x" + , ; / \ ^ + 7 | ^.r', etc, Non secuB ponendo OB esse x. et radium OA esse- I, vnvenies Aroum LD esse x + \x* + -fox* I , \ ■,■>■' 1 etc. Hc,(i notandum est, quod wnitas ista quae pro momenta ponitur. est Superficies cum de SoUdds^ et li/nea cum de superficiebus^ et punctum cum fir lineis (ut in hoc exemplo) agitu/r. Nee vereor loqwi de imitate in punctis. sine Iwieis infinite parvis. si quidem proportiones ibi jam contemplantur Oeometrae^ dum utwntur me~ t/i odis IndwisthiUum. Ex Ids fiat conjectwra de superficiebus et quantitations soUdorum^ an de Centris Gravitatum. LTnder the headings — hi.ve.ui re, pruilietorn m eon eersum. 24 THE INVENTION OF THE DIFFERENTIAL CALCULUS. and Inventio Bast's ex Area, data, together with Inventio Basis ex Longitudine Curva, Newton proceeds to give that which is announced under these titles, and closes the compendium with the demonstration of the two leading propositions. Bespicienti, says he, duo pros reliquis demonstranda oc- currunt. I. DEMONSTRATE QUADRATURAE CURVARUM SIMPLICIUM IN REGULA PRIMA. PRAEPARATIO PRO REGULA PRIMA DEMONSTRANDA. Sit itaque curvae alicujus ABB Basis AB ' = x, perpendicular iter ap- plicata BB = y, et area ABB = z, ut prius. Bern sit B(3 = o 1 BK=v, et rectangulum Bj3HK(ov) aequale spatio B/3SB. Est ergo A (3 = x+o, et A8/3 = z + ov. His prae- missisj ex relatione inter x et z ad arbitrium assumpta quaero y isto, quern sequentem vides, modo. Bro lubitu sumatur §#* = z, sive %x* = zz. Turn x + o (A/3) pro x, et z-\-ov(A8/3) pro z suhstitutis, prodibit $ in x' 3 + 3x 2 o + 3xo 2 + o 3 = (ex natura Curvae) z 2 + 2zov + d 2 v 2 . Et siddatis (±x 3 et zz) aequalibus, reli- quisque per o divisis, restat $ in 3a?' 2 + Sxo + 6 l = 2zv + ov 1 . Si jam sup- ponamus B/3 in infinitum diminui et evanescere, sive o esse nihil, erunt v et y aequalesj et termini per o multiplicati evanescent, quare restabit - ( x\ $ x Sxx = 2zv, sive \xx (= zy) = f#;r 2 , sive x- ( = — J =y. Quare e contra si x* = y, erit fafy = z.* * Newton chooses here a complicated case; but if we take the most simple example, which he gives somewhere s = x 3 ; then, (if we suppose again that, in order to retain the figure, x = AB; x + o = AB; z = ABB) we have (x + o) s = a? + 3ox 2 + 3o 2 x + o 3 = s + ov; consequently 302* + 3o*x + o 9 = ov; therefore 3x* + 3ox i o* = v. Now o = zero gives 3*» = v = y. THE INVENTION OF THE DIFFERENTIAL CALCULUS. 25 DEMONSTRATE. Vel qeneralitef. si x ax " = z : sive. nonendo = c, et J m + n ' J m + n ' V on + n =p 1 si ex" = z sine cx p = z n : turn x + o pro x y et z + &v [sive, quod perinde est, z + oy) pro z, substitutis, prodit c n in x p +p>ox p ~ l , etc. = z n + noyz' 1 ' 1 ; etc. reliquis nempe terminis, qui tandem evanescerent, omissis. Jam sublatis c n x p et z" aequalibus, reliquisque per o divisis, . n o-i «-i / nyz n nyc'x p \ . ,. ., 7 „ restat c px = nyz \ = — — = — — — ) sive dividendo per c # p , erit ex' -i ny P -ir 7 7 na px = — ^ sive pcx = mi ; vet restituendo pro c, et m + n L £ ± ° m +n r ex pro p, hoc est, m pro p — w, et na pro pc, Jiet ax n = y. Qua,re e contra, si ax 1 =?/, erit — - — ax ™ =z. Q. e.d. *' m + n This is therefore Newton's compendium of the Differential Calculus, which in 1669 was sent from Cambridge to the President of the Society in London, as likewise to Collins. To this Newton added his Tangential Method in the letter which he wrote to Collins on the 10th of December, 1672, in which he said, as the reader will recollect, that he was glad that Barrow's " Lectiones" had met with so much approbation, and that his (Newton's) Method of Tangents which was indicated in the letter by an example, belonged to his universal method of Quadratures. Respecting the sensation which this discovery gave rise to, see the letters (printed in the Comm. Epist.) in which it was announced in Italy, France, and Holland. When Leibnitz had heard of it, and, speaking of some results of it which had come <to his notice, wrote, ( rem gratam feceris si demonstrationem transmiseris (letter to Oldenburg, of 22nd May, 1676), Newton wrote concerning his method and his mventipn, not yet published in detail, the letters to Leibnitz, which are printed in the Commercium Epistolicum. Newton begins by V)*\ ? 26 THE [NVENTTOfl OF THE DIFFERENTIAL CALCULUS. saying, Quamquam Dom. Leibnitii modestia in excerptis quce ex epistola ^a * ejus (ill me nuper misisti, nostratibus multum tribuit circa Specula- Cy^ / tioncia quondam infinitarum serierum de qua jam ccepit esse rumor quoniam tamen ea scire pervelit qua ab Anglis htic in re inventa sunt, — he is therefore willing to communicate some information. With regard to infinite series, the idea, he intimates, through which they were at first discovered by him, was that of extracting from algebraical expressions, in the same manner as from decimal fractions, the roots which could not be otherwise obtained. But inasmuch as all expressions of roots and divisions could be regarded as powers with fractional or negative u exponents; since, instead of V«, vV, etc., we could write the forms J& v i*" i a 11 _i \^ a\ a*, and instead of — j , j-= , the forms a 3 , a 4 , and so, in complicated * I X cases, instead of . the form a 2 (a 3 + bx*) 4 : therefore all the rules ^ Va 3 + bx l ^br the extraction of roots, were comprised in the one theorem which \ i> jt (he says) he will impart to Leibnitz, viz.* (P+PQf =F* +- AQ + 7 ^ BQ + ^^ CQ etc. n 2n 4n The generality, continued Newton, of his method could be easily shown by examples ; since,f for instance, the expression v c 2 + x 1 was by this method changed into the series c -f ( — ) # 2 — ( — = ) x* + ( = ) x* — &c. \2cJ V8c7 \16cV As a further example Newton chooses the expression : ' c 5 + c 4 x — x b . * By this means every curve, however complicated might be its equation, pro- vided it was not an implicit one, was brought under Wallis's rules of Quad- rature (summations) which in the letter were presumed to be known and were known to Leibnitz, as to all geometers, since 1659. f This refers, as Leibnitz knew, to the expression for the hyperbola, of which the quadrature (just given by Mercator) had appeared so difficult, and of which Wallis had said hie haeret aqua. Waliis' rule of summation is applicable to the series and not to the finite expression Vc" + x*. THE INVENTION OF THE DIFFERENTIAL CALCULUS. 27 or (c 6 + c*x — x 5 )\ Of fractional expressions also Newton annexes a few examples, viz. the cases in which there might come into the equation 1 1 a of the curve terms such as or or a + x {a + xy " y'b 3 + 3b 2 x+Bbx* + x 3 ' But if the equation were yet more complicated, that is to say, "affected,"* then the roots could be sought by approximation, of which Newton gives two examples. He then continues, Quomodo ex cequationibus sic ad infinitas series reductis arece et longitudines curvarum contenta et superficies solidorum vel quorum libet segmentorum figurarum quarumvis eorumque centra gravitatis determinentur nimis longum foret describere ; sufficiat specimina qucedam talium problematum recensuisse, inque lis brevitatis gratia liter as A, B, C etc., pro terminis seriei sicut sub initio nonnunquam usurpabo. With this introduction Newton gives, in nine examples, the entire results of his Analysis.f * This expression refers to what is now called an implicit equation, Thus Newton having heretofore brought under the Wallisian theorems, or (as we may say) method of Integration and rule of Quadrature, all curves with equations such as y = Vc 2 + x 2 ; y = tyc b + &x - x b , etc., or in general y =/(#), turns his attention here at last to the implicit functions, f{x,y) = 0, i.e. to expressions such as y 3 i axy + x % y - x 3 - a s = 0. f The difference between the contents of the Analysis, and that which is here communicated to Leibnitz, will be rendered evident by the following example. In the Analysis we read, (compare above, page 23) Longitudines curvarum invenire. Sit ADLE circulus (where AB = x, there- fore since AE= 1, {DBf = x (1 - x), or DB = V#(l - x) = V# - x 2 ) cujus arcus AB longitudo est indayanda. Ducto tanyente DHT et complete indefinite parvo rectangulo HGBK, et posito AE=l = 2AC; Erit ut BK sive GH, momentum Basis AB (x) ad 1' HD momentum Arcus AD :: BT: DT= BD {sive v'.r - x") : DC {sive £) = 1 {BK) : 2 V* - x s {DH)Adco- K 2 28 THE INVENTION OF THE DIFFERENTIAL CALCULUS. After Leibnitz bad answered tbis letter, and requested further ex- planations, Newton says in bis letter of 24th October, 1676, that the way in which he had hit upon a part of his method in the commencement of his studies was this, that he had endeavoured to interpolate the series, the interpolation of which Wallis had declared to be necessary for Quadratures that were too difficult for him.* He had written [he says] a compendium of his whole method, which had been communicated through Barrow to Collins, in quo significaveram Areas et Longitudines Gurvarum omniwm et Solidorum Superjicies et Contenta^ ex datis Rectis et vice versa ex Ms datis Eectas determinari posse. When he afterwards wished to make a treatise out of this, he had added [he states] other things, and in particular the method of drawing Tangents, which Sluse also had discovered, (but with this difference, that the method of Newton was applicable to complicated curves), but this method of Tangents and other things he preferred [he says] not to communicate to Leibnitz. To this Leibnitz answers in these words, in which it is said is con- tained his independent discovery of the Differential Calculus, viz. that ■ r x que , — , site — 3 est momentum arcus AD. Quod reduction Jit -111* 2 Quare per regulam secundum (one of the "Wallisian rules of Quadrature or Summation, which Newton makes use of) long Undo arcus AD est I .1 4 7 9 x + qX + 4 a; + Yi 2^" 'ii 62^ etc. Instead of these words and the accompanying figure in the Analysis, Newton in his letter to Leihnitz gives only the result : Si ex dato sinu vel sinu verso arcus desideratur et d diameter ac x sinus versus, erit arcus a 4 11 x* 3r* lis = d^x* + — + — (or *' + §** + / z J if d = 1). * Newton's words are : Sub initio studioriun tneorum mathematicorum ubi incideram in opera JVallisii, considerando Series quorum inter calatione ipse circuit Aream etc. This is the place where Wallis, in his preface, had said, hie haeret aqua. THE INVENTION OF THE DIFFERENTIAL CALCULUS. 29 whereas JSluse's method of tangents was not sufficient, he had discovered another, and to this he adds, arbitror quce celare voluit Newtonus de Tangentibm ducendis ab Ms non abludere. Leibnitz's words are : Claris- simi Slusii Meihodwm Tangentium nondum esse absolutam celeberrimo Newtono assentior. Et jam a midto tempore rem t Tangentium longe generalius tractavi ; scilicet per differentias Ordinatarum. Nempe T IB [intervallum Tangentis ab Ordinata in Axe sumptum) est ad IB lC Ordinatam, ut 1 CD [differentia duarum Abscissarum A \B, A 2B) ad D 2(7 (differentiam duarum Ordinatarum \B 1 0, 2B 2 0). Nee refert quern angidum faciunt Ordinatae ad Axem. Unde patet, nihil aliud esse invenire Tangentes, quam invenire Differentias Ordinatarum, positis differentiis Ab- scissarum (seu \B 2B = 1 CD) si placet aequalibus. Dine nominando (in posterum) dy differentiam duarum proximarum y (nempe A IB et A 2B) ; et dx seu D 2 C differentiam duarum proximarum x (prioris lBlC, posterioris 2B2C); patet dy* esse 2ydy ; et dy s esse Sy'dy, etc. et ita porro. Nam sint duae proximae sibi (id est, differentiam habentes infinite parvam) scilicet A \B = y ; et A 2B = y + dy. Quo- niam ponimus dy* esse differentiam quadratorum ab his duabus rectis, Aequatio erit dy* = y' 2 + 2ydy -\-dydy — y 2 . Seu omissis y 2 — y l quae se destruunt, item omisso quadrato quantitatis infinite parvae (ob rationes ex Methodo de Maximis et Minimis notas), erit dy' 2 = 2ydy. Idemqae est de caeteris potentiis. Dine etiam haberi possunt differentiae quantitatum ex diver sis indefinitis in se invicem ductis factarum : ut dyx erit=ydx+xdy; et dy' 2 x = 2xy dy + y l dx. Dine si aequatio a + by + cx + dyx + ey 2 +fx 2 + gy 2 x + hyx 2 etc. = / statim habetur Tangens Curvae ad quam est ista Aequatio. Nam ponendo AB = y, et A2B = y+ dy (scilicet, quia IB 2B seu lCD = dy) ; Itemque ponendo \BlC = x, et 2B 2C=x + dx (scilicet, quia2CD = dx), et quia 30 THE INVENTION OF THE DIFFERENTIAL CALCULUS. eadem aequatio exprimit quoque relationem inter A 2B et 2B2C, quae earn exprimebat inter A \B et \B 1 C ; Tunc in aeguatione ilia pro y et x substituendo y + dy, et x + dx,Jiet a+ by + ex + clyx + ey 2 + fx' + gy 2 x + hyx 2 etc. ■ uif t i* t ui/u, t ty t jtio t yy < c -r/iyx eic. bdy + cdx + dydx + 2eydy + 2fxdx + 2gxydy -\-2hxydx etc. + dxdy + gy 2 dx + hx 2 dyetc. = 0. + ddx dy + edy dy -\-fdx dx + gx dy dy + hy dx dx d est quantitas communi more. + 2gydydx + 2hxdxdy etc. d est nota Differentiae. + g dx dy dy + h dy dx dx Ubi, abjectis Wis quae sunt supra primam lineam, quippe nihilo aequa- libus per aequationem praecedentem : et abjectis Mis quae sunt infra secundum, quia in Mis duae infinite parvae in se invicem ducuntur : hinc restabit tantum aequatio haec />dy + cdx + dydx + dxdy etc. = 0, quicquid scilicet reperitur inter lineam primam et secundam. Et mutata aequa- tione in rationem seu Analogiam, fiet dy _ c + dy + 2fx + gy 2 + 2h xy etc. dx b + dx-t 2ey + 2gxy ■+ hx 2 etc. ' Id est . dy -\B2B,seu-\CD T lB\ .c+dyetc. T \B qma _ £ seu » = _ ^^ emt dx D2G IB 2 C) b+dx etc. lBlC Quod coincidit cum Regula Slusiana, ostenditque earn statim occurrere hanc Methodum intelligenti. Sed Methodus ipsa {priore) nostra longe est amplior. Non tantum mini v.rhiberi potest, cum plures sunt lite roe indefcrminatae quam y et x [quod saepe jit maximo cum fructu) ; Sed et tunc utilis est cum inter- rm in nt irrationaleS) quippe quae earn nullo morantur modo, neque idlo modo necesse est irrationcdes tolli, quod in Methodo Slusii necesse est, et calculi difficultatem in immensum auget. Quod ut appareat, tantum utile erit in irrationalitatibus simplicio- ribus rem explanare. Et primum sit in simpUcissimis generaliter. Si sit aliqua potentia aut radix x : erit dx' = zx e ~ l dx. THE INVENTION OF THE DIFFERENTIAL CALCULUS. 31 Si z sit £, seu si x sit \jx, erit d# z , seu hoc loco d \/x = \x * dx — r- ; ut notum aut 2 *Jx ' Sit jam Binomium, ut seu — 7- : tit notum aut facile demonstrabile 2sjx' J V 3 : a ■+■ by + cy 2 etc. quaeritur d V 3 • a + by + cy* etc. seu da/, posito iy = 3, et a -f by + cy 2 etc. = x. Est autem dx — bdy + 2cydy etc. Erqo dx" seu — = ^—^- — . Eadem Melhodus adhiberi 3x 3 3 xa + by + cy 1 etc. \ ■potest etsi Radices in Radicibus implicentur. Sine si detur aequatio valde intricata, ut a + bx^y 2 + b^ : 1 +y + hx' 2 y^y* + y Vl -y = o, ad aliquam Curvam cuius Abscissa sit y (AB), Ordinata x (BC) 1 tunc Aequatio proveniens utilis ad inveniendam Tangentem TC, statim sine calculo scribi poterit ; et erit liaec bx , bdy ' OX ■ y 2 + b\/ 3 : 1 + y + — - - = x 2y dy + 2 V + b V 3 : 1 + y 3 x 1 + y~f + hx 2 dy + 2hxydx x ^ y l + y*J\—y H V X = x 2ydy + dy*Jl-y f~^~ =0 - 2Vf + y*/l-y 2Vl-y SeU) mutando Quotientem hanc inventam in Analogiam^ erit — dy ad da?, seu T \B ad \B 1 (7, ?t£ omnes provenientis aequationis termini per dx multiplicati, ad omnes ejusdem terminos per dy multiplicatos. Ubi sane mirum et maxime commodum evenit, quod dy et dx semper extant extra vinculum irrationals. Methodo autem Slusiana omnes ordine irrationales tollendas esse nemo non videt. Arbitror, quae celare voluit Newtoniis de Tangentibus ducendis, ah his non abludere. We see here that Leibnitz avoids making any mention of a 32 THE INVENTION OF THE DIFFERENTIAL CALCULUS. Newtonian method of tangents. We remember Barrow's method of tangents in which he had given : T a figure and a process quite si- milar to those of Leibnitz's letter : in which figure, (as may be seen above Chapter I., page 5) Barrow took MR = a, NR = e 1 and gave the rules: Inter convputandwm omnes nhjicio terminos, in quibus ipsarum <7, vel e, pofr.s/V/.s habetur vel in quibus ipsae ducuntur in se, and we remember that Newton, supplementing Barrow, said in his letter of 10th December, 1672, that his (Newton's) method of Tangents could be apprehended from an example, in which he mentions the ride and then says: hoc est union particulare methodi generalise quae extendit se non mode ad ducendum tangentes verum etiam ad resolvendum alia abstrusiora problemata de curvitatibus. We repeat that, if Leibnitz was acquainted with this letter of Newton's, in which was contained the Newtonian method, which was Leibnitz's method, he ought not to have avoided the mention of this method, and ought not therefore to have written, arbitror qua) a lire voluit Newtonus de Tangentibus ducendis ah his non abludere, but to have written: quo? celare voluisset Newtonus mihi cognita sunt, nam litems ejus \Qmi Decembris, 1G72, ins/ ><.<•/'. "the invention of the differential calculus. 33 Nor let it be said, that it was Newton's business to remember in 1676 that he had in 1672 written that letter on Tangents to Collins, and that Collins might perhaps have communicated it to a friend ; or that, if Newton, after the lapse of several years, did not remember this fact, then Leibnitz too might forget it; for although indeed it was four years since Newton had written the letter to Collins, yet it was not four years since Leibnitz had been acquainted with the letter, and in the next place Leibnitz was the learner, who would pay attention to all that came new to him ; whereas Newton was not, like a learner, attentive to the extent of what he communicated, and could quite forget having made this communication which after all he had sent only to Collins and not to Leibnitz. That Newton's letter on Tangents had actually been known to Leibnitz, has been only rendered evident since 1849 by Gerhardt's discovery of the abstract which Collins had got ready for Leibnitz, containing the last part of this letter of Newton's upon tangents in the words : quod scilicet Dn. Newtonus cum. in Uteris suis 10 Decembris, 1672, communicaret nobis methodum, ducendi tangentes ad curvas geo- metricas ex aequatione exprimente relationem ordinatarum ad Basin, subjicit hoc esse unum particulare, vel corollarium potius, methodi gene- ralise quae extendit se absque molesto calculo, non modo ad ducendas tangentes accommodatas omnibus curvis, sive Geometricas sive Meclianicas, vel quomodocunque spectantes lineas rectas, aliisve lineis curvis ; sed etiam ad resolvenda alia abstrusiora problematum genera de curvarum jlexu, areis, longitudinibus, centris gravitatis etc. Neque (sic pergit) ut Huddenii methodus de maximis et minimis, proindeque Slusii nova Methodus de tangent ibus, (ut arbitror) restricta est ad aequationes, Sur- darum quantitatum immunes. Hanc methodum se intertexuisse, ait Neivtonus, alteri illi, quae aequationes expedit reducendo eas ad infinitas series; adjicitque, se recordari, aliquando data occasione, se sigirijicasse Doctori Barrovio, lectiones suas jam jam, edituro, instructum se esse tali methodo ducendi tangentes, sed avocamentis quibusdam se praepeditum, quominus earn ipsi describeret. F 34 THE INVENTION OF THE DIFFERENTIAL CALCULUS. In the next place Edleston attests that a copy of the whole letter was sent to Tschirnhaus in Collins's paper " about Descartes," and Gerhardt confesses that Leibnitz and Tschirnhaus, in 1675, worked together, and were so intimately connected that they used the same paper and the same ink and pen.* Even before these most recent investigations, it was concluded from the silence of Leibnitz that the Newtonian letter on tangents was known to him ; for this much had been distinctly affirmed in the Commercium Epistoliciim, and Leibnitz did not deny it. Prof. De Morgan has discovered that in the first edition of the Comm. Epist., which appeared in the lifetime of Leibnitz, this distinct affir- mation and statement is only found in the judgment pronounced by the Committee, nusquam mentionem reperimus Methodi ejus Differ enticdis * Edleston's Correspondence, etc., page xlvii, and Gerhardt (I., page 91) and Tract II. of 1855, page 68. For superabundance we have Leibnitz's own words to prove his intimacy with Tschirnhaus, see Leibnitz's letter to Oldenburg of 28th De- cember, 1675: Quod Tschirnhausium ad nos misisti, fecisti pro amico — inventa mini ostendit non pauca Analytica et Geometrica. Leibnitz moreover, if Tschirnhaus had hesitated to show him what he had, could most easily see the whole letter when he was in London, in October, 1676. Gerhardt, in his recent volume (Leibnitz Mathem. Schr. Band IV., 1859, page 420) says: "In May, 1675, Tschirnhaus was not in Paris, but either in London "or on his way to London." This remark, instead of assisting Leibnitz, as Gerhardt thinks, corroborates on the contrary Edleston's statements. Indeed what was sent to Tschirnhaus at that time, being altogether fourteen folio leaves, (Edleston, I.e.) it would be difficult to suppose that this was sent to Paris, but not difficult to conceive that it was sent to Tschirnhaus, while he was in London. Tschirnhaus received them in London and returned with them to Paris. Gerhardt states (ibidem, Vol. V., page 421) that Leibnitz's mathematical manuscripts of the year 1675 prove the intimacy of Leibnitz and Tschirnhaus : " on the same leaf (says Gerhardt) we find " the pen and handwriting of Tschirnhaus, together with the pen and handwriting " of Leibnitz." Tschirnhaus's answer to Collins's paper is quoted in the Commercium Epistolicum as received June 8, 1675. Collins, in the "Extracts from Mr. Gregory's Letters," says, that Tschirnhaus Mas "here a quarter of a year in the summer of 1675." (I am in- debted to the kindness of Mr. Edleston for this memorandum made by him when he examined (lie papers at the Koyal Society.") THE INVENTION OF THE DIFFERENTIAL CALCULUS. 35 ante litems ejus {Leibnitii) 21 Junii, 1677, hoc est Anno integro post- quam D. Newtoni Epistola, 10 Decembris, 1672, scripta Parisios ipsi communicanda transmissa fuit, while in another edition which appeared after Leibnitz's death the remark was also found in a second more conspicuous (?) place in the Commercium Epist. (on the occasion of the printing of this letter, Missum fuit apographum hujus Epistola} ad Leibnitium mense Junto, 1676). This is an unheard-of thing! exclaims De Morgan. This Prof. De Morgan could fairly do, as long as it was assumed, while he made this great discovery (Philos. Magazine, June, 1848) that Leibnitz had not seen the letter of 10th December, 1672 ; for then it was felt that Leibnitz's silence proved more against him, if the statement of the fact appeared twice in the Commercium Epistolicum than if it was only once there. But since 1849, now that we can no longer doubt that Leibnitz received the entire letter from Tschirnhaus, and the reference to the same from Oldenburg, this discovery of a variation in the statement about it, by which Prof. De Morgan has become celebrated, obviously loses all that immense importance which it may ever have had. But instead of now giving up his attempt of proving a case against Newton, Prof. De Morgan asseverates yet more strongly (in the Companion to the British Almanac of 1852) that it is clear from Gerhardt, that it was not the letter of 10th December, 1672, but an abstract of the same, that was sent to Leibnitz direct, while the whole letter was sent to Tschirnhaus, and that hence the deceitful design of Newton and of the Committee was manifest. Prof. De Morgan here perverts the case entirely. There is indeed an error in the Comm. Epist. There were in fact at the time of the Comm. Epist., 1712, on the shelves of the Society, two rolls of Collins's bound together, one with the heading, Extracts from Mr. Gregory's Letters, to be lent Mr. Leibnitz to peruse, who is desired to return the same to you, (in which was contained the whole letter of 10th December, 1672), and another in which this letter was found F'2 36 THE INVENTION OF THE DIFFERENTIAL CALCULUS. only in abstract, as Gerhardt gives it, this also having been inscribed by Collins, To Leibnitz, 14 June, 1676, about Mr. Gregory's remains; and the Committee, as well as Newton, assumed erroneously that the tirst roll had been sent to Leibnitz, instead of the second. But into this error Newton fell even in the lifetime of Leibnitz, because already in the first edition of the Gomm. JEpist. it was registered that the first named collection was to be lent to Leibnitz; the smaller collection, which gave this letter only in abstract, was not referred to at all. Therefore Leibnitz's death has nothing at all to do with the matter, and that which was said before Leibnitz's face may have been mis- stated through error, but cannot have been misstated with a deceitful intention. " Something can he allowed for harry" says Prof. De Morgan himself, "for the Committee was appointed in parcels on March " 6th, 20th, 21th, and April llth, and their report was read as early as "April 2±th." So important is and has always appeared the question, whether Leibnitz read the letter of 10th December, 1672, that in order to excuse the silence of Leibnitz, Prof. De Morgan only a short time before Gerharclt's discovery, and so also Biot in the Journal des Savants following strictly De Morgan, made a great stir, because the affirmation that Leibnitz had seen the letter, came only once in that edition which appeared in Leibnitz's lifetime, and not twice, as in the second edition. Moreover we see even now Prof. De Morgan catching at a straw, and saying, that it was only an abstract, not the whole, of the letter that had been communicated to Leibnitz. But now that since the year 1849, through the investigations of Gerhardt and Edleston, this whole matter is seen in a pretty clear light — Leibnitz having been so intimate with Tschirnhaus, that they worked on the same paper, and the whole letter having been sent to Tschirnhaus, and an abstract to Leibnitz, Leibnitz's advocates turn round, and say, after all the strife they themselves raised upon the subject before L849, that it is of no consequence whether Leibnitz did read this letter of Newton's on tangents — u a sheet of blank paper, after THE INVENTION OF THE DIFFERENTIAL CALCULUS. 37 " what Shise had published, would have done just as well as the abridgment " or the ichole" says Prof. De Morgan at the end of his essay of 1852. The reader will smile at here seeing, that the same Prof. De Morgan would first have us consider the fact as so important, that an increase of reputation might be gained by it, and secondly so unimportant ! 11 Only one [document) is undated, and this is that on which the whole "turns" said Prof. De Morgan in 1848; and in 1852 he exclaims, " a sheet of blank paper would have done as well." It is impossible to contradict one's self more glaringly. But let us quit Prof. De Morgan, and cast a look backwards. It cannot have escaped observation, that we have hitherto not employed the word fluxions. It will be asked why. The answer is because Newton in his Compendium on the Calculus in 1669 has not once used this word. Also those supporters of Leibnitz are therefore mis- taken, who sometimes introduce him as the discoverer of the Differential Calculus, and Newton as the discoverer of something else which they prefer to call " FluxionaV calculation; (as in Gerhardt's Tracts we find two separate chapters under the headings, Discovery of the " Fluxional 1 ' calculus by Newton, and discovery of the higher analysis by Leibnitz). The thing which the second comer did or did not discover, is not different from that which the first comer discovered. Newton gives (see above pages 21, 24) in his Compendium 1669 the rules of the calculation : Regula 1 ; Si v, . cm ^ ax = ii ; erit x = area ; * ' m+n ■ he works this out in all examples, and says (under the heading : Applicatio praedictorum ad reliqua hujusmodi problemata) : Jam qua ratione superficies ex momento suo perpetim dato per praecedentes regulas elicitur, eadem quaelibet alia quantitas ex momento suo sic dato elicietur. Exemplo res fiet clarior. Then follow the examples under the head- ing, Longitudines curvarum invenire, among which we have the small (afterwards so called, Differential) triangle (see the figure above, page 22), 38 THE INVENTION OF THE DIFFERENTIAL CALCULUS. then we have in general the title, Invenire prcedictorum conversum. The — (XYl =£— first proposition, Si ax = y ; erit x " = area =z, is at last proved (at the end of the whole Treatise), where we read, Si ax n = area = z. m7 (we have Newton's proof, the same as is now given, effected by giving the increment o to x and then putting o = zero), erit ax n = y. Then Newton sa) r s, quare e contra si ax n = y, erit ax " = z. In this Newtonian discovery of the Differential Calculus, which is contained in the Com- pendium, the word fluere does not occur. It is in 167G that Newton, in his letter of the 24th October to Leibnitz, in the passage which was not legible for the latter, data ozquatione quot- cunque fluentes quantitates involvente fluxiones invenire et vice versa, first uses the word fluere, in order to indicate succinctly the whole method of calculation employed in the Compendium. CHAPTER IV. BIOT'S JUDGMENT ON THE DISCOVERY OF THE DIFFERENTIAL CALCULUS. M. Biot, as is well known, has expressed himself very specifically on these questions in the Journal des Savants, and 1856, in the new edition published by him and Lefort of the Commercium epistolicum. We are sorry to find that M. Biot everywhere quite echoes the words of Prof. De Morgan, which can only be explained by supposing, either that M. Biot purposely refuses to see how the thing stands, or that holding fast by the customary differential form and style of writing of the present day, he is unable to throw himself back into the past. It is an extraordinary thing, that Wallis is not mentioned in that part of the Commercium Epistolicum in which M. Biot and M. Lefort give the names of those, " whose labours paved the way for the discovery " of the infinitesimal Analysis [" dont les travaux ont prepare Vinvention " de V analyse infinitesiniale 1 ^. " On s'etonnera pent etre, says M. Biot or his co-labourer M. Lefort, " page 254, de ne rencontrer ni Wallis ni Huygliens. Cependant " Wallis et Huygliens ne me paraissent avoir aucun droit direct de paternite " sur les nouveaux calculs, quHls ont tous deux meconnus, le premier plus " encore peut etre que le second. [The reader will perhaps be astonished, " says M. Lefort, or M. Biot, page 254, at not meeting with either "Wallis or Huygliens. But Wallis and Huygliens do not appear to "me to have any direct paternal right in the new calculations, which 40 mot's judgment on the discovert " thev have, both of them, misappreciated, the first perhaps even more " so than the second."] Thus we see that Wallis is made of so little consequence, that while persons remotely interested in the matter are named, such as Fermat, Cavalleri, Hudde, Sluse, and while these and even Descartes and Ricci are quoted from, in order to give specimens from those whose travaux ont prepare V invention au dix-septihne siecle, Wallis is not even admitted. We on our side affirm that it is, above all, the works and the labours of Wallis, that the whole Commercium Epistolicum and Newton's letters to Leibnitz, as well as Leibnitz's letters, always presuppose and refer to. This Messrs. Biot and Lefort ought not to have overlooked, were it only for the Becensio. Therein Newton says : Per infinitas aequationes intelJiguntur illae, quae involvunt Seriem terminorum con- vergentiuni et ad veritatem propius propiusque accedentium in infinitum ; ita ut postremo a veritate distent minus xdla data quantitate ; et, si in infinitum continuentur, nullam omnino difi'erentiam relinquant. Wallisius in Opere suo Arithmetico, piihUcato A.D. 1657, Cap. 33. Prop. 68. reduxit fr -actionem - y. per perpetuam Divisionem in seriem A + AR + AB 2 + AS* 4 AM* + etc. Vicecomes Brounker quadravit Hyperbolam per hanc seriem 1111 + o 7 + ~r — ~, + ; — 7, + etc. 1x2 3x4 5x6 7x8 hoc est per hanc 1 — ^ + ^ — 4 + 5 — <j + 7 — »+ etc - conjungendo singulos binos Terminos in unum. Et haec Quadratura publicata est in Actis Regiae Societatis, mense Aprili 1668. Paulo post Dominus Mercator evulgavit Demonstrationvm J/i/jus Quad- rntttrae per Divisionem Domini Wallisii ; et deinceps I/and multo p>ost Jacobus Gregorius Geometricam ejusdem Demonstrationem in lucem edidit. And further [ibidem): [Xcu-foni compendium 1669] — continet praedictam generalem Methodum Analyseos, monstrant&m qaomodo resolvendae sunt finitae Aequationes in infinitas ; utque per Methodwm Momentorum wppli- OF THE DIFFERENTIAL CALCULUS. 41 candae sunt Aequationes tarn finitae quam infinitae ad omnium proble- matum solutionem. Incipit vero, ubi finem fecit Wallisius. et methodum Quadraturarum super tres Regulas struit. Wallisius Anno 1655, Arithmeticam suam Infinitorum in lucetn dedit ; per cujus libri Propositionem LIX. si Abscissa cujusvis Curvilinearis figurae vocetur X, et n atque m sint Numeri, et Ordinatae ad rectos m+n angulos erectae sint X " ; Area figurae erit X " . Atque hoc assumitur a D. Newtono, tanquam prima Regula, super quam fundat suam curvarum Quadraturam. Wallisius autem propositionem hanc de- monstravit gradatim, per multas particidares propositiones / tandemque omnes in unam collegit per Tabulam Casuum. Newtonus vero omnes casus in unum reduxit^ per Dignitatem cum indefinite Indice : et sub extremo Compendii, semel simulque demonstravit per Methodum suam Momentorum ; primusque indefinitos dignitatum Indices in Operationes Analyseos introduxit, Ceterum per 108 Propositionem Arithmeticae Infinitorum Wattisii, perque plures alias propositiones quae sequuntur ; Si ordinata com- posita fuerit ex duabus vet pluribus ordinatis cum signis suis + et — acceptis ; Area composita erit ex duabus vel pluribus areis cum signis suis + et — acceptis respective. Atque hoc a D. Newtono assumitur, tamquam Regula secunda, super quam instituit suam Quadraturarum methodum. Hence we see that it is not by mere accident, like almost all those whom Biot and Lefort mention as being important, but much more directly, that Wallis comes into the question, if we look back at the position of this branch of knowledge immediately before the discovery, as Messrs. Biot and Lefort wish to do, at least as they tell us. In the next place it becomes very clear from the commencement of Leibnitz's letter to Newton of the 27th August, 1676, that the Wallisian simple beginnings of quadratures, (Wallis's method, we might say, of integration) are the foundation of the correspondence between G 42 biot's judgment on the discovery Leibnitz and Newton. For after the exposition of Leibnitz's method of transmutation, which then appeared to him a correct one, Leibnitz says, Unde ad quadraturas absolutas, vel hypothetical Gcometricas, vel serie infinitd expressas Arithmetics jamjam multis modi's potest perveniri. "What other are these methods of quadrature, which are nowhere de- scribed, and which are therefore assumed as known, but those of Wallis ? Indeed Leibnitz immediately after says downright . positd fi infinite par vd % an expression and an idea which Wallis has, and if any one will not admit that Leibnitz took this out of Wallis, it would be for that very reason more necessary for him to mention at least that Wallis has also this expi'ession. In the second letter to Leibnitz, Newton again says expressly that it was through the interpolation problem, which Wallis had instituted, that he himself had been led to his new invented method. Here Newton continues speaking to Leibnitz of these opera of Wallis's for many pages together, and takes it for granted that they are known to Leibnitz, and speaks of them as the foundation of the process for effecting the quadrature. Finally Leibnitz, to name him too once more, begins hi* letter to Newton of the 21st June, 1677, with the words, Egregie placet, quod descripsit qua via in nonnulla sua elegantia Theoremata in- cident (Newtonus), et qua' <!<■ Wallisianis interpolationibus habet, vel ideb placent, quia hac rations dbtinetur harum interpolationum demonstration cum res ea antea (quod sciam) sola inductione niteretur. It is obvious that the inductions and the quadratures of Wallis are everywhere pre- supposed by Newton as well as by Leibnitz as a foundation. Messrs. Biot and Lefort, who name and quote from Format, Cavalleri, Iliulde, Sluse, and even from Ricci and Descartes, as auteurs qui ont prepari Vinvention au dix-septihne siecle make thus no mention of Wallis, and the grounds on which they justify themselves are, that " Wallis et Huyghens^ ont le premier plus encore que le second, meconnu " ces nouveaux calculs [Wallis and Huyghens have misappreciated, " the first even more than (he second, the new calculations].'''' This is doubly incorrect. Even if it is substantiated, that Huyghens, after the OF THE DIFFERENTIAL CALCULUS. 43 discovery had been made, did not immediately understand it, or that he did not acknowledge it, yet he ought not on account of this ex post facto behaviour of his after the discovery of the Differential Calculus to be excluded, but for another reason, namely, because he had not before taught anything that in the main or somewhat nearly resembled the Differential Calculus. Descartes and Fermat also misappreciated the Differential Calculus, that is, they did not understand it at all, because they were not at all acquainted with it, and yet these writers are quoted from. The question, does not turn upon what an author said after the discovery, but upon what he said beforehand that was like it. But further, though it be true of Huyghens, that he was not favorably disposed to the Differential Calculus, (though this is not the ground on which he should be excluded,) yet the fact does not stand thus with Wallis, because it was he himself in 1693, who in his treatise de Algebra highly extolled the calculus invented by Newton, and published it before Newton himself. The justice of Wallis's title is, we hope, distinctly clear to the reader, because we have given in Chap. II. Wallis's quadratures by integrations and summations in the simple cases which he could master (of the others he said honestly hie haeret aqua, thereby ex- citing and forcing Newton). We can admit that Biot, amidst his present multifarious occu- pations, is not answerable for the book which appeared in 1856 under his name and the name of the real author M. Lefort, this Biot tells us in the Preface: U V execution appartient tout entiere a Mr. Lefort; " et il s'est acquitte de. cette tdche — avec une puissance de travail, que u je suis keureux de reconnoitre, mats quHl m'aurait ete imptossible d y y " apporter. [The execution belongs entirely to M. Lefort, and he has " acquitted himself of this task with a diligence, which I am thankful " to acknowledge, but which it would have been impossible for me to "have exercised."] But M. Biot himself in the Journal des Savants for 1832, forgets the claims of Wallis; his words are (page 267, line 2): " la premiere G 2 44 biot's judgment on the discovery " lettre de Newton a Leibnitz contient les resultats de Newton sur les " series, notamment la formule du binome, le tout sans aucune demon- " stmt ion ni indication de methode quelconqne, disant settlement, quHl u in possede une a Vaide de laquelle ces series etant donnees, il pent " obtenir les quadratures des courbes dont elles derivent. [The first " letter of Newton to Leibnitz contains the results arrived at by " Newton in reference to series, particularly to the binomial theorem, " the whole without any demonstration or indication of a method, " saying only that he is in possession of one by which these series " being given, he can obtain the quadratures of the curves from which " they ore derived."] This is not correct. The parts of Newton's method there in question are not only, I. the reduction of complicated equations to series of the powers of x which he openly commu- nicated, and II. a method which he concealed of effecting the Quadra- ture of these series, for the Quadrature of equations reduced to simple powers was a thing understood of itself, and that could not be kept back, because Wallis had long ago given this in his works, which were well known to everybody. So it happens that Biot, because the method of tangents of Bar- row and Newton does also not seem important to him, inasmuch as Descartes and Fermat, had had another, comes to this perverse con- clusion in reference to the whole matter, that we must assume " trots " phases de V invention Men marquees: 1° emploi des evanouissans comme " methode aux fonctions rotionelles, ceci appartient specialement a Fermat ; " 2° extension aux fonctions irrationnelles par le developperneM en serie, " surtout au moyen du theoreme du binome, voila la part speciale de " Newton ; 3° reduction de cet artifice particulier en methode generale " de calculj c<>il<t Leibnitz, [three well-marked phases of the inven- tion, 1. the employment of vanishing quantities as a method for " rational functions, this belonging especially to Fermat ; 2. the ex- " tension of this to irrational functions by development in the form " of a series, especially by means of the binomial theorem ; this being "the special part of Newton; 3. the reduction of this particular OF THE DIFFERENTIAL CALCULUS. 45 " artifice to a general method of calculation, in which lies Leibnitz's " invention."] Thus Newton is made to have discovered only une part specielle and un artifice particulier, [a special part and a particular artifice], but Leibnitz une methode qenerale, a general method ; in other words, Biot will allow Newton only the developpement en serie, and not the general calculation data aiqiiatione^ quotcunque Jluentes quantitates in- volvente, jluxiones invenire et vice versa. We see that Newton is taken into high company and must content himself, according to the judgment of M. Biot, with only that modest place which Fermat and Leibnitz just leave between them. This judgment of Biot's however is quite unsound, because Fermat hardly deserves to be named at all, and secondly, because Newton's Compendium, the Analysis of 1669, contains not only the developpe- ment en serie but also the methode generate^ which consequently is not in any way left for Leibnitz, if we leave out of consideration that he was perhaps not acquainted with Newton's discovery. Indeed, if Biot had said only that Newton's discovery of a general method was not to be taken into consideration, because Leibnitz had not seen it, then the question would turn upon this, whether the communication of the method of tangents, and whatever else Newton communicated to Leibnitz, was or was not sufficient to deprive Leibnitz of the honour of the discovery. But M. Biot not only says that nothing whatever but the method of series was communicated to Leibnitz, but he says that Newton had also discovered nothing else, which is tan- tamount to saying, that Newton in his letter to Leibnitz concealed nothing, and did not write his anagram, data aequatione, etc. The ground of the exclusion of Wallis and of the higher eulo- gium of Leibnitz, lies evidently in the design of making Fermat a co-discoverer of the Differential Calculus. It is necessary to throw dust in people's eyes, in order to make this long ridiculed crotchet of the French wear again for a moment a serious appearance. It has been ridiculed, I repeat it, not only in England, but also always 46 biot's judgment on the discovery in Germany ! But evidently this was Biot's design ; with this view he pronounces an unfair judgment against Newton, and with this view he and Lefort publish a comic edition of the Commercium Epist. with a French sauce ; M. Biot sees that the strife between Eng- land and Germany is being renewed ; " therefore we in France must " put in our word", said M. Lefort to M. Biot, or said M. Biot to M. Lefort, and so they got up an edition with clever addeuda, in order to thrust in Format in an unsuspicious manner. Biot gave the capital of his reputation, and Lefort the capital of his labour for this joint-stock concern, in which the little various readings of the first and second editions of the Commerc. Epist. are taken advantage of, in order to have an excuse to set aside Wallis* and to push in Descartes and Fermat. * The excuse on s'etonnera peut etre de ne voir ni Wallis ni Hvyghens, qui out tons deux meconnu las nouveaux calculs is characteristic of the whole book. The idea, which this phrase is designed to convey, is that Huyghens (who was certainly not a Frenchman, but who -having lived in Paris and having been received into the Academie des Sciences at the time of its foundation — in order that it might not be composed of only minor celebrities— has been ever since looked upon by the French as a fellow-countryman,) might appear in the same degree as Wallis, an Englishman, to have prepared the way for the discovery which was then im- pending, and that the omission of the one in Biot and Lefort's book balances that of the other. Even connoisseurs, if they do not keep a strict watch over their memories, arc liable to be lulled asleep by these smooth phrases, especially because Huyghens was so very celebrated on other accounts. If the reader does not allow the phrase to domineer over his consciousness, then he thinks that the less important Huyghens was left out in order to justify the omission of Wallis. At the third stage of reflection one gets hold of the idea, that Huyghens never came under consideration, not even in Biot and Lefort's idea, and that he is only empha- tically named here ; so then one perceives that to be the best interpretation which credits the author of these lines with the greatest share of French ingenuity; for although Fermat and Descartes are thrust in, yet because the date of their dis- coveries is so very remote, it might annoy a Frenchman to find all more modern countrymen of theirs, (Members (!) of the Parisian "Academie des Sciences", which had been in existence since 1666,) not even mentioned by name in Messrs. Biot and Lefort's Book;— on this account they here at least introduce Huyghens, who was celebrated on other accounts, and who lived in Paris. Thus we kill two birds with one stone ; we get rid of Wallis and have mentioned Huyghens. OF THE DIFFERENTIAL CALCULUS. 47 Biot's conclusion is therefore false, and the case remains thus, that Newton's invention and Leibnitz's are the same, and that Newton made the invention earlier than Leibnitz. The celebrated Frenchman Montucla, who weighs more in the scale of historical mathematics than any Frenchman, says, after having discussed all the facts in detail, (page 109, vol. 1), " il est temps de nous resumer, et d'abord on ne peut u douter, que Newton ne soit le premier inventeur des calculs dont il u s^agit ^ les preuves en sont plus claires que le jour." ["It is time we " should sum up, and in the first place it cannot be doubted that Newton " was the first discoverer of the Calculus in question ; the proofs of " this are clearer than daylight".] Why does not M. Biot cite this French writer? We believe, however, that we have proved, not only that Newton was the first discoverer, but also that Leibnitz, inasmuch as Newton's method of tangents was known to him, had no right in his letter of 1677, to write arbitror quce celare voluit Newtonus de Tangentibus ducendis a meis non abludere, but that he ought to have written, quce celare voluit Newtonus mini nota sicnt, nam literas ejus lOmi Decembris 1672 inspexi. This is the verdict which those who in 1712 edited the Commercium Epistolicum annexed to their edition, a verdict which at the time when it was given was doubtful, because it was new, and because Leibnitz was in possession of the honour of the invention, which grew still more doubtful, as the notation of Leibnitz [dx and fdx) became general, whereby the difficulty of understanding the ques- tion was increased, but which now since the fact of Leibnitz's having read with Newton's open communications, also Newton's letter upon Tangents, has been recently since 1849 established, has become in the eye of impartial readers a safe verdict. CHAPTER V. A NEW FEATURE IN THE QUESTION. We have shewn that Leibnitz's merits, owing to his having seen Newton's letter of 1672, which he professed to be ignorant of, were smaller than he claimed, inasmuch as Newton's method of tangents together with his and Barrow's demonstration of the same, make up that which is called the "independent" discovery of Leibnitz. But we regret to say, that the matter perhaps does not end here. For we are alarmed at hearing Gerhardt naively tell us, that he has upon his table a manuscript, which he got out of the Hanoverian library, in Leibnitz's handwriting, without date, and headed, Excerpta ex tractatu Newtoni Manuscripto de analysi per cequationes numero terminorum injinitas. What are we to think of this ? And still worse, Gerhardt adds hereto that in his opinion Leibnitz cannot have seen these extracts from Newton's compendium between 1672 and 1674. So he forgets alto- gether that Leibnitz ought never to have seen this paper at all, if a tittle of his reputation is to remain with him. If we do not deceive ourselves this is Gerhardt's meaning,* that because in Leibnitz's extract we find the sign [/y], as the now so well-known mark of integration, while however the differential sign [dy] is not found, (for of the latter Gerhardt makes no mention,) ' We give Gerhardt's words as the first Addendum. A NEW FEATURE IN THE QUESTION. 49 therefore Leibnitz may or must have made these extracts certainly not before 1674, but before 1677 (in 1675-76). The French also, with their usual acuteness, have so understood the matter. For Biot and Lefort say at the end of their edition of the Commercium Epistolicum, (page 290) " II est possible que ces extraits " (de Leibnitz) aient etS pris sur le manuscript de Collins, pendant le " sejour [de Leibnitz) dhine semaine a Londres en Octobre 1676 ; mats " leur contexture ne permet pas de douter qu'au moment de+la trans- u cription Leibnitz ne fUt en possession des elements du calcul integral. " [Possibly these extracts of Leibnitz were taken from the manuscript " of Collins, during the week's stay that he (Leibnitz) made in London "in October, 1676; but their context does not leave us room to " doubt, that at the moment he transcribed them, Lebnitz Was in " possession of the elements of the integral calculus."] Mark that the word integral calculus, and not the word differen- tial calculus, is used here. In this very delicate question a word is of consequence, and M. Lefort knew right well that the most natural word was differen- tial calculus; as Gerhardt knew, that if by the side of the sign [/?/], he could have found the sign dy, this ought also to have been mentioned with the other. We may therefore venture to say that Gerhardt and the French editors intimate, that Leibnitz indeed made these ex- tracts, after his discovery of the Integral Calculus* but yet before his discovery of the Differential Calculus. * They imagine that Leibnitz discovered the integral calculus, when they find him put down the sign Ji/. In truth expressions, such as in Gerhardt (page 58, tract of 1855,) omne l—omn. omn. I., and the substituted S pro omni, are nothing more than those we meet with in Wallis and De St. Vincent, ductus plant in planum (quod appellat ille Vincentius) plant in planum ductum in meo ( Wallis) tractatu dicitur ductus rectarum omnium unius plani in alterius rectus. (Preface to the Arith. infinit.) The idea that a surface = omn. y is indeed the entire import of the "Wallisian summations, in order thereby to measure the surface. Hence Wallis or even Cavalleri is the discoverer of the integral calculus, as far as this can he recognized without the accompaniment of a Differential Calculus. In his Mathesis, or Arithm. 50 A NEW FEATUEE IN THE QUESTION. How is this possible? We must here say something of the relations of Leibnitz to Collins, or more correctly speaking, to Oldenburg. The Royal Society in Lon- don had committed the oversight of employing as their secretary, not an Englishman, but a German named Heinrich Oldenburg. This im- prudence could not but soon have its consequence, and this consequence in particular, that when once the right man came, the interest of Eng- land was more or less sacrificed to a German friendship. We say here nothing against Oldenburg, for he knew not what he did, and Leibnitz did but take advantage of this situation. There soon arises a friend- ship between them. We lack the first letters directed to Oldenburg. Leibnitz, supported by his patron, the Baron of Boineburg, whom Oldenburg had long been acquainted with, must have known how to hit the proper tone ; for so early as 5th August, 1671, Oldenburg writes to him,* while committing letters for Germany to a noble friend of his : dimittere harum gerulum nobilisshnum non potui, quin Te salu- tarem. simul et Jidem facerem, me reliqua quae de me exspectas, quam primum fieri id poterit, confecturum. Caeterum cum eximius Hehnon- tius, affectu mihi conjunctissimus, propediem ad nos sit reversurus, poteris si placet, ipsi tuto committere, quaecunque forsan mihi scribenda vel communicanda occurrtrint. What are these reliqua qua de me exspectas ? and what could Leibnitz have to say to Oldenburg, for which it ap- universalis of 1657, Wallis puts down for such additions of progressions term in or urn gumma vel aggregatum = S. See Opera, edition of 1699, Tom. I. page 140. Leibnitz may now and then have believed that he had made a discovery by speaking of omn. omnl. and S pro omni, but he must soon have perceived the contrary, be- cause he does not again speak of his sign S pro omni, that is, of his and other people's idea of summing series either in one of his letters to Newton of these dates or anywhere else. To add to Wallis's ideas of summation the differen- tial calculus, that Mas the discovery. The sign [Jy] therefore means nothing if the sign \_dy~\ and the idea which can be laid into this latter sign, and in others of its kind, is not found. * We of course quote everywhere from Gerhardt's Edition of the Letters. A NEW FEATURE IN THE QUESTION. 51 peared expedient to name Helmontius as his friend, to whom anything could be communicated by word of mouth or by letter so securely. Leibnitz, like many young men of his age, was over-desirous of acquiring a great reputation, and the Secretary being a German was to assist him. This was the bargain that was soon and perhaps almost naively ^> struck between them. Thus at once in the letter of the 12th June, 1671, we see the Secretary so interested in the "circulation" of a Leibnitzian paper, that he has it printed. Cceterum, vir amplissime, movent gessi desiderio tuo, et pro commodiore distributions scriptum tuum hie recudendum tradidi. (Gerh. page 22.) That Oldenburg in 1671 did not yet believe that Leibnitz was already a great man is proved by his words in the letter of December, 1670. (Gerh. page 16) : Finem hie facerem nisi ad Epistolae tuae calcem de Motus perpetui procurandi ratione perquam facili, a Te inventa, non- mdla innueres Ais Te rei demonstrationem expedivisse — Facile, puto, credes, me in Anglia peregrinum, sine palpo et assentatione de Anglis pronuntiaturum. Sunt inter eos viri complures, subacto in rebus Mathe- maticis et Mechanicis judicio praepollentes, quorum de invento isto tuo sententiam ut exquiras, prius quam id evidges, ejusve Actorem te scri- bas, omnino et amice suaserim. Si consilium allubescat, meque hac in re parario opus fuerit, provinciam non detrecto, omnemque quae virum bonum decet candorem spondeo. How delicately Oldenburg here labours to avert the danger of his friend falling into ridicule, and offers him- self as a pararius, or agent, whom, as not being an Englishman, Leibnitz can trust. But Oldenburg was appointed by the English, in order that he might protect the honour of the English, and not that he might be the agent of a foreigner, and give him clan- destine support. Leibnitz followed the advice of Oldenburg, and escaped the ridicule of introducing himself as the discoverer of the impossible perpetuum mobile; but when he came to London in 1672 he had the lighter misfortune of twice giving out as his own invention what was already to be seen in print. We refer to the well known anecdote, that Leibnitz H2 52 A NEW FEATURE IN THE QUESTION. in a party at the house of Boyle, ventured to say that he had dis- covered a certain method for employing the subtractions of square roots, whereupon Dr. Pell cited to him Mouton, in whose works this was to be read. Oldenburg contrived a defence for Leibnitz, in which the latter at last added, (Gerhardt, page 31) that he had something else, namely, a method (methodum Jiabeo) of summing fractional series [sum- mam inveniendi seriei fractionum in infinitum decrescent ium 1 quorum numerator unitas, nominatores vero numeri triangulares aut •pyramidales aid triangulo-triangulares). We cannot help saying that the next passage in the correspondence, excites a suspicion that Leibnitz here commits a plagiarism, in the close of that very representation, by which he defends himself from the suspicion of another plagiarism. For it can scarcely be supposed that Leibnitz was not acquainted with the book of Mengolus, published in Germany, (and at Bonn,) in which these summations are given ; the fame of which work had penetrated as far as England ; and Leibnitz's words, when he was taunted with this plagiarism: cum nondum mihi inquirendi in Mengo- lum otium fuerit, (page 46) et cum Mengoli liber non sit ad manus, page 48, do not even contain a downright affirmation, that he was not acquainted with it. Oldenburg here again contrives his defence, and as Leibnitz had now quite become his pet and favourite, he exerted himself for his fame more than for his own, (very naturally, for Oldenburg himself could certainly not pretend to be a great mathematician, it was difficult enough to get his countryman and friend into reputation) and so we see with astonishment the endeavours of the two friends quickly crowned in the access of the young man to the honour of becoming a member of the Royal Society. The unusual request, which Leibnitz had addressed to the Royal Society, was couched in the terms that had been agreed upon between him and Oldenburg, (voti coram Te expositi, page 33) ; and so this weighty transaction was concluded just as they had wished, and Oldenburg was further to promise, by word of mouth, that Leibnitz would make every exertion in order that the Society might never repent of having complied with his request. A NEW FEATURE IN THE QUESTION. 53 So it was by Oldenburg's exertions that Leibnitz had been received into the Society, without, at the time of his reception, having been preeminently qualified by his merit. This Leibnitz himself allows and admits when he says (see Gerhardt's Tract of 1848, pages 29 and 30, line 2; cum Parisios appidissem anno Christi 1672 erani in superbd pene dixerini Mathescos ignorantia, and in Desmaiseaux, (Recueil II., pages 5, 114) " au premier voyage en Angleterre, je n y avois " pas encore la moindre connaissance de la Geometrie avancee ; [on my u first journey to England I did not yet know anything of advanced " Geometry."] We see that it was not in the few months that he stayed in Paris, before his first journey to London, 1672, but afterwards, that Leibnitz learned what was necessary. In order, nevertheless, that he might continue to play the part of a great man which he had begun to play in 1672, we see him in London and after his return from London come always upon the stage as a discoverer. Even Guhrauer, the professed biographer of Leibnitz, is not fascinated with this trait in his hero's character, and says in one case (Vol. 1, page 329), "one cannot help " placing the universal Characteristic, or the Philosophical Calculus, " which Leibnitz was in search of and attempted to discover, on a "level with the finding of the philosopher's stone, and the manu- " facture of gold. And with regard, so continues Guhrauer, to another " but purely mathematical project of Leibnitz's, the ANALYSIS SITUS, " Kant, whom no reputation (Guhrauer throws this in) could dazzle, " leaves it an open question, whether the cause of its non-completion, "was that Leibnitz thought his attempts as yet too imperfect, or that " the case was with him as ic has been, according to Boerhaave, with " several great chemists, who gave themselves out to be possessed of " secrets, when they had really nothing but a persuasion and a con- " viction of their capacity for acquiring such ; and thought that they "could not possibly fail in the execution, if they would once choose " and attempt it ; at all events it appears as if the mathematical " discipline, to which Leibnitz gave by anticipation the title of Ana- 54 A NEW FEATURE IN THE QUESTION. " lysis situs, and of which Buffon has bewailed the loss, had never " been anything but a chimera." Man kann nicht nmhin, " die allge- u meine Characteristic, oder den philosophischen Calcul, (den Leibnitz " suchte und ertinden wollte), nait dem Steine der Weisen und der " Goldbereitung auf eine Linie zu stellen; bei einem andren, aber rein 11 mathematischen Entwurfe Leibnitzen's der analysis situs, lasst Kant, " welchen kein Name blendete, [sagt GuhrauerJ, ' es dahingestellt, ob " ' die Ursache der Nichterflillung dahin zu setzen, dass dem Leibnitz " ' seine Versuche noch zu unvollendet schienen, oder ob es ihm ge- " ' gangen sei, wie Boerhave von grossen Chemisten vermuthet : dass 11 c sie ofters Kunststticke vorgaben, in deren Besitze sie wiiren, da sie " ' eigentlich nur in der Ueberredung und dem Zutrauen zu ihrer " ' Geschicklichkeit standen : dass ihnen die Ausfuhrung derselben nicht " ' misslingen konnte, wenn sie einmal dieselbe iibeniehmen wollten j " ' wenigstens habe es den Anschein, dass jene mathematische Disciplin, " ' welche Leibnitz im voraus Analysiti situs betitelt, und deren Yerlust " ' unter Andern BufFon bedauert hat, wohl niemals etwas mehr als " ' ein Gedankending gewesen sei.' " This is Kant's verdict. With a character of this kind — " a burn- " ing desire for fame [einer brennenden Begierde nach Ruhm"], as Gerhardt terrns it (Vol. 1, page 3), it may be eas) T for a man to think himself the discoverer of something which has already been discovered, and one knows not where this rage for discovery will be arrested. In England Leibnitz, as we know, had already twice been unlucky with his inventions, (hence perhaps a certain spite against the country) ; we refer to the discovery with which he came out in the Soiree at Boyle's, and to the other with Mengolus. But Oldenburg had de- fended him, and he had retui'ned to Paris. The first communication from Paris of any importance is contained in Leibnitz's letter of the 26th October, 1674, in which he writes to Oldenburg, just after mentioning some general investigations which appeared to him of not much value : majoris ad usum vitae momenti est Prqfectus Geometriae ; et inprimis Dimensio Curvilinearum : wide saepe A NEW FEATURE IN THE QUESTION. 55 praeclara Problemata Meckanica pendent. In ea Geometriae parte rem memorabilem miki evenisse nimcio. Scis B. Vicecomitem Brounkerum, et CI. virum Nic. Mercatorem exkibuisse Infinitani Seriem numerorum rationalium^ spatio Hyperbolae aequalem. Sed hoc in Circulo efficere kactenus potuit nemo. Etsi enim III. Brounkerus et Wallisius dederint numeros rationales magis magisque apjpropinquantes ; nemo tamen dedit progressionem numerorum rationalium 1 cujus in infinitum continuatae summa sit exacte aequalis Circxdo. Id vero mihi tandem feliciter suc- cessit: invent enim seriem Numerorum valde simplicem^ cujus summa exacte aequatur Circumferentiae Circidi ; posito Biametrum esse Uni- tatem. Et kabet ea series id quoque peculiare^ quod miras quasdam Circidi et Hyperbolae exkibet harmonias. Itaque Tetragonismi Circu- laris Problema, jam a Geometria traductum est ad Arithmeticam Brfi- nitorum, quod kactenus frustra quaerebatur. Restat ergo tantum, ut Boctrina de Serierum seu Brogressionum numericarum summis perficiatur. Quicunque kactenus Quadraturam Circidi exactam quaesivere, ne viam quidem aperuere per quam eo pervenire posse spes sit, quod nunc primum a me factum dicer e ausim. Ratio Biametri ad Circumferen- tiam, exacte a me exkiberi potest per Rationem, non quidem Numeri ad Numerum [id enim foret absolute invenisse) ; sed per rationem Nu- meri ad totam quandam Seriem. Oldenburg answers Leibnitz (who, as we perceive, stated himself to have discovered a special quadrature of the circle by approxima- tion) saying that the English had not only this, but also a general method, by which to discover the same, and much else that was therewith connected. Oldenburg says : ignorare te nolim, Curvarum dimetiendarum rationem et metkodum a Gregorio nee non ab Isaaco Newtono ad curvas quaslibet, turn Meckanicas, turn Geometricas, quin et eircidum, se extendere ; ita scilicet ut si in aliqua curva ordinatam dederis, istius metkodi benejicio possis lineae curvae longitudinem jigurae aream et alia invenire. Now the full extent of this general method is no other than that which was afterwards termed the Differential Calculus; and no one 56 A NEW FEATURE IN THE QUESTION". denies that if Leibnitz knew the Compendium on that subject of 1669, which Collins and Oldenburg possessed, this would be sufficient to convict him of plagiarism And so indeed Leibnitz's curiosity was in the highest degree excited by the notion of the existence of this general method in the reach of his friend Oldenburg. That Leibnitz himself had no sort of general method, of which his single quadrature was a special application, is evident from his letter, Besides which Huyghens writes just at this time to Leibnitz, Cerhardt II. S. 16 : wms . M si . '■' . - - hant la Quad - - lit - - stratum. Chas Leib- had no method of d - ig . f demonstrating quadrat. - but Oldenbmg s : bis cori - : and described to him almost the ■ paper which Newton . and of which not only Collins but also Oldenburg had copies in their desk, we mean the - ' ~ -.in which the whole method and the Differential Calculus is found. Also in other letters of Olden- bur g's - the year 1669, Oldenl _ tes not onb sJ - with lysis, No. XJJLL] Wo see tha: Newton exception to tl fas - tempera: - of publishing only V salts, he gave away his whole method in this compendium. What now could I do. when he bee.. this immense wealth of the English? He might either beg thai this thod be communicated to him. or he might answer. u I will not "have your methods." But a man like I . who is "bnrnn _ with the les to g fame, does not act in such a simple manner: he d - not to learn these methods at whatever price, but . - he did not stand in need of them : he accordingly : i Oldenburg in the letter which Oldenburg could show | Newton Gerhardt, N XXIII. ' - •• N< fomtw he .\luin ex) . rrarum s eiermm et S .'forum Dimmsi ltrvrum A NEW FEATURE IN THE QUESTION. 57 Grravitatis inventiones, per ap>propinquat ioncs scilicet, ita enim interpretor. Quae Methodus si est universalis et commoda, meretur acstimari ; nee dubito fore ingeniosissimo auctorc dignain. Addis tale quid Orcgorio innotuisse ; but on the same date he wrote the subsequent No. 24, Mittara T1BI inventwm meum, satis certe mcmorabile, quod magnitudincm Circuli per seriem numerorum rational ium infinitum mire simplicem expriniit : si mihi vicissim duo vestratiuni inventa Geometrica pollicearis, unv/m Collinii, de quo aliquando mentionem fecisti, de summis serierum numericarum Jinitarum, quorum termini sint primanorum, secundanorum, tertianorum etc. reciproci; alterum G r eg orii circa methodum appropin- quandi ad veram Circuli et Hyperbolae magnitudinem per series con- vergenteS) cujus in Exercitationibus Geometricis exempla dedit. Et vero si Collinianum mihi consensu Clarissimi autoris, cui plurimam salutem a me dicas rogo, mis&ris quamprimum (nam etiani editum prostaf : nisi fallor in libro quodam Anglico) statim transmittam meum et Gregorian inn praestolabor, dum TIBI commoditas oblata fiterit obtinendi db autore ; neque enim credo Londini agit. Intelligo autem non inventa tantum, sed et demonstrationes mitti debere. Meum exactissime demon stratum, sed et numeris comprobatum habeo, et visum est ita memorabile insignibus quibusdam Geometris, ut invenforum Cgclometricorxim hactenus cognitorum apicem appellare non dubitaverint. This matches very well for Leibnitz who had now already learnt that series like his own were already printed in Gregory, (only without the method,) and could not therefore hope to shine by the side of Newton with his one poor quadrature, however he might dress it up ; before the secretary however it was still feasible to glorify this in- vention ; and it was this accordingly that he wanted to exchange with Oldenburg for the general method, about which Oldenburg- was to enquire, not from Newton, but from Gregory, [cui innotuit Juec methodus,) that is, he was to get oral information, for which reason the business was to be adjourned awhile, viz. till Gregory came to London. Thus it was most cleverly pre-arranged that Oldenburg was i 58 A NEW FEATURE IN THE QUESTION. not to write upon the subject, but to serve Leibnitz by oral enquiries. It agrees perfectly with the diplomatic address of the Advocatus and Staatsrath Leibnitz, in writing to his friend Oldenburg, who himself had been a consular agent, that he does not bluntly say, " you know "on what terms we are with one another; my invention is a small •' thing ; get me secretly the greater one that your people have ;" Leibnitz was not writing to a spy, whom he had bribed with money, but to a friend, of whom he only required, that he should do a small service to a fellow-countryman, without knowing how great was that service. But the import is not the less unfair ; and we must ask whether after that No. 23, which Oldenburg deposited in the archives of the Society, Gerhardt's No. 24 could be anything but a private letter of the same date; for in No. 23 and in No. 24, the same subject is brought forward, though indeed in different ways. Compare the expressions in the two letters. At no other time was the position of things in regard to this single quadrature on the one side, and the Method of Quadratures on the other, such as is pre-supposed in this letter, which Gerhardt has therefore inserted in its present place, and could not have inserted elsewhere. We will now at once remark, that any one, who would serve Leibnitz better than Gerhardt serves him, may here say, Gerhardt does not understand these things; he gives us documents, which are not that for which he passes them. No. 24 cannot be, as Gerhardt makes out, a letter supposed to be despatched, but only a draft, which never went to England. But with all the partiality that this view implies, it makes the case for Leibnitz not better than before ; for it shows that Leibnitz had the design to get himself syste- matically informed about this method, i.e. the Differential Calculus, by means of oral enquiries, (according to the draft letter No. 24) ; while instead of it he sent off No. 23, by which he concealed that desire ; perhaps because he did not feel quite sure that Oldenburg, to whom he could not yet speak by word of mouth, would serve him. The curiosity and the unallowable disavowal of this curiosity, and A NEW FEATURE IN THE QUESTION. 59 the wish of obtaining through Oldenburg a secret information, remain proved, though our interpretation has tried to assist Leibnitz better than Gerhardt. On the 20th of May, 1675, Leibnitz writes: Cum nunc praeter ordinarias curas Meclianicis inprimis negotiis distrahar, non potui exam- mare series quas misisti ac cum meis comparare, which would again be a falsehood, meant to conceal Leibnitz's desire to get informed about the English method, if it be (as we think it is) true, what is proved by Gerhardt's documents, (Tract of 1848, p. 23, note **,) that Leibnitz, at this very time, was not at all occupied with mechanical labours*. * In order not to break the thread of our investigations we will here just cursorily introduce an example of Leibnitz's anticipated discoveries out of the Correspondence, which confirms Kant's severe verdict respecting him, and shows at the same time that Leibnitz wanted to exchange that which he had not got hold of for something more substantial. He writes 12th June, 1675. Ego rem molior, et satis credo in numerato habeo, qua nescio an ad usum major possit sperari in Algebra, methodum scilicet, per quam omnium Aequationum radices instrumento qnodam, sine ullo calculo (j)ost Aequationum praeparationem non diffi- cilem) in numeris pro instrumenti magnitudine quantumlibet veritati propinquis, haberi possint. Si Collinius ant Parius inventum supradictum communicare voluerint, ego meum inventum, nemini hactenus a me monstratum, vicissim ipsis patefaciam. Oldenburg answers and gives in six long numbers, all of what Collins had supplied him with, and this : Dn. Newtonus (ut hoc ex occasione literarum suarum, meaning Collins, addam) lenejicio Logarithmoricm graduatorum in scalis TrapaWnXoo? locandis ad distantius aequales, vel Circulorum Concentricorum eo modo graduatorum admi- niculo, invenit aequationum radices. Tres JRegulae rem conficiunt pro Cubicis ; quatuor, pro Biquadraticis : In harum dispositions, respectivae coejficientes omnes jacent in eadem linea recta, a cujus puncto, tarn remoto a regula prima, ac graduatae scalae sunt ab invicem, linea recta Us super extenditur, una cum iiraescr^it'is con- sentaneis genio aequatlonis, qua in regularum una potestas pura datur radicis quaesitae. Lubenter equidem cognosccremus, num Tu, Vir Doctissime, et Newtonus noster in artificium idem incideritis. But now that Leibnitz has got all he wants out of the English, and can profit nothing further by them, he breaks off the subject with the words, Methodum Celeberrimi Newtoni, radices Aequationum inveniendi per Instrumentum, credo differre a mea. Neque enim video in mea quid aut Logarithm! aut Circuli Concentrici conferant. Quoniam tamen rem vobis non ingratam video ; 12 60 A XEW FEATUEE IN THE QUESTION. In the letter of 12th May, 1676, Leibnitz had once more an oppor- tunity of enquiring, quite without offence, just cursorily, what the English method might be ; and when Newton had thereupon written to him his first letter, he could say to Oldenburg, whom he thereupon visited in London ; " you see Newton himself has written to me ; so "now you can tell me all and just a little more about it;" and then Oldenburg may have given him, and has given him, as Biot and Lefort tell us, the Analysis of Newton, but only to make extracts. We have been obliged to elucidate this transaction, it however may have taken place in many other ways ; in any case it lies before us fearfully as a naively told fact, that Leibnitz has secretly read the Analysis of Newton, for he made extracts from it. Every one must immediately feel that he can only have made those extracts in London, that is, when he for the second time, we do not know for what reason, went there, being bound to Hanover; even the date is of no primary consequence ; all that matters is the secrecy with which Leibnitz held possession of these extracts, for as that Analysis of Newton's was to be had printed as early as 1711 in Jones's edition, and 1712 in the Comm. Epist., Leibnitz's extracts of it as of a Newtonian " manuscript" must certainly bear date at least before 1711; and now let us ask if in Leibnitz's confidential corre- spondence with Bernoulli, which lasts to 1712, Leibnitz had not on every page occasion to say that he had made extracts form Newton's important paper, which furnished the key to all Newton's publications, to the Pnncipia, the Memoirs upon Light, the Linece Tert. Ord., and the Quadratures. But Leibnitz did not so act ; he concealed these Newtonian extracts from Bernoulli and from all his other friends, from Tschirnhaus and from all the world, — and from Newton — and yet he had here everything; for if he excerpted the analysis, abconabor solvere, ac tibi cummunicare, quamprimum otii sat erit. Just like the Chemists, of whom Kant and Boerhaave speak, who boast of discoveries which they are not yet possessed of! And what sort of a discovery was this? It is one which Leibnitz never again recurs to. A NEW FEATURE IN THE QUESTION. 61 he surely did not neglect that which was best therein, even if he did not copy it, but only kept it in his memory, and extracted figures. But as all our readers will not possess the correspondence of Ber- noulli and Leibnitz in the edition of 1745, of which the index is here conclusive, we will, in order to give some conception of the frequent occasion that Leibnitz had to mention the fact to Bernoulli, if he had not been all too conscious of the necessity he was under to conceal it, copy verbatim from the index to this extensive and highly confiden- tial correspondence (which lasted from 1694 to Leibnitz's death), the rubrics Newtonus et Calculus infinitesimal is promotus : Newtonus, ejus Opuscula quaedam in Wallisii Operibus inserta 1. 185 — ejus Calculus Fluxionum in quo differat a differentiali . .191 an sit primus illius inventor, . II. 111. 283. 297. 308. 309. 313. 364. 375 — — unde ilium desumsisse suspicatur BEENOULLIUS I. 191. 195 — ejus errores ab HuGENlO notati, . . . 208. 211 — opus aliquod vult in lucem emittere, . . . 241 — ex eo expectatur Problema Celerrimi descensus, . 247. 253 illud solvit ..... 262. 266. 269 — quid ei tribuatur circa Corporum Attractionem . . 390 — gravitatem Corporum, extra Terram, esse reciproce in dupli- cata ratione distantiarum a Centro, sed, intra Terram, directe in simplici distantiarum ratione statuit, . .411, 415, 420. 424 — — et litem cum Fatio habuisse dicitur, . . I. 475. 483 qua de causa secundum Leibnitii conjecturas, . . 480 — quaedam ad eum spectantia, II. 31. 55. 86. 106. 111. 137. 154. 247 290. 302. 357. 361 — an ei tribuenda Serierum Methodus, ... 97 — ejus Lunae Theoria, .... 106. 124. 153 — ejus Optica : Enumeratio Linearum tertii ordinis : Quadratura Curvarum Geometricarum publicae fiunt, . . 123. 124. 180 — ejus Optica Anglice scripta, Latine vertitur, . . 159. 347 62 A NEW FEATURE IN THE QUESTION. Newtonus, ejus Arithmetica Universalis in publicum Cantabrigiae emissa, ..... 182. 185. 189 de ea Leibnitii judicium, .... 182 — ejus excerptuin quiddam Nicolao BERNOULLIO Nic. fil. mittitur 210 — colorum experiuienta quaedam a Mariotto facta Newtoni tentatis non eongruunt, .... 213. 216. 234. 235 — secunda ejus Principiorum Philos. Editio, 223. 226. 229. 291. 299 — in ejus Principiorum Phil, loca quaedam Bernoullianae adni- madversiones, . . . 240. 241. 253. 294. 299 — Regiae Societati Bernoullium proposuit, . . 299 — commcrcium Litterarium cum BERNOULLIO habuit, . . 302 — differentia inter ejus Philosophiam et Leibnitianam, . 364 — inter eum, (vel potius Clarckium) et Leibnitium Philoso- phica controversia, . . . 381. 382. 384. 390. 396 Plura vide in Calculi injinitesimalis historia. Calculus infinitesimalis, 7. 9. 14. 15. 26. 28. 35. 40. 41. 46. 53. 55. 57. 62. 65. 67. 75. 76. 81. 84. 91. 104. 127. 129. 179. 201. 202. 217. 218. 223. 226. 227. 231. 298. 306. 319. 321. 331. 332. 334. 367. 401. 461. — propagatus, ..... 12. 28. 30 — in eum difficultas proposita, . . . . .377 soluta, ....... 382 — idem est ac Methodus Fluxionum, . . . .190 — illius adversarii, II. 23. 25. 39. 40. 69. 71. 78. 148. 150. 153. 170. 172. 177. 178. 211. — ejus historia 151. 154. 155. 161. 283. 286. 291. 299. 300. 302. 308. 313. 315. 320. 323. 325. 327. 330. 334. 337. 340. 343. 351. 358. 361. 364. 367. 375. 377. 378. Not in one of all the above passages in this correspondence, (which lasted from 1694 later than 1712) is it stated that Leibnitz possessed anything from Newton, which the. public in general did not possess. We shall not be expected to prove, that Leibnitz must have commu- A NEW FEATURE IN THE QUESTION. 63 nicated with Bernoulli about these extracts of this Newtonian pamphlet, if he did not feel himself obliged to keep it secret. We should be glad, if we had deceived ourselves in this last chapter, and if Leibnitz's extracts from the Analysis could be other- wise explained; but then the other four chapters of this Enquiry- would still require no alteration. Even if we are right in our last chapter, it need not absolutely follow that Leibnitz is a plagiarist in the worst sense ; but perhaps Oldenburg did not allow him time enough; or he did not extract everything, or he saw the analysis only through Collins, and might subsequently believe that he himself was the discoverer, and that he had no need to mention what he had seen. But beyond these grounds of excuse nothing can be alleged for him. We have been urged to this investigation by the unworthy attacks that have been made on Newton, whom we were accustomed to revere as one of our nation ; and we are ready to apologize to Leibnitz, if, in the last chapter, but it is only the last we refer to, we have spoken too immoderately against him or against Oldenburg. ADDITIONS AND REMARKS. GERHARDT'S WORDS RESPECTING THE (LEIBNITZ) NEWTONIAN MANUSCRIPT, IN GERHARDT MATHEM. SCHRIFTEN LEIBNITZENS 1. P. 7. " I have found in the collection of Leibnitz's manuscripts in the library in Hannover a manuscript with the heading : Excerpta ex tractatu Newtoni Msco. de Ana- lysi per aequationes numero ter- minorum infmitas, without, as I am sorry to say, the date at which Leibnitz wrote the same. The first line of this manuscript reads as follows : AB n x ; BD n y ; a, i, c, quantitates datae ; w, n numeri integri. Si ax n y, erit x " n [fy] areae*. Further " Wir haben in der Sammlung " der Handschriften Leibnitz its auf " der Koniglichen Bibliothek zu Han- " nover ein Manuscript gefunden i/iit " der Aufschrift: Excerpta ex trac- 11 tatu Neutoni Msco. de Analyst per " aequationes numero terminorum in- u Jlnitas J aufdem leider der VermerJi u der Zeit fehlt, in welclier Leibnitz " es schrieb. Die erste Zeile dieses " Manuscripts lautet: AB n x; 11 BD n y ; t?, J, c quantitates datae ; m " m, n numeri int/gri. Siax n n y ; na m + n erit - — x m + n n[/y] areac- hi * "In his Excerpta it was Leibnitz's *" In seinen Excerpten pflegte Leibnitz "fashion to include his remarks as here " die eigenen Bemerkungen durch Klam- " in parentheses. — Gerh." " mern einzuschliessen — Gerh." 60 ADDITIONS AND REMARKS. " Leibnitz has only noted down the " Folgenden hat nich Leibnitz nut das "example —, = 3/, and the develop- " ment of- — — ; , in a series, and the " Newtonian Extraction of Roots ; " but the chapter De Resolutione " aequationum affectarum, in which " Leibnitz seems to have interested " himself particularly, is almost com- " pletely written out." " Beispiel — = ?/, ferner die Ent- " wickelunq von -, in eine Reihe 11 unddie Wurzelausziehung Newton's " angemerkt ; dagegen ist fast voll- " standig der Abschnitt : De Reso- " lutione aequationum affectarum, " ausgeschrieben ,fiir welchen Leibn itz " sich besonders interessirt zu haben " scheint."'' i It is easy to prove that the attacks of Biot on Newton's character are unfounded. With regard to the vehement controversy between Leibnitz and Newton, Biot forgets that it did not originate with Newton. For if we were even to assume with M. Biot, that Leibnitz had more right to the discovery than he really has, still it is at all events he that had got something from Newton, and not Newton from him ; for which reason Leibnitz in publishing ought to have said, that it was known to him that Newton had the very same thing which he (Leibnitz) published in 1684. Newton did not take offence at Leibnitz's thus ignoring his equal claim to the discovery ; but only in his next publication, in the Principia, of which the printing was completed in 1687, added the well known Scholiam: In Uteris quae mihi cum Oeometra peritissimo G. G. Leibnitio annis abhinc decern inter 'cedebant, cum signijicarem me compotem esse methodi determinandi maxima et minima, ducendi Tangentes, et similia peragendi, quae in terminis surdis aeque ac in rationalibus procederet, et Uteris transpositis hanc sententiam involventibus eamdem celarem ; Rescripsit Vir Clarissimus se quoque in ejusmodi methodum incidisse, et methodum suam communi- cavit a mea vix abludentem / praeterquam in verborum et notarum formulis. Utriusque fundamentum continetur in hoc Lemmate. M. Biot justly remarks in the Journal des Savants (1855, p. 603) that Newton hereby "recognises the independence of the rights of Leibnitz," " reconnait " V independance des droits de Leibnitz" whence indeed twenty years later, in 1711, when Newton was no longer amicably disposed to Leibnitz, he found it inconvenient that there should exist a scholium so favourable to the latter, and implying such a recognition of his claims ; " en 1711 lorsque "Newton etait exasperS" says M. Biot, u Ce scholie devenait pour lui " une piece a decharge fort embarrassante .*" "in 1711, when Newton K2 68 ADDITIONS AXD REMARKS. " was exasperated, this Scholium was for him a very embarrassing " piece." But how did Leibnitz act? He ought, as we see, to have named Newton in his publication of 1684, and the latter might have taken amiss this ignoring of his claim ; still Newton mentioned in his first publication, by the side of his own, the contemporaneous right of Leibnitz. Ought not Leibnitz now at least to have confirmed this, or at all events done something rather than commence an attack upon Newton? But in truth he just now with a premeditated design injured Newton, by inserting just after Newton's Principia had been published a memoir in the Acta Enid. 1689, in which he, under the pretence of being as yet unacquainted with Newton's Principia of 1687, gave the most important propositions thereof on his own part. We will not dwell upon this affair; let it suffice that Biot strongly and sharply censures Leibnitz for it ; and that even in their extraordinary edition of the Commercium EpistoliGum, (1856), Messrs. Biot and Lefort repeat the censure, saying at p. 209, Cette publication , (dans les Actes de Leipsig i Mens. Web. 1689) est a mes yeux le seul tort que Leibnitz ait en envers Newton, jusqu 1 <tu moment de In deplorable controverse qui a empoisonne I, urs derhiers jours. So this, according to our good French friends, is (le tort) the wrong of Leibnitz ! but herewith the affair began, for though they would gladly mystify us by using the politic phrase, c 'est a mes yeux h seul tort que Leibnitz ait en envers Newton jusqu'av &c, yet no one will be so far misled by this as to forget the dates, which prove that in this only wrong Leibnitz committed also the first wrong, for before 1689 all that had been said or printed by Newton, or the hitter's friends, was in no way calculated to irritate Leibnitz, but on the contrary purely amicable. How must it have wounded Newton, that after the gigantic work of the Principia, on which he had laboured so much, and which he published in 1687, Leibnitz with French levity took the credit of the whole work to himself, publishing the interesting theorems of it on his part in 1689, as if he had at that time not seen the Principia? ADDITIONS AND REMARKS. 69 It is justly said in the Epistola ad Amicum that if Leibnitz had really not seen the Principia two years after its publication, yet nevertheless he had seen the Epitome of the Principia which was published in the Acta Erurt. of 1688. Qua beta, says Newton, in the Epistola ad Amicum, Ds. Leibnitzius sch diasmata sua de motimm coelestium causis — compositit et in Actis Lipsicis ineunte anno 1689 imprimi cura- vttf quasi Ipse quoque praceipuus Newtoni de his rebus Propositiones vmetdsmt idque mcthodo diversa, et librum Newtoni nondum vidisset. Qua licentia concessa Authores qitilibet inventis suis facile privari possunt. Quam prmtum Liber Newtoni hocem vidit exemplar ejus D. Nicolao Fatio datum est ut ad Leibnitium mitteretur (1687). Viderat Leibnitius (1688) Epitomen ejus in Actis Lipsicis. Per commercium epistolicum quod cum viris doctis passim habebat, cognoscere potuit Propositiones principalis in libra iUo contentas into et librum iptsum procurare. Sin Librum ipsum non vidisset, videre tamen debuisst t antequam sua de iisdem rebus cogitata publicaret, idque ne festinando erraret in subjecto novo ac difficili et Newtono injurius esset auferendo inventa ejus, et Lectori molestus repetendo quae Newtonus antea dixerat. We repeat that it is Biot, who most severely censures Leibnitz's conduct ; for he says (Article on Leibnitz in the Biographic uni- versale and Biot's Com. Epist. Colliusii, p. 209.) Ainsi Vimmortel ouvrage des Principes avait paru depuis deux ans et Leibnitz ne V avait pas regarde: il ne V avait pas regarde meme apres que lies decouvertes monies qu?il off rait pour la premiere fois au mondc, avoir ut ete annoncees dans les Actes auxquels Leibnitz renvoie ; et il assure u'en avoir jamais eu connaissance que par cet extra/it. Sans doute il faut le croire, car il serait trop desesperant pour Vhonneur de Vesprit humain de supposer un si grand genie capable de la plus vile imposture* : mat's * In the Journal des Savants of 1852, p. 136, IJiot adds: "qttviqtte d'apres V ordre des prohlemes que Leibnitz attaque et d'apres V usage qu'il fait des lot's de Kepler pour etablir ses deductions il est a peine croyable qu il n'ait pas daigne se renseigner plus surement." Thus he (Biot) does not believe from internal evidence (a peint) that Leibnitz had not also read the Principia. 70 ADDITIONS AND REMARKS. alors il faut bldmer un dedain si aveugle ou une si condamnable insouciance. When then for ever so long a time not a word about Newton's contemporaneous discovery of the Differential Calculus was to be wrung from the mouth of Leibnitz, and he was gradually acquiring the reputation of being the only discoverer, Newton's friends began to vindicate his right ; and when Leibnitz not only went on, without Newton's having spoken, to state the case for himself, but also in the Act. Erud. of 1705, referring to Newton's publication of 1704, de quad rut urn curvarum et de lineis tert. ord., came out in quite an irritating manner against Newton with the celebrated words semper adhibuit, for which Leibnitz was never able to justify himself, and with the comparison of Newton to Honoratus Fabrius, and of himself to Cavalleri, by which he got up the semblance of a charge of plagiarism against Newton — then the cup was filled, and Newton, who, not having as yet in the publication of 1704 said a word against Leibnitz, now finding himself with insidious phrases set down as a plagiarist, was at length — no man could have commanded himself longer — provoked to publish the documents disputing the latter's claim to the exclusive discovery. We will give the leading passage out of this politic memoir of Leibnitz's. Leibnitz says : Ingeniosissimus demde Autor anteguam ad Quadraturas curvarum vel potius Curvilinearum veniat, praemittit brevem Isagoqen. Quae ut melius intclligatur, sciendum est cum magnitude /aliqua continue crescit, veluti Lima {exempli gratia) crescit fiuxu Puncti quod earn describit, incrementa ilia momentanea appellari difterenttaSj nempe inter magnitudinem quae antea erat, et quae per mutationem momentaneam est producta; atque kinc natum esse Calculum Diffi r< nt'udem, eique reciprocum Summatorium ; cujus elementa ab inventore D. Godefrido Guilielmo Leibnitio in his Actis sunt tradita, variique usus turn ab ■ipso, turn a D. D. Fratribus Bernoulliis, turn et D. March ione Hospitalio, [cujus nuper extincti immaturam mortem omnes magnopere dolere debent, qui profundioris doctrinae profectum amant) sunt ostensi. Pro differentiis ADDITIONS AND EEMAEKS. 71 igitur Leibnitianis D. Newtonus adhibet, semper que adhibuit, Fluxiones, quae sunt quam proxime ut Fluentium augmenta aequalibus temporis particidis quam minimis genita, ; usque turn in suis Principiis Naturae Mathematicis, turn in aliis postea editis eleganter est usus, quemadmodum et Honoratus Fabrius in sua Synopsi Geometrical motuum progressus Cavallerianae metlwdo substituit. We see that Leibnitz had not really the courage to say openly, that he was the first, and Newton the second discoverer, but that he only gave a glimpse of this intimation by the concluding word substituit, and through the word adhibet, to which he mysteriously appended the words semperque adhibuit ; moreover Leibnitz had not even the courage to sign his name to the article, but denied stubbornly, even till his death, that he had written this Review ; which however every one looked upon as having proceeded from him, and which, as is now proved, he had really written. How long then, in order to satisfy M. Biot, ought Newton and his friends to have been silent ? So far was the Gommercium Epistolicum from being an act of aggression, that we have much rather reason to say, that it would have been no longer fair to fight a disguised battle, and with politic phrases to cover the case thinly over, while the documents could be brought forward. Thus it cannot be maintained that Newton was the irreconcileable strife-loving party ; Newton was mild and loved tranquillity ; he even abstained from editing Kinkhuysen's Algebra, ne quietem suam perderet, (Com. Ep. No. 23, and 57), and he wrote to Leibnitz even in 1693, Spero me nihil scripsisse, quod tibi non placeat, aut si quid sit, ut Uteris id mihi significes, quoniam amicos pluris facio quam inventa mathematica. The individual small attacks of Biot on Newton have as little foundation. II ne faut pas, says Biot in 1832, (Jour, des Sav. p. 271), vouloir Justifier Newton d" 1 avoir supprime dans la troisieme Edition des Principes le celebre sckolie quHl avait insere dans les premieres et qui reconnaissait les droits de Leibnitz. That is, M. Biot wants Newton, after he has engaged in a dispute with Leibnitz, to acknowledge once 72 ADDITIONS AND REMARKS. more that which he now denied ; a singular demand indeed ! It is here again to be remarked in Newton's favour, that he simply withdrew the scholium favourable to Leibnitz, without replacing it by a hostile one. Enjin, continues M. Biot, il ne faut pas trouver beau, ni juste ni honorable, h Newton d'avoir encore poursuwi son rival dans la tombs, par une nouvelle edition du Oommercium Epistolicum augmentee de deux nouvelles httres de Leibnitz, qiC il s'etatt procurSes, et quHl accompaffna ./'/>,/■ refutation tres amere. The Coram. Ep. was made public during the lifetime of Leibnitz, and the lettres que Newton s' eta it procurers are bv no means private letters, but one of them is the charta volans mathematica 7 JuMi 1713, which Newton had certainly no need to get, as Biot's words would lead us to imagine, by any unallowable means, because Leibnitz liimself had everywhere circulated it, in order thereby to assail Newton ; and the other is the reply of Newton's friends thereunto, (the " refutation" being the Ad leetorein of this edition : for the Recensio reprinted there, had appeared in the Phi- losophical Transactions one year and eight months, and in the Journal Literaire, in French, one year and seven months before the death of Leibnitz).* M. Biot further makes mention of an account-book of the year 1659, kept by Newton (who was at that time sixteen years old) which has been brought to light, of which Sir David Brewster (Biot's authority) thus speaks : u At the end of tin booh there is a list <>f his expenses, entitled Impensa u propria, meupyimg fourteen pages. On the ith page the expenses are ie summed up thus: — * Even Newton's " Remarks" on the last Leibnitian pleadings before Conti (of 9th April. 1710) w< re written before Leibnitz's death, that is, before the 14th Nov., 171<>. It i-. not Dcs Maizeaux's collection of 1720, in which (see pages 78, 98) those Remarks dated 18th May, 1716, were first printed, but the History of Fluxious by Raphson, of which book pages 97' — 123 are as it were a second edition. ADDITIONS AND REMARKS. 73 Totum .... £3 5 6 Habui .... 400 Habeo .... 14 6 " On the 5th page there are fourteen loans of money, extended thus : Lent Agatha . . £0 11 1 Lent Oooch ..100 " and he then adds at the bottom of the page, lent out 13 shillings more u than £4. " Among the entries are Chessemen and dial . £0 1 4 Effigies amoris ... 010 Do 10 " and on the last page are entered seven loans, amounting to £3. 2s. Gd. " There is likewise an entry of l Income from a glasse and other things to tl ' my chamber-fellow, £0 9.' Another page is entitled Otiose et frustra expensa. Supersedeas. Sherbet and reaskes. China ale. Beere. Cherries. Cake. Tart. Bread. Bottled beer. Milk. Marmelot. Butter. Custards. Cheese.' 1 '' M. Biot is very facetious on the subject of such details being offered to the public in England, but he might be facetious against himself for he adds still more detail, saying : Ne voulant pas imiter le singe de la fable qui prenait le Piree pour un nom d'homme, fai eu recours a V obligeance de M. le professeur De Morgan, le priant de vouloir bien m" 1 interpreter les mots dont le sens me semblait douteux, ou qui ni 1 etaient tout a fait inintelligibles. Grace a hit. je vais ici me prevaloir de son erudition archeologique dans la langue de Cambridge, en faveur des lecteurs frangais, peut-etre meme anglais qui voudraient connaitre an, juste, en quoi consistaient les exces de Newton. Marmelot equivaut evidemment au mot actuel marmalade, en frangais L 74 ADDITIONS AND REMARKS. marmelade ; Reaskes, maintenant Rushes, designe une sorte de biscuits Ugers. China ale, litteralement Vale de Chine. Tout le rnonde sait que Vale est une sorte de Mere legere de couleur jaune pale. Mais qu 1 est-ce que V ale de Chine? M. de Morgan a ingenieusemsnt devinS que ce devait Ure la une locution employee alors parmi les etudiants de Cambridge pour designer le the. We see that M. Blot and M. de Morgan have made some exertions In order to be able to impart those details to the public with still more precision than they were given with before them. We quote from the same article of Biot's, in 1855, the following passage about Newton's character: Aux occasions rares ou il lui arrivait [Newton) d\issister a des banquets publics dans la sallc commune du college, si V on n" 1 avait pas la precaution de Vy /aire penser, il arrivait en desordre, les souliers abattus sur les talons, les bas non attaches, les cheveux non peignes, et un surplis sur le tout. D' ] autres fois il sortait le long d'une rue sans songer qu : il n'etait pas convenablement habille ; puis s' en aper- cevant il regagnait bien vite son logis tout honteux. D' auditeurs il n'en avait que trls-peu ou pas du tout, et il faisait le plus souvent ses lecons devant les murailles. On ne le voyait jamais non plus prendre aucun amusement, aucun exercice, se meler a aucun jeu. U se delassait a" une Hude par une autre, toujours pensant, toujours meditant. H etait rare qii'il se couchdt avant deux heures du matin, pour se lever vers cinq ou six ; dormant au plus quatre ou cinq heures. Quant a son caracte're moral dans le peu de commerce qu'il avait avec le reste des homines, on le represente doux, pose, inoffensif, ne se mettant jamais en colere ; de plus charitable et genSreux dans V occasion. Ces derniers penchants, on sait qu -1 il les garda toujours, et V accroissement de sa fortune ne jit que lui donner les moyens de s'y abandonner plus librement. That M. Biot cannot understand such a character we perfectly comprehend, yet this is not the fault of Newton, but of Biot himself. Altogether Newton as inventor is in his right, even by the estab- lished conventions of the literary world, according to which an act of publication is necessary to establish one's title in a discovery, because mere thoughts may have been by the thinker himself considered valueless. For Newton in 1669 sent his Differential Calculus to the President of the Royal Society in London ; he had given it previously to Barrow, the Secretary of the Society Oldenburg and Collins had also copies. Newton had not forbidden these persons to speak of it, and Collins as well as Oldenburg made this discovery everywhere known as early as the year 1669, as is evinced by letters in the Comm. Epist., from Oldenburg, 14th September, 1669, to Sluse in Leyden ; from Collins, 25th November, 1669, to James Gregory ; from Collins, December, 1671, to Borelli in London; from the same, 26th December, 1671, to Vernon; from Oldenburg in several communica- tions to Leibnitz before 1675; but Collins and Oldenburg of course did not come forward with the detail, because it was proper that this should proceed from Newton, who meant to publish it in Kinkhuysen's Algebra, but was prevented by the controversies which arose from the publication on his Theory of Colours, and the labours which he bestowed upon the Principia. Newton thus made an exception to the practice of the Geometers of his time, because he did not keep to himself the method that he had discovered, but had written it down completely, and without reserve, and at once put it in circulation amongst his friends. It is commonly thought, that Newton eagerly concealed that which he had discovered, while Leibnitz, more magnanimous than Newton, communicated to the world all he knew, but Leibnitz cannot even claim the honour of this communicativeness ; for his publication in l2 76 ADDITIONS AND REMARKS. the Acta Eruditorum in 1684 was not made to be understood, but to be not understood. The publication was even not understood by mathematicians, such as Bernoulli — and James Bernoulli requested Leibnitz to expound to him that which was unintelligible therein. And Leibnitz did not answer. In the Memoires de VAcademie of 1705 we read : Mr. Jac. Bernoulli pSnetroit deja dans la Geometrie la plus abstruse, et la perfectionnoit par ses dScouvertes, a mesure quHl Vetudioit, lorsqu 'en 1684 la face de la Giometrie changea presque tout a coup, li 1 illustre M. Leibnits donna dans les Actes de Leipsic quelques essais de son nouveau Calcul differentiel, ou des Infiniment petits, dont il cachoit V art et la methode. Aussi-tot Mrs. Bernoulli, car M. Bernoulli Vun de ses freres, et son cadet, fameux GSometre, a la mime part a cette gloire, sentirent par le peu quails voyoient de ce calcul quelle en devoit itre Vetendue et la beaute, ils s'ap- pliquerent opiniatrement a en chercher le secret, et Venlever a Vinventeur, ils y reussirent, et perfectiomi&rent cette Methode au point que M. Leibnits par une sincerite digne oVun grand liomme a declare quelle leur ap- partenoit autant qxCa lui. Gerhardt endorses the above assertion, and we do so with him. He namely tells us, Leib. Math. Works, III, p. 5, 1855. "In the " Acta Enid., in 1684, Leibnitz had made known his new method; "James Bernoulli could readily imagine of what importance it was; " vet he was unable to raise the veil which, as it appeared, concealed " almost impenetrably the very concisely enunciated principle. At "last in the year 1687 au opportunity presented itself to James " Bernoulli to enter upon a correspondence with Leibnitz, the author " of the new method, and request him to furnish explanations and " directions by which to understand it. This letter of Bernoulli's was " delivered, while Leibnitz was absent upon a long journey ; and " so it happened that Leibnitz did not furnish James Bernoulli with " an answer to it, till after his return in the year 1690, when however " he had no longer any need to instruct Bernoulli in the principle of •• this higher analysis. For Bernoulli had, proprio Marte, and by a ADDITIONS AND REMARKS. 77 " persevering study, penetrated the mystery, and had already manifested "the proficiency he had acquired by the solution of the isochronous- " curve-problem, which Leibnitz had proposed to the Cartesians." "In den Actis erud. hatte Leibnitz 1684 seine neue Methode bekannt " gemacht ; Jac. Bernoulli mochte wohl ahnen, von welcher Wichtigkeit " sie sein kbnnte ; dennoch vermochte er den Schleier nicht zu liiften, " der das in grosster Kurze dargestellte Princip derselben, wie es schien, "fast undurchdringlich verhitllte. Endlich im Jahre 1687 bot sich Jac. " Bernoulli eine Gelegenheit dar, mit Leibnitz selbst, dem Verf. jener " neuen Methode, eine Correspondenz anzuknixpfen und ihn um die " Aufklarung und Anleitung zum Verstandniss zu bitten. Dieses Sch- " reiben von Bernoulli traf indess ein, als Leibnitz auf einer grossen " Reise begrifFen . So geschah es, dass Leibnitz erst nach seiner " Riickkehr im Jahre 1690 eine Antwort darauf an Jac. Bernoulli " iibersandte, in der er jedoch letzteren nicht mehr liber das Princip der "hbhern Analysis zu belehren nbthig hatte. Demi derselbe war durch " eigne Kraft und durch ein beharrliches Studium in das Mysterium " eingedrungen und hatte bereits seine erlangte Meisterschaft durch die " Lbsung des von Leibnitz den Cartesianern vorgelegten Problems der " isochronischen Curve bekundet." Thus far Gerhardt, the warm friend of Leibnitz. It is therefore entirely incorrect to say, that Leibnitz openly published the Differential Calculus. The contrary is manifest. Bernoulli had to find it out by his own ability; in doing which he was certainly assisted by Leibnitz's notice of 1684 and 1686, but perhaps not less by the publication of Newton's Principia in 1687. Such is the case of the "publication." And it is Gerhardt himself who cannot help admitting this against Leibnitz. It is well known that the Marquis de l'Hdpital was nominally the publisher of the detail of the Leibnitzian Differential Calculus, and it is a matter of course that he over and over again names Leibnitz as the discoverer of this method. It is however this very man's statement which makes against Leibnitz ; Newton has remarked this at the end of the Recensio in the words : Nondum, inguit Hospitalius, tarn simplex erat (Tangentium methodus) quam a Barrovio reddita est, naturam Polygonorum propius consider ando, quod sponte menti objicit parvulum Triangulum, compositum ex particula Curvae inter duas ordinatas sibi infinite propinquas jacentis, et ex differentia duarum istarwm Ordinatarum, duarumque itidem correspondentium Abscissarum. Atque hoc Triangulum illi simile est, quod ex Tangente et Ordinata et Sub- tang ente fieri debet: adeo ut per tmam simplicem Analogiam omnis jam Calculatio evitetur, quae et in Cartesiana et in hac ipsa prius Methodo necessaria erat. Quo tamen vel haec vel Cartesiana revocari ad usics ftosset, necessario tollendae erant Fractiones et Radicales. Ob huius itaque Calculi imperfectionem, introductus est ille alter Celeberrimi Leibnitii, qui insignis Oeometra inde est exorsus, ubi Barrovius aliique desierant. Porro hie ejus Calculus in Regiones hactenus ignotas aditum fecit : atque ibi tot et tanta patefecit, quae vel doctissimos totius Europae Mathematicos in admirationem conjecerunt, etc. Hactenus Hospitalius. Non viderat nimirum Newtoni Analysing neque Epistolas ejus 10 Dec. 1672, 13 Jun. 1676 et 23 Oct. 1676 datas : quarum nulla ante annum 1699 typis publicata est: nescius itaque Newtonum haec omnia effecisse atque indicasse Leibnitio, Leibnitium ipsum arbitrates eat inde incepisse ubi desierat Barrovius. Instead of naming the Marquis de l'H6pital as the author of the Analyse de* infiniment petits, published in 1696, it is at last time that ADDITIONS AND REMARKS. 79 we should attribute this important book to its true author John Bernoulli, though 1' Hopital represents himself to be its author: In Bernoulli's opera IV. p. 387 — 558, we read : Johannis Bernoullii Lectiones mathematicae de methodo integralium aliisque conscriptae in usum Illi March. Hospitalii cum auctor Parisiis ageret Annis 1691 et 1692 Lectio prima : De natura et Calculo integralium. Vidimus in praecedentibus [Lntelligit, says the note, Lectiones in calculum differentialem quae praecesserunt, quasque supprimendas duxit, siquidem omnia quae in lectionibus istis continentur ab Hospitalio relata fuerunt in librum suum quern inscripsit Analyse des infiniment petits). Upon this same subject John Bernoulli himself writes to Leibnitz, (compare the edition of Gerhardt, p. 480; for all the other editions do not contain this passage ;) De suo aliud nihil addidit [Hospitalius) nisi quod tres quatuorve paginas repleat. Sed nolim quicquam ip>si de hisce referas, alias qui jam amicissimus mihi est, eum haud dubie infensissimum haberem. It was thus that Bernoulli published the Differential Calculus, and allowed a rich French Marquis to designate himself as the author of the publication, 1696. The conclusion that the Recensio is the work of Newton, is one that De Morgan was not the first to arrive at, as M. Biot makes out in compliment to his fellow-labourer Mr. De Morgan, who simply repeated this from a positive statement of Wilson's and only added some " internal evidence," saying : u throughout the whole [of the Recensio) " there is not one compliment to Newton [except in quotations introduced " in proof of assertions) not one word expressive of admiration, and not " one reference to any thing he had done which he might not in perfect good " taste have been the author of. Who could have written thus about " Newton, except Newton himself." It was not then uncommon to write anonymously as Newton has done in the Recensio ; the practice was not merely innocuous, but so far useful, as the matter of the work was thus left to speak for itself. Also Leibnitz often wrote anonymously in the Acta Erud. and other places. His biographer Guhrauer says somewhere, Observations on page 186, V. II : " Dass diese Schrift aus Leibnitzens eigner Feder geflossen, " lehrt Inhalt and Schreibart : das Leibnitz darin beigelegte Lob bildet "keinen Einvvand; er war in solchen Dingen ganz objectiv." "That " this piece came directly from the pen of Leibnitz, is told by the " style alike and the import ; the praise therein bestowed upon "Leibnitz constitutes no objection to this view; he was in such things " altogether objective." We see that Newton in writing anonymously was habitually more modest and subjective, in the opinion not of his biographer, but of his eager opponent. The new edition of the Commercium Epist., which appeared in France, cannot be said to possess much value. Of the variations of the first from the second edition, there was but one, to which Professor De Morgan himself, who has discovered their existence, could attribute the smallest consequence, (and since 1848 now that the question about the letter of the 10th December 1672 depends no more upon the silence of Leibnitz, even this variation is no longer worth any attention) ; all the others have never been of the slightest importance. What are we then to understand, when the French edition makes these petty variations the pretence for its publication? Why do not Messrs. Biot and Lefort tell us, that which was remarked here by that very Professor De Morgan to whom they and we are indebted for these various readings? " Those who are acquainted with " the bibliographical habits of the beginning of the last century will not " impute wilful unfairness even to such additions and suppressions as " some of those I shall have to describe." These are De Morgan's own words, and it is readily comprehended that the very number of the additions makes them so easy of detection, and therefore a disingenuous intention about them quite impossible. The French edition complains naively, that all these additions are not favourable to Leibnitz. Messrs. Biot and Lefort should not have found this surprising, for indeed the whole Commercium Epistolicum is unfavourable to Leibnitz; but there is an inexactness even in their statement which has been already previously acknowledged by Professor De Morgan, who has remarked that the non-mention in the first edition of the year of Collins's death was more serviceable for an attack upon Leibnitz than the citation of this date in the second edition. The French editors of the Comm. Ep. in 1856, Messrs. Biot and Lefort, misunderstand moreover what is M 82 ADDITIONS AND REMARKS. written in the German language and so give for instance as a post- script to Bernoulli, what was never a postscript, (Gerh. Works of Leibnitz, Pt. 3, p. 66 to 73, and Biot and Lefort, loc. cit., p. 266) which has a very comic effect, for in Biot and Lefort's edition the letter now runs : Ceterum an earn mihi animi parvitatem tribuis, ut tibi vel fratri tuo succenseam, si quos in Barrovio usus perspexistis quos mihi 1 inventionum contemporaneo, ah eo pc-tere necesse non fuit; and the Postscript runs: P. P. An earn in me animi parvitatem pittas, ut vel tibi, vel D. fratri tuo, succenseam, si vos in Barrovio usus perspexistis, quos mihi, inventionum contemporamo, ab eo petere necesse non fuit. M. Lefort accordingly believes that Leibnitz repeated his letter in his postscript. We know whence this proceeds. M. Lefort did not understand the two German lines, which Gerhardt introduced at p. 71. This new edition would have been somewhat useful, if it had furnished the correspon- dence of Leibnitz with Newton after the first edition of the Commer- cium after 1712, but though this is promised in the Table of Contents, the text gives, in lieu thereof, merely little politic abstracts of these later letters, and there is nothing about Gerhardt's Tracts, because unfortunately the French editors do not read anything which is German.* * It may be also presumed that the principal editor, M. Lefort, does not understand English, for otherwise M. Biot would not have signed the single citation at p. 45 with the initials J. B. B., while all such other citations are signed Lefort. M. Lefort says, in a particular observation at page 248 : Je ne me crois pas oblige de suivre V orthographe de Vouvrage de sir D. Brewster. Q.uand on voit ecrit, pur exemple tome II. p. 429 cog nitam pour cognatam et p. 435 tres-semble pour tres-humble, on pent craindre que les epreuves n' client pas ete revues par une personne assezfamiliere, avec les langttes latine et francaise. M. Lefort as we see piques himself upon his knowledge of his native French language. But if he is so strict with the orthography of the two languages, Latin and French, of which the latter may in some degree be known to him, we remark that the person to whom he entrusted the correction of the only five German words that appear in his edition, should, in the word Schriften, p. 287, have made the first letter a capital, because we have here a noun- substantive, like the others, in which this person has employed a capital letter, and that M. Biot might have instructed M. Lefort, that though Sir David Brewster is here reprimanded for an orthographical error, yet through this whole reprimand it would orthographically have been correct to have written Sir, with a capital S. ADDITIONS AND REMARKS. 83 We repeat that people in German) 7 will know better how to defend Leibnitz, which also is not the object of Messrs. Biot and Lefort. Ferraat's name must positively not be forgotten; it is on this account that the French will have their say in this controversy ; on this account clear water must be muddled; on this account they put themselves on the weaker side, because it would look too extraordinary, if France were to designate Newton as the sole inventor, and slip in Fermat. At this people would smile still more, as also now they smile ; for whatever is done, no one relishes that Fermat sauce. The controversy about the discovery of the Differential Calculus is a question between England and Germany, from which the French must keep their finger away ; let them come in honourably, if they like it, as judges ; but if they want to make an independent party in the contest, we must shut them out, and fight by ourselves ; the weaker party even disdains such equivocal succours. What Leibnitz did for the Differential Calculus, even if he did not discover it, is at any rate infinitely more than Fermat has done, as, indeed, no French- man before the middle of the succeeding century, achieved anything at all therein; for all that L'Hopital, the rich and influential, appropriated to himself, he had in reality taken from Bernoulli, who was silent because his Lectiones were splendidly paid for by the Marquis. As L'Hdpital and Bernoulli, 1696, did not know what the Coram. Epistolicum communicated to them, so in 1712, the Comm. Epist. and Newton himself were ignorant of what Gerhardt has com- municated to us, viz. that Leibnitz had perused Newton's Analysis. Where would this question now be, if Newton had been able to lay before the public the clandestine Leibnitzian excerpta without date out of his Newton's Analysis? Then would Bernoulli, the honest Bernoulli, whom Leibnitz betrayed, have been unable to strive in his behalf, and Leibnitz himself would have been obliged to couch his lance. And while this is the question, Messrs. Biot and Lefort go collecting little various readings of a book, which being but too moderate did not once intimate what Newton scarcely suspected. This edition of M 2 84 ADDITIONS AND REMARKS. .Messrs. Biot and Lefort's is indeed everywhere a singular one: for instance at p. 196, M. Lefort has revealed to us that Leibnitz had found the exact quadrature of the circle. Every one who reads what is there quoted, will understand us. On page 199 M. Lefort says : En retablissant encore (/) id tin paragraplie omis ou tronque, fat voulu montrtr V esprit qui a preside aux extraits du Commercium Episto- licum (de 1712) et rSduire a sa juste valeur le certijicat d ) invpartialite delivre par V AbbS Conti aux editeurs. Here that which is evident is only the malice of M. Lefort. He himself, or anybody else would have left out what is here missing, because it speaks irrelevantly about the solutions of equations, as irrelevantly as if it had spoken about M. Lefort. At page 204 Biot and Lefort say that Newton's second letter of 24th October was nine months in reaching Leibnitz, "par suite de ses nombreux voyages" "in consequence of his nume- rous travels." Leibnitz merely travelled from London passing through Holland to Hanover, in not quite two months, for he arrived (Gerhardt I. p. 27) at his destination (Hannover) in December. The words of Oldenburg (in Gerhardt's Math. Works of Leibnitz, I. p. 151,) dites done, s^il vous plait, si je dots bailler la grande hJtrede Newton, and the suspicious word Hodie (Gerh. ibidem, p. 154, cf. Comm. Ep. of 1712, No. LXVI.) even gives us reason to apprehend that Leibnitz had already read the letter of 24th October in London, and that it was but officially that afterwards, nine months after it was written and five months after his arrival in Hanover, he had it sent to him once more. At p. 285, M. Lefort says: Si la publication du Commercium Epistolicum en 1712 fut une oeuvre de parti, que dire de sa rSimpression en 1722, six ans apres la mort de Leibnitz? Dans cette pretendue i- ('impression, le nouvel editeur corrige, ajoute, retranche, interpole, com- mente; et la passion Vaveugle au point qiCil ecrit, sans Vy voir, sa propre condamnation dans VStonnante piece de polemique qui resume le livre auquel elle sert de preface. So says M. Lefort without further additions. We leave to the reader ADDITIONS AND REMARKS. 8 the satisfaction of discovering for himself, what piece htonnante this is, in which Newton has done so much towards his own prejudice, for M. Lefort has clearly enough designated the pidce, but the incredibility of his verdict forces one to be a long time in search of what he has meant. M. Gerhardt is completely thrown out of his saddle, by his supposition that when in a paper Leibnitz's writes S pro omne he thereby invents the " Integral" calculus, and that this had preceded the later invention of his Differential Calculus. Thus M. Gerhardt gets quite into an ill-humour with John Bernoulli, and says of him, " that he " has all his veins filled with unmeasured and extravagant pride and pretensions" " er strotzt durch und durch von ungemessenem Stolz " und hochster Anmassung" (Leib. Works, III p. 113), adding at p. 115, " in general John Bernoulli is considered as the discoverer " of the Integral Calculus," and that Leibnitz discovered the Integral and afterwards the Differential Calculus, " But for the reason, that " as regards the Differential Calculus, he was able to exhibit general " propositions, he therefore made this publicly known and kept back " the Integral Calculus, in which he was unable to exhibit any general " method." " Allgemein halt man Joh. Bernoulli fur den Entdecker " der Integralrechnung," Leibnitz habe die Integral-Rechnung und spiiter erst die Differenzial-Rechnung entdeckt. " Aber aus dem Grunde " wahrschcinlich, dass er fur die Differenzial-Rechnung allgemeine " Lehrsatze aufstellen konnte — aus diesem Grunde machte er allein die " Differenzial-Rechnung bekannt und hielt die Integral-Rechnung, fur " welche er solche allgemeine Methoden nicht aufstellen konnte, zuruck." We have sufficiently remarked that the first discovery of the Literal Calculus, as summation, did not wait for Leibnitz, inasmuch as Wallis had written a book upon the same as early as 1657. On the other hand the invention of the Integral Calculus, in the higher sense, as is commonly and justly supposed, was not only not first achieved by Leibnitz, but in this sense was only achieved by John Bernoulli. Gerhardt falls into a glaring contradiction after this violent attack upon ADDITIONS AND REMARKS. 87 Bernoulli: for at p. 114, loc. cit., Gerhardt says, that the first letter (from Bernoulli) to Leibnitz is full " of the most adulatory praises of " the latter ; " and hence " because Leibnitz was at no time inaccessible " to such offerings," (voll " der schmeichelhaftesten Lobeserhebungen des " letztern," und daher " weil Leibnitz fiir dergleichen durchaus nicht " unempfanglich") this correspondence between these two became, says Gerhardt, the most voluminous of all. How does this agree with Gerhardt's just now quoted statement about Bernoulli, that " his veins "were filled with unmeasured pride and extravagant pretensions?" The fact is, that Bernoulli does not manifest either of these extremes in his character, and that Gerhardt is merely disconcerted, without exactly knowing why, but in the feeling that his Theory, of Leibnitz having first invented the Integral Calculus, and then the Differential, will in what he here has to say about Bernoulli not suit at all — an idea, according to which one should cease to designate Leibnitz as the inventor of the Differential Calculus, and since even the word Integral is an invention of Bernoulli, one would have to make Leibnitz the inventor of Wallis's summatory idea, which new view though it would at first have astounded Leibnitz, might perhaps on closer reflection have suited him just as well as it now suits Gerhardt. The only thing wanting is that M. Gerhardt should get out of temper, not with John Bernoulli alone, towards whom he is quite ill-disposed, but also against Leibnitz, because the latter supposed that he had invented something else, which does not suit M. Gerhardt. Again John Bernoulli at last, (Works of Leibnitz, III., p. 132,) is praised by Gerhardt even more than there is reason. " After his (Leibnitz's) death," we read, "the controversy" (about the discovery of the Differential Calculus; cf. p. 131, from the words nicht ojfentlicJi,) " was openly taken up by " John Bernoulli, and maintained triumphantly to the signal discomfiture " of the English." " Nach seinem (Leibnitzens) Tode wurde der Kampf " (iiber die Erfindung der Differenzial-Rechnung cf. S. 131. die Worte " ' nicht offentlich') von Seiten Joh. Bernoulli's offen aufgenommen und " siegreich mit grosser Demiithigung der Englimder gefiihrt." Now 88 ADDITIONS AND REMARKS. this again is incorrect. On the contrary John Bernoulli, after Leibnitz's death, apologized to Newton, as appears from the letter of his to Newton which is so well known and quoted also by Lefort (page 250.) In this and all his last letters (cf. Brewster, II. 504, Edleston, page 169, note) John Bernoulli courts the friendship of Newton, assuring him, that it was not true, as Leibnitz treacherously said, that he (Bernoulli) had ever written anonymously on the question against Newton, though this was true. Where have we here a controversy with the at last discomfited Newton ? No ! No ! Newton's claims are too firmly established. " All controversy about the discovery is at an end," cries Gerhardt in behalf of Leibnitz. Gerhardt begins triumphing too soon, and this is our excuse for speaking too strongly perhaps against Leibnitz, whom clever Frenchmen extol so high, and for Newton whom Gerhardt courageously defending a German great man, could and dared not appreciate, lu order to show how clear, one might almost say how over- clear, if this were possible, a question can be made in France, when there is no deliberate intention of perplexing it, let us quote at full length Montucla's judgment (in his History of Mathematics, III. p. 109). II est temps, says Montucla, de nous resumer, et cV abord on ne jpeut douter, que Neuton ne soit le premier inventeur des calculs dont il s 'agit. Les preuves en sont phis claires que le jour ; mats Leibnitz est-il cou- pable d' avoir piddle comme sienne une decouverte qu 1 il auroit puisee dans les ecrits mhne de Neuton f c' est ce que nous ne pensons pas. Dans les deux lettres de Neuton, comniuniquees a Leibnitz, on ne voit que des resultats de la rniihode ou des deux metliodes employees par Neuton ; mats non leur explication. Un homme doue d'une sagacite transcendante tel qii 1 etoit Leibnitz, n? a-t-il pas pu etre excite par la a rcchercher les moyens employes par Neuton et y reussir ; d 'autant que Ferrnat, Barrow et Wallis avoient ouvert la voie. En effet si Von considere combien peu il y avoit a /aire pour passer de leurs metliodes au calcul differ entiel ; il paroitra, ce semble, superjlu de rechercher ailleurs V origine de ce dernier : car ce que Barrow designoit par e et a n" 1 etoit que Irs incrimens on de- cremens simultanes de V abscisse et de VordonnSe, hrsqu'ils etoient devenus assez petits pour pouvoir retrancJier du calcul leurs puissances supSrieures a la premiere : or en supposant, par exempjle, cette equation x 3 = by 2 , le calcul de Barrow donnoit 3x*e — 2bya ; de merne V equation .//* = bSy donnoit 4x 3 e = 3ba. U analog ie conduisoit done a remarquer que si Von avoit x" =y on devoit avoir nx~ l e = a, quelque fut le nombre n, entier ou fractionaire, posit if ou nigatif, et consequemnunt V incriment, par exemple, de \x ou x^ devoit se trouver -x^~ l e : ou au lieu de e, mettant une carac- 2 teristique qui donne a reconnoitre son origine, comme dx (vest celle qu 1 a N 90 ADDITIONS AND REMARKS. choisie Leibnitz) voila VScueil des irrationaliUs decline et le passage du calcul de Fermat, Barrow et Walk's au calcul differentiel de Leibnitz, et de cette seule observation dependent toutes les operations de ce calcul. Ajoutons, quant au calcul inverse, que Wallis avoit deja designe les eUmens des aires des courbes par le rectangle fait de Vordonnee et dhme portion infiniment petite de V abscisse qu'il nommoit A, de sorte que VeUment de Vaire du cercle etoit, par exemple, A Naa — xx. II avoit aussi reduit a de semblables expressions les elSmens des longueurs des courbes, et ineme par une analogie fondee sur la ressemblance du petit triangle caracteristique avec celui de la soutangente, de la tangente et de /' ordonnee. It is clear MontucLa does not do the same thing with Biot, (whom however, since he has united on his head the three crowns of the Academy, an honor that falls to the lot of few mortals, one must look upon as the greatest man in France) — for he counts Wallis among those qui ont prepare" V invention au dixseptieme siecle, and we may therefore choose to let it pass that Fermat is here also named in too good company perhaps rather conspicuously. Dutens's edition of Leibnitz's Mathematical works, (Pt. 3 of the Opera), published after Leibnitz's death 1768, is prefaced, as is but reasonable, with an eulogium upon Leibnitz by Joucourt ; we never- theless in this very shrine of Leibnitz's highest glories read not that which none would have ventured to say except the Journal des Savants, viz. that Newton had not as yet discovered all ; but Joucourt, the geometer, the biographer and panegyrist of Leibnitz, says here in Leibnitz's works, Preface p. xxxix. at the end of his history of the invention — Neivtonum fateor, pro med cestimatione, primum inventorem fuisse calculi differ -entialis ; and thus these Opera Leibnitii of Dutens or Joucourt, which is as much as saying Leibnitz himself, do not go so far in the praise of Leibnitz as Biot, but only so far as Montucla does. N2 The first letter which Leibnitz wrote to G alloys, one of those inferior geometers in Paris, at the time when he himself was there, is significative for the illustration of his doings in mathematical affairs in general. We give this letter (see Gerhardt's Edition, I., p. 177) : I m' indisposition m 'a empeclie de faire ma cour cette semaine comme je me Vestois propose. C est pourquoy je Vous supplie de suppleer par vostre bonte au defaut de ma presence, si V occasion se presente de parler utilement de V affaire qui vous est renvoyee, est f espere que vos fareurs seront bientost suivies d?un succes favorable. Je rf ay pas ose ecrire a Mons. Je Due de Cheureuse, de peur d? abuser de la grace qu'il me fait de ne me pas rebuter entierement, lorsque je viens quclquesfois buy faire la reverence. Mais je scay que Vos recom- mandations serviront bien mieux a me conserver Vhonneur de la protection que tout ce que je pourrois ecrire. Comme je ne veux pas abuser de vostre temps, qui est du au public. et a des personnes pour lesquelles le public s'interesse; je ne veux adjouter que le recit dhine petite conqueste que je viens de faire sur V Hyperbole. Tout le monde sgait qii 1 Archimede a donne la dimension de la Courbe du Cercle en supposant la quadrature de la figure. Messieurs Hugens, Wallis et Heuraets out fait voir que la Courbe de la Parabole depend de la Quad rat are de V Hyperbole. Mais personne a don>a : encor la dimension de la Courbe de V Hyperbole par la Quadrature de son espace ; non pas mvme de celle de V Hyperbole principale, qui a les asymptotes a angle dmit et les costez rectum et transversum egaux, et qui est entre les Hyperboles ce que le Cercle, est entre les Ellipses. J" 1 en suis venu a bout a la fin par un effort d? esprit sur ce que Mons. Oldenbourg m'avoit icrit depuis peu que Messieurs les Anglais C avoicnt clicrchee, et la cherchoient encor sans succes. Cela m'anima a faire une petite tentative, d'autant ADDITIONS AND REMARKS. 93 plus que je sgavois que Mons. Gregory [qui est grand Geometre sans doute) y avoit renonce en quelque fagon publiquement dans sa Geometric des Courvilignes. Mais je vous en parleray plus amplement, quand J auray Vhonneur de vous saltier , cependant je me dis etc. This letter is dated Paris, 2nd November, 1675. Oldenburg's letter [que M. Oldcnbourg m'avait ecrit depuis pea,) which Galloys indeed was not acquainted with, is known to us ; it had just come fresh from England, and is dated 30th September, 1675; it runs as follows: Oldenburg to Leibnitz : — scire cupis, an dare Nostrates Geometrice possint dimensionem Ciirvae Ellip>seos aid Hyperbolae ex data Circuit aut Hyperbolae quadratura. Respondet Collinius, illos id praestare non posse Geometrica praecisione, sed dare eos posse ejusmodi approximationes 1 quae quacunque quantitate data minus a scopo aberrabunt. Et speciatim quod attinet alicujus arcus Circuit rectificationem, impertiri Tibi poterit laudatus Tschirnhausius methodum a Gregorio nostro itiventam, quarn^ cum ille apud nos esset, Collinius ipsi communicavit. Thus it is not true, that Oldenburg had written : " que Messieurs les Anglais le cherchaient sans " succ 'es /" for Leibnitz himself had not sent to Galloys that quadrature or rectification of the circle or hyperbola, which now remains and for ever will remain an impossibility, but only an approximation to it ; but that very thing which Leibnitz pretends to have discovered, sneering at the English for not having done so, he obtained through the English, and rediscovered it after them. Even here the excuse remains, that what Leibnitz sent to Galloys, was perhaps not the same thing as he got from Tschirnhaus ; but he concealed the fact that he had got something from that quarter. This was his system. " Messieurs les Anglais" says Leibnitz, while on the contrary (see Gerhardt, I. p. 55,) he calls his French geometers nostros geometras characterizing himself as a Frenchman, as indeed he was. From Messieurs les Anglais Leibnitz gets his wisdom, petitioning them for that which as yet they had only in an imprinted form, and then he writes, Je ri? ai pas pu vousfaire la cour, and je viens de faire wie petite conquete sur V hyperbole. In this Messieurs les Anglais the whole matter 94 ADDITIONS AND REMARKS. is comprehended. Let it not be said that Tschirnhaus, to whose English mathematical documents Leibnitz was referred, perhaps kept these back — so that Leibnitz could not in this way make his conquete of what the English had conquered ; for of the intimacy between Tschirnhaus and Leibnitz we now first learn from Gerhardt that it was as close as possible (see page 34, where this intimacy is already mentioned). Thus Leibnitz writes, petitioning favours and returning thanks to England ; but when he has to do with his friends on the Continent he assumes the pretension of having no need of the English, calls himself the pupil of Huyghens only, and sneers at Messieurs les Anglais, while he is paying court to such people as Galloys. If people will not admit the correctness of Boerliaave's expression as applied by Kant to Leibnitz's character, then they must aver, that Leibnitz was the most fortunate man in literary matters that the world has ever seen. For, although he corresponded with the original discoverer of the Differential Calculus, he invented it independently after it had been discovered, being in no way influenced or assisted by the fact that his bosom friend had a tract on it in his desk. Again, at a later period, when Leibnitz, at once desirous of concealing his discovery in reality, and of appearing to disclose it to the world, was requested by James Bernoulli to explain what he had or had not invented, this letter of Bernoulli's did not come into his hands till three years (!) afterwards, when Bernoulli had discovered by his own diligence that which Leibnitz chose not to tell him: and in 1689 Leibnitz wrote in the Acta Eruditorum de motu corporwm coehstium 1 without noticing Newton's Principia, in which this matter was treated of; the work having existed for the rest of the world since 1687, but for Leibnitz not till after he had given its contents, as discoveries of his own, in his Memoir. Thus Leibnitz must have made himself, if not purposely yet de facto Lord of Time, and if a fact took place too early for him he let it lie, and did not take it up until such time as suited him. Newton's letter of 24th October, 1676, was especially submissive and obedient to the fortune of Leibnitz, for not only did this letter not reach him, until after he had committed the Differential Calculus to paper as his own discovery, but in the interval (of nine months !) between the date of this letter and its delivery, Leibnitz was expressly asked whether it was his pleasure that it should come : dites done, si je dots vous baffler la grande lettre de Newton ; the person to whom it was entrusted, considering even a copy of this 96 ADDITIONS AND REMARKS. letter so precious, that it could not be confided to the post, although the original was on every account to remain in London. Thus Leibnitz gained a considerable space of time, and had it in his power, when his own invention was quite ready to come out, to answer Newton in a grandiose style on the very day of the arrival of that nine-inonths'-old letter; Accept \Jiodie (!)] literas ttias sane pulcherrumas / e vesttgto remitto in- ventum meum, quod a tuo, quod celdsti, non abludit. How much less considerable would have been the glory, if the letter, instead of thus doing homage to the fortune of Leibnitz, had not enquired when it might be allowed to come. Now there are in the life of Leibnitz many such lucky incidents. Of a last quite trifling piece of good fortune, — that Leibnitz was able to make extracts from Newton's manuscripts, without Newton's knowing it, and that this, after the lapse of a century, can be so innocently narrated, and should not at all look as if it could not readily be explained as one of the miracles of the fortune of Leibnitz, — of this we need not speak. M. Gerhardt says that Leibnitz, without being obnoxious to any blame, could make extracts from his competitor's manuscripts. NEW ADDITIONS AND REMARKS. I will add in this English Edition a short remark on Oldenburg's position of the 28th October, 1676, — in reference principally to two letters of Newton, which are given by the Rev. Mr. Edleston, (p. 257, seq. Com. Ep. with Cotes,) and which are therefore here copied from Edleston's book : NEWTON TO OLDENBURG. S 1 ' Octob 26. 1676. Two days since, I sent you an answer to M. Leibnitz's excellent Letter. After it was gone, running my eyes over a transcript that I had made to be taken of it, I found some things w ch I could wish altered, & since I cannot now do it my self, I desire you would do it for me, before you send it away. 5 In pag : 3. Sect : Pudet dicere.] for a D. Barroio tunc Matheseos Professore write only per amicum, Pag: 5. Sect: At quando.] After quibuscum potest comparari ; write ad quod sitfficit etiam hoc ipsum unicum jam descriptum Theorema si debite concinnetur. Pro Trinomiis etiam et aliis qui- 10 busdam Regulas quasdem concinnavi &c. Pag : 6. Sect : Quamvis multa.] Where you find y c words Gregorianis ad Circidum et Hyperbolam editis persimiles, for per- similes write affines, Pag : 9 or 10. Sect : Theorema de.l for error erit h — - — + &c. 1 5 90 140 . v 3 v 4 p write error erit 1 1- &c. 90 194 Pag: 6 vel 7. Sect: Quamvis multa.] about ye end of y e section turn plenariam into plenam or rather blot y e word quite out. 98 NEW ADDITIONS AND REMARKS. Pag: ult. vel penult. Sect: Ubi dixi]. write soluhilia for solutUia. 20 And if you observe any other such scapes pray do me y e favour to mend them. So in pag 5 or 6. Sect. Quamvis multa.] It may be perhaps moi'e intellig{ib}le to write evdvvaei for euthunsi. Pag 8 or 9. Sect : Per seriem.] After y e words product ad mullets figur as : you may if you please add these words, ut et ponendo 25 summam trmmorum 1 — f + i — tV + tt ~ aV + ¥5 — 3V + 3*3 <& c esse ad totam seriem 1 — \ + \ — \ + ^ — j\ + &c tit 1 + V 2 ad 2. Sed optimus ejus usus &c. I feare I have been something too severe in taking notice of some oversights in M. Leibnitz letter considering y e goodnes & 30 ingenuity of y e Author & y 1 it might have been my own fate in writing hastily to have committed y e like oversights. But yet they being I think real oversights I suppose he cannot be offended at it. If you think any thing be exprest too severely pray give me notice & I'le endeavour to mollify it, unless you will do it w tL a 35 word or two of your own. I believe M. Leibnitz will not dislike y e Theorem towards y e beginning of my letter pag. 4 for squaring Curve lines Geometrically. Sometime when I have more leisure it's possible I may send him a fuller account of it : explaining how it is to be ordered for comparing curvilinear figures w th one 40 another, & how y e simplest figure is to be found w th w eh a pro- pounded Curve may be compared. S r I am Yo r humble Servant Is. Newton. 45 Pray let none of my mathematical papers be printed w th out my special licence. Some other things in M. Leibnitz letter I once thought to have touched upon, as y e resolution of affected ^equations, & y e impossibility of a geometric Quadrature of y e Circle in w ch M. :><> Gregory seems to have tripped. But I shall add one thing here. NEW ADDITIONS AND REMARKS. 99 That y e series of sequations for y e sections of an angle by whole numbers, w ch M. Tschurnhause saith he can derive by an easy method one from an other, is conteined in y* one ^equation w ch I put in y e 3 d section of y e Problems in my former letter for cutting an angle in a given ratio, and in another gequation like 55 that. Also y e coefficients of those ^equations may be all obteined by this progression n — Oxn—1 n—2xn—3 n— 4 x w — 5 n — 6 X n -=■ 7 „ 1 x — — — — - x — — x — — - x — — x &c. lxn-l 2xn — 2 3xn — 3 4xw-4 The first coefficient being 1 . y e 2 d n—Oxn—1 „ „ n—Oxn—l n—2xn—3 D 1 x — — . y e 3d 1 x — — x — — . &c. 60 Ixn— 1 J lxn—l 2xn—2 & n being y e number by w ch y e angle is to be cut. as if « be 5. i '..,5x4 3x2 1x0 , . « „ « o then y e series is 1 x - — - x - — - x - — - that is 1x5x1x0 & J 1x4 2x3 3x2 consequently y e coefficients 1.5.5. So if n be 6 y e series is 6x5 4x3 2x1 „ ,, , , , „ „ _ n . 1 x x - — - x - — - x that islx6x|x|x0& consequently 1x52x43x3 * y l J y e coefficients 1.6.9.2. This scrible is not fit to be seen by any 65 body nor scarce my other letter in y* blotted form I sent it, unless it be by a friend. For Henry Oldenburg Esq: at his house about y e middle of y e old Pal-mall in Westminster London. NEWTON TO OLDENBURG. S r I am desired to write to you about procuring a recom- mendation of us to M r Austin y e Oxonian planter. We hope yo r correspondent will be pleased to do us y* favour as as{sic} to recommend us to him, y* we may be furnished w th y-e jjggf sor t s f Cider-fruit-trees. We desire only about 02 100 NEW ADDITIONS AND REMARKS. 30 or 40 Graffs for y e first essay, & if those prove for o r purpose they will be desired in greater numbers. We desire graffs rather than sprags that we may y e sooner see what they will prove. They are not for M r Blackley but some X. other persons about Cambridge. But M 1 ' Austin need only direct his letters to me or to M r Baiiibrigg fellow of o r College. In y e mean time we return o 1 ' thanks to you & your friend for y e good will you have already shewn us. M r Lucas letter I have received, & hope to send you \ V . an answer y e next Tuesday Post. I thank you for your care to prevent their prejudicing me in y e Society, as also for giving me notice of y e things miswritten in my late letter. In pag 3 y e words you cite should run thus. Cujus 2x2x2 triplo adde Log. 0. 8, siquidem sit —tt^ — *" = 10. But in X X . pag 8 y c signes of y e series 1 + ^ — £ — | + ^ + &c are rightly put two + & two — after one another, it being a different series from y l of M. Leibnitz. But in y e next two or 3 lines, to prevent future mistake you may if you think fit, after y e words res tardius obtineretur per tangentem 45^, add XX V. these y?ov&sjuxta seriem nobis communicatam. Seing y e letter is still in yo r hands, you will do me y e favour to make these further amendments Pag. 3 Sect [Pudet dicere] cum D. Collinsio for ad D. Collinsium XXX. pag. 5. Excmpl. 4 after y c words vel quibus Tibet dig- nitatibus binomii cujuscunq: add licet non directe ubi index dignitatis est Humerus integer. pag 6 or 7 in y e end of y e section quamvis multa I desire you would cross out y e words adeo ut in potestate habeam XXXV. descriptionem omnium curvarum istius ordinis quce per 8 data puncta deter minantur. And in y e 2 d sentence of y e next section I could wish these words also numero infinite multas were put out. NEW ADDITIONS AND REMARKS. 101 pag 9. Sect [Prceterea qucei] for mihi quidern haud ita clara sunt put nondum percvpw. And after a line or two where XL. you see y e words et eerie minor est labor, put out certe. By these alterations S 1 ' you will oblige Yo r humble Servant {Tuesday} Nov. 14 167G. Is. Newton. From these two letters, and particularly the first, it becomes very probable, that on the 28th October, 1676, when Newton's letter of the 26th arrived, Leibnitz was actually in London. We know that Leibnitz was in London for " one week in October /" (Collins writes: " Aderat Me Dom. Leibnitius per unam septimanam in mense Octobris ; " in reditu suo ad Ducem Hannoverce" see Collin's letter to Newton, dated 5th March, 1677, given in the Commercium £Jp. : No. LXV ;) and the word send in line 6 of Newton's first above cited letter of the 26th October, shows I think, that Newton at that time had no idea of Leibnitz's intention of visiting London, and no knowledge of Leibnitz's presence in London at that time. Now it is but fair to suppose that Oldenburg would have mentioned to Newton the personal appearance of Leibnitz in London, if not immediately, at least a few days after the fact. Newton's not being aware of Leibnitz's presence on the 26th October, agrees well therefore with the supposition, that Leibnitz had not yet arrived in London, or had only just arrived at that date : the week of October spent by Leibnitz in London is hereby consequently proved to have been the last week of that month. On the contrary, Newton's letter (of the 26th October) does not well agree with the supposition that Leibnitz's week in London could have fallen in an earlier part of October; for Newton would not, while he knew that Leibnitz was bound for a further journey, have spoken of sending the letter at once away ; and would not, in his letter of the 14th November, line 26, have used the words " seeing" &c, if already, when he wrote his former letter, such knowledge had been in his possession. 102 NEW ADDITIONS AND REMARKS. We may therefore say that "Leibnitz's week falls at the very end of October, which also agrees with Leibnitz's presence in Amsterdam on the 18th or 28th November, 1676, (his letter from Amsterdam bears that date — see Com. Ep., loco cit.) and (agrees) with Guhrauer's words, "In October, 1676, Leibnitz quitted Paris, where it was not his fate " to return." (See Guhrauer's Life of Leibnitz, I. p. 170.) Now, under this supposition, we think that Oldenburg may be excused for showing, nay, was almost obliged to show to Leibnitz Newton's above mentioned letter of the 26th October, and consequently also Newton's letter of the 24th October, intended for Leibnitz. That friendly disposition of Newton's, which Oldenburg is desired or permitted to testify to Leibnitz in the 34th line of the letter of the 26th October, could not have been better expressed to Leibnitz, than by Oldenburg's confidentially giving to him this letter ; moreover in the 55th line Newton adds something intent ively for Leibnitz. Let any one reflect on Oldenburg's position, I do not say as a friend of Leibnitz, but as a friend of Newton, and I think it will appear to have been very natural, nay, only right perhaps, on the part of Oldenburg, to have shown to Leibnitz Newton's letter of the 26th October. For Oldenburg was not a great Mathematician, and he had no reason to suppose that there could be any slight difference between what Leibnitz might be able to derive from Newton's letter to Leibnitz of the 24th, and what Leibnitz could get out of Newton's letter of the 26th October, nor is there perhaps any difference between the two. But Leibnitz by reading the letter of the 24th and of the 26th October, was enabled to take a strong position for the purpose of pressing Oldenburg to show him Newton's manuscript De Analyst. For Leibnitz could with literal truth say, that the blotted condition of Newton's letter to him (see the last line of Newton's letter of the 26th) had prevented his reading it, and Leibnitz might infer from the 37th line of Newton's letter of the 26th, that it was only want of "leisure" that NEW ADDITIONS AND REMARKS. 103 had prevented Newton from giving other details (contained in the Analysis). Here then the false position in which Oldenburg had put himself by showing the letter of the 26th brought him into a disagreeable dilemma, namely between refusing Leibnitz's request (to see the Analysis) bluntly, and without those excuses, which Newton had used in his letter to Leibnitz of the 24th : (" quoniam jam non possum explica- " tionem ejus pi'osequi,") and, on the other hand, complying with his request. The excuse Newton pleaded was not an untruth in that higher sense of the excuse, in which Leibnitz (but not Oldenburg) was competent to view it ; for Newton also, when speaking confi- dentially (to Oldenburg) in his letter of the 24th October, had said : " I hope that this will satisfy M. Leibnitz, for, having " other things in my head, it proves an unwelcome interruption to " me to be at this time put upon considering these things." [See Edleston, p. LIL] But Oldenburg did not comprehend in what sense this excuse was meant, and must have half supposed that he was only requested by Leibnitz to do what Newton, if he had had the time, would have done himself. In this dilemma between having to say to Leibnitz more bluntly than Newton might wish, that something essential was kept back from Leibnitz, or else of overstraining the powers granted to him by Newton, Oldenburg erred, we think, by choosing the latter alternative, namely, of showing to Leibnitz Newton's manuscript De Analyst. It may here be remarked that Leibnitz's so called invention of the new calculus, in his letter to Oldenburg of the 21st June, 1677, need not have appeared to Oldenburg (the word " hodie" being omitted) an act of piracy in regard to Newton, on account, I mean of Oldenburg's friendly act in showing him Newton's manuscript De Analyst; for Leibnitz had chosen for his invention the tangential side of the problem, which to Oldenburg must have appeared unconnected with his (Oldenburg's) friendly action. 104 NEW ADDITIONS AND REMARKS. We have already said in a former note, that the nombreux voyages of Leibnitz, which Messrs. Biot and Lefort mention, as explaining why Oldenburg did not sooner send Newton's letter of the 24th October to Leibnitz, are one of those French fictions, which those gentlemen introduce into the case ; for Leibnitz arrived in Hanover in the latter part of December, [see Guhrauer, I. p. 188,] and it is with a bad grace that Oldenburg tells us, that the mere copy (!) of Newton's letter, (the original was to remain in London) could not have been sent before May, (four month's after Leibnitz's arrival in Hanover,) because the mere copy was in Oldenburg's eyes so valuable, that it (the copy !) could not go by post (!) though Newton's first letter had gone by post, and though Leibnitz, if he had not already secretly received a copy of the same in London, would, we suppose, have been in some small degree desirous to receive it soon, and might have friendly blamed Oldenburg, when he answered him, for keeping a copy of it so long out of his sight. Oldenburg's position to Leibnitz indeed was not such, that Oldenburg should have hesitated to risk the very small trouble of having to get a second copy made, (which was the sole misfortune that could have ensued,) if by the will of God the first copy given to the post had been destroyed or lost. Indeed we cannot believe this, nor need we believe it, for Oldenburg's little intrigue here in his own opinion was innocent. All that we have said agrees well with Oldenburg's French postscript, in which he says, after having kept back the letter several months, " dites done si je dots vous baffler la grande lettre de Newton" (and with Leibnitz's over hasty word " hodie" in his first draft of the letter, when at last he did answer). It is interesting to add here some curious words of Leibnitz's answer to the Commercium EpistoUcum^ 1714, and some eager words of Newton, drawing consequences from Leibnitz's answer. We cite from des Maizeaux's Recueil II. page 5, seq., and Raphson's History of Fluxions, page 97, seq. NEW ADDITIONS AND REMARKS. 105 Leibnitz sa) T s, 1714 : " Je fis connaissance avec Mr. Collins dans raon "second Voyage d'Angleterre a mon second Voyage Mr. Collins me "fit voir une Partie de son Commerce; j'y remarquai que Mr. Newton " avoua aussi son ignorance sur plusieurs choses, et dit entre autres, "qu'il n'avait rieh trouve sur la Dimension des Curvilignes celebres, " que la Dimension de la Cissoide." Newton answers : " Mr. Leibnitz instances in a Paragraph concerning my ignorance, " thinking that the editors of the Gommercium Epistolicum omitted it, " and yet you will find it in the Gommercium Epistolicum, page 74, "line 10, 11, and I am not ashamed of it. He saith, That he saw " this Paragraph in the hands of Mr. Collins when he was in London " the second time ; that is, in October 1676. It is in my Letter of the "24th of October, 1676, and therefore he then saw that Letter." What now does Leibnitz say ? He is so far from denying that he had seen in London, in October, Newton's letter of the 24th October, 1676, that he actually asserts that he has seen at that time something more : " Comme "je n'ai pas," he says, answering Newton's words, " daigne" lire le " Commereium Epistolicum avec beaucoup d'attention, je me suis trompe' "dans l'Exemple que j'ai cite, n'ayant pas pris garde, ou ayant oublie "qu'il s'y trouvoit ; mais j'en citerai nn autre: M. N. avouoit dans " un des ses Lettres a M. Collins, qu'il ne pouvoit point venir a bout " des Sections secondes (ou Segments seconds) de Spheroi'des ou corps "semblables: mais on n'a point insere ce Passage ou cette Lettre dans "le Gommercium Epistolicum ; il auroit ete plus sincere par rapport a " la Dispute, & plus utile au public, de donner le Commerce litteraire de " M. Collins tout entier, la ou il contenoit quelque chose qui meritoit "d'etre lu; & particulierement de ne pas tronquer lcs Lettres, car il y "en a peu parmi mes Papiers, ou dont il me reste des Minutes." Leibnitz died soon after writing this rather confused answer, but Newton was so much astonished at reading it, that he said in his "observations" upon the preceding epistle (Raphson, page 111, des MaizeauXj page 75 ; : P 106 NEW ADDITIONS AND REMARKS. " Mr. Leibnitz acknowledges, that when he was in London the second " time, he saw some of my Letters in the hands of M. Collins, especi- " ally those relating to Series ; and he has named two of them which he "then saw, viz. that dated the 24th of October, 1676, and that in which " he pretends that I confessed my Ignorance of second Segments. And " no doubt he would principally desire to see the Letter which contained " the chief of my Series, and particularly that which contained those " two for finding the Arc by the Sine, and the Sine by the Arc, with the " Demonstration thereof, which a few months before he had desired "Mr. Oldenburg to procure from Mr. Collins; that is, the Analysis per " cequationes numero terminorum infinites . But he tells us, etc." Here we see that Newton, from the curious admissions of Leibnitz, began at last half to suspect that Leibnitz might have " made extracts" from his " Analysis." Gerhardt and Biot and Lefort now tell us that this half suspicion of Newton is well founded. I take the liberty of copying here Edleston's Synoptical view of Newton's Life: — 1642 Dec. 25. Isaac Newton born at Woolsthorpe, near Grantham, Lincolnshire. 1664 Feb. 19. Observations on two halos about the Moon. 1665 May 20. Paper on fluxions,* in which the notation of point is used. Nov. 13. " Discourse" on fluxions and their applications to tangents and curvature of curves.f 1666 May 16. Another paper on fluxions. Octob. Small tract on fluxions and fluents with their applica- tions to a variety of problems on tangents, curvature, areas, lengths, and centres of gravity of curves.^ Nov. Small tract similar to the preceding, but apparently more comprehensive.il (Notation by points in first and second fluxions. Basis of his larger tract of 1671). 1669 July 31. His De Analysi sent by Barrow to Collins. Dec. Writes notes upon Kinkhuysen's Algebra sent by Collins. 1671 July 20. Letter to Collins. (Prevented by a sudden fit of sick- * Shewing how to take the fluxion of (or to differentiate) an equation connecting any number of variables. It is referred to in a paper which seems to be part of a draft of his observations on Leibnitz's letter of Apr. 9, 1716. (Rigaud's Appendix, p. 23, compared with Raphson's History of Fluxions, p. 116). f Rigaud and Raphson, u. s. \ In this tract his previous method of taking fluxions is extended to surds. The area of a curve, whose ordinate is y, is denoted by [] y. (Rigaud's Append, p. 23). || Raphson, p. 116. Wilson's Appendix to Robins' Tracts (II. 351 — 356). P2 108 NEW ADDITIONS AXD REMARKS. ness from visiting him at the Duke of Buckingham's installa- tion as Chancellor. Will not, he fears, have time to return to Discourse of infinite series before winter). 1672 May 25. Letter to Collins (does not intend to publish his lectures).* Dec. 10. Letter to Collins, containing an account, requested by Collins in a letter received two days before, of his Method of Tangents.f * " Finding already, by that little use I have made of the press, that I shall not enjoy my former serene liberty till I have done with it, which I hope will be so soon as I have made good what is already extant on my account." He adds that he may possi- bly complete his method of infinite series, " the better half of which was written last Christmas." Mace. Corr. II. 322. t This part of the letter is cited in the 3rd edition of the Principia, p. 246, instead of the letters to Leibnitz referred to in the two first editions. Its contents were sent to Leibnitz July 26, 1676, along with Newton's letter of June 13 of that year. There is a copy of it at the Royal Society (Miscell. MSS. LXXXI.) written in a tremulous hand, a consequence probably of the endeavour of the copyist to imitate Newton's writing. It has an address in Newton's hand, '• These to his ever honoured ffriend M r . John Collins...," and bears the post-mark of May 27 (probably 1676). This transcript may be conjectured to have been made at Collins's request for the purpose of accompanying the other papers which he was preparing to send through Oldenburg to Leibnitz. See Commerc. Eplst. p. 47. (128, 2nd ed.) Doubts have been expressed whether these papers were actually sent to Leibnitz. We have however Collins's own testimony that they were sent as had been desired (Comm. .Epist. p. 48, or 129, 2nd ed ), besides Leibnitz's and Tschirnhausen's acknowledgements of the receipt of them. (lb. pp. 58, 66, or 129, 142). It may also be observed that the papers actually sent (in a letter dated July 26, 1676) to Leibnitz by Oldenburg have been recently printed from the originals in the Royal Library at Hanover (Leib. Math. Schrift. Berlin, 1849), and that in them, as in Collins's draft, which is preserved at the Royal Society ("To Leibnitz the 14th of June 1676 About Mr. Gregorie's remains" MSS. LXXXI.), we find the contents of Newton's letter of Dec. 10, 1672, except that instead of the example of drawing a tangent to a curve, there is merely allusion made to the method. Collins's larger paper (called " Collectio" and " Historiola" in the Commerciiim Epistolicum), of which the paper just quoted " About Mr Gregories remains" is an abridgement, and which con tains Newton's letter of Dec. 10 without curtailment, is stated in the second edition of the Cotmnercium to have been sent to Leibnitz, but whether that was the case may be fairly questioned. This paper was intended by Collins to be deposited in the NEW ADDITIONS AND REMARKS. 100 1676 June 13. Letter to Oldenburg, containing a general answer to Lucas with a promise of a particular one, and also " some communications of an algebraical nature for M. Leibnitz, who by an express letter to Mr. Oldenburg had desired them." (read to the Soc. June 15 : the part for Leibnitz* was sent to him at Paris, July 26). Sep. 5. Letter to Collins. (Infinite Series of no great use in the numerical solution of equations. The University press cannot print Kinkhuysen's Algebra : the book is in the hands of a Cambridge bookseller with a view to its being printed : shall add nothing to it. Will alter an expression or two in his paper about infinite series, if Collins thinks it should be printed). 1676 Oct. 24. Latin letter to Oldenburgf for Leibnitz, who desired archives of the Royal Society, where it is still preserved, with the title " Extracts from Mr Gregorie's Letter" (MSS. Lxxxi.), consisting of thirteen sheets. A copy of Newton's letter was sent to Tschirnhausen in May, 1675, in Collins's paper " About Descartes" (14 folio leaves, Roy. Soc. MSS. lxxxi). * It was afterwards printed in Wallis's Opp. in. 622—629. (Oxf. 1699), and, from that work, in the Commercium Epistoliciim, where the typographical error of 26 Junii for Julii, which is corrected in "Wallis's errata, is also copied in the heading of the letter. t The original letter extending over 14 folio pages is in the British Museum (MSS. Birch 4294). It was accompanied by a note to Oldenburg (Mace. Corr. II. 400; in a postscript to which he observes: "I hope that this will so far satisfy M. Leibnitz that it will not be necessary for me to write any more about this subject; for having other things in my head, it proves an unwelcome interruption to me to be at this time put upon considering these things." Newton sent some corrections by the next post (Appendix, p. 257). A copy of the Letter so corrected was not despatched to Leibnitz until May 2 of the following year, the delay arising from Oldenburg's anxiety to send this " Thesaurus Newtonianus" by a safe hand. Leib. Mathem. Schrift. I 1. 151 (Berlin, 1849). On Nov. 14 he desired Oldenburg to make some further corrections, (Appendix, No. XVII.) which, however, were not introduced into the copy sent to Leibnitz, which was made ten days before. This letter, like its predecessor of June 13, was printed in the 3rd Volume of 110 NEW ADDITIONS AND REMARKS. explanation with reference to some points in the letter of June 13. Oct. 26. Letter to Oldenburg, with corrections for his letter of Oct. 24, &c* Nov. 8. Letter to Collins, thanking him for copies of the letters of Leibnitz and Tschirnhaus, with remarks shewing that Leib- nitz's method is not more general or easy than his own.f 14. Letter to Oldenburg (cider-fruit-trees : Lucas's 2nd letter: further alterations of his letter of Oct. 24)4 We have omitted in this copy of Edleston's Synoptical view all those other valuable notes and dates which are irrelevant to our special subject. Messrs. Biot and Lefort can learn from what we give, how much an honest and elegant investigation in difficult matters differs from their sophistical and untrue pleadings. Wallis's Opera, from which it was copied into the Commercium Epistolicum. Wallis says that he obtained his copies of the two letters from Oldenburg. Leibnitz wrote two letters in answer (June 21, July 12, 1677); in the former of which he gives examples in differentiation. Oldenburg acknowledged the receipt of these Aug. 9, observing, "Non est quod dicti Newtoni vel etiam Collinii nostri responsum tam cito ad eas expectes, cum et urbe absint, et variis aliis negotiis distineantur." (Leibn. Math. Schrift. I. i. 167, Berlin, 1849). Oldenburg died the following month, but there is no reason to think that, if that event had not taken place, Newton would have departed from his intention of not continuing the corre- spondence. Leibnitz's answers will be found in Wallis's 3rd volume, the Commercium Epistolicum and his Works. * Appendix, No. XVI. t Mace. Corr. II. 403. \ Appendix, No. XVII. In his papers on the early history of the Differential Calculus, particularly on Newton and Craig in the London, Edinburgh, and Dublin Philosophical Magazine of 1852, p. 321, Professor De Morgan makes the following statements : (the usual signs " " will denote that I use Professor De Morgan's words, while my own will be included in parentheses.) " My present object [says Professor De Morgan] is the early " history of the principle of the Differential Calculus in England : " I mean the principle of infinitely small quantities, as distinguished " from that of ultimate ratios or limits." Up to 1704 Newton always " used infinitely small quantities." The method of fluxions translated by Colson from Newton's latin, written in the period 1671 — 1676, is "strictly infinitesimal," and so also in the first edition of the Principia, 1687, the description of the fluxion " ' is founded on infinitesimals.' " This will be seen in the following extract from the first edition of Newton : " Cave tamen intellexeris particulas finitas. Momenta quam pri- " mum finite sunt magnitudinis, desinunt esse momenta. Finiri autem " repugnat aliquatenus perpetub eorum incremento vel decremento. " Intelligenda sunt principia jamjam nascentia finitarum magnitu- " dinum." [We cannot, I think, agree with Professor De Morgan, who tells us, that these words of Newton are " strictly infinitesimal." On the contrary we see how Newton protests against these ideas]. " The treatise De quadratura was written by Newton long before "1704; — it appeared in its essential features in Wallis's Algebra of " 1693 — and here we now see the subsequent abandonment of uncloaked " infinitesimals." For Newton wrote, 112 NEW ADDITIONS AND REMARKS. 1639 in Wallis: and 1704 in his De quadrat ura \: " Quantitas infinite parva ... Et " Quantitas admodiim parva... "ha? quantitates proximo temporis " Et si quantitates fluentes jam sunt " momento per accessum incremen- " z. y, et a?, hse post momentum tem- " torum momentaneorum, evadent " poris incrementis suis oz, oy^ ox, " z + <?.».. .terminos multiplicatos per " auctae evadent z + oz ... Minuatur " tanquam infinite parvos dele, et " quantitas o in infinitum, et neglec- " manebit Eequatio..." "tis terminis evanescentibus..." [Here Professor De Morgan thinks we see Newton's "subsequent "(1704) abandonment of uncloaked (1693) infinitesimals." But why has Professor De Morgan cut the phrases of Newton, and thereby prevented us from seeing that the word " fluentes," which he gives us (1704) occurs also in Newton's phrases (1693) in Wallis? With this word in 1693 we see much less of the cloak thrown over infinitesimals in 1704; the other little differences of expression are indeed inessential ; at all event3 Newton did not throw (1704) a new borrowed cloak over his idea, but only perhaps somewhat more clearly than 1693 repeated his oldest idea of 1687 : " principia (nascentia et) evanescentia."] Professor De Morgan then speaks of Craig's book, and of Newton's behaviour in this respect; of Craig's and Newton's supposed clandestine measures and intentions, by which they wish to lead the public away from truth. Craig, he intimates, wrote three separate tracts with almost the same title; the first was edited in 1685, the second in 1693, the last in 1718. "In the preface of 1718 Craig informs us, that in 1685 he " was a resident at Cambridge, and that Newton at his request read " his work. Here however we have reason to think, that he spoke " of the wrong tract ; [why should, we ask Professor De Morgan, Craig, speaking on his own tracts, speak of the wrong one?] "After " some exemplifications of Barrow and Sluse, not referring to Newton " as having any method of quadratures, but only to the Binomial " Theorem, Craig proceeds to say, that nothing is wanting to extend " his method to all but transcendental curves, except only the removal NEW ADDITIONS AND REMARKS. 113 " of two difficulties. The first difficulty is the extraction of roots, " which he gets over by a Series of Newton's, which he hears that " Dr. Wallis has sent to press, [the first edition of Wallis is dated " 1685] which Newton has had the goodness to communicate in manu- " script ." The second of Craig's Tracts is of 1693. " That this was the tract," [though Craig in the Preface of 1718 says, that it was the Tract of 1685] "which Newton examined before it was printed, I infer as " follows. In the Preface of 1718, Craig states that Newton proposed " two curves, of which he gives the equations [these equations mentioned in the Preface of 1718 by Craig are : m-y* = x* + aV, and my 1 = x 6 + ax*] " as examples in corroboration of Craig's objections against Tschirn- " hausen." [Professor De Morgan withholds from us the knowledge, that in Craig's Tract of 1685 the first of these examples appears at page [ill 1 + x 2 ) X X* 42 in the form Z 2 = J- P " Now the attack upon D. T. (Tschirnhausen) is at the end of the " second tract of 1693, and the curves specified are the first two examples " at the beginning." [Professor De Morgan withholds from us the knowledge that Tschirnhausen is also attacked in the first edition of Craig of 1685, and not only attacked as in the second tract, but this with one of these very two equations, which Craig must consequently have received in 1685]. "Moreover in the first tract Craig was no " deeper in the Differential Calculus than to imagine, that Pdy = Qdx " always gives Py = Qx, which we may undertake to say Newton could "not have passed over without detection." [Craig's book was not written on the Differential Calculus, but treated of quadratures as found by Craig's particular method 5 Newton had therefore no reason to speak to Craig of this method]. "It is also to be noticed, that in the second " tract the name of Newton does not occur once, though it is full of " the Differential Calculus, and though Leibnitz, Sluse, Barrow, Gregory, " are frequently mentioned. This, under all circumstances, Ave may " suspect was Newton's own doing." [Here Professor De Morgan Q 114 NEW ADDITIONS AND REMARKS. kills his enemy : Newton is not mentioned in the book ; that must under all circumstances be Newton's doing ! Newton is detected, as Professor De Morgan always detects him, playing a foul game. Newton had cleverly " cloaked over" with the cloak of fluxions and ultimate ratios Leibnitz's infinitesimals, and so Newton here again, as Professor De Morgan tells us, has hidden and cloaked himself over, for he must, as Professor De Morgan makes out, be in this book in which he is not mentioned. The lie is given to every witness before us, the truth being the contrary of what we read]. "And I am strongly inclined " to think that it was this very tract of Craig's, which immediately " suggested to Newton the progress which the views of Leibnitz were " making, and induced him to forward to Wallis the extracts from his " De quadratura Curvarum." [There is in the date no difficulty]. " I conclude therefore that Newton, seeing the progress the Differential " Calculus was likely to make in England, procured the entire suppression " of his name in Craig's tract, and made up his mind to insert a part " of his treatise in the forthcoming work of Wallis." So far we have principally cited Professor De Morgan ; we shall now, in two words, give the fact as it actually stands. Craig did not as yet in 1685 understand either Leibnitz's Differential Calculus, of which only a little had transpired, or Newton's fluxional form, of which nothing till then was known, but he showed Newton something about another method of quadratures, and Newton for the sake of assisting Craig in an attack against Tschirnhausen, suggested to Craig two equations ; one of which is still in our own days to be read in Craig's book printed 1685, although Professor De Morgan says that these examples were given 1693. In 1603 Craig had got hold, as he thought, of the whole theory of the Differential Calculus, and he here speaks a good deal of Leibnitz, but not of Newton ; Newton indeed had in L693 not yet published on the subject. But in 1718 when he wrote his third tract, Craig was fully acquainted with the discoveries of Newton and of Leibnitz, and he there says in his preface, that what he wrote in 1685, was for so early a date a good tract on quadratures, and he NEW ADDITIONS AND KEMARKS. 115 was proud of saying that Newton saw this his first tract before it was printed in 1685. Every witness in the matter is here correct, with the exception of Professor De Morgan, who gives them all the lie. But they do not lie. Firstly Craig, speaking of his own tracts, says that it was his tract of 1685 that Newton saw. The tract itself, printed in 1685 (!) says that Newton gave the author a manuscript (see page 27 ! !). Professor De Morgan says no ! it was your second tract in which Newton has assisted you. Now it happens that in this second tract Newton is not mentioned. For this very reason, says Professor De Morgan, it is this tract which Newton saw. We ask, is it not absurd to treat history in this manner? Professor De Morgan " infers''' 1 and " supp>oses" and " is much inclined u to think 1 '' that of all facts which are before us, the contrary is true, for Newton's honour is at stake. Now may we not "infer" and " suppose" and be " inclined to think' 1 '' that Professor De Morgan is the last person whom we can trust in matters concerning Newton. This is very serious, for Professor De Morgan carries things at this moment in England with a high hand respecting Newton and respecting the history of fluxions, and clever Frenchmen eagerly avail themselves of his remarks. We will add here the whole short preface of Craig's book of 1718, which says : Prsefatio ad Lectorem. Hahes hie B. L. quce multos ante annos de Calculo jluentium sum meditatus, & cvjus prima Elementa, cum Juvem's essem, circa Annum 1685 excogitavi : Quo tempore Cantabrigian commoratus D, Newtonum rogavi, ut eadem, priusquam prailo committer entur, perlegere dignaretur : Quodq ; Ille pro summa sua humanitate fecit : Nec-non ut Objectiones in Schedulis meis contra D. D. T. allatas corroboraret, duarum Figurarum Quadra- turns sponte mihi obtulit ; erant autem harum Curvarum JEquationes m' 2 y 2 = x 4 + a 2 x 2 & my* = x 3 + ax 2 ; Meque interim certiorem fecit se posse hujusmodi innumeras exhibere per Seriem Infinitam, quce in datis con- ditionibus abrumpens Figuroz propositce Quadraturam Geometricam deter- Q2 116 NEW ADDITIONS AND REMARKS. minar, t. In Patriam postea redeunti magna mihi infsrcedebat familiaritas mm Eruditissvmo Medici) D. Pitcairnio & D. D. Gregorio ; quibus significavi qualm pro Quadraturis Seriem haberet D. Newtonus, quam penithd ipsis ignotam wterq ; fatebatur. Post aliquot tferb imnses narrabat mihi D. Pitcairnius D. Gregorium Seriem similiter abrumpentem invenisse. Ego nullus dubitans, quin tandem ex duabus prcedictis Quadraturis ipsi a me communicatis deduxerit, per Litems D. Newtonum rogavi, ut Seriern sua m mihi transmittere vellet, ut an eadem esset cum Gregoriana pers- . ' rem : Rogatui meo an unit Vir illustrissimus per Literas 19 Sept. 1688 datas : Nee mirum si parva esset inter utramq ; Seriem discrepant ia, cum Gregorius, ex duobus Mis Exemplis & indicatd . a me Seriei Newtonianae indole, suam facile deducere potuisset ; quamq ; statim in Tractatu D. Pitcairnii De Inventoribus publicandam curavit. Hanc kistoriolam Lectoribus impertire cequum videbatur, ut soli Newtono Sen Mam tribuendam esse cognoscerent. Satius quidem multo fuisset, si ipse [dum vivus esset) Gregorius eandem Orbi Mathematico communicasset, quodq ; se facturum promisit per Literas dat. Londiui 10, Oct. 1691. Me interim in iis hortatus est, ut, si quid haberem ad Mt murium ejus in hoc negotio jnvandam, id ego quam citissime ad Mum transmits rem / quod sine mora a me rem onxnem fideliter ab initio narranh factum erat. Opus enim erat mihi facillimum, utpote qui omnes ejus cU Pitcairnii Literas hanc rnuti riam spectantes turn apud me hetbuerim, & adhuc habeo. Ego interim (ob plures rationes non jam enumi randas) nihil per quam generate in Quadraturis per hujusmodi Series expectandum jure ratus, ad propria i n Meihodum promovendam Studia mea convertebam : Nee irritos jirursus f'nisse conatus colligere potes ex Tractatu Ann. 1603 edito, & Sped mine in Actis Philosophicis Anni 1697 de Spatiorum Transcen- dent inm Quadraturis, quo? in Geometria omnino tum noveeerant. Ejusdem Anno 17<i-_>, longe ultra omnium a/iarum Umites promotee, Theoremata aliquot generalia in Act. Phil. Anni 1703 erant publicata. Et magnopere mihi pJaeuisse fateor, cum perciperem, quod proedicta Series Newtoniana Casum tantum simplicem Theorematis nostri primi comprehenderet. Integram jam Methodum cum aliis huic ajflinibus in sequenti libro NEW ADDITIONS AND REMARKS. 117 explicatam B. Lector inveniet. Et si qaidpiam in his ad Geome- triam promovendam sibi occur rat, turn me finem in his edendis proposition obtinuisse sciat. This is Craig's preface of 1718. We will also give the first lines of Craig's second book of 1693, which are as follows: "In actis philosophicis specimen exhibui methodi generalis " determinandi Figuraruni Quadraturas ; cuuique postea plus otii nactus " fueram, credebam me non posse illud melius, quam in eadem materia " ulterius perficienda, collocare ; plurima enim turn deerant, quseque me "jam feliciter obtinuisse spero. Ne antem nimium mihi adscribere, vel " aliis detrahere viclear, libenter agnosco Leibnitzii Calculum differentialem " tanta mihi in his inveniendis suppeditasse auxilia, ut sine illo vix " assequi potuissem " We add a remark, which we certainly cannot introduce without saying that it is a pity that Professor De Morgan has not found it out, for he would have made another use of it than we do. It is supposed that Newton's " De Analyst" such as it was printed in the Commercium Wpistolicum^ and consequently such as it is now before us, was sent in 1669. But this perhaps is not the case. The best witness in this matter is Oldenburg, especially if we attend to what he said in this respect in 1669, the very year in which Newton's treatise was written or sent to London. Oldenburg's letter of 14th September, 1669, ad Franciscum Slusium, though inserted as No. XIII. in the Commercium EpistoUcum, has not yet been read with attention respecting this question. Oldenburg says at that date, that Newton's Analysis ("universalis "methodus Analytica") has been sent [to himself, to Lord Brounker, or to Collins] ; Oldenburg then adds : " Auctor sic incipit. u De Analyst per jffiquationes numero terminorum infinitas. " Methodum gencralem, quam de Curvarum quantitate per Infinitam " terminorum seriem mensuranda olim excogitaveram, etc." " Et ad calcem sic ait : " Nee quicquam hujusmodi scio ad quod haec Methodus, idque variis " modis, sese non extendat. Imo Tangentes ad Curvas mechanicas (si "quando id non alias fiat) hujus ope ducuntur. Et quicquid vulgaris " Analysis per requationes ex finito terminorum numero constantes " (quando id sit possibile) perficit, hsec per iEquationes infinitas semper " perficiat. " Et hsec de Areis Curvarum investigandis dictas ufficiant. Imo cum " Problemata de Curvarum Longitudine, de quantitate & Superficie NEW ADDITIONS AND REMARKS. 119 " Solidorum, deque Centre Gravitatis, possunt eo tandem reduci ut " qugeratur quantitas Superficiei plana? linea curva terminatae, non opus " est quicquam de iis adjungere." Here we have the beginning and the closing words of Newton's De Analysis such as Oldenburg had it before his eyes, 14th September, 1669. Consequently what Oldenburg possessed cannot be that which has been given in the Commercium Epistolicum, for the latter only begins as Oldenburg says, but it does not end so. We may suggest, that Newton's Analysis from page 83, line 18 in the Commercium Epistolicum, edition of 1722 [in Biot's edition page 67, line 15] was read some time in the year 1669 as follows: Denique si index potestatis ipsius x vel y sit fractio, reduco ipsum ad 1 4 II integrum : ut in hoc exemplo y* — xy" + x 3 = o. Positio y = ?>, & x 3 = «, resultabit v 6 — z 3 v + z 4, = o, cujus radix est v = z 4- z 3 , &c. sive (restituendo i I 2 4 valores) y 1 = x 3 + x, &c. & quadrando y=x 3 -\- 2x 3 . Et haec de Curvis Geometricis dicta sufficiant. Quinetiam curva, etiamsi Mechanica sit, methodum tamen nostrum nequaquam respuit. "Exemplo sit Trochoides ADFG, cujus (etc. :" those 45 lines which are now read page 88, 89 — in Biot page 71 [from line 10] 72 [to line 11] — up to the words " determinabilis est," with Oldenburg's finishing sentence as follows) : Determinabilis est. Nee quicquam hujusmodi scio, ad quod haec Methodus idque variis modis, sese non extendat (etc. : thirteen further lines just mentioned, page 118, as the end in Oldenburg's letter of 14th September). We consequently believe, that page 84, 85, 86, (?) 87, and (?) page 90, 91, 92,9 3, (in Biot page 67, [from line 23] 68, 69, 70, [to line 8] 72, [to line 17] 73, 74, 75,) were introduced into the manuscript in or about the year 1672. Newton, in fact, writing to Leibnitz 1676, 24th October, calls his treatise a " compendium" and says of it : " Eo ipso tempore quo (Mer- " catoris Logorithmotechnia) prodiit, (1669) communicatum est ad I). 120 NEW ADDITIONS AND REMARKS. " Collinsium (raeum) Compendium quoddam methodi harum serierum, in " quo significaveram Areas et Longitudines Curvarum omnium, et " Solidorum superficies et Contenta, ex datis Rectis ; et vice versa, ex " his datis Rectas determinari posse deinceps Collinsius non destitit " suggerere ut baec publici juris facerem : Et ante annos quinque cum " suadentibus amicis consilium ceperam edendi Tractatum de Refraetione " Lucis et Coloribus, quem tunc in promptu babebam ; coepi de his " Seriebus iterum cogitare ; et Tractatum de iis etiam conscripsi ut " utrumque simul cderem. Sed " Newton here speaks of that Treatise (Tractatus) which is not in the Commercium Epist. but which was published by Colson 1736, and we see that "ante annos quinque" (that is 1672) he meditated on these matters. We know that Newton was, after 1673 till 1683, engaged in labours of a different kind, and a copy of what we now read as Newton's Analysis has existed in Collins' handwriting, who died 1682. Therefore Newton's additions to what he first sent 1669 and what Oldenburg possessed 1669 were made at once in 1669 for Collins, or between 1669 and 1673, and this is what Collins possessed. It is a pity, we repeat, that Professor De Morgan has not found out this fact ; perhaps he will still think it not beneath his honour to make use of it in his fashion, and to draw from it with Anti-Newtonian instinct a good conclusion. NEWTON'S SUPPOSED ERROR. In a scholium at the end of the Tractatus de Quadrature:, Newton says : " Quantitatura fluentium fluxiones esse primas secundas tertias " quartas, aliasque dixinius supra. Hae fluxiones sunt ut termini serierum "infinitarum convergentium. Ut si x sit quantitas fluens, et fluendo " evadat [x + o) n deinde resolvatur in seriem convergentem „ „_, nn — n „_„ n 3 — Snn + 2n ., „_.. x + nox -i oox H ox -\- etc. 2 6 " terminus primus hujus seriei x n erit quantitas ilia fluens, secundus " nox n ~ x erit ejus incrementum primum, seu differentia prima, cui " nascenti proportionalis est ejus Fluxio prima ; tertius — -— oox n * erit " ejus incrementum secundum, seu differerentia secunda cui nascenti ,, • i- . „, , , n 3 — Snn + 2n „ „_„ " proportionalis est ejus t luxio secunda ; quartus o x erit " ejus incrementum tertium seu differentia tertia, cui nascenti Fluxio "tertia proportionalis est; et sic deinceps in infinitum." John Bernoulli caught hold of this, and wrote to Leibnitz (7th June 1713) : " vides hanc regulam (Newtoni) falsam esse. Nam excepto " primo et secundo termino, reliqui omnes alludunt a differentialibus " superioribus potestatis x n et hoc est, quod in nupero meo Schedi- " asmate Actis Lipsiensibus inserto jam notavi. Animadverti New- u tonum in suo errore perseverasse usque ad annum 1711 cum libellus " ejus fuit recusus. Sed in exemplari quod mihi dono misit per " Agnatum meum, ibi" (meaning that scholium) " calamo adscripsit K 122 NEW ADDITIONS AND REMARKS. " altera vice voculani ut — ubi habebantur haec verba ' tertius (terminus) — - — c?x n ~ 2 erit ejus inereinentuin secundum, et quartus nn — Snn + 2n „ „_. 6 " ' erit ejus inerementuni tertium " ' erit ut ejus 1 etc." interseruit c ut,' scribendo nunc. In the Acta Eruditorum of Leipsig, Bernoulli made a great noise about this error of Newton. Montucla (in his Histoire des Mathematiques III. p. 105) speaking of the matter, says: "Bernoulli insera dans les " actes de Leipsick, sous un nom deguise une lettre fort amere, ou '•'Newton e'taitpeu manage; il y pretendait que Newton n'avait jamais " connu les regies de la seconde differenciation, ou celle de prendre la " fluxion d'une fluxion, et il se fondait sur ce que Newton dans son traite" " de quadratura curvarum, dit, que les fluxions des differents degres " sont representees par les termes de son binome m (m — 1) x 1 ' x + \_rnx J x + 1.2 x' + "or cela n'est vrai qu'en supprimant les de'nominateurs numeriques. "Car si l'on prend la fluxion, ou la diflfe'rentielle, de ma;"* -1 , on aura pour "seconde diffdrentielle seulement [m (m — l)a?" 1-2 ] a; 2 . Mais il est eVi- " dent, que c'^tait une pure inadvertence de Newton, seduit un instant " par une analogie qui regne entre sa formule et les fluxions successives. " Cette piece (in the Acta of Leipzig) etait si aigre, que Bernoulli a ^te " long temps sans l'avouer, mais il etait facile de l'y reconnaitre."* Now this famous Newtonian error, which almost Newton himself took for an error, writing the word " ut" with his pen in the copy of the second edition, when he presented the same to Bernoulli, (for Bernoulli * Montucla quotes the formula as as™*, etc. Bernoulli as z"\ etc. To make our remark clear, we have also in our quotation of Bernoulli written the letter x. NEW ADDITIONS AND EEMARKS. 123 at the time and before he made his anonymous attack in the Acta of Leipsig, had through his nephew Nicholas Bernoulli, who made a journey to London, sent a friendly message on this matter to Newton), now we believe this famous Newtonian error is in truth no error. The series (x + 6) n = x n -f nx n ~ x o + etc. is, as is well known, Taylor's theorem in one of its cases. What we wish to say will therefore at first be in reference to this theorem. The Differential or Fluxional Calculus has to solve the following problem and question : how great, if any (simple or complex) algebraic expression be augmented in only one of its constituent parts, is the augmentation of the whole expression, compared to the augmentation of the part. If, for instance, the algebraic expression (function) is (a;" 1 ), and it is the part x, to which we give an augmentation, and if we here again take the simpler case, by supposing the function (or algebraic expression) to be (a? 2 ), then the question or problem is: how great is the augmentation of (as*) if x receives a certain augmentation, which we will call dx? It is false to suppose, that in a general manner the augmentation accruing to the whole by an augmentation of one part, can be indicated by one of the seven algebraic operations (addition, subtraction, mul- tiplication, division, elevation into powers, extraction of roots, and logarithms.) On the contrary, the comparing of the augmentation of a whole function (or algebraic expression) to the augmentation given to a part of it, is such a complicated thing, that all the seven operations, and all their possible combinations are not able to make this comparison, and consequently not able to give us the answer. This is the difficulty in which we are placed by the question. The Fluxional or Differential Calculus overcomes the difficulty, by in- troducing an eighth mode of algebraic operation, namely, by introducing to us first, the idea of a multiplication, such as (a x J), to which secondly, we have to add this new idea, that, namely, the one of the two R2 124 NEW ADDITIONS AND REMARKS. factors or coefficients does not remain unchanged during the operation of the multiplication, but changes : is variable. We ask : how is this possible ? how can we multiply with a certain and clear result a by b, if one coefficient (let us say a) does not hold its value during the operation, being what is called a variable? The Fluxional or Differential Calculus overcomes this new dif- ficulty, which it has itself created, by telling us not only abstractly that the one coefficient (let us say a) is variable, but by telling us also which kind of variation there is in a. In the above case, for instance, the algebraic expression or function being (aj a ), and our question being: how great is the augmentation of (x 2 ) if x receives a certain augmentation, which we will call dx? — the answer given by the Fluxional or Differential Calculus tells us firstly, that the augmentation is a multiple of dx (of the augmentation given to the part of the function), namely, that it is 2x x dx, but with this understanding, that the one of the two factors or coefficients of this multiplication, namely, the expression 2x, does not hold its value, but is variable, adding that the nature and kind of its variation is enunciated and indicated by the second differential, namely, that it is again a multiplication (2 x dx) dx, which has to be considered as laying in the first. In other words : since it is not possible to say by any common combination of the seven algebraic operations in a general manner (for a single case is not in question) what augmentation accrues to (x' t ) when x (the one part of that function) is augmented by dx, therefore a new mode of operation is introduced, which is a multi- plication in which the one coefficient holds in itself a second multi- plication. We repeat, the rule is : multiply dx with 2x, which 2x flows during the multiplication, its fluxion being the augmentation of 2x when x is augmented. This (eighth) mode of algebraic operation is, as we see, so complicated, that it cannot be expressed in one rule, but that two rules (or two NEW ADDITIONS AND REMARKS. 125 differentiations) are necessary, the first rule saying that the augmentation of (x 2 ) is dx multiplied with a thing, which is itself augmented, [2x] and the second rule (or differentiation) saying, how this 2x is augmented. If the function, or algebraic expression is simpler, namely, if it is addition or multiplication, for instance, if it is mx, then the Differential or Fluxional Calculus can say by one rule with one of the old seven operations what augmentation mx receives if one part of it (say x) is augmented by dx. Namely, the augmentation of mx is m x dx. But if the function be (sc a ), two rules are necessary, first and second differentials. The second rule is the correction of the first rule. The first rule speaks of the function, the second rule speaks of the first rule, and thereby indirectly of the function. Sometimes the number of rules is infinite. For instance, the number of the rules for finding the augmentation of the function [x*) or \'x is infinite, for after saying that that augmentation is dx multiplied with £(»"*) (or with — —J wherein a; is a variable, and after the second V 2 \jxl rule telling us, that the variability in that first rule is —£(#"*), or ( - ] , we still find this variable x in our answer, we must therefore give a third rule (third differential) respecting the variability of the second, which rule will again give us that variable x, and so on, without end. In the case of (x 2 ) there is that end. Its first differential is 2x x dx and the differential of this differential is 2dx x dx, where all coefficients and factorials are constants, for dx is (what few people seem to admit) a constant, whereas x is variable. If we have till now before us the seemingly difficult idea of a multiplication regulated by a new multiplication, Taylor in his theorem brings new light into these ideas. It is as if Taylor said: you may theoretically and in the abstract 126 NEW ADDITIONS AND REMARKS. speak of one rule (the first differential) and of its modification by a second rule (the second differential), and in your fancy you may think that one multiplication is hidden or lies in the other, but if you leave with me the world of abstraction and theory, and enter into nature and into reality, letting dx actually be something, a foot, or an inch, or some other reality according to the actual problem, then those abstract rules will require one very slight modification. It is easy to show what this modification is. We suppose there will be no possible objection if we say, that the series 1, 4, 9, 16, 25, 36... is one case of the variable function (a? a ), namely, that case, In which the varied and variable part of the function [x) has been supposed to be in one instance 1, and in which the variability of x, or the augmentation dx given to x, is supposed to be also 1. For in this supposition we have x= 1, x + dx = 2, x + dx + dx = 2 + dx = 3, x + dx + dx + dx = 3 + dx = 4, etc. Consequently in this case all the values which (x' 2 ) can have, and therefore (x*) itself (in the full extent of its variability) is contained in the series ...1, 4, 9, 16, 25, 36,... 3, 5, 7, 9, 11, 2, 2, 2, 2. We have put under the principal series of all these numbers their differences. Thereby we see that it is not sufficient to give to x one augmentation dx, but that two such augmentations must be given and calculated before a difference of the first differences can appear. Between two NEW ADDITIONS AND REMAKKS. 127 numbers of the series there is but one difference. Three numbers have two differences, and now (not when we have but two numbers) we can speak of a difference of the differences. For instance, in our series the numbers 4, 9, 16 have the two differences 5 and 7, and now we can speak of the difference of these differences. It is in this sense that Taylor, as it were, says : theoretically and abstractedly your rules can tell me that the difference of the function x 2 from its next augmented expression (x + dx) 2 is of a flowing nature, (not every- where the same) and you can theoretically and abstractedly at once, in addition thereto, tell me, what the flowing nature (or fluxion) of that first difference is, but in reality we cannot find a second difference in one first difference, we can only find it in two first differences. Therefore, Taylor wishing really to do in this function (x"), or rather in all functions, what theory tells us, could not content himself with calculating one difference [(x + hf — x 2 ] or [{x + dx) 2 — x 2 ], but finds him- self obliged to calculate two successive differences [(x + h + Kf — (x + /?)*] together with [{x + h) 2 — x 2 ], before a difference of these differences (or the second differential) appears, and wishing nevertheless to give his formula only for one difference [only for : f(x + h) —f(x)]j he consequently has to divide that member, which contains the second differential by 2. We see that he consequently in his formula has to divide the member containing the third differential by 2x3, the following member by 2x3x4, etc. Taylor's series contains therefore in truth, so as Newton said, the subsequent differentials : in other words, Taylor's or Newton's series is the Differential or Fluxional Calculus, and nothing else, and Taylor's series needs not to be proved, but proves the new Calculus, and is thereby proved itself. Newton knew this, and in his constant practical applications and calculations from his youth till 1711, he had been so used to consider the terms of such series as the subsequent differentials, or fluxions practically, that also theoretically and abstractedly in the scholium which we have copied, he very properly said that they were both the same. 128 NEW ADDITIONS AND REMARKS. In 1711, when Newton was seventy years of age, he was (we may say this with the greatest veneration for Newton) not sufficiently intent on abstract mathematics to wish to dispute about the greater or smaller coincidence of the terms of those series with the subsequent fluxions; he therefore left the question open by adding the word ",«#" to the scholium, which thereby is not essentially altered, not so much as it may appear altered by the word " very" in the third section of Newton's letter of 20th April, 1714 (in Edleston's Corresp,., page 173). In a German treatise inscribed Versuch die Different ialreehnilng auf andre als die bis herige Weise zii begrunden, I have explained this more at length. Leibnitz, on his entrance into society, having jnst taken his degree at the University, became attached to the interests of Boineburg, who was on a small scale, we may say a Metternich of those ages, in the service of the Elector of Maintz. This Boineburg asserted from the time of the last election to the German Empire, a claim for money due to him from the King of France. The Elector permitted Leibnitz, who was in the Elector's service, to go to Paris, 1671, for the purpose of secretly and privately inducing the King of France to conquer Egypt, Leibnitz supposing that this scheme would avert the danger of the King's threatening position and preponderance in Europe. But the Elector did not pay for Leibnitz's stay in Paris, it being ingeniously arranged by Boineburg that this expense was charged to the King of France, whom Leibnitz had to advise and to persuade. Boineburg's own object was to realise through Leibnitz's presence in Paris his pecuniary claim. A strange errand (Leibnitz's moral " character" would necessarily be thereby affected ; namely, " strengthened" and " advanced" Guhrauer says). Compare Guhrauer's Biography of Leibnitz (1846) I., page 49, line 10, et seqq.; page 52, line 12; page 55, line 1; page 56, line 17; page 59, line 12; (page 62, line 9, conf. page 82, line 6); page 95, line 3; page 98, line 26; page 105, line 27 (" entretenu") ; page 125, line 12, etc. No biographer or other author, I believe, has enquired : for what purpose did Leibnitz go to England towards the end of October, 1676 ? We presume that there were frequently in Havre or in Calais ships going via London to Amsterdam, and that the direct intercourse between France and Amsterdam was not common. Perhaps some persons know of other reasons; to suppose him actuated by a desire to learn what mathematical secrets the English were yet withholding from the rest of the world, would suit this inquiry, but I have no grounds for believing it. That Leibnitz was not in possession of the Differential Calculus on the 27th August, 1676, shortly before his second journey to London, is proved by his letter of that date, because in that letter he operates with the infinitely small quantity, called /3, vulgari more, using the method of Transmutations, in the same kind of manner as not only Leibnitz, but everybody did at Leibnitz's time and before him. Some passages of that interesting letter of the 27th August, 1676, are explained by a first draft of the same, which (imitating Gerhardt, who has edited first drafts of Leibnitz's several writings), we will give in extenso (the original is found in the bookcases, containing Leibnitz's manuscripts at Hanover) : (By using parentheses, such as [ ], we have indicated that Leibnitz NEW ADDITIONS AND REMARKS. 131 after finishing the letter struck out, what these parentheses contain, writing over the line what then follows) : This letter of the 27th August, 1676, which Leibnitz sent to England, is an answer to two letters, to one coming from Newton, and to another coming from Collins. His first draft has reference only to that part in which he answers Newton. We therefore, in what here follows on the left side of the page, have given verbotenus the first half of Leibnitz's letter, such as it went to England, and opposite to it on the right side, Leibnitz's sometimes longer, sometimes shorter first draft : 27 Aug. 1676. Literae tuae, die 26 Julii datae, plura ac memorabiliora circa rem Analyticam continent, quani multa volumina spissa de his rebus edita. Quare Tibi pariter ac Clarissimis Viris, Newtono ac Collinio, gratias ago, qui nos participes tot medita- tionum egregiarum esse voluistis. Inventa Newtoni ejus ingenio digna sunt, quod ex Opticis Ex- perimentis et Tubo Catadioptrico abunde eluxit. Ejusque methodus inveniendi Radices iEquationum & Areas Fi- gurarum, per Series Infinitas, pror- sus differt a mea : Ut mirari libeat diversitatem itinerum per quae eodem pertingere licet. Literae tuae novissimae, Pellii Newtonii Gregorii Collinsii inventis et meditatis plenae plura ac me- morabilia circa rem Analyticam continent, quam multa volumina ingentia impressa ; vellem explicata essent, sed quomodo poterit id fieri in literarum angustia. Newtono et Collinsio multas a me gratias agas rogo ; Newtono quod Methodi suae circa series specimina mihi com- municare voluit, plurimum me illi obligatum profiteor. Digna sunt omnia ingenio ejus, quod ex opticis experimentis et libro catoptrico abunde eluxit. Mirari vero subit in varietatem itinerum per quas perveniri potest ad interiora rerum ; mea enim me- thodus longe a Newtoniana diversa est ; in nonnullis ad eadem per- venimus; in pluribus alias plane S2 132 NEW ADDITIONS AND REMARKS. scries 1 exhibco. Universalitatera ex ipsa Methodi descriptione existima- bitis, quam vobis exhibeo. Meroator Figuras Rationales, Equidem fateor 2 Mercatori me 1 Newton's words " Method of Series" are here also used by Leihnitz, who calls not only one, hut all his manifold methods, the method of "Series;" very properly, for quadratures etc. are, generally speaking, always given in infinite series, and are only exceptionally found in finite algebraic expressions. 2 Leibnitz here says, that the invention which Newton's letter communicated to him (Newton's generalization of Mercator's divisions) had already before he received the letter been invented by himself, this being quite an easy matter. This assertion of Leibnitz is not true, nor did he on second consideration venture to tell Newton this untruth, which we see in embryo only in this draft of the letter. Leibnitz had just finished his work De Quadrature! ("'jam anno 167<j compositum habebam opusculum Quadraturae Arithmeticae, ab amicis ab illo tempore lectum," says Leibnitz, in Act. Erudit., April, 1691, page 178). This he had completed in at least forty propositions in a beautiful manuscript ready for the p: ess, which remains to this day in the Leibnitz shelves of the King's library in Hanover. But in the same, written with a later hand, namely, not in Leibnitz's beautiful handwriting, in which the first part of this magnificent work is written, but in Leibnitz's hurried and quick handwriting, and squeezed in a narrow margin, we read the following : " Scholium ad propositionem 29 quod ope progressions Geometricae demon- " stravimus, poteramus et demonstrare per [aequationes] divisiones [pulcherrimas] "in infinitum continuatas pulcherrimas N.J. Mercatoris, Holsati, e societate Regia " Britannica (quae) coincidunt cum prop. 26 si ponamus prorsus etiam " exprcssiones (?) ac demonstrationes (?) propos. 28; deberet autem scribi istas esse " decrescentes. Simile quiddam ad radicum purarum vel affectarum extractiones " accommodari potest in numeris literisve, nam et in illis divisio quaedam locum habet, " quod jam dudum exemplis quibusdam expertus sum (ohe eos qui ex mea Circuli " expreBsione sequi putabant Circulum esse quadrato diametri commensurabilem) " etiam quant itates irrationales, e.g. diagonalem in quadrato per infinitam seriem " rationalium numerorum efferri posse. Sed hoc Clarissimum Virum Isaacum "Newtonum ingeniose ac feliciter prosecutum nuper acccpi, a quo praeclara multa "theoremala expectari possunt. Porro si contra ponatur c acq. AB* et b aeq. BF S D7W » . a BF* , AB* AB' , , " manente a aeq. hi' net - — aeq. — — - — =, aeq 1 - — — - + — — etc., quemadmodum of c ^ BI'iAB l BI* BF* BF* BF* BI H "ante aeq. -j™ - ~Tn\ + "T/js etc -> slve ponendo AB constantem sive permanentem. NEW ADDITIONS AND REMARKS. 133 sen in quibus Ordinatarum valor ex [primam occasionem] partem deberi datis Abscissis rationaliter expriml inventionum [Nam] circa series in> FC t 2 "aeq. 1, et BF vel BC aeq. t, tunc priore modo supra posito sive , fiet 2 1 + t FG i . "aeq. t* - t* + t a etc., secundum propositionem 28 et summa omnium sive area t 3 t b f . FO " dimidii spatii BFGfS erit - - - + - , et ad summam omnium — - seu aream spatii o 5 7 ^ '•dimidii BFGfi mutatis mutandis evadit series - - — 4 t-j etc., ut probari potest ex 1 It oi " corollario 2. ad prop. 25. ex quibus expressionibus per series cum t minor est " quam 1 prodierit series cum major est, et omnimodo sufficit prior sola, quoniam "si arcus BF3I sit major quadr. 2 tunc sufficit comparari excessum EM." So Leibnitz's great work De Quadratura, of which he speaks so often, was nipped in the bud by the first letter of Newton. For though in a short epitome of this Leibnitzian work, which Gerhardt has ventured to publish, the prop. 24 and 25, as Gerhardt admits, as well as the prop. 46, 47, 48, are of a later period, (see Gerhardt. Leibn. Math. Schr. V. Band 1858, page 86, line 27, and the note at page 105) still this later epitome could not efface all the weak points of this interesting work, and in particular not its twice repeated phrase : " oportet autem AB non esse minorem BF and oportet autem arcum BOD non esse quadrante majorem," (Gerh. 1. c. page 107). So then Leibnitz finds, when Newton's letter arrives, that with Newton's invented general rules like those of Mercator, " poteramus" as he says, " demonstrare propositiones" 28, 29, etc., and that then these propositions would have been general, namely valid also, " si arcus major quadrante." On the whole, we see by that scholium that the work De Quadratura was disturbed and changed by what Leibnitz could take from Newton's letter " quam nuper accepi." Consequently it is not true, that Newton's invention (the generalization of Mercator's divisions, etc.) was known and familiar to Leibnitz before Newton's letter; and not true, though Leibnitz uses the word " fateor," that he had taken " primam occasionem inventorum suorum" from this generalization of Mercator's idea. Luckily only what Leibnitz wished to answer, not his actual answer, contained that untruth. We beg Leibnitz's pardon for using such harsh language, for it is quite possible that Leibnitz was naive, and cheated himself, believing in his having invented what Newton's letter contained, for some people (and Leibnitz may have been of these) having rather lively imaginations, find no distinction between what they learn and what they invent, but take some old idea of their own, which bears on the subject, for the mother of the new things which they learn, and then cheating themselves, say : " these are my children." 134 NEW ADDITIONS AND REMARKS. potest, (ut scilicet indeterminata Quantitas in vinculum non ingre- diatur,) quadravit ; & ad Infinitas Series reducere docuit, per Divi- siones. Newtonus autem, per lladi- cuni Extractiones. finitas. Nam cum is Hyperbolain per infinitam seriem more suo [ex- tricasset] quadrasset utique facile erat judicatu, posse quamlibet figu- ram rationalem eodem modo per infinitam seriem quadrari. (Another sentence appears to be written over this sentence above its lines, but very- small, and to me unreadable and un- intelligible.) Figura rationalis enim naturali aequatione explicari potest sIt y aeq - r+v fiet x s — x 5 -f x 1 — x° + x 11 etc., • et sum ma omnium y seu area figurae erit 4 6 Vel etiam methodo 3 a Mercatoris diversa, nam si aliter dividas ex x 2 x" x"- To + 12 etc - fln _i 1 1 1+x 2 x* + x* x G etc., s We are saturated through and through with the word " method" in Leibnitz's several writings, which occurs in this draft thirteen or eighteen times. Even this trifling remark on Mercator's divisions Leibnitz calls his method, and thinks that it differs from that of Mercator. Newton says in the Recensio (page 14 of the second edition of the Com. Up. Collinsii, and page 17 in Biot and Lefort's edition) on a similar occasion : " Commercium cum Oldenburgio renovavit Leibnitius scribens se mirificum habere Theorema, quod daret Circuli vel ejus Sectoris cujuscunque Aream accurate in serie numerorum rationalium ; Octobri autcm insequente scripsit se invenisse Circumferentiam Circuli in serie simplicissimorum numerorum. Eadem Methodo, sic enim Theorema illud nominat :" Newton also then, as we see, was struck with Leibnitz's " methodo-mania" and with his desire of inventions, which made him always speak of " methods" when the thing itself was much more humble than what the word " methodus" would suggest. NEW ADDITIONS AND REMARKS. 135 et summa omnium ex cognitis Hy- perboloeidium quadraturis habebi- tur. Sed quoniam pluriraae figurae non sunt rationales, ut Circulus, Ellipsis, aliaeque innumerabiles, ideo opus erat methodo nova, qua de- monstretur (?) figuras rationales inventas, esse aequales vel propor- tionates portionibus vel pendentibus figurae non rationalis quaesitae. Mea methodus 4 Corollarium est 4 The note of the editors of the Com. Ep. here says : " Leibnitius hanc Methodum vulgari more prolixius hie exponit, quam Analysis ejus nova paucis exhibere potuisset, ideoque Analysin illam novam nondum invenerat." " Hie modus transmutandi figuras Curvilineas in alias ipsis aequales, ejusdem est generis cum Transmutationibus Barrovianis & Gregorianis. Et Conica? Sectiones hac Methodo semper ad Series Infinitas reduci possunt per divisiones. Generalis tamen non est : Nam si Curva sit secundi generis, incidetur in a?quationem quadraticam : si tertii generis, in cubicam, si quarti, in quadrato-quadxaticam, si quinti, in quadrato- cubicam, &c. prseterquam in casibus quibusdam valde particularibus. Per extractiones vero Radicum Problemata facilius solvuntur absque Transmutationibus." It should be remarked, that Leibnitz, in speaking of this invented method, here praises it, as reducing any high equations to equations " ubi dimensio ordinatae non ascendat ultra Cubum aut Quadratum aut etiam Simplicem dignitatem." It is untrue that Leibnitz possessed such a method, and that he could reduce the higher equations into Cubic and Quadratic equations. The note of the Com. Ep. Collinsii, which we have just cited, remarks very quietly that the conic sections or equations of the second degree can be reduced thereby to a form applicable to Mercator's division, but that with this method " De Transformationibus" neither Leibnitz nor any one else could arrive at any result in the third nor in any higher degree, " praeterquam in casibus quibusdam valde particularibus." (It is in our days well known to mathematicians that sometimes a few particular cases of a higher order can be solved by the simplest " methods," to use Leibnitz's favourite word). The note adds, with modesty : " per extractiones Radicum Problemata ' facilius' solvuntur absque Transmutationibus," which does not say, that all equations even of the highest degree could be squared and reduced by these Newtonian " extractiones radicum." Newton never pretended, either that his generalization of Mercator's divisions was his only method, or that it was already general by itself. 136 NEW ADDITIONS AND REMARKS. tantum doctrinae generalis de Trans- fbrmationibus ; cujus ope Figura proposita quaelibet, quacunque /Equatione explicabilis, transmit-* tatur in aliam analyticam aequipol- lentem; talcm lit, in ejus JEquatione, ordinatse dimensio non ascendat ultra Cubum aut Quadratum, aut etiam Simplicem Dignitatem, seu Infimum gradum. Ita fiet ut quae- libet Figura, vcl per Extractionem radicis Cubieaj vel Quadraticse, Newtoni more ; vel etiam, methodo MercatoriS) per simplicem Divi- slonem ; ad Series Infinitas reduci queat. On the contrary, he says to Leibnitz in his first letter : " Quomodo ex Aequationibus sic ad infinitas series reductis cetera determinantur, et quomodo etiam Curvae omnes Mechanicae ad ejusmodi Aequationes infinitarum serierum reduci possunt longum foret describere : and : Non tamen omnino universalis evadit nisi per ulteriores quasdam methodos;" for this was what he did not wish to communicate in the letter, this being that part of the invention which is the Differential Calculus. 1 1 is generalization of Mereator's divisions, together with his Differential Calculus, Newton, Wallis. Leibnitz and everybody would at that time call the method of Series. Moreover, in his second letter to Leibnitz, Newton gave (in enigmate) that part of his method which is the Differential Calculus, and there, in his second letter flinging out of his abundance a great number of theorems in 6 lines, (page 157. 173; Ed. of Biot, page 133) he modestly, but with some confidence says: "aliqua de his evadunt compositissima adeo ut vix per Transmutationem figurarum, quibus Jacobus Gregorius et alii usi sunt, absque ulteriori fundamento inveniri posse putem." It is almost repugnant to us to read after these words again the second untruth of Leibnitz's letter, pretending that he could reduce all higher equations to cubic or quadratic or simple equations. Indeed, if that was feasible, nobody needed to invent the Differential Calculus. NEW ADDITIONS AND REMARKS* 137 Ego vero, ex his Transmuta- tionibus, Simplicissimam ad rem praesentem delegi. Per quam sci- licet unaqiueque Figura transfor- matur in aliani aequipollcnteni ra- tionalem ; in cujus sequatione, Or- dinata in nullam prorsus ascendit Potestatem : Ac proinde sola Mer- catoris Divisione per Infinitam Seriem exprimi potest. Ipsa porro generalis Transmu- tation nm methodus, milii inter po- tissima Analyseos censenda videtur. Neque enim tantnm ad Series In- finitas, & ad Approximations ; sed & ad solutiones Geometricas, aliaque innumera vix alioqui tractabilia in- servit. Ejus vero Fundamentum vobis candide libereque scribo ; per- suasus qure apud vos habentur pra> clara mihi quoque non denegatum iri. Transformationis fundamentum hoc est : Ut figura proposita' rectis innumeris utcunque,modo secundum aliquam regulam sive legem ductis, resolvatur in partes ; quae partes, autalise ipsis aequales, alio situ, aliave 5 This sentence is very curious. Newton indeed must have smiled to see Leibnitz cull " meam methodum"' Avhat he here describes, and what was extremely common, not only at that time but at any time, namely, to turn one figure into another figure by " sotne means or other" or, as Leibnitz has it, " ufcunque (!) secundum " aliquam regulam sire legem." 138 NEW ADDITIONS AND REMARKS. forma reconjunctae, aliam componant figuram prion aequipollenteni, sen ejusdem areae ; etsi alia longe figura constantem. Unde ad Quadraturas absolutas, vel hypotheticas Geome- tricas, vel serie infinita expressas Arithmeticas, jainjam inultis modis perveniri potest. Ut intelligatur ; Sit Figura AQCDA. Ea, ductis rectis BD parallelis, resolvi potest in Trapezia X B J), J5 S D, &c. Sed ? ductis rectis convergentibus ED, resolvi potest in Triangula E X D J), E J) 3 Z>, &c. Si jam alia sit Curva A X F jF 3 F, cujus Trapezia % B JF, JB S F sint Triangulis E X D 2 Z>, E J) 3 D ordine respondentibus sequalia, tota figura AE J) J) J) A, toti figura? A X F Z F S F J3A erit aequalis. Quinetiam Trapezia, Trapeziis conferendo, fieri potest ut t N 2 P, vel quod eodem redit, Eectangulum .N a P : sit asquale Trapezio respon- dent X B 2 P/, sive Eectangulo X B J) ; tametsi recta X N X P non sit aequalis rectae X B x D, modo sit t N fl ad X B ,5 ut X B X D ad X N jP; quod infinitis modis fieri potest. Quae omnia talia sunt ut cuivis statim ordine progredienti, ipsa na- tura duce, in mentem veniant ; con- tineantque Indivisibilium Mcthodum NEW ADDITIONS AND REMARKS. 139 generalissime conceptam, nee (quod sciam) hactenus satis universaliter explicatam. Non tantura enim Pa- rallels & Convergentes, sed & alias quaecunque certa lege ductal, rectae vel curves, adhiberi possunt ad Re- solutionem. Quanta autera & quam abstrusa hinc duci possint, judicabit qui methodi universalitatem animo erit complexus. Cert urn enim est omnes Quadratures hactenus notas, absolutas vel hypotheticas, nonnisi exigua ejus specimina esse. Sed nunc quidem suffecerit ap- plicationem ostendere ad id de quo agitur; Series scilicet Infinitas, et modum Transforraandi figuram da- tam in aliam aequipollentem ratio- nalem, Mercatoris metliodo trac- tandam. AQCA sit Quadrans Circuli, Radius AQ = r, Abscissa A l B = x, Ordinata 1 B l D = y, Aequatio pro Circulo 2rx — x 2 = y 2 . Ducatur recta AJ)\ producaturque donee ipsi QG etiam productae occurrat in l N: Et Q jiV vocetur z. Et erit A t B seu 2r 3 . 2zr 2 x = —, 5 , et ,B ,1) sive y = -= 3 . Eodeni modo, ducta A J) 2 A 7 ; si Q. 2 N=z-@ (posita scilicet x JV 2 iV 2r 3 = /9) erit A 2 B. r 2 + z 2 -2z/3 + /3 2 Id vero hac methodo sum con- secutus. Sit quadrans circuli ABCD, AD aequalis a ; AE aeq. a;, EB = y A erit aequatio pro circulo 2ax — x 2 aeq. y 2 . Haec aequatio in numeris resolvatur indefinite [methodo Dio- phantea] ut si ponatur : y aeq. zx — fiet aequatio 2a 3 x — a i x 2 aeq. z 2 x 2 sive 2a !i — a 2 x aeq. z 2 x vel x aeq. — z j aeq. AIL et y aeq. a -f- z T2 HO NEW ADDITIONS AND REMARKS. et A 2 B— A ,P sive recta ,/>' ,5, erit 2r 3 2/- s . r* + a" - 2zB + yS 2 r 2 4- is 2 sita /3 infinite parva, (post destruc- tions et divisiones) erit X B ^B 4r»z/3 ' \J_\r* + z*' Habita ergo recta X B X D, et recta , B n B 1 habebitur valor Rectanguli X D X B 2 Z>, multiplicatis eorum valo- ribus in se invieem ; habebitur in- 8r*zzft quain - — — , pro valore Rect- &ngu\i J) X B ,B. Sit jam Curvae X P 2 P 3 P etc. natura pro arbitrio assumpta talis, ut Ordinata ejus^AJ^P (ex data abs- cissa U ,JS sive z\ sit Wr l -\z 2 Ideo, quoniam X X 2 X= 8, erit rect- 7, *T AT • 8/-V/3 angulum,/ A A^itiam — . MJ /- + a 2 Ac proinde aequale Rectangulo J> x Bf et S patmm t P x N a N a P,P ,P aequale spatio Cireulari respon- dent! J) Jiji B D 2 D ] Z>. Est autcm quaelibet Ordinata XP rationalis, ex data abscissa QX; quia, posita QN=e i Ordinata .VPest 8>V ' ~~- — jj aeq. Eli. Sed quoniam ad aream Cireuli habendam opus est summa omnium reetangulorum quale est BE{E) et vero rationa- liter inveniinus ipsam y vel EB superest ut inveniamus ipsam E (E) quod fiet subtrahendo AE ab ^.4 (is) est autem AE aeq. -g r 2 2« 3 ^ ideo ponendo etA(E)a, q . (iV+{z) ipsas s indefinitas, pro arbitrio assum- tas, esse progressionis arithmeticae, et differentiam omnium constantem, seu imam infinitesimam ipsius a esse, tunc sequens (z) erit z — 8 ita ut differentia inter z et , s — 8 sit /?, ergo A[E) erit -s 5 — -—3 — , et: -^E4-^(#) erit: 4- \J_\ r* + z 2 ' sive r" + 3rV + 3rV + Ergo a' + z ■ a 2 '+ z i -2zB+B i1 aeq. E (E) sive reductis omnibus ad unum denominatorem rcjectisque illis, quae ccterorum comparatione . n . „ 4c/3a 3 sunt infinite parva, net y-z =r« v [a* + z*y aeq. E (E) quam quantitatem du- , . „_ 2za* 8z*a b B cendo in Eh aeq. ., ..net „ .... 1 a 2 + z 2 a 2 4-z) s area rectanguli BE[E). Cumque eadcm sit ratio de ceteris id genus NEW ADDITIONS AND REMARKS. 141 ipsa per infinitam Seriem integro- rum exprimi potest, dividendo. Et Spatium talibus Ordinatis compre- hensum, aequipollens Circulari, in- tinita Serie numerorum Rationa- lium, Methodo Meroatoris quadrari potest. Quod cum facillimum sit facere, hie omitto. Neque enim elegantiae suae, sed Methodi Gene- ralis explicanda? causa, hoc exem- plum assumpsi. Ita siquis loco Circuli mihi de- disset Curvam, in qua Ordiuata ascendisset ad gradum Cubicum, potuissem earn reducere ad Curvam, in qua Ordinata non assurrexisset ultra Quadratum, vol etiam ne quidem ad Quadratum. reetangulis exiguis omnibus, patet summam infinitarum quantitatum (differentias infinite parvas haben- tium) quarum una est [a' + z da- turam esse aream circuli, quare posito y3 esse ut diximus infinitesi- mam ipsius «, et ipsas z esse arith- metice proportionates, seu differen- tiam habentes constantem /3, patet figuram curvilineam cujus abscissae 2«V sint z ordinatae vero ?— « „-,, futu- + z 2 ) 3 ram esse circulo aequipollentem quoniam ordinatae ejus in yS (diffe- rentiam ipsarum z) ductae reetan- gulis 6 circuli clementaribus BE (E) aequantur, ergo summa earum or- dinatarum 7 in constantem /3 ducta- rum, seu area figurae curvilineae novae summae omnium eoruni rect- angulorum, seu areae portioni circu- lari respondenti aequabitur. Sufficit ergo invenire summam omnium {<? + *', seu omnium 8sV a 6 + 3aV + 3aV + z" ' quam fractionem Methodo Merca- toris in infmitorum paraboloeidum 6 & ' All these are not Leibnitian rules, but the well-known Theorems of Quadra- tures of Wallis and Cavalleri. 142 NEW ADDITIONS AND REMARKS. ordinatas resolvendo, earumque gummas ipsis subjiciendo, habebitur series rationalis infinita exprimens magnitudinem semisegmenti seu portionis circularis ut AEB. Sit a aeq. 1, 8s 2 aeq. b ) 3s 2 + 3s 4 + s* aeq. c, fiet 8a" b seu 1+ 3z 2 + 3z 4 + z« ' 1 + c ' aeq. b — bc + be 1 — 5c 3 etc. ; et si tribus prirais terminis contenti simus ipsasque b et c rursus explicemus fiet h aeq. 8s 2 - 24s 4 -24s 6 - 8z 8 l+c l + 12s G + 18s 8 + 158s 10 + 48s 12 + 8s 14 8z 2 etc. et ordmando 1 + 3s 2 4 3s 4 + s 6 erit aeq. 8s 2 - 24s 4 + 48s 6 + 10s 8 + 158s 10 + 48s 12 + 8s 14 , etc. et sum- 8s 2 ma omnium 7> 7 -, erit 1 + 3s 2 + 3s 4 + s 6 8s 3 24s 5 aeq. — — etc. quae (si conti- o o nuetur series modo praescripto) erit area spatii circularis ABBA posito . „ 2a 3 , „_ 2s« 2 Ah aeq. ., — ^ et BB aeq. —. r, . H a' + z 2 l a' + z' Atque baec est methodus generalis, quae omnibus omnino curvis analy- ticis, et suo modo etiam transcen- dentibus applicari potest, utcunque aequationes earum sint implicatae aut affectae, re ad puram analysin NEW ADDITIONS AND REMARKS. 143 Itaque semper, sive Extractioni- bus Radicum Neivtonianis (gracilis cujuslibet dati) vel Divisionibus Jfer- catortS) poterit cujuslibet Figurse spatiura inveniri, interveutu alterius fequipollentls. Multum autem ad Simplicitatem interest quid eligas. Omnium vero possibilium Cir- euli, & Sectoris Conici Centrum habentis cujuslibet, per Series In- finitas quadraturarum, simplicissi- mam hanc esse dicere ausim quam nunc subjicio. Sit QA Jf' [Vid. Fig. prece- dent.] Sector, duabus reetis in cen- tre Q concurrentibus & Curva Conica A X F, ad Verticem A sive Axis extremum perveniente, com- prchensus. Tangenti Verticis AT occurrat Tangens X FT. Ipsum AT vocemus t) & Reetangulum sub Semi-latere Recto in Semi-latus Transversum sit Unitas. Erit Sec- tor Hyperbola?, Circuli vel Ellipseos, rcducta: tantum enim opus est inveniri modum, quo aequalitas curvae naturam explicans rationa- liter atque indefinite diophantes more solvatur ; quod vero hie sem- per fieri potest secus ac in proble- matibus numericis, quoniam hie possunt irrationales etiam caleulum ingredi modo ipsae indefinitae y et x in vinculis non comprehendantur. Ista methodus generalis varios habet casus compendiaque innu- mera quae circulum examinanti sese obtulerunt, quorum ununi, velut non inelegans ascribam, ip- sius Harmoniae causa quam in ea deprehendo : series h i i 1 i i i i i i i etc. etc. TSo etc. — i — 3 etc. tjo etc, rex- primit aream .. 1 cuius quad- circuh I J \ , , , ratum in- hyperbolae Jscriptum = ^ quod mutatis mutandis ad quasli- bet etiam circuli portiones applicari potest. Quemadmodum etiam gene- ralem habco seriem pro area sec- tionis conicae centrum habentis cujuslibet, id est Circuli. Hyper- bolae et Ellipseos per expressionem omnium ni fallor possibilium sim- plicissimam. 144 NEW ADDITIONS AND REMARKS. per Semi-latus Transversum divisus, t t 3 £ f = _4.__f._+_ & c . Signo ambiguo • valente + in Hyperbola, — in Cir- culo vel Ellipsi. Undo, posito Qua- drato Circnmscripto 1, erit Circulus | — i + 4 — f , etc Quae expressio, jam Triennio abhinc & ultra a me eommunicata amicis, hand dubie omnium possibilium simplicissima est maximeque afficiens mentem. Undo duco Iiarmoniam sequen- tem ; 3 111 1 1 11 I 1 _T p+p _ 3 8 rS 24 35 48 63 gO 90 120 CL *-" — II II 1 pf P _ 2 etc. = ] 1 g 25 4S 80 TI>0 5 s i 9 ete. lExprimit ! etc. ( aream eujus quadratum inscriptum est \. S 48 120 f circuit ABGD hyperbolae aequilaterae I CBEFi : Numeri 3, 8, 15, 24, etc. sunt Quadrati Qnitate minuti. Vicissim, ex Seriebus llegres- suum pro Hyperbola banc inveni. Si sit nuinerus aliquis Unitate minor 1 - ///.ejusquc Logarithmus ITyper- bolicus 1, erit l 3 r Eadem certis artibus ad eurvas non analyticas sive transcendentes possunt applicari : [sed in] [ubi vero] : et methodum habeo propo- siti! longe generaliorem, de qua infra, per quam arbitror quantitatem incognitam possibilem determina- tam quamcunque per seriem ratio- nalem infinitam cxprimi posse [quo- niam] quamvis tarn nominator quam numerator sit compositus. _1 V m_ l~lx2 + lx2x3 _ 1x2x3x4 etc. Si numerus sit major Unitate, ut 1 + n, tunc pro eo inveniendo niihi Compendia autem reperi pecu- liaria pro regrcssu [ex arcu ad sinum aut sinum complcmcnti, et pro regressu a logarithmo ad nu- merum] primum autem inveni re- gressum ex logarithmo ad nume- rum, ut indc etiam ab arcu ad sinum eomplementi. Easdem plane series inveni, quas in Uteris suis NEW ADDITIONS AND REMARKS. 145 etiain prodiit Regula, quae in New- toni Epistola expressa est ; scilicet erit l 3 l 4 1 1* n—- H H 1 1x2 1x2x3 +-. etc. 1x2x3x4 Prior tamen celerius appropinquate Ideoque officio ut ea possim uti, etiam cum major est Unitate Hu- merus 1 -f- n. Nam idem est Lo- garithmus pro I ■+ n et pro — - . Unde, si ] +n major Unitate, erit minor Unitate. Fiat ergo 1 + n \ — m — - — ac inventa m, habe- 1 + n ' bitur et 1 + «, Humerus quaesitus. Quod regressum ex Arcubus at- tinet, incideram ego directe in Re- gulam quae ex dato Arcu, Sinum Complementi exhibet. Nempe, Sinus Complementi a a ~ 1 x 2 + 1x2x3x4 etc. ISed postea quoque deprehendi, ex ea illam nobis communicatam pro inveniendo Sinu Recto, qui est a a a I ~ fx2 x~3 + 1x2x3x4x5 etc. posse demonstrari. Quod tribus Verbis sic fit. Summa Sinuum Complementi ad Arcum, seu ora- exbibet Newtonuspro regressu ex lo- garithmo ad amussim et pro regressu ab arcu ad sinum supplemcnti vel si- num versum, cujus differentia a radio est sinus complementi. Cujus [me- tliodi vobis] compendii inventi de- monstrationcm tibi scribam,ut videas quam diversis rationibus ad eaudem seriem venerimus : si sit' numerus 1 + n et logarithmus I erit n aeq. I + 172 + 1,2, 3 etC ' qime Serl0S est in epistola gratissima Newtoni- ana, sed ego alia uti malo ejusdem originis, quae procedit per + et - alternative ac proinde celerius ap- propinquat. Nimirum quia idem est logarithmus pro numero 1 + n 1 u- ! ut pro numero ; nine ponendo r l+n 1 f fc aeq. 1 — m net m aeq. 1 r ? - h y- etc. 1 ^ 1, 2 ^ I, 2, 3 unde facile ex invento m babebitui 1 + n seu numerus Regressu utor ex arcu ad sinum complementi," nam posito arcu a radio 1 erit sinus complementi aeq. a J a I~M + 1,2,3,4 1,2,3, 4,5, G etc. vel ut cum Newtono loquar u 146 NEW ADDITIONS AND REMARKS. Ilium 1 — + a 1x2 1x2x3x4 etc. a a a est 4- 1 1x2x31x2x3x4x5 etc. Porro, Summa Sinuum Com- plementi ad Arcum (seu Arcui in locis debitis insistentium) aequatur Sinui Recto, ducto in Radium ; ut notum est Geometris. Id est, aequa- tur ipsi Sinui Recto, quia Radius hie est Unitas. Ergo Sinus Rectus 3 5 n, a ft + 1 1x2x3 1x2x3x4x5 &c. Hinc etiam, ex dato Arcu & Radio, sine ulla prorsus aliorum notitia, haberi potest Area Seg- menti Circularis duplicati : quae est a 3 a 5 1x2x3 1x2x3x4x5 &c. + 1x2x3x4x5x6x7 Unde optime Segmentorum Tabula ad Gradus & Minuta &c. calcula- bitur. Pro Trigonometricis autcm ope- rationibus, percommoda mihi vide- tur hrec expressio : Ut Sinus Com- plement c ponatur g 8 a 4 _ ~1x2 + 1x2x3x4' quoniam sola, memoria retenta, (nam res eodem redit) sinus versus a 2 a" a* 1?2 + etc. 1,2,3,4 1,2,3,4,5,6 Ex qua serie pro sinubus compk- nienti facile demonstrari potest al- tera pro sinubus rectis a Colliusio nobis per Mohrium transmissa, ut postea animadverti, quoniam summa sinuum complementi ad arcum dat sinum rectum (ut facile demonstrari potest, et facile 8 ab illis depreben- ditur, qui in his versati sunt) et summa omnium sinuum comple- menti ad arcum, seu omnium 1,2 + 1, 2, 3, 4 etc. est a — + 1, 2, 3 1, 2, 3, 4, 5 ect. ergo arcu posito a et radio 1 sinus- rectus est + etc. 1 1, 2, 3 1, 2, 3, 4, 5 quamquam idem etiam recta consr qui liceat, [quod initio 11011 animad- verteram.] Fundamentum autem demonstrationis talium omnium quae advidi simplicissimuin est : exempli causa pro inventione nu- meri ex logarithino i r r n aeq. j + — + -_ etc. 8 These series of which Leibnitz here speaks with so much prolixity in 16TH are as Newton shortly remarks in the Recensio (page 15, ed. of Biot and Lefort, page 18) the same which Leibnitz had received 1675 through Oldenburg. NEW ADDITIONS AND REMARKS. 147 omnibus casibus & operationibus, directis scilicet siraul & reciprocis, sufficit; Quod ideo sit, quoniam »n „ a 2 « 4 , ^quatio c = 1 — — H est plana. 2 24 l Unde si vicissim quseras Arcum ex Sinu Complementi, radix extrahi potest ; adeoque fiet Arcus a — V 6 — v 24c + 12 exacte satis ad usura eorum qui in itineribus Tabu- larum commoditate carent ; quia a 6 error sequationis non est 720 Innumera alia possunt dici, quae his fortasse elegantia et exactitu- dine non cederent. Sed ego ita sum comparatus ut plerumque, Me- thodis Generalibus detectis, rem in potestate habere contentus, reliqua libenteraliisrelinquam. Neque enim ista omnia magnopere aestimanda sunt, nisi quod artem inveniendi perficiunt, mentemque excolunt. Si quae obscuriora videbuntur, ea li- benter elucidabo : Et illud quoque explicabo, quomodo hac methodo Aequationum quoque, utcunque affectarum, Radices per Infinitam Seriem dari possint, sine ulla Ex- tractione ; quod mirum fortasse vi- debitur. Sed desideraverim ut Clarissi- mus Newtonus nonnulla quoque ergo r 1, 2 1 summa omnium n est aeq^ r I 4 ,2,3 + etc. 1, 2, 3, 4 ergo n — summ. n aeq. I quaeritur ergo curva, in qua si ab n ordinata novissima assumta in unitatem seu parametrum constantem ducta, au- feras summ. n seu aream figurae, residuum aequetur abscissae I in eandem a unitatem ductae, quam curvam certa analysi deprendetur solam ex omnibus possibilibus cur- vis esse Logarithmicam, ejusque constmctione deprehendetur 1 + n esse numerum posito 1 logarithmo ; simili methodo sinus complementi vel recti inventio ex dato arcu de- monstrabitur nimirum in locum sum- marum substituendo summas sum- marum. Quae Methodus a New- toniana ita longe lateque differt, ut mirer quomodo itinera usu adeo diversa eodem ducere potuerint vel uno in casu. Porro quoque cujus- libet aequationis sive finitae sive infinitae radicem methodo mea extrahere possum, finitae quidem, transformando problema Geome- triae communis in problema tetra- gonisticum, cujus incognita semper infinita serie haberi potest ; infi- nitae autem et finitae simul per quandam methodum non quidem U2 148 NEW ADDITIONS AND REMARKS. amplius explicet : Ut, Originein Theorematisquod initio ponit : Item, Modum quo quantitates p, q 1 r, in suis Operationibus invenit : Ac denique, Quomodo in Methodo Re- gressuum se gerat ; ut, cum ex Logarithmo quaerit numerum. Ne- que enim explicat quomodo id ex Methodo sua derivetur. Nondum mihi licuit ejus Literas qua merentur diligentia legere : Quoniam tibi e vestigio respondere volui. Unde non satis nunc quidem affirmare ausim, an nonnulla eorum quae suppressit, ex sola earum lec- tione consequi possum. Sed optan- dum tamen foret, ipsum ea potius supplere Newtonum : Quia credibile est, non posse eum scribere, quin aliquid semper praeclari nos doceat Vir (ut apparet) egregiarum medi- tationum plcnus. Ad alia tuarum litcrarum venio, quae Doctissimus Collinius commu- nicare gravatus non est. omnium simpUcissimam,sed omnkim generalissimam, quae hoc funda- mento 9 nititur, quod datis duabus aequationibus finitis vel infinitis eandem incognitam continentibus, semper aequatio alia nnita vel in- finita reperiri potest in qua omnes dictae incognitae potestates sunt ablatae ; quae methodus eo in casu servire potest, quo ceterae omnes deficiunt. Habes origines eorum omnium quae a me in hoc argumento de- prehensa sunt, candide prorsus et quantum sufficit illis qui nihil in his versati sunt expositas. Saepe [Newtoni met.] porro saepe Xewtoni methodus ad elegan- tiores ducet expressiones, saepe etiam mea ut res docet. Mult as alias habebam in ea ista meditationes et mittam (?) eas (?) quam primum Newtonus scire (?) poscit (?) nam etiam radicum ex- tractiones per infinitas series coepe- ram et in affectis methodum Vietae 9 This is Descartes' invention, his method of assuming an equation with unde- termined coefficients, (see Schooten, the Geometria of Descartes, page 49, princ. 247, 262, and Gerhardt. Leibn. Math. Schrift. III. Band, page 727.) Leibnitz of course calls it his method (" mea methodus" hoc loco) because, he just made use of it, and applied it (as Newton had done before him) to the newly-invented infinite series, (see Newton's letter of 24th October, 1676, [in Uteris transpositis] "altera tantum " in assumptione," etc.) NEW ADDITIONS AND REMARKS. 149 decimalem reddere nitebar genera- lissimam, idque me credebam om- nium primum instituisse, sed aliud ex Newtoni Uteris didici non in- vitus. Praecipitatam 10 vides epistolam turn quia responsum postulas, turn ne qua iniqua suspicione teneamini, quasi occasione [Newtoni] vestrarum literarum [adjutus fuerim] in hac re adjutus beneticium dissimulare vo- luerim ; itaque gratas hodie, die lunae, in Germano pharmacopae redux domum forte praeteriens ac- cipiens literas, nam lator earum, Regius, quern nominas, nondum domum meam invenerat, illis primo tabellione, ipso die mercurii, res- pondere volui. 10 Leibnitz we see likes to appear very prompt in replying when a letter from Newton arrives, "ne injusta suspicione teneamini quasi occasione Newtoni literarum adjutus fuerim." He appeared so prompt in the most critical moment, namely when he answered Newton's second letter (" Hodie" accepi, as Gerhardt reads, Leibn. Math. Schr. I. Band, p. 154.) It is quite true, as Newton says, that summations of infinitely small quantities having already been made by Wallis, and tangents drawn by Slusius in all those cases, where there was not any irrationality in the equation (in all hyperboloids and paraboloids) — the whole in- vention in fact depended upon carrying on one of the tangential methods through those cases where irrationalities occur. Newton does this by taking in his work De Analyst *Jx in the i 1 form of x 1 and - = x x and saying, elegantly and most clearly, Regula I. si ax n = y\ erit -x " =Area. Quod exemplo patebit; si x l -f x 1 = y ; erit ±x 3 -f fa?* = Area, etc. 1 Again, in his letter of 10th Dec, 1672, he says : " mea methodus tangentium etc. non (quemadmodum Huddenii " methodus de maximis) ad solas restringitur aequationes illas, quae " quantitatibus surdis sunt immunes." This was the invention, and Leibnitz did not make it, but he took it out of Newton's manuscript; for, supposing that all Gerhardt's documents are true, not one of them, in which d vx occurs, is dated before Leibnitz's second journey to England. The proof that Newton and not Leibnitz is the inventor, is therefore given by these documents, which may also serve to show us Leibnitz's conduct. The first of the two Leibnitzian documents, namely, in which d ^x is 1 He thereby substituted a general calculation, in the place of isolated solutions. NEW ADDITIONS AND REMARKS. 151 mentioned, is dated November, 1676, and given by Gerhardt (Append. IV. to his tract of 1855). In this Leibnitz commits the fault of writing d Vaj = -7= • On the whole Leibnitz is in this first document somewhat wx embarrassed with the new idea. But rejoiced at having mastered that form V x he writes to Tschirnhaus some words, (which we have not) meaning that in Tamesis ostio, just after his second visit at Oldenburg's, he had got hold of a paradoxical idea. We say Leibnitz wrote this because Tschirnhaus in a letter first published by Gerhardt, (Mathem. Schr. Leibn. IV. Band p. 431) begs Leibnitz to tell him what that idea might be, and at the same time, what the expected second letter of Newton might contain. The words in Tschirnhaus' letter, dated Rome 1677, are "quas series nescio mini " per Methodum Gregorii possint terminari, et posses Dno Newtono " proponere saltern series hasce terminandas methodum quoque, " qua haec inveniuntur, si desideras, sequentibus communicabo, nee " credo, qua es facilitate, sententias tuas paradoxas admodum, quas " eruisti in Tamesis ostio, nee non quaecunque se tibi memorabilia " offerunt celaturum. P. S. Endlich ersuche, so was wiirdiges in Mons. " Newton briefen mir zu communiciren" (" lastly, I beg that if in " letters of Mr. Newton there be anything remarkable, you will com- " municate the contents"). We have to notice, that although Tschirnhaus begged Leibnitz to tell him what the paradoxical idea might be, and at the same time what Newton's letters might contain, still not before 1679 did Leibnitz communicate to Tschirnhaus that he could master the form v 'x (Gerh. ibidem, page 479 : " sine sublatione fractionum et irrationalium — itaque "nunc opinor," compared with the end of page 470). Nor did Leibnitz ever communicate to Tschirnhaus Newton's second letter intended half for Tschirnhaus, nor his second answer to Newton, (Gerh. loco cit. p. 505, where Leibnitz makes reference only to his first letter to Newton). So then " in Tamesis ostio" in the moment of departing from London, the paradoxical idea, the Differential Calculus, the pushing of the 152 NEW ADDITIONS AND REMARKS. rules for tangents through irrationalities, came to Leibnitz just attc- his second visit to Oldenburg as an extraneously learnt matter, namely with a mistake, Leibnitz writing d\j^ c — -jz-. This is what the first v x document of Gerhardt contains, and Leibnitz at once spoke of this as of his own invention. This is almost excusable here. For in Newton's Analysis, supposing that Leibnitz had inspected the same, there is no rule of tangents, or let us rather say tangents and their rules are in the Analysis everywhere, but still they are nowhere ; they are in the paragraph " Longitudines " curvarum invenire," they are in the next following paragraph " In- " venire praedictorum conversum ;" they are in the paragraph " Appli- " catio praedictorum ad Curvas Mechanicas," and in the words " Hinc in " transitu," etc. after the Demonstratio ; but because they are every- where and still have no particular place in the Analysis, therefore Newton added them to his Analysis in the letter of 10th December, 1672, in which he says " my method of tangents goes through irra- " tionalities," adding " hoc est unum particulare vel potius corollarium " gcncralis methodi quae extendit se ad omnia." Now Leibnitz reading Newton's Analysis, saw how the letter of 10th December, 1672, which he had also read, was to be understood. Leibnitz was puzzled with this for a little while, and at first fell into an error, but ho afterwards succeeded. Something in the matter therefore is his own work. For the Analysis did not contain tangents, and the letter which did contain tangents, did not say how they evaded irrationality. Leibnitz therefore reading the Analysis had to deduct' from it the tangential rule. Now people may call this inventing, T call it the proof of a non-invention. For if in my letter which you clandestinely read, it is said I have the thing which is the great difficulty, namely to get over irrationalities, adding it is " una particnla Methodi meae quae extendit " sc ad omnia;" and if then you make extracts out of my Analysis in which my method is so extended "ad omnia," you are not the inventor NEW ADDITIONS AND REMARKS. 153 of my method, although you have just a little difficulty in adjusting my tangents 2 to my Analysis, to do which the geometers Oldenburg and Collins were not clever enough. You may therefore, in some degree, think that you are the in- ventor, but you will have a certain disagreeable feeling within you, and will wish to avoid speaking of what you have seen. Thus did Leibnitz avoid acknowledging that he had read Newton's letter of 10th December, 1672. Taxed with it, in the first edition of the Commercium Epistolicum, in the most conspicuous place of the Corn. Ep., namely in the last document, in the judgment of the Committee of the Royal Society, Leibnitz did not choose to answer ; and his friends, including Professor de Morgan, deny that he had seen it till they are pushed into a corner by Edleston's new state- ments. Also Gerhardt avoids speaking out clearly, for only hesitatingly does he tell us, that Leibnitz saw Newton's manuscript Treatise dp Analyst. 2 Gerhardt is quite mistaken, if he thinks the signs to he of consequeuce. On the contrary I will admit, that in scraps of Leibnitz's hand, dated before his second voyage to London, the signs Sdx and dx occur, as abbreviations, not as inventions- If we had Huyghens's or Wallis's scraps (as we have through Gerhardt those of Leibnitz) we might also in their calculations see, that in trying to find new quadratures, the calculator (we mean Huyghens or Wallis) would sometimes write down an abbre- viation, perhaps dy or Sx*, if at that state of the calculation it suited him, not to calculate what the sum or the difference (according to the nature of the formulae) might be. But therewith no progress was made. Leibnitz and Wallis could not differentiate a single irrational form, not the form Var, and Wallis confessed, " hie haeret aqua.'' Irrationality occurs, unfortunately for Leibnitz and for Wallis, in all not quite elementary formulas, and that irrationality alone was to Wallis, as to all Geometers, the obstacle in their calculations. The signs S and d are therefore mere abbrevia- tions if the theory had made no progress, and here it is proved by Gerhardt, that, Leibnitz only just after his second voyage to London, and not before the same, learnt with difficulty to master this obstacle. He took Newton's general calculations out of Newton's Analysis, and therewith filled up (see the words " itaque nunc opinoi" on p. 470, loco citato) those signs which were before but empty. X 154 NEW ADDITIONS AND REMARKS. Almost, in the same manner, Leibnitz avoids Newton's name in the second document containing d*>Jx, which Gerhard t gives us. In this second manuscript, namely, dated some months later, it is not the name of Hudde which is struck out, but that of Newton. Compare Gerhardt's edition of this Leibnitian manuscript and Gerhardt's note to it, (both of which I give at the foot of this page 3 ) and Daguerreotype Copies of its first lines, (which I have had taken in Hanover, and have deposited for inspection in Cambridge at the office of the Editor of this work, and in London, at Messrs. Macmillan and Co., Henrietta Street, Covent Garden.) 3 The document is given in Gerhardt's Tract of 1855, page 143, in Appendix V., as being entirely in Leibnitz's hand-writing, and reads as follows : 11. Julii 1677. Methode genet-ale pour mener les touchantes des Lignes Courbes sans cahul, et sans reduction des quantites irrationelles et rompues. Monsieur Slusius a publie la methode pour trouver sans calcul les touchantes des lignes courbes, dont 1' equation est purgee des quantites irrationelles ou rom- pues. Par exemple une courbe DC estant donnee, dont 1' equation exprime la relation de BC ou AS que nous appellerons y, a AB ou SC, appellee x, soit a + bx + cy + dxy + ex* + fy* + gx*y -+ hxy % + kx* + ly* etc. = on n'a qu'a ecrire = bl + cv + dxv + 2exi + 2fyv + gx*v 4 %*£ + 3/br*£ + Zhfv + dy£ 4 2gxy£ + 2hxyv NEW ADDITIONS AND REMARKS. 155 It is thereby evident that Leibnitz knew that he had taken the Diffe- rential Calculus out of Newton's Analysis, and out of Newton's tangential letter; for, wishing to publish what he had so taken, in this second document of Gerhardt's, he struck out the name of Newton, and 4 mx 2 y 2 4 mx 3 y + pxy 3 4 qx* 4 ry* 4 2mx 2 yv + nx 3 v 4 pffi f 4qx 3 B 4 4ry 3 v 4 2imfZ 4 3nx 2 y^ 4 Zptfxv. c'est a dire changeant 1' equation en analogie : £ _ c + dx 4 2fy + gx 2 4 Ikxy 4 3/y 8 4 2mx~y v b 4 dy + 2ex 4 2gxy 4 Ay 4 3/cx 2 etc. r r VT> C*Z et supposant que - exprime la raison -=-?- — ou — ^-^- , l'on aura TB, ou SO, en 1 v BC, x oG supposant BCet SC donnees. Lorsque la valeur des grandeurs determinees b, c. d, e, etc. avec leur signes, fait de la valeur - une grandeur negative, la touchante ne sera pas CT, qui va vers A commencement de l'abscisse AB, mais C(jT)qui s'en eloigne. Voila tout ce qu'on en a publie jusqu'icy, aise a entendre a celuy qui est verse en ces matieres. Mais lors qu'il y a des grandeurs irrationelles ou rompues, qui enferment x, ou y, ou toutes deux, en ne peut se servir de cette methode, que par reduction de l'equation donnee a, une autre delivree de ces gran- deurs. Mais cela grossit horriblement le calcul quelques ibis, et nous oblige de monter h des dimensions tres hautes, et a des equations, dont la depression souvent est tres difficile. Je ne doute pas que ces Messieurs*) que je viens de nommer ne sachent le remede, qu'il y faut apporter, mais comme il n'est pas encor publie, et que je croy qu'il est connu de peu de personnes, outre qu'il donne la derniere perfection au probleme que M. des Cartes disoit avoir le plus cherche de tous les autres de la Geometrie, a cause de son utilite, j'ay juge a propos de le publier. Soit une formule ou grandeur ou equation, comme par exemple celle que dessus a 4 bx + cy + dxy + ex 2 4 fy 2 etc. appellons la par abrege w et ce qui proviendra lors qu'elle sera traite comme ci-dessus : scavoir b% + cv 4 dxv 4 dyg etc. sera appelle dw, de merae si la formule seroit X ou /u, le provenu serait d\ ou d/u et ainsi dans toutes les autres. Soit maintenant la formule ou equation ou grandeur to * Leibnitz hatte zU Anfang : Iludde, Slusius, et autres, geschrieben ; spater hat er das Uebrige, ausser Slusius, durchgestrichen. ("Leibnitz having at first written: Hudde, Slusius et autres, struck this out, and left Slusius." Gerhardt's note, page 154, line 16). X2 156 NEW ADDITIONS AND REMARKS. while he said "je ne doute pas que ces Messieurs Hudde et Slusius" (of whom he knew that they could not get over irrationalities) "ne "sachent le remede, quand il y a des grandeurs irrationelles et rom- " pues," he had in his mind Newton, whose name he struck out, of egale & - , je dis que dtv sera egale a ~ 2 M . Cela suffit pour manier les fractions Enfin soit w egale a Vw, je dis que dw sera Sgale a — ce qui suffit pour traiter comme il faut les grandeurs irrationelles. z \j w Algorithm de V analyse nouvelle de maximis et minimis, ou des touchantes. Soit AB - x, BC=y, TeC la touchante de la courbe AC, et la raison TB ou SG = x gera a ppell^e -^. Soyent deux ou plusieurs autres courbes BC = y SO fy AF, AH, et posant BF = v, BH=w, et la droite FL touchante de la courbe LF dx MH dx .. - „ , AF, et MH de la courbe AH, et ■= = - et -=-5. = — , je dis que dy ou dvw sera egal a vdiv + wdv ; estant v = w = x et y = vto = x', alors pour v et w sub- stituant x, nous aurons dviv = 2xdx. NEW ADDITIONS AND REMARKS. 157 whom alone he knew this to be the case, and of whom alone it was true. (Tout cela reussira aussi si Tangle ABC est aigu ou obtus, item s'il est infi- niment obtus, c'est a dire si TAC est une ligne droite.) This is Leibnitz's manuscript, as given by Gerhardt, which has to be compared with the Daguerreotype Copy. The same Daguerreotype contains a part of Leibnitz's first letter to Newton with almost the same apothecary-excuse mentioned above page 149. Gerhardt's words, which we give at the foot of page 155, refer to our page 155, line 20, and indirectly to what we have printed page 154, line 16, and the Daguerreotype Copy proves, that the name of Newton fell there in a somewhat particular manner out of the author's pen, though Gerhardt's note, which we have here given, has not spoken of the same. The full length titles of the most modern works quoted by us in the present work are : Sir David Brewster's Memoirs of the life, writings, and discoveries of Sir Isaac Newton. Edinburgh 1855. J. Edleston's Correspondence of Sir Isaac Newton and Professor Cotes, including letters of other eminent men, now first published, etc. London 1850. Leibnitzens mathematische Schriften, herausgegeben von G. J. Gerhardt. Erster Band. Berlin 1849. Zweiter Band. 1850. Dritter Band. Halle 1855 u. 1856. Vierter Band. 1858. Funfter Band. 1859. C. J. Gerhardt, Dr., die Entdeckung der Differenzialrechnung durch Leibnitz. Halle 1848. Cited as Gerhardt's I. (first) Tract or as Gerhardt's Tract of 1848. Derselbe, die Entdeckung der hohern Analysis. Halle 1855. Cited as Gerhardt's II. (second) Tract, or as Gerhardt's Tract of 1855. H. Weissenborn, Dr., die Principien der hoheren Analysis, als historisch-kritischer Beitrag zur Geschichte der Mathematik. Halle 1856. Other citations are indicated with sufficient precision in the work. Gerhardt's Tract of 1855, p. 38, speaks of a book entitled, " Gregorius " Vincentius Curvilineorum amcenior contemplation nee non examen circuit " quadraturce. Lugo 1 . 1654." No such book exists, but only a " Cur- " vilineorum amcenior contemplatio necnon examen circuit quadratures a 11 R. P. Gregorio Vincentio propositaz, Autlwre Vincentio Leotando /" which book has not therefore, as Gerhardt makes out, Gregoire de St. Vincent for its author. CAMBRIDGE: PRINTED SY W. MBTCA1 FK. GREEN STREET. ^fi^cr Y OF 1>lO Lo 6eV OF ~6 -^ CO K CALIFORNIA ==9=SI RETURN TO the circulation desk of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY Bldg.400, Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS • 2-month loans may be renewed by calling (510)642-6753 • 1-year loans may be recharged by bringing books to NRLF • Renewals and recharges may be made 4 days prior to due date. DUE AS STAMPED BELOW rvi APR 2 1999 SEMTONILL AUG 2 q 2001 OF CALIFORNIA U. C. BERKELEY OCT 1 S> iOW SENT ON ILL AHK 2 2 2005 U.C. DCRKELEY 2,000(11/95) ^^ TY OF CALIF Y OF CALIF OF CALIFORNIA t%&£±§**s LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CAL ■ X< 2 LIBRARY OF THE UNIVERSITY OF CALIFORNIA UBH»RY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY it tl^o: WZl //86&1 v^ir = m "- vtf* .^iSySrrS^ ^V/AVV' ^-^^fr^N. I •—*... ,^»* Qi/ LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA ^ ^IWK 'fe. Wffl P