Unarranged fragments, mostly relating to the dispute with Leibniz

Out of Mr Gregories Letter of the 17th of May 1671

This method of infinite Series hath no good successe in the second Segments of round Solids, at least so farre as I can improoue them yet such as it is you shall haue it

Sit igitur Ellipsis ADCE, cuius Axis transversus AC. Coniugatus DE; Centrum B, supponatur nunc Sphærois generari ex revolutione Semiellipseos ADC, circa Axem AC, seceturque Sphærois a duobus Planis parallelis, per Puncta K, B, Axi DB normalibus, et a duobus Planis Parallelis per Puncta B, F, axi BC normalibus:

Sit $BF = a$ , $BK = b$ ; eritque pars Sphæroidis KIFB a quatuor dictis Planis comprehensa = $= \frac{1680 r^{6} b a - 280 r^{4} b^{3} a - 42 r^{2} b^{5} a - 15 b^{7} a - &c}{840 r^{5}} - \frac{1680 r^{6} b a^{3} + 280 r^{4} b a^{3} + 280 r^{4} b^{3} a^{3} + 126 r^{2} b^{5} a^{3} + 75 b^{7} a^{3} + &c}{5040 r^{5} c 2} - \frac{16 r^{6} b a^{5} + 8 r^{4} b^{3} a^{5} + 6 r^{2} b^{5} a^{5} + 5 b^{7} a^{5} + &c}{320 r^{5} c^{4}} - \frac{48 r^{6} b a^{7} + 40 r^{4} b^{3} a^{7} + 42 r^{2} b^{5} a^{7} + 45 b^{7} a^{7} + &c}{2688 r^{5} c^{6}} - &c$ in infinitum. This Series is nothing but a Congeries of other Series's all of them being infinite, yet is the best I can haue to this Purpose, I hope Mr Newton hath better, and that with his Leaue yee will informe mee, If you would haue it agree to the Sphære, yee shall only put c in place of r, which will render it more Simple, I can giue such a Series as this for any the second Segment of any round Solid, and if you like this, I shall giue a Series for the second Segments of an Hyperblolick Spindle, which I imagine is of greater Consequence than any thing else for Guaging –

ADD 3968.412

AB = x

BC = o = CD

BE = y

HF = \dot{y} o = IK

KG = \overset{..}{y} o o

GI = \dot{y} o + \overset{..}{y} o o

{EF}^{q} = o o + \dot{y} \dot{y} o o

2 EF, EF = {}^{2}{\dot{y}} \overset{..}{y} o o

EF = \frac{\dot{y} \overset{..}{y} o o}{EF} . G

{GF}^{q} = o o + \dot{y} \dot{y} o o + 2 \dot{y} \overset{..}{y} o^{3} + \overset{..}{y} \overset{..}{y} o^{4}

{}^{2}{GF}, \dot{GF} = 2 \dot{y} \overset{..}{y} o o + 2 \overset{..}{y} \overset{..}{y} o^{3} + 2 \dot{y} \dot{\overset{..}{y}} o^{3} + &c

\dot{GF} = \frac{\dot{y} \overset{..}{y} o o + \overset{..}{y} \overset{..}{y} o^{3} + \dot{y} \dot{\overset{..}{y}} o^{3}}{GF}

\frac{\dot{y} \overset{..}{y} o o}{EF} GL = 2 o

GL = {}^{2}p = 2 \dot{y} o + \overset{..}{y} o o

EL = 2 \times o

FN = q . \frac{\sqrt{o o + p p - 2 p q + q q}}{\sqrt{x}} + \frac{\sqrt{o o + p p + 2 p q + q q}}{\sqrt{x} + o} = Min = R + S

R^{2} = \frac{o o + p p - 2 p q + q q}{x}

S S = \frac{o o + p p + 2 p q + q q}{x + o}

2 R \dot{R} = \frac{−2 p q + 2 q \dot{q}}{x}

2 S \dot{S} = \frac{+2 p \dot{q} + 2 q \dot{q}}{x + o}

\frac{- p \dot{q} + q \dot{q}}{R x} + \frac{p \dot{q} + q \dot{q}}{S x + S o} = 0

- S p \dot{q} o + S q \dot{q} o - S p \dot{q} x + S q \dot{q} x + R x p \dot{q} + R x q \dot{q} = 0

\frac{\sqrt{o o + p p + {}^{2}p q + q q}}{\sqrt{x + o}}

- p o + q o - p x + q x = \frac{\sqrt{o o + p p - 2 p q + q q}}{\sqrt{x}} \times x p + x q

\sqrt{o o + p p + 2 p q + q q}

- p + q \times x + o \times \sqrt{x + o} = \sqrt{o o + p p - 2 p q + q q}

p + q \times x \times \sqrt{x}

\sqrt{o o + p p + 2 p q + q q}

\sqrt{x + o} = \sqrt{o o + p p - 2 p q + q q}

\sqrt{x}

o o \underset{.}{x} + p p \underset{.}{x} + 2 p q x + q q \underset{.}{x} + o^{3} x + o p p + 2 p q o + q q o = o o \underset{.}{x} + p p \underset{.}{x} - 2 p q x + q q \underset{.}{x}

2pq

4 p q x + o^{3} x + o p p - 2 p q o + q q o = 0

o o + p p + 2 p q + q q

q q - 2 p q + p p

{x + o}^{3} = o o + p p - 2 p q + q q

p p + 2 p q + q q

x^{3}

o o p p - 2 o o p q + o o q q

x^{3} + 3 x x o + 3 x o o + p p q q - 2 p^{3} q + p^{4}

\begin{array}{r} + o o q q - 2 o o p q o o p p \\ - 2 - \end{array}

x^{3} \begin{array}{r} + o o p p, + p^{4} \\ - 2 o o p q \end{array}

\frac{1}{16} b^{4} - \frac{4}{8} {\dot{b}}^{3} v - \frac{1}{4} {\dot{b}}^{4} + \frac{3}{2} {\dot{b}}^{3} v + \frac{1}{4} {\dot{b}}^{4} - \dot{b} 3 v + \frac{1}{4} a a b b - a a b v + a b b h - 2 a b h v = h a b b

\frac{1}{16} b^{4} + \frac{1}{4} a a b b - a a b v - 2 a b h v = 0

v = \frac{4 a^{2} b^{2} + b^{4}}{16 a a b + 32 a h b} = \frac{a^{2} b + \frac{1}{4} b^{3}}{4 a a + 8 a h} = \frac{a a b + \frac{1}{4} b^{3}}{8 a h}

\frac{a a + \frac{1}{4} b b}{v} = \frac{8 a h}{b} = \frac{8 a h}{b} = FH

BC = a

CG = \frac{1}{2} b ∷ \frac{8 a h}{b}

4 h = HL

\sqrt{o o + p p + 2 p q}

q - p

\frac{x + o}{x} \frac{3}{2} = \sqrt{o o + p p - 2 p q}

q + p

o o + p p + 2 p q

p p - 2 p q

in x

1 + \frac{3 \times o}{x} = o o + p p - 2 p q

p p + 2 p q

o o p p - 2, o o p q + p^{4} \mp 2 p^{3} q

1 + 3 \frac{o}{x} = o o p p + 2, o o p q + p^{4} \pm 2 p^{3} q

\frac{3, o^{3} p p - 6 o^{3} p q + 3 p^{4} o}{x} = 4, o o p q

3 p p o o + 3 p^{4} = 4 p q x o + 6 p q o o

q = \frac{3 p \times EF q}{4 x o}

\frac{4 x o}{3 p} = GR = \frac{EF q}{q}

p. 11. l. 4. 3dly We do not dispute about the antiquity symbols of the symbols used by Newton for of fluents fluxions & moments Summs & Differences. And yet tho the symbol

\frac{a a}{4 x}

be used by Mr Newton for fluents or s is summs is be older then the symbol

\int \frac{a a}{4 x}

used in the same sense by Mr Leibnitz, in the same sense & all the symbols of fluxions used by Mr Newton be older then any symbols of fluxions used by Mr Leibnitz & Mr Newtons had symbols from fluxions like before he wrote his Letter dated 24 Octob 17676, but Mr Leibnitz has none to this day & the rectangles under the fluents fluxions & the mom letter o be older s used by Mr Newton for moments be are older then the symbols dx & dy used in yethe same sense d by MrLeibnitz. in the same sense. But These are only ways of Notation & signify nothing th to the method it self wchwhich may be without them. [All the use that can be made of them is to shew that Mr Newtons Leibnitz method was as old as the use of the symbols dx & dy & Th Mr Newtons as old as his use of any symbols for the same things.] — as in the Principia. In the first Proposition of his book De quadratura Curvarum he used prickt Letters but did not make them necessary to his method. for in the Introduction to that book he described thatis method at large & illustrated it with various examples without making any use of prickt letters. That book was printe first printed in the year 1714 but the a copy of the first Proposition was copied out of it & thereof was copied & at yethe request of Dr Wallis was copied & sent to yethe Dr Wallis by Mr Newton in his Letter of 27 August 1692 & printed the next year in the second Volume of the Doctors works. It was made use of upon all occasions, by Mr Newton when he wrote his Principia Philosophiæ And so many things are mentioned cited out of this book it in Mr L Newtons Letter of 24 Octob 1676 that one may reasonably conclude that the book was writ before that time. In his Analysis per æquationes &c — — — — — trifling. And if it were not, yet Mr Newton has the advantage as was mentioned above. For prickt letters are put for fluxions or veloci wchwhich are motions & finite quantities & a & never signify moments w or differentces which are parte infinitely small parts generated by fluxion. unless wWhere the symbols of fluxions (whether prickt letters or others are multiplied by the letter o either to expres or understood the rectang make them infinitely little the rectangles are put for moments, or where fluxions are considered as the exponents of moments, & accordingly the factor o in the factor o being understood, the symbols of fluxions are put being are put for the exponents of moments, but in such cases the factor o is always understood. This symbol factor Mr Newton always expresses when he would is demonstrating a Proposition but when he is only investigating a Propopsition he usually neglects to write it down Now Mr Leibnitz has no symbols of fluxions in his method & theref. all Mr Newtons symbols of fluxions are the oldest in the kind. And as for Mr the3 the dx & dy of Mr Leibnitz they are not to be compared wthwith the symbols of fluxions but wthwith the symbols of Moments that is with the rectangles under the symbols of fluxions & the letter o used by Mr Newton And those rectangles are the oldest being used by Mr Newton in his Analysis A. O wchwhich sent by Dr Barrow sent to Mr Collins in Iuly 1669. But whereas And tho our great Mathematician tells us that Mr Newton by putting o for the increment of x loses all the advantage of the differential method: the contrary is true. Mr Leibnits by putting dx for the letter o has lost a considerable part of the advantange of Mr Newtons method without adding any thing new to it besides his symbols. For the Method by Mr Leibnitz's way of Notation is not Geometrical nor demonstrative,. Nor y It is only fitted for investigation, & is not so expedite for that poss as Mr Newtons method when he who in such cases neglects to write down the letter o, & so uses but one letter to represent moments the other other letter being understood. And These advantages it has lost, & hads added no new ones. For there is nothing that can be done by it but what may be done by Mr Newton's method & that wthwith as much or more dispatch. But whereas Mr Leibnitz has told us in the Acta Eruditorum had changed the Letters a & e used by Dr Barrow into d dx & dy he tells us in the Acta Eruditorum mensis Iunij 1686 pag 297 that this change did not create a new method was not necessary to yethe method Malo autem dx & simila adhibere saith he quam literas pro illis quia istud dx est modificatio quædam ipsius x, ut sola quando id fieri opus est litera x cum suis scilicet potestatibus & differentialibus calculum ingredieatur, & relationes trancendentestranscendentes inter x et aliud exprimantur: Qua ratione etiam lineas transcendentis inter x et aliud exprim æquatione Explicare licet. So then by the confession of Mr Leibnitz the by his own confession might have used letters as Dr Barrow did & the method would have been the same, but he chose rather to use the letters dx symbols dx & dy for as more convenient. And yet he from these symbols he gave the method the name of the differential method as if it was had been a new method. At wchwhich rate a man might put any letter for yethe Abscissa of a curve & modify the letter variously to represent the Ordinate, the le area, the length, the subtangent, & the radius of curvity & call this a new methodu, & give it a new name to extinguish the memory of former authors & make himself famous for inventing a new method of solving all sorts of Problemes relating to curve lines. — as in the Principles. Mr Newton seldome uses prickt letters till he has brought Problems to equations & by those equations is to deduce fluxions from fluents or fluents from fluxions. How he deduces fluxions from fluents he has told isus in the first Proposition of his book the Quadratura Curvarum & there he uses prickt letters & by them shews how from æquations involving fluents to deduce the first econd & third fluxions of the fluents & so on in infinitum & this he does by one very short rule comprehended in five lines How Also in explaining how he extracts a fluent out of an equation involving the its fluxions he uses prickt letters & extends the Proposition to yethe 2d 3d & fl following fluxions. In his Letter dated 242 Octob 1676 he set down this his method comprehended his method in these two sentences Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire & vice versa. And Vna methodus consistit in extractione fluentis ex æquatione fluxionem ejus involvente, altera &c. When Dr Wallis was printing thise second Volume of his works he descr wrote to Mr Newton to explain those sentences to him & Mr Newton & Mr Newton did it by prickt letters coppying sending him a copy of his book the first Proposition of his book the Quadratura Curvarum & of another Paper This was in yethe year 1692. In yethe year 1686 Mr Newton wrote in the Scholium upon the second Lemma of the second book of his Principia Philosophiæ that th ten years before upon his writing to Mr Leibnitz about his Method of fluxions Mr comprised in this sentence Data æquatione quotcunque fluentes quantitates involvente Fluxiones invenire & vice versa. Mr Leibnitz wrote back that he had also fallen into such a method & communicated his method scarce differing only in from Mr Newtons except in the forms of words names & symbols. And Mr Newton's symbols here related unto were those of prickt letters with pricks. And indeed in his Letter of 24 Octob 1676 so many things are quoted out of book of Quadratures that one may thence conclude ytthat this book was writ before that time letter. But in the Principia Philosophiæ & in the Letters of Mr Newton to Mr Collins & Mr Oldenburg & thhis Analysis published in the Commercium Epistolicum there was & in his Analysis he had no where occasion to treat of Equations involving fluents & fluxions. In yethe Introduction to the book of Quadratures Mr Mr Newton explained the method of fluxions at large & illustrated it with various examples & yet made no without making any use of prickt letters wthwith pricks. And will orour great Mathematician say that Mr Mr Newton did not then understand the Method of fluxions because there are no such Letters in that Introduction 4

r x - \frac{r}{q} x x = y y

r \dot{x} - \frac{2 r}{q} x \dot{x} = {}^{2}y \dot{y}

r \overset{..}{x} - \frac{2 r}{q} \dot{x} \dot{x} = 2 y \overset{..}{y} + 2 \dot{y} \dot{y}

r - \frac{2 r}{q} x . 2 y ∷ \dot{y} . \dot{x} ∷ PN . MN

\dot{x} = 1

\dot{x} \dot{x} = 0 = \overset{..}{x}

\frac{r - \frac{2 r}{q} x, \dot{x}}{2 \sqrt{r x - \frac{r}{q} x x}} = \dot{y} = \frac{r \frac{1}{2}, q - 2 x, \dot{x}}{2 \sqrt{q q x - q x x}} - \frac{2 r}{q} = 2 \overset{..}{y} \sqrt{r x - \frac{r}{q} x x} + \frac{r r - \frac{4 r r}{q} x + \frac{4 r r}{q q} x x}{2 r x - \frac{2 r}{q} x x}

\overset{..}{y} = \frac{- \frac{2 r}{q} - \frac{\frac{1}{2} r r}{r x - \frac{r}{q} x x} + \frac{2 r}{q}}{2 \sqrt{r x - \frac{r}{q} x x}}

\overset{..}{y} = - \frac{r \dot{x} \dot{x}}{x - \frac{x x}{q} \sqrt{r x - \frac{r}{q} x x}}

. the

PS q - PH q = 2 SHN

\frac{q q - 2 q t}{2 SH} = HN

SH = \sqrt{q q - q r}

HN = \frac{q^{\frac{1}{2}} in q - 2 r}{\sqrt{q - r}} = \frac{q q - {}^{2}q t}{2 \sqrt{q q - q r}}

CN = \frac{{}^{2}q q - 2 q t - q r}{2 \sqrt{q q - q r}}

. CA

AN = \frac{−2 q q + 2 q t + q r + q \sqrt{q q - q r}}{2 \sqrt{q q - q r}}

\frac{2 d y}{MN} = \frac{2 d \sqrt{r x - x x} \times d - 2 x}{2 d x - 2 x x}

\frac{2 y y}{r - \frac{2 r}{q} x} = MN = \frac{2 r x - 2 \frac{r}{q} x x}{r - \frac{2 r x}{q}} = \frac{2 q x - 2 x x}{q - 2 x}

\frac{2 d y}{MN} = \frac{d \times d - 2 x \times r^{\frac{1}{2}}}{d^{\frac{1}{2}} \sqrt{d x - x x}}

. d

d \frac{1}{2} \times p \frac{1}{2}

d^{\frac{1}{2}} r^{\frac{1}{2}} = \frac{d^{\frac{1}{2}}}{d^{\frac{1}{2}}} . \frac{p^{\frac{1}{2}}, \dot{x}, d^{\frac{1}{2}}}{2 \sqrt{d x - x x}}

\begin{array}{l} 3477∟142857 \\ 1042∟5 (0299815 \\ 6954285714 \\ 3470714286 \\ 3929428571 \\ 341285715 \end{array}

\begin{array}{r} 341285715 \\ 312942857 \\ 28342858 \\ 27817143 \\ 525715 \\ 347714 \end{array}

\begin{array}{r} 178001 \\ 173857 \\ 144 \end{array}

\begin{array}{r} 19∟63 \\ 126 \\ 7∟56 \end{array}

If the net d value of yethe goods were 100li the Queens duty would be 52.2.6 as above, and the summ of the value & duty augmented by the 7th part thereof would be namely

[\begin{array}{r} 152 . & 2 . & 6 \\ 21 . & 14 . & \frac{5}{7} \end{array}] \overset{li}{173} . 17 . \frac{5}{7}

would be the gross pr value by the candle For the allowance of 6 &

6 \frac{1}{2}

per cent in the eighth part of yethe whole. Say therefore: As in this case the gross value by the candle 173.17.01

\frac{5}{7}

is to the Queens duty 52li.2.6d in this case, so is that is, as 1 to 0∟299815, & so is all other cases the gross value to yethe Queens duty in all other cases. Wherefore if you multiply the gross value by the ratio 0∟299815 you will have the Queens duty. As if the Q gross value be 100li this Queens duty w sum multiplied by the Queens duty said ratio will produce the Queens duty 29li∟9815 or 29l.19.07

\frac{1}{2}

The truth of which method will thus appear China ware sold for 100li

\begin{array}{r} 3477143) & 1042∟5 & (0.2998151 \\ 0.3010299 & 6954286 \\ 0.0752575 & 31294287 \\ b 1. 1∟1892 a & 31294287 \\ 0.1892 a - b \\ 0.0946 5 a - 5 b & 27817144 \\ 0.054 6 b - 5 a & 34771 \\ 0.162 18 b - 15 a & 17386 \\ 189 21 b - 17 \frac{1}{2} a & 1042499628 \\ 18 \frac{1}{2} a - 22 b & 348 \\ 37.44 .// 5 \frac{2}{7} . 6 \frac{2}{7} . & 499976 \end{array}

The right method of Computation as is conceived by the Auditors is to find the ratio of the Queens duty to the gross price in any one case & then in all cases to multiply the gross price by that ratio. Suppose Let the net value of the goods be 100li & the in this case the Queens duty will be 52l.2.6 as above; & the summ of this value & duty augmented by the 7th part thereof (we (the the allowedance for prompt payment & warehouse room), namely 173li.17s.1d

\frac{5}{7}

will be the gross value by the candle. For the allowance of 6 &

6 \frac{1}{2}

per cent eighth part of the whole is allowed for 7th part in And as this value is to the Queens duty in this case so is 1 to ,299815, & so is the gross vales to the Queens duty in any other case. As if the gross value be were 100li, this multiplied by ,299815 will produce the Queens duty 29,9815 that is 29li.19.7

\frac{1}{2}

. The truth of wchwhich method will thus appear. And if the gross value were 3748li.10s, this multiplied by ,299815 will produce the Queens duty 1123li∟8565, that is 1123li.17s.1d

\frac{1}{2}

. Now the truth of this method may be thus proved. China ware sold for 100li Now that 29li.19.7

\frac{1}{2}

is the Queens just duty will further appear by the following computation. The western Churhes in their external profession of faith were now macedonians, except that they still used the language of one hypostasis as equipollent to that of one usia & substance. There might be many who believed the holy Ghost to be God but they did not yet begin to speak out. I The easter Chur / & began to bye shy of one anothers communion. For the little Council of Alexandria had agreed that the bishops who had abolished t consented to the abolishing of the use of the word usia with its compounds, ed (except the ringleaders) should be looked upon as p bishops who had been only deceived & should not lose their bishopricks if they would b in time they would retract & come over to the Nice Council of Nice. But now the Latines finding themselves strong enough resolved to deprive them all bisho the bishops who would not retract so soon as they could get an Emperor for their purpose. And began also to break break friendship wthwith the Macedonians & declare for the party of Athanasius against them. The friendship between the Macedonians & the Churches of Italy lasted about seven five or six years. F [And all this time the bishop of Rome & taat torton Auxentius governed the diocess of Millain. For when Athanasius & the bishop & Lybia wrote to the Africans (wchwhich was in the year 371) Auxentius was not excommunicated by the bpbishop of those of his party in Italy. nor Valent V Vrgantius by those of P but] For Basil was made bishop of Cæsarea in Iune A.C. 361 & soon after the Coun dieity of the holy ghost was published by the Councile of Rome & Illyricum wchwhich met at one & the same time. afterwards wrote his epistole to 73d epistole in wchwhich he saith saith that when he met with Sabinus the deacon he wrote by him to the whom yethe western bishops had sent with letters into yethe east, he wrote [by him] to yethe bishops of Illyricum & to those of Italy & Gallia. The Concils therefore of IRome & Illyricum met in yethe year 371 or 372 (for they met before the death of Athanasius. And in these Councils the deity of the holy Ghost was proclaimed, & Auxentius condemned. The Council of Rome consisted of 73 bpsbishops & The friendship & communion between the Macedonians & the Churches of Italy & the west lasted, I think, till Pope Damasus called at Rome a Council of 93 bishops at Rome whicoh sent Elpidius & wthwith a letter to a Council convened at the same time in Illyricum, in wchwhich letter they thus describe their faith. When the evil . . . . . . erring bishops. By these words they signify that the communion between the Greek & Latin Churches was not yet broke off. The Churches of the whole Empire whether for or against the consubstantiality of the Son, for or against the Deity of the Holy Ghost, were hitherto of one communion composed hitherto one visible catholick Church, notwithstanding the differences of opinions that many of its members were corrupt in their opinions ☉ & the Latines of the were meditating to excommunicate & throw out of their communion all those bishops. And C to shew that they were in earnest they excommunicated Auxentius bishop of Millain but had yet done it, except in excommunicating ted The Council of Illyricum having recieved this letter — — — . . . others used that of three. They Basil wrote also at the same time by Sabinus to the bishops of Illyricum. For Basil Tinhiishis 73d epistle writes in his 2373d Epistle he writes thus: Ego enim cum in Sabinum inciderim in Sabinum diaconum quo nuncio huc ad nos illi [occidentales] usi sunt, ad Illyricos scripsi, ad Italos præterea & Gallos episcopos & nonnullos privatim. These Councils therefore of Rome & Illyricum & Gallia were convened after Basil was made bishop, that is, after Iune 371, & so may be placed in the year 372. For Athanasius heard of them before his death and died in May 373. The controversy therefore wchwhich about the deity of the holy Ghost wchwhich Athanasius had been hitherto labouring to kindle brake out into a flame in the year 372 In another Epistle written this year to the Churches of Italy & Gallia Basil exprest himself in this manner. We are seized wthwith a persecution . . . . . . . . . . . t / during the reign of Valens the Christians of the Greek Empire brake into several parties. The churche of wchwhich were most numerous generality were were those who for the sake of peace contended for the language of the scriptures & in conformity to the Apostles rule of holding fast the form of sound words & to yethe decree of the Church catholick in yethe Council of Antioch & Church catholick against Paul of Samosat rejected the use of the word usia with its compounds as novel & tending to faction & sabellianes Paulinianism & forbore the curious inquiries of the Gnostick o Nicolaitans into the metaphysical nature of the deity God & Christ & the holy Ghost as novel & tending to Montanism Paulinianism & Sabellianism, & forbore the curious inquiries of the Nicolaitans or men of science falsly so called, into the metaphysical nature of the supreme Beings: co contenting themselves with the declaration that the Son was like the father according to the scriptures & for doing so were called falsly called Arians by the Saintworshippers For they anathematized as well the novel language of Arius on yethe one hand as that of the Gnosticks on the other. Next to them in number were thae Camausiæ hem homousiams who owned not the deity of the holy Ghost & were falsly called Macedonians by the Saint worshippers, as if Macedonius was the author of that opinion. 5

{a + x}^{m} \times {a - x}^{n} = y^{p}

a + x = z . \dot{x} = \dot{z} . a - x = v . - \dot{x} = \dot{v}

z^{m} v^{n} = y^{p} . m \dot{z} z^{m - 1} v^{n} + n z^{m} \dot{v} v^{n - 1} = {}^{p}y^{p - 1} \dot{y}

m z^{m - 1} v^{n} - n z^{m} v^{n - 1} . p y^{p - 1} ∷ \dot{y} . \dot{x} ∷ y . subtang. = subp

\frac{m y}{z} - \frac{n y}{v} . p ∷ y . subtang = \frac{p m}{z} - \frac{p n}{v} = subperpendic

p m a + x - \frac{p n}{a - x} . y ∷ \dot{y} . \dot{x} ∷ \dot{y} . 1 . \frac{p m}{a + x} - \frac{p n}{a - x} = y \dot{y}

\frac{p m}{a + x} - \frac{p n}{a - x} = \frac{1}{2} y y

. 6 Pag. 12. l 6. + The Postscript not being sent to Mr Newton he did not think himself concerned to meddle with it till at length Mr l'Abbé Conti pressed him to write an Answer that the Pos the Post Postscript & the Answer might both be shewed to the King & continued to press him after the King had seen the Poscript & the Answer was not writ till after the King had seen the Postscript. This Answer was a follows. Bettween Letter 6 & 7 vinsert this Paragraph Introduction to the Observations. The Answer of Mr Leibnitz being sent open to Mr Remond at Paris to be sent from thence to Mr l'Abbe Conti, & copies of Mr Newton's & Mr l@Abbe Conti's Letters being sent also thither. Mr Newton declined to return an answer in writing & only drew up the following Observations to satisfy his frends privately that he could easily it was easy to have returned an Answer had he thought it been fit to let Mr Leibnitz go on with his politiques. At the end of all the Letters add the Paragraph in yethe German Elogium of Mr Leibnitz.

\frac{1}{30 a^{3}} + \frac{1}{112 a^{3}} + \frac{1}{18 a^{3}} \times \frac{z^{7}}{a^{10}} = \frac{4}{45} + \frac{1}{112} = \frac{493}{448}

\frac{560}{5040}

Tempus autem quæ quantitas est hic fluens hic expono per lineam rectāam uniformiter fluentem crescentem vel ac temporis momentum per particulam lineæ eodem temporis momento genitam, et inde lineam voco quantitatem uniformiter fluentem, et particulam ejus voco momentum ejus, et quantitates alias omnes simul augescentes voco fluentes et earum particulas eodem temporis momento genitas voco fluentium momenta et velocitates augescendi voco quantitatūum fluxiones] et particulas singulis temporis momentis genitas nominando momenta — et has motuum vel incrementorum velocitates nom (ex fluxione temporis) nominando fluxiones × (i.e. fluxiones pro differentijs) ✝ ubi Leibnitius Inventor nominatur, pag. 44. lin 17 — de qua sequentia paulatim e anno 1676 ex chartis antiquioribus extraxi. Anno vero 1676 priores decem olim excogitavi et anno 1676 Propositiones in formam sequentem redegi, undecimam vel eodem anno vel paulo post adjuxi, et duodecimam ex Epistolis anno 1676 scriptis jam addidi. Add 3968 No 416 bis ut videre licet in Lib. 11, Prop. XIV cas. 3. Quoniam methodus meus generalis ex methodo fluxionem et methodo Serierum convergentium convergentium componitur, idque et quantitatum in series resolutarum momenta prima secunda, tertia cæteraque sunt t terminis serierum correspondentibus serierum terminibus momento temporis genitis proportionalisa ideoque terminis s in resolutione Problematum nonnunquam usus sum terminis serierum. Sit A quantitas fluens, &

B = A^{\frac{m}{n}}

quantitas alia fluens. Augeatur quantitas A momento O, et quantitas B evadet

{A + O}^{\frac{m}{n}}

. quæ in seriem resoluta fit

A^{\frac{m}{n}} + \frac{m}{n} O A^{\frac{m - n}{n}} + \frac{m m - m n}{2 n n} O O A^{\frac{m - 2 n}{n}} + \frac{m^{3} - 3 m m n + 2 n n}{6 n^{3}} O^{3} A^{\frac{m - 3 n}{n}} + &c

. Et hujus terminorum fluxione primæ sunt

\frac{m}{n} A^{\frac{m - n}{n}}

. Et ipsius

A^{\frac{m}{n}}

fluxio prima est

\frac{m}{n} A^{\frac{m - n}{n}}

& hujus fluxio est

\frac{m m - m n}{n n} A^{\frac{m - 2 n}{n}}

, Et hujus fluxio

m^{3} - 3 m m n + 2 m n n

To SrSir Isaac Newton at the Lower Side of Leicester fields Westminster. Office C suposd Golden square. For SrSir Isaac Newton at his house in St Martins lane street nigh Leîsterffields These Honored Sir In obedience to my fathers Commands I am oblig'd to continue here at london for sometime longer SrSir Wm Quintin haueing promis'd my friends in the Countrey that I shall be preferd in the Customes, now seeing that I haue Qualified my self for that purpose, & that Iam in expectation dayly to come into Mr Iohn Selbys place at Whitby, he being to be promoted to a Collection in the South, & that now Sir the Welfare of our Family seems much to depend on my good success. YrYour kindnes has been extraordinary to me, for otherwise Imight haue suffer'd very much. I apply dayly to the Treasury & hope that Mr Iohn Selbys Warrant & mine will be granted us & that how glad I shall be to be in a Condition to support my Dear Father, that has liu'd well in yethe world SrSir you must not think much with me for being so free with you Imust Confess Iue been a great trespasser upon you but Ihope yryour Goodness will pardon me, & that my Dear father prays dayly you may continue my friend, & not let us sink now, when there is so fair a probability of my getting now into business, Iue' taken abundance of pains & the great fatigue Iue' undergone & hardships here as ar almost inexpressible. I hope I shall demonstrate, when please god I am in business my gratitude to you & how much I am Honored Sir AugtAugust yethe 7th 1716 YrYour most humble & most obliged Servant Wm Newton His patet me anno 1676 & annis minimùm quinque vel septem prioribus methodum & Analysin methodum generalem habuisse reducendi Problemata ad æquationes fluxionales. Et ubi ex hujusmodi æquationibus tam affectis quam simplicibus eliciendi quantitates fluentes, et quanpperin et in hac methodo quandoque assuma series pro quantitatibus ignotis assumere et assumptas terminorum coefficientes & indices dignitatum ex conditionibus Problematis determinare, Et inter computandum fluxiones et momenta ex fluentibus & fluentes vicessim ex fluxionibus & momentis deducim & in huic finem me Theoremata plura pro Quadratura figurarum & inventione fluentium per hanc methodum concinnasse, & nomen Analyseos huic methodo impossuissem methodos augentium General et Barrovij methodo meæ me nomen Analyseos hui methodo olim ab anno 1669 imposuisse propter usum ejus in Solvendis problematis propter generalem usum hujus methodi in solutionionis problematibus me nomen Analyseos eidem ab anno 1669 imposuisse & huic. affines fuisse methodos Tangentium Gregorij, et Barrovij, propterea quod Methodum Slusij similiter producerent, sed [eos ubro de me nunquam conquestos fuisse licut sed a] & Barrovium Analysin meam ut methodum novam cum Co ad Collinium mississe misisse. Brevitate verborum effectum est ut Scholium præcedens male intellectūum fuerit, et propterea rem totam fusius enarrabo. Cum vero D. Leibnitius post annos septem anno 1684 elementa hujus Analyseos in lucem emitteret & silentio præteriret literas meas pæ o ea omnia quæ vead hanc methodum spectantia quæ vel ab Oldenburgo accepterat vel in manyu Collinij viderat: posui Scholium superius ut inde constaret me primum de hac methodo scripsisse et [elementa ejus in Lemmate præcedente posita synthetice demonstrata non habuisse aliunde] & Lemma superius ab editis Leibnitianis non fuisse desumptum. Terminos serirum pro momentis nonnunquam usus sum propter analogiam. Nam hi termini per terminos 1. 1× correspondentes seriei numeralis

1.1 \times 2.1 \times 2 \times 3.1 \times 2 \times 3 \times 4

multiplicati convertuntur in momenta. Et his admitus D. Leibnitius qusam Collinium consulebat de compendium in man hocce in manu Collinij videre potuit. Is enim hoc tempore Collinium de commercio Gregorij et meo consuluit et partem litererum nostrarum in ejus manu vidit & ab Oldenburgo paulo ante postulaverat ut demonstrationem mearum serierum a Collinio procuraret: quæ Demonstratio extabat in hoc Compendio Analysis vero de qua hic locutus sum eadem est cum Analysi per series cujus compendium Barrovius anno 1669 ad Collinium misit Ad hæc His respondit D. Leibn. 27 Aug 1676 in hæc verba. Hoc compendium est Analysis illa per series fluxiones et momenta quam Ionesius edidit. Hæc est Analysis illa per series, fluxiones et momenta cujus specimen Barrovius noster anno 1669 cum Collinio communicavit ut supra. Vbi symbolum o ut quantitates fracta In Epistolis meis 10 Decem 1672 & 24 Octob 1676 datis dixi quantitates surdas methodum meam non morari, Et hæc hanc resm exemplo habere explicuio in Analysis mea a Ionesio edita, pag. 14. Substituatur utique in æquatione pro quantitate radicale symbolum ubi quodvis; Tractetur symbolum illud ut quantitas fluens; Et completo opere pro symbolo et ejus fluxione scribatur quantitas radicalis et ejus fluxio. 9 Newtonus quantitatem aliquam ut uniformiter fluentem spectat Newtonus [fluxionem exponentis temporis exponit per unitatem fluxiones aliarum quantitatum per alia symbola,] momentum temporis per exponit per Newtonus designat Tempus utique per quantatemquantitatem quamcunque uniformiter fluentem momentum fluxionum ejus designat per literam a Leibnit unitatem & momentum per literam o Newtonus designat: Aliarum quantitatum fluentes N designato per alia symbola et momenta earum per symbola illa ducta in sym momentum o, e etst aAreas vero curvarum designat per ordinatas ductas in quadrato inclusas. Leibnitius pro fluxionibus nulla habet symbola, pro momentis præfigit literam t symbolis fluentium literam d, pro et pro Areis præ figit Literam s symbolis O Ordinatarum literam. Newtonus caculo suo usus est in Analysi quam Barrovius cum Collinio mense Iulio anni 1669 communicavit; et Leibnitius hanc methodum tum in Analysi ailla tum alias in Epistolis 10 Decem. 1672, 13 Iun 1676 & 24 Octob 1676 valde generalim esse & ad endas curvarum areas longitudines curvitates (sc. per fluxiones secundas descripsit, designavit significavit & verbis partim apertis partim fig descripsit partim figuratis celavit, [eamque ad Curvarum areas longitudines, centra gravitatis soliditates & curvaturas (sc. per fluxiones secundas) sese extendere, ut et ad inversa Tangentium Problemata aliqaque difficiliora, sino vero ad omnia pene dixerat problemat si forte numeralia quædam Diophantæis similia excipiantur.] Leibnitius suam communicare cœpit anno 1677 mense Iunio Et ut suam faceret scripsit Et in hæc verba. Newtono assentior Clarissimi Slusij methodum tangentium nondum esse absolutam celeberrimo Newtono assentior: Et jam a multo tempore rem tangentium longe generalius tractavi scilicet per differentias Ordinatarum. Sed anno tamen superiore methodum differentialem minime invenerat. Scripsit enim 27 Aug. 1676 multa esse adeo mira et implexa ut neque ab æquationibus pendeant neque a Quadraturis: qualia sunt (ea multis alijs) Problemata methodi Tangentium inversæ; quæ etiam Cartesius in potestate non esse fassus est. [Annis 1675 & 1676 D. Leibnitius quadraturam quandam circuli componebat limabat et poliebat vulgari more: sed postquamea Analysin graisse novam invenerat prolixius exponere vulgari more quæ Analysis sua nova paucis exhibet non satis operæ pretium videbatur. Leibn. in Actis Eruditorum Anno 1691 p. 178.] sed acceptis Newtono Literis rescripsit anno sequente mense Iunio methodum differentialem ut ut o olim a sc inventam communicare cœpit & ut his verbis. Clarissimi Slusij methodum tangentium nondum esse absolutam celeberrimo Newtono assentior: et jam a multo tempore rem tangentium longe generalius tractavi scilicet per differentias Ordinatarum &c. Quod methodum hanc a multo tempore invenerat probandum est. Nam D. Leibnitius pro se testis esse non potest. Iniqus esset Iudex qui in rebus controversis hominēem quemvis pro seipso testem admitteret. Et vires more considerat hase considerat, non ut qualitates primaries, primas causis destitutas aut rebus corporibus essentiales, d et causes destitutas considerat sed ut qualitates lats patent quarum causes ignoramus. Et vires considerat ut qualitates primas causis destitutas aut corporibus essentiales sed quarum causes ignoramus. Sir Isaac Newton These Hon'd Sir, I take leave to put You in mind of bringing with You to the Royal Society Your Key of the Iron Chest which contains the Common Seal, the Lease being ingross'd and ready for the Seal to be affixt to it. Crane-Court, Nov. 9th 1713. I am Your most Obliged and Obedient Servt.Servant Io: Thorpe Galileo argued that uniform gravity by acting equally in equal times upon a falling body would produce equal velocities of descent in those times, or that the whole force imprest, the whole time of descent & the whole velocity acquired in falling would be proportional to one another; but the whole descent or space described would be arise from the time & velocity together & there be in a compound ratio of them both, or as the as the time of descent & the velocity of descending together that is as the squares of the times of in ther of there descent. And hereby he demonstrated that the a Projectile in a space void of resistance describes a Parabola. And Mathematicians unanimously agree that he was in the right, And Mr Leibnitz himself being one of them. And yet he measures the force imprest, not by the velocity acquired to wchwhich it is proportional, but by the space of descent to wchwhich it is not proportional.10nal. And if a body ascend, grav the gravity of the body by acting upon it equally in equal times will take of equal velocities of ascent in those equal times, or that ② the whole force imprest, ① the whole time of ascent to any height ③ & the whole velocity taken of in that as time are proportional to one another but not to the space of ascent. But The whole space of ascent will be as the velocity & time together arise from the time & velocity together & be in compound ratio of them both, that is, as the square of either of them. And upon these rules of desce ascending & descending, Galileo demonstrated that projectiles would, in spaces void of resistance, describe Parabolas. And all Mathematicians acquiesce (not excepting Mr Leibnitz himself) unanimously agree that he was in the right. And it is proportional but by the space of ascent to which it is not proportionall. 11 et in epistolis supra impressis significavit methodum suam ad tangentes directe & inverse et lqud ad alia etiam problemata genera extendere, & method Inter Leibnitius et elis [et methodum suam in tangentibus directe et inverse per exempla exposuit.] Leibnitius negaverat Tangentes per inverse per æquationes trac vel & quadraturias tractavi posse sed Lecta Newtoni literis animum advertit ad methodos tangentium & consideratio diffentiarumdifferentiarum in methodis illis primam lucem ipsi affundit. Leibnitius hic fatatur methodus determinandi Tangentes per differentiams linearum primam ipsi lucem affunderitisse Leibnitio id est methodums Archimedis a Fermatio, Gregorio, Barrowo restitutuma & promotam, Scripserat Newtonus methodum suam ad tangentes determinisse in omni problemat genere. a Newtono ad æquationes is quasvis in indeterminatas & motuum velocitates applicatam. Vidit hoc Fermatious antea, determinando punctum flexus contrarij. Nulla est hæc analogia. Differentiæ non sunt summarum differentiæ, nec relationem habent habent ad summas habent nisi quatenus sunt nisi quantitatumes infinite parvorum D Leibnitius methodum suam minores aliunde habe ad Pam D. Leibnitius aliunde accepit. Methodus igitur determinandi Tangentes per differentias lineárum primam lucem affuderat Leibnitio (p id est methodus Archimedis a Fermatio Gregorio Barrowo restituta & promota, a Newtono ad quantitatum a augmenta seu momenta generaliter applicata. Hujus methodi exempla in problemate tandgentium directo (p) & inverso (p) & in Quadratura curvarum p et applicatione methodo serierum p Newtonus dederat. [Et Lebnitius subinde postea de methodo tangentiūum amplianda cogitare cœpit (p) & cum ipsius Hudenio de eadem locutus est (p) & methodum tangentium per differentias ut omnium optimam in animo revolvens (p p) invenit methodū incidit in differentialem (p) et eandem cum Newtoniana statim conferens similitu similitudinem de contulit & per omnia similem deprehendit p cum t ta primam lucem intde habuit p & methodum differentialem differentialēem sic reperiam cum Newtoniana statim contulit idque in problemate tangentium directa p & inversao p et in Quadraturis p & similem deprehendit animadvertit p sed in Actis Lipsicis prætendit se hoc non prius animadvertisse quam opera Newtoni et Wallisij prodiere. Sic etiam oblitus est literarum Oldenburgi per quas s] Leibnitius ex his deprehendit similitudinem in methodorum p sed] Et ex his D. Leibnitius statim deprehendit similitudinem methodorum ab initio p Sic et Et His admonitus D. Leib D. Leibnitius in methodum tangentium per differentias animum advenit intendit (p 46, 47 87, 88) & methodum differentialem Newtonianæ similem esse statim ab initio deprehendit p. 90, 91, 93.

\frac{NH}{CN + FG} = \frac{e e n o - a n o o \times e^{5}}{e^{3}, e e n n o o - \frac{a n n o^{3}}{2}} = \frac{e^{4} - a e^{2} o}{e e n o - \frac{1}{2} a n o o} = \frac{e e}{n o} - \frac{a}{2 n}

\frac{CF}{2 FG} = \frac{n o}{e} \times \frac{e^{5}}{e e n n o o + a n n o^{3}} = \frac{e^{4}}{e e n o + a n o o} = \frac{e e}{n n} - \frac{a}{n}

CP . CG ∷ \sqrt{CN} . \sqrt{GF}

& arcus HC, CP erunt synchroni, et

HC - CP

erit decrementum momentaneum ex resist & grav. Est

HC - HN

increm momentaneaum ex grav. Ergo

2 HC - CP - HN

decrem. moment. ex grav. resist.

= HC - CQ

a D Leibnitius hic fatetur se methodum differentialem a relatione tangent Differentiarum ad Tangentes didicisse habuisse, id est a methodo determinandi Tangentes per Differentias didicisse Sothiant Archimedes, Fermatius, Gregorius, & Barrowus methodusum s Differentiarum ad Tangentes applicuerante: Newtonus se Fermatio se a Fer se primam hic Lucem primam a Fermatio hic accepisse t fatetur pe esse atos et hanc methodum a Fermatij s acceptam G nead æquationes abstractas & proportiones motuum applicando, et per æquationes & proportiones motuūum ad op aia quævis problemata quævis quæ per æquationes tractari possunt generalem redtus Et l sgt per proportiones motuum et hoc cum method & proportiones motuum & series tractari possunt applicu sic reddidit valde generalem & se talis methodi compotem applicando aum esse & methodum Slusij ejusdem corollariūum esse significavit in epistolis supra impressis pag & Deinde d Leibnitius Deinde cum scriberet & quibusdam methodū illustravit p pag & Et cum D. Leibnitius scripsissit methodum inversam tangentium impotestate esse et methodum Slusij esse corollarium ejusdem ab æquationibus & quadraturis non pendere, Newtonus respondit esdem hanc etiam in potestate esse Et tum dem et exemplum generale dedit solutionis per quadraturams Curvarum ubi datur relatio inter latera duo quævis trianguli quod ab ordinata, tangente & subtangente constituitur. Et his ominibusomnibus Et tum demum D. Leibnitius his omnibus admoteitus methodum differentialem methodum generalem quæsivit per relationem differentiarum ad Tangentes. habuit quæsivit & in Newtonianam invenit incidit quæ methodo Newtoni responderet, & se in ejusmodi methodum incidisse mox significavit scripsit ad Oldenburgum significavit in ult sd per Epistolā pag de quam videas pag supra impressam pag Methodus tangentium a Slusio publicata a Dixerat Newtonus methodum tangentium a Slusio ascriptam, a principijs suis methodo sua generali facillime fluere. Slusius Facillime fluit hæc methodus a Differentijs. Et Leibnitius hic fatetur se methodum differentialem a relatione Differentiarum ad Tangentes habuisse. Et tum demum com D. Leibnitius animum advertit ad methodos tangentium, & consideratio Differentiarum in methodis illis primam lucem ipsi affundit Et met problemata tangentium inversa ali aliaque similia a tang ab æquationibus & quadraturis pendere se primam negasse & subinde a Newtono didicisse statim oblitus est Vide p. 65, 85, 86, 93. Vidi b mox b Hoc antea Fermatius qui, per differentiam primam evanescentem id est per osculum determinavit punctum flexus contrarij. c Et notavi c mirabilem &c c Nulla est hujusmodi analogia Differentiæ non sunt summarum differentiæ; non sunt ejusdem generis quantitates cum summis; [Leibnitius primam lucem aliunde habuit.] sunt ad summas ut momentum ad tempus, vel punctum ad lineam, non ut radix ad potentiam. Leibnitius primam lucem aliunde habuit. Sic etiam oblitus est literarum Oldenburgi per quas series suas omnes accepit p. 12 he saith that when he was in London the second time he saw some of my Letters in the hands of Mr Collins cheifly those wchwhich related to series & observed that in one of them I acknowledged my ignorance in the dimensions of the vulgar figures except the Sphæroid Cissoid. & & that the Committee had omitted this. But being told ytthat this was in my Letter of 24 Octob. 1676 & was published in the Commerc. p 74 he acknowledges that it was, & addesdadds that he would cite another instance. iIn another of my Letters wchwhich he then saw (meaning a Letter dated 1672 & not yet published said that I acknowlededacknowledged my ignorance that I could not find the second sections (or segments of Sphæroids, & that the Committee acknowledged had omitted this. were as much as was proper in that ere I it being of that book to enter into disputes about Mr Pell at that notice of Mercators series for the Hd Mr Collins had some communited mine & Gregories series to the Mathematicians at home & abroad, might meet with some of them either at London or Paris without being with Mr Collins.] & might be able to give him notice of

x^{n - 1} \dot{x} = \dot{y} . {}^{n}x^{n} \dot{x} = \dot{y} x

. Provided nevertheless that over & above the summs above mentioned there may be issued yearely out of the coinage monnys of England, the summ of 300li per an' for the salary of the General of the Mint in Scotland, 50li per an' for the salary of the Clerk of the Bullion & 50li per an' in part of the salary of the Warden of the said Mint untill the next voidance of their places. in the place Scholium Passage Paraph there referred unto I do not find one word to this purpose. On the contrary I there represent that I sent notice to Mr Leibnitz of my Letter bef Method sent notice to me of his method & left him to make it appear that he had method before the date of my Letter, I that is, eight months at least before the date also And by referring to the Letters wchwhich passed between Mr Leibnitz & before, I left the Reader to consult those Letters & interpret the Paragraph thereby. For by those Letters ld understand see that I of that Method & the Method of Series five years before those Letters that is, in the year 16 Mr Leibnitz knew nothing 13 quæque beneficio serierum ad omnia pene problemata se extenderet, etiam ad inversa tangentium problemata ab aliqdque difficiliora se extenderet & methodum exemplis illustrarem literis vero transpositis hanc sententiam involventibus [Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire, et vice versa] fundamentum ejus celare, at exemplar epis Epistolæ cujusdam ad Anno 1672 add Collinium dat accepisset etiam exemplar Epistolæ a me anno 1672 ad Collinium scriptæ in qua in dixeram methodum Tangentium Slusij Corollarium esse hujus methodi generalis, Cum methodus Tangentium Slusij fundaretur in di in proportione differentijæ Ord Ordinatarum duarūum sibi proximarum ad differentiam Abscissarum et anno 1672 ad Collinium scripsissem hanc methodum esse Corollarium methodi generalis quæ citra molestum ullum calculum ad resolventdum abstrusiora problematum genera de curvitatibus, Areis, longitudinibus, centris gravitatis, curvarum &c & ad quantitates surdas minime hæret, deinde exemplar autem hujus Epistolæ ad D Leibnitium, mittere anno 1676 ad D. Leibnitium mitteretur; et eodem anno literis alijs ad ipsum missis significarem me compotem esse methodi ducendi tangentes determinandi maximas & minimas ducendi tangentes quadrandi curvilineas et similia peragendi quæ in terminis surdis æque ac in rationalibus procederet, et cujus beneficio quaque beneficio serierum ad omnia pene problemata [etiam ad inversa tangentium aliaque difficili ora] se extenderet, [et methodum exemplis illustrarem] respondet Le D. Leibnitius id sibi non videri; esse enim multa usque ad eo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturs, qualia sunt ex multis alijs problemata methodi tangentius inversæ. Cum autem rescripsissem inversa de Tangentibus problemata esse in potestate, aliaque illis difficiliora & methodum tangentium Slusij ex meis Principijs statim consequi, quætionesquæstiones sol & similiter quæstiones de maximis & minimis et resolvi & quadraturas reddi falciliores & alia similia peragi, et Analysin meam asd surdas quantitates aut ad Curvas suas vocant Mechanicas non hærere, eandem vero exemplis illustrassem sed fundamentum ejus literis transpositis hanc sententiam involventibus [Data æquatione quotcunque fluentes quantitates involvente fluxione invenire et vice versa] fundamentum ejus celassem: Respondit D. Leibnitius in hæc verba. Clarissimi Slusij Methodum Tangentium [id est methodum per differentias Ordinatarum] nondum esse absolutam Newtono assentior Celeberrimo Newtono assentior &c et jam a multo tempore rem Tangentium longe generalius tractavi, scilicet per differentias Ordinatarum. Deinde methodum suam communicavit a mea vix abludentem præterquam in verborum et notarum formulis. Vtriusque fundamentum continetur in hoc Lemmate. Hoc fecerunt Gregorius in ejus Prop 7 Geometriæ universalis anno 1668 impressæ Prop. 7 et Barrovius in ejus Lect 10 anno 1669 impressa, idque methodo consimili. Idem D. Leibnitius facere potuisset jam a multo tempore. Sed inversa tangentium Problemata differentiales & quadraturas reducere jam anno superiore minime noverat. Vide pag 65 l. 14, 15. Cum vero a Newtono didicerat Clarissimi Slusij methodum Tangentium nondum esse absolutam, sed Corollarium esse methodi generalis quæ extenderet se citra molestum ullum calculum ad abstrusiora problematum genera, etiam ad inversa tangentium problemata aliaque diffilciliora, & quæ ad quantitates surdas & Curvas Mechanicas minime hæreret, & cujus ope Quadraturæ redderentur faciliores: cœpit is methodum Tangentium per differentias Ordinatarum ab alijs traditam jam longe generalius tractare quam antea, & novo nomine novisque symbolis ut suam ornare. Quo effectum est 14 Id est,

y = \frac{x \dot{x}}{\sqrt{a x' - x x}}

vel

\dot{y} = \frac{x \dot{x}}{\sqrt{a x - x x}}

. Et nota quod Differentiæ rectius decerentur partes. Sunt enim partes summarum non et & non autem differentiæ, neque aliaquam habent relationem ad summas nisi quatenus sunt earum partes. idque & n idque generaliter per s quadraturas sine ejus methodo ge idque in t semper per quadraturas solas absque methodo generaliore quando datur relatio inter latere trianguli TBC. Hæc Isagoge et Corollarium Propositionis ultimæ scripta sunt ubi liber prodijt: reliquus in MS reliqua ex MS antiquo impressa sant. idque nonnunquam per quadraturas nonnunquam geraliusgeneralius methodois generalioreibus. aliaque difficiliora. Vide etiam Simplicior est expressio

\dot{y} \sqrt{a x - x x} = x \dot{x}

. maximas et minima dedterminari quæstiones de maximis et minimis aliasque quasdem determinari, & Problem Vide etiam pag 30 & pag 47 lin 4 Gregorius methodum serierum ex unitca tantum serie Newtonianam methodoum invenit & Newtonum tamen inventorēem primum libere agnovit: Leibnitius pluribus edoctus, fuit. se nihil ab alijs Vide pag 30 & pag 47 lin 4, 8. & 71, 72. LM

\frac{2 a n n o^{3}}{e^{5}} \times \frac{2}{8 n o} \times \frac{2 e^{3}}{n n o o}

\frac{4 a n n}{8 e n^{3}} = \frac{a}{2 e n}

\frac{2 a n n o^{3}}{e^{5}} \times \frac{n o}{e}

\frac{8 n^{4} o^{4}}{4 e^{6}}

a. n 15 Hic est tractatus de Analysi per æquationes numero terminorum infinitas supra impressus. Hinc liquet methodum fluxionum non tantum Newtono ante annūum 1669 Newtono involuisse sed etiam ad magnum perfectionis gradum eo tempore provectam fuisse. 16 ② In the year 1671 I wrote a Tract concerning the methods of converging series & the method of fluxions jointly andbut did not finish it, that part of it being wanting which related to the solution of the inverse meth Problems wchwhich could not be reduced to quadratures, as I mentioned in my letter to Mr Oldenburg dated 24. Octob 1676 &which was published by Dr Wallis in the third Volume of his works. From this Tract I extracted in the year 1676 I extracted the following book of Quadratures; & therein I copied from the former Tract without any alteration the Tables set down in the tenth Proposition for squaring the simpler some Curves, or reducin & comparing them with the Conic Sections. In my Letter of to Mr Collens to Mr Collins dated 8 Novem. 1676 found by MrIones a mongst his Papers & published by hism & dated 8 Novem. 1676 I had relation to this Book in saying Nulla extat Curva — — — adeo generaliter. This mentioned in set down in relates to Coroll. 2. Prop. 10 of this Book. And in the second Lemma of the 2d Book of Principles I demonstrated the Elements of the method of Fluxions. ③ ∥ This Book was in the hands of Mr Ralpson & Dr Halley in the year 1691 as the former has beft attested in print before his death & the latter still attests. And in the year 1692 at the request of Dr Wallis I sent hi to him the first Proposition of this Book with examples in first & second fluxions & he published before the end of the year he printed the the same in the second Volume of his Works wchwhich Volume came abroad in April 1693. And this is was the first time that any Rule was published for finding second third fourth & other differences. But the Book continued in MS till the year 1704. And then in publishing it I wrote a Preface in wchwhich I affirmed that I found the method of fluxions gradually in the years 1665 & 1666. For I thought then that I might safely write this because Dr Wallis in the Preface to the thir first Volume of his works published in spring. ④ In the abovementioned Letter of the 24 Octob 1676 at the request of Mr Leibnitz I described how before the Plague wchwhich raged in London in the years 1695 & 1696 by considering how to interpole the series of Dr Wallis I found the method of converging series together wthwith the Rule for converting the powers & dignities of Binomials into such series, & that Dr Barrow about the time that the Logarithmotechnia of Mr Mercator came abroad sent to Mr Collins a compendium of thiese series. This compendium was A copy of this Compendium was in the handwriting of Mr Collins was found by Mr Iones in the among the papers of Mr Collins & published after it had been collated wthwith yethe original wchwhich Mr Iones borrowed of me for that end. The title thereof was Analysis per series numero terminorum infinitas. And in this Tract there are instances of calculating by the method of fluxions, this method being interwoven with that of Series. & it is therin affirmed that by this method of the method of series extends to all Problems & that by the help thereof ejus beneficio curvarum area & longitudinis &c (id modo fiat) exacte & Geometrice determinantur And therefore I then understood the method of Fluxions so far as it is conteined in the first five or six Propositions of the book of Quadratures. And by the Testimony of Dr Barrow & Mr Collins I understood it thus far some years before that time For Mr Collins in a Letter to Mr Strode And how this is done is explained in my said Letter of 24 Octob 1676. & illustrated with examples. And this is not to be done without the method of fluxions so far as it is conteined in yethe first five or six Propositions of the book of Quadratures. And therefore the method of fluxions so far as it is conteined in those Propositions was known to me when I wrote the said Letter of 24 Octob 1676 & before that even when I wrote the Analysis per Æquationes numero terminorūum infinitas wchwhich was in the yeare 1669 it was known to me & choun by the Testimony of Dr Barrow & Mr Collins it was known to me some years before that. For Mr Collins in a letter to Mr Strode dated 26 Iuly 1672 & published by Order of the R. S. in the Commercium Epistolicum wrote thus. Mense Septembri 1668, Mercator Logarithmotechniam suam edididt suam, quæ specimen hujus methodi (i.e. serierum infinitarum) in unica tantum figura, nempe Quadraturam Hyperbolæ continet. Haud multo post quam in publicum prodierat Liber, exemplar ejus Cl. Wallisio Oxonium misi qui suam de eo judicium in Actis Philosophicis statim fecit; aliumque Barrovio Cantabrigiam qui quasdam Newtoni chartas (qui jam extemplo remisit: e quibus et ALIIS, quæ OLIM ab Auctore cum Barrovio communicata fuerant, patet illam Methodum a dicto Newtono aliquodt annis antea ALIQVOT ANNIS ANTEA excogitatam & modo universali applicatam fuisse: ita ut ejus ope in quavis Figura Curvilinea proposita quæ una vel pluribus proprietatibus definitur Quadratura vel Area dictæ Figuræ, accurata si possibile sit ACCVRATA SI POSSIBILE SIT, sin minus infinitè vero propinqua, Et Evolutio vel longitudo lineæ curvæ, Centrum gravitatis Figuræ; solida ejus rotatione genita, & eorum superficies; sine ulla radicum extractione obtineri queant. o So then by the testimony of Dr Barrow founded upon papers communicated by me to Dr Barrow him before the Logarithmotechnia came abroad from time to time, I had brought the method to the perfection here described, some years before the Logarithmotechnia came abroad. In another Letter, Mr Collins saith, above two years before. So then By these testimonies it appears therefore that I had the method therefore in the perfection here described before September 1666, & by consequence then understood the method of fluxions that is so far as it is described in the first five or six Propositions of this Book of Quadratures to squar square curves exactly when it may be done, or at least by continual approximation, & by consequence that I then understood the method of fluxions so far at least as it is conteined in the first five or six Propositions of the book of Quadratures. Which together wthwith the testimony of Dr Wallis I recon sufficient to justify me in what I said in the Introduction to this Book. ① In December 1672 upon notice from Mr Collins that Mr Iames Gregory had a method of drawing Tang improved the methods of tangents of Dr Barrow so as to draw tangents without calcuclation & that Mr Slusius had such another Method wchwhich he intended to communicate to Mr Oldenburg, I wrote the following Letter to Mr Oldenburg Collins dated 10 Decem 1672. Ex animo gaudeo D. Barrovij — — — — reducendo eas ad series infinitas. These last words you have here a general description of the great extent of the method of fluxions, with an example of it in drawing of Tangents. The wor the method here spt described was interwoven with the another method of resolving converging Series in wchwhich I reduce equations to converging series. ② For in the year 1671 I wrote a Tract concerning the method of 17 NB. Hunc Librum de Quadraturis MS Halleius & Ralphsonus Anno 1691 manibus suis tractarunt uti posterior publice testatus est et posterior adhuc testatur. Propositionem primam cum exemplis in differentijs primi fluxionibus primis & secundis inveniendis D. Wallisius anno 1672 a me accepit & eadem in secundo ejus Volumine operum ejus impressa fu anno proximo lucem vidit, & prima fuit hujus generis Regula quæ lucem vidit; estqus Regula illa verissima et maxime universalis. In Lemmate secudosecundo libri scdisecundi Principiarum mathematicorum Philosophiæ, Schol Regulam illam synthetice demonstra & in Scholio quod Lemmati subjunxi Propositionem dixi me Propositionem eandem totidem syllabis posui dixique me eandem totidem syllabis prius in Epistola ad 24 Octob. 1676 ad Oldenburgum data posuisse. Hanc Epistolam D. Wallisius in tertio operum suorum edidit volumine edidit, Et ibi habetur hæc Propositio totidem literis tanquam fundamentum methodi cujusdam ducendi tangentes, determinandi maxima & minima, quadrandi figuras & similia peragendi, In qua metho & in hac m et hic in qua non hæretiur ad Æquationes Radicalibus unam vel utramque Indefinitam Quantitatem involventibus utcunque affectas. Et eodem fundamento dixi me ad Theoremata quædam generaliora quadrandi curvas pervenisse. Et Theorema primum (sc. pro ibubiubi posui et exemplis illustravi addidique me pro Trinomijs etiam et alijs quibusdam Regulas quasdam concinasse. Est autem hoc Theorema primum idem cum Propositione quinta libri de Quadraturis, Et Theorema primum pro Trinomijs idem cum Propositione sexta ejusdem libri. Et hæ duæ Propositiones pendent a Propositionibus quatuor primis ejusdem libri. Et propterea methodus fluxionum quatenus continetur in Propositionibus sex primis Libri de Quadraturis mihi innotuit anno 1676. Sed et Propositio septima et octava, ejusdem sunt generis. Nona autem ac decima requirentur ad solutionem Problematis quod posui in Epistola ad D. Collinium Novem. 8 1676 data & ad Ioneso edita verbis sic Latine redditis Nulla extat Figura curvilinea cujus Æquatio — — — haud tamen adeo generaliter. ☉ ☉ Et in eadem Epistola Ordinatæ CuvilinearumCurvilinearum quæ cum Ellipsi et Hyperbola comparari possunt eædem ponunturhabentur ac in Tabula secunda Prop. X libri de Quadraturis. Ideoque Tabula illa (et propterea methodus fluxionum) mihi innotuit cujus ope Tabula constructa fuit) mihi ante innotuit anno 1676: immò et annis aliquot antea. Nam dixi in Epistola illa me Theoremata pro Comparatione Curvarum cum Conicis Sectionibus in Catalogum dudum retulisse. Anno igitur 1676 Methodum fluxionum intellexeram quatenus in hoc Libro hocce de Quadratureis Curvarum exponitur. Sed et anno 1671 eandem intellexerāam Namque in Epistola mea prædicta ad D. Oldenburgum 124 Iunij Oct. 1676 scriptam, dixi me in Tractatu quem tunc ante quinquennium scripseram de Seriebus conscripseram me etia alia præter methodum serierum haud pauca congessisse inter quæ erat methodus Tangentium ducendi Tangentes quam solertissimus Slusius ante annos duos tresve tecum communicavit; de qua tu (suggerente Collinsio) rescripsisti eandem mihi etiam innotuissi. Dein Deinde Diversa ratione in eam incidimus. Et subinde dico me in hanc methodum tangentium incidisse per methodum quandam quæ ad quæ ad quæstiones de maximis et minimis, de que quadraturis & alijs se extendit & quantitates surdas non moratur, & cujus fundamentum continetur in hac sententia, Data æquatione fluentes quotcunque quantitates involvente, Fluxiones invenire; et vice versa. De his omnibus hic loquor tantquam mihi cognitis ante quinquenniusm cognitis. Sed et in Epistola [ad D. Collinium 10 Decem. 1672 scripta data hanc methodum sic descripsi. Ex animo gaudeo — — — ne grave ducas. Methodus fluxionum hic perspicue describitur. Et præterea, in Epis] eadem 19 — nec prius cum amicis communicare cœpit quam ab Anglia discesserat, & postqꝫuam ab Oldenburgo acceperat, remisit acce ut Anglis ignotam. Methodum exhibendi Arcum cujus sinus datus Leibnitius ab Oldenburgo postea quæsivit, Maij 12 16121672, ideoque nondum intellexit p. 38. a Collinius jam ante quadrennium series Newtonianas ante triennium Gregorianas cum amicis communicare cœpit. Leibnitius in Anglia diversabatur Aanno superiore & hujusmodi series nondum communicaverat, nec prius cum amicis communicare cœpit quam ab Anglia discesserat, Et subinde cum series aliquas ab Oldenburgo acceperat, quas a suis diversas esse tum fatebatur, tandem unam bum acceptarum remisit Oldenburgo ut Anglis ignotam. Nullas autem communicavit nisi quas ab Oldenburgo acceperat. b Methodum exhibendi arcum cujus sinus datur Leibnitius ab Oldenburgo postea quæsivit, Maij 12 1676, ideoque nondum intellexit habuit. Vnde nec methodum jam habuit perveniendi ad seriem alteram numerorum rationalium quam se primum invenisse hic jactabat quæā utique eandem esse dicit Sola methodus transmutatoria quam postea communicavit non dat arcum ex Sinu, ideoque non est methodus de qua hic agitur, sed post inventa fui fuisse videtur. Series forsan duas habere potuit sed methodum qua inventæ sunt nondum habuit. p. 41. Hanc seriem D. Collins initio anni 1671 a Gregorio acceperat ut supra; D. Leibnitius cum amicis in Gallia opusculum de eadem hoc anno communicare cœpit, celata hac Epistola. p. 42. a His verbis Leibnitius series acceptas a suis diversas esse testatur confessus est testatur et earum tamen unam pro sua unam tanentamen acceptarum quasi Anglis ignotam præ sa anno proximo pro sua remisit. Miror quænam fuerint ipsius series quibuscum non potuit series ab Oldenburgo missas jam comparare non potuit, Si aliquas ab communicatas diversas habuit certe b eas nunquam communicavit mirum est quod eas nunquam communicavit in lucem producit protulit et quare eas ab acceptas lucem nunquam viderunt. siquis habuit ab acceptis diversas p. 42. b Hoc nunquam fecit Leibnitius . . . . . . vindicandi, cum incidisset tandem in transmutatione figurarum cujus beneficio series illa inveniri potuit prætendens hanc quas longe diversam ab ijs quas per Mohrum acceperat, Anno & quasi alios multos [Anne Leibnitius ignorabat se hanc seriem a] quasi a nemine acceptam quia longe diversam ab ijs quas per Mohr Georgium Mohr acceperat & ab alio nemine acceptam. Certe series hæc Leibnitio ipso teste non est Leibnitiana quia non diversa ab ijs quas ab Oldenburgo acceperat. Certe Leibnitiana non est quia seriem meam In Epistola 26 Octob. 1674 data dicebat seriem seriem suam suam inventam esse eadem methodo inventam esse cum serie pro endo arcu ex dato sinu (Hæc series prima...earum quas per Mohrum mox accepitat jam methodum hanc ignorat Ex Quomodo invenietur jam ignorat methodum inveniendi sibi mitti suar postulat et ut eandem obtineret sua ab his longe diversa circa hanc rem meditata vicissim promittit seriem scilicet quam ab Oldenburgo acceperat, Vide ejus Epistolam Maij 12, 1676 non eadem methodo (ne sua non esset) sed alia nova aliqua inventam demonstratam.] Series hasce mirabatur ut valde ingeniosas sed demonstrare nondum potuit Postulat igitur demonstrationem sibi mitti, et ut eandem obtineat, sua ab his longe diversa circa hanc rem meditata promittit. *** ad verba [Via quadam sic satis singulari] notetur. Leibnitius in Epistola 26 Octob 1674 data dicebat series suas una et eadem methodo inventas esse. & hanc methodum jam vocat viam quandam satis singularum. Series pro arcu ex dato sinu prima est arum quas per Mohrum mox accepit. Series hisce per Mohrum acceptas ut valde ingeniosas laudabat sed demonstrre nondum potuiterat. Postulabat igitur demonstrationem sibi mitti, et ut eandem obtinereat sua ab his longe diversa circa hanc rem meditata promittebat.] Sed necrium suo series suas ab Oldenburgianis diversas, nec et viam singularem qua ejusmodi series prodirent, nunquam misit communicavit.

DG = e

\begin{array}{r} e e - 2 a o - o o (e - \frac{a o}{e} - \frac{n n o o}{2 e^{3}} \\ - 2 a o - o o \\ - \frac{2 a o + \frac{a a o o}{e e}}{o - \frac{n n o o}{e e}} \end{array}

☾

\frac{\frac{a a o o}{e e} + o o}{2 e - \frac{2 a o}{e}} = \frac{n n o o}{2 e^{3}}

2 2) o o + \frac{a a o o}{e e} = FG = \frac{n n o o}{2 e^{3}}

E Quasi Seriem Gregorianam ab Oldenburgo acceptam Mense Maio a suis distinuerat. Opusculum de hac serie compositum hoc anno cum amicis in Gallia communicare cœpit celata Oldenburgi communicatione. Seriem eandem quasi Anglis ignotam jam promittit se missur cum Oldenburgo communicaturum, & proximo anno communicavit. Et prætendit se cum amicis in Gallia ante biennium id est anno 1672 communicasse. Prætenditebat etiam ante menses quatuordecim Series quas mense Aprili O suas in Series mense Aprili ab Oldenburgo missas, Leibnitius a suis diversas esse distit diversas esse agnovit Inter Missarum una erat series Gregorij pro arcu circuli ex tangente altera alia erat series Newtoni pro tangente ex arcu ex sinu. Has pro suis Leibnitius e agnovit Opusculūum De serie priore Leibnitius ho compositum Leibnitius hoc anno cum amicis in Gallia communicavitre cœpit, Methodus inveniendi seriem posteriorem Leibnitius anno proximo postulavit ab Oldenburgo Celata Oldenburgi epistola. Et jam promittit se hanc seriem quasi Anglis ignotam cum Oldenburgo communicaturum, quasi et eandem simili fide dicit se eandem jam plusquam biennio abhinc cum Geometris in Gallia communicasse. Seriem alteram easdem laudabat ut novas & valde ingeniositas & sibi tum primum cognitas postulandbitat et inde occasionem accipuit exempl & ideo postulabat. earum demonstrationem ad se mitti, quasi series nullas ab Oldenburgo prius accepisset. Et hoc parto . . . . . acceperat. a Quasi Leibnitius nesciret hanc s hanc Quadraturam Anglis Gregorianam esse &c innotuisse. vel ab Ol b Anno 1673 Leibnitius in Anglia commorans de hujusmodi seriembus aliquid audire potuit. Annūo proximio sscripsit jactabat se hujusmodi seriem unam atque alteram habere sed nullam cum Anglis communicavit, anno tertio mense Aprili vel Maio accepit ab Oldenburgo ejusmodi series aliquot quas a suis diversa, esse agnovit. Eodem anno opusculum de serierum acceptarum una cum amicis in Gallia communicare cœpit, celata Oldenburgi epistola. Et jam promittit se hanc seriem quasi Anglis ignotam Oldenburgo remissurum. b Quasi ante Annum easdem non accepisset ab Oldenburgo. c Opusculum prædictum de Quadratura arithmetica D. Leibnitius polire perrexit. 20 When Mr Leibnitz published the series o Gregory at his ow & such other things as he had notice of from England for preventing disputes & doing justice to every body: he should have acknowledged who the correspondence he had kept wthwith Mr Oldenburg & what he had received from England or seen in the hands of Mr Collins relating to the things wchwhich he published. a He omits the greatest part of my method of series & its connexion with the method of Fluxions. b He gives Mr Fatioo the lye who had seen my ancient papers, & pretends that when I represented that I wro in wh in the year 1671 I wrote a tract on the method of series & another. ytthat method founded on this Proposition Data æquatione fluentes quantitates involvente invenire fluxiones & I did not dream of the calculus of fluents & fluxions c He affirms falsly that I use prict letters for the differential characters For fluxions & differences are quantities of a different kind. d He places the invention of the methods in the invention of the the symbols as as if used in them as if new names & new symbols ought to pass for new inventions. e He affirms falsly that in my Principles I had frequent occasion to use my calcuscalculus of fluxions for a after I had invented the Propositions by Analysis I demonstrated them by composition. f He affirms falsly that in this book there is no footstep of this Calculus for yethe 2d Lemma of the second book conteins the elements of this calculus And the Scholium upon this Lemma asserts this Calculus to my self. g Prickt letters appeared in the second volume of the Works of Dr Wallis wchwhich was printed in the year 166931693 & mentions that he had the what he there printed of this matter he had from me in the year 1692. My Tract of Quadratures was handed about among in London in 1691. I made much use of it in writing my book of Principles & composed it many years before but never placed the method in the use of prickt letters: for the Introduction to this book was writ in yethe year 1704 without p & conteins a description of yethe method without yethe use of prickt letters. h Incrementum constans ipsius x nunquam notabam per x punctatam notatu uno puncto. Sed p Illud noto per

\dot{x} o

; & ubi

\dot{x}

est unitas, per o i Notatio mea per o sub falso discitur calculi differentialis commoda destruere. Notatio est elegantior et utilior. k Regulam circa gradus utiliores falsam non dedi Mathematicus ille eminens erravit. Annon hic est Bernoullius. l Apparet me ab Autore nostro falso accusatum Leibnitiusm Anno 1689 ubi scripsit Tentamen De motuum Cœlestium causis, rectam methodum differentiandi differentiali non intellexisse. Rectam methodum Wallisius noster e literis nostris anno 1692 ad eum missis descripsit anno in secundo Operum volumine anno 1693 in lucem emisso, Eandem recte descripsi in Propositione prima libri de Quadraturis; et hic liber manibus amicorum Londini terebatur anno 1691, & subsidio mihi fuit annis 1689, 184684 1683, 1684, 1685 & 1686 ubi scribebam Principia Philosophiæ et multa ante conscriptus fuit. Et recte capiendo fluxiones fluxionum et momenta momentorum inveni Demonstrationem Theorematis Kepleriani anno 1677 & Curvaturam Curvarum ante annum 1673 ut ex epistola mea 10 Decem 1672 ad Collinium data manifestum est. a literas punctatas uno duobus tribus &c punctis superpositis non adhibit pro dx, ddx, d3x; dy, ddy &c nunquam adhibuit. Hæ sunt quantitates diversi generis. Ex verbis ejus nunc adhibet incidit suspicio quod is fuit author Epitomis libri De Quadratura Curvarum in Actis Erudit. anno 1705 ips impræessæ unde nata est hæc controversia. b In Principijs Naturæ Mathematicis, calculo fluxionum utendi nulla erat occasio. Inventa sunt Propositiones per hanc Analysin, demonstratæ vero per synthesin. c 1 Methodum Serierum a me inventam minuit.2 Somnia narrat. Ex Epistola 24. Octob. 1676 data certissimum est me fluxionales æquationes per ea tempora habuisse. 3 fallitur. Pro symbolis differentijarum non utor symbolis fluxionum. Hæ sunt quantitates diversi generis.4 Fallitur. Methodi non consistunt in formulis verborum et symbolorum. In tractatu de quadraturis olim scripto literis punctatis utor, in ejus Præfatione nuper scripta literis punctatis utor et tamen methodum fluxionum liabsque literis punctatis describo. 5 Falitur. In libro Principiorum nullam habui occasionem utendi calculo fluxionum Propositiones inveni per Analysin demostravidemonstravi per synthesin. Analysis al tamen ita per Demonstrationes syntheticas ita elucet ut Marchio Hospitalius scripserit librum pene totum ex hac Analysi constare et ipse Leibnitius agnoverit me omnium primum specimine publice dato ostendisse quod partem hujus methodi nobilissimam et latissime patentem haberem: eam utique qua solidum minimæ resistentiæ, Curva celerrimi descensus, Catenaria et Vellaria inventa fuerunt. 6 Fallitur. Nam Lemma secundum libri secundi & Scholium ejus continet elementa hujus calculi. 7 Fallitur. Nam Volumen tertium seri editum fuit anno 1699. Extant literæ punctatæ in volumine secundo quod prodijt impressum fuit anno 1693. Liber autem de Quadraturis in manibus amicorum terebatur anno 1691 et deinceps & in meis anno dum Principia Philosophiæ scripsiberem & multo ante scriptus compositus fuit. ut ex Epist 8. Fallitur. Incrementum constans ipsius x nunquam notabam per

\dot{x}

punctatum uno puncto. Illud noto per

\dot{x} o

: et ubi

\dot{x}

est unitas, per o. 9. Fallitur. Notatio mea per o non destruit commodo calculi differentialis se e contra, commodior est. 10 Fallitur. Regulam circa gradus ulteriores veram dedi. 11 Annon Ber Mathematicus ille eminens Bernoullius est? Anne author Epistola seipsum hic citavit. 12 Accusationem probare debituit. Non probata pro calumnia haberi debet. 13 Rectam methodum Wallisius noster e literis nostris anno 1692 ad eum missis in secundo Operum volumine anno 1693 impresso descripsit. Eandem in libro de Qua Proportione prima Libri de Qua olim compositis de Quadraturis habetur. 13 Methodum veram Leibnitius ipse non habuit Anno 1689 ubi scripsit Tentamen suum de motuu motuum cœlestium causis. Post annum 1690 methodus illa cœpit a Leibnitianis intelligi. Anno 1696 Dn. Marchio Hospitalius eandem edidit. Methodum veram Wallisius a me accepit anno 1692 & anno proximo in secundo Operum Volumine an edidit. Eadem describitur in Propositione prima libri de Quadraturis olim scripti. A fluxionibus secundis A momentis secundis multæ Propositiones in Libris Principiorum pendent. & Hujus generis est determinatio Problematis Kepleri Propositionis Keplerianæ quam anno 1677 inveni. Dixi etiam in Epistola 10 Decem 1672 me per ad Collinum data methodum de qua ibi locutus sum, id est per methodum fluxionem curva ad curvaturas cCurvarum invenis se extendere,. Hæ autem pendent a fluxionibus secundijs. etiam Regulam utiqutiutique Fluxionisūum utique primasrūum intelligenti Regula secundarum latere noti potest cum sit eadem cum Regula primarum ut Propositionem primam Libri de QuadurisQuadraturis consulenti statim patebit 21 complaint that Newton in his Principles & book of Quadratures had used fluxions for the differences of Mr Leibnitz the inventor of ean thor as Honoratus Faber substituted progressions of Motion as for the method of Cavellerius. There have hitherto been printed only a few copies of the Commercium Epipstolicum relating to the infinitesimal method The Commercium Epistolicum relating to the Infinitesimal method 1. It was hoped that a few copies of the following Commercium EpistoliumEpistolicum printed off & sent sen printed off & sent to Mathematicians who were able to judge of these things, But might hit have silenced the dispute about about the dispute about the Infinitesimal Method the complaint that Mr Newtons book of Quadratures was a peice of Plagiary: but Mr Leibnitz & his friends have d declined answering it, & endeavoured to run the dispute into a wrangle & squabble about other matters such as occult qualities, miracles, gravity, attraction sensoriums, the perfection of the world, a vacuum atoms the solving of problems & the like: all wchwhich are nothing to the purpose which are nothing to the purpose. & some of them still continue to squabble And tho Mr Leibnitz is dead, yet this squabbling humour is still continued by some of his friends. And therefore it has been thought fit to publish this Commercium Epistolicum together wthwith the given Account given thereof in the Philosophical Transactions for Ianuary 171

\frac{4}{5}

& so leave it to posterity to judge of this matter by the ancient Records. 3 Against the credit of these Records it has been objected that the Committee of the R. Society published every thing that made against Mr Leibnitz & omitted every thing that made against Mr Newton. In my s But Thise objection was only a pretence made by Mr Leibnitz but had nothing more in it then clamour. For in his a Postscript of his first Letter to Abbe Conti, for proving this he wrote that in his second voyage into England Mr Collins shewed him part of his correspondence & he observed there that Mr Newton avowed acknowledged his ignorance in many things & said (among other things) that he had found nothing about the dimension of the celebrated curvesilinears besides the dimension of the Cissoid But saith he they have supprest all this. And Mr Newton in his Letter of to Abbe Conti dated 26 Feb. 171

\frac{5}{6}

replied that this passage was not suppressed but published in the Commercium pag 74 lin. 10 & 11. Mr Leibnitz in his Answer written to Abbé Conti Apr. 9 1716 acknowledged that he had been mistaken, but saith he, I'le cite another instance. Mr Newton affirmed acknowledged owned in one of his Letters to Mr Collins that he could not find the content of second sections (or second segments) of Spheroids or such like bodies: but they have not inserted this passage or Letter in the Commercium Epistolicum & that the Commmitte had omitted that. And Mr Newton in his Remarks upon the Letter of Mr Leibnitz replyed that whether he (Mr Newton) could solve that Problem or not was nothing to the purpose point in question, & that if the Committee had omitted it they would have done right, it being nothing to the point in question, but on the contrary, Mr Collins in a Letter to Mr Bertet Iames Gregory yethe 24th of December 1670, & in an other to Mr Bertet the 21th of Feb. 1671, both printed in the Commercium Epistolicum pag. 254, 26, wrote that his metho his (Mr Newtons) method extended to second segments of round solids. And that Mr Oldenburg wrote the same thing to Mr Leibnitz himself the 8th of of December 1674. See the Commercium Epistolicum pag 39. Another thing objected againste the credit of those Records ha was this So that the accusation [amoounts to nothing more than a Cavil] was groundless & amounts to nothing more then a piece of railery. In a flying paper dated 29 Iuly 1713 Mr Leibnitz called in question the letter of Mr Oldenburg sent to him with 8 or 9 series & the Answer to it in the han date by wchwhich he [Mr Leibnitz acknowledged the receipt. of that Letter. 2 The Letters of Mr Leibnitz to Mr Oldenburg are wthwith Mrare dated 3 Feb. 20 Feb. 30. Mar. 26 Apr. 24 May. 8 Iune, 1673; 15 Iuly, 26 Octob. 1674; 30 March, 15 Apr. 20 May, 12 Iuly, 28 Decemb. 1675; & 12 May, 27 OAug. 18 Novem 1676 & 21 Iune & 12 Iuly 1677. And these are were are all except the 3d & the the ferive last entered entered in in the Letters books of the R. S. No 6 pag 35, 34, * 101, 115, 137; & No 7 pag. 93, 110, 213, 235, 149, 189, And the Originals of all these letters except except that of 27 Aug. 1676 wchwhich was published by Dr Wallis, & that of 18 Novem following are stilll extant in the hand writing of Mr Leibnitz. several of these letters were also published by Dr Wallis as The Letters of 15 Iuly & 26 Octob. 1674; 12 Iuly & 28 Decem. 1675 & 21 Iune & 12 Iuly 1677 were also published by MDr Wallis. And all these Letters together wthwith the Letters of Mr Oldenburg to Mr Leibnitz wchwhich are copies of wchwhich are still preserved make up the continued correspondence between Mr Leibnitz & Mr Oldenburge from Feb 3 167

\frac{2}{3}

to the death of Mr Oldenburg wchwhich in excepting that the Letter by wchwhich Mr Leibnitz desired the conception out a collection of Gregories Letters & that of Mr Oldenburg to Mr Leibnitz wchwhich accompanied those Excerpta by wchwhich he sent to Mr Leibnitz that collection together wthwith Newtons Letter of 13 Iune 1676 & that of Mr Collins dated 14 Iune 1676. The Letters of Mr Leibnitz ran upon numbers till Iune 8 1673. Then Mr Leibnitz began to study the higher GeomGeometry & the correspondence ceased both for a time. And the next year in Iuly 15 Mr Leibnitz renewed the correspondence beginning his Letter of Iuly 15th with these words Diu est quod nullas a me habuisti litteras. And from thence forward the correspondence continues without intermission to the death of Mr Oldenburg except that the aforesaid two Letters are wanting [the latter of which is extant but was not published?] having nothing in it material to the controversy.] So then the Commercium Epistolicum of Mr Oldenburg wthwith Mr Leibnitz the English & Mr Collins Mr Oldenburg Mr Collins & l Mr Newton is here printed entire except those two Letters. And For Mr Collins & Mr Newton had no correspondence with him except only by means of Mr Oldenburg Leibnitz. 4 When the Commercium Epistolicum came abroad wchwhich was in the end of the year 1712. Mr Leibnitz to avoid answering it pretended ding all the two next years that he had not seen it, but had desired an Eminent Mathematician to examin it & nor was at leasure to examin it, & therefore but had referred it the judgment of an able & impartial Mathematician. And the judgment of this Mathematician dated 7 Iune 1713 was inserted into a scurrillous paper dated 29 Iuly 1713 & deispersed over all Europe, without telling the name of the Mathematician or that of the Printer or City where it was printed. [But in the end of the year 1715 & beginngbegining of the year following Mr Leibnitz to give the per judgment of the Mathematician more credit began to tell the world that it was Mr Leibnitz Iohn Bernoulli & to make this probable he translated thise Letter of 13 Iune 1713 into French & sent the transla in the translation omitted the sentence quemadmodum ab eminente quodam Mathematico dudum notatum est, by wchwhich words the author of the Letter cited Mr Iohn Bernoulli as a person different from himself.] This And this translati [And this translation he sen inserted into his letter of to Madam Pelniz & Madam Kilmanseg dated Apr 18 Apr. 1716 & caused it it to be printed in Holland Decem. 1715] 6 In this flying paper it is said that prickt letters did not appeared first in the third Volume of Dr Wallis his works. H And that Mr Newton did not understand how to find second differences till it was commonly understood by others. But this is a mistake misrepresentation. The third Volum e came abroad in 1699. But prickt letters appeared in the second Volume wchwhich came abroad in Spring 16983. In this Volume Dr Wallis published the first Proposition of Mr Newtons book of Quadratures illustrated with examples in prickt lettersfirst & second differences. And this is the first line that any Rule came abroad for finding 2d, 3d, & fourth differences & & at & fift differences & so on. He Dr Wallis published also at the same time the fift Proposition of the Book of Quadratures & reprentedrepresented that he had received these things from Mr Newton by Letters dated Aug 27 & Sept 17 1702. This Proposition depends upon the four preceding Propositions & therefore the first five Propositions of that Book were in being in the year 16 17921692. This Book in MS was in the hands of MDr Halley & Mr Ralpson in the year 1691 as Mr Ralpson has publickly attested & Dr Halley still attests. And this was before the differential method came abroad began to make a noise abroad & may suffice to clear this Book from the accusation of plagiary wchwhich gave a beginning to these disputes. In the XIVth Proposition of the second Book of Principles you have an instance of solving Problemes by second Differences. And this was the fore the differential is the first instance of that kind made publick. It is true that Mr Leibnitz was the first who published a Rule for first differences. But he should then have acknowledged (as he did in his Letter of 21 Iune 1677 written to Mr Oldenburg) that he knew by Letter Mr Newtons Letters that Mr Newton had the a Method wchwhich did the same things. Or rather, he should have acknowledged that Mr Newton gave him light into the method. For Dr Wallis in the Preface to the two first 22 first volumes of his works published in spring 1695 wrote that the method of fluxions & differential Calculus is one and yethe same d method differing only in some modes of expression & that Mr Newton in his Letters of 13 Iune & 24th Octob 1676 expainedexplained to Mr Leibnitz this Method found by him ten years before that time or above; that is, in the year 1666 or before. The editors of the Acta Eruditorum the next year in giving an account of these two Volumes cited a sentence out of it this Paragraph & therefore knew of it. And Dr Wallis gave notice of it to Mr Leibnitz himself b in a Letter to him dated & printed in the third Volume of his works The Original Letters & & B Letter Books wh from whence the Commercium Epistolicum was printed are still kept in the Archives of the R. S. & [have been sometimes shewed to strangers & some of them compared with the Comercium Epistolicum. [And particularly the Letter of Mr Oldenburg dated Apr 15 1675, & the Answer to it in the handwriting of Mr Leibnitz dated 20 May have been have been viewed & collated with the print by many strangers some of wchwhich & coll altogether some of whom knew the hand of Mr Oldenburg.] & collated by them collated with the print. & no question is made in England of their sincerity Mr They have also upon occasions been shewed to strangers [& some by of them been collated with some of the printed Letters before the by who have viewed them & collated some of them with the printed Letters to their satisfaction. It has been ObjctedObjected with clamour that Mr when Mr Newton wrote the Scholium at the end of the Book of Quadratures he did not understand second differences: but the Objector knew that by the first Proposition of the Book ytthat he did understand second differences. An Account of what has been done since the publishing of the Commercium. The Commercium Epistolicum was published before the end of the year 1712 & hno answer has hitherto been given to it: for it is unanswerable. Mr Leibnits to avoid answering it p p pretended the Mr Leibnitz pretended the two first years that he had not seen the Book being then at Vienna he had not seen the Book, but had written to an able & impartial Mathematician to give his judmentjudgment upon it. And the Answer or pretended answer of the Mathematician dated 7 Iune 16713 was inserted into a fl scurrilous paper dated 29 Iuly following & printed without the name of the author or printer or Mathematician or place where it was printed & the paper was dispersed over Europe: & it was two years & an half before Mr Leibnitz began to tell the word that the Mathematician was Iohn Bernoulli. The Letter T ascribed to him was in these words. Videtur N...sis occasionem nactus serierum opus multum promovisse per extractiones radicum, quas primus in usum adhibuit et quidem in ijs excolendis ut verisimile est ab initio omne suum studium posuit; nec credo — — — alijs fuisset familiaris &c Thus far the Mathematician. His first argument against Mr Newton is founded upon a ridiculous supposition that prickt letters are essential to the method of fluxions, wchwhich is not so. In the Introduction to the Book of Quadratures heMr Newton teaches the method of fluxions & yet there is illustrates it with examples & yet there is not a prickt letter in all that Introduction. In the second Lemma of the second Book of Principles Mr Newton demonstrated the Elements of the Method of fluxions: & yet there is not a prickt letter in all that Lemma. In his Letters of Octob. 24. 1676 10 Decem 1673 & 24 Octob 1676 he d wrote that he had a general Method wchwhich readily gave the method of of solving Problems of wchwhich the method of Slu Tangents of Slusius was a Corollary & wchwhich stuck not at surds & was founded in the latter of these two letters he added that this foundation of this Method was founded in the sentence Data æquatione fluentes quotcunque æquata quantitates involvente fluxiones invenire & vice versa. And will any impartial man affirm that when Mr Newton wrote those two Letters, he did not so much as dream of the method of Fluxions because there deducing fluents from fluxions & fluxions from fluents because there are no prickt letters in those two Letters. In his Book of Quadratures he used prickt Lett letters, in his o In h Sometimes he used prickt letters & sometimes letters without pricks or other symbols & sometimes he wrote of it wthwithout any symbols. What symbols he used in his Analysis per series numero terminorum infinitas has been shewed above. There he used the symbol

\frac{a a}{64 x}

in the very same The MathematianMathematician affirms same sence in wchwhich Mr Leibnitz uses the symbol

\int \frac{a a}{64 x}

, & therefore in those days he dreamt of the summatory method before Mr Leibnitz wchwhich is the inverse of bing the differential. And when Mr Leibnitz was in England the second time & saw in the hands of Mr Collins many he saw my Lette Mr Newtons Letter of 24 Octob. 1676 as he has confest in his Letter of in wchwhich this Analysis is mentioned by the name of Compendium serierum, & in the hands of Mr Collins he saw many of the Letters of D Mr Newton & Mr Gregory & had especially that wchwhich ran principially upon series & had then an opportunity to ask for this Compendium serierum & see this symbo. The Mathematician says further that Mr Newton in his Principles of Philosophy had frequent occasion to use prict letters & yet used them not: but the truth is, he had no occasion to use them in that book. For that book is written by composition after the manner of the Ancients: but yet the Analysis by wchwhich it was invented shines through the composition.; & this was acknowleged formerly both by the marq. de l' Hospital & by Mr Leibnitz himself as has been mentioned above The second argument of the Mathematician against Mr Newton to prove that the calculus of fluxions was not older then the differential Calculus is The Mathematician saith further that prickt letters appeared first in the third volume of the works of Dr Wallis many years after the differential calculus had obteined every where. But he hath misrepresented the matter matter. Prickt letters came abroad The third Volume of Dr Wallis Works came abroad in yethe year 1699 & prickt letters appeared in his second volume wchwhich came abroad in spring 1693, wchwhich was two years before Dr Wallis heard that the Differential Method began to be celebrated abroad The second argument of the Mathematician to prove that the calculus of fluxions was not older then the differential calculus is that Mr Newton did not understand how to find the fluxion of fluxions or difference of differences. And yet if you be the Mathematician knew by the first Proposition of the Book of Principles Quadratures but Newton did know understand how to find the difference of differences or second difference & the difference of second differences or third difference & so on in infinitum. And if you look into the Book of Principles lib. 2 Prop. 4 you will see that he the there knew how to find the difference of moments or second difference. And I do not meet with any ancienter instance made publick of working in second differences. 23 Pag. 1. lin. 20. — prima (p. 37, 38.) Add: Et symbola Leibnitij nondūum obtinuerunt in Anglia. Newtonus Tractatum suaum de Quadraturis edidi die ante quam ederetur dudum scrip tusits erant quippe Nam plurima ex eo citata sunt in Epistolis 24 Octob. & 8 Novem 1676 (p. 34) In Epistola priore ponuntur recitantur Ordinatis Curvarum quarum collationem cum Sectionibus Conicis Newtonus tum ante annos quinque (seu (seu anno 1671) in Tabulam retulerat (pag. 178. In Epistola posteriore anno 1711) a Ionesio edita Corollarium secundum Propositionis decimæ verbis fusius enarratur, his verbis. Nulla extat Curva cujus Æquatio ex tribus constat terminis — haud tamen adeo generaliter. Hæc sunt omnium quæ in Libro de Quadraturis habentur difficilima & ababsabsque methodo fluxionum aut simili inveniri non potuerunt & abunde satis demonstrant, methodum Fluxionum & Momentorum quatenus in illam quatenus in in Propositionibus decem primis Libra illius habetur, Newtono innotuisse anno 1676 & aut & antea. Object. 1. Newtonus introducist attractiones & vult grav i.e. qualitates occultas & vult attractionem gravitatem est attractionem id est qualitatem occultam ejusmodi esse. Resp. Quo sensu Newtonus voce attractionis utitur, ostenditur ex ejus supra (pag. 35, 36): Princ libro Principiorum: & plenius patet ex sequentibus. In Introductione ad Sect. XI Lib. 1 Newtonus hæc habet verba sic scribit. Qua de causa jam pergo motum exponere corporum se mutuo trahentium, considerando vires centripetas tanquam aAttractiones, quamvis fortasse si physice loquamur, verius dicantur impulsus. In Mathematicis jam versamurs, & propterea missis disputationibus Physicis, familiari utimur sermone, quo possimus a Lectoribus Mathematicis facilius intelligi. Et in petulo post in Scholio S sub finem hujus Sectionis: Vocem Attractionis hic generaliter usurpo pro corporum conatu quocunque accedendi ad invicem sive conatus ille fiat per ab actione corporum vel se mutuo petentium, vel per spiritus emissos se agitantium, sive is ab actione Ætheris, aut Aeris Medijve cujuscunque seu corporei seu incoprporporeiincorporei oriatur corpora innotantia in se invicem utcunque impellentis. Eodem sensu generali usurpo vocem Impulsus, non species virium & qualitates Physicas sed quantitates & proportiones Mathematicas in hoc Tractatu expendaris, ut in Definitionibus explicui In Mathesi investigandæ sunt virium quantitates & rationes illæ quæ ex conditionibus quibuscunque positis consequuntur: deinde ubi in Physicam descenditur, conferendæ sunt hæ rationes cum Phænomenis ut innotescat quænam virium conditiones singulis corporum attractivorum viribus competant Object. 2. Newtonus asserit Deum habere sensorium Resp. Newtonus per Sensorium nihil aliud intelligit quam spatium infinitum tanquāam locum sensetis in quo Deus omnia sentit. Sic enim liquitur de Deo: Is totus est sui sui similis, totus oculus, totus auris, totus cerebrum, totus brachum, totus vis sentiendi intelligendi et agendi: sed more minime humano, more minime corporeo, more nobis prorsus incognito. Vt cæcus non habet idæam colorum, sic nos ideam non habemus modorum quibus Deus sentit & intelligit omnia. Corpore omni & figura corporeo destituitur: ideoque videri non potest, nec sub specie rei alicujus corporei coli debet. Object. 3 Newtonus introducit miracula in Philosophiam Resp. Leibnitius vocat miracula, quæ quotidie eveniunt nemine admirante. 24 And whereas he claimed one of the methods of regression wchwhich Mr Newton sent him Mr Newton sent to him at his own request Mr Newton sent to him & wchwhich upon the first reading he did not know to be his own nor understand he claimed afterwards claimed as his own invention by pretending that he had had forgot it till he found it in one of his old Papers: it lies upon him in point of cadourcandour & justice either to prove thethat he had found it before Mr Newton, Or, & forgot it before he wrote to Mr Newton for it or else publickly & in express words to renounce his claim to it for preven for preventing future disputes about it. It is therefore expected that Mr Leibnitz de renounce all right to this method as first Inventor, as well as to the Differential method of Mouton as second Inventor. For second Inventors have no right. The sole right is in the first Inventor untill another finds out the same thing apart: in wchwhich in which case & then to take away the right of the first Inventor & divide it between him & that other, would be an Act of Injustice. Endeavours have been used to make the learned world beleive that Mr Newton has been introducinesg occult qualities into Philosophy, & the Editors of the Acta Leipsiensia have published that Mr Newton asffirms that gravity cannot be explained mechanically. And if this be done to prejudice the Cartesians & such other Philosophers as place all natural Philosophy in Mechanical hypothetical explications of Phænomena by mechanism, at, against Mr Newton, it is an indirect way of proceeding & shews the w Mr Leibnitz in one & the same Letter (his Letter of 1711 I has opposed his own candor to the arguments of Dr Keill as if it were injustice to question it, that is to refuse his own testimony for himself, & td questioned in the same Letter he has in an arbitrary manner without who much as one argument cal to justifie himself called the candor of Mr Newton for publishing that he found the diff method of fluxions gradually in the years 1665 & 1666; & now it mutually comes upon him to satisfy the worl satisfy the world about his own candor & integrity in the ten or twelve particulars last above mentioned. Nec lex est justior ulla. And whereas he has in his Theodicee accused Mr Newton of making gravity an occult quality & a miracle & thereby introducing & of thereby introducing occult qualities & miracles into Philosophy thereby such opinions & yet Mr Newton has no where declared his any opinion about gravit the cause of gravity: or it lies upon him in point of candor & justice to beg Mr Newtons pardon in publickly for endeavouring by theiss indirect such indirect & unfair practises to prejudice the world against him. The Philosophy wchwhich Mr Newton in his Principles & Opticks has pursued is experimentall & meddles not wthwith therefore it & experimental philosophy teaches not it is not the business of Experimental Philosophy to teach the causes of things any further then they can be proved by experiments. And if Mr Leibnitz would forbear to trouble the world with his opinions whereas Mr Leibnitz is of opinion that nothing is done by election without a reason, or in other words that that all things are governed by fate & necessity & that mak nothing more then matter & motion is requisite to produce all the Phænomena in nature & that man himself is a meer machine; untill he can prove these things by experiments, his Philosophy would be of better credit. God is a supramundane intelligence & the souls of men do not act upon their bodies & that but all animal motion is performed without any other cause than mechanism if opinion is & man himself is a meer machine & God is a supramundane intelligence: if he would forbear to trouble the world with such irreligious opinions & teach nothing more then untill he can prove them conjectures & teach nothing more then he can be Proved by experiments his Philosophy would be of better credit. And whereas Mr Newton sent him at his own request a method of regression wchwhich upon the first reading he did not know to be his own nor understand, but so soon as he understood it he claimed it as invented his own by pretending that he found it had forgot it untill he found it amongst his in on in his old papers: it lies upon him in point of candor & justice to prove either to prove that he was the first inventor of this method, or to renounce his claim to it for preventing future dispute about it. None fuller of Miracles & occult qualities than that. None more occult mysterious and miraculous & trifling than that wchwhich [destroys all the arguments for a Deity taken from Phænomena &] & supposes every thing, proves nothing, & excludes the proof of a Deity from phænomena. Mr Leibnitz in his Theodicee has accused Mr Newton of introducing occult qualities & miracles into philosophy upon a supposition that he make gravity an occult quality miraculously seated in the will of God. And yet Mr Newton has writte in his Optiqus (Quest 22 has written expresly, to the contrary. Quam ego attractionem appello, saith he Qua causa efficiente hæ attractiones [gravitatis, virtutisque magneticæ et Electricæ vel alterius cujuscunque attractiones], hæ peragantur, saith he, in id vero hic non inquiro. Quam ego attractionem appello, fieri sane potest ut ea efficiatur impulsu vel alio aliquo in universum solummodo vim aliquam significare intelligatur qua sit illa vis. Nam ex phænomenis Naturæ illud nos prius edoctos leges & proprietates illstius attractionis; quam Et in id inquirere par sit quanam efficiente causa peragatur attractio. And a little after he mentions the same forces attractions as forces wchwhich by phænomena appear to have a being in nature & tho their causes be not yet known & distinguishes them from occult qualities wchwhich flow from are supposed to flow from the specific forms of things And in the beginning of his Principia Philosophiæ speaking of the centripetal forces & attractions he saith: Virium causas et sedes physicas his jam non expendo. And a little after: Voces autem attractionis impulsus vel propensionis in centrum cujusque cujuscunque in centrum, indifferenter & pro se mutuo promiscue usurpo, has vires non Physice sed Mathematice tantum considerando. Vnde cæveat Lector ne per hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem physicam alicubi definire, vel centris (quæ sunt puncta physica mathematica) vires vere et physice tribuere, si forte aut centra trahere aut vires centrorum esse dixero. And whilst Mr Leibnitz has taught that God is intelligentia supramundana & that all animal motion (even that of man) is purely mechanical, & that there no election of any thing without a reason, that is, none all at all, but all things are by fate & necessity: it lies upon him to satisfy the world that it may be proved by phænomena that there is a God in whom we live & move & have our being. 25 And whereas Mr Newton sent him at his own request a method of Regression, wchwhich upon the first reading he did not know to be his own, nor understood it, but so soon as he understood it he claimed as his own by pretending that he had found it long before, & had forgot it, as he peceivedperceived by his old papers: it lies upon him in point of candor & justice, laeither to prove that he was the first inventor of this method, or to renounce his claim to it for preventing future disputes. It is therefore expected that Mr Leibnitz do renounce all right to this method as first inventor, as well as to the Differential Method of Mouton as second Inventor. For second Inventors have no right. The sole right is in the first Inventor untill another finds out the same thing apart: in which case to take away yethe right of the first Inventor & divide it between him & that other would be an Act of injustice. And whereas he has in his Theodiceus accused Mr Newton of making gravity an occult quality, & & an immediate Act of the Deity wchwhich he calls a miracle immediately seated in the will of God & thereby of introducing occult qualities & miracles into Philosophy; & yet Mr Newton has no where declared any such opinion ror any opinion about the cause of gravity or about the cause of any other sort of attractions: but speaks of them as forces whose causes are not yet known: it lies upon Mr Leibnitz in point of candor & justice, to beg Mr Newton's pardon publickly for endeavouring by such indirect & unfair practises to prejudice the learned part of Europe against him. The Philosophy wchwhich Mr Newton in his Principles & Opticks has pursued is experimental, & it is not the business of experimental Philosophy to teach the causes of things any further then they can be proved by experiments. And the Philosophy of Mr Leibnitz would be of better credit freer from miracles & of better credit if he would prove his Propositions by experiments & be silent where experiments fail him. No Philosophy can be freer from occult qualities mysteries & miracles then that in wchwhich Propositions are proved by experiments, & wchwhich is silent which where experiments are wanting, is silent, or at the most speaks doubtfully, & by way of inquiry leaves the matter to further inquiry. None more trifling then that wchwhich supposes every thing & proves nothing. ✝ but on the contrary declared his opinion very expresly about it in these words. For this r For this reason Mr Newton is silent about the causes of gravity, & has there occurring no experiments or phænomena by which he can discover it prove it appered it could be proved what it is was the causes thereof. And this he has abundantly declared In his Principles, (nd neare the beginning, in these words: Virium causas et sedes Physicas jam non expendo. And a little after: Voces attractionis impulsus vel propensionis cujuscunque in centrum, indifferenter & pro se mutuo promiscue usurpo, has vires non physice sed Mathematice tantum considerando. Vnde caveat Lector ne per hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem physicam alicubi definire vel centris (quæ sunt puncta mathematica) vires vere & physice tribuere si forte aut centra trahere aut vires centrarum esse dixero. And in the end of his Opticks,Quest 22.: Qua causa efficiente hæ [gravitatis, attractiones [sc. gravitas visque magnetica et electrica aliæque] attractiones peragantur, in id vero hic non inquiro inquiro. Quare ego attractionem appella, fieri sane potest ut ea impulsu efficiatur impulsu vel alio aliquo modo nobis incognito. Hanc vocem attractionis ita hic accipi velim ut in universum solummodo vim aliquam significare intelligatur qua corpora, ad se mutuo tenduent, cuicunque demum causæ attribuenda sit illa vis. Nam ex phænomenis naturæ illud nos prius edoctos oportet quænam corpora se invicem attrahant & quænam sint leges & proprietates istius attractionis; quam in id inquirere par sit quanam efficiente causa peragutur attractio. And a little affter, he mentions the same attractions as forces wchwhich by phænomena appear to have a being in nature & may be made usefull in explaining Phænomena tho their Nature tho their causes be not yet known & which may produce by the causes of various effects & distinguishes them from occult qualities wchwhich are supposed to flow from the specific forms of things. But And in the Scholium all the end of his Principles, after he has mentioned the properties But of gravity he adds: Rationem vero harum gravitatis proporietatum ex PhæmenisPhænomenis &c . . . . non habent. Notwithstanding all this Mr Leibnitz has accused Mr Newton of making gravity an occult quality & a miracle an immediate act of the Deity God wchwhich act he calls a miracle. And therefore It lies upon him therefore in point of candor & justice to beg Mr Newton's pardon publickly for endeavouring by such indirect & unfair practises to prejudice the learned part of Europe against him. And whereas theis this Philosophy tends to destroy of Mr Liebnitz is thought by some to weaken destroy weaken all the arguments for a Deity taken from Phænomena, it lies upon him in point of religion to beg pardon of all mankind who beleive a Deity God either to show that his Philosophy is not of such a nature & to declare that he is not offended wthwith Mr Newton for insisting upon such arguments, or to beg pardon of all mankind who beleive a God And yet yethe Editors of the Acta Eruditorum have accused him of denying wthat yethe cause of gravity is mechanical tho he is silent about that cause & have compared an Agent wchwhich he calls a subtile Spirit to Dr H Mores hylarchic Principle. tho he is silent about the nature & cause of that Agent, & Mr Leibnitz has accused him of making Gravity an occult quality & a miracle. The Philosophy which Mr Newton has published in his Principles & Opticks has pursued is experimental, & it is not the business of experimental Philosophy to teach the causes of things any further then they can be proved by experiments. We are not to fill this Philosophy wthwith opinions wchwhich cannot be proved by Phænomena. C I In this Philosophy Hypotheses have no place unless by way Quære or as as conjectures or Questions proposed which may deserve to be examinatiedon. for this r by experiments. We are not to fill this Philosophy with opinions wchwhich cannot be proved by experiments. For this reason Mr Newton in his Optiques distinguished those things wchwhich were made certain by experiments from those things wchwhich remained uncertain & wchwhich he therefore proposed in the end of his Opticks in the form of Queres. For this reason isn the Preface to his Principles, when he had mentioned the Motions of the Planets Comets Moon & Sea as deduced in this book from gravity, he added: Vtinam cætera Naturæ Phænomena ex Principijs Mechanicis eodem argumentandi genere derivare liceret. Nam multa me movent ut nonnihil suspicer ea omnia ex viribus naturæ quibusdam pendere posse quibus corporum particulæ per causas nondum cognitas vel in se mutuo impelluntur & secundum figuras regulares cohærent, vel ab invicem fugantur & recedunt: quibus viribus ignotis Philosophi hactenus Naturam frustra tentarunt And in the end of this book in the second Edition, he said that he forbore to describe the effects of this attraction for want of a sufficient number of experiment to determin the laws of its acting. And for the same reason he is silent about the cause of gravity, there occurring no experimtsexperiments or phænomena by which he might prove what was the cause thereof. And this he hath abundantly declared in his Principles, neare the beginning thereof, in these words: Virium causas et sedes Physicas jam non expendo. And a little after: Voces attractionis, impulsus, vel propensionis cujuscunque in centrum, indifferenter & promiscue se mutuo promiscue usurpo, has vires non physice sed Mathematice tantum considerando. Vnde caveat Lector ne per hujusmodi26 hujusmodi voces cogitet me speciem vel modum actionis causamve aut rationem phyiscam alicubi definire vel centris (quæ sunt puncta mathematica) vires vere et physice tribuere, si forte aut centra trahere aut vires centrorum esse dixero. And in the end of his Opticks: Qua causa efficiente hæ attractiones, [sc. gravitas visque magnetica et electrica aliæque] peragantur, hic non inquiro. Quam Ego attractionem appello, fieri sane potest ut ea efficiatur impulsu vel alio aliquo modo nobis incognito. Hanc vocem attractionis ita hic accipi velim ut in universum solummodo vim aliquam significare intelligatur qua corpora ad se mutuo tendant cuicunque demum causæ attribuenda sit illa vis. Nam ex phænomenis naturæ illud nos prius e doctos oportet quænam corpora seinvicem attrahant, & quænam sint leges & proprietates istius attractionis; quam in id inquirere par sit quanam efficiente causa peragatur attractio. And a little after he mentions the same attractions as forces by wchwhich by Phænomena appear to have a being in nature tho their causes be not yet known, & distinguishes them from occult qualities wchwhich are supposed to flow from the specific forms of things. And in the Scholium at the end of his Principles after he had mentioned the properties of gravity, he added: Rationem vero harum gravitatis proprietatum ex Phænomenis nondum potui deducere, & hypotheses non fingo. Quicquid enim ex phænomenis non deducitur Hypothesis vocanda est; & Hypotheses seu Metaphysicæ seu Physicæ seu Qualitatum occultarum seu Mechanicæ in Philosophia experimentali locum non habent. — Satis est quod Gravitas revera existat et agat secundum leges a nobis expositas, et ad corporum cœlestium & maris nostri motus omnes sufficiat. And after all this one would wonder that Mr Newton should be reflected upon for not explaining the cause of gravity by an Hypothesis, as if it were a crime to content himself with certainties established by experiments & phænomena & let uncertainties alone. And yet the Editors of the Acta Eruditorum have accused him of denying that the cause of Gravity is mechanical tho he is Mr Leibnitz silent about it & of introducing a spirit that cause, & Mr Leibnitz has accused him ✝✝ In principio Theodicæ & in Epistolis quibusdāam of making gravity an occult quality & a miracle It lies upon him Mr Leibnitz therefore in point of candor & justice to beg Mr Newtons pardon publickly for endeavouring by such indirect & & unfair & tricking practises to defame him & prejudice the learned part of Europe against him. In the Acta ErudorumEruditorumErudorum for March 1714, the pag. 1402 & t its said that Mr Newton denies that the cause of gravity is mechanical; & yet he neither affirms nor denys nor affirms it, in any of his writings, nor gives any opinion about it. Its said also that he lays down a certain new Hypothesis concerning a subtile spirit perrading the pores of bodies, perhaps the same with the Hylarchic principle of Dr More Henry More, or some other spirit of less value then Hypotheses unless it be the Æther or subtile matter of the Cartesians. It appears by certain experiments tryed by Mr Hawksby before the Royal Society that electric bodies attract constantly at small distances even without rubbing. The Agent by which this attraction is performed Mr Newton calls a subtile spirit,. And because he does not explain it by an exp Hypothesis, the but leaves its nature cause & properties to be discovered by experiments; they turn his silence to ridicule & make Hypotheses for him. And by such like representations they endeavour to persuade the world that in point of Philosophy he is much inferior to Mr Leibnitz & in point of Mathematicks not only to Mr Leibnitz but also to Mr Bernoulli & Mr Tschurnhause, & therefore not likely to be the first inventor of the method of Fluxions. And whereas Mr Leibnitz calls God tells us that God (the God in wichom we live & move & have orour being) is Intelligentia supramundana, & under the applys the name of miracles not only to extraordinary & supernatural events but even to the laws of nature if imprest upon her by the will of God, & under the name of miracles excludes all thise actions from being ca of the first cause from being considered in Nature experimental Philosophy, & teaches that nothing happens without a reason or in other words that there is nothing done by election choise but all things are governed by fate & in his Hypothesis of an Harmonia præstabilita asserts that the soul acts not upon the body, but that all animal motion, even in man himself, is performed by meere mechanism it lies upon him in point of religion either to satisfy the world that he has not been at work to his philosophy doth not tend to enervate & undermine & enervate & explode all the arguments for a supreme governour of the Vniverse taken from Phænomena; or else to satisfy beg pardon for this of all makind mankind who beleive a Deity God [& that he beleives that Man is some animal motion in Man is not merely mechanical.] Extraordinary events wchwhich cannot be referred to the natural causes are not within the limits of Natural Philosophy: but] & that it not in pursuit of this designe that he make him cry out against Mr Newton cries out against Mr Newton for introducing occult qualities & miracles into Philosophy, & & applys the name of miracles not only to extraordinary or supernatural events but even to the laws of Nature if imprest upon her by the will of God. Anno Christi finiente Apogæum Solis Motus medius Longitudo Solis ab Ap lelogæo Motus medius Solis ab Apogæo. Motus medius ☉is ab Æquinoxio.

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Here he represents that he knew nothing by Mr Newton's Letters but that he had a certain method of Tangents, but but when his Principles came abroad he undo abundantly understood that he had gone much further in these matter methods: now he But he is since gone back from what he acknowledged here & tells us that the Principles are written in the manner of the ancients & have nothing of the new Analysis in them, nor make it appear that Mr Newton knew any thing of these methods when he wrote that book. It lies upon therefore in candor & justice to acknowledge therat when Mr Newton wrote his Principles he had a method of resolving such difficult & valuable difficillima et pulcherrima quæque etiam mistæ Matheseos Problemata quæ sine sine sine calculo differentiali AVT SIMILI non temere quisquam pari facilitate tractabit. For that book is full of such Problemes & therefore could not be invented in the jaudgment of Mr Leibnitz himself could not be written have been invented sine cCalculao Differentiali AVT SIMILI It lies upon him also in candor & justice to acknowledge that theat before he wrote his Letter of 21 Iune 1677 in wchwhich he began first to mention describe propose the Differential Method, he did understand by Mr Newtons three Letters above mentioned, that when he Mr Newton in those days had a method of drawing tangents to Curves determining maxima & minima, drawing tangents to Curves analytical or mechanical, determining the & finding & the tangents lengths areas, lengths sol curvities, solid contents, centers of gravity of curves & curvilinear figures analytical or mechanical & this without taking away surds, & that this method in conjunction with gave the Areas of Curvilinear fuguresfigures in Series which in certain cases brake off & became finite, & compared figures wthwith yethe Conic sections & readily gave the Method of Tangents of Slusius as a branch or Corollary thereof, & gave also the Areas of Curvilinear figures by the help of finite & infinite equations together extended to inverse Problemes of Tangents & others more difficult & even to almost all Problems except perhaps some numeral ones like those of Diophantus. He is also to acknowledge that when Mr Newton told him how general his methods were he could not beleive it nobut forbear to in his Letter of 27 Aug. 1676 represented it improbable. Quod dicere vi [saying Quod dicere videmini plerasque difficultates (exceptis Problematibus Diophantæis) ad series infinitas reduci; id mihi non videtur. Sunt enim] And then it lies upon him to make the world understand why he did not upon publishing his own method he concealed all this & did not candidly tell his Readers what he knew of the MET Methodus SIMILIS & whose it was. & that it was invented before his own & gave in conjunction with Dr Barrows method of Tangents gave him light into his own, [And he is further to acknowledge that the Principia Philosophiæ is a Book full of such Problemes as were not to be invented witho sine calculo differentiali AVT SIMILI.] & seemed incredible to him when he was first told of it, & put him upon considering how to improve the methods of tangents then extant & particularly thatose of Slusius & Barrow wchwhich gave him light into it. For he tells us upon another occasion has told Dr Wallis in his Letter of 28 May 1697) that he found this method by observing that Differences answered to Tangents & summs to Quadratures. Epist. ad 27 And whereas Mr Leibnitz tells us that God (the God in whom we live & move & have orour being) is INTELLIGENTIA supra SVPRAMVNDANA, & applys the name of miracles not only to extraordinary & supernatural events but even to the laws of Nature if imprest upon her by the will of God, so that for atoms to be hard or heavy by the will of God without a mechanical cause would be a miracle & for atoms by consequence for atoms or any thing else to exist by the standing will of God would be a standing miracle & thus by the name of miralesmiracles excludes all the actions of the first cause from being considered in experimental Philosophy, & teaches that nothing happens without a reason, or in other words that there is nothing done by choise, but all things are governed by fate, & in his Hypothesis of an Harmonia præstabilita asserts that the soul of acts not upon the body but all animal motion, even in man himself, is performed by mere mechanism: it lies upon him in point of religion to satisfy mankind that his philosophy doth not tend to enervate & explode all the arguments for a supreme governor of yethe Vniverse taken from the Phænomena of nature; & that while he asserts that nothing happens without a reason, or in other words that there is nothing done by choise but all things are governed by fate, he doth not mean mechanically, & that his complaining of accusing Mr Newton for introducing occult qualities & miracles into Philosophy proceeds not from notions of this kind. It is true that Mr the Philosophy & M of these two Gentlemen differ very much. The one beleives teaches that God, (the God in whom we live & move & have our being) is every where Omnipresent: the other that he is INTELLIGENTIA supramundana SVPRAMVNDANA a God that's no where in the world. Whence it follows that he cannot act upon the world without a miracle The one beleives that animal motion in man is not purely mechanical; the other that it is., The one that the soul or mind (according to the Hypothesis of an Harmonia præstabilita) not never acting upon the body. The one teaches that a Philosophers are to prov argue from Phænomena to there first immediate laaest next causes thereof & from those causes to the next & so on till we come to the first causse: the other that all the actions of God are miracles & miracl & all the laws imprest upon Nature by the will of God are both perpetual miracles & occult qualities & therefore not to be considered by a Philosophers. But why must it go for a miracle or wonder if God has any thing to do with the world & why must all the arguments for a Deity taken from phænomena be exploded by such new hard-names? D Is it philosophical to rail & scold instead of arguing For certainly Philosophers are to reason without railing, & not to rail without reasoning. The one beleives that sees no reason why God may not chuse by the power of his will in things indifferent: the other teaches that nothing is done without a reason, meaning (I think) that all things are by fate. The one teaches that God (the God in whom we live & move & have orour being) is omnipresent: the other &c The one for want of experiments to decide the question doth not affirm whether the cause of gravity be mechanical or not mechanical the other that it is a miracle if it be not mechanical. The one by way of inquiry attributes it to the will of the creator that the least particles of matter are hard: the other attributes the hardness of matter to conspiring motions & calls it a miracle if the cause of this hardness be other then mechanical. The one doth not affirm that animal motion in man is purely mechanical: the other that it is purely mechanical; the soul or mind (according to the Hypothesis of an Harmonia præstabilita) never acting upon the body; I think he means or never without a miracle. The one The one proceeds upon the evidence of Experiment & Phænomena & stops where such evidence is wanting: the other propounds is taken up with Hypotheses & propounds them not to be examined by ExperimtsExperiments but to be received wthwithout examination The one doth not presume to say that God cannot chuse to act by the power of his will in matters indifferent: the other affirms that nothing himself is done without a reason And why must the laws of Nature be call'd miracles & occult qualities (that is to say wonderfull improbabilities if not impossibilities absurdities) if derived from the will of God Mr Leibnitz tells us in his Tentamen de motuum Cœlestium causis after he had mentioned Dr Barrows indefinitely little triangle for drawing of tangents & said (according to the Scholium upon Mr Newtons Xth Lemma) that there were infinite degrees of infinites & infinitely littles he tells us: Si motus exponatur per lineam communem quam dato tempore mobile absolvit, impetus seu velocitas exponetur per lineam infinite parvam, & ipsum elementum velocitatis, qualis est gravitatis sollicitatio, vel conatus centrifugus per lineam infinities infinite parvam. Atque hæc Lemmatūum loco annotanda duxi pro methodo nostra quantitatum incomparabilium & Analysi infinitorum, tanquam doctrinæ hujus novæ Elementa. And this is all one as to say that if the space described be considered as the fluent, the velocity must be considered as the first fuxionfluxion & the variation of the velocity as the second fluxion, & so on Theis was the foundation of Mr Newton's arguing in his Principles & this (wthwith a little variation of the Language) Mr Newton makes the foundati one of the a Leibnits recceons among the Elements of his Analysis. And while Mr Newton's in his book of Principles is founded upon these Elements & in that book he argues right. about the: it is certain that when he wrote that book, he had not ynthen forgotten the method of second fluxions. understood the elements of of that Analysis & upon wchwhich Mr Leibnits founded the cal that Analysis & of the infinitesimal Analysis & had not then forgotten the method of second fluxions. And Mr Leibnitz himself in his Letter of 21 Iune 1677 in explaining how his Method gave gave shewed how it gave shewed how it gave the method of Tangents of Slusius & proceeded without striking at taking away surds & extended to Quadratures, & then & then delaclared himself of opinion that since Mr Newtons method did these things it was of the same kind. Arbitror saith he, qua celare voluit Newtonus saith he de Tangentibus ducendis, saith he, ab his non abludere. Quod addit, ex hoc eodem fundamento Quadraturas quoque reddi faciliores me in sententia hac confirmat, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. And before this Mr Newton had told him further in his three Letters above mentioned, that his method determined Maxima & Minima, gav gave the lengths, & curvaities &of centers of gra Q Curves, & centers of gravity of curvilinear figures, & compared such figures curves with the Conic Sections & squared them by infinite series series or æquations wchwhich in certain cases brake off & became finite, & by the help of finite & infinite series æquations together reded to resolved inverse problems of tangent & others more difficult, & even to almost all Problems except perhaps some numeral ones like those of Diophantus. And When Mr Leibnitz was first told all this the great extent of the Method, he could not beleive it, but represented it improbable saying Quod dicere videmini plerasque difficultates ad (exceptis Problematibus Diophantæis) ad Series infinitas reduci; id mihi non videtur. Sunt enim multa usque mira & implexa ut neque adb æquationibꝰus pendeant neque ex Quadraturis: [Qualia sunt (ex multis alijs) Problemata methodi tangentium inversæ. And Mr Newton replied that his Analysis extended to such inverse Problemes & others more difficult] It lies upon him therefore in candor & justice to acknowledge that before he found the Differential method or at least before he wrote his Letter of 21 Iune 1677, he knew all this & wondred at it: And to give an Account why he concealed it his knowledge thereof when he published the Differential method as his own & why he did not then let the Germans know what he meant by the Differential methodus SIMILIS, t or whose it was, or what he had learnt of it by his correspondence wthwith Mr Oldenb. concerning it. And — Analysis extended to almost all sorts of Problemes & represented it so general that Mr Leibnitz himself could not beleive it in answer exprest his disbeleif of it. untill before he found the differential method. It lies upon him therefore in candour & justice to aknowledge this & to give an Account why he did not acknowledge any thing thereof when he first published the differential method. For it was not enough to mention a methodus SIMILIS without saying whose it was & of what antiquity & extent according to the notice he had from England. 28 — I do not know. But this I knotw that [after Mr Leibnitz had appealed to yethe R S. MrO there lies no appeal from the Committee of the R. Society to Mr Bernoulli. Mr Berno Mr Leibnitz] Mr Bernoulli is a party man & can be no judge. And Mr Leibnitz objected against can be on Iudge in this matter. d He is Mr Leibnitz He is too much of a party, & what Mr Leibnitz objected against Mr Dr Keill namely that he is Homo novus & rerum anteactarum parum peritus, & the same objection lies against Mr Bernoulli Mr Leibnitz allows that in the differential method letters may be used instead of the diff But this I know that Mr Bernoulli had the differential Method from Mr Leibnitz & is the chief of his disciples, & his writ & gave his opinion in the Acta Leipsica before he saw the Commercium Epistolicum & that what Mr Leibnitz objected against Dr Keill may b he is at wchwhich time he was homo novus & rerum anteactarum parum peritus, as Mr Leibnitz objected against Dr Keil, & that what he wrote afterwards he saw the Commercium was in his own defense. [And for Mr Leibnitz to appealed from the Committee of the Royall Society to his confident whose opinion he knew the chief man of his party who had] & his skill in Mathematicks will not mend the matter. He h is skilful enough to know that in taxing me with ignorance of in second differences he was mistaken & that the some errors in Mr Leibnitz his Tentamen de motuum cœlestium causis proceeded from his want of skill in those differences but he has not hitherto been too partial to acknowledged at what he knows. — & that before the end of the year 1666 I [sometimes used letters with one or two pricks for quantities involving first or second fluxions &] wrote a small Tract on this subject wchwhich was the grownd of that larger Tract wchwhich I wrote in the year 1671 both wchwhich are still in my custody & are founded upon the fistrst Proposition of the Book of Quadratures explained & demonstrated by letters without pricks tho at the same time I sometimes used & that when I wrote in a Geometrical manner without prickt letters or other symbols; & that when I wrote the larger of those two Tracts I had made my Analysis composed of the methods of Series & Fluxions together so universal — And therefore he is the Agressor. In both both his two Letters to Dr Sloan that pi of (that dated 4 Mart. 1711 & 1711 & that dated 29 Decem. 1711, he pressed the R. Socity to condemn Dr Keil & & before I medled in this matter challenged me to declare my opinion. in this matter. His words in his last second letter are: Itaque vestræ æquitati committo, annon coercendæ sint variæ & injustæ [Keilij] vociferationes, quas ipsi Newtono, Viro insigni et gestorum optime conscio, improbari arbitror; ejusque sententiæ suæ libenter daturum Iudicia mihi persuadeo. And therefore he is the aggressor. and ought to prove should have proved P. S. He Mr Leibnitz seems to say that what he formerly allowed concerning my having found a method like his own proceeded from his beleiving me, but since what I have retracted what I allowed acknowledged formerly he may be allowed to do the like. Here especially since the probabilities rem observed by Bernoulli, Here he accuses Bernoulli of calling my candor in question which is the ver, vizt in the Libel above mentioned. Mr Bernoulli may pretend its probable that no Probleme can be solved by the method of fluxions without the use of prict letters & theat in the Introduction to the Book of Quadratures the Method of fluxions is not explained because there are no prickt letters in it that Introduction, & that in the second Lemma of the second book of the Principia Philosophiæ the Elements of the method of fluxions are not demonstrated because there are no prick letters in it, & that from the sentences [ set down in my letter of 24 October 1676] viz Data æquatione fluentes involvente fluxiones invenire & vice versa, et and Vna methodus consitsitconsistit in extractione fluentis ex æquatione fluxionem involvente it cannot be inferred that I ohad then knew how then knew the first Proposition of the Book of Quadratures had at that time a method of reducing Problems to fluxional equations or of resolving such æquation because there are no prickt letters in those sentences. And that in the Analysis per æquationes numero terminorum infinitas there are no specimens of the Method of fluxions because there are no prickt letters. And that to the Schediasma the Proposition in the Schediasma hereunto annexed is no part of the Method of fluxions because it is explained without prickt letters tho it be the very same with the first Proposition in the book of Quadratures upon wchwhich the Method of fluxions is founded. And that its improbable that any caln solve no Problemes can be solved by the method of fluxions without the use of prickt letters. But if these things are only probable then its probable possible that they it may be otherwise & that I might have the Method of fluxions above 50 years ago notwithstandnotwithstanding any thing wchwhich hath been said to the contrary. But other men will rather inferr from these things that I had the method of fluxions when fifty years ago putting such symbols for fluxions as I thought fit, & Mr Leibnitz was of the same mind before he knew what symbols I used, as in his Letter of 21 Iune 1677 & where he allowed me a Methodus similis & in the Acta Eruditorum where he spake again of for October 1684 p 473 where he obtiner again mentioned a Methodus sil similis extending to the higher sort of Problemes & in his Letter to me dated 17 Mar 1673 st. n. hereunto annexed wherein he acknowledged of his own accord that by the Principia Philosophiæ it appeare I had shewed that I had such a methodus similis & added that he also had endeavoured to reduce sthe difficulter Problems to such an Analysis. And in the Acta Eruditorum for Iune 1686 he left me at liberty to use pag 297 he allowed that in the Differential method one might use letters instead of dx & such like charactersymbols, & thereby left me at liberty to use what symbols I thought fit. And further, I do not put prickt letters for differences or moments as Mr Bernoulli supposes as Mr Bernoulli supposes, but for fluxions wchwhich are finite quantities of another kind, the first Differences l the one being infinitely little the last finite & fluxions quan ma magnitudes parts & fluxions the other finite motions velocities of motion. And whereas Mr Leibnitz pretends that since I have retracted what I follrmerly alleged him, therefore he may be allowed to do the like; he should have proved that had retracted, & if I had done an ill thing yet that would not authoriz him to do the like. And whereas he pr questions my credit in what I said of Dr Wallis, he will find the truth of what I said in the Paragraph of Pthis Preface hereunto annexed 30 Leibnitius eadem fide chartam totam volantem scribere potuit qua citationem prædictam delevit, & verisimile est quod Menkenius eandem imprimere prælo committere commisit. Nam Leibnitius chartas imprimendas ad ipsūum mittere solebat. Et Bernoullius Epistolam sine nomine imprimendāam ad Leibnitium non misit In prima sua ad Abbatem de Comitibus Epistola Leibnitius scripsit — — — Et hæc. Quæstio dirimatur notandum est clarius alligatur De Ad hanc Questionēem spectat quod D. Wallisius noster [Celeberrimus Professor Oxoniensis, Propositionem primam Libri de Quadraturis exemplis in fluxionibus prim & secundis in veniendis illustratam. edidit anno 1693 in Volumine secundo Operum suorum pag 392. Et hæc fuit Regula omnium prima antiquissima quæ lucem vidit pro differentijs differend ut fluxionibus secundis tertijs quartis cæterisque in infinitum inveniendis Eandem Newtonus demonstravit synthetice in Lemmate secundo Libri secundi Principiorum: cum Propositionem sine Demonstratione prius posuisset in Epistola ad Oldenburgum 24 Octob. 1676 ut ad Leibnitium mitteretur & ibi significasset eandem esse fundamentum methodi generalis de qua scripserat tum ante decem annos quinque, id est anno 1671. In hujus Propositionis solutione habetur Algorithmus Methodi fluxionum.Gregorius scripsit ad Collinium 2 5 Sept. 1671, se ex Barrovij methodis tangentes ducendi methodum generalem ducendi Tangentes & Geometricam ducendi tangentes ad omnes Curvas sine calculo: & Slusius se similem methodum habere mense Novembri 1672 scripsit ad Oldenburgum. Et Newtonus ad Collinium 10 Decem 1672 scripsit in hæc verba: Ex animo gaudeo D. Barrovij nostri reverendi Lectiones Mathematicas exteris adeo placuisse, neque parum me juvat intelligere eos [Slusium et Gregorium] in eandem mecum incidisse ducendi Tangentes methodum & And . Et subinde methodum suam tangentium descripsit, & addidit hanc methodum esse corollarium Methodi generalis solvendi abstrusiora Problemata & non hærere ad quantitates surdas. Epistolas totas Gregorij & Newtoni habes infra in Commercio, & earum Exemplaria Oldenbusrgꝰus misit ad Leibnitium inter Excerpta ex Gregorij Epistolis 26 Iunij 1676; et Leibnitius Prælectiones Barrovij mense Novembri proximo secum tulit in Germaniam. Sunto jam ipsarum

AB = x

BC = y

utim Epistola Newtoni seu AbissaAbscissa

AB = x

& Ordinata

BC = y

. Et sint earum fluxiones p et q & momenta op et oq, et in AB producta capiatur

BE = o p

, erigatur Ordinata EF parallela BC & occurrens Curvæ in F et compleatur parallelogrammum BCGE et erit

FG = o q

. Proponatur æquatio quævis quantitates duas fluentes x et y involvens, puta

x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} = 0

ut in Epistola illa. Multiplicetur omnis æquationis terminus per indicem dignitatis x et productum divisum per x (videlicet

3 x^{2} - 4 x y + 2 b x - b b

,) vocetur R. Multiplicetur omnis æquationis terminus per indicem dignitatis y & productum divisum per y ,(videlicet

−2 x x + 2 b y - 3 y y

) vocetur S. Et per Regulam in Epistola illa Newtoni traditam erit subtangens

BD = \frac{S y}{R}

, vel potius

= \frac{−5 y}{R}

propterea quod AB et BD ducantur in partes contrarias. Sed ex methodo tangentium Barrovij est

FG . GC ∷ BC = y . BD

. adeoque

BD = \frac{o p \times y}{o q} = \frac{p y}{q}

. Ergo

\frac{p y}{q} = - \frac{S y}{R}

et facta reductione

R p + S q = 0

Hæc est æquation involvens fluxiones. Et si plures sint quantitates fluentes eadem operatio instituta in singulis dabit æquationem involventem fluxiones exhibat solutionem omnium. Hæ Et hæc Regula illud omne comprehendit quod Leibnitius ad Newtonum rescripsit in Epistola sua anno 1677 ad rescripsit, ut et illud omne quod in Actis Eruditorum anno 1684 in lucem edidit. Exhibet enim hæc Regula solutionem Propositionis, [Data æquatione fluentes quotcunque quantitates involventes invenire fluxiones; quam posui in Epistola 24 Octob 1676 ad O] primæ Libri de Quadraturis. Propositionem quintam libri de Quadraturis Wallisius edidit — — — — ab ipso excogitatam, i.e. anno 1666 aut antea; nec non teste N. Fatio de Duillier Et his præmissis qui chartas antiquas Newtoni viderat, & contra seipsum testimonium perhibuit. Et his præmissis legatur jam Recensio Commercij Epistolici & Consulatur Commercium ipsum ubi de factis dubitatur. Hæc Regula Hoc Theorema tur Sic inventio fluxionum ex æquatione fluentis involvente ex Epistola Newtoni cf methodois Tangentium Barrovij & Gregorij a Newtono citati facillime fluit. [Idem vero exhibet solutionem Propositionis primæ Libri de Quadraturis.] deducitur Et s hujus Theorematis sic ex Newtoni Epistola derivatum inventumio illud omne comprehendit illud omne quod Leibnitius ad Newtonum rescripsit anno 1677 rescripsit, ut et illud omne quod in Actis Eruditorum anno 1684 in de hac methodo in lucem edidit. Exhibet enim hoc Theorema solutionem. Propositionis primæ Libri Newtoni de Quadraturis Newtono Idem vêro Newtono innotuitsse anno 1686 quemad ut manifestum est ex Lem. 2 Lib. 2 Princip. ut ut et anniso 1676, manifestum est 1672 & 1671 quemadmodum manifestum est ex Epistolis Newtoni 13 Iunij & 24 Octob. anni ianni illius, ex quibus utique Wallisius scripsit Newtonum Leibniti methodum exposuisse, et Leibnitio. Idem Newtono innotuisse etiam anno 1672 manifestum est ex Epistola ejus Epistola jam explianta ijs quæ jam dicta sunt de Epistola ejus hoc anno scripta, sunt jam dicta sunt jam dicta fuerunt. Iudex sive celando et nomen suum & nomeina eorum quos judicat, libellum potius quam fd sententium judicis dictitat. a Hæc Epistola, cum celetur et Iudicis et Iudicatīim nomen, Libellum defamatorum magis sapit quam æqui probi judicis sententiam. b Algorithmus habetur & c In Analysi per series Newtonus utitur d Nullam habuit occasionem. Propositiones non invenit per synthesin sed per Analysin inventas demonstravit synthetice ut in Geometriam admitterentur. e Volumen tertium. f Eodem sensu g In libro h Eminens ille i Recta methodus ubi Ordinatim Applicata (ceu momentum areæ) fluentis) deducitur ex æquatione Abscissam et Aream involvente deducitur. Recueil Tom. 2. p. 4, 5, 36, 52, 53. In Epistola ad Cometissam de Kilmanseg Agnovit postea Oldenburgium ad se mississe des Essays Ad hanc Quæstionem spectat quod Leibnitius differentias & methodum differentialem vocat quæ Newtonus momenta, Et Methodus differentialis eadem est cu momentorum: Et quod methodus fluxionūum Newtoni non magis differt a methodo differt a methodo deffentialijdifferentialij quam differt a methodo momentorum. [Fluxiones sunt velocitates quibus mnd mome Vna et eadem est methodus nisi quatenus Newtoni methodus sit amplior Momenta vel differentiæ sunt partes genitæ quantitatem genitarum, fluxiones sunt velocitates quibus partes illæ generantur. Leibnitius & Newtonus consideratnt solas partes genitas, Newtonus considerat etiam partes velocitates quibus partes generantur. Newtoni methodus complectitur methodum Leibnitij.] Vna et eadem est methodus quoad partes differentius vel momenta. Hæc sunt partes quantitatum genitarum. Vterque considerat has partes Fluxiones sunt velocitesvelocitates quibus partes generantur. Vterque considerat partes. Newtonus considerat etiam partes quibus velocitates quibus partes generantur. Newtoni methodus complectitum methodum Leibnitij. Leibnitius itaque series a se missas Epistolam Oldenburgij 15 Apr 1675 ad se missam accepit seriebus refertam, seriebus refertam accepit Sed Series nunquam si fas est credere nunquam contulit cum suis. Ad hanc Quæstionem spectat quod Leibnitius differentias & methodum differentialem vocat quams Newtonus momenta & methodum momentorum quod methodus momentorum & methodus fluxionum eadem sit. Momenta sunt partes quas Leibnitius differentias vocat, fluxiones sunt velocitates quibus partes generantur. Leibnitius considerat partes, Newtonus considerat etiam velocitates. & Newtoni methodus est amplior & Leibnitij methodum complectitur. Et hæc est Regula ducendi tangentes quam Newtonus posuit in Epistola illa posuit ut partem aliquam vel specimen vell Corollarium vel Specimen mea Methodi suæ generalis: Ducatur jam secundum methodum Barrovij a Gregorio promotam Ordinata nova EF priori BC proxima & compleatur parallelogrammum BCGE et pro momentis BE vel CG & GF scribamus p et q et erit

q . p ∷ y . BD

. seu

\frac{p y}{q} = BD = - \frac{S y}{R}

et facta reductione prodit

R p + S q = 0

. Hæc æquatio ubi duæ tantum sunt fluentes involvit earum momenta. Et ubi plures momenta. vel ut Leibnitius loquitur, omnium differenteias. Et Theorema hocce quod sic ex specimine in Newtoni Epistola posito facillime deducitur, totum illud omne comprehendit quod Leibnitius anno 1677 ad Newtonum rescripsit, ut et illud omne quod in Actis Eruditorūum anno 1684 in lucem edidit. [et illud omne quod Newtonus in Lem 2 Lib. 2 Princip. et Prop. 1 Lib. de Quadraturis edidit.] Hoc Theorema exhibet solutionem Propositionis primæ Libri de Quadraturis, ideoque solutio Propositionis illius anno 1672 Newtono innotuit. In libro de Analysi per Series Fluxiones ac Differentias anno 1711 a Ionesio edito extat Fragmentum Epistolæ D. Newtoni ad D. Collinium Nove. 8 1676 data ubi Newtonus scribit in hæc verba. Nulla extat Curva cujus æquatio — possint comparo. — Eadem methodus Æquationes quatuor terminorum aliasque complectitur, haud tamen adeo generaliter. Hactenus Newtonus. Hæc autem absque Methodo fluxionum fieri non possunt. Indicant vero methodum quadrandi Curvlineas in libro de Quadraturis expositam, eousque productammotam fuisse ante 8 Novem 1676. In Epistola Newtoni ad Oldenburgum 24 Octob. 1676 ha bd citantur ponuntur Ordinatæ Curvilinearum in Tabulam tum olim Catalogum tunc olim re qusarum collationes cum Conicis sectionibus Newtonus in cCatalogum tunc olim in retulerat. Earundem Curvarum et Eodem ordine et ijs literis literis cCollationes cum Con. Sect. describuntur in Tabula posteriore in libro de Quadraturis: ideoque Tabula illa composita fuit est methodus quadrandi Curvas eousque producta, fuit annis aliquodt ante annum 1676. Id quod absque methodo fluxionum fieri non potuit. In prædicta Analysi prædicta per series prædicta quam Barrovius anno 1669 ad Collinium misit Newtonus exponit tempus per abscissam Curvæ fluentem per aream Curvæ ejus fluxionem ejus fluentis per Ordinatam & momentum ejus per fluxionem ductam in temporis momentum o. Et sub finem Tractatus illus dat specimen calculi. Et his præmissis Leibnitius methodum Newtoni serierum Newtoni in Tractatu illo descriptam postulabat a Collinio per literas ad Oldenburgum 12 Maij 1676, potuitque videre in manibus Collinij proximo mense Octobri ubi in Angliam venit & in manibus Collinij vidit epistolas plures Newtoni Gregorij & aliorum, eas præsertim quæ circa series versabantur ut ipse in epistolis ad D. Abbatem de Comitibus & Cometissam de Kilmansegger agnovit. Et his præmissis legatur jam Recensio Commercij Epistolici & consulatur Commercium ipsum ubi de factis dubitatur. ✝ Tandem agnovit iterum in Epistola sua ad tCometissam de Kilmansegger, ut supra in Præfatione ostenditur — Et Mathematicum esse Bernoullium ipsum scripsit, & chartas illas voluntem denuo dispersæ & Gallice in Hollandia imprima auctoritate Bernoullij imprimi curavit & Problemata Bernoullij Analystis Anglis solvendam proposuit Et auctorite Bernoulij Bernullij Indivisa N constituti amicos suos rerum mathematicarum inscios passim contra Newtonum passim ciere conatus est, cum tamen Bernoullius. Commercium Epist. p. 32, 530, 3, 39, 42, 43, 44, 45, 49 58, 87, 88, 96. Charta volans p in Commer p. In Epistola Charta illa 29 Iulij 1713 data Leibnitius Epistolam Oldenburgi qua 15 AprilAprilis 1675 datam qua Oldenburgus series aliquot & inter alias seriem Gregorij quam L ad Leibnitium misit et inter alias seriem Gregorij quam Leibnitius postea ut suam edidit in dubium vocare conatus est, dicendo: Tale quiddam Gregorium habuisse ipsi Angli & Scot. Wallisius, Hookius Newtonus & junior Gregorius ultra triginta sex annos ignoraverunt & Leibnitij esse inventum crediderunt. At hæc Epistola ut et E in Libro Epistolico Regiæ Societatis asservata, ut et Epistola autographa Leibnitij q se series missas recepisse agnoscentis, cum ijsdem epistolis in CommerioCommercio editis, coram exteris plurimus collata sunt in qud Comite de Kilmansegger, Abbate de Comitibus, ministris aliquot aut publicis exterorum quot Principum & et alijs exteirs non paucis Anno 1715 collatæ sunt et ab omnibus agnitæ; & Leibnitius ipse a anno proximo in Epistola sua ad D. Cometissam de Kilmansegger 18 Apr. 1716 data & a D. Deso Maiseaux edita, idem agnovit narrando historiam epistolis in Commercio editis conformem. Narrat enim quomodo Londinum venit initio anni 1673, cum Oldenburgo amicitiam contraxit, a Pellio nostro apud D. Boyle audivit quod Mercatorum seriem invenisset pro HypolaHyperbola, librum comparavit & secum asportavit in Galliam, ibi sub Huygenio cœpit meditationes Geometricas gustare, parvo tempore multum profecit, & invenit seriem suam pro circulo, dein sic pergit Nous crûmes que j'etois le primier, qui avois fait quelque chose de tel sur le circle; & j'en écrivis sur ce ton-la a [15 Iulij & 26 Octob] a M. Oldenbourg en 1674 avec qui auparavant je ne telles choses, quoique nous eussiouns échangé deja pleusieurs [Feb. 20, Mart. 30, Apr 26 Maij 24 & Iunij 8] plusieurs Lettres. M. Oldenburg m'ecrit [15 Apr 1675] m'envoya des & essays. Cependant le mieu [Londinum missa 27 Aug. 1676] fut asser applaudi par M. Newton même [24 Octob. 1676] per Mr Newton même. Il est trouvé par apres [anno 1712] qu'une nomme M. Gregory avoit trove justement la même series que moi. Mais c'est ce qu' j'appris tard. Hic Leibnitius agnoscit se recepisse ab Oldenburgo de essays exempla serierum Newtoni, et in eadem Olde inter hæc exempla erat series Gregorij ut in Commercio videre licet. At Newtonus Wallisius & Gregorius junior et Hugenius hanc seriem a Gregorio ad Collinium ab Collinio Oldenburgo ad Leibnitium missam fuisse, per ea tempora ignorarunt. Contra fidem epistolarum in Commercio editarum scripsit insuper Leibnitius — videntur Altamen ut accusationem probaret — — — in Commercio pag. 39. Cæterum Leibnitius in prima sua ad Abbatem — — — . ostendit Subinde in prima sua ad D. Leibnitius autem in proxima — — — — Leibnitius. Altamen post ejus mortem — — — — referri debet. Et hæc Quæstio est utrum Leibnitius sit inventor Methodi & pro differentijs igitur Leibnitianis Newtonus adhibet semperque [ex quo usus est hac methodo] adhibuit fluxiones, quemadmodum Honoratus Fabrius motuum progressus Cavallerianæ methodo substituit. Ad hanc Quæstionem spectat quod Algorithmus methodi habiatur in Propositione prima Libri de quadraturis & quod D. Wallisius Propositionem primam Libri de Quadraturis illustratam illam cum exemplis inveniendi fluxiones primas et secundas edidit annno 1693 in Volumine secundo operum suorum pag. 393. — id est anno 1671. Eodem spectat quod Iacobus Gregorius scripsit ad Collinium 5 Sept 1671 — — — — — Propositionis primæ libri de Quadraturis. Propositionem quintam Libri de Quadraturis — de factis dubitatur. — si fieri potest. Hoc artificium anno 1669 Newtono innotuitsse patet per anno 1669 uti patet ex Analysi per series,vid Com pag Epist. pag immo et annis aliquot antequam Mercatoris Logarithmotechnia prodiret (id est anno 1666 aut antea) teste Barrouꝰus uti patet ex Epis per testimonium ex Epistola ✝Vid Comm: Epist. p. Collini i ad D. Strode. per testimonium Barrovij Propositio illa quinta pendet a quatuor prioribus; ideoque Methodus fluxionusm quatenus continetur in Propositionibus quinque primis libri Principiorum de Quadraturis Newtono innotuit anno 1666 aut antea. Newtonus incidit in Theorememata suam de refractionibus, & coloribus lucis ineunte anno 1666. Sub idem tempus invenit etiam methodumos serierum et fluxionum. De his omnibus scripsit Tractatus anno 1671 ut in lucem im mitterentur. Sed anno proximo subortæ per d cum nonnulla communicasset de natura lucis, subortæ statim per diversorum Epistolas objectionibus alijsque refertas, crebræ interpellationes, ipsum prorsus a consilio deterruerunt, et effecerunt ut seipsum argeret imprudentiæ (ut ipsa olim locutus est) quod umbram captando, eatenus perdiderat quietem suam, rem prorsus substantialem. Ab his omnibus igitur edendis abstinuit igitur usque ad annum 1704 anno 32, & anno 1704 usque ad annum 1704 et interea aliqua tantum communicavit privatim cum amicis. Librum vero MS de Quadraturis literis punctatis refertum Halleius noster & Ralphsonus anno 1691 manibus suis tractarunt ut alter testatum reliquit & alter adhuc testatur. Eodem libro Newtonus annis 1685 & 1686 co inveniendis Principijs Philosophiæ se plurimum usum fuisse profitetur. Et quæ in Epistolis tribus Newtoni anno 1676 scriptis ex eodem citantur, indicant ipsum eundem jdem tum in MS extitisse. Et his præmissis — SrSir Isaac Newton Bar. St Martins Street Kill

- \frac{15 x^{4}}{4096 a}

\begin{array}{r} 524 \\ - 15 \\ 509 \end{array}

+ \frac{131 x^{4}}{1024 a}

Dominus Brunkerus qu quadravit Hyperbolam per hasce duas series

\frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + \frac{1}{7 \times 8} + \frac{1}{9 \times 10}

&c et

\frac{1}{2 \times 3} + \frac{1}{4 \times 5} + \frac{1}{6 \times 7} + \frac{1}{8 \times 9}

&c. HEt hæ series cædam sunt cum seriebus

1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} + \frac{1}{8}

&c et

\frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6}

&c ut colligendo summas binorum terminorum patebit. Impressa est autem Brunkeri Quadradtura Anno 1668 mense Aprili idque in Actis Philosophicis Num. 34. Et paulo post impressa est Mercatoris Logarithmotechnia cum eadem Quadratura promota. Et mox Gregorius Mercatoris Quadraturam eandem Geometrice demonstravit postita hac prima Propositione. Si fuerint quantitates continue proportionales A, B, C D, E, F, &c numero terminorum infinitæ, quarum prima et maxima A; erit A–B ad A ut A ad summam omnium; hoc senim passu demonstratur apud Geometras. Hæc Gregorius. Et Per hanc Propositionem Geometricam prodit quadratura Hyperbolæ absque Divisione Arithmetica. Mercator vero Propositionem probavit per Divisionem Arithmeticam. Sed methodum generalem. Quadrandi Curvas per hujusmodi series nec D. Brunker nec Mercator nec Gregorius inveneritunt. utere Gregorius hujusmodi methodum diu quæsivit & vix tandem sub finem anni 1670 invenire potuit. Newtonus hujusmodi methodum aliquam primus iis omnium prodtulit. &c Proposuerat Wallisius in Arithmetica sua Q seriem Curvarum cujus ordinatæ sunt

1, \sqrt{a x - x x}, a x - x x, a x - x x \sqrt{a x - x x}, a a x x - {}^{2}a x^{3} + x^{4}

&c et areæ

x, *, \frac{a x x}{2} - \frac{x^{3}}{3}, *, \frac{a α x^{3}}{3} - \frac{2 a x^{4}}{4} + \frac{x^{5}}{5}

, &c, existente Abscissa x Et quòd si hæc series posterior in locis alternis quapravi interpolari possit habebitur area circuli: Invenit Newtonus methodum interpolandi hanc seriem hanc seriem. Et eadem methodo interpolavit seriem potestatum binomij

1 . a + x . a a + 2 a x + x x . a^{3} + 3 a^{2} x + 3 a x^{2} + x^{3}

&c. Et ponendo Ponatur utique literam aliquam puta n pro indice binomij, et per Methodum Newtoni t prodibit x+ hujusmodi seriems

{a + x}^{n} = a^{n} + {}^{n}a^{n - 1} x + n \times n - 1 \times a^{n - 2} x^{2} + n \times n - 1 \times n - 2 \times a n - 3 \times x^{3} + &c

. Et per hujusmodi seriem invenit Regula inde pervenit ad methodum generalem ubi n potest esse namerus integer, vel fractus vel surdus affirmativus vel negativus, et a et x nomina quæcunque designant. Et hæc series eadem est cum Regula generali quam Newtonus in Principio Epistolæ suæ primæ cum Leibnitio communicavit. Sic Newtonus pervenit ad methodum generalem quadrandi Curvas quarum orditnatæ sunt binomiorum potestate dignitates quæcunque, sive nomina qu sint quantitates simplices vel sive compositæ, sive et indices dignitatum sint numeri intiegri vel fracti vel surdi et affirmativi vel negativi, ut ex epistolais ejus ad D Oldenburgum cum Leibnitio communicandis intelligo. Deinde animadvertit idem per divisiones et extractiones radicum fieri & methodum promovit ad radices æquationum affectarum sive æquationes illæ sint affectæ solas quantitates fluentes involvant sive etiam fluxiones, Et methodum etiam primus invenit assumendi terminos serierum et eosdem per collationem terminorum homogeniorum determinandi. Hæc invenerat antequam ut ex ijsdem epistolis et alijs ejus scriptis patet. Slusius methodum ducendi tangentes ab his ex tribus Lemmatibus consequiqui derixesse dixit, & Lemmata impressa sunt in Actis Philosophicis. At quomodo methodus ab istis Lemmatibus derivetur consequatur nondum sd constat. Annon Slusius Anglis iiserit methodum illam aliunde habuerit hauser habuerit. Leibnitius vero ne unam quidem seriem earum quas sibi arrogorare conatus est primus invenit: ala Newtonus Methodum perveniendi ad has Series anno 1676 a Newtono accepit, acceptæ nihil unquam additdit. Newtonus methodum et primus invenit et perfecitam reddidit. Quinetiam Regula jam descripta methodum fluxionum quam Leibnitius differentialem vocat in se involvit. [Si terminus primus binomij sit quantitas fluens & terminus secundus Binomij sit momomentum termini primi, terminus primus seriei erit momentum dignitas um primi seu quantitas quantitatis fluenstis, terminus secundus erit momentum primum dignitatis, tertius quartus quintus & reliqui datas habebunt rationes ad ter momentorum secundum tertium quartum & reliqua in infinitum.] Si momentum dignitatis quantitatis fluentis desideretur s] Siquidem Seriei cujuscunque terminus secundus sit momentum termini primi et tertius quartus quintus et reliqui sint ad momenta termini primi in datis rationibus, si modo Binomij nomen secundum sit momentum primi nominis primi. Verbi gratia sit o momentum fluentis x et quæratur momentum dignitatis

x^{n}

, id est quæratur differentia inter

x^{n}

{x + o}^{n}

et resolvendo

{x + o}^{n}

in seriem

x^{n} + n o x^{n - 1} + &c

, secundus seriei terminus

n o x^{n - 1}

erit differentia illa quæsita. Pr Sit jam p momentum quantitatis fluentis y et quæratur momentum quantitatis fluentis

x^{m} y^{n}

et resolve id est quæratur differentia inter

x^{m} y^{n}

{x + o}^{m} {y + p}^{n}

et resolvendo binomia in series et multiplicando series in se mutuo prodibit series cujus secundus terminus est diff

m o x^{m - 1} \times n p y^{1 - 1}

est differentia quæsita. Et hæc est methodi momentorum et fluxionum quam Leibnitius differentialem vocat Demonstratio omnium brevissima & maxime naturalis. Communicavit igitur Newtonus cum Leibnitio per Epistolam suam primam fundamentum verissimum verum methodi differentialis a quo methodus illa sponte fluit et per quod brevissime et optime demonstratur. [Sed et in Epistolais sua secunda alijs methodum illam tantum non demonstra se habrehabere dixit & verbis disertis tantum non descripsit: Leibnitius autem methodum illam nondum demonstravit neq nunquam melius demonstrare potest. Sed et Newtonus methodum intram que serierum & methodum fluxionum sob maximam affirmitatem semper inter se sconjuxit et simul tractavit. Et Leibnitius affirmitatem illam agnovit in Act Lips. Anno 1693 mense Apr. ubi series infinitas a methodo differentiali derivate cotnatus est. Scribit ibi Leibnitius in hæc verba. Cum antea series infinitæ fuerint quæsitæ cum primo inventore Nicolao Mercatore Holsato per divisiones et cum summo Geometra Isaaco Newtono per extractiones; visum mihi fuit posse easad eas perveniri per commodius et universalius per suppositionem ipsius seriei quæsitæ tanquam inventæ ita ut terminorum coefficientes ex successu definirentur. Rectius dixisset seriem infinitam continue proportionalium ejusque summam veteribus innotuisse, D. Brunkerūum p esse primum inventorem Quadraturæ per seriem infinitam. Mercatorem propositionem veterum per sum de summa seriei infinitæ continue proportionalium demonstrasse per divisionem et inde derivasse & ampliasse Quadraturam Hyperbolæ inventam a Brun per seriem infinitam a Brunkero inventam. Newtonum regulam invenisse generalem reducendi binomia & binomialia omnia ad series infinitas & per hanc Regulam & extractiones radicum affectarum methodum generalem tractandi curvas trancendentes Analytice & quadrandi curvsilineas omnes & similia peragendi primum omnium invenisse. Rectius Nullibi agnovistse Leibnitusius in Actis Lipsiensibus se series qua series aliquot infinitas ab Oldenburgo primum accepi Anno 16753 accepisse & Regulam Nuetoni reducendi binomia in series anno proximo a Newtow accepisse didicisse eas præsertim quæ ad circulum spectant. Nullibi agnovit se Regulam Newtoni accepisse reducendi binomia in series infinitas quæ prim fuit methodum generalem aperiat quadrandi curvilineas & similia peragendi. Nullibi agnovit Se se vidisse quasdam Newtoni epistolas ex quibus didedixit Newtonum vel Methodum differentialem vel huic methodo simillimam primum invenisse, Nullibi agn et Sententia aliqua literis transpositis designata celasse. Nullibi agnovit Newtonum in Epistola 24 Octob. 1676 data significasse Inversa Problemat de Tangentibus Pro Problemata esse in potestate aliaque illis difficiliora: ad quæ solvenda se usum esse duplici methodo una concinniori alteræ universaliori. et utramque sententia literis transpositis designata celasse. Sed his silentio traditis Leibnitius LCe jam visum esse jam scribit quod visum sibi esse fuit posse series Newtonianas perveniri commodius et universalius per suppositionem ipsius seriei quæsitæ tanquam inventæ ita ut terminorum cofficientescoefficientes ex successu definiantur. Quinetiam sub finem epistolæ secundæ sub finem 24 Octob 1676 datæ Newtonus Le inversa de tangentibus Problemata esse in potestate aliaque illis difficiliora ad quæ solvenda duplici usus est methodo et una concininniore altera generalionegeneratione & methodum utramque tentatijs deprimit hisce designavit sententijs. Vna methodus consistit in extractione fluentis quantitatis ex æquatione simul involventeventeinvolvente fluxionem ejus. Altera tantuum in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possunt; et in collatione terminorum homologorum æquationis resultantis ad eruendos terminos assumptæ seriei. Hæc Regula verbis alijs ita sonat. Ex Binomij F dignitate Fluentem ere Hanc seriem anno 1669 Newtono innotuisse patet ex Analysi supra impressa pag 19 lin 19, 20 1 Methodus tam fluxionum quam serierum hanc infirmitarum fundatur in operationibus quatuor vel quinque generabllibus ex Analysi supra impressa pag 19 lin 19, 20 quarum hæc est prima. Et per hanc operationem Fluens ex æquatione quacunque non affecta extrahitur s eruitur. 2 Operatio secunda generealis qua fluens ex æquatione non affecta extrahitur. p. 636. l. ult. Id est, Data æquatione fluent quotcunque fluentes quantitates involvente invenire fluxiones et et vice versa. Solvitur vero sumendo secundos ter terminos serierum pro momentis primorum. B Sit æquatio

x^{n} = y

. [Fluat x uniformiter & sit ejus momentum

\dot{x} o

fluat etiam y et sit ejus momentum

\dot{y} o

. Resolvatur binomium

{a x + a \dot{x} o}^{n}

in seriem et series

a x^{n} + {}^{n}a \dot{x} o x^{n - 1} + &c

. Et secundas seriei t

= y + \dot{y} o

et .] Solvitur vero per operationem primam in Epistola superiore descriptam sumendo secundos serierum terminos pro flu momentis primorum. Sit æquatio o

x^{n} = y

. Fluant x et y uniformiter & sit ejus fluxio earum momentumans

o \dot{x}

& ipsius y momentum

o \dot{y}

. Reslvatur binomium in serie et fluendi velocitates.

\dot{x}

\dot{y}

Et erit

{x + o \dot{x}}^{n} = y + o \dot{y}

. Resolvatur binomium in seriem et

x^{n} + n o \dot{x} x^{n - 1} + &c = y + o \dot{y}

et ablatis æqualibus

x^{n}

et y manebunt æqualia

n o \dot{x} x^{n - 1} + &c = y + o \dot{y}

et ablatis æqualibus

x^{n}

et y manebunt æqualia

n o \dot{x} x^{n - 1}

etij et

o \dot{y}

, quibus per o divisis fit

n \dot{x} x^{n - 1} = \dot{y}

. Est igitur

n \dot{x} x^{n - 1}

Et similiter Sit æquatio

x^{m} z^{n} = \dot{y}

et simili operatione prodibit

m x x^{m - 1} \dot{z} z^{n - 1} = \dot{y}

. Et idem fit in singulis terminuis æquationum compositarum. Et hoc etst fundamentum verum methodi fluxionum, Newtonus inven et d fundamentum verum et demonstratio optima brevissima et maxime generalis et Newtono ab initio inotuit ut ex ejus Analysi per æquationes infinitas (pag 3 Reg. 1. & pag 19) manifestum est. Newtonus communicavit invenit methodum seriersum infinitas Leibnitius invenit secundos terminos serierum Newtonus communicavit cum Leibnitio methodum suam perveniendi ad series infinitas, Leibnitius secundos serierum terminos invenit. p 634 a Tempus hyberno inter annos 1664 et 1665 p. 634 l. 11 a Regula prima. l. 30. b Regula secunda. p. 636 l. 19 Analysis per æquationes infinitas supra impressa l 45. Ad verba: Diversa ratione in eam incidimus, nota: [Imo Slusius per Lemmata suæ quæ in Epistola quæ in Actis Philosophicis ut fundamentum methodi suæ hujus impressa sunt in eam incidere non potuit, sed methodum aliunde habuit.] Slusius i prbus methodum Riccio sibi minime arrogavit sed Riccio cuejdam tribuit & methodum a se abjudicaret Lemmata posuit ponendo a quibus methodus derivari non potest. Qu Extant Lemmata in Actis Philosophicis Newtonus methodum primus communicavit & verum ejus fundamentum solus posuit. Qui aliter sentit doceat quomodo Leibnitius methodus a Lemmatibus illis derivari possit. p. 635 l. 1. Exemplum generale quadraturæ curvarum per methodum fluxionum. p 635 l. 1 Exemplum generale quadrandi curvas per methodum fluxionem & series infinitas conjunctim. Vid. pag. 18. lin 31. p. 639. Pag 639 l. 1. Ex his patit Series Newtoniana exeorum quantitatem indefinitam (id est fluentem) unam vel plures in se semper involvunt, aideoque ad methodum fluxionum perinent. Ib. lin. 20. Ex his patet Propositiones de Newtoni de Quadraturam Curvarum ante annum 1676 inventas fuisse. P. 640. l. 7. NB D. Brunker Hyperbolam per hanc seriem minus

\frac{1}{2} + \frac{1}{12} + \frac{1}{30} + \frac{1}{56} + &c

id est seu per hanc

1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5}

(conjunctis binis terminis) primus quadravit. Mercator quadraturam hanc ampliavit. Gregorius circulum per hanc seriem

1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7}

&c quadravit. Leibnitius hanc Quadraturāam sibi ab Oldenburgo accepit. & sibi arrogavit. Newtonus quadravit Circulum per hanc seriem

1 + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - &c

quadravit seu

1 + \frac{1}{15} - \frac{1}{63} + \frac{1}{143}

&c quadravit P. 644. l. 17. Id est, Vna methodus ...... assumptæ seriei. Prior methodus est Operatio tertia vel Regula tertia generalis qua methodus Analysis per seriersum infinitas et fluxionēes s perficitur. Posterior est Regula quarta et ultima. Duæ Primæ Regulæ in principia epistola priora epistol fitæ f traduntur et prima quidem dat quantitatem fluentesm ex equationibus non affectis, secunda dat Fluentesm ex æquationibus affectis fluxionem ejus non involventibus, tertia dat fluentesm ex æqu. affectis fluxionem ejus involventibus. Quarta dat fluentes ex omnibus. Hiasce quatuor operationiebuss perveniturdi ad fluentes Newtonus ante annum 1676 invenerat. Ab eo tempore his nihil additum est. Leibnitius ex tempore de his cogitare cœpit Et Newtonum esse harum omnium Regularum inventorem primum nemo dubitat. Et in Regulis hisce quatuor methodus tota comprehenditur moment fluentium et momentorum tam in Æquationibus infinitis quam in finitis comprehenditur, cum inventio fluxibus ex momen per Reg. 1 cum momenta fluentium facile inveniantur per Reg 1. ut supra per Reg 1 ut Supra expositum est. In Leibnitij Epistolam secundam. Pag. 648. lin 20. E His verbis patet Leibnitium jam primum calculum differentialem cum amicus jam primum communicare cæpisse et methodum de d differentias didicisse ex secundis termi potes ponendo secund fluentem cum ejus momento pro binomio habuisse secundos terminos dignitatum binomij pro momentis dig (vel (ut ipse loquitur) differentijs) dignitatum, perinde ut Newtonus antea fecerat. Vide pag 19. Inveniri possunt plures hujusmodi series assumendo seriesm arearum pro lubitu, et invdeniendo computando seriesm Ordinatarum & inter methodo Newtoniana sub initio hujus Epistolæ exposita interpolando seriem utramque; vel assumendo seriem quamlibet ordinatarum ad Curvas quæ per finitas æquationes quadrari possunt et interpolando seriem arearum. Et Methodus prior est tertia in his Epistolis tradita posita Regula tertia generalis (in his Epistolis tradita) extrahendi quantitates indeterminatas & fluentes & et posterior est Regula quarta. Quantitas fluens stbilis Per Regulam primam fluentses et equs momentum ex æquationibus non affectis, per secundam ex æquationibus affectis fluxionem non involventibus, per quartam ex tertiam ex æquationibus affectis fluxionem simul involventibus, per quartam ex conditionibus Problematis eructur et extrahitur. Et momentum fluentis invenitur per Regulam primam invenitur ut supra, [ vel etiam dicendo quod si fluens sit ut area curva momentum ejus et fluxu erit ut Ordinata] vel etiam per Ordinatam Curvæ cujus area cujus Abscissa uniformiter est ut fluens? Et Et sic Newtonus Regulis hisce quatuor totum flu Analysi suam totam per fluentes & earum momenta idque æquation et Analysin idque in æquationibus tam infinitis quam finitis hat in Epistolis suis Epistolis suis complexus Est cum Leibnitio communicandis jam complexus fuit est. & methodum fluxionum tantum Newtonum Et eum esse harum quatuor Regularum inventorem primum nemo dubitat. Et [Analysin eandem his Regulis complexam nemo ampliorem dd reddidit alij aliter exponsu explicuerunt, nemo ampliorem reddidit, Leibnitius ex literis jam primum communicabat dedt anno sequente suis verbis cum amicis ca Leibnitius anno sequente 1677 sequente proferre cœpit & nondum totam protulit, Newtonus Anno 1669 cum Collin Collinio Collinius deinceps cum amicis communicabat ut supra. p. 645 lin. 259. Ad verba a aliter se habere solet. a Id est, ad solutionem Problematis methodus generalis jam descriptæa sæpe requiritur. p. 648 lin 19 Ad verba a in posterum. a Cœpit agitur Leibnitius methodum suam differentialem jam primum cum amicis communicare p 649 lin 24. Ad verba celare voluit Pluribus argumentis Leibnitius se jam methodum differentialem Methodum p 648 lin 248. IAd verba. Idemque est de cæteris potentijs. Posita serie potentiarum binomij Leib Ex secundis terminis potentiarum binomij Leibnitius didicit hic colligit differentias. suas. Newtonus quandoque ex secundis terminis potentiarum quandque ex secundo termino seriei g potentias omnes involventis momenta sua colligebrat. Eadem est utriusque computandi ratio ut conferenti computationem Leibnitianāam hic positam cum computatione Newtonimana sin Analysi sua pag. superius impressa pag 19, statim patebit. p 649 lin 24 Ad verba celare voluit notetur. Leibnitius Multis arguments docet Leibnitius se in methodum incidisse quæ cum Newtoniana con per omnia congruit. Analysin per fluentes et earum momenta Newtonus in æquationibus tam infinitis quam finitis Newtonus in his Epistolis ad Regulas quatuor reduxit. Pr primam est trachtiohitur fluentsis & ejus momenta ex BinomiojsūBinomijs atque adeo ex æquationibums quibuscunque non affectis et momentum fluentis. simul prodit. Per secundam extrahitur fluens ex æquationibus affectis fluxionem non involventibus, Per tertiam extrahitur fluens ex æquationibus affectis fluxionem simul involventibus. Per quartam extrahit fluens ex conditionibus Problematis. Regulæ duæ primæ in principoio Epistolæ superioris duæ anno 1665 inventæ sunt pstis gruentis apertæè tradantur traduntur & anno 1665 inventæ sunt (pag: duæ ultimæ in fine hujus celabantur hic occultantur tradebantur. Harum Regularum Newtonum esse primum inventorem nemo dubitat. Analysin eandem alij aliter explicuerunt, nemo ampliorem reddidit. Collinius anno 1669 a D Barro accepit 34 In the same Letter of 27 Aug. 1676 after Mr Leibnitz had done with had described his Quadrature of the circle & Hyperbola equilateral Hyperbola, he added: Vicissim ex seriebus Regressuum pro HyperbaHyperbola hanc inveni — — — — — quæ in Newtoni Epistola expressa est; scilicet erit

n = \frac{l}{1} + \frac{l^{2}}{1 \times 2} + \frac{l^{3}}{1 \times 2 \times 3} + \frac{l^{4}}{1 \times 2 \times 3 \times 4} + &c

. .... Quod regressum ex arcubus attinet incideram ego directe in Regulam, quæ ex dato arcu Sinum complementi exhibet. Nempe sinus Complementi

= 1 - \frac{a^{2}}{1 \times 3} + \frac{a^{4}}{1 \times 2 \times 3 \times 4}

&c. Sed postea quoque deprehendi ex ea illam nobis communicatam pro inveniendo Sinu recto qui est

\frac{a}{1} - \frac{a^{3}}{1 \times 2 \times 3} + \frac{a^{5}}{1 \times 2 \times 3 \times 4 \times 5}

&c posse demonstrari. And yet in the same Letter he desired Mr Newton to explain further to him the extraction of affected roots & the method of regressiōon by wchwhich alone these series were to be found. [By this one would think that Mr Leibnitz had found these four series or at least the three first of them by the inverse method of series or method of Regressions, sometime before he received that method in Mr Newtons Letter, & that he now understood that method very well, & yet [& yet he wanted both the invers & direct method of series when he wrote his Letter of 12 May 1676 desiring Mr Oldenburg to procure him the Demonstration of the direct & invers series expressing the relation between the Arc & the Sine, & by the Demonstration meaning the direct & inverse methods of finding those series And when at his request Mr Newton in his Letter of 13 Iune 1676 had sent him those methosds, he did not understand those methods but w in his one would think that in point of candour he should not have been so forward in turning them against the Author, as if the Inventor of the method did not know how to make use of it. And much more would one think that he understood the method of regressions by which he pretended to have found the inverse method of series the three first of those four series. For when had And yet in the same Letter of 27 Aug 1676 afte [when he had put in his claim to that Method the three first of those series as found by that method, he forgot himself & after a few lines] he desired Mr Newton to explain it further to him.] His words are. Sed desderaverimdesideraverim ut Newtonus Clarissimus Newtonus nonnulla quoque amplius explicet: ut Originem Theorematis quod initio posuit,: Item modum quo quantitates p, q, r in suis operationibus invenit: Ac denique quomodo in Methodo Regressuum se gerat ut cum ex Logarithmo quærit numerum Neque enim explicat quomodo id ex methodo sua derivetur. He pretended to two series for the Two of the four series wchwhich he pretended to were for finding the Nūumber from yethe Log. & yet in the same Letter he desired Mr N. to tell him yethe method of finding them. When Mr Newton had received this Letter, he wrote back that all the said four Series had been communicated to Mr by him to Mr Leibnitz, the two first being one & the same series in different circumstance, & the third being the excess above of the Radius above the versed sine which was communicated. Whereupon Mr Leibnitz has desisted from his claim. AndHe wWhen Mr Newton in the same Letter had explained what Mr Leibnitz desired he added two series for faciliating the method of Regressions And Mr Leibnitz in his Answer dated 21 Iunij 1677 desired that Mr Newton would send him in the continuation of those two series against him But a month three weeks after in a Letter dated 12th Iuly 1677 he added: Relectis Newtoni literis video id facile non tantum ex ejus extractionibus derivari sed et altera illa methodo sub finem literarum ejus exposita inveniri qua me quoque aliquando usum in veteribus meis schedis reperio. Sed cum in exemplo quod forte in manus meas inciderat sumpseram, nihil prodijsset elegans solita impatientia eam porro adhibere neglexisse. If That is, had he had found the method of Series so long ago, that before the year 1676 he had forgot the inverse method before the year 1676 when the two series shewed him by Mohr would have made him understand the use of it it had then remembred it & before the year 1675 when the several eight series sent him by Mr conteined many instances of the use of it if he had then remembred it & before the year 1674 when the series Theorem for finding the arc by the sin whose sine given would give afforded him an instance of the usefulness of it if he had then remembred it. And before the year 1673 when the series for finding the Arc whose tangent is given would have afforded him an instance of the usefulness of it if he had then remembred it. [He had therefore invented the method of series so early that he had forgot one half of it before he left L (the method of Regression) before he left London.] He could not find the inverse method before he had the direct method. Mr Newton at his request had sent him both methods. As soon as he had them & understood them, he replied that he had found them so long ago as to have forgot that he had ever found the inverse method. In the year 1668 Mr O In the year 1676 Mr Oldenburg Mohr had given him an elegant Instance of this method of Regressions as above. In the year before that Mr Oldenburg had sent him three of or elegant instances of it. In the year before that he pretended to have had a Theoreme for finding the Arc whose sine was given & wanted a Theoreme for finding the sine whose Arc was given wchwhich would have been an elegant instance. In yethe year 1668 Vicount Brounker & the Mr Mercator published a Theoreme for finding the Logarithm whose Number was given, Mr M Leibnitz & the inverse Theorem for finding the Number from the Logarithm given would have been an elegant instance. This Theoreme he wanted & yet he threw away the method method of finding it He wanted it his Theorem so much as to write three lines to Mr Oldenburg for the method of finding such Theoremes the method of finding this & such like Theorem, & the last time & the last desired expresly that Mr Newton would explain to him quomodo in methodo Regressuum se gerat ut cum ex Logarithmo quærit Numerum. But as soon as he understood it the method, it was his own: for he used it long ago then found in his old papers that he had used it long ago, but had neglected it for want of an elegant instance. And if he had found the inverse method of series so long ago as to have forgot that he had found it, much more had he found the direct method long ago, A man may forget things & find them again in his old papers, but he t to in cases of right such pretences are not to be regarded wthwithout proof. Mr Newton deserved to have been treated wthwith more candour & gratitude. In the Acta Eruditorum Mensis Ianuarij 1689 pag 37 Mr Leibnitz making mention of the method of Series, wchwhich Mercator represents that Mercator found them by division & Newton enlarged the method by extractions of roots both pure & affected & then adds. A me ut obiter hic dicam, methodo serierum promovendæ, præter transformationem figurarum irrationalium linearum in rationsales symmetras (voco autem rationales quarum Ordinatæ semper ex abscissis haberi possunt in numeris rationalibus) excogitata est ratio pro curvis transcendenter datis, ubi ne extractio quidem locum habet. Assumo enim seriem arbitrariam, eamque ex legibus problematis tractando obtineo ejus coefficientes. This Transformation of Figures is no part of the method of series. It's only a Lemma for towards doing that sometimes by the division of Wallis, (not always but in a very few cases,) wchwhich might may be always always done more readily by the extraction of roots without it The other method of assuming an arbitrary series is Mr Newton's. In his Letter of 1676 he set it down in this sentence. Atl Altera [methodus consistit] tantum in assumptione seriei pro quantitate qualibet incognita ex qua cætera commode derivari possunt et in collatione terminorum seriei resultantis homologorum æquationis resultantis ad eruendos terminos assumptæ seriei. So then Mr Leibnits has no right to the invention of any part of the method of Series. Yet there is a sort of series the invention of wchwhich is due to him, & those he mentions in yethe end of his Letter dated 3 Feb. 1673. Modum habeo, saith he, summam inveniendi seriei fractionum in infinitum decrescentium, quarum Numerator unitas, nominatores vero numeri Triangulares, aut Pyramidales, aut Triangulo-triangulares &c. These are found in the following manner 35

\begin{array}{r} 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \\ - \frac{1}{2} - \frac{1}{3} - \frac{1}{4} - \frac{1}{5} - \frac{1}{6} \\ \frac{1}{2} + \frac{1}{6} - \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + &c = 1 . \end{array}

\begin{array}{r} \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \\ - \frac{2}{2} - \frac{2}{3} - \frac{2}{4} - \frac{5}{5} - \frac{2}{6} \\ + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} \\ \frac{1}{3} + \frac{1}{12} - \frac{1}{30} + \frac{1}{60} + &c = \frac{1}{2} \end{array}

And these are found in the following manner. De serie From the series

1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}

&c aufer os terminos omnes post primum et manebit take all the terms but the first & there will remain

1 = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + &c

. De hac serie aufer omnes terminos post primum & manebit From this series take all the terms but the first & there will remain

\frac{1}{2} =

\frac{4}{12} + \frac{6}{72} + \frac{10}{600} + \frac{12}{1280} + &c

seu =

\frac{1}{3} + \frac{1}{12} + \frac{1}{30} + \frac{1}{60} + \frac{1}{105}

& sic deinceps so on. And in generall, from even any regular series by subducting all the terms but the first or two first or three or four first will give a new series equal to yethe first or two first or three or four first terms. This method seems to consist in subducting all the terms of from any regular series exce all the terms except the first or two first, or three or four first. From the series

\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

, &c subduct all the terms but the first & there will remain

1 = \frac{1}{1, 2} + \frac{1}{2, 3} + \frac{1}{3, 4} + \frac{1}{4, 5}, + &c

= \frac{1}{1} - \frac{1}{2} + \frac{1}{3} -

And from this series take all the terms but the first & there will remain

\frac{1}{2} = \frac{2}{1, 2, 3} + \frac{2}{2, 3, 4} + \frac{2}{3, 4, 5} + \frac{2}{4, 5, 6}

, &c. And from the first series take all yethe terms but the two first & there will remain

\frac{3}{2} = \frac{2}{1, 3} + \frac{2}{2, 4} + \frac{2}{3, 5} + \frac{2}{4, 6}

&c And from this series

\frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9}

, &c take all the terms but the first & there will remain

1 = \frac{2}{1, 3} + \frac{2}{3, 5} + \frac{2}{5, 7} + \frac{2}{7, 9}

, &c. He pretended to two series for finding the number whose Logarithm was given, & yet wāanted the method of finding those series. And when Mr Newton had further explained it he still desired him tagain to explain it further & tat length when he understood it he wrote back that he had it long ago as he found in his old Papers, but had neglected it for want of an elegant example. 36

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By his In the year 16875 when he wrote the letter letter to Mr Collins printed by Dr Wallis I gather that he had did not then use the differential method as he would have done had he then known it. It does not appear that he used discovered his knowleding any thing of it before the receipt of SrSir Is. Newtons two Letters from Mr Oldenburg A.C. 1676 SrSir Isaac then In those letters letters SrSir Isaac discovered that he knew it & had written a treatise of it five years before. He represented that it was a method wchwhich did not stick at fractions & surds, wchwhich extended to the Problemes of Quadratures direct & inverse tangēents & others more difficult & gave an instance of its great extent in the problemes of Quadratures setting down a general series which ran in infinitum & gave the quadrature in an infinite series when it was not expressible by a finite æquation but when it was expressible by a finite equation, brake off & gave that equation & w wrote in the same letter. that th he other ser that this was the first of Theoreme of those wchwhich he there spake of & therefore he had at that time other Theoremes of a higher kind derived from that method. This Theoreme is the fift in his book of Quadratures & thence it is certain that the is for squaring the sixt is of the same kind for & dother the same thing in trinonomialstrinomials wchwhich the fift doth in binomials & the third & fourth are introductory to the fift & sixt & therefore these four Propositions were then known to him as the result of the method of fluxions: & so were the sd ninth & tenth with the Tables of quadratures set down in athe Scholium of the 10th. For he tells in the second of the two letters above mentioned he did set down the Ordinates of these curveds & ts gave saith said that he had tables of all their Quadratures: at wchwhich Tables could not have been computed wthwithout the knowledg of the 9th & 10th Propositions. In the same letters he sets down ænigmatically this Probleme Data æquatione fluentes quotcunque quantitates involvente invenire fluxiones & contra vice versa. And the first part of this Probleme is the first Problem in the book of quadratures & the second Probleme is a very easy & obvious corollary of the first Probleme or rather it is nothing else then the first pro Probleme applied to the case of Quadratures & therefor it must be allowed that SrSir I. N. had before the Propositions in the book of Quadratures were known to SrSir Isaac before the writing of the said two letters to Mr Oldenburg & by consequence that he had then not only invented the method of fluxions but also carried it to a higher pitch then Mr Leibnitz was able to carry it under the name of the differential method during the 30 years wchwhich followed before yethe publishing of the book of Quadratures notwithstanding the light wchwhich Mr Newton gave him into it this method it by those two letters. And by all these considerations it is manifest that Mr Newton was the first inventor of the method. And the same thing is further manifest by his little Tract de Analysi per series numero terminorum infinitas written in the year 1669 ✝ ✝ & mentioned in the Mr Newtons second Letter of Mr L to Mr Oldenburg & lately published by Mr Iones. & by the letters of Dr Barrow & others found amongs the papers of Mr Collings & p published wthwith that book Tract by Mr Iones. SrSir Isaac was therefore the first inventor of this method, & the next question is whether Mr Leibnit & how far Mr Leibnits was a collateral inventor, or had it from SrSir Isaac. And that he might have it from SrSir Isaac is [And if he had it not from SrSir Isaac by means of the lett correspondence wchwhich he kept with Mr Collins yet he before the writing of the Mr Newtons two letters above mentioned to Mr C receipt of Mr Newton's two letters above mentioned yet a man of his parts might easily have it from those two letters.] For the infinite series described in those letters are nothing else then the method the summ of the indefinite or fluent quantity & all its differences. The first term or g is the fluent quantity, the second is the first difference, the fthird is the second difference, & so on in infinitum. [Let x be a fluent quantity of & dx & let o be its difference & the binomium quantity

{x + o}^{\frac{m}{n}}

being by Mr Newton's rule set down in these let first of resolved into those two letters resolved into an infinite series, becomes

x^{\frac{m}{n}} + \frac{m}{n} o x^{\frac{m - n}{n}} +

m−n &c, where

x^{\frac{m}{n}}

is the first term

x^{\frac{m}{n}}

is a the fluent quantity &

\frac{m}{n} o x^{\frac{m - n}{n}}

the second term

\frac{m}{n} o x^{\frac{m - n}{n}}

is the its first difference. Now instead of o if wthwith Mr Leibnits you put the symbol dx the series becomes

x^{\frac{m}{n}} + \frac{m}{n} d x x^{\frac{m - n}{n}} + &c

wchwhich

\frac{m}{n} dx x^{\frac{m - n}{n}}

is the in the language of Mr Leibnitz is the difference of the quantity

x^{\frac{m}{n}}

. A] Let the binom For By the rule set down in Mr Newtons first Letter Letter the binomium

{x + d x}^{\frac{m}{n}}

is resolved into this infinite series

x^{\frac{m}{n}} + \frac{m}{n} x^{\frac{m - n}{n}} d x + \frac{m m - 2 m n}{n n} x^{\frac{m - 2 n}{n}} dd x + &c

where the first term

x^{\frac{m}{n}}

being the fluent the second term

\frac{m}{n} x^{\frac{m - n}{n}} dx

is the its first difference & the third term

\frac{m m - 2 m n}{n n} x^{\frac{m - 2 n}{n}} ddx

is its second difference & so on in infinitum. And wherein I pray does Mr Leibnitz his differential method consist but in the invention of these differences 37 And in maintainging it declined to give any reasons for his claim, insisted upon his own candor as if it were jnjust to question it, pressed that Mr Newton should declare his opinion as the only Man now alive who sufficiently understood this matter, refused to contend with any man but Mr Newton as if younger men were novices & uncapable of understanding the ancient Letters & Papers wchwhich remained upon this subject & under the person of a nameless Mathematician & two other pretended nameless correspondents, set on foot a triple Libel against Mr Newton full of indecent railing & injurious accusations without any proof. Whether that Libel was writ by himself or his correspondents is not material. If by his correspondents, they were imployed by him & he is answerable for what they have writ untill he produces their names. And if he refuses to produce their names he will deserve to be reputed the Author. As for the symbols used in his method 4 As for the symbols used by him Mr Leibnitz they are of a later date then Mr Newtons. He has no symbols for fluxions: Mr Newtons are the oldest & the only symbols of that kind. His The differ symbols of differences ax & dy are later by eight or ten years then the rectangles under the fluxions & the moment o used by Mr Newton in the same sense. And thes symbols of summs

\int x

, f

\int y

are later by 16 or 20 years then MrNewtons of including the Ordinate or Fluxion in a square to signify the Area or fluent. 5 And if the Methods be compared, Mr Newtons is the more elegant because he uses but one infinitely or indefinitely small quantity signified by the letter o. It is the more Natural & Geometrical because founded upon the rationes primæ quantitatum nascentium wchwhich have a being in Geometry, whilst indivisibles infini & quantitates minimæ or primæ nascentes upon wchwhich the method differential method is founded have no being. There are rationes primæ quantitatum nascentium but not differentqiauantitates prima nascentes. Nature generates quantities by continual flux or increase & the ancient Geometers admitted such a generation of areas & solids where they drew one line into another by motion to generate an area & the area into a line to generate a solid. But the summing up of indivisibles to generate an area or solid was never yet admitted into Geometry. Mr Newton's method is also of greater extent being adapted either for finding out a Proposition or for demonstrating it: Mr Leipbnitzes is only for finding it out. When the work succeeds not in finite equations Mr Newtons method has recours to converging series & thereby his method becomes incomparably more universal then that of Mr Leibnitz wchwhich is confined to finite equations. And where the law of the fluxions is not known F but the fluxions are had only in a few particular cases Mr Newton finds that Law quam proxime by drawing a Curve line through any number of given points & thence deduces the fluents & solves solution of the Probleme; & on this Account also his method is more universal then that of Mr Leibnitz. [So then the Differential method is only a part branch of Mr Newtons general method, & if Mr Leibnitz would have that part to be his own, it lies upon him to prove that he had it before the year 1677. All other arguments are in vain.] Mr Leibnitz indeed pretends to a share in the method of series The method of M. Fermat de maximis & minimis Mr Newton in M. yo retained & made universal without usiningusing any more symbols then one for infini indefinitely small quantities. Dr Barrow began to use more symbols then one for infinitely small quantities drawing of Tangents & Mr Leibnitz changed the Symbols of Dr Barrow into dx & into dx & dy & from this Notation gave the method a new name without ever acknowledging himself obliged to the Doctor. And Mr Newton made the method general in the years 1665 & 1666, Mr Leibnitz in the year 1677. And if he Mr Leinits pretends to have done found it earlyier, it lies upon him to prove & to give the world an fair account when & by what means he found it. Mr Newton publishinedg his Treatise of Quadratures in the year 16704. This Treatise had lain by him many years & that it might not be taken for a new piece was written long before many things being cited out of it in his Letter of 24 Octob 171 1676. That it might not be taken for a new piece Mr Newton repeated what Dr Wallis had published nine years before without being then contradicted when by the Editors of the Acta Lipsiensia or by Mr Leibnitz himself, Mr namely that he found the found the method of fluxions was invengrteaduallyd by degrees p in in the years 1665 & 1666. Herupon the Editors of those Acta who composed ( in Ianuary 1705 in the style of Mr Leibnitz (who in those days made extracts of books for the) represēend that Mr Leibnitz was the first inventor of the method & ytthat Mr Newton had substituted fluxions for differences which was all one as to tax him with falshod & plagiary.. this was in Ianuary 16705 And Mr Keil in an Epistle published in the Philosophical Transactions for May & Iune 1708 replied: retorted the accusation, saying: Fluxionum Arithmeticam sine omni dubio primus invenit New D. Newtonus ut cuilibet ejus Epistolas a Wallisio editas legenti facile constabit. Eadem tamen Arithmetica postea mutatis Nomine & Notationis modo, a Domino Leibnitio in Actis Eruditorum edita est. And this was the beginning of the prsentpresent controversy. Mr Leibnitz in a Letter to Dr Sloan dated 4 Martij st. n. 1711 appeald to Mr Newton & gave his reasons agtagainst MrKeill & desired that the R. Society would make cause Mr Keil to make a publick recantation. Mr Keill shewed shewed the Acta Lipsiensia to Mr Mr Newton twheo Acta Lipsiensia had not seen then & chose rather to explain & defend what MrKe he had affirmed then to rittract. And Mr Leibnitz in a second Letter to Dr Sloan dated 29 Decem 1711 insisted upon his own candor as if it would be injustice to exprest that he should defend it instead of making good his accusation insisted only upon his own candor, as if it would be unjust to question it & against MrKeil;, [justified what had been published in the Acta Lipsiensia, & yet [& yet allowed that Mr Newton had found the method of fluxions by himself, apart though though not so early;] called Mr Keil a novice unacquanted with things past & one that acted g without hi authory frōom Mr Newton, & a clamorous man, & sdsaid ytthat yethe Acta Lips. had given every māan his due & appealed to the judgment of Mr Newton himself as the only man (now Barrow, Gregory, Oldenburg, Collins & Wallis were dead) who understood those matters rembred who last knew what had been done formerly. But Mr Newton had given his opinion before in the Introduction to his Book of Quadratures & Mr Leibnitz had a mind that he should retract it desired that Mr Newton himself would give his opinion in this matter. He knew ytthat Mr Newton had already given his opinion in the Introduction to his Analysis book of Quadratures: but Mr Newton must retract that opinion retract that opinion & allow that he had substituted fluxions for yethe differences of Mr Leibnitz was the first Inventor, & then Mr Leibnitz would allow him to be that Mr Newton was the second Inventor & sufer him to be quiet. or not be quiet. The R. Society therefore having as much authority over Mr Leibnitz as over Mr Keil & being now twice pressed by Mr Leibnitz to interpose & seeing no reason to condemn Mr Keill wthwithout inquiring into yethe matter & that Mr Leibnitz declined to make good his accusation dered appointed Committee & that neither Mr Newton nor Mr Lebnitz (the only persons alive who remeembred what had passed in these matters 40 years ago) could be witnesses for or against Mr Keil, & that Mr Leibnits instead of making good his accusation declined to make good his accusation appointe & that there were no others alive who remembred what had passed in these matters 40 years ago, appointed a Committee to search old Letters & Papers & report their opinion thereupon, And ordered the Letters & Papers to be published together with the opinion of their Committee, by wchwhich opinion Mr Newton was the oldest inventor had the method [above five years before] the year the writing of his Letter dated 13 Iune 1676 & sometime] before the writing of his Letters dated 10 Decem 1672 & th above five years before the writing of his Letter dated 13 Iune 1676 & even before the writing of his Analysis communicated by Dr Barrow to Mr Collins in Iuly 1669; & so was the first inventor in the year 1669 or before & it appeared not that Mr Leibnitz had it before the year 1677. Its objected that Mr the R. Society heard have not heard Mr Leibnitz reasons, & therefore their judgment is void. But this is a sophistical Objection this is a sophistical objection. They If it be pretended that Mr Leibnitz has not yet produced his reasons for himself: I answer that in all the Letters wchwhich passed between him & Mr Dr Wallis about these matters he was no [wherein he defended himself wherein he defended himself as well as as well as he was able], he was not a produced not aney one reason to prove either that he had the differential method before the year 1677 or that Mr Newton had it not be above ten years before, as Dr Wallis had affirmed. that time. He produced not any one argument against Mr Fatio to prove that Mr Newton was the oldest inventor by many years. He did not then deny that Mr Newton was the first inventor In his first Letter agtagainst Mr Keill he produced what his reasons against what Mr Keill had said. ButAnd after Mr Keil had explained himself & answered those reasons: he produced no more reasons but in his next Letters for want of reasons cried out cried out. Quæ D. Ioannes Keillius nuper ad Te scripsit, candorem meum apertius quam ante oppugnant: quem ut ego hac ætate, post tot documenta vitæ, Apologia defendam, & cum homine docto sed novo, & parum perito rerum anteactarum cognitore, nec mandatum habente ab eo cujus interst, tanquam pro Tribunali litigam nemo prudens æquusque probabit. That is, he told the R. Society that they would be unjust unless they would allow him to be judge a witness in his own case. 38 And am not I as good a witness that I invented the methods of series & fluxions in the year 1665 & improved them in the year 16766, & that I still have in my custody several mathematical papers some of written in the years 16764, 1665 & 1666 some of wchwhich happen to have been dated & that in one of them dated 13 Novem 1665 the direct method of fluxions is illustrated with Examples & demonstrated, & that down in this it is set down in these Problemposite words: An Equation being given expressing the Relation of two or more lines x, y, & z & described in the same time by two or more moving bodies A, B, C &c, to find the relation of their velocities p, q, r &c. And that the resolution of this Problem is there set down in these words Set all the terms — — — — — relation of p, q, r &c & that this Resolution is there illustrated with examples & demonstrated And that in another Paper dated 16 May 1666 a general method of resolving Equations is Problems by motion is set down in seven Propositions the last of which — — — — quantitatum inter se. And that in confirmation of all this I am ready to produce the original papers at this time I had made my Analysis composed of the methods of series & fluxions together so universal as to reach to all almost all sorts of Problemes as I mentioned in my Letter dated 13 Iune 1676. These things tdo not rely meerely upon my own testimony, but the ancient papers Manuscripts themselves are ready to be produced. — And that in this Tract when any of the the area arising from any of the terms in the Valor of the Ordinate cannot be expressed by Vulgar Analysis I represent it by prefixing the symbol ▯ to yethe term. As if the Abscissa be x & the Ordinate

a x - b + \frac{b b}{b + x}

, the area will bee

\frac{1}{2} a x x - b x + ▯ \frac{b b}{b + x}

. And that in this This the same Tract I sometimes used a Letter with one prick for f quantities involving first fluxions & the same Letter with two pricks for quantities involving second fluxions. And that the larger Tract wchwhich I wrote in the year 1671 was founded upon this smaller & Tract & began with the Reduction of finite quantities to converging Series & with these the solution of two Propblems & the s set down in these words. 1. Relatione quantitum fluentium inter se data fluxionum relationem determinare. 2. Exposita Æquatione fluxiones quantitatum involvente invenire relationem quantitatum inter se. The original Manuscripts are still extant to have justify what is here represented & have been seen been seen by Mathematicians many years ago. Mr Fatio mentionsed them inat his the end of his book of abo entituled Fruit walls improved, & which book was published in the year 1699. And in my Letter wh of 24 Octob 1676 I mentioned the last of these Tracts sas written five years before that Letter & about the method of series & another method which was founded in this sentence Data æquatione fluentes quotcunque quantitates involvente invenire fluxiones & vice versa. And in the end of that Letter I added that I had a method of extracting fluents out of equations involving their fluxions: which implies that I then knew how to reduce Problems to fluxional equations. And by my Letters of 10 Decem. 1672 & 13 Iune 1676 it appears that I had then made my Analysis composed of the methods of Series & fluxions together so universal as to reach to allmost all sorts of Problems. But Mr Leibnitz has no acknowledged that in Feb 1673 he knew & for some months after, he knew little or nothing of the higher Geometry but at length learnt it afterwards of Mr Huygens at Paris, [& pretends only not that he found the Differential Method in before the year 1676 but nor brings not so much as one argumtargument to prove that he found it before the year 1677, except that wchwhich I am still going to answer at the wchwhich is drawn from his solving the Problem of Beaune the pretence that he solved the Probleme of Beaune by his method had s unles certa Anslysis in his Letter of 27 Aug. 1676 is y must signifies the differential Analysis] & that he had not the differential method till after the year 1675 The first of these shews what was the opinion of was writ before Mr Leibnitz before he knew any thing more of my method then what he had from my Letters & Pap writ in or before my the year 1676 & from the Principia use ytthat use that I made of this Principia Philosophiæ the second s mathematica,. & the f The second shews was the notion of the English Mathematicians tradition in England before they we heard that this Method began to be celebrated in Holland as invented by Mr Leibnitz. Andmong the said Letters & Papers were also several papers written by my self in the years 1664, 16565 & 1666 some of which happen to be dated Et Me Commercium Epistolic oblivione minime respondetum olivionioblivioni tradere conæretur tanquam nimis longo prolixo cui responderet, nec teamen satis prolixūmprolixum ob Epistolas qui ob b epistolas t. Commercio Epistolico tanquam respondere recusaret, & convitijs uteretur 1Cum D. Leibnitius 5Epistolas præcedentes prius mitteret in Galliam quam earum tertia in Angliam veniret, 2Commercio Epistolico respondere aperte recusaret nulla pro se argumenta licet provocaturs afferret, & convitijs utere verbose ser & convitijs uteretur jus s 3causam suam probare licet provocaturs minime aggrederetur, & a4verbose scriberet, et 4convitijs uteretur &: 6Newtonus minime rescripsit, sed Observationes quasdam sequentes in tertiam illam scriptas cum amicis solummodo communica Cum D. Leibnitius convitijs uteretur et addue non posset ut vel Commercio Epistolico responderet vel argumenta pro se afferret; s a convitijs nam abstineret, et cumque nec cumque præcedentes Epistolas præcedentes is in Galliam prius prius mitteret in Galliam quam earum tertia in Angliam veniret & prætenderet se hoc facere ut testes haberet et alias etiam pro & alijs utetur convitijs argumentis contumelas adhiberet: Newtonus minime rescripsit sed Observationes sequentes in tertiam illam Epistolam scriptas cum amicis solummodo communicavit. Observationess in Epistolam præcede upon the preceding Epistle. 39 P. S. When the Committee of the R. S. published the Commercium Epistolicum the Papers in my custody were not produced, & it will be more usefull to read the progress by which the method was invented. Among my papers written in the years 1664, 1665 & 1666 some happen to be dated: amongst wchwhich is that which follows 13 Novem. 1665 Probleme. An Equation being given &c In a little Tract written in the In a paper dated 16 May 1666 my method of resolving Problems by my motion is set down in 7 Propositions the last of which is the same with that conteined in the Paper of 13 Novem 16665 tho exprest in other words. In a little Tract written in October 1666 the same method is set downin the same seven Propositions but the seventh is enlarged & an eighth is added to them. That which is added to the seventh is in these words Note that if there happen to be in any equation either a fract or surd quantity or a mechanical one (i.e. wchwhich cannot be Geometrically computed but is expressed either by the area or lenghlength or gravity or content of some curve line or solid &c) to find in what proportion the unknown quantities increase or decrease do thus. Take two ........ required Exempl. 1. To find p & q the motions of x & y whose relation is

y y = x \sqrt{a a - x x}

........ wchwhich was required ExamleExample 2. If

x^{3} - a y y + \frac{b b^{3}}{a + y} - x x \sqrt{a y + x x} = 0

is the relation is the relation sought. Example 3. If x=AB upon the

\dot{x} = AB + BC = \sqrt{a x - x x}

, [that is if

BC = \sqrt{a x}

to the line

AB = x

the line BC

= \sqrt{a x - x x}

be ordinately applied.] & the superficies be called z Suppose that

z z + a x z - y^{4} = 0

is the relation between x y z Example 3. If to yethe line

AB = x

, the Ordinate

BC = \sqrt{a x - x x}

be appl ordinately applied at right angles, & y be put for any indeterminate line BE & the area superficies ABC be called z: Suppose that

z z + a x z - y^{4} = 0

is the op relation between the lines x y & z whose motions are p, q, & r, & that p & q are desired. And The Equation

z z + a x z - y^{4} = 0

gives (by Prop. 7)

2 r z + r a x + p a z - 4 q y^{3} = 0

. Now erecting the perpendicular

AD = 1

& completing the parallelogram ABHD I consider that the superficies

ABHD = AB \times BH = x \times 1 = x

, that is, 1.

\sqrt{a x}

\sqrt{a x - x x} ∷ p . r

. Or

r = \sqrt{a x - x x}

. Which valor of r being substituted into the equation

2 r x + r a x + p a z - 4 q y^{3} = 0

, gives

2 p z + 2 p x \times \sqrt{a x - x x} + p a z - 4 q y^{3} = 0

, wchwhich was required. The eighth Proposition was this. If two bodies A & B with their velocities p & q describe the lines x & y, & an equation be given expressing the relation between one of the lines x & the ratio

\frac{p}{q}

\frac{q}{p}

wchwhich or the ratio of their motions q & p: to find the other line y. And the resolution of this Proposition is here set down by in these steps Rules. 1 If the valor of

\frac{q}{p}

1. Get the valor of

\frac{q}{p}

& if it be rational & its denominator consists but of one term, multiply that valor by x & divide each term of it by the logarithm of x in that term [that is by the index of its dignity] the Quote shall be the valor of y. As if

a x^{\frac{m}{n}} = \frac{q}{p}

then is

\frac{n a}{n + m} x^{\frac{n + m}{n}} = y

. 2 But if the denominator of the valor of

\frac{q}{p}

consist of more terms then one, it may [usually] be reduced to such a form that the denominator of each part of it shall have but one term so that y may be then found by the precedent Rule Which Reduction is thus performed. 1. If If all the terms of the Denominator be not multiplied by x or xx or x3 &c Increase or diminish x untill the last term of the Denominator vanish. 2. And when all the terms of the Denominator are multiplied by x, xx or x3 &c divide the numerator by the Denominator (as in decimal numbers) untill the Quotient untill the Quotient consist of such parts none of whose Denominators are so multiplied by x, xx x3 &c, & begin the division in those terms in wchwhich x is of the its fewest dimensions: unless the If then the terms in the valor of

\frac{q}{p}

be such as was before required the valor of y may be found by the first Rule only it must be so much incre diminished or increased as it was before diminished or increased by increasing or diminishing x. But if the denominator of any term of more terms then one, find those the parts of y's valor wchwhich correspond to severally. Example 1. If

\frac{x x}{a x + b} = \frac{q}{p}

, then by division is

\frac{x}{a} - \frac{b}{a a} + \frac{b b}{a^{3} x + a a b} = \frac{x x}{a x + b} =

\frac{q}{p}

(as may appear by multiplication) Therefore (by the first Rule)

\frac{x x}{2 a} - \frac{b x}{a a} + ▯ \frac{b b}{a^{3} x + a a b} = y

. Examp. 2. If

\frac{x^{3}}{a a - x x} = \frac{q}{p}

. I suppose

x = z - a

, or

\frac{z^{3} - 3 a z z + 3 a a z - a^{3}}{2 a z - z z} =

\frac{q}{p}

And by division

- \frac{a a}{z z} - z + a + \frac{a a}{4 a - {}^{2}z} = \frac{q}{p}

, as may appear by multiplication. And substituting

x + a

into the place of z, you have

x - \frac{a a}{2 x + 2 a} + \frac{a a}{2 a - 2 x} = \frac{q}{p} =

\frac{x^{3}}{x a - x x}

And therefore by the first Rule,

\frac{x x}{2} + ▯ - \frac{a a}{2 x + 2 a} + ▯ \frac{a a}{2 a - 2 x} = y

. 40 He saith that he formerly beleived me & on that account acknowledged that I had found the method of fluxions apart, but now now M. Bernoulli has given him reason to suspect me, he is at liberty to retract. But his Letter of to Mr N. dated

\frac{7}{17}

Mar. 1693 & herunto annexed was written before he had any other notice of my method then from Mr Oldenburgh Mr. Collins & the Principia Philosophiæ, & therefore he is not at liberty to retract. He questions my credit in what I say about Dr Wallis & therefore I have subjoyned the Paragraph in the his Preface of Dr Wallis together with theis Doctors Letters relating to it. The DrDoctor there plainly tells him that he had the Meth that Mr N. in the year 1676 I explained to him the Method of fluxions invented by me ten years before those days or above. Dr Keil has not affirmed so much Mr L. beleiv did not think fit to contradict M Dr Wallis in those days but is now very angry angry at Dr Keill. P. S. When the Committee of the Royal Society published the Commercium Epistolicum, the Letters & Papers in my cCustody were not produced. Among them the were was the following Letter of Mr Leibnitz, & a Letter of Dr Wallis, dated both said which upon both wchwhich upon a fresh occasion have been produced & in more left in the Archives of the R. Society,. Among them were also several papers written by my self in the years 1664, 1665 & 1666 some of wchwhich happen to be dated. And one of them writ in the year november 1665 v hereunto subjoyned. Illustri vViro ISAAC NEWTON Goth Godefridus Gulielmus Leibnitius S.P.D. Quantum tibi . . . . . . Vale. Dabam Hanoveræ

\frac{7}{17}

Martij 1693. 13 Novem. 1665 Probleme An Equation being given &c In another Paper dated 16 May 1666 my method of resolving Problems by motion is set down in seven Propositions the last of wchwhich is the same with that conteined in this paper of Nov 13 Nov. 16665 And in a third paper da small treatise written in November 1666, the same seven Propositions are set down again & the seventh is improved by shewing how to proceed without sticking at fractions or surds or such quantities as are now called transcendent. And an eighth Proposition is added conteining the inverse method of fluxions so far as I had then attained it, namely by the methods of Quadratures & particularly by the three Rules upon wchwhich the Analysis per æquationes numero terminorum infinitas is founded & by most of the Propositions Theorems set down in the ScholumScholium to the tenth Proposition of the Book of Quadratures, [most of which are here set down in this Paper.] I And in the Tract which I wrote in the year 1671 the two first Propositions are And that in this Paper I sometimes used a letter with one prick for quantities involving f first fluxions & the same letter with two pricks for quantities involving fi second fluxions. And that in the Tract which I wrote in the year 1671 the two first Propositions were these. 1 Relatione quantitatum fluentium inter se data fluxionum relationem determinare. 2 Exposita æquatione fluxiones quantitatum involventes, invenire relationem quantitatum inter se. All wchwhich are to be compared with my Letters of 10. Decem. 1672, 13 Iune 1676 & 24 Octob. 1676. published in the Commercium Epistolicum & with the Scholium to the second Lemma of the second Book of Mathematical PriciplesPrinciples of Philosophy. And in my Letter of 24 Octob. 1676 I represented that I had written a Tra Dr sent to Mr Collins a Tra compendium of the method of series about that time that Mercator published his Logarithmotechnia & this Compendium is the Analysis per Series now extant. I mentioned also that token five years before the writing that Letter that I had is in the year 1671 I had written a Tract concerning the method of series & another method together, & that the other method was founded in this sentence Data æquationes flentesfluentes quotcunque quantitates involvente fluxiones invereinvenire & vice versa. I suppose he means be that because he finds no prickt letters there. And by the same way of arguing he may say & Mr Bernoulli may pretend that they find nothing of that Analysis in the method of fluxions in the Introduction to the Book of Quadratures, tho nothing else be there described that whole Introduction be entirely taken up in describing that this Method & illustrating it with Examples [& my saying there that I found this Method in the years 1665 & 1666 gave occasion to this controversy]. I there In that Introduction I say that I invented (not prickt letters, but) the method of fluxions in those years the years 1665 & 1666, & describe it in that Introduction there without the use of prickt letters, as I did long about thirty years twe & long before in my Analysis above mentioned. Let the world judge whether the method described in this Introduction & said here to be invented by me in the years 1665 & 1666 gradually in those years be not one & the same method with that described in the said Analysis wchwhich was & communicated by Dr Barrow to Mr Collins 47 years ago & in my Letters of 10 Dec 1672 & 24 Octob 1676 & by me concerning which I say in my Letter of 13 Iune 1676 that Analysis by my method of series thereby extends to almost all sorts of Problems (except perhaps some numeral ones like those of Diophantus) but becomes not altogether universal without the assistance of this other method it the help of some further methods, wchwhich in next Letter dated 24 Octob 1676 I describe to be the direct & invers methods of fluxions the method I explain to be the direct & invers method describe by these sentences Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire, et vice versa of extracting fluents out of equations involving their fluxions, & the method of assum assuming arbitrary series. Although I said in that Letter that the foundation of the method of wchwhich I wrote a Tract in the year 1671 was founded in this sentence Data æquatione fluentes quotcunque quantitates inolvente fluxiones invenire & vice versa — & Dr Wallis (Vol 2 Opper. pag 393 lin. 32) will has told him that that this method needs no further explication then that wchwhich I gave of it in my Letter of 24 Oct. 1676. That of DMr Leibnitz shew that he understood that I had such a genera Methodus similis as he men before he knew any thing more of it then what he had from my Letters in the year 1676 & from the Principia Philosophiæ: that of Dr Wallis that the tradition in England news tradition that I invented the method describ [mentioned there by the DrDoctor] was much older in England then the tradidtion newes that Mr Leibnitz invented it the first beginning in the year 1676 as old in england as the year 1676, at wchwhich time copies of my Letters were sent to him by Mr Oldenburg. Cum D. Leibnitius Epistolas præcedentes in Galliam prius mitteret quam Responsem ejus in Angliam veniret et Responsum illud scurriliter scriptum esset convilijs sentiret l abundaret & A Autho; Newtonus minime rescripsit, sed Observationes sequentes in Responsum illud a se scripsittas et cum amicis tantum communicavit. & convilijs abundaret, et Author Commer antiq Commercio Epistolico minime responderetur; Newtonus non rescripsit, ne monumenta antiqua sed seq. Observationes se in Responsum illud a a se compositas cum amicis tantum communicavit. Cum D. LeibnitiasLeibnitius Epistolas præcedentes in Galliam prius misisset mitteret quam Res c earum tertia in Angliam veniret, & Newtonus minime rescripsit sed sed Observationes sequ suas sequentes in tertiam illam scriptas cum amicis tantum tantum communicavit. Observationes in Epistolam præcedentem. And that in or before Se 1665 I invented the first Proposi That in the year 1665 I invented the firs methods of series & fluxions & still have in my custody a paper dated 13 Novem. 1665 in which the first Proposition of the Book of Quadratures is propounded & Resolved illustrated with examples & Demonstrated Geometrically, the letters p, q, r being here put for the fluxions of the fluents x, y, z. And That in the year 1666 I improved this method (extended it to second fluxions & & extended it to second fluxions & before the end of the year & sometimes & used sometimes letters without pricks & sometimes letter without pricks letters with one or two pricks for for firs quantities involving first or second fluxions & before the end of the year wrote a small Tract on this subject which was the ground of that larger Tract which I wrote in the year 1671, both which Add 396841 to be added the to complete the series (for he has no where said t said in all that book that he uses the word phrases second & third differences in the sence of Mr Leibnitz) the Scholium will be true wthwithout any correction.] & the Objection will vanish. This account of the Com C The credit force & evidence of the Commercium Epistolicum an & of the account here given of that matter stands upon the authority of the ancient Letters & papers there partly by Dr Wallis & partly by order of the R. Society the originals of those published by order of the Society are still preserved. If Mr Leibnits or his those ancient Letters & Papers are altogether against Mr Leibnitz it is his own fault. If the Notes upon them for enabling the reader to compare them & understand them at one reading, have in any thing misrepresented them, Mr & h Leibnitz & his friends are at liberty to set shew wherein they are misrepresented & to set the matter right set right the representation. [If they only tell a contrary story If in any thing they make for him they are at liberty to explain it. But Mr without proving what they affirm, Mr Leibnitz can be no witness in his own cause & his friends at in Germany know nothing of the matter but what they have from Mr Leibnitz or from the Records themselves.] The friends of Mr Leibnitz in Germany know nothing more of what passed between him & Mr Oldenburg when he was in England or France d then what he tells them or what they find in the Letters wchwhich then passed between them. He cannot be a witness in his own cause, & therefore what stories they affirm tell in his favour concerning those things, is not to be regarded unless they prove their affirmations by t any further then the truth thereof appears out of ancient records] But they are not at liberty to tell stories without at discretion without proving what they say. Mr Leibnitz cisan not to be ad be an evidence in his own cause & his friends know nothing of the matter his correspondence with the English 40 years ago in his but what they travells 40 years ago but what they have have from him or from the ancient Letters & Papers records. the Letters & papers then written & still preserved. 41Add 3968 No 41 It has been said that Mr Newton in the Scholium at the end of his Book of Quadratures has put the third fourth & fift terms of a converging Series respectively equall to the 2d 3d & 4th differences of yethe first Term, & therefore did not understand the method of second third & fourth differences. But in the first Proposition of that Book he shewed how to find the first second third & following fluxions & therefore did in infinitum & therefore when he wrote that Book, wchwhich was before the year 1676, he did understand the method of all the fluxions & by consequence of all the Differences. And if he did not understand it when he added that Scholium to the end of the book, wchwhich was in the year 1704, it was must have been because he had then forgot it. And so the Question is only whether Mr Newton had forgot the method of second differences before the year 1704.In the tenth Proposition of the second Book of the his Principia Philosophiæ in describing some of the uses of the terms of a converging series for the solving of Problemes he tells us that if the first term of the series represent the Ordinate BC of any Curve line ACG, & DG be another Ordinate infinitely neare to the former. CBDI be a parallelogram infinitely narrow whose side DI cuts the Curve in G & the its tangent CF in F: the second term of the series will represent the line IF, & the third term the line FG. Now the line FG is but half the second difference of the Ordinate And therefore Mr Newton when he wrote his Principia, did put the third term of the series equall to half the second difference of the first term & by consequence had not then forgotten the method of second differences. ‡‡ In writing that book he had frequent occasion to consider the increase or decrease of the velocities wthwith which quantities are described & that he generated & argues right about it That increase or decrease is the second fluxion of the quantity, & he argues righ & therefore he had not then forgotten the method of second fluxions. Nor is it likely that when he added the Scholium to then end of his book of quadratures he had the meaning of the fi forgot the first Proposition of the Book. *Much less had he forgot it in the year 1692 when at the request of Dr Wallis he sent him the explication of that Proposition & added in all sorts of fluxions, as you may see in yethe second Volume of the Doctors words pag. 391 392 & 393. If the word [ut,] which has been in that Scholium has been accidentally omitted between the words [erit] and [ejus] be restored; that Scholium will agree with the rest of his writings, & the objection will vanish. [He that knows how to find the first fluxion of any line & to expose that fluxion by a another line, knows how to find the fluxion of that line & to expose it by a third line & to find the fluxion of this line & expose it by a fourth line & so on perpetually, that is, he knows how to find the second third & fourth fluxions of the first line The method is one & the same in all the fluxions. The method is yethe And there is the same reason of the method of moments or differentias: so that And he who understands it in the first differences understands it in all yethe rest.] In the year 1692 when at the request of Dr Wallis, he sent to him a copy of the solution of explication explication of the Proposition, Data æquatione fluentes quotcunque quantitates invenire, he extended it to gave examples af thereof in first second & third fluxions as you may see in the second Volume of the Doctors works pag 391, 392, & 393 & 396. And therefore he had not then forgotten the method of second fluxions. & differences. Nor is it likely that in the year 1704 when he added the aforesaid Scholium to the end of the book of Quadratures, he had forgotten both the first Proposition of yethe book & the last upon wchwhich that Scholium was written. If yethe word [ut] wchwhich in that Scholium has bee may have been accidentally omitted between the words [erit] & [ejus] be restored; that scholium will agree with the rest of his writings, [supposing that he used the language of second third & fourth differences in the language sence of Mr Leibnitz. But if he used that language in some other sense of his own, suppose to signify the second third & fourth differences wchwhich rem remain To SrSir Isaac Newton SrSir I beg you will not think mee impertinent in troubling you with this. it is upon yethe account of yethe Medall Proceribus populisque concentientibus I think it will be better thus Senatu Populoque sancientibus Proceres is but a Metaphor & is properly Off yethe Off Beams in buildings, and expresses rather cheif Officers than Noblemen Populos is read in yethe plurall number but I think ofter in yethe singular sancire leges in Tully's word I am yoryour most obedient Serv S: Garth 42 pag. 4. l. 7, add. It has been said that Mr Newton in the Scholium upon yethe Proposition of yethe 2 book of his Principles put the third term of a converging series equal to the secon the second difference whereas it is equa of the first term whereas it is but half that difference, & therefore Mr Newton did not then understand the method of second differences. But the Objector is mistaken himself it may with as much reason be retorted be retorted that the Objector is in an error an charging Mr Newton wthwith an error is in an error himself & therefore did not understand the method of second differences himself: And Mr Newton was For Mr Newton in that Scholium puts the third term of the series equall to but half the second difference of the first term. thereof. It has been said that Mr Newton in the Scholium at the end of this book of Quadratures did has put the terms a second third fourth & fift terms of a series converging series respectively equall to the 2d 3d & 4th differences of the first term, respectively, & therefore when he wrote that Scholium that is, in the year 16 1704 he did not understand the method of second third & fourth differences. That Scholium was added to yethe book in yethe year 1704 but the book of Quad except that Scholium & the Introduction was But Mr Newton in the first Propositiōon of the Book wchwhich it self was writt before the year 1676 explained how to find & in the first Proposition thereof he has shewed bey an a generall Rule how to find the first second third & following fluxions in infinitum & therefore did then understand the method of all the fluxions. He that knows how to find the first fluxion of a any line & to expose that fluxion by another line knows how to find that that other line the fluxion of that other line wchwhich is the second fluxion of that other the first line. O on perpetually the second fluxion of the second line is the third fluxion of the first line, & so on perpetually And if this fluxion be exposed by a third line the fluxion of this line will be the third fluxion of the first line & so on perpetually. And therefore he that understands the method of first fluxions understands the method of all the rest. And there is the same reason of differences. Mr Newton in the tenth Proposition of the second book of his Principia Philosophiæ, in describing the use that may be made of the severall terms of a converging series tells us that if the first term of the Series represent the Ordinate of a Curve BC the second will represent the line IF, & the third the line FG. & Now the line FG is but half the diffe the second difference of the A Ordinate, & therefore Mr Newton in applying in applying these series to the solution of Problems did then puts the third term equal to half the second difference, [& by consequence t to reconcile this with the scholium at the end of the Book of Quadratures the word [ut] wchwhich has been twice accidentally omitted is to be restored in the line 11 & between the words erit and ejus is to be restored :[unless you say that Mr Newton when he wrote his Principia understood the method of second differences, but did not u did not understand it when he wrote the Scholium.] as it really is. Mr Newton therefore when he wrote thies Principia Philosophiæ & s book & long before that, when he wrote the first Proposition of his book of Quadratures understood the method of second differences. And if he did not understand it ten years ago when he wrote the Scholium, it was which was but tenn years ago, it must be because he has forgot it. But there is no need of saying that he had forgot it: for in that Scholium if the word [ut] wchwhich has been twice accidentally omitted between the words [erit] & [ejus] in that scholium be restored, that Scholium will agree with the rest of his writings. It has been represented that the R. Society have passed sentence without hearing both parties & therefore their sentence is voyd. Bu And indeed Mr Leibnitz pressed them to condemn Mr Keil without hearing both parties; But aAnd told them that his own candour could not be questioned without injustice: Which is the same thing as to tell them that they would be unjust if they did not allow Mr Leibnitz to be a witness in his own cause & condemn Mr Keil without hearing both parties. But the R. Society do not take upon th act as a Court of Iustice Iudicature, nor have they been desired to give their opinion in this matter unless by Mr Leibnitz against Mr Keill Keill. And Some months After the Commercium Epistolicum was published Mr Leibnitz pretendinedg that he had was not at had not seen it nor was at leasure to answer consider it, & that he had therefore desired an able & impartial MathematianMathematician to examin the matter, & had received his answer dated 7 Iune 1713. And this answer was published in Germany with additions. It is full of affirmations without any proof. The able Mathematician & the author of the additions are not named & so the whole stands upon the credit of Mr Leibnitz, & even the style is reuoed to be his reputed his We h As his Letters were full of inflexions against Mr Keill so this paper is full of reflexions against Mr Newton. And M That you may compare it with the Acct above written we have here subjoyned it Several accounts of this Commercium having been published abroad all of them very imperfect: its has been thought fit to publish the Account wchwhich follows. It has been represented in Germany that Mr Leibnitz first found out the differential Calculus in numbers, & then by inventing the Analysis of infinitesimals translated this Calculus to Geometry. [And indeed Mr Leibnitz in the year 1673 pretended to the calculus differentialis of Mouton in numbers & was reprehended for by Dr Pell for intruding into other mens inventions & wrote an Apology for himself wchwhich is printed in the Commercium Epistolicum. And In this Apology he represented that he had found out several things apart without knowing what Mouton had done before, & had added some things of his own, but unluckily instanced in a property of numbers [natural triangular pyramidal triangulo, triangular] observed before by Monsr Paschal whose book he had seen, He re & yet represented that Monsr Paschal had not observed it. He tells us also that from given numbers multiplied after a certain manner he could produce very y the numbers of very many series proceeding in infinitum, & that he could solve many Problemes in Progressions by adding subducting multiplying & or dividing the Progressions, as by dividing an unit by a series of natural, triangular, Pyramidal, or triangulo-triangular numbers; & principally that he could find the summ of a series of fractions so produced. See the mystery. From the series

\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}

&c subduct all terms but the first ter & the first term

\frac{1}{1}

will remain

\frac{1}{2} = \frac{2}{1 \times 2 \times 3} + \frac{2}{2 \times 3 \times 4} + \frac{2}{3 \times 4 \times 5} + \frac{2}{4 \times 5 \times 6} + &c

And from the first series take all the terms but the two first & there will remain

\frac{3}{2} = \frac{2}{1 \times 3} + \frac{2}{2 \times 4} + \frac{2}{3 \times 5} + \frac{2}{4 \times 6} + &c

When one of Mr Newtons se From the series

\frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + &c

subduct all the terms but the first & there will remain

1 = \frac{2}{1 \times 3} + \frac{2}{3 \times 5} + \frac{2}{5 \times 7} + &c

. And by subducting this series from the last but one there will remain

\frac{1}{2} = \frac{2}{2 \times 4} + \frac{2}{4 \times 6} + \frac{2}{6 \times 8} + &c

that is

1 = \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + &c

as above. And these were the great inventions of Mr Leibnitz about Numbers in those days wchwhich Mr Leibnitz pretended to in those days.] But this is to tell us that he first found out the differential calculus of Mouton & then by inventing the Analysis of infinitessimals translated the Moutons calculus to Geometry. The inventi His pretending to The invention of Moutons method is nothing to the purpose. It lies upon him to prove that he invented the Analysis of infinitesimals before the year 1677. In a letter to Dr Wallis dated 28 May 1697 he tells us that the consideration of differences & summs in series of numbers gave him the first light by observing that Differences answered to tangents & summs to quadratures: that is, it put him upon considering Dr Wallis's summatory method of Tangents Quadratures & Dr Barrow's differential method of Tangents. & beginning where Dr Barrow left off, as the Marquess de L'Hospital has observed. But this consideration alone could not give him light into the method of drawing Tangents without sticking at surds & into the universal extent of this method for solving all sorts of Problems. That light he had by his correspondence wthwith Mr Oldenburg as is in the years 1676 & 1 as is manifest by what has been said above. It has been pretended suggested by Mr Leibnitz himself, that the Committee of R. Society published only those papers wchwhich made against him & suppressed those that made for him. B the symbol o, tThe proportion of the fluxions (wchwhich are finite quantities) giving all all the other moments. It is more natural & geometrical And whereas Mr Leibnitz has represented that the use of the letter o is vulgar & destroys the advantages of the Differential Method 44 The reason of his inconsistency in this matter was his computing by a wonderfull unphilosophical errour, the quantity of impulsive force from the quantity of matter acquired by a falling body from the quantity of its matter & of the space delscribed by it in falling; recconing the force acquired to be in a compound ratio of the matter & the space together described by it. Now matter is as the weight thereof & the space described is as the square of the line of its falling, & therefore according to Mr Leibnitz the force acquired in falling is as the in a compound ratio of the matter falling & in a falling body weight of the falling body & the square of the time of its falling. And by consequence, where the weight remains the same, the force acquired in falling will be as the square of the time So that if the time of falling be divided into equal parts, & in the first part of yethe time four degrees of force will be acquired, in the three first parts of time nine degrees of force will be acquired, & so on perpe in the four first parts of time sixteen degrees of force will be acquired, & so on b if in th And therefore if in the first part of the time one degree of force impulsive force be acquired in the second part of the time three degrees of force will in the third will be acquired, in the third part of the time five degrees of force will be acquired in the fourth part of time seven degrees of force will be acquired, & so on. And if so the weight or gravity of the body which is supposed to be uniform & by wh by its action impresses these impulsive forces upon the body, acts with three times more force in the second parts of the time then in the first & with five times more force in the third part of the time then in the first & with seven times more force in the fourth part of the time then in the first, & so on. Which is as much as to say that the th falling falling body grows heavier & heavier as it falls, & becomes three times heaveier in the midle of the second part of the time then in the middle of the first & five times heavier in the middle of the third part of the time then in the first middle of the first, & so on. Or that the weight of the body is proportional to the time of its falling: And by consequence that in the beginning of the first part of the time the body would hathve no weight at all. Which is contrary to the Hypothesis of uniform gravity & to experience it self. The Theory of Projectiles invented by Galilæo is founded upon the Hypothesis of uniform gravity, & is generally approved by Mathematicians. Now uniform gravity is that which acts with an uniform force & in equal times equally upon it & by acting in by acting equall by acting with equal forces upon the body communicates equal forces to it. If in the first part of time it communicates one degree of force & gives one degree of velocity to the falling body; in the second part of time by acting as much as in the first, it will communicate another degree of force & give another degree of velocity to the falling body; & in the third part of time it will generate a third part of force & a third degree of velocity & so on perpetually: & therefore the time of falling the time of falling, the force imprest, & the velocity of descent will be always proportional to one another one another into the time of falling & to one another. But the space described by the falling body arises partly from the time of descent & partly from the velocity of the falling body & therefore will be in a compound ratio of them both or as the square of either of them them, & by consequence as the square of the force. And so if two equal bodies be thrown directly upwards tho one with double a velocity to double to that of yethe other, the swifter body will rise four times higher then the other in four three time twice the time & in that time the action of gravity upon it for taking away the force by wchwhich it ascends will be double to the action of gravity upon the other body, & therefore the force by wchwhich it ascended & wchwhich is taken away by that action is only double to the force by wchwhich the other body ascended. A body therefore of one pound weight is not (as Mr Leibnitz supposes in the Acta Eruditorum ad Annum 1686 pag. 162) throw in Vacuo four ft times as high but sixteen times as high by the same quantity of impulsive force wherewith a body of four pound weight is thrown one foot high. The ground of his error is that he confounds [Acta Erudit: ad Ann. 1686 pag. 162; & ad Ann. 1690 pag. 234; & ad Ann. 1691 pag. 439; & and Ann. 1695 pag. 155] the spaces described in unequal times by bodies falling downwards or thrown upwards with the spaces described in equal times by bodies rising & falling in equal times at the ends of the unequal arms of a ballance. Whereas in the first case the spaces described are as the time & the velocity together, that is, as the square of the velocity: in the second case the times being equal the spaces described are only as the velocityies. And because tis true that in an horizontal place, where gravity neither adds to nor takes from the Velocity, a body of four pound weight will at the same be carried one foot by the same force imprest at once whereby a body of one pound weight will at in yethe same time be carried on four feet: therefore he supposes [Ancta Erudit: ad Ann. 1686 pag 162,] that in a perpendicular plane wherein the force is not imprest at once but Gravity continually augments or diminishes the force impr velocity in proportion to the time of its acting; a body of four pounds weight will be thrown upwards one foot by the same force imprest at once whereby a body of four one pound weight will be thrown upwards carried up four foot; or that a body of One pound weight will in falling four foot, will acquire the same impulsive velocity force as a body iof four pounds weight in falling one foot. Then wchwhich nothing can be more contrary to experience both to reason & experience.

BD = AB = a

BC = x

BDEC = y

Cc = o

Bc = x + o

CEec = o x \frac{a a}{a + x + o} = \dot{y} o

\frac{e + f x + g x x + h y}{h o + m x + n y} = z

. Or

z = e + f x^{m} + h x^{p} y^{q}

. 109dt.17gr+41.8=151.33 46 p. 38 l. 14. The Principle upon wchwhich he founded his Dynamique is erroneous The forces of equal bodies are not as the spaces by them in unequal times but as the spaces described by them in equal times P. 44. l. 1. For fluxions write moments. For fluxions are not differences. P. 46. l. 15 3. Mr Newton in 1704 in the Introduction to his book of Quadratures published that he found the Method of Fluxions gradually in the years 1665 & 1666, the next year in giving an account of this book in the Acta Lips. for this was called in question & Mr Newton was accused of Plagiary. Mr Keill in 1711 defended him. p. 49. l 10. In the Acta Eruditorum. for P 49. l. 10 This was written by Mr Leibnitz himself in pursuit of the charge of Plagiary published in the Acta Lips 1705 as above & conteined a letter of Mr Bernoulli to Mr Leibnitz dated 7 Iulij 171 Iune 17313 written to the same purpose. Mr Leibnitz was then at Vienna & the Copies of the Commercium were sent to him thither by several hands from England. And thereupon he wrote to Mr Bernoulli to examin the Commerciusm to appealed from the judgmtjudgment of the Committee to that of Mr Bernoulli, desiring him to examin the Book Book, & inserted Mr Bernoulli's Answer into & pretending that he had not yet seen it himself; & Mr Bernoulli's answer dated 7 Iune 1713 he inserted into anoth a Paper of his own dated 29 Iulij 1713 & caused them to be published without their names, as if written by other persons unconcerned. And Dr Keill answered this Paper in the Iournal Literair. P. 49. l. 19. The pretence that Mr Leibnitz wo intended to write a Commercium is a sham. The P. 52. l. 7. Hoc factum est anno 1673 ad finem vergente. P. 53. l. 6. Hæc Newtoniana Leibnitiuss non diditcit vidit ante annum 1676. P. 53. l penult. Mr Newton scripsit Lemma II Lib. II Princip. & Scholium subjunctum, non ut methodum Leibnitio concederet sed ut eandem sibi vindicaret. 48 before his Lectures came abroad, — — — — — because they did many more things of this kind. [He has not produced the least proof that he knew the method before the year 1677, but on the contrary in his Letter of 27 Aug. 1676 he questi declared his disbeleif that my methods were so general tas I had described & said that many Problems, & among others the inverse Problems of Tangents, could not be reduced to æquations or quadratures; & placed the perfection of Analysis in Analytical Tables of Tangents & the Combinatory Art, saying of the first, Nihil est quod novim in totaTOTA Analysi momenti majoris; & of the second, Ea vero nihil differt ab ANALYSI ILLA SVPREMA ad cujus intima Cartesius non pervenit: est enim ad eam contstituendam opus Alphabeto cogitationum humanarum. And in a Letter from Amsterdam to Mr Oldenburg 28 Novem. 1676 he was thinking to perfect the improve the Method of Tangents of Slusius by such a Table of Tangents.] In that Letter he placed the perfection of Analysis in other methods then the differential, & represented it improbable that my methods should be so general as I had affirmed 6. Post verba [Metaphysica in Idæis, Idææ et Philosophia omnis vera in Phænomenis fundantur, & incepit Newtonus a Phænomenis: Momortus ab idæis (clausis oculis) incipiendum esse & e clausis oculis & ab hujusmodi somnijs ad phænomena pergendum esse contendit.] adde: Sed cum audiamus quid Leibnitius ipse quam cui Author noster cum Bernoullijs cum Bernoullijs duobus tantopere favet, he de de Metaphysicæ sanæ notionibus olim scripsit. in In charta enim quam [in Actis Leipsicis anno 1694 mense Martio ] De Primæ Philosophiæ Emendatione & de Notione Substantiæ scripsit quæque in Actis Eruditorum Lipsicis anno 1694 mense Martio impressa fuit hæc habet. Video plerosque qui Mathematicis doctrinis delectantur — — — — hoc commodo caremus. Hactenus Leibnitius. 11. Post verba [rimari philosorumphilosophorum est.] adde. In Principiorum editione nova Newtonus secunda sub finem, Newtonus scripsit se rationem proprietatum gravitatis ex phænomenis deducere nondum potuisse & Hypotheses se non fingere. Et postquam descriptionem Systematis mandani motus Planetarum Cometarum Æquinoxionem & maris nostri a vi gravitatis per phænomena et mathematicas demonstrationes derivasset subjungit motus particularum corporum a spiritu quadam s electrica quodam quo attractio electrica perficetur quam maxime pendere, sed hæc paucis exponi non posse. Et Editores Actorum Lipsiensium (Anno 16 1714 mense 6. A In Metaphysicæ sanæ notionibus.] Metaphysicam sanam intelligit Cartesianam qua utique ab ideis inmatis ad existentiam verugm pergitur. Audiamus igitur autem quid Cartesius Leibnitius de hæc Metaphysicæ dque sanæ notionibus olim scripsit. Is enim in tdisserationedissertatione de Metaphysica sanæ notionibus De Primæ Philosophiæ emendatione & de Notione substantiæ in Actis Lipsicis anno 1694 mense Martio impressa, hæc habet. Video, ait, plerosque qui Mathematicis doctrinis delectantur .......... hoc commodo caremus. Hactenus Leibnitius. Metaphysica utique in Idæis, Ideæ & Philosophia omnis vera in Phænomenis fundatntur, et incipit Newtonus a Phænomenis: Momortus ab Idæis Idæas [ Idæas non a Phænomenis deducendas esse sed clausis oculis formandas, esse [& ab h hujusmodi somnijs] ab Idæis ad phænome pergendum esse contenditur deinde oculos deinceps apperiendos esse ut phænomena beneficio Idæarum l videamus idæarum clare & distincte videamus. qua utique Ideæ non deducuntur a phænomenis sed innatæ esse finguntur, non pergitur disputatur a Phænomenis a de existentiam reum sed sed ab innatis Idæis existentia Dei res atori spatium extensionem esse corpus & res non extensis existere probatur ab Idæarum innatarūum figmentis. — materiam in forma quacunque positam, ope legum naturæ formas omnes quarem est capax successive induere assumere tandemque ad illam quæ est hujus mundi devenire; philosophiam omnem naturalem &c Vires attrahentes, tanquam qualitates occultas a Newtono introductas, Leibnitius in Tractatu de Bonitate Dei explod alibi hostilite animo explodit. Idem hic facit Momortus [quasi hæ vires in Gallia damnatæa essent Newtonus induceret a] idque fraudulanter quasi vires attrahentes in Gallia damnatas: attrahentes a Newtono essent quas Newtonus induceret At Quatates occultas & Vires attrahentes in Gallia damnatus Newtonus non int inducit — — — per fractonem excitatis. Memorial of the Master & WorkrWorker of the Mint about the Forms of new money. 2B. Demonstratum Vacuum se dedisse.] Mediaonum maxime fluidaoꝝrum qualia sunt Aer Aqua et Argentum vivum, resistentiam prope omnem a vi intertiæ partium fluidi oriri & & hujusmodi resis vim inertiæ proportionalem esse densitati fluidi Newto adeoque resistentiam fluidorum Mediorum summe fluidorum proportionalem esse densitati eorundem Mediorum quamproxime Newtonus et ratione experimentis ostendit, et legem resistentiæ in talibus fluidis exposuit multisque experimentis confirmavit. Spatia autem cœlestia cum In Vacuo Boyliano conferit, in quo resistentia corporum minime sentitur cadentium longe minor epe reperitur quam in aere & si forte aliqua sit, sentiri tamen non potest In spatijs cœlestibus quæ supra Atmosphæram sunt & aere omni magis sunt vacua, sunt resistentiam min adhuc minorem esse statuit, Et et ad Philosophiam cœlorum sufficere quod spatia illa medijs resistentibus vacua sint., Newtonum ad Philosophiam cœlorum sufficiter credit, ad Quod Cœlo Medio prope omni corporeo vacua sint esse Newtonus credit sed credit sed Philosophiam cœlorum in hac sententia non fundat Sufficit quod mMedio omni resistente prope destituuntur. Ad Medio corporeo non resistente plena sint, Quæstio est ad Philosophiam cœlorum minime spectants sSed ejusmodi tamen, medium corporeum dari Newtonus tamen non credit nisi forte duo sint corporum genera alterum cujus partes vim inertio habent inertiæ, alterum cujus partes vim talem non habent. 5B. A In experientia universali nixa] Quanam experientia memb constat motus omnes animalium et hominis ipsius mere mechanicas esse & ab actione voluntatis minime oriri nisi quatenus omnes cogitandi et volendi actioneis a quibus oriunturs motus isti oriuntur, in motu corporum consistetare? 6. In metaphysicæ sanæ notionibus] Metaphysicam sanam intelligit Cartesianam: Qua utique . . . . . . devenire statuit. Sed nec Ideas substantiarum nos habere aut Sed neqcꝫ Cartesitus alicubi probavit, facultatem cogitandi rem cogitantem esse atque corpus omne quiescere quod in corpore ambiente quiescit aut vim et aut neque aut rem omnes extensam extensionem esse, aut nec motum corporūum in sola translatione relativa causis tere sine vi inertia consistere, aut neque aut rems cogitantes nullspatio præse adesse, aut Deus non esse omnipræsentem per substantiam suam Cartesius alicubi probabvit. Hæc omnia sunt meræ Hypotheses. spatio non adesse, seu nullibi esse, nequeaut Ideas nobis innatas non posse esse aut nos aut nos ideas substantiarum nos habere. Hæc omnia sunt Hypotheses. Metaphysica in Ideis abstractis fundatur, Philosophia vera in Phænomenis. In Philosophia vera oculis apertis disputamus, In Metaphysica clausis oculis disputamus, in Philosophia vera videmus et a rebus visis phænomenis veritatis deducimus. Metaphysica a Th Theogonia Gentium antiquarum originem habui habuit qua utique Mundum totum animasque mortuorum aut partes esse Dei aut ejus potentias esse disputarunt somniando statuerunt seu finxerunt id est Deum esse rerum omnium naturam: Physica de rebus cognitis et earum causis tantum tracta [causas rerum visarum et cognitarum a per argumenta certa missis religionibus deducit] missis gentilum religionibus incipit a rebus visis et cognitis et earum rationes & causas investigat, idque per argumenta si fieri potest mathematica. 72 be found by the same method, & then subjoyns: Nec quickquam hujusmodi scio ad nquod hæc methodus idque varijs modis sese non extendit. Imo tangentes ad Curvas Mechanicas (siquando id non alias fiat) hujus ope ducuntur. Et quicquid vulgaris Analysis per æquationes ex finito terminorum numero constantes (quando id sit possibile) perficit, hæc per æquationes infinitas semper perficit: Ut nihil dubitaverim etiam nomen Analysis etiam huic tribuere. 74 Tempus utique per quantitatem quamcunque uniformiter fluentem, fluxionem ejus per unitatem, et momentum per literam o Newtonus designat. Aliarum quantitatum fluentes fluxiones designat is per alia symbola, et momenta earum per symbola illa ducta in momentum o, et areas curvarum per ordinatas quadrato inclusas. D. Leibnitius pro fluxionibus nulla habet symbola, pro momentis literam d præfigit symbolis fluentium, et pro areis præ fig literam

\int

præfigit symbolis Ordinatarum. Newtonus calculo suo usus est in Analysi quam Barrovius cum Collinio mense Iulio anni 1669 communicavit; et hanc methodum tum in Octob. 1676 valde generalem esse significavit, & verbis partim apertis differentialem vocat, anno 1676 minime invenerat. Scripsit enim ad Oldenburgum eo anno, Augusti 27, in hæc verba multa esse adeo mira et implexa ut neque ab æquationibus pendeant neque a Quadraturis: qualia sunt (ex multis alijs) problemata methodi Tangentium inversæ, quæ etiam Cartesius in potestate non esse fassus est. At acceptis Newtoni literis, anno sequente in Literis ad Oldenburgum 21 Iunij datis, methodum differentialem ut olim a se inventam communicare cœpit his verbis. Clarissimi Slusij methodum tangentium nondum esse absolutam celeberrimo Newtono assentior: et jam a multo tempore rem tangentium longe generalius tractavi scilicet per differentias Ordinatarum. — Hinc in posterum nominando dy differentiam duarum proimarum y &c. Quod methodum hanc a multo tempore invenerat probandum est. Nam D. Leibnitius pro se testis esse non potest. Iniquissimus esset Iudex qui in rebus controversis hominem quemvis pro seipso testem admitteret. 75 GEORGIVS LVD. D. G. MA. BRIT. FRAN. ET HIB. REX. FIDEI DEFENSOR. Ecclesia tuta. And such Letters were writ in Council the name of the Elders & people; together the people being present at the writing of them, & giving their assent (Act xv. 12 6, 12, 22, 23:) for the Elders were did not to act as Lords in did were not to act as take the oversight of the flock of God as Lords over Gods heritage but, as ensamples to the flock. The President was to act by the advice & with the consent of the Council & the President & Council were to advise wthwith the peop acquainte the people therewith & to advise the & teach them & procure their assent for avoiding ill will discord faction & schism. 76 Fluxionum eique cognov serierum infinitarum incidisse. In the winter between the years 1664 & 1665 upon reading Dr Wallis's Arithmetica Infinitorum & trying to interpole his series progressions for squaring the circle, I found out first an infinite series for squaring the circle & then another infinite series for squaring the Hyperbola & soon after. This latter series was the same with that published by Mr Mercator between about 3 or 4 years after .[We found out this series by different methods & Who found out this series first I do not em know But And two years before he published it I had found out the general method of finding squaring curves by such series by the help of division & extraction of roots.] & two years before he published it I had a a general method of squaring curves by such series by the help of division & extraction of roots. pag tertia, et specimen ejus methodi cum Gre Specimen Sed et specimen ejus methodi Collinius cum Gregorio communicavit. Et Anno 1676 Newtonus eandem in Epistola quadam quoad series infinitas in epistolais quadam ad celeberrimum virum G. G. L D. Leibnitium literis duabus duabus ad D. Oldenburgum datis et ad D. Leibnitium missis, quoad series infinitas fuse explicuit, quoad Fluxiones vero f subobscurie insinuavit. Scripsit enim se tunc compotem fuisse in epistolarum secunda Die 24 Octobris data, se tunc compotem fuisse — Et ibidem subnectit Theorema subnectit pro Binomijs quod quadraturam ad finita exhibet æquatione ubi fieri potest, ad infinita verò ubi quadratura per finitam æquationem exhiberi exhiberi nequit. Quo exemplo satis ostendit methodum utramque et fluxionum, et illar serierum infinitarūum una et eandem esse qutd quodad fundamentum, sed illam fluxionum generaliorem esse & t lla ad illam serierum infinitarum se habere ut genus ad speciem vel totum ad partem fluxionum absque methodo serierum infinitarum perfici non posse Hoc Theorema primum vocat hoc est primum Seriei Theorematum — patebit. Scripsit vero Newtonus methodum ex dictis haud difficile colligi, se vero impræsentia ne fuse exponere cogeretur, eandem his literis transpositis celare (6accdæ13eff7i3l9n404qrr4st12vx. Quæ quidem ordinatæ hanc conficiunt sententiam Data æquatione Fluentes, quotcunque quantitates involventes invenire Fluxiones et vice versa. Et hisce autem Literis anno 1677 acceptis Leibnitius respondit se in parilem methodum incidisse uti Princip Phil. Natur. Math. pag 253 &c videre licet: cujus tamen explicationem (in cæteris suis inventis publicandis non raro nimis præ properus, uti in Act. Erudit. plus semel conspici queat) ad Annum usque 1684 distulerit. D. Iacobus Gregorius ex serie aliqua unica Newtoni serie quam a Collinio acceperit, methodum ipsam serierum infinitarum collegit. D. Leibnitius circa annum 1675 Londini fuit, & postea cum Collinio commercium Epistolicum habuit acet tum dem subinde epistolas Newtoni ab Oldenburgo accepit; & tum demum rescripsit se methodūmmethodum Newtonianæ non dissimilem habere. Qua ratione in hanc methodum inciderit, ipse p novit: nobis sufficit quod Newtonus sit primus ejus inventor. Ipse Leibni Is inventionem terminorum serierum infinitarum primus docuit, et secundus seriei cujusque terminus est quantitas illa quam Leibnitius differentiam vocat primi termini & cujus adeo inventionem ipse sibi arrogat Sed et Newtonus hanc methodum longius provexit quam alius Ipsam enim ad quadraturam Curvarum per series infinitas infinitas quæ in æquationes finitas vertuntur ante annum 1676 applicuit. I Ipsam etiam ad æquationes fluxionales — nequeunt. Et methodum generalem tradit jam invenerat extrahendi fluxiones ex æquationibus fluxionalibus. Methodos insuper universaliores ad comparationes Curvarum quæ geometrice quadrari non possunt eodem tempore tradiderat — subministraverit. Et has methodos insuper ad enodanda naturæ phænomena in Principijs suis mathematicis p adeo feliciter applicuit ut ejus inventis cultores methodi differentialis vix aliquid addiderint. Motus autem Planetarum in Ellipsibus Tandem verò per gravitatem versus umbilicum Planetas autem in Ellipsibus per gravitatem versus Solem revolvi & ar Planetas autem in se mutuo graves esse & maxime in solem, & gravitate ab sua versus circum solem re in Ellipsibus revolvi ac radijs ad solem ductis areas describere temporibus proportionales, Newtonus ante annum 1677 demonstravitre didicit, & demonstrationem cum Societate Regia anno 1684 communicavit. Produeijtrunt hæc demonstratio in lucem anno 1687, et Leibnitius anno 1689 Theorema a Newtonianum, sed in Actis etiam seipsicis descripti differentiali sed errante, calculo suo differentiali ad incudem revocavit semel aliquats atque iterum erravit nec tamen non dubitavit asserere sit quoque propro marta i idem Theorema incidisse Theorema idem invenisse ut suum faceret. Asseruit enim se Theorema illud invenisse, sed c in calculo aliquoties erravit, Nondum enim ideoque nihil invenit. Sed nec methodum jam dicice satis didicerat qua Newtonus Principia sua sua mathematica invenstitgavit, Calcul Operari quidem in differentijs primis jam ante didicerat sed mentem Newtoni q modum operandi in differentijs secundssecundis nondum ut ex erroribus ullos ejus manifestum est. At ex eo tempore Leibnitius et alij hanc methodum ad Problemata difficiliora applicare ceperunt et methodum differentialemis in quæstionibus difficilioribus a Leibnitio et Bernoulli enodandis encoleri cœpit, facto et initia a resolutione Problematum de resistentijs mediorum & motibus Planetarum corporum cœlestium, quæ Newtonus antea in Princip tractaverat per methodos fluxionum antea tractaverat enodat antea confecerat & tractaverat, in lucem ed ut ex ejus Principijs Mathematicis manifestum est Methodus igitur fluxionum quam Leibnitius differentialem vocat, non prius apud exteros excoli et celebrari cœpit, quàm Newtonus eandem apud nos ad summum fastigium exvexerat. et 77 Iustitia hic reddenda est d Newtono (cui Geometria Optica et Astronomia multum debent) qui etiam propria Marte aliquid hic [sc. Methodo differentiali] simile habuit quem uti postea didici [videlicet ex Algebra Wallisij Anno 1693 impressa]. Verum est quod alijs utitur symbolis: sed quemadmodum ipsa characteristica, ut it dicam, magna pars est artis invenienti inventoris, credo quod notra [symbola] plus luminis præbent. Hæc Leibnitius in the Iournal des Sçavans 23 August 1694. D. Barrowus in ejus Lect 10 anno 1669 impressa methodum tangentium longe generalius tractavquam tractaverat quam Slusius idque per differentias linearum Ordinatarum et aliarum linearum. Gregorius in Prop. 7 Geometriæ universalis anno 1668 impressæ et Barrovius in in ejus Lect 10 anno 1669 impressa rem tangentium tractaverant per differentias ordinatarūum. Idem facere potuisset Leibnitius jam a multo tempore. Cum vero a Newtono didicisset Clarissimi Slusij methodum Tangentium nondum esse absolutam Methodus Slusij cCorollarium est methaodi Tangentium per differentias Ordinatarum. Nam prim dignitates in primo ejus Lemmate c Etenim in primo ejus Lemmate differentia dignitatum duarum ejusdem gradus applicata ad differentiam laterum, est differentia infinite parva ordinatarum duarum applicata ad differentiam infinite parvam C Abssarum Abscissarum. Cum autem igitur Newtonus dixisset Methodum Slusi Tangentium Slusij Cor esse unum particulare vel Corollarium potius methodi generalis quæ extenderet se citra molestum ullum calculum . . . . et exemplum dedisset hujus methodi in Serie generali pro quadraturis cœpit Leibnitius methodum Tangentium per differentias Ordinatarum ab alijs traditam jam longe generalius tractare quam antea, et novo nomine novisque symbolis ut suam ordnare. Et ut eandem ad se raper ne eandem a Newtono didicisse videretur scribit in hæc verba. Clarissimi Slusij methodum Tangentium nondum esse absolutam Newtono assentior. Et jam a multo tempore rem Tangentium longe generalius tractavi scilicet per differentias Ordinatarum. Et cum hanc Methodum mense Septembri Octobri Anni 1684 edere cœpisset, scripsit postea se inventum plusquam nonum in annum pressisse. Vnde Quod perinde est ac si dixisset se invenissetum habuisse ante mensem Novem Octobrem anni 1675. Cum tamen Et postea cœlatis Ne postea methodum solo suo nomine edidit cœlatis Newtoni literis donec a D. Wallisio ederentur. D. Deinde ne parum candidus fuisse Et ne parum candides egisse videretur, prætendit se nihil omnino ex his literis didicisse nisi quod Newtonus habuerit methodum tangentium cum Elementa calculi sui edidit anno 1684, nihil aliud de Inventis Newtoni in hoc genere ipsi innotuisse quam quod ipse olim significaverat in literis, posse se Tangentes invenire non sublatis Irrationalibus. sed cum Cum tamen ex literis hic impressis constet [D. Leibnitium Anno 1676 mense Maio ubi Demonstrationem quadruræ circuli poliba poliebat, & mense Augusto finiente ubi scriebat multa usque mira et implexa esse ut neque ab Æquationibus pendeant neque a Quadraturis qualia sunt (ex multis alijs) Problemata methodi Tangentium inversæ, methodum suam differentialem minime invenisse: sed acc method et hanc methodum in epistolis tribus Newtoni ad ipsūum hoc anno et initio jam missis abunde satis delineatam fuisse & D. Leibnitium ex ijs jam tam didicisse Methodum Newtono jam tum innotuisse Differentiali similem qua et hanc methodū Newtoni in utrāque methodo quadraturas per Æquationes quas Leibnitus differentiales vocat faciliores reddi.] methodum Differentialem anno 1676 Leibnitio minime innotuisse, anno vero 1677 Leibnitium didicississe cognovisse quod Newtonus methodums Leibnitianæ Differentiali similemis haberet Newtono innotesceret Nam et anno 1684 mense Octobri ubi principia elementa methodi differentialis edidit subjunxit sub hoc titulo: Principia Nova methodus pro maximis & minimis itemque tangentibus quæ nec fractas nec irrationales quantitates moratur; & subjuxit: Et hæc quidem initia sunt tantum Geometriæ cujusdam multo sublimicoris difficillima & pulcherrima quæque etiam mistæ Matheseos pertingentis quæ sine calculo nostro differentiali, aut SIMILI, non temere quisquam pari facilitate tractabit. Quid Leibnitius per methodum similem his intelligat senties Conferaturando hacnc methodi differentialis descriptionem cum simillime descriptione methodi in epistolis tribus Newtoni ut methoduorum similitudinem videas Indtroduction Pag 7. Instead of necessary to salvation should it not be necessary to commumunioncommunion P. 17. should it not be. Put this note in. P. 21. lin 2, 3. Let the note φ come at yethe end of the sentence without a parenthesis. 79 Mint Office. 13 Apr. 1714 SrSir I have considered the inclosed Estimate made upon a Proposal offered to the Lords Commissioners for Trade & Plantations for coyning 1500 Tonns of Copper into half pence & farthings in three five years time Problems solved wthwithout prickt letters in yethe Introduction to the book of Quadratures After minusculis a et b designat dd He might haved added further [that in [one & the same book, the book of Quadratures, [Mr Newton sometimes uses letters with pricks sometimes not. For] in the Introduction to that book of to the book of Qu of Quadrature some Problemes are solved by the method of fluxions wthwithout the use of such such Letters wthwith pricks with pricks In the first Proposition of the book letters with pricks are made use o th & in the body of the book letters wthwith pricks are made use of: And yet the method is one & the same. Mr Leibnitz confines his method &c In A little after before, vizt In the second Lemma of yethe 2d Section of yethe book Mr Newton demonstrated synthetically the Elements of the method of fluxions & added this Scholium. In literis — Vtriusque continetur in hoc Lemmate. This Scholium referrs to t Mr Newtons Letter dated 24 Octob. 1676 Where he distinguishes between the method of series & the method of fluxions & readily gives yethe method of Tangents of Slusius & th represents that yethe method of series gives t fluxions readily gives the fluxions proceeds in surd quantities as well as rationals & readyily gives the Method of Tangents of Slusius &c And in his Letter dated 10 Decemb. 1672 he represents that the method whereof the method of Tangents of Slusius was but a Corollyary or branch & wchwhich proceeded in æquations affected wthwith surd quantities, was very general & reached to the abstruser sorts of Problems, & particularly to Problemes about the curvature of Curves. Now in solving such Problems by the method of fluxions depend upon the second fluxions come under consideration & therefore Mr Newton understood had extended his method to the second fluxions in those days or before the yeare before the 1672, that is, before Mr Leibnitz endeavoured to make the initial method. But whereas it doth not appear that Mr Leibnitz understood the second differences before the year 1684.

\begin{matrix} \begin{array}{r} 11970 \\ 2394 \\ 143680 \\ 287∟371 & (5.4.9 \\ 12.240 0 \\ 20 \\ 400 \\ 495 \end{array} \begin{array}{r} 11970 \\ 2394 \\ 14365 . & 5.5.7∟441 \\ 287311 & 5.4 \\ 275 \\ 12∟311 \\ 24622 \\ 49244 \\ 295464 \\ 275 \\ 20464 \\ 409∟280 \\ 385 \\ 2428 \end{array} & \begin{array}{r} 11971^{} \\ 2394^{} \\ 14365^{oz} \\ 287311^{dwt} \\ 5^{dw} . 5^{gr} . {7 \frac{1}{2}}^{mi} \end{array} \begin{array}{r} 55000) 287311 & (5^{dwt} \\ 275 \\ 12311 \\ 24622 \\ 147732^{} \\ ) 295464^{gr} & (537207^{} \\ 275 \\ 2046 & 7^{g} ∟ 44 \\ 165 \\ 3964 \\ 385 \\ 114 \\ 40 \end{array} \end{matrix}

\begin{array}{l} 22^{car} \\ 88^{gr} . 1 \frac{1}{2} \\ 176 \\ 1056.13 \end{array}

\begin{array}{l} 287311 \\ 1723866 \\ 6895464 ( \\ 1379092,8 \\ {125∟372,07}^{gr} W^{t} \\ 5.5.7∟441 \end{array}

\begin{array}{r} 125∟37207 \\ 37 611621 \\ 1629∟83691 & (1∟543406 \\ 5738 & 12537207 \\ 5280 & {123∟828664}^{gr} sta w^{t} . \\ 4583 \\ 4424 \\ 3596 \\ 3168 \\ 4289 \\ 4224 \\ 651 \end{array} 120^{gr} = 1^{li} - \frac{{3 \frac{1}{4}}^{d}}{4} \begin{array}{l} + 7^{d} ∟ 6571 = 1^{li} . 6^{d} \frac{5}{6} or \frac{6}{7} \\ - 8125 \\ 6∟8446 = 6 \frac{1}{2} \frac{1}{4} \frac{1}{10} \end{array}

\begin{matrix} 1^{li} . 0^{s} . {6 \frac{6}{7}}^{d} & \overset{34.946}{} \end{matrix}

And therefore their standard wtweight is 5dwt. 3gr.

16 \frac{1}{2}

miles. And their value 1li. 0s.

6 {\frac{5}{6}}^{d}

246∟818706 = 1^{li} . 0^{s} . {\frac{5}{6}}^{d}

\begin{matrix} 651 & 247^{d} ∟ 657128 . & 320 \times 3.3 \frac{1}{4} : 3840.13 . \\ 1^{h} . 0^{s} . {7∟657128}^{d} = & 742971384 \\ 3840) & 3219∟542664 & (0.838422 \\ 30720 \\ 14754 \\ 1152 \\ 3234 \\ 3072 \\ 16226 \\ 1536000 \end{matrix}

\begin{array}{r} 247∟657128 \\ - 838422 \\ 246∟818706 \end{array}

\begin{array}{r} 866 \\ 768 \\ 980 \end{array}

80 was a copy of Mr Iames Gregories Letter of 15 Feb. 167

\frac{0}{1}

wherein was the above mentioned series for finding the arc whose tangent was given There was also a copy of Mr Newtons Letter above mentioned dated 10 Decem 1672. 24 12 11. 13. When Mr Leibnitz wrote to Mr Oldenburg for the Demonstration of the series for yethe arc of a circle whose sine was given, Mr Oldenburgh & Mr Collins des wrote to deire Mr Newton to describe his own method of finding it. And thereupon Mr Newton wrote his Letter of 1676 13 Iune 1676: wchwhich is printed at large in the Collection of Mr Iones wchwhich conteined his method of Series & was sent to Paris Iune 26. [& neare the end of it when he had described his method of Series wchwhich he calls infinite equations, he addsed that by Analysis by the assistance of there equations extends to almost all equations Problems except some numeral ones like those of Diophantus, but becomes not altogether universal without some further methods of reducing Problems to infinite series & equations & of infinite equations to finite ones when it might be done] This Letter together with the extracts of Gregonis Letters was sent to Paris 26 Iune 1676 25 13 Mr Leibnitz in his Answer dated 127 Aug 1676 in recompense for Mr Newtons method of Series sent back a Theoreme for transmuting figures into one another, b & thereby Demonstrated yethe series of Mr Gregory for finding the arch whose tangent was given. [But this Demonstration was composed wthwithout the assistance of the Differential method, & therefore Mr Leibnits had not yet found out that method. For in the Acta Eruditorum Mensis Aprilis pag 178 he wrote that in yethe year 1675 Iam Anno 1675 compositum habebam opusculum Quadraturæ Arithmeticæ ab amicis ab illo tempore lectum, sed quod materia sub manibus crescente ad limare ad editionem non vacavit postquam aliæ occupationes supervenere; præsertim cum nunc exponere prolixius exponere vulgari more quæ Analysis nostra paucis exhibet, non satis operæ pretium videatur And in his Letter of 12 May 1676 he wrote to Mr Oldenburg that he was then polishing this Demonstration to send it h to him. And in October following he began to left Paris being called home by his Prince to be imployed in publick affairs wherby he was no longer a leasure to file & polish this work for yethe the Press vulgari more. 26 In his said Letter of 27 Aug 1676 he added further: Quod dicere videmini plerasque difficultates (exceptis Problematibus Diophantæis ad Series infinitas reduci; id mihi non videtur. Sunt enim S multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturis. Qualia sunt (ex multis alijs) Problemata methodi tangentium inversæ. And this also makes it very certain that he did had not yet found out the differential method. 27 Mr Newton in his Answer dated 24 Octob. 1676 at the request of Mr Leibnitz gave an A account how before yethe plage wchwhich happen began to rage in Summer 1665 he found the method of these series, & upon the coming out of Mercators Logarithmotechnia communicated to Mr Collins by Mr Barrow a Compendium of these series & two (meaning the Analysis above mentioned) & two years after at yethe suggestion of Mr Collins wrote a larger Tract on yethe same subject, joyning wthwith it a Method from wchwhich yethe determination of Maxima & minima & the method of Tangens of Slusius & some others readily flowed, & wchwhich stuck not at surds, & was founded upon this Proposition: Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire; et vice versa. This Proposition he wrote down enigmatically & then added that it faciliated the Quadratures of Curves & afforded him infinite series wchwhich brake off & became finite when the Curve was capable of being squared by a finite equation. And of this he gave an instance by such a Series. By wchwhich circumstance it is manifest that this was one of the methods mentioned in his former Letter of 13 Iune where he said that Analysis by the help of series extended to almost all Problemes but became not universal wthwithout some other further methods methods of finding infinite series & reducing them back to finite equations 28 In his Letter of Octob 242 1676 he said further that his method extended to yethe inverse Problems of Tangents & others more difficult, & that by a double method which he comprehended in this senstence wchwhich he exprest enigmatically: Vna methodus consistit in extractione fluentis quantitatis ex æquatione simul involvente fluxionesm ejus: altera cætera commode derivari possunt, & in collatione terminorum homologorum æquationis resultantis ad cruendos terminos assumptæ seriei. 29 Mr Leib At length Mr Leibnitz after in his Letter of 21 Iune 1677 after all this description of Mr Newtons method, began to put in for it as coinventor in these words. Clarissimi Slusij methodum Tangentium nondum esse absolutāam Celeberrimo Newtono assentior. Et jam a multo tempore rem tangentium longe generalius tractavi; scilicet per differentias ordinatarum — positis differentijs Abscissarum, si placet, æqualibus. Hinc nominando in posterum proximarum x, patet

d y^{2}

esse

2 y d y

esse

3 y^{2} d y

&c. Which is the first Lemma of Slusius. What Dr Barrow 8 years before called a & e Mr Leibnits begains now to call

d x

d y

. Hinc nominando in posterum, saith he. And then by a calculation perfectly like thosate of Mr Gregory & Dr Barrow, he gives an example of drawing Tangents shews how to proceed without sticking at surds, & adds: Arbitror quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere. Quod addit ex edem fundamento quadraturas quoque reddi faciliores me in hac sententia hac confirmant, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationes differentialem. About seven years after Mr Leibnits published this method of Tangents in yethe differentiall characters in the Acta Eruditorum & added mensis Octob. 1684, under this Title: Nova methodus pro maximis et minimis itemque tangentibus quæ nec fractas nec irrationales moratur & added: Et hæc quidem initia sunt tantum Geometriæ cujusdam multo sublimioris ad difficillima quæque & pulcherrima quæque etiam mistæ matheseos Problemata pertingentis, quæ sine calculo nostro differentiali aut simili, non temere quisquam pati facilitate tractabit. Compare this with the description wchwhich Mr Newton gave of his method in his Letter of 10 Decem. 1672 & you will find the methodsdescriptions very much alike. And hitherto And hitherto Mr Leibnitz put prætended only to be coinventor of this general method, as appears by his words, aut simili. Why he used the symbols

d x

d y

rather then the letters a & e of Dr Barrow, he tells us in the Acta Eruditorum mensis Iunij 1686. Malo autem, saith he,

d x

et similia adhibere quam literas pro illis quia istud

d x

est modificatio quædam ipsius x, et ita ope ejus talibus & differentijs calculum ingrediatur & relationes transcendentes inter x et aliud exprimantur. Qua ratione inter etiam lineas transcendentes æquatione explicare licet. Which is as much as to say that he used these might have used letters but to avoyd defining their signification of upon every new occasion he chose rather to use lett these symbols. And how he began to put in for the method as sole inventor In the win Autum 1683 Mr Newton sent the Principal Propositions of his Principia Mathematica to London where they were communicated to the R. Society & in yethe year 1686 he sent up the rest of the book, & two years after an Epitome thereof was printed in the Acta Le Eruditorum, & yethe year following Mr Leibnitz published three papers relating to that book The one was an Epistle de lineis Opticis In What Mr Newton had said about reflexions refractions M & needed nothing more then to apply it to reflexions then to change put yethe sines of inc refraction equal to yethe sine of incidence & to change its signe: Mr Leibnits applied to reflexions. Another was Schediasma de resistentia Medij et motu Projectilium in Media resistente. In this 81 pricks & secondly because Mr Newton when he wrote his Principia did not understand second differences but gave a instead of f represented the constant difference of x by th not by a prickt letter as he doth at present but by the letter o (in a vulgar manner wchwhich destroys the advantages of the differential calculus) but has also given a false Rule about the higher degrees of differences. And hetherwhether this Mathematician be Mr Leibnitz himself or any other person is not material. He was imployed by Mr Leibnitz & his p Mr Leibnitz so far approved his paper & as to caused it to be published & thereby has made it his own untill he tells the names of the author, & therefore we shall consider this paper at present as writ by Mr Leibnitz As to the first argument it amounts to this that tho Mr Newton when he wrote his Letter dated 24 Octob 1676 had a m had a method which he comprehended under these sentences: Data æquatione fluentes quotcunque quantitates involvente invenire fluxiones et vice versa: And Una Vna Methodus [inversa] consistit in extractione fluentis quantitatis ex æquatione fluxionem involvente yet he had not the method of fluxions because he did not in that calculus use letters with pricks. And the answer is But The sophism is evident at the first sight. For Mr Newton doth not place his method in a certain forms of symbols nor confine himself to any particular sort of symbols for fluxions. Where he puts the Areas for fluents of Curves for fluents de denotes puts the Ordinates for fluxiones & denotes the fluxions by the symbols of the Ordinates. And where he puts the letters x, y, z fluents, he either puts denotes their fluxions either by other letters, as p, q, r, for fluents for AB, CD, EF, or by the same letters in other forms as X, Y, Z, or

\dot{x}

\dot{y}

\dot{z}

for fluxions. Mr Leibnitz has no symb This evident by his Book of Qua And this is evident by his b And this he does to this day as is evident by his book of Quadratures, where he uses prickt letters in the first Proposition, the Ordinates of Curves in the last Proposition & other symbols in the S Problem solving several Problems in the Indtroduction. When Mr Leibnitz changed the Letters of Dr Barrow into other symbols he did not invent a new method of tangents: he only invented new symbols, & if in the differential method he had used Letters (as he tells us he might have done) the method would have been the same by his own confession: for he tells us the that he might have used Letters. There are no harder Problems yet solved by any man then those found in the Principia Philosophiæ, & Mr Leibnitz himself has told us that the Differential method extends to yethe abstruser Problems in Geometry such as cannot be solved without this method or another like it & the Mrarques de L'Hospital has told us that the Principia are full of this calculus. Mr Leibnitz has no fluxions in this method. All Mr Newton's symbols of fluxions are the oldest in that kind. Mr Leibnitz had no symbols of Differences before the year 1677, Mr The nameles Mr Leibnitz represents in the represents pretends that Mr Newton uses the prickt letters instead of

d x

dd x

d^{3} x

d y

dd y

&c, & upon this pretence claims the grownds his cla objection & made used the same pretence in the Acta Lipsiensia of Feb A.C. 1705 pag. 1 But this pretence is a gross mistake if not a wilfull one. Mr L Newton does not use prict letters for differences unless the Co with the coefficient o either exprest or understood. H Wherever he uses prickt letters they signify differences fluxions & Mr Leibnitz has no fluxions in his method. All Mr Newtons symbols for fluxions are the oldest in the kind. He The differential symbols appear no older then the year 1677. The the rectangles under fluxions & the letter o were used for m by Mr Newton for moments in his books of Analysis & are still used Quadratures & are still used in or before yethe year 1669 & are still used by him in the same sense. The symbols for summs

\int x

\int y

\int z

&c do not appear to have been used by Mr Leibnitz before yethe year 1686: Mr Newton in yethe year 1669 inscribed the Ordinate or Fluxion in a square or rectangle to denote the area or fluent. And thus much in answer to the first Objection. As to Itnhe second Objection, Mr Newton in his Analysis, in his Principles, & in his Treatise of Quadratures & to this day, did put the letter o for the constant increase of the fluent x & uses the same notation to this day. He has not changed the letter o into the mark

\dot{x}

as is erroneously suggested nor ever did use the make

\dot{x}

for the letter o,. These symbols represent but quantities of a different sort. The symbol

\dot{x}

represents not the velocity of increase the letter o a particle generated by increase: the first is always a finite quantity, the second is a quantity may be supposed may be usually supposed infinitely little The first is an unite when x flows uniformly, the second & must be multiplied by the second either exprest or understood to signify what Mr Leibnits calls

d x

. But Mr Leibnitz add But Mr Leibnitz adds that Mr Newton formerly represented the constant increase of x by the letter o in the vulgar manner wchwhich destroys the advantages of the differential method. the If If this be true it proves either that Mr Newton has not yet found out the Method of fluxions because he still uses the letter o in the same sence as formerly, or else that the Method of fluxions wants the advantages of the differential, & is nothing more then a vulgar method known to Mr Fermat no above sixty years ago. An the world before Mr Newton pretended to it. If t If this be so then Mr Leibnitz has done nothing but in invented nothing but a new name & a new Notation to make a vulgar method his own. For the Method of fluxions as used by Mr Newton has the advantage of the differential in all respects. It is more elegant, because he uses but one infinitely or indefinitely small quantity signified by the letter o. it is more natural & geometrical because founded upon the rationeis praimæ quantitatum nascentium rationes wchwhich have a being in Geometry whilst indivisibles upon wchwhich the differential method is founded have no being either in Geometry or in Nature. There are rationes primæ quantitatum nascentium but not quantitates praimæ nascentes. Nature generates quantities by continual flux or increase & the ancient Geometers admitted such a generation of areas & solids when they drew one line into another by local motion to generate an area & the area into a line to generate a solid. But the summing up of indivisibles to generate an area or solid was never yet admitted into Geometry. Mr Newton's method is also of a greater extent, being adapted either to the ready finding out of a Proposition or to the Demonstrating it: Mr Leibnits's is only for finding it out. When the work succeeds not in finite equations Mr Newton has recourse to infinite converging series & thereby M his method becomes incomparably more universal then that of Mr Leibnits wchwhich is confined to finite equations. And where the law of the fluxions is not known but the fluxions are had only in a few particular cases Mr Newton finds that law quamproxime by drawing a Curve line through any number of given points, & thence deduces the solution of the Problem And to this degree of perfection Mr Newton had brought his method before the year 1676 as appears by his Analysis & his Letters of 10 Decem 1672, 13 Iune & 24 Octob. 1676. Its added that Mr Newton in his Principles of Philosophy gave a false rule about the higher degrees of dDifferences as had been observed sometime ago by B an eminent Mathematician., This Mat meaning Mr Iohn Bernoulli who in the Acta Lipsiensia of Febr. 170 1713 has published that Mr Newton in the Scholium to the 10th Prop of the 2d Book of his Princ. took the terms of a converging series for yethe first second third & following differences respectively. But Mr Bernoulli is mistaken. Mr Newton tell affirms with no such thing nor meddles with second differences in all that Scholium.. In the Scholium of the last Proposition of his book of Quadratures, the word ut is should has by some mistake been omitted. But the mistake (whether a press-fault or a slip of the pen) might have been easily corrected either by the last Proposition of that book where the series of the first second & following fluxions is described or by the first Proposition thereof where thereof of that Book which teaches how to find all the differences successively, or even by the Scholium in the Principia wchwhich is found fault with. For there Mr Newton puts the third term of the series equal to half that part of the Tangent wchwhich Ordinate wchwhich lyes between the Arc & the Tangent & by consequence th to half the second difference. 82 Mr Leibnitz began his last Letter to Dr Sloan secretary to yethe R. Society wthwith these words. Quæ D. Ioannes Keilius nuper ad Te scripsit, candorem meum apertius defendam, & cum homine docto, sed novo, et parum perito rerum anteactarum cognitore, nec mandatum habente ab eo cujus interest, tanquam pro tribunali litigem nemo prudens æquusque probabit. Thisus is alslo one as to Mr Leibnitz excused himself from making good his charge against Mr Keil & to tolled told the R. Society that they would be unjust unless they would allow him to b trusted if they did not trust his candor & allowed him to be a witness in his own cause. Mr Leibnitz has since upon pretence in opposition to the Report of the Committee, upon a pretence that he was not at leasure to examin the matter himself & had not seen the Commercium, imployed an nameles eminent Mathematician as he pretends, to examin the matter & give his judgment thereupon & the IudgmtIudgment Letter of the nameles & therefore pretended Mathematician he dated 7 Iuly 1713 he caused to be published. And by this judgment Mr Newton had not the method of fluxions in the year till after the writing of his Principia Philosophaiæ, & that for these two reasons. First because in all the Letters publi in Commercium Epistolicum from wchwhich the Committee draw their reasons there is not any instance of Mr Newtons letters wthwith pricks, & secondly 83 I consider time as generated flo flowing or increasing by continual flux & other quantities as increasing in time continually in time & from ytthat flowing fluxion of time, I give the name of fluxions to the velocitys wthwith wchwhich all other quantities increase. Also from the moments of time I give the name of moments to the parts of any other quantities generated in moments of time. I expose time by any quantity flowing uniformly & represent its fluxion by an unit, & the fluxions of other quantities I represent by any unit other fit proper symbols, & the fluxions of their fluxions by other fit symbols & the fluxions of those fluxions by others, & their moments generated by those fluxions I represent by the symbols of the fluxions drawn into yethe letter o & its powers

o^{2}

o^{3}

, &c: vizt their first moments by their first fluxions drawn into the letter o, their second moments by their second fluxions drawn into

o^{2}

, & so on. And when I am investigating a truth or the solution of a Probleme I use all sorts of approximations & neglect to write down the letter o, but when I am demonstrating a Proposition I always write down the letter o & for proceed exactly by the rules of Geometry without admitting any approximations. And I found the method not upon the summs or& differences, bu but upon the solution of this probleme: By knowing the Quantities generated in time to find their fluxions. And this is done by finding not prima momenta but primas momentorum nascentium rationes. And the inverse This Probleme determins the Tangents of Curves Geometrically, & the Inverse thereof determins their Areas & this is done without summing up of their Ordinates. This Method is derived immediately from Nature her self, that of diffe indivisibles not so. For Leibnitian differences or of infinitely small quantities not so. For there are no quantitates primæ nascentes or ultimæ evanescentes, there are only rationes primæ quantitatum nascentium or ultimæ evanescentium. Add 3968(41)85 was with difficulty that I answer Mr Leibnitz, was induced to write the letters of 26 Feb. 1716. So I do not consequence that I should meddle with this controversy any further part with a quiet life for the sake of it, & therefore I intend to meddle with this controversy no further. / But whatever is done, I do not think this business of such consequence that I should meddle wthwith it any further. If it be asked why I did not publish this book sooner, it was for yethe same reason that I did not publish the theory of colours sooner, & I gave the reason in my Letter of 24 Octob. 1676. — And the testimony of these two ancient, knowing & credible witnesses may suffice to excuse me for saying in the Introduction to the book of Principles that I found the Method by the degrees in the years 1665 & 1666. In the beginning of the year 1665 I found the Method of approximating series & the ge Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & so before in November had the direct method of fluxions & the next year in Ianuary had the Theory of colours & in May following I had entrance into yethe inverse method of fluxions. And the same year I began to think of gravity extending to yethe orb of the Moon & having deduced (having found out how to estimate the force with wchwhich a globe revolving within a sphere presses the surface of the sphere from Keplers Rule from Keplers rul of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their Orbs. I having I deduced that the centripeta forces wchwhich keep the Planets in their Orbs tend to are a about the centers of must are reciprocally as the square of their distances from the centers about wchwhich they revolve: I on & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665 & 1666 For in those days I was in the prime of my age for invention & minded Philos Mathematicks & Philosophy more then at any time since. What Mr Hugens has published since about centripetfugal forces I suppose he had before me Atnd so year in the winter between the years 16676 & 1677 I found the Keplers length Proposition that by a centrifugal force the reciprocally as the sequare of the distance a Planet must revolve oin an Ellipsis & about the center of the florce placed in the lower umbilicus of the Ellipsis & with a radius drawn to that center describe equal areas proportional to the times. And in the winter between the years 1683 & 1684 this Proposition wthwith the Demonstration was entered in the Register book of the R. Society. And this is the first instance upon record of any Proposition in the higher Geometry solved found out by the Method in dispute. In the year 1689 Mr Leibnitz endeavouring to rival me published a Demonstration of the same Proposition upon another supposition but his Demonstration was proved erroneous, & therefy it was discovered that he was for want of skill in the Method. And Mr Collins in a Letter to Mr Bertet of Feb 21 16

\frac{70}{71}

said that about four years before that time I found a general Analysis about four years before for squaring of figures all cirvilinear spaces & doing what ever depended upon Quadratures. And in his Letter to Mr Strode Pag. 219. lin. 22. dele: [Obtinet hæc ratio quamproxime ubi co in Medijs rigore aliquo præditis tardissime moventur.] et scribe [In M quæ rigore omni vacant, & quorum vis resistendi oritur a sola vi inertiæ partium, resistentiæ corporum sunt in duplicata ratione velocitatum &c]Pag. 252. lin. 14. Adde Scholium sequens. In my Letter of 2 It has been said that in the old & papers published in the Commer: Epist there are no prickt letters. In And indeed where I But the Marques de l'Hospital acknowledge that the Book of Principles was f And indeed I seldom used prickt letters when I considered only first fluxions as in the Introduction to the Book of Quadratures but when I considered also second third & fourth fluxions, as in the body f the book I distinguished them by the number of pricks. The first In the year 1692 At the request of Dr Wallis I sent him the first Proposition of the Book of Quadratures copied with th its solution & examples printed that ye in first & second fluxions copied almost verbatim from the Book & the Dr printed them the same year in the second Volume of his works And wchwhich came abroad the next year. And thence it may be understood that the Book of Quadratures was then in Manuscript. [In my Letter of 24 Octob 1676 I set down the first Proposition of this Book verbatim in an Enigma & copied also the Ordinates of the Curves in the other second table at the end of the 10th Prop & upon the 7th 8th 9th & 10th Propositions wrote to Mr Collins my Letter of 8 Novem 1676 printed by Mr Iones. & And from all this it may be understood that this book was then in Manuscript. And the Notation by prickt letters & sometimes by the letter o used in this Book as it is the oldest so it is the most expedite best]. In my Letter of 24 Octob 1676 I sent down the first Proposition of this Book verbatim in an Ænigma, & said that it conteined was the foundation of the method, & that this foundatio there mentioned & concealed & gave me general Theorems for squaring of figures by series wchwhich sometimes brake of & became finite & & how it gave me such Series is explained in the first six Propositions of this Book. of Quadratures. I know no other way of finding them I copied also In the same Letter the I copied also many Ordinates of Curves from a Table in the end of the 10th Proposition the many Ordinates of Curves, wchwhich I had may be compared with the Conic Sections ther & upon the 7th 8th 9th & 10th Propositions I wrote to Mr Collins my Letter of 8 Nov. 1676 printed by Mr Iones. And from all this it may be understood that this Book was then in manuscript. And as the notation used in this book is the oldest so it is the best. The Commercium Epistolicum being plain unanswerable matter of fact Mr Leibnitz constantly Endeavoured to avoyd it & ② run the dispute about into a squable about a Vacuum & Atoms & universal gravity & occult qualities & Miracles & the Sensorium of God & the absolute perfection & & everlastingness of the world & the definitions of time & space & the solving of Problems & the Question whethe he did not find the Met differential Method proprio Marte: all which are nothing to the purpose ×× Mr Iames Gregory after a years study found the method of converging series proprio Marte but did not claim it because before he found he had notice of it from England before he searched for it & by that notice knew that he was not the first inventor. Gregorius autem, said Collins, Newtonum primum ejus inventorem anticipare haud integrum ducit. There is but one proper Question & that is: Who was the first Inventor? Let it be proved I that Mr Leibnitz had the Method before he had any notice of it from England & then let it b let it be further proved that he had it before the date of my Letter of 10 Decem 1672 & by consequence before he learnt the hith. In the mean while . If he could have made a good objection he might have done in a short letter without writing another book as big. There is but one pteria proper Question & that is: Wiho was the first Inventor? All other Questions are prevarications. & ten to cloud this question. And untill the Book is answered fairly directly & without prevaricatiionsng, I see no need of setting pen to paper any more about this matter. ① If Mr Leibnitz could have answered the made a good objection against the book Commercium Epist he might have done it in a short Letter without settin writing another book as big. But this Book being unanswerable matter of fact & unanswerable he treated it with opprobious language & indoustrious avoided Answering it & for he tant by endeavoured to indoustriously avoided answering it by several excuses & [endeavoured to rund the dispute into a it & for that end] endeavoured to Pay it aside by runing the dispute into a The Committee of the R. S. said that I had this method above 15 years before Mr Leibnitz began to publish it: [that is, when he I wrote the Analysis per æquationes numero terminorum infinitas] And Mr Collins in his Letter to Mr Strode of 26 Iuly 1672 affirmed by the testimony of Dr Barrow t by papers communicated before that by the Analysis per æquationes numero terminorum infinitas & other papers communicated before to the Doctor Barrow it appeared that I had the method & applied it generally some years before the Doctor sent thyat Analysis per æquationes to Mr Collins, that is, some years before Iuly 1669. And I see no reason why the testimony of Dr Barrow should be questioned in this matter. The Marquess de l'Hospital had not then seen the recommended the Differential Notation, but had And especially since iIt was with difficulty that I was drawn to write & answer Mr Leibnitz. He is dead & The method it self is not of such consequence that I should meddle with this controversy any further. 1 In the Introduction to the following Book I said that I invented the Method of fluxions gradually in the years 1665 & 1666. The first Proposition thereof at the request of Dr Wallis I sent to him in 1692 & he it was printed that year in the second Volume of his works, & came abrodad in Spring following, & this was the first time that any Rule for finding f second third & fourth fluxions & moments came abroad. ③ In the Preface to the first & second Volume of the works Wallis Dr Wallis of Dr Wallis he affirmed that in my Letters of 13 Iune & 24 Octob. 1676 I explained to Mr L. the Method of fluxions found by me ten yeasyears before that time or above. In the Letter of second a Lemma By this Method I invented the greatest part of the Book of Principles in the year the Demonstration of Keplers Proposition in the year 1679 & almost all the rest of the Difficulter Propositions of the Book of Principles in the years 1684, 1685 [& the beginning part of ofof the year 1686] I in ② The MS of the Book of Quadratures Mr Raphson & Dr Halley had in their hands in the year 1691. In the year 1676 in the Summer I composed this book out of former writings as the one as one of them has left attested publickly & the other still attests. 4 And in the second Lemma of the second Book of Principles I demonstrated the set down & demonstrated the Elements of this Method & added mentioned the correspondence between me & Mr L. ten years before about these things to put him in mind of making such an acknowledgement the thereof publickly as he had made privately in his Letter of 21 Iune 1677 wherein he sent me a specimen of his improvemtimprovement of Dr B.s method of TangtsTangents. a Scholium wherein I declared that in the year 1676 I wrote of this Method to Mr L. & set dow in one of those Letters set down the foundation of it in this sentence, Data æquatione fluentes quotcunque æquationes quantitates involvente, fluxiones invenire; et versice versa.: the first part of wchwhich sentence is the very first Proposition of the book of Quadratures And in the Demonstrating the XIVth Prop. of the second Book of Principles I argued in made use of second moments calling them by the name of differentiala momentorūum & momentaum differentiæ. In the year 1676 in summer I composed the Book of Quadratures & then in my Letters of Octob. 24, 1676 men & Novem 8 1676 published by Dr Wallis & Mr Iones, I cited severall things out of it, parti & other Papers from whence I had extracted it: particularly the very words of the first Proposition; the Theorems for squaring figures by converging series wchwhich break of & become finite equations when the Curve can be squared by finite such equations; the orbdinates of Curves wchwhich can be I could compared with the Conic Sections; & the squaring of all Curves of ex exprest by equations of three terms, or comparing them wthwith the simplest Curves wchwhich they can be compared: each of wchwhich are a sufficient proof that I had the method of fluxions when I wrote that Letter. And in that Letter I represented that I wrote in five years before, that is, in the year 1671 I wrote a Tract of that Method & the method of series together, wthwith a designe to have published it together with another Tract wchwhich I had written about the nature refractions & colours of light. But finding my self intangled that by what I had already communicated I began to be entangled in disputes, I forbore to publish them for the sake of quiet for the sake of quiet; & as soon as I could get rid of the disputes I was engaged in I forbore to publish any thing about the method of Fluxions & Quadratures & the Theory of refractions & colours till the year 16 1704 wchwhich was above thirty years together. In From Dr Barrows method of Tangents compared with his own Methods Mr Iames Gregory deduced a method of drawing Tangents without calculation & gave notice of it to Mr Collins by a Letter dated 5 Sept. 1670, & upon notice from Mr Collins that Mr Gregory & Mr Slusius had such Methods I wrote to Mr Collins the following Letter dated 10 Decem 1672. Ex animo gaudeo D. Barrovij amici nostri reverendi Lectiones Mathematicas exteris adeo placuisse — — — ne grave ducas. Thus far my Letter, in wchwhich the method of fluxions is wthwith its large extent is sufficiently described, & illustrated wthwith an example of drawing Tangents. And by sying: that [Hanc methodum intertexui alteri isti qua Æquationeum eExegesin instituo reducendo eas ad series infinitas.] I refer to the method described in the Tract wchwhich I wrote in the year 1671, in wchwhich I joyned the methods of series & fluxions together as I mentioned in my Letter of 24 Octob. 1676. In the same Letter I mentioned also that when Mr Mercators Logarithmotechnia came abroad I Dr Barrow communicated to Mr Collins a Compendium of S my method of Series. And this is the Tract entituled Analysis per series numero terminorum infinitas. This Tract together wthwith some Letters of Dr Barrow rela was sent to Mr Collins in Iul The Logarithmotechnia came abroad in September 1668 Mr Collins sent a few months after sent a copy of it to Dr Barrow who replied that the Method of Series was invented & made general by me about two years before the Logarithmotechnia came abroad publication of Mercators Logarithmotechnia & at the same the same time sent back to Mr Collins my Tract the said Tract of Analysis per series. This was in Iuly 1669. In a letter to Mr Bertet dated 21 Feb. 167

\frac{0}{1}

Mr Collins represented that about four years before that time I had invented a general method of Analysis meaning the method described in the Analysis per series. [For by the litle of Analysis I understood that only merely the reduction of quantities into converging series but much more the method of solving all sorts by æquations involving c of Problems working in æquations wchwhich involve converging series when they cannot be solved by finite equations: And this is the method of moments.] And in a Letter to Mr Strode dated 26 Iuly 1672 he said that after he had sent a copy of the Logarithmo- to Dr Barrow at Cambridge, the DrDoctor quasdam Newton's chartas extemplo remisit: e quibus et alijs quæ olim prius ab Auctore cum Barrovio de eadem methodo communicata fuerant, patet illam Methodum a dicto Newtono aliquot annis antea excogitatam & modo universali applicatam fuisse: ita ut ejus ope in quavis figura ob curvilinea proposita quæ una vel pluribus proprietatibus definitur, Quadratura vel Area dictæ figuræ ACCVRATA siSI POSSIBILE sit, sin minus infinite vero propinqua, Evolutio vel longitudo lineæ curvæ, Centrum gravitatis figuræ, Solida ejus rotatione genita, et eorum superficies, (sine ulla radicum extractione obtineri queant. THere the words [ACCVRATA SI POSSIBILE SIT] relate to the words in the Analysis [Cujus [methodi] beneficio Curvarum areæ et longitudines &c (ID MODO FIAT) exacte & Geometrice determinentur. Sed ista narrandi non est locus.] How this is done is described in the first six Propositions of the Book of Quadratures. And without the method described in those Propositions it cannot be done. And thus it appears by the testimony of Mr Collins Dr Barrow & Mr Collins grounded upon this & former papers communicated to Dr Barrow that I had the method of moments & fluxions & made it general some years before Iuly 1669 or that is in the year 1666 & or before the end of the year 1666, as Mr Collins mentions in other Letters or (as he Mr Collins explains himself in his aforsaid Letter to Mr David Gregory,) about two years before the publication of Mr Mercators Logarithmotechnia, that is, in the year 1666. And all this together with the testimony of Dr Wallis mentioned above & that of Mr Fatio published in the year 1699, & the acknowledgemtacknowledgement wchwhich Mr Leibnitz made in his Letters of 7 Iune 1677 & 7 March 169

\frac{2}{3}

may suffice to justify my saying in the Introduction to the book of Quadratures that I found this Method gradually in the years 1665 & 1666 The method of converging series I found first by interpolation & thesn by division & extraction of roots in the beginning of the year 1665 & as at the request of Mr L. I described in my Letter to Mr Oldenburg 16 24 Octob. 1676 & the same But Dr Wallis had before that given an instance of reducing fractions into converging series by Division. The same year I got some light into the method of moments & fluctions. And its probable that Dr Barrows Lectures might put me upon considering the generation of figures by motion, tho I not now remember it. In the beginning of the year I 1666 I found out the different refrangibility of the rays of light & the Theory of colours depending upon it as is mentioned in the Philosophical Transactions87 Transactions for A.C. 1672. Mr N. Fatio had published A.C. 1699 that he invented the same method of fluxions in the year 1687 without knowing at that time that any body else had the same: but after he saw my Letters & Mr N. Fatio also in a Tract published A.C. 1699 gave this Testimony Ejus [Methodi] fundamenta universa, ac plerasque Regulas, proprio Marte, Anno 1687, circa Mensem Aprilem & sequentes, alijsque deinceps aAnnis inveni; quo tempore neminem eo calculi genere præter meipsum uti putabam — Newtonum tamen primum ac pluribus annis vetustissimum hujus calculi Inventorem, ipsa rerum evidentia coactus, agnosco coactus agnosco And a little after explaining what he means by rerum evidentia he names Newtoni Literæ, alijsqꝫque ejusdem manuscripti codidces. The direct method is sufficiently perfect. The Inverse Method is not yet perfected nor perhaps ever will be. [The first step therein is equipollent to the Quadrature of Curves & this is the subject of the following back.] It may be carried much And I write of it here no further then it is conteined in the following Tract & the methods of converging series & leave the further improvements of it to others who have made or shall make any. I At the first I s usually put the letters a, b, dc, d &c for determinate quantities, x, y, z, v for quantities increasing or decreasing according to any Rule, p, q, r, s for the velocities wthwith wchwhich they increase or decrease, op, oq, or os, for their moments or particles generated in a moment of time, &

p

q

the ordinates of any Curve included in the area a rectangle or wthwith a rectangle prefixed for the area of the Curve. And sometimes I used letters wthwith pricks or lines upon them & sometimes other marks. the but seldome But in the following So in the second Lemma of the second Book of Principles I put A, B, C for quantities increasing Or decreasing & a, b, c for the velocities of their increase or decrease, & in the [following Tract I put

\dot{x}

\dot{y}

\dot{y}

&c for the velocities of the increase or decrease of x, y, z, &

o \dot{x}

o \dot{y}

o \dot{z}

for their moments] latter end of the Tract Analysis per series in demonstrating the first Rule of that Tract I put x & z for increasing quantities ι & υ for the velocities of their increase & o & oυ for their moments, & in the following Tract I put

\dot{x}

\dot{y}

\dot{z}

for the velocities of the increase of x, y & z &

o \dot{x}

o \dot{y}

o \dot{z}

for their moments & x, y, z for increasing quantities whose moments are ox, oy, oz. And In sending the first Proposition of this Book to Dr Wallis in a Letter dated 1692 I used this notation, & It he published it after he had published it I seldome used any other. And in the same Tract I use

x'

y'

z'

for the areas of Curves whose ordinates are x, y, z. * NB. Hunc Librum in M. S. nostrates Halleius et Ralphsonus in manibus suis habtriuverunt anno 1691, uti Ralphsonus publice testatum reliquit & Halleius adhuc testatur. Propositionem primam exemplis illustratam D. Wallisius in secundo Operum suorum Voluminae pa g anno 1692 imprimi Curavit, et sub initio anni 1693 sequentis in lucem emisit. Et hæc fuit Regula omnium prima pro fluxionibus secundis, tertijs, et ulterioribus inveniendi quæ lucem vidit, estque Regula verissima brevissima et optima. Eande Eandem Propositionem ijsdem verbis, tanquam fundamentum methodi fluxionum, e Libro quem anno 1672 conscripseram desumptum, posui in Epistola mea 24 Octob. 1676 ad Oldenburgum data, et a Wallisio edita. In eadem Epistola posui Ordinatas Curvarum quas per Methodum fluxionum quadraveram cum Sectionibus conicis contuleram et in Calogum Catalogum anno 1672 retuleram, qui Catalogus in hoc Libro de Quadraturis habetur, et absque methodo momentorum construi non posset. In eadem Epistola posui Propositionem quintam hujus Libri pro Quadratura Curvilinearum quarum Ordinatæ sunt dignitates binomiorum, eandemque exemplis aliquot illustravi, dixique Regulas hujusmodi ad Trinomia et alia magis composita se extendere, et has Quadraturas Regulas Quadraturam accureatam dare quoties fieri potest. Sed et in Analysi per series numero terminorum infinitas mense Iulio anni 1669 a D. Barrovio ad Collinium missa, descripsi methodum momentorum & quomodo per Problemata per eandem ad series convergentes deduci possint, et hquod hujus beneficio Curvarum areæ et longitudines &c (id modo fiat) exacte et Gemometrice determinantur. Ideoque Propositio quinta Libri hujus de Quadraturis tunc mihi innotuit. Et propterea etiam methodum fluxionum et momentorum quatenus in Propositionibus quinque primis habetur, tunc intellexeram. Nam Propositio quinta a quatuor primis dependet. Sed et testimonio Collinij et Barrovij et Collinij hæc methodus mhi innotuit annis aliquot antea. Nam Collinius in Epistola sua ad Thomam Strode 26 Iulij 1672 data, sic scripsit. Mense Septembri 1668 Mercator Logarithmotechniam edidit suam, quæ specimen edit suam hujus methodi [id est serierum infinitarum] in unica tantum figura, id est nempe Quadraturam Hyperbolæ continet. Haud multo postquam in publicum prodierat liber, exemplar ejus Cl. Wallisio Oxonium mihi — aliumque Barrovio Cantabrigiam, qui quasdam Newtoni Chartas — extemplo remisit: e quibus et alijs ex alijs quoæ olim ab Auctore cum Barrovio communicata fuerant, patet illam Methodum a dicto Newtono aliquot annis antea excogitatam et modo universali applicatam fuisse: ita ut ejus ope in quavis Figura curvilinea proposita, quæ una vel pluribus proprietatibus definitur, Quadratura vel area dictæ figuræ ACCVRATA si possibile sit, sin minus infinite vero propinqua; Evolutio vel longitudo Lineæ curvæ; centrum gravitatis Figuræ; solida ejus rotatione genita, & eorum superficies; sine ulla Radicum extractione obtineri queant. Hic verba [aAccurata si possibile sit] relationem habent ad series quæ abrumpuntur & finitæ evadunt & sic dant Quadraturam accuratam si sit possibile; adeoque methodus quadrandi curvas descripta in Propositione quinta et sexta Libri huius de Quadraturis & propterea etiam methodus fluxionum quatenius in quinque vel sex prioris hujus Libri Propositionibus continetur, annis aliquot annte mensem Iulij 1669, testibus Barrovio et Collinio, mihi innotuitere. Methodum igitur habui ante annum 1667. Id quod volui ostender Nam et Collinius in Epistola ad Davidem Gregorium Iacobi fratrem 11 Aug. 1676 datam scripsit quod ✝ supradictam serierum infinitarūum doctrinam a Newtono biennio ante excogitata fuit, quam ederetur Mercatoris Logarithmotechnia, et generaltergeneraliter omnibus figuris applicata. Methodum igitur habuit generalem applicandi series ante annum 1667. Id qQuod l ostendere erat probandum. The said Doctrine of Infinite series was invented by MrNewton about two years before the publication of Mercators Logarithmotechnia, & generally applied to all Curves. Epist. Collin. ad D. Greg. mMethodum fluxionum anno 1666 aut antea a me inventem fuisse D. Wallisius in Præfatione primi operum suorum Volumineis Anno 1695 in lucem editia testatus est commemoravit. Et librum hunc M S. De Quadraturis nostrates Halleius & Ralphsonus Anno 1691 in manibus suis habuerunt an tractaverat uti Ralphsonus publice testatum reliquit, & Halleus adhuc testatur. 90 Mr Leibnitz agrees with me that the Method of fluxions was obvious. For he saith in his aforesaid Postscript that it would have been easy to have been for me to have found it out if it had been hinted to me. He allows therefore that it would have been easy for him to have found it out if it had bene hinted to him. I affirm that it was hinted to him very plainly, even so plainly that Dr Wallis in the Preface to the two first Volumes of his works published 21 years ago affirmed that in my two Letters of 1676 I explained to Mr Leibnitz the method method of fluxions found by me ten years before the writing of those Letters or above. For I told him that there was a general method of Tan & what was the the extent & characters of the method & made For I made him understand that it was a method of Tangents that I had a general M wchwhich gave the method of Tangents of Slusius made general extended to quadratures of series & other difficulter Problemes set down a series wchwhich it gave me & gave me the series for Quadratures which I there set down. The description For by in my Letters of 10 Decem 1672, 13 Iune 1676 & 24 Octob. 1676 I made him understand that repr For For in my Letters I gav so far described the method as to make Dr Wallis in the Preface to the two first Volumes of his works printed above 20 years ago, say that in my two Letters &c He & his friends pretend now that when I wrote my book of Principles I had not the method of fluxions because I had frequent occasion of using the calculus of fluxions with that book b prickt letters in composing that book & yet nothing thereof appears it. b The ancie But this is a notorious sophism. The Ancients invented their Propositions by Analysis & Demonstrated them by synthesis, & admitted nothing into Geometry before it was demonstrated synthetically. I followed their example that the Propositions in that book might be admitted into Geometry. For the glory of Geometry is its certainty & nothing is to be admitted into Geometry before it be made as certain plane & evident as art can make it. — by Mr Iames Bernoulli who published a specimen of this calculus in May 16 the Acta Eruditorum for May 1690 pag. 218. And from that time the this method began to be celebrated more & more in Germany France & Holland — — in England. And this made Mr Leibnitz in the Acta Eruditorum of May 169 1700 after he had said that no man before me had proved that he had a pub by a specimen made public that he had the method, subjoyn that no man before the two Bernoullis & himself had communicated the method. His meaning th seems to have been that I was the he & BernoullBernoulli were the first who had published specimens of the example differentiall calculusations I & I was the first who by a publicshed specimen without such calculations had proved that I had the method. For the Propositions in the Book of Principles were invented by the Method of Fluxions but composed & published without the calculations that it might be received into into Geometry. For the Ancients admitted nothing into Geometry till it was demonstrated by composition. In this state things continued &c. In the same Reply I said that the Method of Fluxions gave me some certain very general Theoremes for squareding of Curves & there described the first of those Theorem at large & illustrated it with examples. And this Theorem is the fift Proposition of the book of Quadratures & the & it is founded upon the fir & the second of those Theorems is of the same kind & it is is the six Proposition of that book & these two are founded on the four first Propositions And therefore these six Propositions, & by consequence the method of fluxions conteined in them were most certainly knowknown to me wh in the year 1676 when I wrote that Reply. And the D seventh & eighth Propositions of that book are some others two more of those general Theorems. In the same Reply I set down the Ordinates of Curves wchwhich might b I could had compared with the Conick Sections & said that I had long before those days reduced them into a mad a Catalogue of them. And This Cataloge was deduced from the 9th & 10th Propositions of the Book of Quadratures & composed the Scholium upon the 10th Proposition of that Book. And by these things it is evident that the Book of Quadrature & by consequence the method of fluxions described therein was known to me in the year 1676 when I wrote that Letter of Reply & long before. For I had intermitted these studies five years together before I wrote my two Letters of Iune 13th & Octob. 24 1676. And (by what I noted above) it is as certain that Mr Leibnitz when he wrote his Letter of 27 Aug. 1676 knew nothing of the method of reducing Problems to fluxional differential Equations. And if the Manuscript which I wrote in the year 1671 [be compared with the Book of Quadratures it will appear that one & the same method of fluxions is described in them both.] & still held by me, & the Analysis wchwhich Dr Barrow com And ifmunicated to Mr Collins in the year 1669 & wchwhich has been lately published by Mr from a copy thereof in the hand writing of Mr Colling by Mr Iones who purchased his Library, it will appear that the method of fluxions f described in all thr & with anothers Manuscripts which wasere writ before the year 1669 & are still in my keeping, it will be found that one & the same method of fluxions is be compared with the book of Quadratures i it will be found that one & the same method of fluxions is described in them all. And the Method wchwhich Mr Leibnitz calls the summatory method & claims the invention of to himself; is also described in them. For the summatory Quantities wchwhich Mr Leibnits describes by præfixing the Letter

\int

, I represent in this manner,

\int x

\int \frac{a a}{64 x}

, &c I described in my aforesaid Analysis in this manner

x

\frac{a a}{64 x x}

&c & in some of my older Manuscripts in this

▯ x

▯ \frac{a a}{64 x x}

. & in my Book of Quadratures in this

x'

\frac{a a}{64' x x}

& therefore he is not the first inventor of the summatory Method. Knowing therefore that the my method readily gave the Method of tangents of Slusius In his journey therefore from London to Hannover he was considering how to make the method of Tangents of Slusius become general & extend to all sorts as I find by a letter of his dated from Amsterdam 18

\frac{18}{28}

Novem. 1676. And the next year finding that how to deduce the Method of Slusius from the Differential Method of Tangents of Slusius published by Dr Barrow in the year 1670, gave the method of Tangents he wrote bawck in a Letter dated from Hanover 21 Iune 1677 that is by examples of the Differential calculation. For I invented the Propositions in the Book of Quadratures by the Analysis of Fluxions but concealed dem p demonstrated them by Synthesis & published by Composition Synthetically that they might deserve to be admitted into Geometry according to the Rules of the Ancient Geometers, who to make the science certain demonstrated synthetically what ever they found by the Analysis. And in these synthetical Demonstration the Analysis was so conspicuous that Mr Leibnitz allowed that I had this this Book was a proof that To Sir Isaac Newton these 91 calling those things miracles wchwhich create no wonder, & those things occult qualities whose causes are wchwhich are manifest ones whose causes are occult tho the qualities themselves be manifest, & those things the souls of men wchwhich do not animate their bodies. His Harmonia præstablilita is a great miracleous Hypothesis & contradicts the daily experience of all man kind, every man constantly st finding in himself a power of moving his body by his will. He accuses me — — — Epistolicum. If he acknowledged it before the sentences in cyphers were deciphered, & before he knew how by this method the series set down in that Letter for squaring of curves was invented, & or how by the same method the figures whose Ordinates are there, set down were composed with the conick sections, or had seen the Compendium there mentioned wchwhich Dr Barrow th sent to Mr Collins in the year 1669: he ought much more to acknowledge it now. In his Letter of 27 Aug 1676 he acknowledged that he received a p from Mr Oldenburg a packet conteining my Letter of 13 Iune preceding & a parcel of other analytical papers communicated to him by Mr Collins. These other papers conteined Extracts of Gregories Letters & a copy of my Letter of 10 Decem. 1672 in wchwhich the extent the method of fluxions was describdescribed. They were to be sent That Parcel was to be sent back to Mr Oldenburg & accordingly were sent back & are is now in the Custody of the R. Society & therefore werase received & sent back. [And I doubt not but he will still acknowledge that he had did received that extract that parcel parcel & packet & therein a copy of my Letter of 10 Decem. 1672 many many extract of Gregories Letters & particularly a copy of Mr Gregories Letter of 15 Feb. 1671 & a copy of my Letter of 10 Decem 1672.] It conteins many extracts of Gregories Letters & particularly his Letter of 15 Feb 1671 & a copy of my Letter of 10 Decem. 1672. And I doubt not but he will still acknowledge that he received this parcel. L'Abbé Conti SrSir Isaac Newton at his house in S,t Martins Street Near Leicester feilds London 92 Part of a Letter of Dr Wallis to Mr Leibnitz Iuly 30 1697. Optaverim item ut tibi vacet tuam Calculum Differentialem, et Newtono suam Fluxionum Methodum, justo ordine exponere; ut quid sit utrique commune et quid intersit discriminis, et utramque distinctius intelligamus. NB The Doctor here wishes that the common method common to us both & the improvement thereof were stated between us; the first as due to me, & the other as due to Mr Leibnitz. And no Answer was returned to this by Mr Leibnitz; [but under the colour of having improved the Method he went on amongst his friends to call it his own [& And at length, when witnesses were dead, & he thought himself strong enough in disciples; he fell out with Dr Keill for saying nothing more then Dr Wallis had said before without offending him; & in his Letter of 29 Decem 1711 claimed a right to the whole Method; & in order to pick a quarrel with me, pressed that I should declare my opinion in this matter. [A thing which I could not do without either siding with Doctor Keill against him, or retracting what I had published in the beginning of the Introduction to the Book of Quadratures; & what Dr Wallis had said in published many years before in his Preface above mentioned; , without giving any offence to Mr Leibnitz at that time; & what Mr Leibnitz had then yeilded to me by pretending only that he had improved my infinitesimal method as Cartes had improved the metho Analysis speciosa of Vieta. In my Letter of 24 Octob 1676 I said that I wrote a Tract five years before concerning the method of Series & another method together, & that the other method readily gave the Method of Tangents of Slusius & extended to questions about maxima & minima & others, & faciliated Quadratures & proceeded without sticking at surds, & gave the Theoreme for Quadratures there set down, & others of like nature, & was founded upon this Proposition: Data æquatione fluentes quotcunque quantitates involvente, fluxiones invenire: et vice versa] that is, that I should should declare whom I was for, whether for Mr Leibnitz or for Dr Keill. And so I had this Dilemma put upon me, that I must either prepare to engage his army of disciples or publickly retract what Dr Wallis in his Preface, Mr Fatio in Tract de Solido minimæ resistentiæ, & Dr Wallis I in the Preface to my Book of Quadratures had published to the world & [give back that Method wchwhich Mr Leibnitz hath in his correspondence wthwith Dr Wallis he had yeilded & declare all those to me &] confess that I had the ancient Manuscript papers wchwhich Mr Fatio & some others saw in my hands twenty years ago were fourged feigned, & that when Mr Leibnitz acknowledge in his correspondence with Dr Wallis acknowledged that I was the first inventor, he was deceived by me., or else I must prepare to engage him & his army of disciples [& that the sentences set down in my Letter of Octob. 24 1676 Data æquationes fluentes quotcunque quantitates involvente fluxiones invenire ha & vice versa hath no relation to the doth not infer that in those I knew any thing of the direct & inverse methods of fluxions, now called by Mr Libnitz the summatory differential & summatory methods; & that the sentence, Vna methodus consistit in extractione fluentis ex æquationes fluxiones involvente, does not infer that I set down in the same Letter, does not imply that I then knew how to reduce Problems to fluxionsal Equations: & that the universal Analysis mentioned in my Letter of Iune 13 1676 & co & composed of the Method of Series &]; & that the I had the second Lemma of the second book of Philosophical Principles from Mr Leibnitz & yet wrote the Scholium thereupon to assert it to my self. & endeavoured to make it my own by the Lemma Schoium subjoyned to it; & that all the harder part of the Book of Principles of Philosophy was invented either by the Analysis of Mr Leibnitz or (as some would have it) by no Analysis at all, by plane words & s but o] Which gave occasion to Mr Fatio in his Tract concerning the Solid of least resistance to write: Ne published in the year 1699 to write: Newtonum Quæret forsan — — — — — himself] But whilst he durst n But whilst in his Letters to Dr Wallis he found no fault with the Doc him for saying that by my Letters in the year 1676 I explained to Mr Leibnitz the method found by me ten years before or above but yeilded to me so much of the method as was common to us both, & claimed only that he had in the improvements that he had made to it93 made to it: he went on amongst his Friends abroad to call the whole method without distinction his own, & thereby gave occasion to Mr Fatio to insert p write the following Paragraph. Newtonus scit unus omnium optime, satisque indicavit publice cum sua Mathematica Naturæ Principia publiccaret anno 1687, nova quædam inventa Geometrica quæ ipsi communia mecum fuere, neutrum luci ab altero acceptæ sed meditationibus quemque suis debere, et a me decennio ante exposita fuisse. NB. The Lemma Scholium upon the second Lemma of the second Book of Principles is here referred unto. I wrote that Scholium Lemma in the year 16786,. And becasuse Mr I had signified to Mr Leibnitz ten in ten years before that 1676 before that I had a method of tangent wchwhich determined tangents, maxima & minima & other Problems without sticking at surds & included the method of Tangents of Slusius & ately produced in the method of Tangents of Slusius & faciliated Quadratures, & gave me the Series for Quadratures there mentioned & that I had wrote a Tract of this method & the Method of Series together in the year 1671 & that the method compounded of the Analysis composed of these two methods together p it reached to invers Problems of T. & others more diffrcialdifferencial & was so general as to extend to almost all sorts of Problems except perhaps some numeral ones like those of Diophantus: & Mr Leibnitz wrote back the next year that the next year when Mr Leibnitz the next year sent me the Elements of such a Method, & in the year 1684 published those Elements without making any mention of the correspondence wchwhich had been between us: by means I added that scholium to secure the Lemma to my self without saying (whether that Mr Leibnitz had found it apardt & or had received light into it from me ha But whether Mr Leibnitz found the Method apart or had from me light into it from me, is a question I did not there meddle with. Nor was it material, the right thing due to the first inventor. And But Mr Leibnitz proceds. thus NB Whether Mr Leibnitz found the Method apart or received light into it from me I is a questi cujusque specimina quadam dedit in Analysi sua per æquationes numero terminorum infinitas quam Barrovius anno 1669 ad Collinium misit, & in Epistola ad Collinsium 10 Decem 1672 ad Collinium missa. Circa mitium. NB. In the Letters wchwhich hereupon followed between Mr NB. Dr Wallis by his Letter of Dec. 1. 1696 gave notice to Mr Leibnitz of this Paragraph. And Mr Leibnitz made only this exception to it that the methods were not so perfectly alike but had some differences wchwhich the DrDoctor should have mentioned if he had known them. [In one of his Letters dated 28 May 1697 he said. Methodum Fluxionum profundissimi Newtoni cognatam esse Methodo meæ tDifferentiali non tantum adnimadverti postquam opus ejus et tuum prodijt sed etiam professus sum in Actis Eruditorum et alias quoque monui. Id enim candori meo convenire putavi postquam opus ejus non minus quam ipsius merito Itaque communi nomine designare soleo Analyseos infinitesimalis, quæ latius quam Tetragonistica patet. Interim quemadmodum et Vietæa et Cartesiana methodus Analyseos speciosæ nomine venit, discrimina tamen nonnullis. And in his Letter of 29 Mart. 1967 De te autem queri nanquam mihi in mentem venit; quem facile apparet nostra in Actis Lipsiensibus prodita, non satis vidisse. It never came into his mind to complain of the Doctor's Preface, seing he had not sufficiently seen the improvements made to my method.] For as the Analysis speciosa of Vieta & Cartes differed] He did not deny that in the year 1676 I had explained to him the Method found by me ten years before or above but only contended for the m that he had improved the method] See the Letters published in yethe third Volume of the Doctors works. Yet he Mr Leibnitz persisted in amongst his friend to calling the method his own amongst h & thereby gave occasion to Mr Fatio to write what follows.— a celeberrimum Virum Leibnitium literis misit, ,[quæ tamen per aliam quandam methodum generalis redderetur [& hanc aliam in altera Epistola descripsit ait]] Eodem porro anno — — — fusius patebit. Fundamentum vera Problema autem in quo fundabatur huæjus methodis literis quidem transpositis ad hunc modum celabat — — — et vice versa. Mente Newtoni ex his et alijs acceptis constat percepta, Vir celeberrimus rescripsit anno proximo rescripsit se in parilem methodum incidisse (ut ex ejus Epistola in tertio Operum Wallisij Volumine videre licet) cujus tamen explicationem — — — distulerit. 93 Galileo upon a supposition of uniform gravity shewed that if gravity in a falling body produced one part of velocity in one part of tie it would produce another part of velocity in another part of time & a third part of velocity in a third part of time &c. so that the velocity produced would be always equall proportional to the time in which it was produced & by consequence to the force producing it. For if gravity uniform gravity impress one part of force in another one part of time, it will impress two another parts of force in another part of time & a third part of force in a third part of time & so on NB. Dr Wallis by his Letter of Decem. 1. 1696 gave notice to Mr Leibnitz of this Paragraph to Mr Leibnitz; And who made only th [in his Letters published in yethe third Volume of the Doctors Works) made only the exception to its that altho the Methods were were not perfectly alike but had some differences indeed alike & therefore he called them both by the name of the infinitesimal method yet they differed in some improvements wchwhich he had made to it but Mr L & wchwhich the DrDoctor should have mentioned if he had known thesem improvements. See their Letters published in the third Volume of the Doctors works. Yet After this Mr Leibnitz persisteding amongst his friends to call the Method his own & thereby gave occasion to Mr Fatio to write what follows. Out of the Answer of Mr Leibnitz published in the Acta Eruditorum for May. 1700 p. 202. Certe cùm Elementa calculi mei edidi anno 1684 ne constabat quidem mihi aliud de inventis ejus in hoc genere quam quod ipse olim significaverat in literis posse se tangentes invenire non sublatis irrationalibus; quod Hugenius quoque mihi se posse mihi significavit postea etsi cæterorum istius calculi adhuc expers: sed majora multo consecutum Newtonum, viso demum libro Principiorum ejus satis intellexi. Out of the Letter of Mr Leibnitz to Mr Oldenburg dated 21 Iune 1676. NB What is here said acknowledged & was further said acknowledged in his Letter to Mr Oldenburge dated 21 Iune 1677 concerning the likeness of the Methods admits of no retraction. In theis latter part of this Answer he challendged Mr Fatio to solve the ProbleProblem of Be Bernoulli a series of Curves cutting another series at right angles. Which makes it probable that the solution of this Problem has been ever since reserved for a challenge. Mr Fatio returned an Answer but the Editors of the Acta Lips. would not print it, & what it was I do not know. 92 After he had voluntarily in his Letter of 1693 voluntarily k given me the preference speaking of himself as if he also had endeavoured to can pass the same thing. After Dr Wallis had said in his Preface — — & the Editors of the Acta Eruditorum had complained read the Preface complained of noting in the complained of Dr Wallis for nothing more in that the Preface to the two first Volumes of his works besides the words Quod moneo nequis &c & the DrDoctor had excused his represented that he know nothing more of that Method then that it was of the same kind with mine & the Editors & Mr Leibnitz had acquiesced in that excuse & Mr Leibnitz had represented that acknowledged the likeness of the Methods & [said that he therefore called them by the common name of the infinitesimal method, but yet as the Analysis of Vieta & Cartes were called by the common name of Analysis Speciosa & yet differed in some things, so my method & his might differ in some things &] given me the preference as the earlier inventer pretending] pet pretended to nothing more th then that the metho the methods differed in some th] th pretended to nothing more then the improvements that he had made to it: to call the whole method his own That is whether For the For the two parts of the sentence set down in my Letter of 24 Octob 1676, vizt Data æqu.—, et vice versa denote For the sentence ,[Data æquatione fluentes quotcunque quantitates involvente, invenire fluxiones et vice versa, which I inserted into my Letter of 24 Octob 1696 as the foundation of the method of series & flux upon wchwhich & the method of series together, I had written a Tract in the year 1671; a relates to the direct & inverse methods of fluxions, called by Mr Leibnitz the Differential & the summatory methods. & the sentence, Vna methodus consistit in extractione fluentes ex æquatione fluentem involvente, relates also to both methods; to the direct method by wchwhich I brought Problemes to fluxional equations, & to the inverse And both these methods, & especially the second, are for enlarging the bounds of Analysis, & extending it to such quantities as Mr Leibnitz calls trans the transcend the vulgar equations & therefore are by Mr Leibnitz called transcendent; of wchwhich kind are the Ordinates of Curves called by Des Cartes Mechanical. — Wallisij operum volumina primum et secundum prodiere, Hugeniusque curiositati meæ favens locum inde descriptum ad Newtonum pertinentem mihi mature transmisit. NB. The first part of this Answer should have been made to Dr Wallis, had it been agreed with our Letters of 10 Decem 1672, 24 Octob. 1676 & 21 Iune 1677. The second part admits of no retraction upon a pretence that I had deceived him. The third part seems to relate to the place above mentioned in the Preface to the two first Volumes of the Doctors works, concerning the likeness of the Methods. In the same Answer, he acknowledged that I was the first who had proved by a specimen made publick that I had that part of the infinitesimal calculus by which the solidum rotundum minimæ resistentiæ & the Curva celerrimi descensus &c were invented. Mr Fatio returned an Answer to Mr Leibnitz, but the Editors of the Acta would not print it, & what it was I do not know. 94 Mr Newtons method could not be well understood wi The whole design plain meaning of the place was to tell the world that Mr Leibnitz was the first inventor of the method, & that Mr Newtons method could not be well understood without first understanding the Differential method first of Mr Leibnitz & therefore Mr Newton useds & ever did use fluxions for the Leibnitian differences as Fabrius su even as Fabrius substituted motions for the method of Cavallerius. When In the year 16676 he had notice of Mr Newtons method & that Mr Newton had written a Treatise of it five years before & the notice was so plain that he was able the next year to compare the methods & see the likeness. when he had found the differential method to compare the methods & see the likeness as appears by his Letter of 21 Iune 1677. And in the year 1684 when he published the elements of his method, he acknowledged a another method like his own but tho he did not say whose it was, & of what antiquity as he he ought to have done. At that time I was writing my Principia Philosophiæ & the book came out in the year 1687 & was the first instance made publick of applying the Method to the higher sort of Problemes. And the next instances made publick wasere thatose of Mr Leibnitz De Lineis Opticis e de motu projectilium in Medio resistente & de Motuum cœlestium causis, published in the yethe year 1689 & wholly all of them taken out of the Book of Principles. And the next year the Differential method began to be taken notice of abroad & in the year 1693 Mr Newton received a Letter from Mr Leibnitz wherein he acknowledged that in publishing the book of Principles I made it appear that my Analysis I h my Analysis recht beyond the vulgar & said that he also by convenient characters had endeavoured to subject the transcendent Geometry to Anslysis. But h And hitherto he gave me the preference. He d In the beginning of the year 1695 Dr Wallis hearing that the method began to be celebrated in Holland under the name of the Differential method inserted into the Preface of thise two first Volumes of his works athat in the Letter wchwhich in the year 1676 had passed between me & Mr Leibnitz I explained to him the method found by me ten years before that time or above & the next year gave notice to Mr Leibnitz of this passage & in the Letters wchwhich followed between them Mr Leibnitz did not deny it nor find any fault wthwith what the Doctor had said. In the year 1699 the DrDoctor published those Letters in the third Volume of his works having received copies of mine from Mr Oldenburg him self & of Mr Leibnitz's from the Library of Mr Collins, & Mr Fatio published the same year that I was the oldest inventor by many years, I & was not contradicted by Mr Leibnitz the next year Mr Leibnitz returned an answer without contradicting him. Towards the end of the year 16703 Dr Wallis died in October 167 1703 the last of the old men who knew what passed in the days of Mr Oldenburg & hitherto Mr Leibnitz forbore to claim the first invention. This claim was first made in the year 1705 by the Passage above mentioned in the Acta Eruditorum for Ianuary 1705. And thereby therefore Mr Leibnitz & his friends are the aggressors & out to prove their claim. If he pretends he's in possession I can tell him that he never was in quiet possession, & that I never was out of possession in England, & that what possession he has was got by concealing from his Countrimen his knowledge of what passed between him & me in the year 1676. — upon which Mr Leibn so much stress is laid is wrested against me; it was written not to give away that Lemma to Mr Leibnitz but on the contrary to assert it to my self. Because Mr Leibnitz in his in publishing the Elements NB Mr Leibnitz having prepared the way to shew tell the DrDoctor that his what was his method was not fully altogeraltogether the same with mine but that that bef he & that he had improved what & that whadt he had added but some things had been added by him to what was common to us both to the common method he repeats in his next letter what he had said in the Acta Eruditorum of of Iune 1706 concerning the inven Analytical inventions of Galilæus, Cavallerius, Fermat, Cartes, Gregory a S. Vincent, Huygens, Wallis, Wren Iames Gregory, Isaac Barrow, Mercator & me: NB The use wchwhich D Mr Leibnitz makes of this in his following Letters is to tell the DrDoctor that [the methods [ differed more then he understood, [that I had made the method of Series universal & that if I would print what as he understood I had further by me (not excluding the method of fluxions) it would open new ways to the improvement of science.] And that he had improved it] his method was not so perfectly the same with mine but differed from it by the some improvements that wchwhich he had made. He did not complain of the DrDoctor for saying that by my Letters I had in yethe year 1676 I had impr explained to him by my two Letters the method found by me ten years before or above but invited me contented said only contend that he had improved it by shewi teachin how to reducing transcendent quantities & mechanical Curves to equations & by adding the invertion of exponential Equations.to the method. NB. Mr Leibnitz in the Acta Eruditorum for October 1684 published the Elements of the Differential method & there mentioned a Methodus similis without saying whose it was & what he knew of it as he ought to have done in point of candor. Afterwards in the Acta Eruditorum for Iune 1686 he in giving an AcctAccount of what had been done by the Analytical inventions of Galileo Cavallerius Fermat Cartes, Gregory of St Vincent, Huygens, Wallis, Gregory, Barrow, Mercator & me, he mentioned my general method of Series, & but instead of saying acknowledging what he knew of my method infinitesimal Analysis he left invited me to was silent about it & left me to publish it my self giving me the complement that by doing so I, & in this Letter to Dr Wallis he doth the same. with this repeats all this instead of doing me justice. And by the same sort of Candor Dr Wallis if he had been silent. & left Mr Leibnitz to publish them himself without pleading ignorance This fault Mr Leibnitz committed against me at the very same that he & the Editors of the Acta would not allow the like in the like in Dr Wallis towards Mr Leibnitz, but made him plead ignorance before of what he had omitted. all this he had notice from London the year before & yet In the year 1684 Mr Leibnitz published Gregorys series as his own without letting the world know & in the year 1684 he published the Elements of the differential method. Both these he published as his own without letting the world know that he had received any thing from London concerning them. I was then writing the Principia Philosophiæ & this book came abroad in 1687 & was the first instance specimen made publick of applying the method to the higher sorts of Problems. In the year 1689 Mr Leibnitz took the published the principal Propositions of this book as his own in three papers called Epistola de lineis Opticis Schediasma de resistentia Medij & motu Projectilium gravium in Medio resistente & Tentamen de motuum cœlestium causis, pretending that he had found them all the ye before htat book came abroad. And to make the principal Proposition [concerning the revolution of the Planets revolving in Elliptical Orbs & describing with a Radius drawn to the Sun describing equal areas in equal times about the Sun placed in the lower focus of the Orb, described equal areas in equal times he] ada] his own adapted to it an erroneous demonstration. And this was the second specimen made publick of applying the method to the higher Problems. In the years 16796 & 1691, the Differential method began to be celebrated & in the beginning of the year 1695 Dr Wallis hearing that it began to be celebrated in Holland inserted into the Introduction of the two first Volumes of his works as Para that in the Letters wchwhich passed between me & Mr Leibnitz in the year 1676 I explained to Mr Leibnitz thies method found by me ten years before or above that is in the year 1666 or before, & gave notice of this to Mr Leibnitz in a Letter dated 1 Decem. 1696. And the Editors of the Acta Eruditorum in giving an Account of the two volumes of Dr Wallis took cited some words out of this passage But neither they nor Mr Leibnitz cited in the Letters wchwhich followed between him & Dr Wallis denyed the truth thereinof. In And in the year 1696 the Marquiss de 1693 I received a Letter from Mr Leibnitz dated in Ian 17 March wchwhich began with these words in wchwhich he acknowledged that by the book of Principles it appeared that I had a method of doing what the vulgar AnalysAnalysis would not reach & added that he had also by a convenient notation endeavoured to subject the transcendent Geometry to Analysis. But he did not yet begin to give the preference to himself. In the beginning of the year 1695 Dr Wallis hearing that the Method began to be celebrated in Holland under the name of the Differential method of Mr Leibnitz inserted into the Introduction of the two first Volumes of his works, a Paragraph importing that in the Letters wchwhich passed between me & Mr Leibnitz in the year 1676 I explained to Mr Leibnitz him the method found by me ten years before or above, that is, in the year 1666 or before, & gave notise of this Paragraph to Mr Leibnitz in a Letter dated 1 Decem. 1696. In the years 1669, 167 In Iuly 1669 Dr Barrow sent to Mr Collins my Analysis per series numero terminorum infintas conteining my method of series with some specimens of my methods of fluxions intermixed. And in the year 1671 I wrote a larger Tract of on those methods together. grounded In the year 1670 Dr Barrow published his method of tangents & Mr Gregory deduced from it the method of Tangents of Slusius & by a letter dated 5 Nov. 1670 gave notice thereof to Mr Collins & in the end of that year fell into my method of series., & in a Letter dated 15 Feb. 167

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sent him a Series since claimed by Mr Leibnitz & the series wchwhich he had communicated to his received from me & Gregory he freely communicated to his friends at home & abroad. In a Letter dated 10 Decem 1672 I gave no sent the same my method of tangents of Gregory & Slusius to Mr Collins & told him that I took it to be the same with that of Gregory & Slusius & that it was a branch or rather a Corollary of a general method of Analysis wchwhich extended to the abstruser sorts of Problemes concerning the Curvities, areas, lengths centers of gravity of Curves &c & that proceeded without freeing equations from surds & that I had interwoven this method with that other of series, meaning in the tract wchwhich I wrote the year before. In the beginning of the year 1673 Mr Leibnitz was at London & pretended to the Differential method of Mouton & received notice of Mercators series for the Hyperbola & might receive notice of some other series without being acquainted wthwith it & going from thence to Paris was instructed in the higher Geometry by Mr Huygens till & in the mean time wrote Letters to Mr Oldenburg about other matters dated Mar. 30, Apr. 26, May 24, Iune 8 & the next year in Iuly & October 16 sent notice to Mr Oldenburg that he had found the area area or circumference of a circle by a series of numbers, & that by the same method any Arc might be found by the like series whose sine was given tho the proportion to the whol of the Arc to yethe whole circumference was not known. If the proportion was known it gave him the circumference. If it was not known yet it gave him the arc. In April 1675 he received eight series from Mr Oldenburg & amongst which wasere the series of Gregory for finding the arc by the tangent & the tangent by the arc & mine for finding the Arc by the sine & the sine by the arc. And by his Letter of 120 May 1675 he acknowledged the receipt thereof & said that he wa could not at pr was not then at leasure to examin them & compare them with his own which he had found some years before that is, in the year 1673 or before. His own were therefore different from all these., & tho he was not at leasure to examin what was sent him yet he was at leasure to compose a demonstration of one of them & communicate the composition to his friends at Paris without letting them know what he had read. In his Letter of 12 May 1676 he admired the two series for finding the arc by the sine & the sine by the arc & said that because of their elegance Mr Oldenburg would do him an acceptable favour if he would send him the demonstration thereof, that is, the method of finding them & promised him a recompence & added that Mr Collins could easily satisfy his desire. And all this makes it plane that he had not yet any method of finding these two series. In my Letter of Iune 13 1676 I sent him my method of series with many s examples, & said that Analysis by these series extended to almost all Problemes, but became not altogether universal without some further methods, wchwhich I forbore to explain because I began long ago to be weary of these studies, so as to havieng absteined from them about five years. At the same time with this Letter Gregory died in neare the end of the year 1675 & Mr Collins at the request of Mr Leibnitz & some others of the Accademy of Sciences, collected his Letters & made extracts of them from Gregoris Letters & gave the collection to Mr Oldenburg to be lent to Mr Leibnitz to peruse & & M & & to return the same. tal And this extract is still in was returned to Mr & in the tit with this title, Extracts a from Mr Oldenburgh Gregories Letters to be lent Mr Leibnitz to peruse who is desired to return the same to you. And the same was sent to him at the same time with my Letter of 13 Iune 1676 & returned soon some time after to Mr Oldenburg & is still in the hands of custody of the R. Society, & among the Letters conteins thosate of Gregory dated 5 De Novemb. 1670 & 15 Feb. 1671 & that that written by me 10 Decem 1672. Mr Leibnitz in his answer dated 21 Iuly 1676 thanked Mr Oldenburg M Collins & me for what had been sent him, & in requital sent back his method of series by transmutation of figures into other figures in whose equations the Ordinate ascends only to one to no power & therefore may be reduced to an infinite series by the division of Mercator alone, & or to one or two or more powers & then may be reduced to an infinite series by my Extractions95 extractions of roots ② & all this method of transmutation he commended to the skies as one of the most valuable inventions in Analysis & illustrated it with a series for finding the arc of a circle whose tangents was given, saying that he had communicated to his friends that expression for the circle above three years before or above, ① & that he would hereafter explain how by this method the roots of Equations howsoever affected may be given in infinite series without any extraction. ③ But this method without my extractions of roots could not give the ar help him to a series for the arc whose sign sine was given & therefore was t not the method which gave him the series for the circle & mentioned in his Letter of 26 Octob 1674. And if he began to communicate it at Paris before the above three years befo The series for finding the the Arc for by the tangent was sent to him from London twice, first in Mr Oldenburgs Letter of 15 Apr 1675 & then in a copy of Mr Gregories Letter of 15 Feb 1675 inserted sent to him in Iune 1676 amongst the extracts of Gregories Letters. It might give him the series of Gregory, but not above three years before he wrote this Letter of 21 Iuly 1676. At For at that time he was only learning the higher Geometry of Mr Hygens & iIf he began to communicate Gregories series so early as that series at his first coming to Paris he might carry it with him from London. had it before he invented it. For when he received it from London in Mr Oldenburgs Letter of 15 Apr: 1675 he did not know it to be his own. And when he received it again in Gre a Copy of Gregories Letter of 15 Feb 17671 he knew it to be Gregories & concealed his knowledge. extraction of roots,. [& that he would hereafter explain how by this method the roots of equations howsoever affected may be given by an infinite series without any extractions.] And this method of transmutation he commended to the skies as one of the most valuable things in Analysis, & illustrated it with a Gregories series for finding the arc of a circle whose tangent was given, saying that he had communicated it to his friends [that expression for the circle] three years before or above, that is, before or above 27 Aug. 1673, the [or while he was yet learning the higher Geometry frōom Mr Hugens. And yet eight months after when he received it from Mr Oldenburg 15 Apr 1675 he did not know it be his own & ten months after that, when he received it again in a Copy of Gregories Letter of dated 15 Feb 1671, he knew it to be Gregories, never to this day acknowledged & yet after that he published it as his own without acknowledging that Gregory had found it before him & that it had been sent to him twoice from London.] or soon after his coming to London from Paris from London. In the same Letter he said that he did not beleive that my meth most difficulties (except my method of Infinite series was so general as I had described, because many Problems were so wonderfull & perplexed intricate that they could not be reduced to equations or quadratures, such as were (amongst many others) the inverse problems of tangents. And therefore he had not yet found the methods by of reducing Problems to series either by the help of by differential equations or by assuming arbitrary series. In my answer next Letter dated 24 Octob 1676 I replied that five years before I (vizt anno 1671) I wrote a Tract about series & another method together wchwhich readily gave the method of Tangents of Slusius & stuck not at equations involving surds & extended to quæstions de maximis & minimis & others & Quadratures & sothers & couched it in the following sentence exprest a enigmatically, Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire & vice versa. I set down a Theoreme also for squaring of curves wchwhich breaks & was saying that it was found by this method & mentioned other Theoremes for comparing them with the Conic sections. H How these Theoremes follow from this method is explained in the book of Quadratures & without this method it was impossible to find them I said also that my methods extended to inverse Problemes of Tangents & others more difficult & that either by extracting the fluent out of an equation involving its fluxion or by assuming a series at pleasure . I said also that if the relation between any two sides be defined by any of the right anguled triangle conteined under the ordinate Tangent & sSubtangent be defined by any equation, the Probleme may be resolved without my general method but if the Abscissa also entred the vinculum, it used to be otherwise, that is, for the most part to require my general methotd, wchwhich method is compose the method of series made general by the methods of fluxions & arbitrary series as I said in my Letter of 13 Iune 1676 before. This Letter Mr Leibnitz saw in the hands of Mr Collins when he was in London the first second time, as he has lately acknowledged in one of his Letters, & after his arrival at Hannover received a copy of it. Aft In his answer dated 21 Iune 1677 he sa sent back the method of Tangents published by Dr Barrow in 1670 & the improvement with the characteristic changes & how this meethod readily gave the method of Slusius as Gre Gregory had notified, & how in his Letter of 5 Novem 1670 & how the method wchwhich did this might be improved so as to proceed wthwithout in equations involving surds & extended to quadratures as I had notified in my Letters of 10 Decem 1672 & 24 Octob 1676. This is all the Differential method wchwhich he then sent back to me & of complained that it was too long to be answered & yet should have been longer by printing all the Letters entire 96 Mr Newton was therefore the first inventor, & whether Mr Leibnitz invented it afterwards w proprio Marte afterwards or not is not a question of no consequence. The first inventor is the inventor & inventoris jura are due only to him. He has the sole right till another finds it out & then to take right from him & share with another would be an Act of injustice, & a great encouragemtencouragement to pretenders. But however there are great reasons to beleive that Mr Leibnitz did not invent this without receiving some light from Mr Newton. For it must be considered observed that wherever Mr Newton in his Letters spake of his general method he understood his Meth'd of Series & Fluents together taken together as two parts of one general method. The series are applie in som In his Analysis the series are applied to the solution of Problemes by the method of fluents for finding & thereby give new series. In the treatise written A.C. 1671 he year 1671 he wrote of both together. The series in his book of Quadratures are derived from the method of fluents & were derived from that method before the year 1676 p. 72. The method of extracting Fluents out of equations involving their Fluxions comprehends both together. The method of involving assuming the terms of a series & determingdetermining them by the conditions of the Problem is proceeds by means bof the method of fluents. And wWhen Mr Newton represented ytthat his method of series infini series extended to the solution of almost all problemes except numeral ones like those of Diophantus, he included in the method of fluents be included the meth included Problems of inverse problems of Tangents & therefore wchwhich & those problems are not tractable wthwithout the method of fluents. And sometimes series are considered as fluents & their second terms as moments. And Mr Newton sometimes derives this method of fluents from the series into wchwhich the power of a binomium is resolved p 19. lin. 19, 20. In the next place it must be observed that Mr Newton at the request of Mr Leibnitz communicated to him one freely & plainly one half of this general method the method of Series. So far as he could do it without writing openly of yethe other part. p. 45, 49. Mr Gregory by the help of but one series wthwith notice that it was the result of a general method found out the method in within the space of a year. Mr Leibnitz pretended to have two series in the year 1674 & had eight others sent him by Mr Oldenburg in April 1675 & took a years time to consider them: but this method being altioris indaginis he could not find it out, but at length requested Mr Collin Leibnits Oldenb. to procure it from Mr Collins, & at the request of Mr O. & Mr C. Mr N sent it to him. And when he had it he coul understood it with difficulty & desired Mr Newton to explain some things further. And for the other half of the method, Mr Leibnitz had a general description of it in Mr Newton's Letters of 10 Decemb 1672, 13 Iune 1676 & 24 Octob 1676, with an examples of it in drawing of Tangents p 30, another example of it in squaring of curves p 72 a third example of it & solving inverse problems of tangents p. 86. [If he had been told nothing more thaten that Mr Newton had a general method wchwhich extended to inverse Problemses of tangents & others more difficult, it would have been enough to exclude him from the title of in pretense of inventor. But when he was told that it ext] In being told that it extended to Tangents of mechanical curves, & to quadratures, curvities, & centers of gravity of curves in general & to inverse problems of Tangents he was sufficiently told that it was founded upon the consideration of the small partiscles of quantity called particles & moments by Mr Newton & indivisibles infinitesimals & Differences by Mr Leibnitz. For there is no other way of considering any of these Problemes but then by thesese particles of quantyquantity. Mr Newton therefore having told Mr Leibnitz one half of his method & made so great a discovery of the other half are to say said that the foundation thereof was obvious (p. 72. l. 1 & had reason to put it in cyphers to least it should be taken from him. & Mr Leibnitz Now Mr Leibnitz by Mr Newtons example of drawing Tangents, seing his Mr Newton method of tangents the same wthwith that of Slusius, but more general in not sticking at surds, & extending to Mechanical curves, & that this method was of Tangents was but a Corollary of a very general methods, & was founded upon the consideration of infinitesimals: & set his mind upon improving the method of Slusius & that by the differences of the Ordinates to (p. ) & And now Mr Leibnitz knowing by Mr Newtons Letters that the method of Slusius for drawforing tangents was a corollary of Mr Newtons general method (p 39) set hims mind mind upon improving it this method & that by the differences of the Ordinates (p 87, 88) He considered that as the summs of the Ordinates gave the Area in the method of Cavallerius, so their tdifferences gave the tangents & thence received the first light into the Differential method (p. 104) & presently having found it he saw that this method it answered to the description wchwhich Newton had given of his method in drawing of Tangents & that without sticking at surds, & in rendring quadratures more easy & in bringing of inverse Problemes of Tangents to Equations & Quadratures p 88, 89, 90, 91 93. In his journey home by London & Amsterdam he was upon another designe of improving the Slusius's method of Tangents published by Slusius for & extending it to the solution of all sorts of Problemes & this was by calculating a certain Table of Tangents as the most useful & easy method of what he could then think on (pag 87) This was in Novem. 28 1676 & therefore he had not invented the Differential method at that time. But the perusal of Mr Newtons Letters had given him a notion that the method of Slusius had given him him a notion that the method of Slusius was capable of being improved into a general method of solving all sort of Problems. In his last letter against Mr Keil (p. 119) he represetsrepresents that his friends how know how he found the Differential method in a very different manner from what Mr Keil surmises & these words imply that he found it after he got home to his friends after & by consequence in the year 1677 & most probably not before the receipt of Mr Newtons second Letter. For the publishing of his an opusculum upon Mr Greg. series was first suspended upon his co by b by his coming into business, & then laid aside upon his finding the Differential method. p. 42. And now Having Thus much concerning the invention of the Method. And now it may not be amiss to take some notice of the conduct of Mr Leibnitz in this & such like matters. Dr Pell reprehended him for pretending to Moutons differential method. Mr Collins reprehended him for intermedling with a Letter what Mr Gregory & Mr Tschurnhause were eupon, as appears by a Letter not yet published. When Mr Tschurnhause had fallen upon the method of assuming the terms of a series & determining them by the conditions of the Probleme Mr Leibnitz took it from him by pretending (without proving) that he had communicated it to Mr Tschurhause ten years before when they were together at Paris, & that Mr Tschurnhause had forgot the communication. Mr Leibnitz should not have pretended to two series in the year 1674 & afterwards have written to Mr Oldenburg for the method of finding them. He should not have forgotten the eight series sent him by Mr Oldenburgh & Mr Collins & after he had taken a years time to find it out, hav & the next year have written to Mr O He should not have taken a years time to consider them upon pretence that he was busy about mechanical things, nor have written to Mr Oldenburg to procure him the method of finding them from Mr Collins wthwithout M Newtons knowledge the method of finding such series when he could not find it himself. For by his own rule h that if he had gone should afterwards have forgeotten the receipt of the method, Mr Oldenburg & Mr Collins were not to put him in mind of it without authority from Mr Newton. He should not have s He should not have publ communicated to the Ginthent at Paris an opusculum written upon one of the eight series sent him without letting them know that hed received it from Mr Oldenburgh & Mr Collins. He should not have published that series afterwards in Acta Lipsica as his own without L He should not have promised Mr O. & Mr C. su a series of a very different kind from the two wchwhich he had received from the Mohr, & then have sent th in recompence for Mr Newtons meethod & then have sent then when he meant to send them nothing but one of the eight series wchwhich he had received from them the year before. He should not have published that series as his own in the Acta Lipsica without acknowledging whence he had it & that it was o Mr Gregories series sent by him to Mr Collins in the year 31671. 97 He should not have endeavoured to share the method of series wthwith Mr Newton by sending his transmutation of figures for a general method of finding series when he had newly desired Newtons method for want of a general method of his own, & the transmutation was not be not method not being general till Mr Newton made it so by communicating his extraction of roots, & even then being tedious & of no use. He should not have laid claim to some of the reciprocal serie sent him by Mr Newton by changing the signes & subducting the versed sine from the Radius when he had no method of finding reciprocal series besides the method sent him by Mr Newton & did not yet understand it. He should not at the same time have desired Mr Newton to send him his method of reciprocal series & when after he understood it had it & with difficulty understood it have written back that by his old papers he found the he had found it before as he peceivedperceived by his old papers but for want of a good example had laid it aside. He should When he sent his differential method to Mr Newton he should have acknowledged that he had but newly found it out. And not have described it in such a manner as shows that heis aim was then to make himself the first inventor of the method. For men are not to interrupt one anothers proceedings nor to snatch away one anothers inventions. And when he published an Account of Mr Newton's Differential method the f book of Quadratures & represented that the method of fluxions was from the first beginning thereof substituted to & used instead of by Mr N. instead of the Differential method (p ) he had forgot that the very first proposition of the Book was set down in Mr Newton's Letter of 24 Octob 1676 as the foundation of the method of fluents & fluxions upon wchwhich Mr Newton had written a treatise in the year 1671. And the Mr Keil put him in mind of this, & explained this to him & that all the second Proposition of the book was in the Analysis written A.C. 1669 & that the eight following Propositions (wchwhich are all founded upon the method of fluents) & fl were all of them found before the year 1676 the writings of the said Letter in the year 1676 (pag. 113) yet Mr Leibnitz & by consequence before Mr Leibnitz knew any thing of understood the differential method: yet Mr Leibnitz will has not mended the account that he has given of that Book of Quadratures but cryes out represents that his candor is questioned, that he is t at his age after so many p documents of his life it is not to be expected that he should defeddefend it by an Apology, & enter into a debate against a young man unacquainted with what passed & not authorized by Mr Newton, & refuses to explain how he came by the differential method, & affirms that the affir blames Mr Keil for finding fault wthwith the account given of Mr said book, saying that it is free from detraction & has given every man his due, that is that the method of infinitesimals is due to Mr Leibnitz & the rest of the book to Mr Cheyne & Mr Craige: & nothing to Mr Newton but drudgery & dishonesty in prosecuting other mens inventions as his own. Seriem Gregorij mense Maio superiore Anni hujus Leibnitius ab Oldenburgo acceperat & eo tempore suam esse vel non noverat vel non audebat asserere (p 42 lin 8) De hac ser Hanc seriem Gregorius anno 1671 Gregorius qu ante quadrennium hanc seriem Collinio misserat, & & ante finem anni hujus emortuus est, sub id tempus & as Leibnitius jam opusculum de hac sere compositum cum Gallis communicare jam cœpit p 42. l. 25. R Eandem prius cum Gallis ut suam communicasse non constat. Le Communicavit forte sed vix ut suam. Nam Quadraturam quam Leibnitius postea OldergusOldenburgus postea a Leibnitio accepit Leibnitius ut mense Maio Anni hujus ab ipsissimo Oldenburgo acceperat & eo tempore suam esse vel non noverat vel non audebat asserere (p 42 l 8) ideoque nondum cœpeat aperte communicare ut suam ideoque nondum cœperat aperte Gregorius, hoc ante quadrennium communicavit hanc seriem eandem Collinio miserat, & hoc anno mortuus est Sub idem tempus Et Leibnitius opsuculum de hac serie cum amicis Gallis jam communicare cœpit p 42 l 25. Anno 1674 Leibnitius jactabat se hujusmodi series duas habere sed easque a aliunde Collinio accipere potuisset, sed sed quænam fuerint & unde habuit non constat. Vt methoCommunicare potuit sed vix ut suam. Nam quadraturam quam cum Oldenburgo postea communicavit, accepeitat ab Old ipsissimo mense Maio hujus Anni & eo tempore suam esse tunc non novera Hanc Quadraturam Leibnitius mesemense Maio A. Quadraturam quam Leibnitius suam esse. Quadraturam Gregorij Leibnitius ho Mense Maio anni hujus ab Oldenburgo accepiterat ut supra, Eandem et et eo tempore suam esse tunc vel non noverat vel non ausus fuiterat asserere. Eandem ab alijs prius accepere Opusculum de eadem At opusculum de eadem ante finem hujus ann, cCum Gregorius ante finem anni hujus jam emortuus esset, cum amicis Geometricis opusculum de eadem in Gallia communicare cœpit. Si ante bienno Eandem Quænam fuerint series duæ quas anno superiore habuit et unde habuit in dubio est. Certe demonstrationem den habuit non habuit smox de anno proximo utriusque nondum habuit. The method of Tangents first communicated by SrSir Isaac Mr Newton & afterwards printed by Slusius was grounde Slusius grounded his method of Tangents upon three Lemmas the two first of wchwhich were these 1 Differentia duarum dignitatum ejusdem gradus applicata ad differentiam laterum, dat partes singulares gradus inferioris ex binomio laterum ut

\frac{y^{3} - x^{3}}{y - x} = y y + y x + x x

. Quod facile ostenditur. 2 Tot sunt partes singulares ex binomio in gradu quolibet, quot unitates in habet exponens Dignitatis immediate superioris; tres nimirum in Quadrato, quatuor in cubo &c Et hoc vulgo notum. Here Slusius When Slusus applies these demonstrates his method by these Theoremes he takes the Difference of two Dignities & the difference of their latera h for infinitely small Differences. If Mr Leibnitz when his mind ran upon improving this method of Tangents printed by Slusius understood the De had recourse to these Lemmas & at length understood them, he might easily have understanood that Mr Newtons Theoreme for resolving the dignity of a Binomium into an infinite series was of the same kind with these two Lemmas & w but much more general & complete & comprehended them as a General comprehends a particular & by consequence he would easily have understood that this Lemma was the foundation of Theorem had the same relation to Mr Mr Newtons general method wchwhich as those Lemmas some hofad the particular method of Slusius published by Slusius. And since he gave the name of Differences to the infinitely small parts of quantity, its very probable that he had that name from the first of these Lemmas of Slusius. & by consequence understood them Archimedes began the methods of squaring curves & drawing tangents to them by conderingconsidering the infinitesimals of quantity. Cavallerius & Fermat applied this methodsodsmethod to Equations Fermats method was first published by Herigon & Schocten. Wallis, Gregory, Barrow & Slusius improved these from Quadratures Tangents & Tangents then it for Tangents Mr Newton made it general extendeding this it to the solution of all sorts of difficulter Problems Leibnits by means of the metho & notice of his having made it general was sent to Collins & by Collin Gregory Slusius & others in the year 1669 Mr Leibnits having notice thereof & one half of Newtons method being communicated to him wthwith some descriptions & examples of the other half, fell lg of into Newtons method the other half of Newtons method, but A.C. 16767 wrote back to Mr newton as if he had known it long before, pretends to be the first inventor refuses to let the world know plainly the time & manner of his invention & to perswade them that he was the first inventor, he was to have the R. S. take his word qu & quarrels Mr Keill for questioning it with o oring from Mr Newton & concealing what he learnt by his correspondence wthwith Oldenburg & Collins, appeals from young men who were not acquainted with it to being witnesses that is but himself to himself alone the other two being long since dead, & Mr Newton not knowing what they writ to him being not witness to their Letters to be conceals what he learnt by his correspondence with Mr Oldenburg & Mr Collin & will not allow young men to understand the Letters & papers written before their time who were not privy tho that correspondence capable of understanding the letters & papers then written before their time, & yet appeals to yethe judgmtjudgment of yethe R. Society amongst whom there is not a man now to be found besides himself who was privy to what it. And tho he desired had the method of series plainly communicated to him at his own own request, yet he cannot forbear putting in for a share in that invention also. In the year 1673 when he was in London the first time, there was not a Mathematician of note in that city who had not heard of the method & seen some of the series & yet soon after his going thence to Paris he could not forbear putting for the first inventor of such a series for the area of the circle. He could not then forbear pretending to two such series one of wchwhich was for finding the arc by the sine both found by the same method one & the same method tho he wanted the demonstration of them or method of finding them. For he s he said that they both found by the same method one of one of them was for finding the arc by the sine, theis cannot be found by his method of Transmutations & he wrote a little after for the demonstration. either of this series or When he had received eight such series from Mr O & Mr C. he took time to consider them & compare them with with his own p but never produced any other series then those some a what he had received. That After a years consideration he wanted the Demosntration or method of finding them if an dropt the receipt of them & found out a pretence to desire the Demonstration of Mr Oldenberg T & desired Mr O. to procure him the Demonstration or method of Mr Collins without Mr Newtons leave, tho by his own rule, if they had sent him the Method & he had forgot the receipt of it & claimed it as his own, Mr Oldenburg & Collins were not to contradict him without authority from Mr Newton. Mr O & Mr C. pressed Mr N to send his own method himself. Mr N sent it & SrSir A copy of yoryour book entituled Clavis Domus Heber was presented to me in yoryour name some months ago & I then desired the person who brought me the present to return my thanks & told him that I did not understood Hebrew. I have looked over it & find the designe very good. The Hebrew tonge is said to be narrow, & the just signification of several words to be almost lost & few books are extant written in the ancient Hebrew are extant, & the sdesigne of recovering the ancient signification of the words must be very commendable: but for want of skill in that tongue I am unable to make a further judgement of the success in this desgne. then I am perswaded that a person of yoryour abilities has not wanted very good success f succeeded wanted the success desired. I am To Sir Isaac Newton KntKnight Master & Worker of her MatiesMajesties Mint. These. 98Add. 3968 No. 41 When Mr Newton in his Letter dated 13 Iune 1676 had explained his method of series, he pub added: Ex his videre est quantum fines Analyseos per hujusmodi infinitas æquationes ampliantur: quippe quæ earum beneficio ad omnia pene dixeram problemata (si numeralia Diophanti et similia excipias) sese extendit. Non tamen omnino universalis evadit . . . . . tulerit excogitavi. To this Mr Leibnitz replied in his Letter of 27 Aug. 1676 replied answered: Quod dicere videmini plerasque difficultates exceptis problematibus Diophantæis) ad Series Infinitas reduci; id mihi non videtur. Sunt enim multa usque adeo mira & implexa ut neque ab æquationibus pendeant neque ex Quadraturis. Qualia sunt (ex multis alijs) Problemata methodi Tangentius inversæ [By this Answer it is absolutely certain that Mr Leibnitz had not yet found out his Differential equations nor the reduction of problems to Quadrature di æquations or Quadratures by the differential method.] And Mr Newton in his Letter of 24 Octob 1676 returned the following Answer replied Vbi dixi, omnia pene Problemata &c . . . . . . . . ad eruendos terminos assumptæ seriei. By the Answer of Mr Leibnitz it is most certain that he had not then found out the reduction of problems by the Differential method to differential æquations and Quadratures by the Differential method & that much ler whether by by finite or in finite or infinite or to & that Mr Newton had then found it out by the Method of fluxions & infinite series the reduction of Problemes to diff fluxional æquations & Quadratur & finite series [& by a sequenc to Quadratures their means the Reduction of the difficulter inverse met problemes of thangents & other more difficult to such equations series & quadratures] Mr Newton in his Analysis above mentioned wrote in arguing why the method there proposed should be caled Analysis subjoyned: Denique ad Analyticam merito pertinere censeatur, cujus beneficio Curvarum areæ & longitudines &c (id modo fiat) exacte & Geometrice determinentur. Sed ista narrandi non est locus. And in his Letter of Iune 13 1676 he touched upon the same subject saying (as above) neque [vacat] alia quædam tradere quæ circa Reductionem infinitarum serierum in finitas ubi rei natura tulerit excogitavi. And in his Letter of 24 Octob. 1676, he sai that upon by the me explained this method further saying that by the method wchwhich he couched in this sentence Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire & vice versa, he had endeavoured to render the speculation of Curv concerning the Quadrature of Curves more general simple & had arrived at some very general Theorems & there set down the first of those Theorems & illustrated it several examples. And this is the firft Theoreme Proposition in his bBook of Quadratures & the second of those Theoremes is the sixt Proposition. And these two depend on the four first Propositions And of that book & therefore the six first Propositions thereof were understood by him when he wrote in the year 1669 when Dr Barrow communicated the said Analysis to Mr Collins: wchwhich Propositions comprehend the direct method of fluxions & some part of the Inverse method. Hereupon the Editors of the Acta Lipsiensia the next yeare in Iune (in the Stile of Mr Leibnitz) in giving an account of these two first Volumes of Dr Wallis's works, taooke notice of this clause of the Doctor Preface & complainesd not of the Doctors saying that Mr Newton in his two Letters of 1676 explained to Mr Leibnitz the Method of fluxions found by him ten years before or above, but of his only that while the Doctor mentioned the tdifferential method, nequis scilicet, ut ipse ait calculus & said that he did it nequis causetur de calculo differentiali nihil ab ipso dictum fuisse, They say that as he did not explain it as tell the Reader that Mr Leibnitz had this method with infinite calculus at that time when the Letters passed between Mr Old him & Mr Newton by means of Mr Oldenburg. And in the several Letters wchwhich followed thereupon between Mr Leibnitz & Dr Wallis, Mr Leibnitz acknowledged that the two methods agreed concerning this matter Mr Leibnitz in the main denyed not what Dr Wallis had written concerning the meth concerning this matter Mr Leibnitz allowed the methods to be the same in substance, denyed not what Dr W what Dr Wallis had witt that Mr Newton had the method 10 years before the writing of those Letters as Dr Wallis had written affirmed, pretended not that he had the diffentialdifferential method so early, brought no proof that he had the differ Differential method it before the year 1677 no other proof besides the concession of Mr Newton that he had it so early, pretend ⊡ represented that tho the method, agreed in the main, yet they might differ in some things & challenged to himself only those things wherein (as he conceived) they might differ, namely the Notation

d x

d^{2} x

(

dd x

d^{2} x

ddd x

d^{3} x

d^{\frac{1}{2}} x

&c. 2ly Differential equations, (tho he had allowed those to Mr Newton in his Letter of 21 Iune 1677) & 3dly exponential Equations. But after the death of Dr Wallis, Mr Newton repeating what Dr repeating what Dr Wallis had affirmed published nine years before & Mr Leibnitz & the Editors of the Acta Lipsiensia had not then contradicted ten nine years before, those Editors in the style of Mr Leibnitz, began to began to pretended that Mr Newton had borrowed the Differe deduced the Diffe Leibnitz was the first Inventor & Mr Newton had substituted fluxions for the Leibnitian differences, And when Mr Keil defended Mr Newton, Mr Leibnitz refused to argue the matter with him pretending that he was a novice unacquainted wthwith the mat what had formerly & pastsed, & a clamorous fellow & had no authority from Mr Newton & appealed to Mr Newton himself, intending thereby endeavouring to pick a quarrel with him, or force him to retract what he & Dr Wallis had published, & to acknowledge himself a lyar & a plagiary. As for The Question lies in a little room. Mr Leibnitz has affirmed in the year 1676 affirmed that the inverse Problems of Tangents were not reducible to equations & was polishing the series of Gregory vulgari more & conjecturing that the method of Slusius might be made general by a Table of Tangents & therefore had not then found the differential method calculus. And yet in the beginning of his Letter of 21 Iune 1677 he affirmed that he had found this differential calculus jam tum a multo tempore. It lies up No man as He might say this or al Mr Newton. It lies upon him to prove it that he had it before the year 1677. For no man is a witness in his own cause. He has published the series of Mr Gregory as his own. It lies upon him to prove that he had it before he received it from Mr Oldenburg. He In yethe year 1674 he pretended to have Theorema found a series for finding the arc by the sine & two years after he desired Mr Oldenburg to procure from Mr Collins the Method of finding that that series. It lies upon him to prove that he had found out that series by himself in In Mr Newtons Letter of 13 Iun or before the year 1674. Mr Newton in his Letter of 13 Iune 1674 sent him several series with the methods of finding them & he pretended to have found some of them before the receipt of that Letter: It lies upon him to prove that he had methods of finding them & had found them out before. Mr Leibn At his own desire he Mr Newton sent to him two his inverse methods of series & he understood them with great difficulty, & as soon as he understood them he pretended to have found one of them long before & to have forgot it as he perceived by his old papers. It lies upon him to prove that he had every found it before. When A year or two after Mr New A year or two after Mr Leibnitz Newton had published his Principia Philosophiæ Mr Leibnitz a year or two after published three papers relating to the principal Propositions in that book, pretending that he had found them before the publishing of that Book: it lies upon him to prove that he had found them before. He pretended that the Propositions concerning the motion of bodies in resisting Mediums were for the most part found out by him twelve years before while he was yet at Paris that is, before he had the Differential method: it lies upon him to to prove it. He pretended that he had demonstrated to the Proposition that the a body revolving in an Ellipsis & wthwith a radius drawn to the lower focus describing equall Areas in equal times, is drawn to attracted towards yethe lower focus by a force which is reciprocally as the square of that radius; his demonstration is erroneous & it lies upon him to prove that he did not some time after the publishing of Mr Newtons Principia try to adapt a Demonstration to that Proposition in order to make himself with a designe rival Mr Newton in the P invention of the Proposition. [To commit an error is a very pardonable fault incident to humain nature:, but to l claim from another man to the invention of a Proposition from another man by an erroneous demonstration adapted to it, is a fault of another higher kind, & Mr Leibnitz is complained of for doing this to make himself a coinventor. His Tentamen de motuum cœlestium causis is full of faults. He is not complained of not for those faults but for writing that piece faulty piece in hast before he understood the subject & doing it as a coinventor without acknowledging that Mr Newton went before him in those things matters & gave him light into them. He is complained of for making Dr Barrows method of Tangents his own by a a new changing the notation & giving the method a new name without ever acknowledging himself beholden to Dr Barrow. Now that you may know what kind of calculation Mr Newton used when in or before the year 1669 when he he wrote this Compendium of Anlysis By the same way of working the second Rule may be also demonstrated. Or if any æquation whatever be assumed expressing the relation between the Abscissa & Area of a Curve & the Abscissa Ordinate may be found as is mentioned in the next words. And if this new equ Ordinate drawn into an unit be put for the Area of a new Curve the Ordinate may of this new Curve may be found by the same method,. And so on perpetually. wchwhich And these Ordinates are repre represent the first second third fourth & following fluxions of the first Area. This was Mr Newtons way of working in those days when he wrote the said Compendium of his Analysis; And by the same way of working he demonstrated the two first Propositions in his Book of Quadratures: & the very same way of working is used by him to this day. Among the examples &c It is the more Geometrical & natural because there are in Geometry quantitates primæ rati founded upon the rationes primæ quantitatum nascentium wchwhich have a being in Geometry but whilst indivisibles & quantitates primæ nascentes upon wchwhich the Differential method is founded have no being. Nature generates quantities by continual flux or increase & the ancient Geometers admitted such a generation of n areas & solids where they generated areas by draewing one line into another by continual motion, & solids by drawing their bases into their altitudes to generate an area, & the area into a line to generate a solid. But the summing up of indivisibles to generate an area or solid was never yet addmitted into Geometry. Mr Newtons method is adapted either for finding out a Proposition or for demonstrating it, Mr Leibnitz's is only for finding out. Mr When the work succeeds not in finite equations Mr Newtons method has recours to converging series & thereby becomes incomparably more universal then that of Mr Leibnits wchwhich is confined to finite equations. And where the Law of the fluxions is not known but the fluxions are had only in a few particular cases Mr Newton finds that law quamproxime by his Differential method, & thence resolves to deduces the fluents & solves the Problem, & on this account also his method is more universal then yethe method of Mr Leibnitz. So then Mr Leibnitz has only a part of Mr Newtons general method, & if he would have that ꝑpart to be his own, it lies upon him to prove that he had it before the year 1677. All other arguments & pretences of are in vain. Several Accounts of this Commercium having being given abroad all of them very imperfect: it has been thought fit to publish the Account wchwhich follows. SrSir I received your Book of the second Edition & return you my hearty thanks for the favour of such a present. Its now above eighteen years since I left off the styudy of Mathematicks & the disuse of thinking upon these things makes it difficult to me to take them into consideration: but however I have run my eye over yitour&it & performd & cannot but applaud it much. I wish heartily that France may always flourish with men who improve these sciences. I am Mr Leibnitz has since [upon a pretence that he was not at leasure to e & had not seen the Commercium] in opposition to the judgment Report of the Committee [ ] imployed a able nameless Mathematician to examin the matter & give his judgment thereupon. And the Mathematician in a paper Letter to Mr Leibnitz dated 7 Iuly 1713 has dropt the consideration of the ancient records & given his opinion that when Mr Newton wrote his Principia he used not letters with pricks Mr Newton had not the method of fluxions till after the writing of his Principia Philosophiæ & that for these reasons. First becaus in all the Letters of the Commercium Eipstolicum from wchwhich the Committee draw argume their argentsarguments there is not any instance of Mr Newtons using letters with pricks 100 — For this is one & the same Proposition wthwith that of deducing the Ordinate from the Area, & vice versa, & the Area from the Ordinate & therefore was known to Mr Newton when he wrote this Analysis. [And he that considers Mr Newtons Book of Quadratures will For better understanding this Analysis I should tell see that that Book was written as the first step towards the solution of the inverse of method of this Proposition this Propositions. The first & second for solving this fundamental Proposition, the four first Propositions For better of that book conteining the direct method, the rest conteining the inverse )] method. so far as Mr Newton had then proceeded But for better understanding this Analysis In this Analysis Mr Newtons represents a moment of time by the or of any exponent of time, by the letter o, a momen In this Analysis Compendium Mr Newton represents the uniform fluxion of time or of any exponent of time, by an unit, the moment of time or of its exponent by the letter o, the fluxions of other quantities by any other symbols & the moments of those quantities by the rectangles under those symbols & the letter o, & the area of a Curve or fluent by the Ordinate inscribed of fluxion inclosed in a square. When hise is demonstrating any Proposition he uses the letter o for a finite moment of time or of its exponent & performs the whole calculation in finite figures by the Geometry of the Ancients without any approximation; And & so soon as the calculation is at an end he & the Equation is reduced, he supposes that the moment o decreases in infinitum & vanishes. But when he is not demonstrating but only investigating a Proposition, he for making dispatch, he uses all man supposes the moment o to be infinitely little, & uses all manner of a forbears to write it down & uses all manner of approximations wchwhich he conceives will produce no error in the conclusion. An example of the first kind you have in the end of this book of Analysis Compendium in demonstrating the first Rule of the three Rules laid down in the beginning of the book, [& another example you have in the beginning of his book of Quadratures in demonstrating the first Propositiō of the book.] Examples of the second kind you have in fi the Analysis same Compendium in finding the big lengths of Curve lines p 15, & in finding the Ordinates areas & lenghts of Mechanical Curves p. 17, 18. And he tells you that by the same method tangents may be drawn to mMetchanical curves. And in his Letter of 10 Decem. 1672 he adds that Problems about the Curvature of Curves geometrical or metchanical isare resolved by the same method. Whence its manifest that he had then extended the method to the second fluxions, if not to the third. For when the Areas of Curves are considered as fluents (as is done in this Analysis) the Ordinates express the first fluxions, the tangents are given by the second fluxions & the curvatures by the third. And indeed the solution of Problem Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire fluxiones & vice versa extends to all the fluxions as is manifest by the examples given thereof of the solution of that Proposition sor Problem published by Dr Wallis Tom 2. p. 391, 392, 396. Thus the method of Fluxions is[ as unvers is as universal b is as universal in the for investigating Propositions for by working] in finite æquations finor the investigationngs of Propositions by finite æquations is as universal as the differential method, & exceeds it in the method of demonstrating, being also on that account more geometrical, besides the advantage of working in converging series wchwhich makes it incomparably more universal. But that you may be fully satisfied that Mr Newton used this Analy Method when he wro Method of calculaticalculating in the year 1669 or before when he wrote the said Analysis Compendium I will haere set down his tddemonstration of the first Rule above mentioned. Sit Curvæ alicujus – – – – – . erit

\frac{n}{m + n} a x^{\frac{m + n}{n}} = z

. Q. E. D. Where note that

c^{n}

x^{p} + p o x^{p - 1}

&c is the ign denotes the series into wchwhich the dignity

c^{n}

{x + o}^{p}

is resolovedresolved &

z^{n} + n o y z^{n - 1}

denotes series into wchwhich the dignity

{z + o υ}^{n}

is resolved. T Among the Examples. And in the Analysis pag. 16 where Mr Newton saith that Momentum est superficies ubi cum de solidis agitur & linea cum de superficiesbus & punctum cum de lineis agitur, its is all one as to say if he had said that when solids are considered as fluents, their moments are superfiessuperficies & the moments of those moments (or second moments) are lines & the moments of those moments (or third moments) are points. or that is to say lines infinitely short. And in his Principia Philosophiæ where he frequently considers lines as fluents decribeddescribed by points whose velocities increase or decrease, the vellocities are the first fluxions & their increase the second And by the Letters & papers dated between the years 1669 & 1677 inclusive the Committee found that Mr Newton had theis method before Iuly 1669 & that nothing appeared of occurred by wchwhich it might could be gathered that Mr Leibnits had any other meth differential method then Moutons before the year 1677. ⊡ allowed that the methods agreed in the main & said that he therefore he used to call them by the common name of the Infinitesimal Analysis, represented that as the Analysis speciosa methods of Victa & Cærtes were called by the common name of Analysis speciosa & yet differed in some things so perhaps the methods of Mr Newton & himself might differ in some things & callengedchallenged to himself only those things wherein as he conceived they might differ; namely first the Notation dxy, ddxy or d2xy, dddxy or d3xy,

d^{\frac{1}{2}} y

, dey; secondly differential equations (tho in his Letter of 21 Iune 1677 he allowed Mr Newton a sh recconed those common to Mr Newton & himself) & thirdly Exponential Equations. The first & third are allowed to Mr Leibnitz, the second Mr Leibnitz rec in his Letter of 21 Iune 1677 Mr Ol recconned common to Mr Newton & himself. This was the state of the dispute between Dr Wallis & Mr Leibnitz at that time but after the death of Dr Wallis &c he He Mr Leibnis compared Mr Newton & himself to Vieta & Descartes in respect of what was common to themir both methods But after the And four years after, when Mr Fatio called Mr Leibnitz the second inventor of this calculus & suggested that Mr Leibnitz the second inventor of this calculus might borrow something from Mr Newton the oldest inventor by many years: Mr Leibnitz in his Answer published in the Acta Lipsiensia in May 1700, did not deny Mrthat Mr Newton was the oldest inventor by many years nor asserted any thing more to himself then that he had found the method apart or without the assistance of Mr Newton. And in making this defence he added Quam [methodum] ante Dominum Newtonum et Me nullus quod sciam Geometra habuit; uti ante hunc maximi nominis Geometram NEMO specimine publice dato se habere probavit, ante Dominos Bernoullios & mMe nullus communicavit. Hitherto therefore Mr Leibnitz did not pretend to be the first inventor Hitherto therefor But after Mr Leibnitz d But after the death of Dr Wallis, (the last of these the old mathematicians who were acquainted with what had passed amongst Mathematicians here between the English & Mr Leibnits 340 or 450 years ago) when Mr Newton in the Introduction to his book of Quadratures repeatinedg wtwhat Dr Wallis had publickly affirmed nine years before & Mr Leibnitz & the Editors of the Acta Lipsiensia had not then contradicted: those Editors in the style of Mr Leibnitz began to pretend that Mr Newton had Leibnitz was the first Inventor & that Mr Newton had substituted Fluxions for the Leibnitsian differences. And when &c Mr Keil defended Mr Newton Mr Leibnitz refused to argeargue the matter with him pretending that he was a novice unacquainted with what had formerly passed, & that he had no authority from Mr Newton, & that he behaved himself clamorously & that Mr Newton understood the matter better then any body & was desired as he hoped, would not refuse to give his opinion: that is, now Barrow, Gregory, Oldenburg Collins & Wallis who understood this matter were dead, he Mr Leibnitz was resolved to attaque Mr Newton himself & force him to try to run him down by the advantage of his correspencecorrespondence with the learned abroad or force him to retract what he & Dr Wallis had published & by consequence to acknowledg himself a lyar & a plagiary. And four or five years after, when Mr Fatio had called Mr Leibnitz the second inventor of this Calculus & had suggested that Mr Leibnitz might borrow something from the Letters of Mr Newton the oldest inventor by many years: Mr Leibnitz in his Answer published in the Acta Lipsiensia in May 1700 did not deny Mr Newton to be the oldest inventor, nor asserted any thing more to himself then that he had found it apart. And in making this defense he added Quam [methodum] ante Dominum Newtonum et me nullus quod sciam Geometra habuit; uti ante hunc maximi nominis Geometram, NEMO specimine data publice dato se habere probavit, ante Dominos Bernoullios & Me Me nullus communicavit. Hitherto therefore Mr Leibnitz did not pretend to be the first inventor. But after the death of Dr Wallis, (the oldest last of those who were acquainted with the history wh of what had passed amongst Mathematicians 40 or 50 years ago) Mr Leibnitz Newton in the Introduction to his book of Quadratures repeating what Dr Whallis had published publickly affirmed nine years before &c 101 Mr Leibnitz deduces the circular motion of the Planets from the Vortex, the motion of ascent & descent from gravity, but has not demonstrated that a Planet will in every revolution will ascend & descend by Gravity just once & not oftner. In a Letter to Mr Newton from Hannover dated

\frac{7}{17}

Martij 1693 Mr Leibnitz wrote thus to Mr Newton Mirifice ampliaveras Geometriam tuis seriebus, sed edito Principiorum opere ostendisti patere tibi etiam quæ Analysi receptæ non subsunt Conatus sum ego quoque notis commodis adhibitis quæ differentias et summas exhibent Geometriam illam quam transcendentem appello, Analysi quodammodo subjicere, nec res male processit. Mr Leibnitz therefore allowed hitherto allowed that Mr Newton had a methodus SIMILIS without pretending that he had it was derived it from the differential. ③ Dr Wallis in the Præface to his work the two first volumes of his works dated published in 1695 made thist this mention of the method. In secundo volumine (inter alia) habetur Newtoni methodus de fluxionibus (ut ille loquitur,) consimilis naturæ cum Leibnitij (ut hic loquitur) Calculo differentiali (quod, qui utramque contulerit methodum contulerit, satis animadvertat, ututut sub loquendi formulis diversis) quam ego descripsi (Algebræ cCap. 91 &c præsertim Cap. 95) ex binis Newtoni Literis (aut earum alteris) Iunij 13 & Augusti Octob. 24 1676 ad Oldenburgum datis, cum Leibnitio tum communicandis (ijsdem fere verbis, saltem leviter mutatis, quæ in illis Literis habentur,) ubi methodum METHODUM HANC LEIBNITIO EXPONIT tum ante decem annos DECEM ANNOS, nedum plures [i.e. anno 1666 vel 1665] ab ipso excogitatam. Quod moneo, nequis causetur, de hoc cCalculo Differentiali nihil a nobis dictum esse. ② At the request of Dr Wallis, Mr Newton in two Letters dated Aug. 27 & & Sept 17 1692 explained at large the solution of these two Problems Data æquationes fluentes quotcunque quantitates involvente fluxiones invenire & vice vers & Ex æquatione fluxionem radicis involvente, radicem extrahere; & Dr Wallis published the same in the second volume of his works. ① In the Iournal des Sçavans of munday 30 August 1694 Mr Leibnitz in writing of the advantages of the differential Calculus above the ordinary Analysis, addsed said: This justice is to be done to MrNewton (to whom Geometry Optique & Astronomy are much indebted) that also in this affair, he has of himself found something like it, according to what I have since understood It is true that he uses other characters: but as the characteristique it self, as I may speak, is a great part of the Art of invention, I beleive that ours are more give more light. Thus far Mr Leibnitz, who did not yet know what characters were used by Mr Newton ④ The next year in the Acta Lipsiensia of Iune an Account was given of the two first Tomes of the Mathematical works of Dr Wallis & thereupon followed several Letters between Dr Wallis & Mr Leibnitz, with & all this wthwithout any complaint against Dr Wallis wthwith frequent mention of the methods of fluxions & differential method & the account given of them by Dr Wallis, & all this wthwithout any complaint against Dr Wallis for saying that Mr Newton in his Letters of 13 Iun. & 24 Octob 1676 explained to Mr Leibnitz this method found ten years before or above. Mr Leibnitz allowed these methods to resemble one another so much that he used to call them by the common name of the infinitesimal method. He only contended that Mr Newton's method & his own might differ in some things, & layd claim to what might be proper to his own but is very confused in describing the same. He pretended to the invention of different the differential notation dy, d2y (or ddy) d3y,

d^{\frac{1}{2}} y

, dey, & it is his own. He But he was not the first who considered the quantities wchwhich he calls second differences. Mr Newton had considered them before in determining the curvatures of Curve lines by his method of fluxions. See his Letter to Mr Collins dated 10 Decem. 1672. See also his solution of yethe two Problemes in the second Volume Tomes of Dr Wallises Works p. 392, 393, 396. Mr Leibnitz pretended also to the sinvention of Differential equations for expressing the nature of transcendent Curves: but these equations were known before to Mr Newton For his fundamental Proposition, Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire, is for finding such equations, & his other Problem ex æquationes fluxionem radicis involvente radicem extrahere is for resolving such equations into converging series. So in his Analysis communicated to Mr Collins in the year 1669 the equation A third thing wchwhich he claims is the invention of exponential Equations. Mr Newton in his Letter of 24 Octob 1676 proposed equations in wchwhich the exponents of the Dignites of indeterminate quantities were frat or surd numbers: Mr Leibnitz in his answer mutualy proposed equations in wchwhich the exponents of the Dignities of indeterminate quantities were indeterminate A quantities. And these æquations were first proposed by him, but have hitherto been of no use. Hitherto Mr Leibnitz therefore giving the name of the Infinitesimal method to the methods of himself & Mr Newton so far as he conceived them common to them both, forbore to claime the whole to them both himself & only claimed what he thou some things in wchwhich he pretended that he & Mr Newton might differ. After these things Mr Fatio in the year 1699 represented that Mr Newton was the first Inventor of the Method & Mr Leibnitz answered Certe cum Elementa calculi meāam edidi anno 1684, ne constabat quidem mihi aliud de Inventis ejus in hoc genere quam quod ipse olim significaverat in literis, posse se Tangentes invenire non sublatis irrationalibus: quod Hugenius quoque mihi se posse mihi significavit postea, etsi cæterorum istius calculi adhuc expers. Sed majora multo consecutum Newtonum, viso demum libro Principiorum ejus, satis intellexi. Calculum tamen differentiali tam similem abeo exerceri, non ante didicimus quam cum non ita pridem magni Geometræ Iohannis Wallisij operum volumina primum et secundum prodiere: Et post aliqua: Quam [methodum] ante Dominum Newtonum & me nullus quod sciam Geometra habuit; uti ante hunc maximi nominis Geometram NEMO specimine publice dato se habere probavit: ante Dominos Bernoullios & ME nullus communicavit. Here it is something acknowledge that no man before Mr Newton gave proof by a public specimen tha he no man before Mr Newton & himself had this method, no man before Mr Newton had proved by a public specimen that he had this method. But he should not have said that in the year 1684 when he published thes elements of his differential method he knew nothing more of Mr Newton's Inventions of this kind then that he could draw tangents without sticking at surds. ‡ ‡ At that time he published nothing more then how to draw tangents & determin maxima & minima without sticking sat fractions or surtds & saying that the same method extended to the abstruser sorts of Problemes in Geometry. And all this was nothing more then what he had from Dr Barrow & Mr Newton. Fermat & Dr Barrow had told him the method of Tangents, & maxima & minima Fermat had told him the method of mo Mr Newton had told him in his Letter of 24 Octob 1676 had told him that by his method he drew tangents determined maxima & minima & solved other Problems without sticking at surds,. & H He learnt afterwards how to do this weth by Dr Barrows method of improving Dr Barrows method of Tangents & published that method it as his own without naming either Dr Barrow or Mr Newton pre HI By Mr Newtons Letters he knew certainly that Mr Newtons methods roa gave the method of Slusius readily as a Corollary & made it more convenient so as not to stick at surds, that in like manner it dermineddetermined maxima & minima & some some other Problems wthwithout sticking there not named, without sticking at surds, that it rendred Quadratures more easy those figurefigures being always squareable wchwhich were at a differential equation; that it & gave yethe Theoreme There described & illustrated wthwith examples for squaring curves by a series wchwhich brake off when the Curve & became finite when the Curve was squarable by a finite equation, & found other Rules for comparing the areas of curves wthwith those of the Conick Sections that He knew also that Mr Newtons methods reached to the Curvature of Curves & to their lengths & centers of gravity &c & to yethe inverse Problems of Tangents, wchwhich his own did not & proceeded in Mechanical Curves as well as others. He knew also that his differential method was nothing else then Dr Barrows method of Tangents wchwhich ytthat Mr Newtons method of series by the help of his other methods extended to almost all sordts of Problems except some numeral ones like those of Diophantus & that in the year 1676 when he wrote his letters of 172 Iune 27 Aug. & Novemb 18 1676 he had not found out the differentiall method, but the nex & therefore Mr Newtons methods for doing these things were the older, & yet the next year so soon as he had found out how to extend Dr Barrows method of Tangents to some of these things, he changes the notation gives the method a new name as if it were his own & writesing back rivals Mr Newton by pretending writing back that he had found ofut this new method jam a multo tempore Clarissimi Slusij Methodum tangentium saith he, nondum um esse absolutam, celeberrime Newtono assentior. Et jam a multo tempore rem tangentium generalius tractavi scilicet per differentias Ordinatarum. And to this day he has not acknowledged himself obliged either to Dr Barrow or to Mr Newton for any thing. When any man has found out any thing & gives notice of his success to 102 his friend, it is not lawfull for his friend to go about to supplant him by pretending that he had found out the same thing bef long before, unless he can prove that he had found it out long before. For af the very notice may put his friend upon trying to find it out. And if hee can find it out & may be allowed to pretend that he had found it out before he received the notice, & rob he he is allowed to supplant & robb the man first Inventor. It lies upon Mr Leibnitz therefore to prove that he invented the Differential method long before the jam tum a multo tempore long before the receipt of Mr Newtons Letters It lies He put in his claim to In the year 1677 he put in his claim to the differential method in these words Et jam a multo tempore rem tangentium generalius tractavi scilicet per differentias Ordinatarum. It lies upon him to prove his claim. In the year 1704 Mr Newton published his Treatise of Quadratures & least the bo The book had been some years before in the hands of some of his friends & thereby he was induced to publish it. And that it might not be taken for a new piece he wrote an Introduction to it & therein represented that it related to yethe meth wherein he described the method of fluxions upon wchwhich it was founded, & represented that he found out that method gradually in the years 1665 & 1666 & the first step to yethe inverse method was the Quadrature of Curves concerning wchwhich he had [olim] long ago written that Tratct of Quadratures. In Mr Newtons Letter of 24 Octob. 1676 there are so many things quoted out of that book as may make it evident that the book was then in being. And that the Method of fluxions was found out in the years 1665 & 1666 is nothing more then what Dr Wallis wrote in the Introduction Preface to the two first Volumes of his works, in the saying that Mr Newton in his letters of 13 Iune & 24 Octob. 1676, methodum hacnc Leibnitio exponit tum ante decem annos, nedum plures, ab ipso excogitatam. Quod moneo nequis causetur de hoc cCalculo Differentiali nihil a nobis dictum esse. The Editors of the Acta Eruditorum & in their Account of these two Volumes (written in the style of Mr Leibnitz take notise of this Prearagraph & complain that Dr Wallis had here omitted what Mr Leibnitz had performed in his two Letters of 27 Aug. 1676 & 24 L 21 Iune 1677 the Letters wchwhich passed at time between & him & Mr Newton, & Mr Leibnitz himself pursue the in the Letters which followed thereupon between him & Dr Wallis, pursued the complaint. But none of them at that time found fault wthwith Dr Wallis for saying that Mr Newton had found the Differential Method 10 years before or above. But now, Dr Wallis being dead, & Mr Leibnitz by an universal correspondence have better established his credit, the Editors of the Acta in Ianuary 16 1705 in the style of Mr Leibnitz, represent that the method of fluxions was derived from the differential method, & the book of Quadratures was a new piece conteining little or nothing but what more than worth mentioning besides what Cheine & Craig had published before. Whereupon Mr Keil in an Epistle published in the Philosophical Transactions in May & Iune A.C. 1708, wrote on the contrary that quod fluxionum Arithmeticam, sine omni dubio, primus invenit Dominus Newtonus ut cuilibet ejus Epistolas a Wallisio editas legenti facile constabit. Eadem, tamen Arithmetica postea mutatis Nomine & Notationis modo, a Domino Leibnitsio in Actis Eruditorum edita est. And Mr Leibnitz in a Letter to Dr Sloane Secretary of yethe Society dated 4 Mart. st. n. 1711 complained thereof to the R. Society, proposing that they should m Mr Keil should make a public recantation. This Letter being shewed to Mr Newton he was at first concerned ynthenytthat a dispute of this kind should arise being of a quiet disposition & very arverse from disputes & careless of fame: but some days after, shewn the Acta Lipsica of Ianuary 1705 being shewn to him & he finding himself there traduced & in an artificial manner & represented a false Lyar & a Plagiary, he gave leave to Mr Keill to pursue the dispute. And Mr Keil thereupon wrote an answer wherein he explained & justified what he had said. This Letter was read in the R. Society 24 May 1711 & then ordered to be sent to Mr Leibnitz. And Mr Leibnitz returned an answer dated 29 Decem 1711 in wchwhich he persisted in his complaint against Mr Keil, justified what was in the Acta Leip Lipsiensia, allowed Mr & pretended that he had found the Differential method above nine years before he published it, that is in the year 1675 or before, & that no body had gone before him in those mattermatters: whence it follows that Dr Wallis & Mr Newton had falsly affirmed that the found the method of fluxions was found in the years 1665 & 1666. He And in both his Letters he appealed to yethe opinion of Mr Newton, pressing in his last that Mr Newton would declare his mind opinion in this matter, that is that Mr Newton would retract what he & Dr Wallis head already declared & what Mr Leibnitz durst not deny while Dr Wallis was in Letter sixteen years before when Dr Wallis published it & [much less durst he deny 365 years before, that Mr Newton wrote a tratisetreatise upon this Method & the method of Series together in the year 1671 as Mr Newton affirmed in his Letter of 24 Octob 1676. bef] that he would retract what he wrote in his Letter of 24 Octob. 1676 namely that he had written a L a treatise upon this method & the method of Series together 3 five years before that is in the year 1671, & that the Analysis he would retract the Analysis published written in the year 1669 & published by Mr Iones & acknowledge it to be a spurious piece. All these things Mr Newton was to retract or be quarrelled by Mr Leibnitz for not giving judgment against Mr Keil. The Royal Society have equal as much authority over Mr Leibnitz & as over Mr Keill, & [were not to condemn Mr Keil upon the accusation of Lei] Mr Leibnitz in accusing Mr Keill obliged himself to justify his accusation least he should g it should be taken for a calumny. And yet Mr Leibnitz in his last letter refused to justify his accusation calling it injustice to expect that he should give his reasons one of his age & reputation should defend his candor & pleadieng to give any reasons for it. And therefore the R. Society considering that the matter Question depended upon things done 35 or 40 years ago ordered a Committee to search out what old p Letters & other Papers were to be met with relating to this matter & report their opinion upon them & then ordered the Papers & the Report of their Committee to be published. [And Mr Leibnitz a few some months after these things came abroat, pretending that was not a leasure to examin himself w] And had invented a great part of Mr Newtons Principia Philosophiæ before he saw that book, & some part of it twelve years before while he was yet at Paris, that is before he had found the Differential method. But such pretenses are of not moment to be regarded. No man is a witness in his own canp & in maintaining it declined to give any reasons waved the decision by ancient letters & papers as unintelligible for his claim insisted upon his own candor as if it were unjust to question it, pressd that Mr Newton should desclare his opinion, as the only man now alive who well understood this matter & refused to contend with any man but Mr Newton as if all other younger men were novice & uncapable of understanding the ancient Letters & Papers upon this subj wchwhich remaind upon this subject, & under the person of a nameles Mathematician & two other nameless correspondents has put about a triple Libel against Mr Newton full of railing & fals injurious accusations without any proof 103 And thus much concerning the methods of tangents used letters & tells us that for certain reasons he chose rather to use the symbols dx & dy. He that shall compare th Dr Barrows method of Tangents wthwith that of Dr Mr Leibnitz will find them exactly the same except the different Notation & the improvements made to yethe method by the suggestions of Mr Newton. And what those improvements were will appear by three of Mr Newtons Letters dated 10 Decem 1672 13 Iune 1676 & 24 Octob. 1676. For Mr Newton in his Letters of 10th Decem. 1672 & 24 Octob 1676 in describing his method of drawing Tangents & determining maxima & minima, represented that it stuck not at surds, & readily gave the method of Slusius & rendred the Quadrature of Curves more easy, And Mr Leibnitz in his Answer when he had shewed how the method of Slusius followed from this method & how it stuck not at irrationals, added, Arbitror quæ celare voluit Newtonus de Tangentibus ducendis ab his non abludere Quod addit ex hoc eodem fundamento Quadraturas quoque reddi faciliores me in sententia hac confirmat, nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. At length Mr Leibnitz published his differential method of tangents & maxima & minima in the Acta Eruditorum mensis Octobris An 1684, pag. 467, with this title; Nova methodus pro maximis & minimis itemque tangentibus quæ nec fractas nec irrationales tollit quantitates moratur, & singulare pro illis calculi genus, per G. G. L. And what he then published is comm generally called the differential method. And yet it conteined nothing more then the method of Tangents published 14 years before by Dr Barrow & d the like method of maxima & minima invented by Fermat & published 25 years before by Schooten & improved by the suggestion of Mr Newton so as improved into a general method wchwhich did not to stick at surds, & by Mr Leibnitz furnished wthwith a new sort of Notation, he not knowing at that time what Notation was used by MrNewton. This meth For Mr Newton at the request of Mr Collins sent & Mr sent to him his method of Tangents in a Letter dated 10 Decem 1672. And it proved to be the same – – – – – – sentences set down enigmatically: the first of wchwhich was this Thus Mr Newton in these three Letters represented that . . . . . . . . . . . . . Thus he concludes that he had now got a method like that of Mr Newton. And at length he published this method in the Acta Eruditorum mensis Octobris An. 1684 pag 467, with this title. Nova methodus pro maximis et minimis itemque tangentibus . . . . . . pari facilitate tractabit. Thus Mr Newton in these three Letters represented that his method was very universal, that it gave the method of Slusius as an obvious Corollary & that it proceeded without sticking at surds & faciliated Quadratures. And after all this information Mr Leibnitz in his said Letter of 27 Iune 16877 proposed Dr Barrows method of Tangents with th some of these improvements shewing how the method of Slusius easily followed from it, & how it might be managed so as not to stick at surds, & then added: Arbitror quæ celare voluit Newtonus ab his non abludere Quod addit ex hoc eodem fundamento Quadraturas quoque reddi faciliores me in sententia hac confirmat; nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. Thus he concludes ytthat he had got a method like that of Mr Newton. And at length he published this method in the Acta Eruditorum mensis Octobris An. 1684 pag. 467, with this title. Nova methodus pro maximis et minimis itemque itemque tangentibus quæ nec fractas nec radicales irrationales quantitates moratur & singulare pro illis caculicalculi genus per G. G. L. And after he had described Dr Barrow the method of Tangents published by Dr Barrow 14 years before & the Fma method of m his new notation & how to find differential equations (as he calls them) without sticking at surds fractions or surds: he & applied this calculus to the drawing of Tangents & the dete finding maxima & minima all wchwhich is nothing more than the method of he subjoyns. Et hæc quidem initia sunt Geometriæ cujusdam multo sublimioris ad difficillima et pulcherrima quæque etiam mistæ matheseos problemata pertingentis, quæ sine calculo nostro differentiali, aut simili, non temere quisquam pari facilitate tractabit. 105 He published thise elements of his Differential method without mentioning any other author then himself or taking any notice of what had passed between him & Mr Newton by means o. And afterwards excused it by saying that before the publishing of Mr Newtons Principia Philosophiæ, he knew nothing more of Mr Newton's method then that it was a method of Tangents wchwhich stuck not at surds. p. 104, 107. Now Mr N Leibnitz denyes that he had all this light from Mr Newton or knew or in the time of this correspondence learnt that Mr Newton had a general method of this kind, or could do any th any further method then to draw Tangents wthwithout sticking at surds:, refuses to argue agai (p. 104, 107) & claims inventoris jura to himself [He appeals to the R. Society, & yet but declines arguing before them against Mr Keil, but and insists upon his unquestionable candour & [will not allow that such young men as Mr Keill can understand the Letter who were not privy to the correspondence between Mr Leibnitz & Mr Oldenburg 36 years ago can now understand it.] the R. Society nor look upon the bet and the R. R. Society look upon Mr K the ancient Letters & Papers here printed to be the best evidence now to be had in the math now Question depending between Mr Leibnitz & Mr Keil.] & affirms that the method of fluents was always used by the Mr Newton in the room of the Differential method. p 108, 119. Mr Keill is of thinks otherwise And the R. Society look upon the ancient Letters & Papers here printed as the best evidence now to be had in the Question depending between them. But Mr Leibnitz now denyes that he had all this any all this light from Mr Newton or in the time of this correspondence learnt that Mr Newton had a general method of this kind or any further method of this kind then to general method of this kind or any further method of this kind then to draw Tangents without sticking at surds which might be done wthwithout knowing any thing more of the method p. 104, 107.) And when h in like manner, after he had conversed wth the mathematicians at London he wrote from Paris as if he had never heard of Mr Newtons method of series R & pretended to be the first inventor of two series for the circle (p 38) & the next year when Mr Collings & Mr Oldenburg sent him eight series & he kn he received eight series from Mr Oldenburg & knew none of them to be his own, he forgot the receipt of them before the end of the year & he forgot the receipt of them & communicated one of them as his own to his friends at Paris (p 42) & the spring following forgo had forgot the receipt of the eight series & endeavoured to get the Method from Mr Oldenburg & Mr Collins without the knowledge of Mr Newton (p. 45) tho by his own rule if he should afterwards have forgot the receipt of the method & taken it for his own, Mr Oldenburgh & Mr Collins were not to contradict him wthwithout authority from Mr Newton p. 118. And for compassing his desire in recompence for Mr Newtons method he promised them his own inventions about the same subject, the very different from two series wchwhich he had received from Collins by one Mohr, tho those inventions were nothing else then an Opusculum or discourse upon one of the eight series wchwhich he had received the year before from Mr Oldenburg but had forgot the receipt of it. p 42, 45, 61. And when Mr Newton at his own request sent him the received from Mr Newton the method of deriving reciprocal series from one another tho he understood it with difficulty yet he wrote back that he had found it before as he peceivedperceived by his old papers but had laid not meeting wthwith an elegant example of its use had neglected it. p. 63 96 And when he published the above mentioned series as his own, (p 97) he had not only forgot that he had received it from Mr Oldenburg but also that the had received the co collection of Le Gregories Letters had been sent him at his own his own request, amongst wchwhich was the Gregories Letter of Feb 15 1671 conteining that series then sent to Mr Collins p 25, 47. He alledges He alledges the authority of M the Marquiss de L'Hospital Mosr Hugenius & the two brothers Bernoulli on his side, but never stated yethe matter ty fully to them he never gave them a true account of what he learnt from England by his correspondence wthwith Mr Oldenburg & they are not to be blamed for judging according to what appeared to them. If Mr Leibnitz received the series of Gregory from from Collins & afterwards published it in the Acta Lipsica as his own they are not to be blamed for taking it to be Leibnitz's till they should see reason to beleive that it was Gregorius. And the case is the same for the infinitesimal method. T As for those Gentlemen who have used or approved the Differential method, & particularly the Marquess de L'Hospital, Mr Hugenius the Mr Varignon & the two brothers the two Bernoulli & there is nothing in these Letters & papers that can reflect upon them. They were strangers to the communic correspondence between Mr Leibnits & Mr Oldenburg & are much to be commended for the use & improvements they have made of the infinitesimal method. And when he Mr Collins in yethe beginning of yethe year 1671 received several series from Mr Gregory with leave, to communicate them to whom he pleased & was very free in doing so. Mr Leibnitz went from London Mr Leibnitz having been at London began soon after to pretend himself the first inventor of two such series, & in the year 1675 had eight series sent him by Collins & Mr Oldenburg, & knew none of them to be his own & yet the same put ab year communicated the last of them to his friends at Paris as his &own, the next year sent it back to Mr Oldenburg as his own, & had a copy of Mr Gregory Letter sent il to Mr Collins sent to him b sent him of the Letter by wchwhich Mr Gregory had communicated that series to Mr Collins A.C. 1671 & yet afterwards published that series in the Acta Lipsica as his own without letting the world know that he had it from Mr O. & Mr C. & who had it from Gregory. The next year he preten After he had A year after the receipt of the eight series he pretended to have forgot them receipt of them & endeavoured to get the method from Mr O. & Mr C without Mr Newtons knowledg; tho if he should also have forget the receipt of the method, Mr Oldenburg & Mr Collins were not to f & assert Mr Oldenburg & Mr Collins were not to tell him that wthwitho & taken it for his own Mr Oldenburg & Mr Collins by his own rule were not to contradict him without authority from Mr Newton. When the Mr Newton sent him his method with some of the series, he desired examples of series: he endeavoured to make some copy of the reciprocall series his own in some cas tho he had no other method of finding them in some cases, tho he did not yet understand the method of deriving reciprocal series from one another but at yethe same time wrote to Mr Newton to explain it. & w And when he ha had it, he wrote back that he had found it before as he perceived by his old papers but for want of a good example of its use had neglected it. p. 96. tho he now pretends that in the time of this correspondence he knew nothing more of Mr Newtons method then that it was a method of drawing tangents wthwithout sticking at surds, p. 104, 107. — tho he now pretends that in the year 1684 when he published the elements of his differential calculus he knew nothing more of Mr Newtons inventions of that sort then what he signified in his Letters, namely that he could draw Tangents wthwithout taking away irrationals, wchwhich any man might do without knowing any thing more of the Differential method, p 104, 107 but afterwards by his Principles found that his method was of much larger extent & my p 14 104, 107. After Mr Leibnitz had conversed wthwith the Mathematicians in London by whom he could not but hear of Mr Newtons method of series he endeavored also to make himself the first inventor of that method by pretending to two series for the circle wthwith the method of finding them (p 38) forgetting the receipt of eight series sent him by Mr Oldenburg (p 40, 41) publishing communicating one of it Fr them as his eight at Paris as his own & sending it back to Mr Oldenburg as his own & publishing it in the Acta Lipsica as his own (p 42, 65) tho he knew it was Gregories Gregories (p 47) endeavouring to get wthwithout Mr Newtons leave to get the method from Oldenburg & Collins tho by his own rule if he should have afterwards forget the receipt of the method & take it for his own, Mr Oldenburg & Mr Collins were not to contradict him wthwithout Mr Newtons authority from Mr Newton (p. 45, 118) promising them by way of recompence his own inventions about series of a very different kind from two series brought him from London: wchwhich inventions proved to be were a discourse upon the series wchwhich he had received the year before106before from Mr O & published at Paris, & a Proposition like those of Dr Barrow & Mr Gregory & Dr Barrow for transmuting of figures into one another for & Theorems squaring them in particular cases (p 61) squaring one figure by another in some cases (p 61) & p. 45) by sending Mr Newton a sh Proposition like those of Mr P Gregory & Dr Barrow for transmuting of figures into one another for & thereby squaring one figure by another in some cases & crying it up for a general method to be recconed amongst & one of the chiefest the chiefest parts of Analysis (p. 58) by desiring Mr Newtons method of reciprocal deriving reciprocal series from one another & when he had & unstoodunderstood it, pretending that he had found it before as he peceivedperceived by his old papers, but for want of an elegant example had neglected to use it (p. 63, 96) by publishing in the Acta Lips. the Gregories series of Gregory for the circle & Hyperbola as his own in tel without letting the world know that he had from M. O & Mr C. & that it was Gregories series found by Mr Newtons method (p. 97) & by numbring himself in those Acta amongst the first inventors of the methods of Series & pretending that Mr Newton allowed that he had a general method p. 98. If it be said that Mr Newton should have published his methods sooner, it appears by the following letters that he was communicated a treatise of it to his friends in 1669 (p. 1, 2) that he was soon after about it (p. 27, 28, 29,) but was discouraged by people contending with him about their p inventions conten people contentions arising (p 71) & that Mr Leibnitz should not have discouraged him further by pretending to his methods. He might have sent him his differential method, but he should at the same time have acknowledged that he had but newly found it out & that he had found it out by means of the light wchwhich he had received into it by his cos f from England by Letters, & not have sent it in such a manner as if he had known it long before (p 88) for cCandid men do not intto interrupt one anothers proceedings & nor endeavour to snatch away one anothers inventions And its further to be considered that although the Propositions in Mr Newtons Principia philosophiæ were first published by himself & yet the chief of them were afterwards claimed by Mr Leibnitz p 97. Mr Newton's writing that he found his general method in the years 1665 & 1666 may go for history where there appears no reason to question it, but it gives him not inventoris jura against any man that can prove he had that general method before the year 1669. Mr Leibnitz's pr writing that he had improved the method of Slusius into a general method a multo tempore before the date of his Letter 21 Iune 1677 is no argument against another that can prove he had done the same thing before the date of that Letter p 88. When Mr Tschurnhause produced a general method of solving Problemes by assuming the terms & of a series & determining them by the conditions of the Probleme, & Mr Leibnitz ought not to have pretended publickly that he com without some proof that he communicated that method to Mr Tschurnhause ten years before when they were together at Paris (Acta Lips. 16876. p. 293) Certainly Mr Leibnitz had not that method when he wrote his Letter of Aug. 27 1676 wchwhich was but a month or six weeks before he & Tschurnhause parted from one another at Paris. And so when Mr Leibnitz pretends to have the first invention of the principal Propositions in Mr Leibnitz Newtons Principia, his pretention without any proof gives him not inventoris jura. It might have gone for history if no body else had published the Propositions before him: but after the R. S. cannot take him any man for an evidence in his own for himself in his own case. Mr Newton was desired to tell the original of his method, & when he had told it, to prevent any em contention wthwith MeratorMercator & his friends added in express words that he looked upon Mercator to have found his Quadrature of the Hyperbola before. And tis certain that Dr Wallis had found the division before & also the quadrature of every part of the Quotient wchwhich Mercator should have acknowledged when he put those two inventions together. Mr Barrow did not thinks himself inured by Mr Newtons Analysis. Mr Gregory after he understood Mr Newtons method acknledgedacknowledged him the first inventor. Mr Leibnitz is the only competitor tho a later man then the other two. He pretends that he had found out the Differ Instead of following the example of Mr Newton who voluntarily yeilded the first invention of a Series to Mercator he pretends to have found out refuses to yeild the first invention of the infinitesimal method to Mr Newton pretends to have found it out long before he communitedcommunicated it to Mr Newton's to him would never say when he first found it out, but claims to himself nor acknow of late denys that he then knew that Mr Newton had such a general method, heards himself with the first Inventors of the methods of series, claims the first invention of the principal propositions of Mr Newtons Principia Philospophiæ, without offering at any proof, appeals to yethe R. S. & at the same time refuses to plead before them or bring any proof of what he pretends to, but insists upon the credit of his candor represents that the his Letters printed by Dr Wall & those of Mr Newton & Mr Collins printed by Dr Wallis & other quoted by Mr Keil can be understood by young men such as Mr Keill ils who when were not privy to what passed when the Letters were written, will not allow any man the liberty of opposing him without authority from SrSir Isaac Newton, & insists upon his own cando the credit of his own candor that ,is, he would have the R. S. to lay aside the consideration of the Le ancient Letters & papers of correspondence wthwith Oldenburgh unlesse there be any witnesses now alive that were then privy to what passed between him & Mr Oldenburg Mr Collins & Mr Newton & thereby capable of interpreting those letters und interpreting those Letters; He wk would the society rely upon the candour of his own testimony for himself without examining into yethe matter & give judgment accordingly unless Mr Newton shall h who has always avoyded disputes shall have the vanity to enter into a dispute with him about an invention, & undertake to produce living evidence who were conscious to the Letter to privi to all yethe correspondance between Mr Leibnitz Mr Oldenburg & Mr Collins Mr Barro & Mr Gregory 4 36 years ago. 107 The first Proposition of the Tract De Quadratura Curvarum (communicated to Dr Wallis in September August 1692) extends to the second third & following fluxions, as is manifest by the examples by wchwhich it is explained. And this Proposition was known to Mr Newton when he wrote his Principia Philosophiæ, & ten years before when he wrote his Letter dated 24 Octob. 1676, as above, & five years before that, when he wrote a Tract upon the Method of Series in conjunction with another method founded upon this Proposition, as he mentions in his said Letter. And the Inverse of this Proposition, namely to extract Fluents out of equations involving their fluxions, extends also to these second third & following Fluxions as is manifest out of by what Mr Newton sent to Dr Wallis in August & September 1692 & the Doctor published concerning this matter in the second Volume of his works pag. 396, & it was known to Mr Newton when he wrote his said Letter dated 24 Octob. 1676, that is before Mr Leibnitz knew any thing more of the Differential method then what he found in Dr Barrows method of Tangents. 108 212s. 240d∷106.120∷

99 \frac{1}{4}

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4li. 7. 5

\frac{1}{2} \frac{1}{8} \frac{1}{16} \frac{1}{32}

et ibi condensari et convertio in aquam et humores spiritus humidos, & subinde per lentum calorem in & sales limum & sulphura et tincturas et limum et limum argillim & lapides arenam et lapides & coralla & substantias alias terrestres coagulare paulatim migrare [paulatim digeri et condensari] Decrescente autem corpore Soleis motus medij Planetarum circaum solem paulatim eto tardescent, & crescente Terra in paulatim augebitur dis motus medium Lunæ circau Terram paulatim augebitur. Colla Et collatis quidem observationibus Eclipsium Babylonicis cum ijs Albategrij et hodier cum hodiernis motūum motum medium Lunæ paulatim accelerari, didicit prumus omnium primus quod sciam, notevit deprehendit Ita privatim ut publice & ita methodum differentialem ut methodum serierum Quæritur unde habuit & quando primum habuit.. Leibnitius Anno 1673 mMethodum differentialem Moutoni L Leibnitius habuit anno 1673 & suam esse voluit ‡ ‡ methodum aliam differentialem nondum habuit.. Series pla postea habuit sed ab Oldenb quas anno 1675 ab Oldenburgo accepitterat Methodum generalem accepit ab Anglis prius accipere potuisset. Methodum generalem perveniendi ad ejusmodi series anno proximo ab Oldenburgo petivit, a Newtono accepit antea non habuit. Eodem tempore methodum extrahendi radices in speciebus a Newtono accepit qua transmutatione Leibnitiana figurarūum in methodum quandam generalem evasit sed inutilem, serierum investigandarum evasit sed inclitem. Per extractiones solas res citius peragitur. & subinde methodum generalem habuit sed inutilem [Anno 1674 scripsit se seriem invenisse numerumorum quorum valde simplicium quorum summa æquatur circumferentiæ circuli, et eadem methodo arcum quemlibet cujus sinus datur, per ejusmodi seriem exhiberi. At series pro arcu inveniendo ex sinūum dato, per transmutationem figurarum absque extractione radicum inveniri non potest. Et Leibnitius methodum aliam nondum perveniendi ad hanc seriem postea quæsi postea quæsivit ab Oldenburgo quæsivit a Newtono accepit Ideoque Proinde series hasce duas aut aliunde acceptterat &, at nondum habuit. Nam ideoque nondum habuit. Dicat tandem unde habuit series istas duas si forte habuit] Anno 1677 Leibnitius smethodum novam differentialem habuit: [sed prius hanc habuisse [& prius habuisse et primum invenisse Newtonus nondum constat agnovit publice, nondum didicit f didicerit: Leibnitius hoc probare debet nondum probavit sed eam se ante visas Newtoni literas literas New in manu Collinij: utrum hanc viderit nondum constat. 10 Decemb. 1672 & 24 Octob 1676 datas, eam habuisse nondum probavit d probavit. Extabat Analysis Newtoni in manu Collinij utrum hanc viderit nondum constat. Vide pag. Leibn Newtonus et Leibnitius non sunt idonei judices. Ex monumentis antiquis jam editis judicium aliorum ferendum est. judicent alij judicium ferendum est. Quasi methodum Moutoni non liceat Moutono asserere & series Grego Brounkeri Wallisij et Gregorij, proprijs non liceat non liceadtat proprijs authoribus, nisi his mandat authoritate ab his accepta, non liceat asserere. Certe doctorum et proborum omnium est hujusm interest hujusmodi injurios a se mutuò depellere Hac lege lata tutus Huiusmodi Lege lata tutiores reddentur erunt plagiarij redderentur plagiarij. Validius est judicium eorum qui partibus non sunt addicti. Newtonus habuit anno 1669 ut ex ejus Anaylysi hoc anno Com hoc anno ad Collinium misse manifestum est Vtrum Leibnitius Le hanc Analysin anno 1676 in manu Collinij viderit nondum constat. Certe methodum serierū hic descriptam a Collinio per Olden Vide pag. Ib: dicitur Newtonum fluxiones pro differentijs adhibere Leibnitianis adhibere semperque adhibuisse fluxiones quemadmodum et Honoratus Fabrius in sua synopsi motuum progressus Cavallerianæ methodo substituit: id est Newtonum ha Leinitium estse primum inventorem methodi infinitesimalis & Newtonum eandem a Leibnitio habuisse Et hinc natæ sunt lites inter Leibnitium et L Keillium lites When A year or two after Mr Newton had published his Principia Philosophiæ Mr Leibnitz published his Schediasma de Resistentia Medij. & Principia conteining Mr Newtons Propositions on that subject put into another dress. Ev He intro To make them his own To make them his own hHe introducesd the discourse by telling his reader that he found them for the most part at Paris twelve years before that is A.C. 1676, & communicated some of them to the Royal Academy & in the end of his discourse he adds: Nobis nunc fundamenta Geometrica jecisse sufficerit in quibus maxima consistebat difficultas. Et fortasseis attente consideranti vias quasdam novas vel certe satis antea impeditas aperuisse videbimur. Omnia autem respondent nostræ Analysi infinitorum, hoc est calculo summarum et differentiarum cujus elementa inquædam in his Actis dedimus. But he had forgot that when he wrote his Letter of 27 Aug. 1676 wchwhich was but a month or six weeks before he left Paris, he had not found the Differential He should have let the world know that several fsix first Propositions in this book & particularly the first & the ninth & 10th were known to wthwith with the Quadratures in the scholium to the 10th wchwhich depend on the 7th & 8th were known to Mr Newtons when he wrote his Letter of Oct. 24 1676, & not & not have represented them novel & conteining borrowed conteining nothing of moment but what had been treated of before by others and de already described in the Acta Lipsica in giving account of their works. And particularly he should not have ascribed the vth Proposition to Mr Sheen when he knew that Mr Newton had sent him communicated that Proposition in his said Letter. He should not have He should not have complained of Mr Keil as a young man who knew not what passed in the correspondenc of Mr Olden formerly passed about these mattermatters For this is to appelal to living wittnesses about what passed 36 years ago in Letters between Mr Oldenburg & Mr Newton Leibnitz. whereas some of Some of the Letters themselves were published by Dr Wallis & most of the rest are now found & published & young men are as able to understand them as old ones. He should not have complained of Mr Keil for acting without authority from Mr Newton Otherwise the whole socitety to whom he appeales will be excluded from being disabled tofrōom giveing judgmtjudgment Mr He should not have complained of Mr Keil for writing without authority from SrSir Isaac Newton. For all men ever mas hath authority to repell injuries from his neighbour without asking leave. He should not lay so great a stress as he seems to do upon inventing without the assistance of the first inventor. For the first Inventor alone hath Inventoris jura till another start up: And & then to take away part of his take away his right & share his right it with another would be an Act of injustice & an encouragemtencouragement to pretenders. But while the Committee have given their opinion for SrSir I Mr Keil this ought not to reflect upon the Marquis de L'Hospital, Mr Varignion Mr Iohn Bournoull i & his brother or any others who have used the Differential Method. Mr They knew not that Mr New nothing of the method of Fluxents till the Mr the publishing of Mr Newtons Principia & there Mr Newton forbore to assert chose rather to be his right Mr least it should beget a controversy about an invention wchwhich Mr Leibnitz in his Letter of 21th of Iune 1677 began to pi put in for & Mr Newton ever after declined contending about. They see that the method was proposed by Mr Leibnitz was of great use & are much to be commended for the use they have made of it & for improving it even beyond wtwhat Mr Leibnits was able to do. They knew nothing of what the correspondence of Mr Leibnitz wthwith Mr Oldenberg & Mr Collings. Mr Leibnitz would never let the world know what he learnt from the English by that correspondence. There is scarce a word of it in the Acta Leipsica, & the blame lies only upon him for concealing it. If it be represented that Mr Leibnitz may have improved the method, & made it more general it what ever he has added to it must be allowed him. But it must be considered that when Mr Newton wrote his Letter of 234 IuneOctob 1676 which was before Mr Leibnitz knew any thing of understood the Method Mr Newton had found out the Propositions in his book of Quadratures, that he had then found out the method of extracting fluents out of Equations involving their fluxions & that he had then found out also the methomethod of assuming the terms of Series indefinitely & determining them by the conditions of the Problemae. 30.

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96.

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438 \times 16 + 104^{gr}

\begin{array}{l} \frac{1752}{7008}) 14120 (470; \frac{2}{3} \\ \frac{52}{7060} \end{array}

\begin{array}{r} 18; \frac{3}{4} \\ 375 \\ 3768; \frac{3}{4} (471; \frac{3}{32} \end{array}

109 That the series for the length of an arch of the circle by the Tangent wchwhich in his Letter of Aug. 1 1676 in his Letter of Aug. 1676 you he sent to Mr Olden Mr Newton as your his own & afterwards published in the Acta Lipsica without mentioning any other Author than himself was se communicated to him the year before by Mr Oldenburgh in his Letter of 1675 & having been communicated to Mr Oldenburgh by Mr C. & to Mr C. by Mr Gregory in his Letter of 1671, & acknowledged by Mr L. in his Letter of 1675 to be different from his own. That whereas Mr Ol Leibnitz in his the Acta Leipsica published he ought not to be reputed the inventer of any of these series having received some of them from Mr Oldenberg in his Letters of & seen others in the Acta in Mr Newtons Letters & in a Paper of Mr Gregorys remains composed by Mr Collins com at the request of Mr Leibnitz & others of the Academy of Sciences & sent by Mr Collins Oldenb. to be communic the Academy of Sciences to be communicates to Mr Leibnitz, & having written to Mr Oldenberg 16776 to know the Demonstration of some of those series, that is how they were invented, that wchwhich gave occasion to Mr Newton at yethe sollicitation of Mr O & Mr C to send him yethe demonstration in his Letter of 14 Iune 1676. That Mr Le Leibnitz in yethe year 1673 pretended to yethe differential method of Mr Newton & was checkt fo Mr Leibnitz pretended to a differential method of a very different sort namely that of Mr Mouton, & was chaeckt for it by MDr Pell & thereupon wrote a Letter of excuse dated & therefore at that time had not the differential method wchwhich he now pretends to. That Mr Leibnits had seen That in all the Letters of Mr Leibnitz there appears not the least mention of his present differential method before yethe year his Letter of Iune 1677 whereas tho he gloried much in other inventions of less note such as were his instrument for sefinding the roots of Equations, his a series for yethe area of a circle wchwhich after he had received other series from hence he did not communicate, his method of Transmutations derived from those of Dr Barrow & Mr Gregory That a year before his he wrote this Letter he had seen a Letter copy of a Letter written by Mr Newton in yethe year 1672 to Mr Collins That Mr Newton in yethe year 1672 wrote represented a Letter to Mr Collins date in wchwhich he rep That Mr newton in a Letter to Mr Collins. datted A.C. 1672 describing a method of Tangents wchwhich proved to be the same wthwith that of Mr Slusius, represented that this method was the same wthwith that o but a Corollary of a general method & A of solving difficult problems & it appears to us that this general method was the method of fluxions & that Mr Collins communicated copies of this Letter to Mr Slusius, Mr Mr Churhause & others & sent it in yethe above mentioned paper to yethe Academy of Sciences to be communicated to Mr Leibnits & that a year before Mr Leibnits communic began to communicate the differential method, & we are of opinion that it was not difficult to collect the method of fluxions from this Letter. That when Mr Newton in his Letter of 24 Iune 1676 24 Octob 1676 mentionedinging his Letter to Mr Collins of 1672 & that he there resentedrepresented his method of drawing Tangents to be more general then that of Slusius as not sticking at surd quantities, & to be a cCorollary of the general method of drawing Tange wchwhich he there spake of, Mr Leibnitz took occasion from thence of describing his differential method, & as a method of the same kingd, but without acknowledging that he had seen the a copy of that Letter to Mr Collins That Mr Leibnitz before he proposed his differential method knew by Mr Newtons Letter of 1428 I Octob. 1676 that Mr Newton had the Method of fluxions above 5 years before & that by his the Analysis Mr TNewton Tract de Analysi per æquationes numero terminorum infinitas communicated by Dr Barrow to Mr Collins in Iuly 1669 it is manifest that he there describes had the method of Fluxio That Mr fluxions at that time. For he there describes the generation of quantities by motion & gives the name of moments to their incrementa momentanea wchwhich Mr Coll Leibnits calls differences & gives an example of calculating by such moments, & 110

125^{gr} ∟37207

gross weight of a new Lewidor.

88^{gr} . 1^{gr}

96^{gr} = 88^{☉} + 8^{♁}

. worth but

1^{li} . 0^{s} . 6; \frac{1}{2} . \frac{1}{4} . \frac{1}{16} . \frac{1}{256} . \frac{1}{512} = 1.0.6; \frac{9}{11}

\begin{array}{r} 88 .) 125,37207 ( \\ 15,67151 \\ 1,4246827 \\ 123∟9473873 \\ 1187236 \\ {sta. w}^{t} 123∟8286637 \\ 320) 247∟6573274 \\ 61,9118318 \\ 0,7738979 \\ 644915 \\ 0,8383894 \\ 246∟8189380 \\ = 1^{li} . 0^{s} . 6∟818938 \\ - 5 \\ 318938 \\ - 25 \\ 0.68938 \\ 62500 \\ 0∟06438 \end{array}

\begin{matrix} a & b \\ 0∟819 & 1 \\ 04∟095 & 4 4 b \\ .095 & = & 5 a - 4 b \\ .0.005 & 50 a - 40 b \\ 41 b - 50 a \end{matrix}

\begin{array}{r} 819 \\ 2457 \\ 10647 \end{array}

\begin{array}{r} 819 \\ 819 \\ 9009 \end{array}

Mr Ke Leibnits to be both Iudge & Witness in his own case, contrary to the laws of all nations. For there the second fluxion is

- \frac{n n}{e^{3}}

oo & the second moment or differen

- \frac{n n o o}{e^{3}}

& the third term & Mr Newton puts the third terme of the series equal to the not to this difference but t as p. 4. l 22. For there Roo

\frac{n n o o}{2 e^{3}}

the second term of the series to put equal to FG wchwhich is but half the second moment or difference. For the differences are

il - CB

CB - DG

& their difference is

2 CB - il - DG = ik + DF - il - DG = lk + FG = 2 FG

. — But the Society have as much authority over Mr Leibnitz as over Dr Keil [And if it Mr Leibnitz was as injust in demanding of the Society that they should condemn Mr Dr Keil without giving offering any reasons against him against him] & could not in justice condemn MDr Keil without any hearing the reasons of yethe Accuser against him & if Mr Leibnitz but refused to make good his ch But the but might in justice condemn Mr Leibnitz of calumny if he refused to give his reasons & make good his charge. accusation., as he really did refuse to [Mr Leibnitz demanded of the Society that they should m] as he really did. The Society are not to be commanded to condemn any man. Vpon a complaint they have a right to examin the matter between both parties before they give their opinions again & for the accuser to decline a fair examination is an argument of calumny. The society And Mr Leibnitz did. And yt But the Society have not yet given their opinion in this matter. They have only appointed a Committee to examin this search Records & give their opinion upon them, & have ordered the Records & the opinion of their Committee to be published. And I may add that their Committee was numerous & had in it as able mathematicians as any are Mr Leibnitz himself. was composed of the ablest Mathematicians & s such as were skilled in the hands of Dr Barrow Mr Collins Mr L Oldenburgh Mr Collings Mr Leibnitz, Mr Gregory But Mr Leibnitz tells us that the case was clear & he expected justice: that is, he will be judge in his own case & the Royal Society must be yethe Executioner. ✝ ✝ He tells us that Mr Keill opposes his reputation wchwhich that he should at such an age & after so many documents of his life that he should defend no man that's prudent or just would approve of; He tells that is that that s he is above all human judicature, & that the Society would be injust if they should allow Mr Keil to accuse him as he accused Mr Keil, that they would be unjust if they should indifferently examin the matter between them, that they would be injust if they should not allow He tells us also that he expected also that Mr Newton himself would have done him justice. He knew that Mr Newton had written in the Introduction to the Treatise De Quadratura Curvarum that he froundfound the method of fluxions gradually in the y ears 1665 & 1666 & therefore for Mr Leibnitz to demand the judgment of Mr Newton was to quarrel him. It was to demand that he should either retract what he had printed or expect that Mr Leibnitz would fall foul upon him. And the event has shewed that this was the designe of his demanding Mr Newtons opinion judgment in this matter. Its well known here that Mr Newton kept off these disputes as long as he was able: but when Mr Leibnitz cha being much more desirous of a quiet life that then of being celebrated for mathematical inventions: but when he was challeged by Mr L shewed the Acta Leipsica & chalenged by Mr Leibnitz in two several Letters, & shewed the was also was shewed yethe Acta Eruditorum for the quar he is not to be blamed if in his own defense he consented to the publishing of the Acta Erudito ancient Letters in the Commercium Epistolicum that the world might see how deceiptertfully he had been treated these forty years together What these words meant no man could understand who was unacquainted wthwith the correspondence wchwhich had passed between him & Mr Newton by means of Mr Oldenburg And seven years before. And yet this was all the mention that he then made of that correspondence, thô he ought in justice to have spoken plain. It has been said that Dr Keill & Mr Newton were the aggressors: whereas the contrary is true. Mr Newton was exprest himself displeased at what Dr Keill had published in the Philosophical Transactions least it should create a controversy. This he did long before Mr Leibnitz complained of it., But & continedued to do so till the comp the complaint arrived. But being then shewed what was published in the Acta Lipsica for Ian 16 1708 concerning his book of Quadratures, & seing himself there accused of plagiary for pretending th affirming that he found the method of fluxions in the gradually in the years 1665 & 1666 & worke understanding that what Dr Keill had published was for defending him from that accusation: he gave the DrDoctor leave to return what answer he pleased to the complaint of Mr Leibnitz. And Mr Leibnitz in his next Letter dated 14 29 Decem. 1711 persisted in justifying what had been published in the Acta Lipsica, & challenged Mr Newton to declare his opinion in the matter, that is, to retract what he had published unless he had r or enter into a dispute about it. I Mr Leibnitz in the letters wchwhich passed between him & Dr Wallis denyed not that MrNewton While Mr Newton knew not what was printed in the Acta Lipsica Mr Keil in an Epistle printed — — — edita est. Mr Newton not yet knowing what had been printed in the Acta Lipsica, was offen having seen what was in th the Acta Lipsica exprest himself offended at Dr Sloan the Secretary of the R. Society for printing of this paragraph of Mr Keills Letter, least it should create a controversy. And Mr Leibnitz aft somet understanding it in a stronger sense — what he had written, & Mr Newton upon being shewed the Acta Lipsica, gave him leave to do so. And Mr Leibnitz in a second Letter to Dr Sloan — — — — — — ☉ ☉ He in that matter. Mr Leibnits He knew that Mr Keil had affirmed nothing more that what Dr Wallis had published thirteen years before without being then contradicted. He knew that Mr Newton had also given his opinion to the same purpose in the matter in the Introduction to his book of Quadratures published before this controversy began: but Dr Wallis was dead, M Dr Keill was a Novice & Mr Newton must now be forced to retract that opinion & allow a kno allow that he had substituted fluxions for differences, or not be quiet. Mr Newton therefore was not the agg book of Quadratures published before the controversy began: but Mr Newton must retract that opinion what he had published — — — — or not be quiet. Mr Leibnitz was therefore the aggressor. The Royal Society having as much authority. — means of Mr Oldenburg. And thereupon a correspondence arising between Mr Leibnitz & Dr Wallis, the Doctor in a Letter dated 1 Decem. 1696 & printed in yethe 3d Volume of his works thus excuses himself Calculi for making that inserting that adve Monitum advertisment without saying more of the Differential method. Calculi Differentialis vel nomen audivisse me non memini nisi postquam utrumque Volumen absolverant operæ, eratque Præfationis (præfigendæ) postremum folium sub Prælo, ejusque typos jam posuerant Typothetæ. Quippe tum me monuerat amicus quidam harum rerum gnarus qui peregre fuerat, tum talem methodum in Belgio prædicari, tum illam cum Newtoni methodo fluxion ūum quasi coincidere. Quod fecit ut (transmotis typis jam positis) id monitum inseruerim. By this Letter compa compared with the Monitum it self, Mr Leibnitz could not but understand that it was at that time the opinion of the Mathematicians of in England that Mr Newton found the method of fluxions about the same time with the method of converging series. And yet in the Letters wchwhich followed hereupon between him & Dr Wallis concerning this matter, he denyed not that Mr MrNewton had the method of fluxions so early as Dr Wallis affirmed, pretended not that he himself had the differentialtial method so early, commended Mr Newton for his candor in this matter, brought no proof that he had it before the year 1677 — — — — exponential equations common to Mr Newton & himself. His words were This was the state of the dispute between Dr Wallis & Mr Leibnitz at that time. And in the year 1699 Dr Wallis by the leave of Mr Leibnitz inserted into the third Volume of his works the said two Letters together with the three answers of Mr Leibnitz dated 27 Aug. 1676 21 Iune 1677 & 12 Iuly 1677, having procured those answers from yethe library of Mr Collins. In the same year (A.C. 1699) Mr Fatio, in his Dissertation on the line of the quickest descent, suggested that Mr Leibnitz the second inventor His words were. Quod [Newtonus] addit, ex hoc eodem fundamento quadraturas quoque reddi faciliores, me in e in sententia hac confirmat; nimirum semper figuræ illæ sunt quadrabiles quæ sunt ad æquationem differentialem. In Ieromes version of the Book of Iudeth, Nebuchadonosor king of the Assyrians is said to have overcome Arphaxed in the 12th year of his reign & to have gone against the western nations in yethe 13th year. And according this recconing the 12th or 13th year of Nebuchadonosor may fall in wthwith the 8th year of Iosiah. Eupole. It has been said that yethe Royal Society gave judgment against Mr Leibnitz without hearing both parties. But this is a mistake. They have not yet given any judgment in the matter. Mr Leibnitz indeed desired the R. Society to condemn Mr Keil without hearing both parties, & wthwith by the same sort of justice they might have condemned Mr Leibnitz wthwithout hearing both parties. For they have an equal authority over them both. And when Mr Leibnitz declined to make good his charge against Mr Keil, the Society might in justice have censured him for not making it good. For he that accuses any man is bound to make good his charge. But they only appointed a Committee to search out & exa search out & examin such old Letters & papers as were still extant about these matters in the le & report their opinion how the matter stood according to those letters & papers, [They were not directed to examin Mr Leibnitz or Mr Keill but the Letters & papers only credit & meaning of the Letters & papers, as to their credit & meaning And he report what they found in them only to report what they found in the ancient Letters & papers]. And he that compares their Report wthwith those Letters & papers will find it just. The Committee was numerous & skilfull & And the Society are satisfied in their fidelity of their Committee in prainting faithfully what they found among th in the ancient Letters & Papers without favouring either party adding altering or omitting any thing in favour of either party. And the Letters & Papers &are sealed up together & by Order of the R. Society kept under kept under lock & key that they Commercium Epistolicum may be reexamined by them may be consulted & compared with the Commercium Epistolicum whenever it shall be desired by persons of Note. And whether the Committee have made put a right interpretation of upon those Letters & Papers all the world is judge. It has been said that Dr Keill & Mr Newton & Dr Keill begun this dispute attacked Mr Leibnitz & begun this dispute were the aggressors: whereas the contrary is true. I It is welll known in England that Mr Newton hats of many years neglected these things & endeavoured to decline thise very dispute. It was with some difd which is now started. When he published his Principia philosophiæ he forbore to reflect upon Mr Leibnitz for concealing what had been sent to him from England. He He was solicited again & again to let the Letters between him & Mr Leibnitz be printed by Dr Wallis A before he gave his consent He exprest himself displeased at what Dr Keill had printed in the Transactions least it should create a controversy. This he did a long time before Mr Leibnitz complained. But at length upon the complaint of Mr Leibn. being shewed what that Account of his Tract de Quadratura Curvarum wchwhich the complaint & ever after that complaint arrived untill he was shewed & what was printed in the Acta Eruditorum for Ianuary 1705, & there finding himself traduced as if in writing he had attempted to supplant Mr Leibnitz accused of plagiary for pretending that the had found the method of fluxions in the years 1665 & 1666, & being made to understand that Mr Keill vindicated him from that accusation he gave Mr Keill leve to return what answer he pleased to the complaint of Mr Leibnits. & And when Mr Leibnitz in his Letter of 29 Decem. 1711 persisted in justifying what was there printed in the Acta Leipsica, as if the Editors had not detracted from any mamn but given every man his due; & challenged Mr Newton to declare his opinion in the smaller that is, to retract what he had published & contradict Dr Wallis or enter into the lists with Mr Leibnits a dispute about it. he the R. Society was moved to appoint a Committee to search out search only & examin the ancient Letters & Papers in their custody L concerning this matter, & & report their opinion upon them. Which being published And now those Letters & Papers are published, it And whether the Committe has framed a right opinion upon them is is left to every to judge of this matter by perusing the Originals frame his own opinion upon them. judge. He forbore to reflect upon him for putting publishing a considerable part of the Prin- Principia Philosophiæ into another dress as if he had found it himself. He forbore to reflect upon him for adapting an erroneous Demonstration to the XIth Proposition of his the first Book of the Principles in order to wi make himself a coinventor thereof. He Mr Leibnitz therefore had notice of the advertisement — he did not make a fuller mention of it — he said so little of it & Unless you had rather say that Iosiah in the eighth year of his reign, upon the flight of his delivery from the the Assyrians began to seek the Lord, & ytthat in the 12th year of his reign upon his delivery from the Scythians, he reformed Iudahea from idolatry This kingdom seems to have been potent & to have had wars with the kings of Touran or Scythia beyond the river Oxus with various success, & to have been subdued112 subdued by Cyaxeres. For while Nebuchadnezzar warred in the west Cyaxeres went eastward against the ProvncesProvinces of Persia. Whether the Pischdadians were whom the Persians reccon to have been their oldest kings were kings of this kingdom of Persia or of the Assyrians I leave to be considered examined. Herodotus tells us that the Medes reigned over all Asia beyond the river Halys 128 years together, extcepting the time that the Scythians reigned. They lost their dominion to the Medes Anno Nabonass, 211. [Subduct 128 years & the reign of the Medes over Armenia & Cappadocia as far as the river Halys will begin Anno Nabonass 83, that is two years after the death of Asserhadon: [at wchwhich time [& that Dejoces reigned 53 years Phraortes 22, Cyaxeres 40 & Astyages 235] about wchwhich time the nations revolted from from the Assyrians. When Cyaxeres expelled the Scythians & subdued Assyria overthrew the Assyrian Empire, it may be presumed that he recovered Armenia & Cappodocia as far as the river Halys before he went eastward against the Persians. 113 Pray let me have the satisfaction of seeing you here on thursday morning next about 10, or 11 a clock, to see me sign some papers of moment, in wchwhich you will oblige Ap: 13th 1714. Your Most Humble servant Inewton Cartesius Menti vim attribuit mutandi determinationem motuum corporeorum. Leibnitius ne hoc in hypothesi Harmon præstabilitatæ ne hoc quidem concedit. set Similiter plane Author noster in sequentibus 1 As to what the Author of the Latin paper saith of Mr Hook Flamsteed & Mr Hook Mr Newton never always acknowledged the use of Mr Flamsteeds Observations Mr Hook being often desired to produce a Demonstration of the Proposition claimed by Mr him was never able to produce one. Mr Leibnitz claimed the same Proposition by an erroneous Demonstration & whether he or Mr Tschurnhause were in the right about a method claimed by them both may be disputed. 2 This Author in the next place complains – – – – – – were fourged. 3 The same Author in the next place claims ascribes a general method of series to Mr Leibnitz wchwhich tho this method was found many years before by Mr Newton. And he tells y H And he tells us also that the English & Scotch, Wallis, Hook, Newton, & Gregory junior, acknowledged 36 yeas ago the series for finding the Arc of a circle by the Tangent to be the Invention of Leibnitz. But he should have complained of Mr Oldenburg for not letting the English & Scotch know that this Series wthwith many others was sent by him to Mr Oldenburg Leibnits in April 1675 But of yethe But of the buisiness of series I will take another occasion to write to you. And that a collection of Gregories papers were sent also sent by him to Mr Leibnitz the next year. 4 And in the next place he magnifies an invention called the Exponential calculus without considering ytthat Mr Leibnitz had the hint from Mr Newton & that this Calculus has hitherto been of no use 5 In the last place our Author tells us that the English & Scotch Wallis Hook Newton Gregory junior acknowledge 36 years ago — — in April 1675. The author of the Remarks tells us that 6 In yethe Remarks it represented that Mr Leibnitz never communicated his reasons to the R. Society of England & so yethe Society has not examined the reasons on both sides for giving judgment. And the so upon this pretence the Author of the Remarks gives a contrary judgment to that of yethe Committee of yethe R. S. But the truth is Mr Leibnitz absolute refused to give any reasons, calling it injustice to expect that he should defend his candor & And the Committee of yethe R. S. grounded their Report upon ancient & unquestionable Records & published the Records to yethe world that that the justice of their Report might appear to the world. But the Author of the Remarks hats [has laid aside the Records of the first seven years &] begins his Report wthwith the year 1676 & 17677, & thereby confesses that he has no way to defend Mr Leibnitz but by laying wa aside the oldest Records of the seven preceding years. But of these things I may take occasion to write to you again hereafter he could not And this makes it necessary to write to you again of these matters & agwe state the matters from the Records themselves in relation to both the Questions that about converging series & that about the differential method He represents But because Mr Leibnitz hath his imploye & his correspondents or some of them have composed & published in Germany a somewhere in Germany a paper without a name whereby they endeavour to defame the Committee accuse the defame Mr Newton accuse the Committee of the Royal Society of partiality, set aside Records, being the afirm & deny things without proof endeavour to bring the matter to a squabble [& make it a dispute between England & Germany. (all wchwhich are very dishonest practises)] I intend to give you hereafter a fuller account of these matters out of the Records themselves. 6 fore aliquando tandem in inter omnes conveniat. At vero hunc in finem in Metaphysicæ sanæ notionibus – – – – Geometræ oportebit] id est sperat Author ut Philosophia Newtoni in Phænomenis per Demonstrationes Mathematicas fundata rejiciatur & omnes tandem conveniant in Philosophia quam Geometræ fundabunt in Hypothesibus super Metaphysicæ so Cartesij ad notiones Metaphysicæ Cartesij sanæ aptatis fundabunt Metaphysicam sanam intelligit Cartesianam. Metaphysica in Idæis, Ideæ et Philosophia omnis vera in Phænomenis fundantur, et incipit Newtonus a Phænomenis: incipiendum esse ab Idæis et HypothethesibusHypothesibus vult Author noster. Et metaphysicam sanam intelligit Cartesianam: Qua utique asseritur At – – – – – devenire statuit. Sed nec vim aut facultatem cogitandi substantiam rem cogitantem esse, aut rem omnem extensam Vacuum extensionem esse, aut extensionem rem mobilem esse, aitut motum corporum in sola translatione relativa consistere au sine vi inertiæ, aut consistere, aut rem eandem & cogitare posse et extendi, aut omnis ideas habere substantiarum, Cartesius probavit. Hæc omnia sunt Hypothesis Metaphysicæ Aut Deum ub et res cogitantes nullibi esse. cogitantem nulli spatio præsentem esse aut Deum nullio & et Deum non esse omnipræsentem per substantiam suam aut nos ideas habere substantiarum Cartesius alicubi probavit. Hæc omnia sunt meræ Hypotheses [Metaphysicæ quas clausis oculis ab ideis ded id est somniando confingimus] Quippe Metaphysicam omnis iab Ideis deducitmus clausis oculis & Ideæ non sunt veræ nisi quatenus cum phænomenis congruunt.] Metaphysicæ ab antiqua gentium Theogonia originem habuit, qua utique Deum esse & hujus vel partes vel p vires & potentias esse mundum totum statuere argumentis fallacibus statuere, Et usque nunc de Deo et Angelis & essentijs rerum oculis clausis disputant, argumenta capiendo ab Idæis rerum capiendo quas oculis clausis formamus, id est somniando. 114 Series itaque tres prædictas, utpote elegantes, per hanc methodum Regressuum minime invenerat; methodūum aliam Regressum no minime habebat non habuerat. Sed nec ipsum se hac, methodo se in veteri aliquando in veteribus in veteribus schedis usum esse in veteribus Schedis concedendum esst. Ipse pro se testis esse non potest. Probandum est aliorum inventa sub candoris prætextu nemini coincedendum est. Tantdem D. Leibnitius in Actis Eruditorum Anni 1693 pag. 178 methodum newtoni solvendi Problemata per assumptionem serierium pro quantitate qualibet incognita D. Leibnitius [quam Newtonus in Epistola prædicta d Anno 1676 in dum agnovit. Aut Probandum est quod invenerat hanc methodum ante annum 1676 aut Certe Anno 1676 ubi scripsit multa esse invenerat, Newtonus antem hoc anno eandem in Epistola prædicta descripseritat ut supra. Constat igitur quod D. Leibnitiums nullum habeat jus in methodum serierum. Nam transmutatio Figurarum quam prætendit nihil aliud est quam Lemma quoddam maxime obvium & minime necessarium superfluum in casu particulare, quo et methodum serierum minime constituit. [Sit ADE curva quævis, ABC tangens ejus, BC momentum Tangentis, P punctum datum, PB, PC secantes rectæ duæ radium curvam] nihil aliud est quam Lemma quoddam ad invenienda arearum momenta (quas jam vocat differentias) quod ex quo methodum differentialem invenit evasit inutile. & methodum serierum Nam transmutatio figurarum quam jactat non est methodus serierum compergentium sed meth Lemma quoddam [quo momentum vel incrementum areæ quadrandæ areæ invenerat antequam sintelligeret methodum differentialem] quo momentum areæ per meth seriem quadrandæ in seriem convertendum invenerat prodijt. Quod Lemma post inventionem methodi differentialis nullius amplius fuit usus. P.S. In Actis Eruditorum anni 16 1713 quæ jam modo advectacta sunt in Angliam, Mense Febr. pag. 94, dicitur quod Newtonus resolvendo dignitatem binomij ex fluente et momento ejus compositi binomij in seriem convergentem sumat terminos seriei successivoe pro differentialibus successive termini primi; nempe secundum terminum pro differentia termini prima primi, tertium secundum pro tertia, differentia secunda ejusdem quartum pro differentia tertia &c Ad. At nNewtonus hoc non asserit neque ibi de differentijalibus vel fluxionibus agit At in Tractatu de Quadratura Curvarum Sit

{x + o}^{n}

quantitas fluens dignitas binomij et series in quam hæc resolvetur erit

x^{n} + n o x^{n - 1} + n \times \frac{n - 1}{2} \times o^{2} x^{n - 2} + n \times \frac{n - 1}{2} \times \frac{n - 2}{3} o^{3} x^{n - 3}

&c ul. Sit jam termin x fluens & Sit jam j fluxio quantitatis Et et o momentum ejus et per ea quæ per ea quæ dicta sunt in Introductione ad Librum Tractatum de quadratura figurarum [& in ipso Tractatus initio dicta sunt] dicuntatur sunt ostenduntur fluentis fluxio

x^{n}

fluxio est

n x^{n - 1}

& fluxentis

n x^{n - 1}

fluxio est

n \times n - 1 \times x^{n - 2}

et fluentis

n \times n - 1 \times x^{n - 2}

fluxio esste

n \times n - 1 \times n - 2 \times x^{n - 3}

& sic deinceps secunda

n \times n - 1 \times x^{n - 2}

tertia

n \times n - 1 \times n - 2 \times x^{n - 3}

&c sic deinceps & ejusdem Fluentis

x^{n}

momentum primum est

n o x^{n - 1}

secundum

n \times n - 1 \times o o x^{n - 2}

tertium

n \times n - 1 \times n - 2 \times o^{3} x^{n - 3}

&c sic porro. Vnde liquet di terminos serierum ex mente Newtoni non esse differentiales termini primi sed differentialibus tantum proportionales. [At si Multiplicentur termini serierum per hanc prog (incipiendo a termino secundo) per hanc progressionem

1.1 \times 2.1 \times 2 \times 3.1 \times 2 \times 3 \times 4

&c d habebuntur differentiales. Et hinc liquet Errorem vel Typographi vel Emanuensis in Scholium Propositionis undecimæ Tractatus illius irrepsisse vbi vox ut bis s quæ in Scholij lin 3 habetur iterari debuisset in lin 8 & 11. Id quod etiam liquet ex verbi sectione proxime sequente ubi dicitur Exponi autem possunt hæ fluxiones per Curvarum Ordinatas BD, BE, BF, BG, BH &c.] Et hoc dicere voluit Newtonus in Scholio Propositionis undecimæ Tractatus illius licet vox [ut] quæ in Scholij lin 3 habetur, iterari debuisset & iterari debuisset desit in lin 8 & 11. Dicit enim Newtonus in Scholio illo quod Exponi possunt hæ fluxiones per Curvarum Ordinatas BD, BE, BF, BG, BH. Et hæ qua quidem Ordinatæ non sunt termini serierum sed terminis tantum proportionales. Multiplicentur utique termini serierum (incipientdo a termino secundo) per hanc numerorum progressionem et

1.1 \times 2, 1 \times 2 \times 3, 1 \times 2 \times 3 \times 4

&c seu 1, 2, 6, 24, 120 &c Et habebuntur differentiales ex mente Newtoni seu momentorum series seu progressio momentorum ex mente Newtoni. Et hinc liquet affinitatem esse longe maximam inter methodum serierum convergentium & methodum fluxionum, ut non mirum sit Newtonum postquam incidisset in invenisset methodum serierum, statim incidisse in methodum fluxionum et utramque ex utraque methodum suam unam universalem composuisse. De utraque enim simul scripsit in Epistolis 13 Iunij & 24 Octob 1676 datis, ut et in tractatu quem composuit anno 1671 & in Analysis per æquationes infintas anno 1669 ad D. Collini missa: et in hac Analysi dixit mo scripsit Momentum esse superficiem cum de solidis, lineam cum de superficiebus & punctum (seu lineam infinite parvam) cum de lineis agitur, quod perinde est ac si dixisset scripsisset solidi momentum primūum esse superficiem, momentum momenti seu momentum secundum esse lineam, & momentum tertium esse punctum momenti secundi seu momentum tertium esse punctu. In yethe year 1689 Mr Leibnitz began to pretended to this method a & notwithstanding that that upon readin has been b told admonished that it is Mr ht forgets to acknowledg the true author d Act Lips Ann 1689 p. 37 & 1693 p. 178. lik b Wallis Opera Vol. 2 p vol 3 p 645 Et Cheynes Fluxionum methodus inversa p. 46. But when he sent his method to Mr Newton he forgot to acknowledge that he had but newly found it, he forg he forgot to acknowledg & that f want it he had of it made him of opinion the year before that the inverse problems of tangents & such like could not be reduced to equations & quadratures & that Mr Newtons method was not gso general as he now perceived it to be & forgot to beg pardon for denying Mr Newtons method to be so ge He forgot to acknowledge his error & that this invention he now perceived ytthat Mr Newtons method wchwhich extended to such Problems to be was more general then he appr was at that time aware of could beleive the year before. He forgot to atcknowledge that in the Collection of Gregories Letters & papers wchwhich at his own request Mr Collins & Mr Oldenburg sent him to to him at Paris to be communicated to him he found the Copy of Mr Newtons Letter of 10 Decemb. 1672, conteining the his Newton his Newtons method of Tangents & that it was but & representing it as a Coroll branch or corollary of a general method of solving all sorts of Problems & that the agreemtagreement of this method of Tangents wthwith that wchwhich Slusius published put him upon considering how to enlarge the method of Slusius & led him into by the differences of the Ordinates. [And in general he has forgot to acknowledge any e any thing wchwhich he received from the English by his correspondence with the English.] And he now so far forgot In like man those things as to tell And has He forgot to acknowledge that he received any light into the method by Mr Newtons Letters of 13 Iune & 24 Octob 1676 gave him any light into yethe Letters Method. And those things are now so far out of his Memory that he has told the world that in the time of his correspondence wthwith Mr Newton by means of Mr Oldenberg, he kn when he published the elements of his differential method he knew nothing more of Mr Newtons inventions of this sort then that he had what he what Mr Newton had formerly signified in his Letters namely that he could draw Tangents without taking away irrationalls: wchwhich Hugens had signified that he could also do before he understood the method of Tangents infinitesimal method. p. 104, 107. In like manner about four or five years after Mr Leibnitz had con when Mr Collins had been bet been talking of begun to make known Mr Newtons method of Series three or four years upon a g to the Mathematicians in London & to & communicatieng it the series even to foreigners & Mr Mr Leibnitz about some months some months after Mr Leibnitz about 23 or 4 years after coming to London where he conversed wthwith the Mathematicians, & going thence to Paris began soon [communicated there a series or two as his own as if he may be bet (as he wrote, (p ) & the next year in his Letters wroteto Oldenburg p as if he head nothing] wrote from thence as if he had never heard of Mr Newton's method & put in for the invention next after Mercator pretendinged to be the first inventor of two series for the circle p 38 p 38 & the next year when he received eight series from Mr C & Mr O for the circle (p. 38) & knew none of them to be his own he& forgot the receipt of them before the end of the year & communicated to his friends at Paris an Opusculum upon one of them as his own series (p. 42) & wrote that he had communicated it to the Geometers at Paris above two years before, that is in the year 1673, presently after his coming from London. p 45. And the spring following he endeavoured to get the method from Mr O. & Mr C. without the knowledge of Mr Newton p. 45) & without the knowledg of Mr Newton promising them by way of recompence thise series wchwhich with seven others he had received from them wthwith seven others the year before & published at Paris as his own but represented it a very different series from theirs & that it was the series of wchwhich he had writ tho them some years before. (p. 45.) But they forbo By this means he endeavoured to get the method wthwithout Mr Newtons knowledge tho by his own Rule if he should have forgot the recipt of the method they were not to reclaim it without authority from Mr Newton (p. 118) The But they [forbore sending the method after the eight series [without Mr Newtons knowled & earnestly desired Mr N. to send his own method himself, & he did so, & Mr Leibnitz sent back the said series wchwhich he had promised as hise own hasd promised. (p 49, 61) M For Mr Oldenburg & Mr Collings were not to question his candor wthwithout authority from the Executors of Mr Gregory. p. 118 He still der- He still wanted requested wanted Mr Newtons method of deriving reciprocal series from one another, & wrote for it & when he received it, tho he understood it with difficulty yet he wrote back that he had found it before . . . . . . . Collins p. 25. 47. He has forgot also that the meth Series for Quadratures wchwhich breakse off in some cases & becomes finite was were deduced by Mr Newton from the method of fluents before the dates Letters writing of the Letters wchwhich passed between them in yethe year 1676, that is, before Mr Leibnitz kewknew any thing of the method differentio infinitesimal metho differential method, p but & wherever he has occasion to met referrs the invention & refers the invention of such series to Mr Cheyne. He has forgot also p. 169. Neither ha & that the method of assuming the terms of a series was assumed & determines them by the conditions of yethe Problem was a part of Mr Newtons method general method before the writing of those Letters p. 86. Wallis Vol 2 p [About 8 or 10 years after the writing of those Letters p. 86 Mr Tschurnhause began to use that method. Mr Leibnitz took it from him pretending that heead communicated it to him w ten years before when they were together at Paris. But About month or six weeks But at that time Mr Leinitz had not the differential method nor the me knew not how to reduce inverse Problems of Tangents to equations & therefore had not the method, no & quadratures nor to derive reciprocal series from one another p. 63, 65, 94, 96. & therefore had not the said method.] Dr Wallis in the year 16993 published the said Letters an some extracts out of the said Letters amongst wchwhich was this method.] And tho hHe pretends that Tschurn he communicated discoursed of it to Tschurnhause at th when they were together at Paris but Tschurnhause remembred nothing of the matter. & Mr Leibnits ot knew not how to reduce inverse Problems to equations Or quadratures nor to derive reciprocal series from one another p. 65, 94, 96 It remains therefore the Seing therefore it appears manifest it appears by the following Letters & papers that Mr Newton If it appears therefore by the following Letters & Papers Seing therefore If it appears therefor by the following Letters & Papers that Mr Newton had the method of infinite series in the year 16969 & then applied it generally to yethe solution of all problemes by the method of fluents & that Mr Iames Gregory having notice thereof found ot out the method of series in the end of yethe next year & in the beginning of the year following sent the series for the to Mr Collins the series for the circle since claimed by Mr Leibnitz: it remains that Mr Leibnitz either prove that he had the method of series & infinitesimals & that Mr Newton sent him & Mr Newton & Mr Collins but acknoledged Mr Newton the first inventor of the method. & seeing that Mr Leibnitz in the years 1675, 1676, 1677, received large communications from London relating to these methods whereof he forgot to make acknowledgments, it will remain that but nothing appears that h & it ndoth not appear that he knew any thing of yethe methods before those communications, & the if the pretence of candour & sincerity doth not make any man a witness in his own cause & the pretence of finding things apart without proving the priority of invention is not to be regarded: it remains that Mr Leibnitz either desist from his pretenstions to the series of Gregory & from numbring himself amongst the first inventors of the method of series & the method of infinitesimals; or prove that he had the said series & methods before his correspondence wthwith Mr Oldenburg began &, before have he came into England, before Mr Gregory sent the said series & Dr Barrow the Analysis to Mr Collins & before Dr Barrow communicated Mr Newtons Analysis to M l. 8. for yethe circle tho he wanted the Demonstration l. 19. promised to send them some series of his own very different from theirs but sent them them above mentioned series having forgotten that he had received it from them the year before (p. 42, 45, 61) And when he sent his differential method to Mr Oldenburg, he fogotforgot to acknowledge that he had but just invented it & by the invention was convinced that Inverse Problemes of Tangents were reducible to Equations & Quadratures p. 116 When I said in my Letter of 13 Iune 1676 that Analysis by the help of infinite æquations extended to almost all sorts of Problems except perhaps some numeral ones like those of Diophantus but yet became not universal without some further methods of reducing Problems to infinite series then by division & extraction of roots, & Mr Leibnitz in his answer replied Quod dicere videmini plerasque difficultates (exceptis Problematibus Diophantæis) ad series infinitas reduci; id mihi non videtur. Sunt enim multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturis; qualia sunt (ex multis alijs) Problemata methodi tangentium inversæ. Its certain that we both spake of d resolving Problems by reducing them to infinite series. And yet ye would now persuade us that he spake of reducing them to vulgar equations, & did not intend to deny that they useth any thing more then that they could be resolved by reducing them to vulgar equations or quadratures other equations then the vulgar., [but knew himself how to reduce them to differential equations. And that a few lines after where he told us that he had solved a Problem by certa Analysis he meant this method.] He I spake of reducing almost all problems to converging series by methods wchwhich I had not yet described, he denied the possibility of this & now would make us beleive that he did not deny it. Mr Leibnitz in his Letter of 27 Aug. 1676 wrote thus: Quod dicere videmini plerasque difficultates (exceptis Problematibus Diophantæis) ad Series infinitas reduci; id mihi non videtur. Sunt enim multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturis Qualia sunt ex multis alijs Problemata methodi tangentium inversæ. sq And when I answered that the such Problemes were in my power he replied (in his Letter of 21 Iune 1677) that he scocnceived that I meant by infinite series but he meant Geometrically. And now phe persists in the same reply saying that he meant by vulgar equations. See I See the Answer to this in the Commercium Epistolicum pag 92. He saith that one may judge that when he wrote the his Letter of Aug 27. 1676 he had some entrance into the differential calculus because he said there that he had solved the Probleme of Beaune certa Analysi a certain Analysis. But what if that Probleme may be solved certa Analysi without the differential method. For no further analysis is requisite then this, That th as the Numbers are in Geometrical progression when their Logarithms are in Arithmetical progression, so the Ordinate of the Curve desired increases or decreases in Geometrical progression when the Abscissa increases in Arithmetical, & therefore the Abscissa & Ordinate have the same relati ōon to one another as the Logarithm & its number. And to infer fro this that Mr Leibnitz had the entrance into yethe metho differentiasl method is as if one should say that Archimedes had entrance into it when because he he drew tangtstangents to yethe spiral squared the PrabolaParabola, & found the proportion between the sphere & C yilyn̄nnder, or that Dr Walli Cavallerius Fermat & Wallis had entrance into it because they did many more things of this kind. Let x be the Abscissa, y the Ordinat, qp the sub perpendicular cut off by the Abscissa, s the subperpendicula r the radius of curvature &

\dot{x} o

the moment of the Axis &

\frac{s o}{y} \dot{x}

will be the mom first moment of the ordinate

\dot{y} o

& p y

\frac{r y}{p}

. &

o o + \frac{s s o o}{y y} = \frac{2 r y}{p} \times 2 y \overset{..}{y} o o = \frac{p p o o}{y y}

. &

\frac{p^{3} o o}{4 r y^{3}}

the second moment y

\overset{..}{y} o o

\frac{p^{3} o o}{2 r y^{3}} = 2 \overset{..}{y} o o

\dot{x} s = y \overset{..}{y}

p^{3} \dot{x} \dot{x} = 4 r y^{3} \overset{..}{y}

In my Letters of 13 Iune & 24 Octob. 1676 I I spake of my methods of Series & Fluxions interwoven with one another, by the name of my general method & said that it extended to almost all sorts of Problems & particularly to inverse Problemes of Tangents, & to the squaring of Curves by series wchwhich breake of & become finite twhen the Curve can be squared by a finite equation & I set down such a series & illustrated it with examples. I gave instansces also of the inv the inverse method of tangents saying that when the relation of any two sides of the right angled triangle were conteined was de conteined under the tangent subtangent & ordinate were was givengiven the Probleme might be solved without my general method, but when the Abscissa also enters the vinculum, that method is usually required. I set down instances also of Theoremes for comparing the areas of figures compound figures with those of the Conick Sections. And how all these things are done by the method of fluxions appears by the my book of Quadratures: but without that method, it & there is no other method yet known by wchwhich they can be done. He complains that the Committee have gone aside from the Question about this method to the business out of the way in falling upon the method of series: but he should consider that both methods are but two branches of one general method. I joyned them together in my Analysis. I interwove them together in the Tract wchwhich I wrote in my the year 1671 as I said in my Letters of 10 Decem 1672 & 24 Octob 1676. In my Letter of 13 Iune 1676 I said that my method of Series extended to almost all Problemes but became not general without some other methods meaning the method of fluxions & the method of arbitrary series — a method wchwhich depends upon that method of fluxions, & now to take those other methods from me is to restrain & stint the method of series & make it cease to be general. Also In my Letter of 24 Octob 1676 I called both method all these methods together my general method. See the Commercium Epistolicum pag. 86. lin. 16. And If Mr Leibnitz has been tearing this general method in pieces & taking from me first one part & then another part whereby the rest is maimed, he has given a just occasion to the Committee to defend the whole. It is also to be considered that Mr Leibnitz is perpetually at work to make himself a witness in his own cause & it's allowed in all cCourts of justice to speak to the character credit of the witness. 7 Mr Leibnitz complains represents that he has been in possession He acknowledgeds that when he was in London the first second time time he saw some of my Letters in the hands of MrCollins, & in his last Letter but one he has quoted two of those wchwhich he then saw One of wchwhich & wchwhich were written vizt those written 1672 & 24 Octob 1676. And n I spring he & no doubt he would principally desire to see the Letter which conteined the chief of my series & the Demonstration of the two series wchwhich a few months before he desired entreated Mr Oldenburg to procure from Mr Collins, that is, the Analysis per æquationes numero terminorum infinitas. But he tells us that he never saw where I explained my method series fluxions & that he finds nothing of it in the Commercium Epistolicum where that Analysis is published & my Letters of 10 Decem 1672, 13 Iune 1676 & 24 Octob 1676 are published. 8 He saith also that he never saw where I explain my the my the method wchwhich he ta wchwhich I claim & claimed by me w where he assumes an arbitrary series. If he pleases he may see to look into the Commercium Epistolicum pag. 55, 56 & 86 he will there see that I had that Method when I wrote my Letters of 24 Octob. 1676 & by consequence when I wrote my Letter of & 13 Iune 1676 & five years before. Mr Leibnitz might also find it apart himself, but not so early; He first mentioned it in the year 1689 & explained it at large in the year 1693. And second Inventors have no right. Mr Leibnitz published his Schediasma de restentiaresistentia Medij et motu projectorūum gravium in Medio restente in Ianuary 1689 & concluded it with these words Multa ex his deduci possent praxi accommodata, sed nobis nunc fundamenta Geometrica jecisse sufficerit, in quibus maxima consistebat difficultas. Et fortasseis attente consideranti vias quasdem novas et satis antea impeditas aperuisse videbimur. Omnia autem respondent nostræ Analysi infinitorum, hoc est calculo summarum & differentiarum (cujus elementa quædam in his Actis dedimus) communibus quoad licuit verbis hic expresso. This was the first specimen made publick by Mr Leibnits of applying the Differential method to the higher sorts of Problems. And by the words here cited it appears that Mr Leibnitz himself at that time acc ounted it so. And yet this specimen was published a yeare & an half after that of the Principia, & conteins was borrowed from the Principia, & answers that the things in the Book of Principles answer as much to the calculus of fluxions as those in the Schediasma to the calculus of differences, & the differential117 differential calculus did not begin to be celebrated before the publication of this specimen. For as the Ordinate is to the sub-perpendicular (or subtangent to the Ordinate) so is the fluxion of yethe Abscissa to the fluxion of the ordinate And if you take a fourth proportial to yethe Radius of Curvity as the f prop Ordinate to yethe Perpendicular, & apply a quarter of the Summ of say, As the summ Perpendicular to the Ordinate so the Radius of Curvity to a fourth proportional & to this 4th Proportional apply a quarter of the summ of the squares of the One fluxions of the Abscissa & Ordinate, you will have the second fluxion of the Ordinate. Schol. As the Ordinate is to the subperpendicular (or subtangent to the Ordinate) so is the fluxion of the Ordinate AbscissAbscissa to the fluxion of the Ordinat. Take a fourth proportional to yethe Radius of curvature as the Ordinate to yethe Perpendicular & to this proportional apply [a quarter of] the summ of the squares of the fluxions of the Absciss & Ordinat & you will have the Latus will be the second fluxion of the Ordinate.

r r = x x + y y

0 = 2 x x + 2 y \dot{y}

\dot{x} \dot{x} + x \overset{..}{x} + \dot{y} \dot{y} + y \overset{..}{y} = 0

1 + \dot{y} \dot{y} + y \overset{..}{y} = 0

1 + \frac{1 + \dot{y} \dot{y}}{y} = - \overset{..}{y}

. Apply Let the summ of the squares of the fluxions to of the AbscissAbscissa & ordinate be applied to a quantity wchwhich is to the Ordinate as yethe Radius of curvity as the Ordinate to the Perpendicular, & the Latus will be the second fluxion of yethe Ordinate. How to re When these fluxions make the Ordinate decrease they must be taken with negative singnses. Ho How to reduce the equations wchwhich result from hence & to separate the unknown quantities is the business not of this but another method. Scholium. Vt Ordinata ad sub-perpendicularem (vel sub-tangens ad Ordinatam) ita est fluxion Abscissæ ad fluxionem Ordinatæ Applicetur summa quadratorum fluxionum Ordinatæ et Abscissæ ad quantita lineam quæ sit ad Ordinatam ut Radius curvitatis ad Abscissam Perpendicularem, & Latus erit fluxion secunda Ordinatæ. Hæ fluxiones ubi Ordinatam diminuunt, negativæ ponendæ sunt. Quodmodo æquationes sic prodeuntes reducendæ sunt & quantitates indeterminatæ separandæ non est hujus sed alius methodi Non hujus sed alius est methodi æquationes prodeuntes reducere & indeterminatas separare. Problema hocce cum nullius fere sit usus in Actis eruditorum annos plures neglictum et insolutum mansit. Et eadem de causa solutionem ejus non ulterius prosequor.

\frac{\dot{x} x + \dot{y} \dot{y}}{o r} p = \overset{..}{y} . o . p ∷ \dot{x} \dot{x} + \dot{y} \dot{y} . r \overset{..}{y}

. & Ordinatua ad partem Radij curvitatis inter Abscissam et punctum contactus (ita Et Vt subtangens ad tangentum ita summa quadratorum fluxionum Ordinatæ et Abscissæ ad rectangulum sub radio curvitatis et fluxione secunda Abscissæ Scholium Fluat Abscissa uniformiter & erit, Vt Ordinata ad subperpendicularem (vel subtangens ad Ordinatam) iuta fluxio Abscissæ ad fluxionem Ordinatæ, Et ut Subtangens ad Ab Tangentem ita summa quadratorum Ordin fluxionum Abscissæ et Ordinatæ Ordinata ad perpendicularem ut fluxio abscissæ ad fluxionem arcus & quadrat fluxionis arcus

y^{3} p^{3} \dot{x} \dot{x} = r \overset{..}{y} . y^{3} . p^{3} ∷ \dot{x} \dot{x} . r \overset{..}{y}

ad rectangulum sub radio curvitatis et fluxione secunda Abscissæ Hæ fluxiones &c Vpon account of my progress in these matters he procured for me a fellowship for me in Trinity College in the year 16767 & the Mathematick Professorship fo two years after. p. 1. l. ult. purchased of Rob. Barker p. 3. l 3, 4 blot out yethe word are or add the word which He pretends that in my book of Principles pag. 253, 254 I allowed him the invention of the calculus differentialis independently of my own method & that to attribute this invention to my self is contrary to my knowledge there avowed: & yet in all that book I do not find one word to this purpose. In the year 1685 (at wchwhich time I was writing that book) Mr Craige brought to hime the Acta Eruditorum for October 1684 & desired me to explain to him the elements of the Calculus differentialis conteined therein & I did so & told him that the method was mine as would appear if the Letters wchwhich had passed between Mr Leibnitz & me were published & Mr Craige is still alive & remembers this, & therefore I could not mean in the Book of Principles to allow that Mr Leibnitz was either the first inventor or had received no light from me. In the I was so far from doing this that in the very place which he quotes I did affirm that before he sent me notice of his method I sent him notice of mine & of some of its performances. By this notice he was able to compare the methods I gave him this notice in my Letters of 13 Iune & 24 Octob 1676 & he understood it so far as to be able to compare the methods. By his own confession he saw my Letter of 24 October 1676 when he first was arriv the second time in London wchwhich was eight months before he sent me notice of his method. He had notice of it also by other copies of other Letters sent to him at Paris in Iune 1676 in a Collection of Mr Iames Gregories Letters which he certainly received. In this Collection wasere copies of atwo Letters of Mr Gregory dated 5 Sept. 1670, & of one & 15 Feb 167

\frac{1}{2}

& a copy of my Letters dated 10 Decem 1672, & by these Letters he had notice that Gregory found the series for the quadrature of the Hyperbola Circle by a se the tangent, that I had a in the beginning of the year 1671 & deduced t the next from Dr Barrows differential method of Tangents deduced as a method of tangents without calculation in the year 1670 & in l the beginning of the next year found the series for the quadrature of the circle by the tangent & that I had a general method of Analysis wchwhich readily gave the same method of tangents of Slusius Gregory & Slusius & extended also to the abstruser sorts of Problemes concerning the curvatures areas lengths centers gravity of lines & figures of curves &c and proceeded without fresing equations from surds & that I had written a Tract concerning this method & the method of series together meaning in the year 1671. as And by my Letter of 13 Iune 1676 he had notice that my Analysis was so universal as to extend to almost all sorts of Problems &. And as the series for yethe Quadrature of the circle wchwhich he sent me back in his Lette as his own in his Letter of 27 Aug was 1676 was Gregories so what he sent me back as his own in his Letter of 21 Iune 1677 concerning the Differential method was nothing more then what the Letters above mentioned had given him notice of a year before I the sp In the spring t In May 1676 And further, Mr Leibnitz by hsis Letter of 12 May 1676 desired Mr Oldenburg to procure from Mr Collins the demonstration of two of my series wchwhich he & that is meaning the method of finding them, & about the same time M he desired also that tahe collection might be made of Gregorys Letters aforesaid collection of Gregory Letters might be made & sent to Paris & in October following when he came the second time to London he applied himself to Mr Collings in London & saw in his hands several of my Letters, as he has acknowledged, & no doubt would then desire chiefly to see my Letter in wchwhich I had communicated to him the said two series with the Demonstration wchwhich he wanted, And that is my Analysis per æquationes numero terminorum infinitas, & therefore saw it. By my Letters of 10 Decem 1672 & 24 Octob. 1676 he could not but know that I had the method in 1671 of fluxions in the year 1671 & by this Analysis he could not but know that I had this method in the year 1669. &And after this to all this to tell me that in my Book of Principles pag 253 & 254 I allowed him the invention without & ytthat to attribute it now to my self is contrary to my knowledg, is very extraordinary & that he found it without receiving any light from me & was the first inventor, is very extraordinary. 118 Mr Mercator lived above 10 or years longer without proceeding further then to yethe single QuadatureQuadrature of the Hyperbola. The progress made by Mr Newton shews that he wanted not Mercators assistance. However, for avoyding disputes, he supposes aa See his Letter in yethe Commer. p. that my L Brunker invented & Mercator demonstrated the series for the Hyperbola three or four some years before they published it. The aforesaid treatise of Analysis Mr Newton in his Letter to Mr Oldenburg dated 24 Octob. 1676 mentions in the following manner. Eo ipso tempore &c When Mr Newton had explained these three Rules & illustrated then wthwith various examples, he layd down the Idea of deducing the area from the Ordinate by considering the Area as a quantity g growing or increasing by continual motion flux & becoming bigger or less or in a given accordingly as the increase or flux is swifter or slower And this flux he measured by the length of the Ordinate supposing this Ordinate to mo the Abscissa to increase uniformly in proportion to time. And from the moments of time he calls the giaves the names of moments to the parts of the Abscissa & Area generated in moments of time momentaneus increases or infinitely small parts of the Abscissa & Area generated in moments of time. The moment of a line he called a point in the sense of Cavallerius tho it be not a gemetrical point but an infinitely short line & the moment of an Area or superficies he callsed a line in the sense of Cavallerius tho it be not a Geometrical line but an infinitely narrow superficies. The Abscissa he supposes to increase uniformly as the exponent of time And when he is demonstrating any Proposition he puts a line drawn into the Abscissa for the exponent of time & a moment of the Abscissa drawn into yethe Ordinate for the a moment of the Area. And tells use that And when he considered the Ordinate as the exponent of time moment of the Area he understanoods by it the rectangle under the Geometrical Ordinate & a moment of the Abscissa, as in the method of Cavallerius; & puts the rectangle under assumed an unit for the ordinate of a rectangle proportialproportional to time. [And then adds: Iam qua ratione superficies [curvilinea] ex momento suo perpetim dato, per præcedentes Regulas elicitur, eadem quælibet alia quantitas ex momento suo sic dato elicietur And of this he gives example] Sit ABD saith he, Curva quævis, et ABH rectangulum cujus latus AH vel BK est unitas. Et Et Cogita rectam DBK (saith he) uniformiter ab AH motam areas ABD [curvilineam] & AK [rectangulam] describere; & quod BK (1) [recta] BK (1) sit momentum quo [area] ABD gradatim augetur; & AK (x) & [recta] BD (y) momentum quo ABD [area] AB curilinea] AK [x] ABD gradatim augetur; et quod ex momentao BD perpetim dato possis, per præcedentes [tres] Regulas, aream ABD ipso descriptam conferre investigare sive cum [ipsius] AK (x) momento 1 descripta conferre. Iam qua ratione superficies ABD ex momento suo perpetim dato per præcedentes Regulas elicitur eadem quælibet alia quantitas ex momento suo sic dato elicietur. Exemplo res fiet clarior. And Then he adds examples of this by finding a series for the arc of & how illustrated this wthwith examples & added his method of Regression from the Area or Arc or solid to the Abscissa, A& shewsed how the same method extendesd to Mechanical Curves for determining their Ordinates tangents areas, lengths, ng &c Now in illustrating this method with examples the Among the examples which hes there sets down for illustrating this method there are these. Let the Radius of a circle be &c. Let the versed sine be x & the equations will be Mr Collins gave Mr Gregory notice of this method In the year – – – – correspondence wthwith Mr Oldenburgh. In Feb. 167

\frac{2}{3}

he meeting Dr Pell at Mr Boyle's & pretending to the differential method of Mouton & he was reprehended for it notwithstanding that he was shewn by Dr Pell & get the next day by b that it was Moutons method, persisted in maintaining it to be his own invention by reason that he he had found it by himself without knowing what Mouton had done before & had much improved it. In the end of that L one of h In Mr When Mr Gregory had none of Mr Newtons series was sent to him Mr Greg. he tried to find out the method fy fir first tried to deduce it from his own series combined together, And as he mentions in his Letter dated 19 Decem. 1670, & at length And by some such method Mr Leibnitz before he left London seems to have found the summ of a series of fractions decreasing in infinitum whose Numerator is an unit given number & denominators are triangular or pyramidal numbers or triangulo Triangular numbers &c. From the series

\frac{1}{1} + \frac{1}{2}

And in the year 1682 he published it as his own in the Acta Eruditorum wthwithout mentioning that he had received it from Mr Oldenburg. He might carry it with him from London to Paris in the 16673; but that he had the demonstration of it before the year Feb 1675 doth not appear. He might have carry this series when he went from London to Paris & communicate it there the to his friends at Paris above 3 years before he sent it back to Mr Oldenburg: but he it doth not appear that he con had the Demonstration before the began to communicate his opusculum written thereof so early. When found the demonstration, then he wrote his opusculum upon it, wchwhich was in the year 1675 & communicated that also to his friends. And he himself has told us that this was in the year 1675. However, it lies upon him to prove that he had this series before he received it from Mr Oldenburg because he did not then know it to be his own. He had therefore several direc series & by consequence a method of finding ymthem before he invented & forgot the inverse method. And if he had searched his old papers diligently he might have it there, But having But found this method also there [but he wrote for Mr Newtons methods because he had forgot his own.] Myrina, & saith that when she was Queen of the Amazons in Libya she there conquered the Atlantides & Gorgons & them made a league with Orus the son of Isis in Ægypt & passing he passing (he should have said with Osiris, the father of hasband of Isis & father of Orus & passing through Ægypt subdued Arabia Syria & Cilicia & came through Phrygia to ymthem mediterranean but attempting Thrace passing over into Europe was slain wthwith many of her weomen by the Thracians & Scythians under the conduct of Sipylus a Scythian & Mompsus a Thracian whom Lycurgus king of Thrace had banished. This was that Lycurgus whom Bacchus slew a little before he received this baffle wchwhich put whereby a stop was put to his victories. This is that Neptune who with Apolloo or Orus the Son of Osiris fortified Troy with a wall in the duque reign of Laomedon the father of Priam, & left many bastard children in Greece & Asia minor some of wchwhich were Argonauts othe as Naplius the son of Amymone, Ancæus the son of Abta, & Erginus, & whose son his son Am others were contemporary to them, as Amycus whom Pollux slew in the time of the Argonautic expedition, Lycus whom Hercules slew. Corynetes & Procrustes slain by whom Theseus slew Eumolpus Theseus Hippothous whom Theseus restored to the kingdom of Cercyon, & Theseus himself This that Neptune who wthwith Apollo or Orus the Son of Osiris fortified Troy wthwith a wall in the reigne of Laomedon, & left many children in Greece some of wchwhich were Argonauts & others were contemporary to them. De rea flourished therefore contemporary to Sesastris one generation before the Argonautsic expedition or about 400 years before Solon went into Egypt Hanc seriem D. Collins initio anni 1671 a Gregorio acceperat ut supra D Leibnitius eandem scripto opusculo ut suam cum amicus in Gallia hoc anno communicare cœpit, celata hac epistola. pag. 42. His verbis patet series quas D. Leibnitius se ante annos aliquot invenisse professus est a communicatis diversas fuisse. Pretends that inby the time of that corresponde he understood learnt nothing more of Mr Newtons method of fluents then that it was he had a method of Tangents wchwhich stuck not at surds, will not allow that M young men who were not privy to that correspondence can understand the Letters & Papers wchwhich pas then passed between the correspondents & some of wchwhich passed between the co have been printed by Dr Wallis. printed the elements of his method itn the Acta Lipsica wthwithout mentioning any thing that he learnt by that correspondence or letteing the world know that MNewton also had such a method, excuses himself by pretending that he then knew nothing more of Mr Newtons method being general or extending further then to the drawing of tangtstangents without sticking at surds. Pro M2

\sqrt{CN \times FG}

scribere licet

CN + FG

. Et hac ratione HN sic resistentia erit ad gravitatem ut

\frac{HN}{CN + FG} - \frac{CF}{2 FG} + \frac{FI}{CF}

ad 1 id est ut

\frac{{}^{2}{HN}, FG - CE, EG - CF, CN}{2 CN, FG + 2 {FG}^{q}}

CF, FG - 4 v, FG - CF, CN . - HN + CF = p . FG - CN = q . 2 p, FG + q, CF . CG - CF = r CF - HN = 2 r . - 4 r, FG + q, CF

CN = R o o - 2 S o^{3}

FG - CN = chord GQ - chord CP = 3 S o^{3} . FG - 3 S o^{3} = CN . HN = CF - 2 r

IF = Q o . FG = R o o + S o^{3} . CN = R o o - 2 S o^{3} . HN = o \sqrt{1 + Q 2} - \frac{2 QR o o}{\sqrt{3}}

CF = o \sqrt{1 + Q Q} . \frac{2 R o + Q S o^{3}}{\sqrt{1 + Q Q}} = CG - CF = \dot{r} . MN = Q o - {}^{2}R R o o + S o^{3}

CN = GF

. Decrementum spatij

- HC + CG = HN + CF

, ex resist & Grav. Incr ex Grav

= CG - CF

. Decr. ex resist.

= CG - HN = CF - HN + \frac{FI, FG}{CF}

, ad Grav decr descens ex gGrav

= GF

. Resist. Grav

∷ \frac{CF}{GF} - \frac{HN}{GF} + \frac{FI}{CF}

. 1 .

\sqrt{c n} . \sqrt{GF} ∷ h n . HN

\frac{CF}{GF} - \frac{{}^{2}h n}{c n + GF} + \frac{FI}{CF} . 1 ∷ \frac{CF}{GF} - \frac{+ {}^{2}{CF} - 4 e}{c n + GF} + \frac{FI}{CF}

{}^{2}c n

c n + GF

\frac{c n + GF}{2} . GF ∷ h n . HN - CF, GF + 4 e, GF + c n, CF

IF = MN + S

\sqrt{o o + {MN}^{2} + 2 S, MN + S^{2}} . \sqrt{o o + {MN}^{2}} . \frac{S, MN}{\sqrt{}} = CF - HN

\frac{CF}{2 FG} = \frac{\sqrt{1 + Q Q}}{{}^{2}R o + {}^{2}S o o} . \frac{HN}{CN + FG} = \frac{o \sqrt{} - 2 Q R o o}{2 R o o - S o^{3}} = \frac{\sqrt{1 + Q Q} - 2 Q R o}{2 R o - S o^{2}}

\frac{HN}{CN + FG} - \frac{CF}{2 FG} = \frac{2 R o \sqrt{} - S o^{2} \sqrt{} - {}^{2}R o \sqrt{} - {}^{2}S o o \sqrt{} + 4 Q R^{2} o^{3}}{4 R R o o + {}^{2}R S o^{3}} = \frac{- {}^{3}S \sqrt{}}{4 R R}

spatium FG cadendo describente generat velocitem qua duplum illud spatium 2FG eodem tempore describi posset ut ex demonstratis Galilæi notum est, id est spatium 2FG velocitatem 2FG quæ exponitur per spatium 2FG applicatum ad tempus

\sqrt{FG}

sive hoc est velocitatem

2 \sqrt{FG}

: et in corpore arcum CG generat tantum velocitatem quæ sit ad hanc velocitatem ut

CG - CF

ad FG vel FI ad CF, id est velocitatem

\frac{{}^{2}{FI}}{CF} \sqrt{FG}

. Addatur hæc velocitas ad decrementum prædictum & habebitur velocitas ex decrementum velocitatis ex gravit resistentia sola oriundum And having now got eight or ten series, he endeavoured in May following to get the method also promising from Mr O. & Mr wthwithout the knowledge of Mr Newton tho by his own rule if he should have forgot the receipt th of the method they were not to reclaim it wthwithout authoryauthority from Mr Newton. (p. 118) And for sending him the method he promised them some series invented by himself & communicated to representing them very different from their s the series of wchwhich he had writtedn to them some years before, representing them his own & very different from theirs p 45 but meant the series wchwhich one of the eight series wchwhich he had received from them the year before together wthwith two or three p 61. But they For the series being Gregories they were not to question his candor wthwithout authority from Mr Newton the Executors of Mr Gregory. However they Mr Collings had the Method but forbore to send the method it after the eight series & he & Mr Oldenburg wrote earnestly to Mr Newton to send his own method himself, & Mr Newton did so, p 49. And when a year or two after Mr N. had published his Principia Philosophiæ, Mr And when he pretended that Tschurnhause had this method from him while they were together at Paris (Acta Lips Ann. 1686 he fotrrgot that at that time he did not know how to continue series or derive reciprocal series from one another (p. 63, 94, 96. When A year or two after Mr Newton had published his system Principia Mat Philosophiæ Mr Leibnitz publishing three papers relating to it, sometimes complements Mr

CG - DH = - \frac{m a a + b b}{n a a} o + &c.

GH = \frac{\sqrt{m m a^{4} - {}^{2}m n a a b b + n n b^{4} + n n a^{4}}}{n a a} o + &c

. LI−KH=

\frac{2 b b o o}{a^{3}} . \frac{6 b b o^{3}}{a^{4}} (∷ a . 3 \times o) ∷ GH . LM = \frac{3 \sqrt{m m a^{4} - 2 m n a a b b + n n a^{4} + b^{4}}}{n a^{3}} o o

Resist. grav.

∷ 3 \sqrt{m m a^{4} - 2 m n a a b b + n n a^{4} + n n b^{4}} . 2 n b b

o . GH ∷ DN . XY = DT . GH = \frac{o \times XY}{a - o} . \frac{n a a}{o} GH = \sqrt{} = n a \times XY

NX = c

Resist. grav

∷ 3 XY . \frac{2 b b}{a} = 2 VY = 2 VG = YG

n a a . - m a a + n b b ∷ a . \frac{- m a a + n b b}{n a} = ZY . ZV = \frac{m a}{n} . VY = b b a . VG = \frac{b b}{a}

DN = a . XY ∷ o . \frac{o \times y}{a} = GH

. Resist. Grav

\frac{2 b b o o}{a^{3}} . \frac{6 b b o^{3}}{a^{4}} ∷ \frac{XY, o}{a} . \frac{3 XY, o o}{a a} = LM

\frac{3 XY, o o}{a a} . \frac{2 b b, o o}{a^{3}} ∷ 3 XY . \frac{2 b b}{a} ∷

Resist. Grav

∷ 3 XY . 2 VH = YG

\frac{3 b b o^{3}}{a^{4}} \times \frac{GT}{a} o \times \frac{a^{3}}{2 b b o o}

velocitas ut

\frac{GH}{\sqrt{HK}}

id est ut

\frac{GT, o}{a} \times \sqrt{\frac{a^{3}}{2 b b o o}} = \frac{GT}{b} \sqrt{\frac{1}{2} a}

sive ut

GT \sqrt{a} . \frac{3 GT, o o}{2 a a}

Densitas ut

\frac{3 GT}{2 \sqrt{H}} \times

\frac{3, XY, o o}{a a, \frac{o o, {XY}^{2}}{a a}} = \frac{3,}{XY}

. Velocitas ut CF

\frac{GH}{\sqrt{KH}} = \frac{XY, o}{a} \times \sqrt{\frac{a^{3}}{2 b b o o}} = \frac{XY}{b} \sqrt{2 a}

id est ut

XY \sqrt{a}

. Velocitas autem est ut

\frac{GH}{\sqrt{HK}}

id est ut × F GT in

\sqrt{a}

et Resistentia Densitas medij direc ut resistentia directe et quadratum velocitatis inverse, id est ut

\frac{{}^{3}{GF}}{2 VH, a, {GF}^{q}}

[sive reciproce ut GT.] ut

3 GT 2 VH

directe et

a, {GT}^{q}

inverse, hoc est inverse ut GT. Et Velocitas autem est corporis in puncto quovis G æqualis est velocitatis corporis Parabolam in vacuo describentis cujus vertex est punctum G, diameter GC et latus rectum

\frac{2 {GH}^{q}}{HK}

\frac{{}^{2}{GT}^{q}, o o, a^{3}}{a a, 2 b b o o} = \frac{{GT}^{q}, a}{b b}

. Et velocitas in puncto quovis G eadem est cum velocitate quarum corpumoris in Parabolam describentis cujus pergerat verticem G, diametrum GC & latus rectum

[\frac{{GT}^{q} \times a}{b b}] \frac{{GT}^{q}}{VH}

\frac{n^{3} + 3 n n + 2 n}{2 a^{n} + 3} b b o^{3} \times \frac{o}{a} \times \frac{a^{n + 2}}{n n + n \times b b o o} = \frac{n + 2}{2 a} o \times \frac{o}{a} GT = \frac{n o^{2} + 2, o o}{2 a^{2}} GF = LM

\frac{3 b b o^{3}, a^{3}}{2 a^{4} b b o o} = \frac{3 GT}{2 a a} o o

. Add. 3968 No 41120 For Isaac newton in German street near St Iames Church I writ to you about a month a goe and fearing that my letter might be not Come safe to your hands makes me so bold as to troubel you with this to lett you know of my misfortuns for I haueing had a great deal of illness which has been uery Chargable to me humbley disire that you will plleas you to asist me in this my great nesesety for my wants is uery great makes me so bold to tro troubel you about a fortnight sence my husband had all his goods seased by the landlord so srsir I humbley disire you that you will be pleased to giue the bearer sumthing for me and she will take care to send it to me my son Thomas marryed her mother srsir Humbley beging the fauer that you will be pleased to Answer this I remain srsir your humble saruant basingthorp oct 19 1714 Katthern Rastall For there is upon all these considerations there is a suspicion that when Mr Newtons Series for finding the Arc whose sine is given was communicated to him in England & that in the year 1673 he began to communicate it as his own to some of his Friends at Paris, & that the next year wrote of it as his own in his Letters to Mr Oldenburgh in order to get the Demonstration or method of finding it. But the next year when Mr Oldenburgh sent him this & Gre the series of Gregory & several others: he dropt his pretence to this any of the S series for want of a Demonstration & took time to compare those series with his own as if he he had others different from those sent him. And when he had found a Demonstration of Gregories series, he began to communicate it to Mr O as his own to his friends & concealed from them the letter by wchwhich he had received it from Mr Oldenburgh & pretended to Mr Oldenburgh that he had it a year or two before the receipt of that Letter And that in his Letters of I Iuly 15 & 16 Octob 26 1674, he mentioned but one series for squaring the Circle, & said that the method wchwhich gave that series gave him also a series for any Arc whose circumference was known sine sine was given tho the proportion th of the Arc to the whole circumf.circumference was not known. What to do upon them, & meeting with a Demonstration of Gregories Series, by a transmutation of figures he supprest Mr Oldenburgh Letter & by vertue of that Demonstration began to communicate to his friends at Paris that Series with a demonst as invented by himself. For he tells us in the Acta Lipsiensia: for April 1671: Iam anno 1675 compositum habebam opusculum. Quadraturæ Arithmeticæ ab illo tempore lectum & A One series he had in the year 1673, & might carry it with him from England to Paris another he began to communicate in the year 1675 The first was for finding the Arc whose sine was given as appears by his Letter of Octob. 26 1674, the second for finding the Arc whose tangent was given as appears by his writings ever since, & it lies upon him to prove that he had this latter series before the year 1675 he received it from Mr Oldenb. in the year 1675 in wchwhich he received it from Mr Oldenburgh. on receiving Mr Newton Series again by one George Mohr, Mr Oldenburgh in such a manner as if he had never seen that series e & desired Mr Oldenburgh to procure from Mr Collins Mr Newtons method of finding it. If Mr Leibnits thinks fit to obviate this suspicion the he is in the first place to prove that he had Gregories series before he received from Mr Oldenburgh. 120 Mr Leibnitz wrote to Dr Wallis in his Letter of 28 May 179971697 th wrote thus to Methodum Fluxionum profundissimi Newtoni cognatam esse methodo meæ differentiali — — professus sum in Actis Eruditorum — Itaque communi nomine designare Soleo Analysesos Infinitesimalis — Interim quemadmodum et Vietæa & Cartesiana methodus Analyseos Speciosæ nomine venit discrimina tamen nonulla supersunt: ita fortasse ut& Newtoniana & Mea differunt in nonnulli. And in the end of the same recconing up the differences or improvemtsimprovements wchwhich he had made to Mr Newtons method, he addsed the Exponential Equations. But these Equations are owing to his correspondence with the English. Dr Wallis insquaring the circle interpoled the indices of Dignities 0. 1. 2. 3. 4. 5. &c with thiese series fractions

\frac{1}{2} . 1; \frac{1}{2} . 2; \frac{1}{2} . 3; \frac{1}{2} . 4; \frac{1}{2}

&c. Mr Newton introduced into his Analytical computations the fract, surd, indefinite & negative indices of Dignities — — its Vsefulness to the world. Mr Newton in his Letter of October 24. 1676, wrote that he had two Methods Here also Mr Leibnitz allows that when Mr Newtons Principles of PhilophyPhilosophy came abroad; he understood thereby the affinity that there was between them methods & therefore called them by the common name of the infinitessimal method, & thought himself bound in candor to acknowledge theis affinity; And & there is still the same obligation upon him to make the in point of candor, to make the same acknowledgement. And But And Mr Leibnitz doth not only here acknowledge the affinity of the Methods but also gives the preference to Mr Newtons method in point of antiquity; representing that the as the Analysis in species was invented by Vieta & augmented by Cartes wchwhich makde some difference between them, so the Mr Newtons method & his own might differ in some things. And this subordination & distinction of the methods which he ynthen acknowledged to Dr Wallis, he ought still to acknoledge. The death of that learned Man doth not acquit him from this obligation. In enumerating the differences which he had added to Mr Newtons method, he names in the last place to D first place the notation dx, ddx, dddx or dx d2x, d3x, dx & this is granted him last place Exponential Equations: but those Equations second place Differential Equations: but the Letters between them in the year 1676 shew that Mr Newton had such equations at that time & Mr Leibnitz had them not. The He names in the third place Exponential Equations: But these eEquations are owing to his correspondence with the English. Dr Wallis in the interpolation of series considered fract & negative indices of dignities. And whereas in his Letter of 28 Decem 1675 he wrote to Mr Oldenburg that he had communicated that series above two years before to his Friends at Paris & written to him sometimes about it, & in his Letter of 12 Maij 1676 said to Mr Oldenburgh that he had communicated writen to him about that Series to him above tw some years before; & in his Letter to Mr Oldenburgh dated 27 Aug. 1676, that he had communicated that series to his friends above three years before that is upon his first coming to Paris from London to Paris:: & yet in his Letters of 15 Iuly & 26 Octob. 1674 spake but of series for squaring the circle & said that the method by wchwhich he found it gave him any whose sine was known & yet he is desired to tell us how it came to pass that when he received Mr Oldenburghs Letter of Apr. 15. 1675 did not he did not know this series to be his own, & w & wrote of it to Mr Oldenburg some years before And since he pretends that he had this series in the year 1673, & in his Letters of 15 Iuly & 26 Octob 1674 wrote of but of one Series for squaring the circle & said of the Method which gave him that series, that it gave him also the a series for finding any arch whose sine was given: it lies upon him to satisfy the world that in the year 1672 he had a method wchwhich gave him both those series, & what was that method, & [why he could not by the same method find the series of Mr Newton] how that Me In his Letters of Iuly 15 & Octob 26 1674, he des tells us that of but one series for the whole circumference of a circle, & saith that the method which gave him this series gave him also a series for any Arc whose sine was given, If the th tho the proportion to of the Arc to the whole circumference be was not known. This method therefore by the sine of 30 degrees gave him a series for the whole circumference. He If he had also a series for the whole circumference deduced from the tangent of 45 degrees he is desired to tell the world what method he had in those days which could give him both those series. For the method by transmutation of fg figures will not do it It lies upon him also to tell the world what m And if in the year 1674 he had the Demonstration of a series for finding any Arch whose sine wasis given: he is desired to tell the world what it was & why in his Letter of May 12. 16756 he desired Mr Oldenburgh to procure from Mr Colling the Demonstration of Mr Newtons series for doing the same thing; & whereing his own series differed from Mr Newton's. For there is a suspicion that the Series for finding the Arch whose sine was given wchwhich he mentioned in his Letters of Iuly 15 & Octob 26 1674 for finding the Arch whose wsine was given was was Mr Newton's & that when he wrote h received Mr Oldenburghs Letter of Apr 15. 1675 he durst not challenge it for want of a Demonstration, & that when he found a Demonstration of Gregories Series he but took time to consider & examin the several Series wchwhich he received in that Letter, And before he would reso Neque expoduntur occultæ qQualitates quia latent earum caussæ, sed quoniam ipsæ finguntur esse ph effectuum manifestorum caussæ occultæ, et ex hoc nomine qualitatibus rerum specificis tanquam caussis phænomenων nondum patentibus imposto deterremur a caussis hisce indagandis quasi quidem illæ penitus essent deploratæ postquam magnus ill ipsæ Philosophus Aristoteles in eas penetrare non poterat. Hoc sensu non ipsa gravitas, sed sola gravitatis causa (de qua Newtonus nihil statuit) dici potest, (secundum Peripateticos), qualitas occulta, secundum Newtonum vero de eadem nihil omnino statuitur gravitas est qualitas manifestissima & Q causa ejus nihil om quæ nos latet potest esse substantia 122 Num. VI, VII, XXII, XLVII, LIII, LVI Iudicio primarij Mathematici opponitur Iudicium Wallisij, anno 1695 ti in Præfatione ad operum suorum Volumen primum, anno 1695 datu editum, sde quo D. Leibnitius per ea tempora minime conquestas est. Eaton bill for 5 hons hong batt — – 0 – 6 – 10 for 3 peckes of Oats – 0 – 2 – 0 for 1 peck half penes – 0 – 2 – 0 for a peck of bran – 0 – 0 – 4 0 – 5 – 2 123 & pretending that after the invention thereof he did not want Mr Newton's extractions. Mr Newton in his Letter of Iune 13, 1676 represented that his methods of series there described in that Letter were not universal without some further methods. And in his Letter of Octob 24. 1676 in describing those further methods wrote that – – – – – thereof & that after he found it he had no further use of Mr Ns Method of extractions. It remains Mr Leibnitz in the Acta Eruditorum for April 1691 pag 179, wrote thus Cæterum ex seriebus infinitis a me alijsque — It lies upon him either to prove that he invented any one of these series, or else to beg pardon publickly for pretending to other mens inventions. 124 3dly We do not dispute about the antiquity of the symbols of fluents fluxions & moments, summs & differences used by Mr Newton & Mr Leibnitz these being not essential necessary to the method, but liable to change. And yet the symbol

\frac{a a}{64 x}

▯ \frac{a a}{64 x}

used by Mr Newton for a fluent or summ, is older then the symbol

\int \frac{a a}{64 x}

used in the same sense by Mr Newton Leibnitz. And some of the symbols of fluxions used by Mr Newton (Some of wchwhich are older as old as his Analysis communicated by Dr Barrow to Mr Collins in the year 1669): back while Mr Leibnitz has no symbols of fluxions to this day. And the rectangles under the fluxions & the Letter o used by Mr Newton for moments are much older then the symbols dx & dy used by Mr Leibnitz for the same quantities. But all these symbols are only ways of notation & signify nothing to yethe method it self, wchwhich may be without them. about such things is downright trifling. Letters wthwith pricks Mr Newton lays no stress upon. He seldome uses them are seldome used by Mr Newton unless where he is teaching how to deduceing fluentsxions from equations involving fluents or fluents from equations involving fluxions. And even in these cases he doth not always use them., The letter o he used when in his but thinks them as convenient as any others BAut whatever letters or symbols he puts for fluxions If o the rectangle under the first fluxion & the letter o he puts for the first fluxion moment called the first difference by Mr Leibnitz & the rectangle under the second fluxion & th

o^{2}

he puts for the second moment & so on. And wherever he puts In this sence he used the letter o in his Analysis communicated to Mr Collins in yethe year 1669 & in his book the Quadratura Curvarum co & uses it in the same sense to this day. And the method by this way of notation is as con advantageous & universal bas by any other. And the method by the letter o as it is the oldest so it is as advantage beautifull as advantageous & as universall abs by the symbols dx & dy. Mr Newton has used the letter o these 45 years & above & still uses it, in the same manner & his method thereby is as beautiful & advantageous as adv advantageous & universall as by any other symbols. the differential & more Geometrical. But where there are equations consisting of many terms (as in the first Proposition of the Quadratura Curvarum) he represents the flowing quantities by the letters x y or z & their Fluxions by the same letters wthwith points or by others equipollent And Mr Leibnitz himself in the Acta Eruditorum has confessed that ( (A1686 pag ) has confessed that instead of the Letters d marks dx & dy he could have used letters. 55000. 88

\frac{- 2 p a θ}{t}, \frac{θ a t}{b r}

2 p r + t t = 2 p r - p p + e e = 4 r r - 2 q r - 4 r r + 4 q r - q q + e e = 2 q r - q q + e e = p q + e e = p q + q q - b b = 2 q r - b b

\begin{array}{r} 1197 \\ 2394 \\ 14365 \\ 287311 (5^{{dw}^{t}} \\ 275 \\ 12311 \\ 49244 \\ 295464 (5^{gr} \\ 275 \\ 20464 \\ 165 \\ 3964 \\ 385 \\ 114 \end{array}

\begin{array}{r} 125 ∟ 372^{gr} \\ 11,39745 \\ 1,42468 \\ 125,372 \\ 123∟947 \\ 129∟25 \\ ∟372 5∟2 \\ 10∟6 . \end{array}

d r = \frac{θ a t}{b r} . \frac{b θ θ a a t t}{b r b r} + b r dd r = - \frac{2 p a θ θ a t}{b r b r t} = \frac{- 2 p a a θ θ}{b b r}

dd r = - \frac{2 p a a θ θ}{b b r r} - \frac{b a a θ θ t t}{b^{3} r^{3}} = + \frac{- 2 p r - t t}{b 2 r 3} a a θ θ = \frac{b b - 2 q r}{b 2 r 3} a^{2} θ^{2}

125 In the Acta Eruditorum for March 1714 pag. 140 & 141, Mr Newton it's said that Mr Newton denies that the cause of gravity is the mechanical, & that he laies down a certain new Hypothesis concerning a subtile spirit pervading the pores of bodies (perhaps the same with the Hylarctick principle of Dr H. More) & this spirit is dsp represented of less value then Hypotheses are unless it be the Æther or subtile matter of the Cartesians. It seems the Secretary of the Editors is whereas And yet Mr Newton has no where declared said denyed that the cause of gravity is not Mechanical [nor declared any opinion about it] nor affirmed whether that subtile spirit be material or immaterial. nor declared any opinion about their names or of w He finds It appears by experiments (lately lately shewed to the R. Society by Mr Hawksby that bodies do attract one another at very small distances in such a sense as he uses the word attraction, & he suspects that this attraction & electrical attraction may be performed by one & the same agent, & L the body constantly attracting constantly at very small distances without friction, & the attraction being extended to great distances by friction; & this Agent he calls a subtile spirit. But what is this Agent & by what laws its actions are performed or spirit & what are the laws by which it acts he leaves to be enquired decided by experiments. [He finds It is found by some experiments shewed before then R. Society that this Agent or spirit when sufficiently agitated emits light & that light in passing by the edges of bodies at small distances from them is inflected; & thence he can has represented in the end of his Principles that light & this spirit may act mutually upon one another for producing causing heat reflexion refraction inflexion & vision, And & that if this spirit may receive impressions from light & convey them into the sensorium & there act upon that substance which sees & thinks, that substance may mutually act upon this spirit for causing animal motion. But these things he only touches upon & leaves them to be further enquired into by experiments. And we should not have touched upon them mentioned all this, had not Mr Leibnitz & some of his friends reflected upon Mr Newton for these things & for supposing these things & for not explaining the causes of gravity & electric attraction by Hypotheses & tby this & other deraction reflexions endeavoured to make the world beleive that Mr Newton in point of in point of philosophy ais much inferior to Mr Leibnits & in point of Mathematicks not only to Mr Leibnits but also to Mr Bernoulli & Mr Tschurnhause & Mr Craige & therefore not likely to be the first inventor of the method of fluxions. AndBut whereas Mr Leibnitz some have told us that gravity is seated in the will of God, & Mr Leibnitz has represented this opinion that according to this opinion gravity is a miracle Mr Newton in his Letter of 24 Octob. 1676 wrote that he had two methods of resolving Inverse Problems of tangents & such like difficult ones, one of wchwhich methods consisted in assuming a Series for any unknown quantity from wchwhich the other unkno all the rest might convenitently be deduced & in the collating of the homologous terms of the resulting Equation for determining the terms of the assumed Series. Mr Leibnitz many years after, published this method as his own, claiming to himself the first invention thereof. It remains that he either renounce this claim publickly, or prove that he invented it before Mr Newton wrote his said Letter. Mr Newton introduced into Analysis the use of fract, surd & indefinite indices of dignities, & in his Letter of 24 Octob. 1676 represented to Mr Leibnitz that his methods extended to the resolution of affected Equations involving tdignities whose Indices were fract or surd. Mr Leibnitz in his his Answer dated 21 Iune 1677 mutually desired M. Newton to tell him what he thought of the resolution of Equations inolving Dignities whose indices were undetermined, such as were these

x^{y} + y^{x} = x y

x^{x} + y^{y} = x + y

. And these Equations he now calls exponential, & has & representesd to yethe world that he was the first inventor of exponential equations there But for the future, whenever thereof. thereof, & magnifies the invention as a great discovery. But he has not yet made a publick acknowledgment that of the light which Mr Newton gave him light into the invention it, nor told us theany use of æquations involving dignities whose indices are fluents thereof where the indices of dignities are fluents. One of their statutes is that if any Fellow of the Society, shall contemptuously or contumaciously disobey the Statutes or Orders of the Society, or shall by speaking writing or printing publickly defame the Society or advisedly or malitiously do any thing to the damage & detriment of the Society thereof, He shall be ejected out of the Society. And therefore it lies upon Mr Leibnitz to beg the pardon of the Society for defaming him, least they put this statute in execution. To the HonbleHonourable Isaac Newton these Add 3968 No 41126 SrSir Your HonrsHonours knowne vnbounded goodness, a Singular Patron & fauorer to all schollars & Learned men, is my sole motive for enterprizeing this small present to your HonrHonour. a Rude & vnpollisht Booke, Begotten in spaine, & brought forth in England. I shall be extremly obliged to your HonrsHonours fauor in countenanceing yethe same, haueing onely a sett N. assigned me for yethe performance, wchwhich I hope to dispose off to those worthy Gent.nGentlemen my Benefactors, who did me yethe fauor, to accept my Translation, yethe Booke of Comm̄onCommon Prayer into spanish, my Indigent circumstances did not permit me to vndertake it on my owne account, I am in all Duty to Command SrSir YrYour most Obed.tObedient servant Felix De Alvarado S.rSir Be pleased to Cast an Eye thereon. there being therein, vncommon Varietie I shall not trouble yoryour Hon.rHonour any more on any accotaccount. p. 12. ✝ The APostill snot being directed to M sent to Mr Newton he did not think it proper to answer return an Answer meddle with it till M l'Abbé Conti sollicited him to write an Answer that the Postill with his Answer might be shewed to the King. P. 71. Mr Newton opbserving that as Mr Leibnitz sent first had appealed from the Commimttee of the R. S. to h Mr I. Bernoullej, so he had & so he now sent his Apostill as well to Paris as to Londōon & therein boasted of his happiness in disciples & had now again sent all the Letters to Paris that his Answer might be read there first & sent from thence to London: & yet pretended that & in his Letters declined the & that de to put a stop to these indirect practises, refused to write write an Answer to be sent to him & only drew answer Mr Leibnitz any further by Letters & only drew up the following Rematrques & shewed them privatsly to some of his friends to satisfy that it was easy to have returned an Answer had it been right to let him go on with his politicks 127 In the following papers you will find whe in the year 1669 Mr Newton in the year 1669 had a method of series wchwhich gave the Areas & lengths of Curves (&c exactly & Geometrically if it might be: & that by the testimony [pag. lin of Dr Barrow & Mr Collins he had this method some years before, that is in the year 1666 or before &. This method is explained & illustrated with examplsexamples in Mr Newtons Letter to M of 242 Oct 1676, in binomials & in binomial equ Ordinates in Curves wthwith binomalbinomial Ordinates, & said to extentd to trinomials &c. And in the fift Proposition of the Book of Quadratures it is extended to explained in a general equation extending to all the N extending to all those cases; & that Proposition involves the knowledge of the four preceding Propositions; & therefore by the testimony of Dr Barrow & Mr Collins, the Method of fluxions, so far at least as it is conteined in the first five Proposititons of the book of Quadratures, was known to Mr Newton in the year 1666 or before And Dr Wallis attested the same thing was is attested by Dr Wallis in the Preface to the second Volume of his works A.C. 1695 & gave Mr Leibnitz notice thereof by a Letter, & Mr Leibnitz did not then deny it nor complain. of In his Letter of 21 Iune 1677 he was searching after the method of wchwhich Mr Newton had given him notice. In yethe year 1684 he published the elements of the Method he had found by that search & acknowledged a Methodus similis wchwhich might do the same things. A The next year but one Mr Newton demonstrated the synthe the elements of his method synthetically in the second Lemma of the second book of his Principles, & in a Scholium represented that Mr Newton called it the inverse method of fluxions above 40 years ago since & you have given it a new name as if you had been the first inventor. He preten After he had been in England the first time where he might easily meet with some of Newtons & Gregories Series he first pretended to have found the series for the Arc by the sine. Then he he relinquished that pretence & pretended to have found the series for the arc by the tangent, which was Gregories then p the d he wrote for an Ent & wrote for Mr Newtons method of finding some of his finding the former series, Then And upon the death of Gregory he wrote also for Co a Collection of Gregorys what Gregory had sent to Collins about these thingthings & received it wchwhich was sent to him. Then he came again into England & consulted Mr Collins aabout these matters & Mr Collins shewed him what he desired his p what h the originals. And as he wrote for Mr N the method of series wchwhich Mr Newton had sent to Collins, so no dev that is, for for the method contained in the Analysis per æquationes numero terminorum infinitas, & desired it so much as to offer a reward for it, so he d had now an opportunity to see the original, And yet this candid Gentleman would never make any publik acknowledgment that pag. 6. Mr Newton's method inverse method of fluxions you have baptized by a new name. I know that at the request of Mr Leibnitz a collection of Gregories papers was draw sent to him in the year 1676 & re by him & that in this collection was a copy of Mr Gregory Letter of 1671 February 1671, & wherein he then sent this & many other series to Mr Collins, & therefore Mr Leibnitz knew that Mr Gregory invented this series in or before the year 1671. I know that Mr Leibnitz soon after coming to London consulted Mr Collins to hade seen seen these Papers or received any light into these things from England. two Let me add Also that two years after Mr Newtons Book of Principles came abroad, he Mr Leibnitz published a great number of his Propositions in other words & pretended to have found them him self, & pretended to have found thethem himself & yet the put an erroneous demonstration to by wchwhich he pretended to have found the chief of them as if he had found it by that demonstration to mak is an Erroneous one. After Mr Newton To gave himself a litle to that PropsitionProposition he tried to adapted a Demonstration to it, but the Demonstration proves an erroneusous one, & the error arises from his want of skill in the Method which he improperly calls differential. After Mr Newton – – – 130 To SrSir Isaack Newton Master of the Mint at his house in Leicester-street Feb: 7th: 1777. Lord Portesmouth. Tale quiddam Gregorius habuisse, multi extranei agnoverunt agnoverunt ex epistolais Oldenburgi series ad Leibnitium et Leibni mittentis 15 Apr & 1675 epistola autore, a Leibnitij ad OldenburgurgumOldenburgum series ab Oldenburg serie se receptas fuisse agnosentis 20 Maij subsequentis: multi inquāam extranei hac id agnoverunt; collatis scilicet his Epistolis in Commercio editis cum monunumentismonumentis antiquis ex quibus edebantur. Et in horum numero erant Comes de Kilmanseggar, Abas de Comitibus & ministri aliquot publici externorum Principum. Et Rex Angliæ Leibnitius ipse agnovit in Epistola ad D. Ramond 15 Aug. 1676 scripsit scripsit quod Rex Angliæ ad ipsum Le Roi m'a fact la grace de dire ici, que l'Abbé Conti viendra un jour in Allem Allemagne pour me convertir. Et posthc in Epistola sua ad Dominam Kilmansegger 18 Apr. 1716. M. Oldenburg [26 Octob. 1674] m'ecrivit qu'un M. Newton a Cambridge avoit deja donné des choses semblables, non seulement sur le sCercle mais encore sur toutes sortes d'autres figures scilicet in epistola 26 Octob 1674] et [15 Apr 1675] m'enevoya des essais. [scilicet in Epistola 15 Apr 1675] Scripsit utique Oldenburgus 26 Octob 16754 Newtonum dedis similes series & 15 Apr. 1675 misit specimina. Cependent le mien fuit sat assez aplandi per M. Newton même sc. in Epist 13 Iun 1676.. Il sest trové par après, qu'un nommé M. Gregory avoit trové justment la même series qu'e moi. Mais c'est que j'appris tard. Hæ Epistolæ extant in commercio Epistolico, et in illa 15 Apr. 1675 in qua Oldenburgus misit Leibnitio des essays specimina serierum erat ill series Gregorij quam Leibni de qua agitur. Hanc seriem L a Leipti a Leibnitio postea anno 1676 acceptam Newtonus laudavit nescius per ea tempora quod Leibnitusius eandem acceperat O ab Oldenburgo, Accepit autem bis: primo in Epistola prædicta deinde in exemplari Epistolæ excerptis ex Gregorij ad Collinium qu inter e Epistolais ad ipsum missis 26 Iunij 1676. Et subinde eandem Newtono misit ut suam 27 Aug. 1676 et post annos sex eandem editdit ut suam in Actis Eruditorum celatio omni suo commercio quod habuerat cum Oldenburgo. Et jam dicit se tart conatur se excusare dicendo quod tande agnovit hanc seriem a Gregorio inventam esse p. Hanc Certain But in cases of controversy its against the law of all nations to admit admit any man to be a witness for himselft, & therefore his saying that he had found it above three years before must go for nothing nog g not be taken in evidence It lies upon him to prove it. And if he had found it so long ago yet Gregory had sent it to Mr Collins before that time & Collins & had begun to communicate it to others before that time. And if he had found it before that time yet the series is extant in Mr Newton's Analysis. pag. 6, lin. ult. And to give himself a char And to title to it, it lies upon him to prove that he invented it before the month of Iuly 16969. All that he has done By his Theoreme of Transmutations He taught how to reduce the Area of a circle to area of a curve whose whose Ordinate is

x + \frac{1}{1 + x x}

Abscissa is x & Ordinate

\frac{1}{1 + x x} = y

. The squaring of this Curve by the series

x - \frac{1}{3} x^{3} + \frac{1}{5} x^{5} - \frac{1}{7} x^{7} + \frac{1}{9} x^{9} - &c

was not his invention. Mr N Communicated it to Mr Collins in the said Analysis. pag 6, His lin ult. pag 6 lin ult. His words are: Eodem modo si sit

\frac{1}{1 + x x} = y

, dividendo prodibit

y = 1 - x^{2} + x^{4} - x^{6} + x^{8}

, &c Vnde (per Regulam secundam) erit [area]

ABDC = x - \frac{1}{3} x^{3} + \frac{1}{5} x^{5} - \frac{1}{7} x^{7} + \frac{1}{9} x^{9}

, &c. And after all this evidence Mr Newton Leibnits 132 He s After Mr Newton at his request had After he had commended Mr Newtons series & desired the method of finding them & he & h Mr Newton had gratified him in his request he & his friend Tschurnhause should not have Endeavoured to blow upon it as if it proc produced series only per accidens & more simple & general principles of reducing Series were to be expected, & pretended to other general methods of reducing quantities to infinite series. & claim He should not have claimed some of Mr Newtons reciprocal series & in the same Letter have dired desired Mr Newton to send him the his method of finding reciprocal series & when the method was sent him & he understood it not & wrote again to have it explained, he should not, as soon as he understood it, have written back to Mr Oldenburgh that by his he found perceived by his old papers that head found it before but laid it aside for want of a good example of its use. He should not have published Mr Gregories series in the Acta Leipsica as his own without letting the world know that he received that Series from Mr O & Mr C. & that Mr G. sent it to Mr C. in the year beginning of yethe year 1671. He should not at yethe same time have amused the world with a harmony of series without letting them know that one of those series was nothing else then this the half of this 1

- \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{11}

theanother nothing else then the half of this

\frac{1}{2} - \frac{1}{4} + \frac{1}{4} - \frac{1}{6} + \frac{1}{6} - \frac{1}{8} + \frac{1}{8} - \frac{1}{10} + \frac{1}{10} - \frac{1}{12}

&c. a third nothing else then the summ of these two & that as you may perceive by subducting every other negative term from yethe term preceding. He should not have published his met differential method in the Acta Leipsica without aknowledging that Mr Newton had a method of Newtonus Multiplicationes Divisiones & Extractiones radicum Newtonus p commun ratione per immutabiles directa per immutabiles dignitatum indices notavit, Leibnitius proposuit indices mutabiles. Vide supra, p. 95. temp. temp.∷R.

\sqrt{R R + 3 R S o} = R + \frac{3}{2} S o

g C = L C

1 + \frac{3 S o}{2 R} = o \sqrt{1 + Q Q} - \frac{2 R o o}{\sqrt{1 + Q^{2}}} + \frac{3500}{\sqrt{1 + Q Q}} + \frac{3500 \sqrt{}}{2 R}

- C G = o \sqrt{1 + Q Q} + \frac{R Q o o}{\sqrt{1 + Q Q}}

c G - C G = \frac{3 S o o \sqrt{}}{2 R} - \frac{2 Q R o o}{\sqrt{}}

\begin{array}{r} P - \frac{1}{2} Q o - \frac{1}{4} R o o - \frac{1}{8} S o^{3} \\ P - Q o - R o o - S o^{3} \\ \frac{1}{4} R o o + \frac{3}{8} S o^{3} \\ \frac{1}{4} R o o - \frac{3}{8} S o^{3} \end{array}

2 R + 3 S o

2 R - 3 S o

2 R + 3 S o ∷ {}^{▯}{temp.} {}^{▯}{temp.}

4 R - 3 S o

4 R + 3 S o ∷ temp. temp. ∷ 4 R . 4 R + 6 S o ∷ \begin{array}{r} R . R + \frac{3 S o}{2} . \\ 1 . 1 + \frac{3 S o}{2 R} . \end{array}

2 R - 3 S o . \sqrt{4 R R - 9 S S o o} = 2 R - \frac{95500}{4 R} = 2 R

2 R . 2 R + 3 S o ∷ 1 . \frac{3 S o}{2 R} .

1.1 + \frac{3 S o}{2 R} ∷ L C = o \sqrt{1 + Q^{q}} - \frac{Q R o o}{\sqrt{1 + Q^{q}}} . g C = o \sqrt{} - \frac{Q R o o}{\sqrt{}} + \frac{3 S o o}{2 R \sqrt{}} - \frac{3 Q S o^{3}}{2 \sqrt{}}

C G = o \sqrt{} + \frac{Q R o o}{\sqrt{}} . g C - C G = \frac{3500}{2 R \sqrt{}} - \frac{2 Q R o o}{\sqrt{}}

. Add

1 + \frac{390}{f}

o \sqrt{} - \frac{e f o o}{\sqrt{}}

- o \sqrt{1 + e e} - \frac{e f o o}{\sqrt{}} + \frac{2 f o o, e o}{o \sqrt{}} = \mp \frac{2 e f o o}{\sqrt{}} + \frac{3900 \sqrt{}}{f} - e

\frac{3 g \sqrt{1 + e e}}{f}

ad 2f, seu

3 g \sqrt{1 + e e}

ad 2ff. 133 And At that time Dr Barrow represented tha to Mr Collins that by this & other papers wchwhich the author had communicated to him by the Author before befo before it appeared that Mr Newton had invented this method & applied it universally some years before. Thales The ancentancient calendar year of the Greeks consisted therefore of twelve Lunar months & every month consisted each of thirty days & these years & months they corrected from time to time by the courses of yethe Sun & Moon, omitting a day or two in the month as often as they found the month of 30 days too long for the course of yethe Moon & adding a month to yethe year as often as they found the twelve lunar months too short for the return of the four seasons. This the ancient Gr Hence Thales called the last day of the month the thirtith. And to the And in allusion to thise ancient calendar year Cleobulus one of the seven wise men alluded had reference in his Parable of one father who had 12 sons each of wchwhich had 30 daughters half black & half white. And according to the number of days in this Calendar year, Demetrius Phalareus had 360 statues erected to him by the Athenians, & Cyrus cut the river Giridus into 360 Channels, & the Egyptians recconed their months to consist of 30 days & their year of twelve such months & five additional days. And in the scriptures the three years & an half re of Daniel are by Iohn said to be 42 months & 1260 days. The ancient Greeks intercaled a month every 134 p. 2. l. 7. ad sexdecim et amplius post . l. 8. Progressus vero quos jam fuerat Newtonus p. 3. l 18, Iones ad fidem exemplaris a se reperti supr inter Collinij chartas, Collinijque manu exarati, postquam exemplaris fidem ad Autographum — l. 38 Ordinata. l. 39, Area. p. 5. l. 4. cujusvis Curvæ, ejusdem Curvæ — l. 9. In hoc Compendio suo. l. 11 referentis per rectangulum sub unitate, et litera o comprehensum, adest vel (quod perinde est) per literam o; subintellectia unitate subintellecta scilicet unitate; aliarumque— b Neque ante jam p. 13. l. 323. transcribendam, eamque in manibus Collinij D. Leibnitius vidit antequam Londino excederet. p. 67. l 13 in lucem emitti curavit. lin 34 dele [Et Collinio]. lin 35 excolens. Anno proximo Geometriam altiorem didicitere cœpit erudiente ad Viro Celeberrimo et ejusdem anni mense Iulio rescripsit ad Oldenburgum mirando quodam se potitum esse — p 8. l 16. singulari. Per series suas quas ab acceptis hic distinguit, & ante annos aliquot adeoque antequam Geometriam aliteorem discere cœperat via quadam singulari inventas prædicat intelligere videtur series jam ante descriptas numerorum fractorum quibus denominatores sunt numeri triangulares vel pyramidales vel triangul-triangulares aut similes. Et quanquam D. Leibnitius setentiam suam de seriebus acceptis sententiam D. Oldenburgo perscribere hic promiserat: cavit tamen posthæc — l. 18 paucis discesserimus si a numericis jam memoratis discesserimus. l. 19 Dele [quas ad – – – – – conciemasse.] p. 9 l 9 Quorum votis annuens Newtonus p. 10. l 8. 1675, et annum fere antequam Geometreiam sublimiorem addiscere cœperat. p. 13 l 9 de Analysi sua (ceu compendio methodi serierum) compendio. l. 33 transcribendam, et Collinius eandem cum Leibnitio communicabat quàm antequam ille p. 18. l. 27 Et Leibnitio cujus candorēem in dubium vocare injustum esset, privilegium concedebatur, candorem aliorum pro lubitu in dubium sine absque injuria vocarendi: Newtonum vero oportebat 135 p. 3. l. 5 Dele a votre Maitre Leibnits. &. Vide p. 33. l. 3 dévoré P. 40 l. 12. for imposturues write P. 41. l. 12, 13. Plagiaries. read you who can so little defend him your self from the imputation it. P. 42. P. 46. l. 12. Dele, d'alleurs vous dites que et scribe dsisant qu'il ne vouloit Pag 6. l. 11. & p. 7. 11 for l'assembleé write le Commis the Committee Vide Pag. 21. lin 7. 8. Pag. 44. l. 1, 2. add, except that where D unless in telling Dr Wallis that the Letters wchwhich had passed between him & Mr Oldenburg were either lost or mislaid in a heap of papers P 7. l 2. par vos amis & attributed to you P 8. l. 9. And therefore you may do well to clear your self from being the author of the abusive Synopsis of the Book of Quadratures published in the Acta Eruditorum for Ian. 1705, wchwhich I has occasioned all this dispute. P. 11. l. 10. add. since you have passed your judgmtjudgment upon it. Pag. 51. lin 17, 18 &c dele Il eest vray — — leur Virginityé. Pag 50 lin 10. — — au Mr Leibnitz. N'y dit il pas — &c. Quære. Add. 3968 No 41136 SrSir Isaac NwtonNewton SIR, YOU are hereby Desired and Summoned to Meet the rest of your Brethren, Members of the Society of the City of London, of and for the Mines, the Mineral, and the Battery Works, at the Mineral-Office-General in Black-fryers, at 2 of the Clock in the after noon on thursday next being the 8th Day of this instant Month of October Anno Dom. 1713. Per Cur' H. H. Stringer Serjeant. Dated at the Mineral-Office-General the 6th Day of October Proposuerat Newtonus resolutionem æquationum ubi indices dignitatum sunt fractæ vel surdæ quantitates, p 86: D. Leibnitius vicissim proponit resolutionem æquationum ubi indices dignitatum sunt quantitates indefinitæ. In hac Analysi quantitates fluentes per symbola quæcunque ut v, x, y z et earum fluxiones per symbola alia quæcunque aitut V X Y Z & vel p q r s & momenta per fluxiones in quantitatem momenta parvam coefficientēem aliquam perparvam o ductas op ut op, oq, or os designantur, p et finita et ubi problema ad æquationem dertur & æquatio reducitur, quantitas o et finita Analysi quantitas o diminuitur in infinitum et evanescit [Et hujusmodi symbolæas Newtonus pro momentis mox denotatri possunt Dein comutam peragitur per Geometriam igerem ut mox fit ex demonstratione Regulæ primæ pro momentis utitur mox utitur demonstrando Regulam primam ut et in Tractatu de Quadratura Curvarum demonstrando Propositionēem primam. [At hic symbolum o subintelligit, et momenta denotat per symbola sola fluxionum symbola subintellecta subintelligendo cofficiente o, & pro fluxione quantitatis u fortiter fluentis usurpat quantitat s et ejus momento ponit unitatem]. ] Sed maxima ex parte coefficiens o plerumque subintelligitur & momenta sola eadem symbola pro fluxionibus symbola pro & momentis etiam ponuntur promiscue usurpantur et pro fluxione unitas ptitur tam pro momento quantitatis uniformiter fluentis et quam pro ejus moment pissime fluxione ponitur, ut hic fit ubi 1 ponitur pro momento BK. Huic patet resolutionem dignitatis indefinitæ binomij cujusvis New in seriem infinitam Newtono jam tum innotuisse: quæ operatio resolutiones reductione line quantitatum omnium in s per multiplicationes divisiones vel et extractiones radicum non affectarum simul in series infinitas, comprehendit. Certe D. Leibnitius Anno 1676 negavit Problemata inversæ tangentium methodi tangentium inversæ et multa alia ab æquationibus aut Quadraturis pendere Certe cum Newtonus (Anno 1676) dixisset aAnalysin suam admodum generalem esse (p. 55,) 56 ad omnia pene problemata sese extendere (p. 55, 56) D. Leibnitius respondit Id sibi non videri; esse enim multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturis; Qualia sunt (ex multis alijs) Problemata methodi Tangentium inversæ. Deinde vero, a Newtono admonitas didicit hujusmodi problemata ab æquationibus pendere. Et & quadraturis pendere: et hujusmodi fundamentum methodi Newtoni Analysim illa generalem Newtoni generalem illam Newtoni Analysin in hujusmodi æquationibus fundari statim vidit & erit intellexit. Vide pag 90. lin 26, 27, 28, 29. Annon D. Leibnitius a Newtono admonitus in hujusmodi æquationes incidit. Vide pag. 30, l. 11. & pag 47. lin 16. & pag 55 lin 29 & 56 lin 1 & pag 65 l 14. pag 72. l. 14. p 85 l 32 & p. 88. l 14. & p. 90 l. 26, 27, 28, 29.

\begin{array}{r} 999736 \\ 264) 10000000 (3800 \\ 792 - 1212 \\ 20800 3787∟88 \\ 2112 \\ - 320 3788 \\ 264 \\ 56 3787 \\ 528 \\ 32 \end{array}

137 The Answer of Mr Tchschurnhause to this paper was received by Mr Oldenbeurgh the 8th Iune 1675, & entituled Responsum ac scriptum Dn Collinij de Cartesij inventis.

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D. Leibnitius series pro Hyperbola & circulo quarūum reciprocæ sunt elegantes, ante quadrennium annos plure habuisse dicitur, series plures reciprocas ante biennium ab Oldenburgo accepit, methodum serierum reciprocarum hoc anno superiore a Newtono postulaveritet, & hoc anno acceptam ægre intellexit, & eandem tamem primus inveniterat: Et quamvis eadem hæc methodus ex arcu daret sinum et tangentem, ex logarithmo daret numerum et serierum omnium daret exhiberet reciprocas, eandem tamen neglexiterat ut inutilem. idque ante annum 1673 quadre. ② D Leibnitius se [ante annos plures seriem habuit Mercatores pro Hyperbola, & hujus seriei reciproca est elegantissima exhibuit Series etiam pro circulo se habuisse dixit quarūum reciprocæ solunt essu eles. Et exemplo tamen eleganti serierum reciprocarum caruit]. ① Series pelures reciprocas ante biennium ab Oldenburgo accepit,erat & Methodum serierum reciprocarum anno superiore a Newtono postulaveratitpostulaverat, hoc anno accepit acceptam ægre intellexierat: & eandem tamen prius invenerat prius. ③ Et quamvis series haberet ante annos plures pro Hyperbola et Circulo et hæc methodus ex arcu daret sinum rectum sinum versum & tangentem, & cotangentem & secantem & ex logarithmo daret numerum & serierum omnium exhiberet reciprocas, eandem tamen olim inventam neglexerat ut inutilem. ægre intellexierat & tam demum intellectam mox didicit se olim invenisse ex chartis antiquis mox didicit ex chartis antiquis didicit se hanc methodum olim invenisse. Et quamvis series haberet ante annos plures pro Hyperbola et circulo et hæc methodus Four penny puncheons 15li, three penny 132

\frac{3}{4}

l two penny 9

\frac{1}{2}

l. Penny 76li. 1. Total 44li. 3 Quamvis D: Leibnitius series reciprocas ante biennium ab Oldenburgo acceperat methodum serierum reciprocarum anno Nam superiore a Newtono postulaverat & & hoc anno acceptam ægre intellexerat tamen et eandem tamen priusprimus invenerat. Anno 1675 Series plures reciprocas ab Oldenburgo acceperitat anno 1675 & unius earum am ut elegantiam singularem habentis Demonstrationem anno proximo ab Oldenburgo postula bavitur. biennio p. 45. Seriem pro Quadratura circuli quam anno 1675 cum Gallis ut suam communicavibat, id est Gregorianam, sibi anno 1673 sibi notam fuisse scripsit (p 45). ‡ sed methodo caruit inveniendi elegentem hujus reciprocam caruit. Toto fere quadrennio D. Leibnitius methodum serierum reciprocarum desiderabat & tamen prius invenerat & se invenisse oblitus fuerat. & huius reciproca satis elegans est (p 25 & 41) sed Leibnitio diu ignota. Toto triennio fere quadrennio Liebnitius methodum serierum reciproctarum desiderabat & tamen prius invenerat, inventionis oblitus fuerat & se invenisse oblitus fuerat. 1. Rogatur D. Leibnitius ut hanc ἑνθυνσιν p communicet. 2. M De methodo generali ex methodis serierum & fluxionum composita Newtonus hi sequen loquitur in sequentibus p 14, 15, 18, 30. 55, 56, 71 85, 86. Anno crediderit D. Leibnitius [methodum negligenda et obliviori tradendā esse quas serierum] serierum omnium reciprocas esse in elegantes? Anne] sed methodum rejiciendam negligendam neglexerat esse quæ Et quamvis ex arcu daret sinum et serierum omniūum reciprocas exueret ex logarithmo numerum & serierum omnium exhiberet reciprocas? anne serierum omnium reciprocas esse ineligantes? eandem tamen neglexerat ut inutilem. 138 What is said in the following letters & in a Letter of Mr Collins to Mr Newton printed by Dr Wallis concerning the method of tangents being known to Slusius Sihen before he printed his Mesolabium was grownded upon a mistake. The letter of Mr Collins dated 18th Iune 1673 runs thus Quod ad Slusij Methodum de Tangentibus spectat: Erat ea ab ipso bene intellecta quum suum de Mesolatio librum edidit, sed noluit tunc publici juris facere ed quod nollet Amico suo Angelo Riccio prævenire. Qui tamen postea declinans ipse studia Mathematica petebat a Slusio scribere, pollicitus est eam D. Oldenburgo transmittere ut Transactionibus Philosophicis inseretur. Ante vero quam huc appulerit scribebam Ego ad Te ut intelligerem quid ea de re tu noveris. Tuumque responsum cum D. Oldenburgo communicabam, ut ipse D. Slusio transmitteret: ut sciat ipse, rem eam esse apud Anglos congitam; utut forte non tam diu neque tam mature ut ipsi fuerit. The Riccius published his Exercitatio Geometrica de Maximis & Minimis of Riccius was published at Rome in yethe year 1666 & two years after reprinted in England, & by his third Theoreme for determining Maxima & Minima he teachs how to draw Tangents to some curves. And Slusius in the fourth chapter of his Miscellanies published wthwith his Mesolabium in yethe year 1668, sets down the Proposition Theorem of Riccius & makes this note upon it. Liceret hujus Propositionis usum prolixius extendere, ad determinandas nempe maximas & minimas applicatarum in Curvis, t angentes & similia Verum cum hanc materiam nuper, in eExerciatione sua Geometrica, feliciter aggressus sit Vir Clarissimus Michael Angelus Riccius, doctrina & humanitate singulari, orbi literato notissimus; & justi operis spem faciat: frustra nunc pluribus insisterem cum meliora ac perfectiora ab ipso propediem expectari debeant. And this is that method of Tangens wchwhich Slusius when he wrote published his Mesolabium which reliquished to his friend Riccius & wchwhich Riccius afterwards desired Slusius to publish & Slusius thereupon promised to send to Mr Oldenburg to be published in the Transactions. But the method wchwhich Slusius sent afterwards to Mr Oldenburgh to be published in the Transactions was very different from this, & doth not flow from the Theorem of Riccius & doth not flow & floweth not easily either from the Theorem published before by Slu Riccius & Slusius or from the three Lemmas upon from wchwhich Slusius pretended to derive it. So Upon notic This year Mr Leibnitz Upon notice that Mr Iames Gregory was dead, Mr Le desired th Letters might be collected & sent to Paris [& about at the same time he desired that Mr Oldenburg would send him the Demonstration of my series that is the method of & told them & told him that Mr Collins could help him to it, & therefore he had headrd that Mr Collins had my method, that is my Analysis per series numero terminorum infinitas. For I sent to to Mr Collins no other Method of Series then that] & they were collected & sent accordingly, & among them wasere copies of MrGregories Letters of 5 Sept 1670 & 19 Decem. 16 mine of 10 Decem. 1672 & by these Letters Mr Leibnitz had notice that Mr Barrows method of Tangents was capable of improvement so as to give the method of Tangents of Gregory & Slusius & that thise Method of Gregory & Slusius might be improved so as to give my general method of Analysis & that this method Analysis proceeded without sticking at surds, & that I had interwoven it with the method of series, vizt in a Tract which I wrote upon this subject in the year 1671. Mr Leibnitz wrote also for t to Mr Oldenburg for the Demonstration of some of my Series that is the method of finding them & told him that Mr Collins could help him to it, & therefore knew that Mr Collins had my mMethod, that is my Analysis per series numero terminorum infinitas; for I sent Mr Collins my method in no other Paper then that. [And in October following Mr Leibn. coming to London saw in the hands of Mr Collins many of mine & Mr Gregories Letters especially those relating to series. He saw also my Letter of 24 Octob. 1676 & met meth with Dr Barrows Letctures] At the same time he received also my Letter of 13 Iune 1676 in wchwhich I represented the Art of that Analysis by my methods of Series & some other methods (meaning principally the method of fluxions) became so general as to extend to almost all Problemes except perhaps some numeral ones like those of Diophantus. & And in October following he came to London & consulting Mr Collins saw in his hands many of mine & Gregories Letters especially those relating to series. He saw also my Letter of 24 Octob. 1676 wherein he had further notice of these things, & while he was in London he met with Dr Barrows Lectures, & in going in his way from London to Hannover was meditating how to improve the method of Tangents of Slusius, as I gather from his Letter of

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Novem 1676 writen from Amsterdam to Mr Oldenburg

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Novem. 1676. There was also at the same time a Paper an abusive Harang printed agains Dr Keill in the Actia Eruditorum for Iuly 1676 without the name of the author. But The Author by calling a formula of Mr I. Bernoulli formulam meam has father it upon that writer but I hear that Mr Bernoulli denies that he wrote it, & I doo not think such papers fit to be answered. [There are some matters of fact not yet fully stated. & I give you leave I have set them down in the inclosed paper, & give you leave to l publish it.] For avoiding answering any o such Letters & Papers & ridding my hands of ths matter I have herewith sent you a general relation of the matter of fact, & give you leave to print it, & every body to think of it what they please. For I do not think the matter of so much consequence that I should concern my self about it any further. In my Analysis per æquationes numero terminorum infinitas I said that by this Analysis Curvarum areæ & longitudines &c (id modo fiat) exacte et geometrice determinantur: sed ista narrandi non est locus. And Mr Collins in his Letter to Mr Strode above mentioned, said thias by this Analysis & other things communicated before to Dr Barrow it appeared that I knew this method be some time years before the Logarithmotechnia of Mercator came abroad so as to find the area of any figure accurately if it may be or at least infinitely neare by approximation in infinitum. And in my Letter of 24 Octob. 1676 I represented that the Quadrature of Curves was improved by the method of fluxions & I had thereby by that Method found some general Theoremes for that end & there set down one of those Theoremes & illustrated it with examples. And in the six first Propositions of the Book of Quadratures I shewed how such Theorems were to be found by that Method, & therefore that Method so far as it is conteined in the first six Propositions of the Book of Quadratures but even before Oat the writing Mr Leib was known to me before I wrote the said Leatter & before I wrote the said Analysis & before Mercators Logarithmotechnia came abroad, there being no other Method then that conteined in those Proposi six Propositions by wchwhich ch such Theorems could be found. 1. — before Mr Leibnitz understood the differential Method. For he never pretended to have did not began to study the higher Geometry before summer soon a a little after the Horologium oscillatorūum of Mr Hugens came abroad wchwhich was in April 1673, & never pretended to have had the differential method before the year 1676, & when he wrote his Letter of 27 Aug. 1676 he placed the perfection of Analysis not in the Differential Calculus as he did after he found it, but in a Method founded on Analytical Tables of Tangents & the Combinatory Art. Nihil est, saith he quod norim in tota Analysi momenti majoris. And a little after: Ea vero non differt ab Analysi illa SVPREMA ad cujus intima Cartesius non pervenit. Est enim ad eam constituendam opus Alphabeto cogitationūum humanorum. 2. In October follow 1676 he went from Paris to London & there met with Dr Barrows Lectures & saw my Letter of 24 October 1676. And Mr Iames Bernoulli in the Acta Eruditorum for December 1691 pag. 14, said that the Calculus of Mr Leibnitz – – — – – – – – he never claimed a right to it. 5 In writing the Book of Principles I made much use of the method of fluxions direct & inverse, but did not set down the calculations in the book it self because the book was written by the method of composition as all Geometry ought to be. And this Book was the first specimen made publick of the use of this method in the difficulter Problemes. This book came abroad in Spring 1687 & in the Acta Eruditorum for Ianuary 1689 Mr Leibnitz published a Schediasma de resistentia Medij & motu projectorum gravium in Medio resistente & in the end of it added Et fortassis attente consideranti vias quasdam novas vel certe satis antea impeditas apperuisse videbimur. Omnia autem respondent nostræ Analysi infinitorum, hoc est, calculo summarum & differentiarū (cujus elementa quædam in his Actis dedimus) communibus quoad verbis licuit verbis hic expresso. This paper was And this was the second specimen of the use made public of the use of this method in the difficulter Problemes. And yet it was nothing else than the two first sections of the second book of Principles reduced into another method order & form of words. 6 At the request of Dr Wallis — — — — necessary to the Method. 4 In the year 1684 Mr Leibnitz published the Elements of the differential Calculus as his own without making any mention of his former correspondence with Mr Oldenburg about these matters. He mentioned indeed a Methodus similis, but whose that Method was & what he knew of it he did not say. And this put me upon a necessity of writing the Scholium upon the second Lemma of the second Book of Principles least I should seem to have borrowed that Lemma from Mr Lei 3 The Marquess de l'Hospital in the Introduction Preface to his Analysis said that Mr Leibnis had done me justice in the Iournal des Scavans du 30 Aoust 1694 vizt in acknowledging that I found the method proprio Marte. But the Marquess did not then know that I was the first inventor. 7 If it be asked why I did not publish – – – – till the year 1704 And for the same reason I intend to meddle with this matter no furthe more. The inverse Method of fluxions is capable of great improvements, & the improvements are his who finds them out. 141 It has been represented in Germany that Mr Leibnitz first found out yethe differentiall method calculus in numbers & then by inventing the Analysis of infinitesimals translated this calculus to Geometry. And indeed Mr Leibnitz in the beginning of the year 1673 pretended to But this is to tell us that he found out the differential calculus of Newto Mouton [But this is to tell us that he derived his differential calculus in infinitesimals from Moutons differential calculus in numbers] For In the beginning of the year 1673 Mr Leibnitz was reprehended by Dr Pell for pretendinged to the invention of Mouton & he pretended to the invention of this differential calculus in numbers & was reprehended told by Dr Pell that it was Moutons method & acknowledged wtwhat Dr Pell said to be true, but & yet defended himself by saying that he found it apart without knowing what Mouton had done before. [And it seems that under this pretense it still passes in Germany for the method of Mr Leibnitz.] But if Mouton was the first inventor he had the sole right to the invention till Mr Leibnitz also found it out & he that has once the sole right to an invention is not afterwards to be deprived of his rig of any part of his right by coninventors sharing it with co afterwards with coinventors. The Editors of the Acta LiEruditorum mense Iunii Octob. anni 1685 pag 298 483 tell us that the quadrature of yethe circle by the series

1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}

&c was found out by Mr Leibnitz ten years before or above that is in the year 1675 before the month of October. And Mr Leibnitz tells us in the Acta Erudit. of the next year mense Iunio pag 298, 299, that Mr he found out this Quadrature presently soon after he began to understand Algebra, but was still forced to do many things by the ambages of figures wchwhich he would have done by the compendium of Analysis untill at length he found out his differential calculus. And in a letter to Dr Wallis dated 28 May 1697 he A tells us that the consideration of edifferences & summs in series of numbers gave him the first light into it, by observing that Differences answered to tangents & summs to quadratures: that is, it put upon upon taking into considering Dr Barrows differential method of Tangents & MDr OWallis's summatory methods of Quadratures. And this was some time after the year 1675. But this consideration alone could not give him light into the method of drawing Tangents without sticking at surds nor into the manner of makin universal extent of this method for solving all sorts of Problemes. That light he had from Mr Newtons Letters as is manifest by what has be said above. by his correspondence wthwith Mr Oldenburg by as is manifest by what has been said above. By that correspondence he received the series of Gregory, & learnt that Mr Newton had a general method, which ga of solving Problems without sticking at surds wchwhich method gave yethe method of tangents of Slusius as an obvious corollary & determined the curvities, areas; convex sufacessurfaces; solid contents, & centers of gravity, & of figures & by the conditions of tangents mutually gave the figures &c. And & for squaring of curves gave converging series wchwhich brake off & became finite as often as the curve might be squared by by a finite equation;, And & gave rules for comparing of Figures with the areas of the conick sections. A This put Mr Leibnitz upon considering how to improve the method of Slusius so as to make it universal as is manifest by his Letter to Mr Oldenburg dated at Amsterdam Nov

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1676. And when he began to see how the differential methods of Barrow & Slusius might be improved, he began where Barrow left off as the Marquess de L'Hospital has observed, & to make himself a coinventor changed the notation of Dr Barrow & wrote back pretended to have found the method long before writing back to Mr Oldenburgh: Clarissimi Slusij methodum tangentium nondum esse absolutam Celeberrimo Newtono assentior Et jam a multo tempore rem Tangentium generalius tractavi; scilicet per differentiat Ordinatarum: [And so then proceeded in that letter to describe this new method as his own & compare it wthwith Newtons as his own invented jam tum a multo tempore,] though it be certain that he was but then beginning to understand it. For when he wrote his Letter of 27 Aug. 1676, had t & therein represented that he did not beleive that Mr Newtons Methods were so general as he represented them & added: Sunt enim multa usque adeo mira et implexa ut neque ab æquationibus pendeant neque ex quadraturis: qualia sunt, ex multis alijs, Problemata methodi tangentium inversæ its most certain that he understood nothing of theis matter. He saith indeed in his Letter to Dr Sloan in his Letter dated 29 Decem. 1711 Inventum plusquam in nonum in annum pressi, ut nemo me præcurrisse quæri possit. By wchwhich words he would have us beleive that he found it before October 1675 & by consequence before the correspondence wchwhich he had wthwith Mr Newton by means of Mr Oldenburg. But he is not a witness in his own cause. It lies upon him to prove it. He had Moutons differential method before th October 1675: but the method wchwhich he now calls the diff differential method now in dispute wchwhich t he had not when he wrote his Letter before the letter end when he wrote his Letter of 27 Aug 1676 & there is great reaso if he affirms that he had it before Iune May 1677 it lies upon him to prove his assertion the May 1677, it lies upon him to prove his assertion. It is not enough t to to say that he first found the differential method first in numbers The Differential method in numbers is not his. Mouton found it before him; T It s & it is but of narrow extent & little use. The question is about the universal general method of Analysis as now called usu called the method of moments & fluxions by Mr Newton & the method of infinitesimals & indivisibles by Mr Leibnitz. 142 To SrSir Isaac Newton ☉ And aA man who knows that he is but the second inventor & endeavours to diminish the right of the first inventor is a Plagiary. And whereas he suppo represents that he supprest the Invention above nine years that no body else might pretend to been before him & by consequence had it in the year 1675 it lies upon him to prove that he had it in the year when he had it wrote his Letter to Mr Oldenburg dated 27 Aug. 27 1676, wherein he affirmed affirmed that Problemes of the inverse method of Tangents & many others could not be reduced to Infinite Series nor depended on Equations or Quadratures. And whereas he further represents that he made all that delay that no body who might pretend to have been before him in it complain of his pretending to step in before them But we in England give no right to second Inventors. The first Inventor has the sole right – . . . . doth afterward. And since he has said (in his Letter to Mr Dr Sloan dated 29 Dec. 1711) that he supprest the Method above nine years, & by consequence before he published it, & it follows from thence that he had it in the year 1675, it lies upon him to prove that he had it in the year 1675 when he wrote his Letter of 2 to Mr Oldenburgh dated 27 Aug. 1676, wherin he affirmed that Problems of yethe inverse method of Tangents & many others could not be reduced to Infinite Series nor depended on Equations or Quadratures. And whereas in the same Letter he has appealed to the judgment of Mr O Hugenius as a most intelligent & most uncorrupt judge: he is desired to tell the world whether he believes in his conscience that Mr Hugenius ever heard the cause between both parties or was made acquainted with the commerce between the English & Mr Leibnitz by Letters while he staid at Paris. It lies upon him also to tell the world what was the series mentioned in his Letter to Mr Old. dated 26 Oct. 1674 method by which he found he found a ser series by which he found any Arc of a circle whose sine was given, & what was as he mentions in his Letter of & what was the method by wchwhich he found that series, & could not find the series of Mr Newton the Demonstration of wchwhich he desired Mr Oldenburg (12 May 1676 to procure from Mr Collins. 142 It lies upon him also to tell the world what was the method, by wchwhich he found the series of regression for the circle & Hyperbola sent to him by Mr Newton 13 Iune 1676 before he claimed them from Mr Newton inby his Letter of 27 Aug. 167 following. And since he has told us that his friends know how he came by the Differential method; it lies upon him to let the world know how he came by it & at what time. he But since he com has complaiend of the R. Society for giving judgment without hearing both parties, he is desired to tell the world whether he beleives in his conscience that Mr Hugenius to whom he appeals as a most intelligent & most uncorrupt judge ever heard the cause between both parties or was made acquainted with the commerce what had passed between the English & Mr Leibnitz by Letters during his stay at Paris between the years 1672 & the years 1677. 143

+

And Mr Leibnitz himself in a letter to Mr Newton dated to Mr Newton dated from Hanover

\frac{7}{17}

March 1693 & still extant in his own hand writing, acknowledges the same thing in these words: Mirifice ampliaveras Geometriam tuis seriebus, sed edito Principiorum opere ostendisti patere tibi etiam quæ analysi receptæ non subsunt. Conatus sum ego quoque notis commodis adhibitis quæ differentias et summas exhibent, Geometriam illam quam transcendentem appello, Analysi quodammodo subjeicere, nec res male processit. ☉ Dr Wallis had received copies of Mr newtons two Letters of 13 Iune & 24 Octob. 1676 from Mr OldendenburghOldenburgh & published many things out of them in the second volume of his Works in his Algebra printed in English 1683 & printed in Latin A.C. 1693. And in the third Volume of his works p soon after upon noti h soon after had notice intimation from Holland to print yethe Letters intire because Mr Newtons method notions of fluxions passed there wthwith applause under by the name of the Differential Method of Mr Leibnitz, & thereupon took notice of this matter in the Preface to his first Volume published two years after the second & in a letter dated April. 10th 1695 he wrote thus to Mr Newton about it. I wish you would print the two large Letters of Iune & August [He means Iune & October.] 1676. I had intimation from Holland, as desired there by your friends, that somewhat of that kind were down; because yoryour Notions (of Fluxions) pass there with great applause, by the name of Leibnitz's Calculus Differentialis. I had this intimation when all but part of the Preface to this Volume was printed off; so that I could only insert (while the Press stayd) that short intimation thereof wchwhich you there find. You are not so kind to your reputation (& that of the nation) as you might be when you let things of worth lye by by you so long, till others carry away the Reputation that is due to you. I have endeavoured to do you justice in that point; and am now sorry that I did not print those two Letters verbatim. ♀ ♀ And in a wchwhich Letter to Mr Leibnitz dated 1 Decem. 1676, he mentions the same thing Dr Wallis thus excuses his inserting the said Paragraph into his Preface without making a further mention of the Differential calculus. Calculi Neque Calcul Differentialis vel nomen audivisse me non memini nisi postquam utrumque Volumen absolverant operæ, eratque Præfationis (præfigendæ) postremum folium sub pPrelo ejusque typos jam posuerant Typothetæ. Quippe tum me monuerat amicus quidam harum rerum gnarus qui peregre fuerat, tum talem methodum in Belgio prædicari, tum illam cum Newtoni methodo fluxionum quasi coincidere. Quod fecit ut (transmotis typis jam positis) id monitum inserueritm. And in the Letter which followed hereupon – – – – Mr Leibnitz at that time. By wchwhich it may appear that it was then a received notion in England that Mr Newtōon had found the method of fluxions ten years before the said correspondence between him & Mr Leibnitz or above. The shordt Intimation of this matter wchwhich Dr Wallis inserted into yethe Preface of his first Volume was in these words. In secundo Volumine, (inter alia) habitur – – – – – – – nihil a nobis dictum esse. Hereupon the Editors of the Acta Lips. – – – – of to Mr Leibnitz at that time. And in the year 1699 Dr Wallis upon publishing thise third Volume of his works by the leave of Mr Leibnitz inserted into yethe 3d Volume of his works the said two Letters together wthwith the three Answers of Mr Leibniz found in the Library of Mr Collins & dated 27 Aug. 1676 & & 21 Iune 1677 & 24 Octob 1676 12 Iuly 1677 together with some other L & found in the Library of Mr Collins. In the same year (A.C. 1699) Mr Fatio, in his Dissertation on yethe Line of the quickest descent, suggested that Mr Leibnitz . . . . . before Ianuary 1705

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144 Videamus jam qua facile fuerit ex haoc methodoi Tangentium et generalis pexempla methodo Tangentium Barrovij collato methodum ipsam generalem deducere Methodus vero generalis ex hoc ejus specimeine vel exemplo sic deducitur Et hinc Wallisius in Præfatione ad Volumen secundum Wallisij operum suorum dixit quod Newtonus Literis suis 13 Iunij & 24 Octob. (scribendo quod Methodus sua prompte daret Methodum Tangentium Slusij generalis prompte et absque æquationum reductione daret methodum Tangentium Slusij & methodūum suam Leibnitio explicuisset. & methodum Tangentium Slusij inde fluere dixisse idque absque æquationum reductione dixisset Leibnitius utique in itinere suo per Hollandiam ad de ex Gallia Anglia in Germaniam Methodo tangentium Slusij amplianda cogitabat, ut ex Epistola ejus ad Oldenburgum Amstelodami

\frac{18}{28}

Novem. data tmanifestum eest. eEt proximo mense Iunij rescripsit Clarissimi Slusij Methodum Tangentium nondum esse absolutam Celeberrimo Newtono assentior. Deinde in eadem Epistola descripsit methodum tangentium Barrovij et quomodo ex hac methodo Barrovij Methodus Tangentium Slusij deduci posset, & quomdodo methodus tangentium Slusiujs ad æquationes sic inventa exhiberi potssestposset ubi plures sunt quantitates literæ indeterminatæ quam duæ (quod sæpe fit maximo cum fructum) et eamæ ratione (methodum irrationales non morari procedere non sublatis irrationalibus. quod fit maxima cum fructu Hoc totum est quod Leibnitius anno 1677 rescripsit & cum Hoc totum est quod Leibnitius ea tempore rdescripsit. Et hoc totum ego Epistola Newtoni 1672 ferille consequi jam ostensum Et hæc sunt vestigia [Viderit itaque Lector annon Newtonus Literis suis methodum Leibnitio expolicuit Leibnitius Newtonus methodum habuerit anno 1672 & Leibnitius annois 1676 & 1677 postea ex Lite, Newtoni aliquid didicerit lucem aliquam acceperit] Viderit itaque Lector annon Leibnitius Newtonus methodum habuiterit anno 1672 & b & annon Leibnitius (qui ex tempore anno 1673 & 1674 altiorem Geometriam non didiceritit cœpit discere) post annos quatuor vel quinque ex Newtoni Litteris lucem aliquam acceperit. Et horum vestigijs institit ubi rescripsit Idem consequitur ex epistolais Newtoni 12 13 Iunij & 24 Octob. 1676 data anno 1676 scripto si modo cum methodo Tangentium Slusij conferantur, ut Wallisius olim notavit in Præfatione ad Operum suorum Volumen primum notavit. Qu. 1. Whether is it right to set a president for discouraging Manufactures by taxing them? Qu. 2. Whether is it worth the while to do this for a tax scarce exceeding twelve or fifteen thousand pounds prper an' Qu. 3. Whether, after yethe LdsLords have rejected a Bill for lowereing the fineness of Plate, is it right to tack this to a money bill, & will not the Qu LdsLords reject it. Qu. 4. Whether some of the GoldsmthsGoldsmiths should not be consulted about the plurality of Diets, standard & the marks thereof.

\begin{array}{r} 4268 \\ 17072 \\ 1846∟9 \\ 210∟7 \end{array}

Et eodem temp & auctoritate Bernoullij amicos suos harum rerum mathematicarum inscios contra Newtonum passim ciere conatus est: cum tamen Bernoullius Eodem spectat etiam quod in Epistola Newtoni ad Oldenburgiūum 24 Octob. 1676 data, extat Catalogam destciptæ exiant Ordinatæ — cum conicis sectionibus referuntur in INewtonus in Epistola sua ad Oldenburgum 24 Octob 1676 data, ubi solutionem Problema aliquot generaliam nominasset subjungit: Fundamentum harum operationum, satis obvium OBVIVM quidem quoniam jam non possum explicattionem ejus prosequie sic potius celavi & 6accdæ &c. Celavit ut obvium ne subriperetur. Quam vero sit fuit obvium & quam facile subripi potuit, sic ostendetu patebit. Iacobus Gregorius scripsit ad Collinium. Et Quæstio est Quæritur, non quis methodum totam invenit, (nam tota nondum inventa est sed quis methodum invenit quatenus in scriptis a NewtoNewtono editis habetur. Qui aliquid addiderit is jus habetbit in partes Theo in Regulas additas in partes additas. Setd et L. ipse anno proximo in Epistola sua ad Con. de Kilm 18 Apr data & a Des Maizeaux edita idem agnovit narrando quod cum ipse per E Oldenburgio scriberet O de serie scriberet quam pro circulo invenerat ad Oldenburgium scriberet m (viz per Epistolas 15 Iulij & 26 Octob. 1674;] Oldenburgius responderet (vizt 8 Decem 1674) quod Newtonus Can quidem Cantabrigiensis jam antea dedisset similia dedisset non solum pro circulo sed etiam pro omni figurarum aliarum genere & ipsi misisset [me envoya & ipsi mitteret des essais,] ipsi mittessret mitteret serierum specimina. Hæc sp His verbis Leibnitius Hoc non obstante, ait, series mea satis laudata fuit p per Newtonum ipsum. Il est trove par apres qu' un nomme Postea inventum est quendam creatum D. Gregorium q quendam eandem etiam seriem mecum invenisse: sed hoc didici tarde. Hæc Leibnitius. Literas utique multa fruge Algebraica refertas acceperat, sed tunc præter ordinarias curas Mechanicis imprimis negotijs distractus, non potuit examinare series quas Oldenburgius miserat, ac cum suis compare: ut ipse tunc rescripsit. Et Newtonus Wallisius & junior Gregorius autem hanc seriem a Gregorio ad Collinium & ab Oldenburgio ad Leibnitium fuisse missam, per ea tempora ignorabat Hæc epistola in Galliam prius missa fuit quam in Angliam veniret, [et [Newtonus qui hoc nesciebat a Comite de Comitibus confusus est ut eidem responderet] et eidem Epistolæ subjunctūum erat Problema Bernoullij Bernoullij quod Leibnitius olim proposierat D. Fatio in Actis Eruitarum et solvendum, et [Iam proposuit Analystis Anglis vero Leibniuj postulabat ab Abbate ut is suo nomine proponeret idem Analystis Anglis. ✝ ✝ et præterea EdemEodem tempore LeibisLeibnitius Epistolam volantur Iudicis Mathematici jam edi curavit in Hollandia tanquāam [jam cœpit amicis suis scribere quod judex ille Mathematicus erat ipse Bernoullius & Ej poistolam vol] a Bernoullio scriptam, et amiscis subinde cœpit mox scribere quod Bernoullius erat ejusdem auctor. Sic Quæstio dirimenda erat authoritate Iudicis Bernoullij contra et Iudicis constituti & Problemata proponentis ab Analystis Anglis solvenda. Newtonus autem qui ignorabat Epistolam ad Abbatem cum hoc Problemate per Galliam in Angliam venisse, impulsus inductus fuit ab Abbate ut eidem responderet et Re- Epistola ad Abbatem 26 Feb 171

\frac{5}{6}

data scripsit quod Consessus R. Societatis Hoc artificium per Newtono innotuit anno 1676 uti patet per Epistolam ejus & eo anno 24 Octob ad Oldenburgum scriptam, immo et anno 1669 ut affirmatur Newtonus autem impulsus est ab Abbate et ejus amicis qui ignorabat Epistolam cum hoc Problemate per Galliam in Angliam venisse impulsus fuit ab ComtAbb te &c eut eidem responderet et respōonso suo ad Abbatem 26 Feb 171

\frac{5}{6}

dato scripsit quod Concessus 145 1 Motum harmonicum Planetarum Assumpsit, dest Artic 76 Tentaminis (in Tentaminis artic non probavit Vide Tentaminis Artic. 6. 2. Vim centrifugam simi verso angulorum s assumpsit semper proportionalem esse Lebn. pitatem assumpsit Artic 11 Artic 12 assumpsit contra veritatem: et inde 12. & et ex hac assumptione errante & excessa vis centrifugæ supra vim centripetalē (qui nullus est) etiam assumpta per In calculo etiam errasse volent aliqui. Et Propositionis mathematiæ sic vir veniri non solent circulationis Motum paracentrium Planetarum assumpsit a a differentia virium centrifugarum & centripetarum (quæ nullæ est) oriri de invvit oriri fixit Artic 15, 21, 25. In calculo errase volent aliquo. Et Propositiones mathematicæ sic inveniri non solent. Et Ex errantibus Articulis 12 et 15 deduxit Artic 19. In l EAt Propositiones Mathematicæ sic inveniri non solent Ita privatim ut publice Quæritur unde habuit & quando primum habuit Methodum differentialem Moutoni Leibnitius habuit anno 1673 & suam esse voluit. Methodum aliam differentialem nondum habuit. Series postea habuit sed quas anno 1675 ab Oldenburgo accepit, ab A alijs prius accipere potuisset. Methodum generalem perveniendi ad ejusmodi series anno proximo ab Oldenburgo petijt, a Newtonao accepit, antea non habuit. Methodum extrahendi radices in speciebus a Newtono simul accept qua transmutatio figurarum in methodum quandam generalem evasit, sed inutilem. Per extractiones solas res citius peragitur. Anno 1677 methodum novam habuit differentialem habuit sed quæritur unde habuit. Et quando primum habuit sed Newtonus methodum infinitessimalem ante hunc annum in Analysi et Literis suis abun descripserat ac tantam esse methodi hujus antiquitatem Editores eam jactantur gts majores non asseritet Leibnitius ipse non asserit non asserant Editores jactant, majorem non asserunt. 3 Inde deduxit vim centrifugum mobilis harmonice circulantis esse in ratione radiorum reciproca trilplicata (Artic 12). Newtonus autem demonstravit hanc vim mobilis in st Ellipsis circa focum harmonice circulantis esse ratione radiorum reciproce duplicata. Par Motum paracentricum . . . . . . (quæ nulla est) Leibnitius oriri finxit (in Artic 15, 21, 25.) Ext ex Propositionibus errantibus Articulis 12 et 15 deduxit Artic. 19 Certo Crenda sunt hæc et Propositiones Mathematicæ sic inveniri non solent. Vortices alicubi harmonice alibi non harmonice moveri pro lubitu finxit. Vide Acta Lips. Ann. 17 pag Harmonicæ circulatitonis obstant Vortices P satellitum Saturni Iovis ac Terræ. Motum Cometarum Leibnitius non attingit, Cometa Planetæ et Cometæ ijsdem legibus vahi Newtonus detit revolvuntur apud Newtonum. Vortices alicubi harmonice alibi non harmonice moveri Leibnitius I pro lubitu finxit: harmon prohit harmonicæ circulationi obstant vortices Satellitum Saturni Iovis ac Terræ Harmonice Motibus Cometarum Vortices non favent.. Motum harmonicum Planetarum Leibnitius non probavit sed in Tentaminis Artic 6 ex Princi assumpsit. a on probavit: Newtonus motum harmonicum corporum omnium in centrum immotum attractorum demonstravit. Motum circulationis & motum paracentricum Leibnitius nunc a diversis causis deducit nunc ab eadem deducit Newtonums utrumque ab eadem causa (pro simplicitate naturæ semper deducit. Vim centrifugam . . . . a A relatione Differentiarum ad Tangentes Leibnitius se methodum differentialem habuisse hic fatetur, id est a methodo determinandi tangents per differentias Archimedes Fermatius Gregorius, Barrowus & Newtonus methodūum differentialemrum ad Tangentes applicueraent Newtonus hanc methodum ducendi tangentes aux eritat & præterea ad Quadraturas Curvarum & alia multa problematūum genera applicando, applicuit & valde generalem reddideritat, & idem significaveritat in Epistolis suis ad Collinium A. C 1672 & Oldenburgum A.C. 167 1676. Scripseritat mox mox Leibnitius methodum inversam tangentium in potestate esse ab æquationibus et quadraturis non pendere & responderitat L Newtonus hanc methodum hanc inversam ex potestate esse hanc inversam tangentium etiam in potestate esse. Quibus omnibus int se collatis admonitas. Leibnitius in eandem methodum incidit, ut ex verbis ejus hic positis cum epistola ejus ad Oldenburgum anno 1677 scripta collatis man petit manifestum est. Vide supra pag. b Nulla est hujusmodi analogia Differentiæ non sunt differentiæ summarum Leibnitiums primam lucem a consideratione, tangentium habuit per Newtonum auctæ ab hac methodi aliunde habuit.

16 \times 438; \frac{1}{2} + 46

\begin{array}{r} 1754 \\ 7016 \\ 46 \\ 15) & 7062 & (470; \frac{4}{5} \\ 106 \\ 105 \\ 12 \end{array}

\begin{array}{r} 15) 7068 (471; \frac{1}{5} \\ 106 \\ 105 \\ 18 \end{array}

438 \times 16 + 52

\begin{array}{r} 1752 \\ 7008 \\ 52 \\ 15) 7060 (470; \frac{2}{3} \\ 1060 \\ 105 \\ 10 \end{array}

e + \frac{a o}{e} - \frac{n n o o}{2 e 3}) a o - o o (\frac{a o}{e} - \frac{a a o o}{e 3} - a a - \frac{a a o o}{e e} + \frac{n n a o^{3}}{2 e^{4}} + \frac{a a o o}{e^{2}} + \frac{a^{3} o^{3}}{e^{4}}

\frac{a o}{e} - \frac{a a o o}{e 3} + \frac{a o^{3}}{2 e^{3}} + \frac{3 a^{3} o^{3}}{2 e^{5}}

\frac{a^{3} o^{3}}{2 e 4} + \frac{e e a o^{3}}{2 e 4} + \frac{2 a^{3} o^{3}}{2 e 4}

e + \frac{a o}{e} - \frac{n n o o}{2 e 3}) a o - o o + ✱ (\frac{a o}{e} - \frac{n n o o}{e 3} + \frac{3 a n n o^{3}}{2 e 5} - a o - \frac{a a o o}{e e} + \frac{a n n o^{3}}{2 e 4} + \frac{a n n o^{3}}{e^{4}} + 3 a n n o^{3}

\frac{a o}{e} - \frac{n n o o}{2 e^{3}} + \frac{a n n o^{3}}{2 e^{5}} + \frac{n n o o}{2 e^{3}} - \frac{a n n o^{3}}{2 e^{5}}

\frac{a n^{3} o^{4}}{e^{6}} \times \frac{e^{6}}{n 4 o 4} = \frac{a}{n}

- \frac{a}{n}

\frac{a o}{e} \times \frac{e}{n o}

+ \frac{a}{n}

146 * Quasi Leibnitius hoc non advertisset anno 1677, ubi primum incidit in methodum Newtoni. Vide Literas ejus supra impressas p. 90, 91. Certe Methodum Newtoni ante annum 1671 inventam fuisse, Leibnitius ex Literis ejus intellexerat; sed in Actis Lipsientibus hoc nunquam agnovit. Vide supra p. 70, 71, 72. Sic et se ab Oldenburgo series Newtonianas et Gregorianas ab Oldenburgo accepisse in eunte anno 1675 accepisse, statim oblitus est. Vide p. 40, 41, 42, 45. Et etiam methodum serierum se ab Oldenburgo postulasse & a Newtono accepisse, statum oblitus est Vide p. 45, 62, 98. Et problemata tangentium inversa & similia ab æquationibus et quadraturis pendere se primum negasse & subinde a Newtono didicisse, statim oblitus est Vide p 65, 85, 86, 93. ✝ Methodum Fluxionum ✝ Methodum fluxionum & methodum Differentialem esse unam et eandem methodum Leibnitius hic agnoscit; ideoque se communi nomine del a se designari Analyseos Infinitesimalis a se sæpe designarie solere, licet in nonnullis differre possint, ut Analysis speciosa Dietæ & Analysis eæ Cartesij in nonnullis differunt. Quæritur quis sit Analyseos hujus Infinitesimalis inventor primus & siquid alter alterius inventis addiderit. ** Methodus Fatetur hic Leibnitius igitur Tangentiusm per Differentias primam lucem ipsi affudisse Leibnitio (vide p. 887, 88) id est methodums Archimedis a Fermatio Gregorio Barrovio restitutam & promotam, a Newtonao ad quantitatum augmenta momentanea generaliter applicatam. Hujus methodi exempla in problemate tangentium directo (p 30) & inverso (p 86) in Quadraturis (p 72) & in ex ils ( Methodo serierum (p 15, 18, 19) Newtonus dederat, e Ex his D. Leibnitius statim ab initio deprehendit methodum suam Newtoniana similem esse similitudinem differentialem methodorum pp. 90, 91, 93. Et his admonitus D. Leibnitius in methodum tangentium animum intendit (p 46, 347, 87 88) & differentias in methodum methodum differentiarum generalem formavit (p. 88 89) & Newtonianæ similem esse Newtoniana similem esse statim deprehendit ab initio vidit, p. 88 89 90, 91, 93. ** Vidit hoc Fermatius antea determinando punctum flexus Contrarij. ** Vidit ho Fermatius determinando punctum flexus contrarij, hoc antea vidit. ‖ Nulla est hæc analogia. Differentiæ non sunt summarum differentiæ sed augmenta infinite ‖ Nulla est hæc analogia. Differentiæ non sunt summarum differentiæ sed quantitatūum augmenta infinite parva. Hanc methodum tangentium Leibnitius animo revolvit (p 46, 47, 87, 88) auget generalem reddit (p 88, 89) & Newtonianæ similem esse statim videt, p. 90, 931, 93. Fermatij Gregorij Barrovij a Newtonao ad quantitatum augmenta momentarea generaliter applicatam. Hanc Methodum tangentium . . . . hoc est methodum quam Archimedes, Fermatius, Gregorius Barrovius coluere, Newtonus ad quantitatum augmenta momentanea generaliter applicuit. Hancce tangentium methodum Leibnitius animo revolvit (p 46, 47, 87, 88) generalem reddit (p 88 89 & Newtonianæ similem esse statim vidit p 90, 91, 93. & in Mercatoris Quadraturæ demonstrata anno 1668 a Gregorio editam fuit a Gregorio prius primus editam anno 1668 in Mercatoris quam etiam Gregorius edidit anno 1668 in Mercatoris Quadratura demonstrata Prop. V.) of as general extent for solving the same sorts of problems, found in or before the year 1671 & that Mr Newtons method was the older being found in or before yethe year 1671. He should not afterwards have pretended that when he published the elements of his differential method th he knew nothing further of Mr Newtons method then that it was a method of drawing tangents without sticking at surds: but when his Principia Philosophiæ came abroad he then began to understand that his method was much more general. He should not two years after this book the Principia came abroad, [three years after it was sent to yethe R. Society to be printed & six ye abo almost six ye above five years after the principal of those Propositions were communicated to yethe R. S. ] have published three papers for making himself the inventor of the principal of those Propositions & the first that had opened new ways of Geometry? For Mathematicians say that he hasd added nothing of truth to Mr Newtons Propositions & that in his that his thre papers are of no use but to make himself an inventor & that he has derived some of Mr Newtons Propositions from precarius Pr Hypotheses & erroneous calculations instead of inventing them. He When he undertook to give an Account of Mr Newton's book of Quadratures in the Acta Leipsica thehe should not have given an AcctAccount of his own method instead of Mr Newtons. He should not have represented his own method the older & Mr Newtons a substitute for he knew by his correspondence wthwith Mr O. & by Mr Newtons the letters published & interpreted by Dr Wallis that Mr Newton had the Method of Fluents in the year 1671. 147 Mr Leibnitz blames me for calling the Author of the Letter of 7 Iune 1713 inse of 7 Iune 1713 tells me that I knew the author of yethe Letter of 7 Iune 1713 inserted into yethe defamatory paper but because I did not & blames me for calling him a Mathematician or pretended Mathematician — but because I did not know whether he was Mr Iohn Bernoulli or was only pretended to be him him. For that Mathematicaian Mr Leibnitz blames me for calling that the Mathematician whose Letter was as if I 148 & understanding Mr Newtons method of by the same Letters that the method of tangents printed by Slusius was w agreed wthwith Mr Newtons & was a Corollary of this same general method Mr Newtons general method (p. 30) he set his mind upon improving this method of tangents so as to bring it to a general method And For in his journey home by London from Paris by London & Amsterdam he he was considering ho upon a project of extending it to the solution of all sorts of problems by calculating a Table certain Table of Tangents as the most easy & useful method way he could then think of (pag 87) & wrote of this designe to Mr Oldenburg in a Letter dated at Amsterdam 28 Nov. 1676 (pag 87) & therefore he had not then invented the differential method but was endeavouring to find out such a general method as Mr N. had described.. In his last letter against Mr Keil (p 119) he represents that his friends know how he found this method in a very different manner from what . . . . . . . & then laid aside upon his finding that method p. 42. Now Mr Newton had told him that his method extended to Tangents of Mechanical Curves & to Quadratures Curvities & centers of gravity & of Curves in general & to inverse problemes of Tangents, & h & he was thereby sufficiently informed that the method was founded upon the consideration of the infinitesimal small particles of quantity called infinitesimals by particles augmentas momentanea & moments by Mr Newton & infinitesimals indivisibles & Differences by Mr Leibnitz. And For there is no other way of handing resolving any of those Prob sorts of Problems but by then by considering these small parts particles of quantyquantity. And this consideration [especially after the receipt of Mr Newtons second Letter, was sufficient to make him lay aside his designe of calculating Tables of tangtstangents &] think upon answering might make him begin to think upon the methods of Slusius Barrow Gregory & Fermat who drew tangents by the proportion of the augmenta of the ordinates to yethe augmenta of the Abscissas. For he tells us that he found out the differential method by considering how to draw tangents by the differences of the Ordinates p. 88 & how thereby to e render the Method of Slusius (wchwhich was more general (p. 88.) He & considered that as the summs of the Ordinates gave the Area so their differences gave the tangents & thence received the first light into the Differential method (p. 104) And wthwith Slusius he gave the name of Differences to the moments of dignities. And when he had found the method he saw that it answered to the description of wchwhich Mr Newton had given of his method to in drawing of Tangents & that wthwithout sticking at surds re in rendring Problems of Quadratures more easy, & in bringing of Problemes of Tangents to Equations & Quadratures p 88, 89, 90, 81, 93. In yethe year 1669 notice was sent to Slusius of Mr Newton's general having a general method of drawin solving Problems, wchwhich in drawing of Tangents extended even to mechanical curves p 21.. In yethe year 1676, a description of that method wthwith an example of it in drawing of tangents was sent to Leibnits, p. 30, 47. And Thise exampl method published by Slusius agreed wthwith this example. And Mr Leibnitz in seeking after Mr Newtons method had this mind upon improving the method of Slusius into a general method. & Mr Newton in his Letter of 24 Sept 1676 told him ytthat his method in drawing of Tangents agreed wthwith that of Slusius but was more general. While he was in pusuitpursuit of Mr Newton's general method by improving the method of of Tan. printed by Slusius, it is reasonable to beleive that he would have recourse to the three Lemmas wchwhich S upon wchwhich Slusius had founded thisat method. of T it. The two first of those Lemmas are these. 1 Differentia In his last letter against Mr Keil (p. 119) he represents that his friends know how he found the differential method in a very different manner from what Mr Keil surmised, & these words imply that he found it after he got home to his friends & by consequence in the year 1677,. And most probably he found it upon the receipt of Mr Newton second Letter which gave him new light into it. For the publishing an opusculum opusculum upon Mr Grego a Series of Mr Gregory sent him by Mr Oldenburg was first suspended by his coming into business & then laid aside upon his finding the differential method p. 42. Archimedes began the method of squaring curves & drawing tangents to them by considering the infinitesimals of quantity. Cavallerius & Fermat applied this method to Equations. Fermats method was first published by Herigon & Schooten Gregory Barrow & Slusius improved it for Tangents. Newton about the same time made it general, extending it (in conjunction wthwith the method of series) to yethe solution of all sorts of difficulter Problemes [& notice thereof was sent to sever Gregory Slusius & others in the year 1669] Leibnitz [& Gregory Slusius & & other had notice thereof in the year 1669] Leibnits was in conjunction with the Method of Series. Leibnitz Collins, Gregory, Slusius & others had notice of this general method in the year 1669. Leibnitz was in London in the year 1673 & there claimed the Differential method of Mouton. At that time there when there was not a methematitionmathematician of note in that city who was not acquainted wthwith Collins & had not heard of the method & seen some of the series produced by it. In London Leib. claimed the Differential method of Mouton was reprehend for it by Dr Pell, & yet persisted in his claim because he had found it apart & improved it. A After he was & After Leibnitz he was gone thence to Paris he began to put in for first inventor of two series for the circle () said they were both found by the same method (p ) & yet wrote afterwards for the Demonstration or method of finding one of them (p. .) [In seeking after the method he first fell upon subducted the negation terms from the preceding affirmative terms in these & such like progressions

1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5}

1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9}

&c & thereby came to these & suchlike series

\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} &c = 1

\frac{2}{3} + \frac{2}{15} + \frac{2}{35} + \frac{2}{63} &c = 1

.] Then he took into consideredation the transmutations of figures, & & fo like those & to the transmutations of Mr Gregory & Ba Barrow added a transmutation wchwhich by the Mr in the year in In April 1675 Collins & Oldenburg sent him eight series found by this method, & he then knoew none of them to be his own, or durst at least did not think it adviseable then to own any of them, but said he took time to consider them & compare them wthwith his own & wro yet this same year he communicated the last of them ✝ as his own to his friends at Paris as his own who knew not that he had received them from London.. The next winter spring one Mohr shewed him two series wchwhich he had from Collins & ynthen Leibnitz wrote to Oldenburg for to procure him the Demonstration from Collins promising them a recompence & this he did without putting them in mind to ask Mr Newtons leave., These tho by his own rule if he ahad got the method Demonstration, that is, Mr Newtons method of Series, & claimed it as his own, Mr Oldenburg & Mr Collins were not to contradict him without authority from Mr Newton. Those two series were two of the eight the same wthwith the two first of the eight sent him by Mr Oldenberg, but he pretended to have forgot the receipt of those, & not to have seen the two before Mr Mohr brought them to him. Mr Oldenburgh & Mr Collins upon condsidering the matter, declined sending him sothe method forbore to send him the method wthwithout Mr Newton's knowledge & wrote earnestly to Mr Newton to send descri do send it himself & MrNewton did so. at Mr Leibnitz had been now two or three years in quest of the method of series about two years or Mr Leibnitz had been now two or three years in quest of the method of series, & in his searche one of his essays was to subduct the negative termes from the affirmative in these & such like Progressions.

1 = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6}

&c.

1 = 1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9}

&c & thereby he came to these & such like series

1 = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}

1 = \frac{2}{3} + \frac{2}{15} + \frac{2}{35} + \frac{2}{63} + \frac{2}{99}

&c. T A quarter of the first series & half the second series, together wthwith the last of the ser eight series sent him by Oldenburg made a that a harmony which he sent back to Mr Oldenburg to be communicated to Mr Newton in recompence for thise his method. For Mr Leibnitz knew that Mr Oldenburg & Mr Collins were not to make a discovery without authority from the Executors of Mr Gregory who had sent that series to Mr Collins in the year 1671, & was now dead. In further recompence for Mr Newtons method Mr Leibnitz sent him h method of his own by transmutation of I figures, recommending it for a149 a very general method of series, tho he wanted Mr Newton's method for finding the two series of Mohr, & MathemationMathematition say that is [not his general without the assistance of the methods sent him by MrN. wchwhich] transmutations are of little or no use & did not amount to general method before Mr Newton made it. And tho he had not th wan still wanted Mr N the method of deriving reciprocal series from one another & desired Mr NewtoNewton to send it to him (p 63): yet he could not forbear claiming some series from Mr Newton wchwhich wil were the result of that method. p. 61, 62 And when Mr Newton sent him the Method & tho he understood it with difficulty, he wrote back that he had found it before, as he perceived by his old papers, but for want of a good example of its use had neglected it. p 96. And as he published Mr Gregories series in the Acta Lipsica for his own without letting the world know any thing of what he learnt from the English while he was in England or afterwards by his correspondence wthwith Mr Oldenburgh or that the series was Mr Gregories: so he published the Differential method in the Acta Lipsica for his own without letting the world know any thing of the letters wchwhich he received from mNewton, or of Mr His first endeavour was therefore the to make himself out the first inventor of the method of series; his next to make himself author the first inventor of the differential infinitesimal method. Newtons knowing any thing of the method, till Mr Newton by publishing his Principles convinced every body that he understood the method very well & how to make use of it & that it scarce differed from the differential method except in words & symbols, & the publishing of the correspondence by Dr Wallis justified that notion & made it appear beyond question that the method of fluents was very like in substance the same with the differential method. And then Mr Leibnitz pretended that when he published the ele he understood nothing of this wnwhen he published the elements of thise differential method, nor then kewknew any thing more of Mr Newton's method then that it was a method of dra TangtsTangents wchwhich stuck not at radicals p. 104, 107 & wchwhich any man might have without having the differential method p. 104, 107. promising to recompence them with his own series very different from those of Mohr & this he did without desiring Mr Newtons leave, tho by his own rule if he had got the Demonstration . . . . . . . . . . & Mr Newton did so, & Mr Leibnitz in requital sent him back a harmony of series, one of wchwhich . . . . . . . . . Mr Newton's method in the year 1671 or before. In further recompence . . . . . . . . . . had neglected it. p. 96. he published the elements of it in the Acta Lipsica that is in the year 1674 or yethe beginning of the year following p. 119, that is before he po when & by consequence had it when he was polishing this And here his memory fails him againe for in those days he wrote his opusculum vulgari more & continued to polish it for the press & was of opinion till he fell into lasiness & found out his infinitesimal Analysis & was then of opinion that ytthat inverse method problems of Tangents will were not be reducible to equations or quadraturies, as has been said above. Now if the pretence of plea of reputation candor & sincerity doth not make any man a witness in his own cause, & the pretence of finding things apart without proving the priority &of invention is not to be regarded, & the allowance of either of these pretences would be unjust & prove an endless encouragement to pretenders: if; If the following Letters & Papers be well attested, & by them it appears that Mr Newton had the method of infinite series & fluxions in the years 1669 & 1671 & then applied it generally to the msolution of Problems by the method of fluents & in the year 1671 wrote a larger treatise upon both those methods conjonedconjoined,] & that Mr Iames Gregory 126) His first endeavour was to print out make himself the inventor of the method of series. For he was in Londo In yethe begining of the year 1673 when there was not a Mathematician in London w of note in the city London who was not acquainted wthwith Collins & had not heard of the method And After of Slusius he went from formfrom L. thence London thence to Paris & pretended there to be the first inventor of two series o & s for the circle both found by one & the same method ( 38) one of them was for finding the arc by the sine & he afterwards wrote for the Demonstration of more a ses (p 38, 45) & therefore had not yet the method & & In April 1675 Collins & Oldenburg sent him eight others: For he then knew none of the eight to be his own but took time to consider them & compare them with his own (p 40, 42) & yet the very same year he communicated to his friends at Paris an Opusculum written upon the last of the eight as his own series without letting them know that he had received it from England. p 41, 42. & the next year pretended to have forgot the receipt of the eight series, For & desired Mr Oldenburg to procure him the Method from Mr Collins, p 46 calling it the demonstration of two series shewed him by one Mohr p. 45, & promised to recompence them with his own invention of series very different from those of Mohr, saying that they were the series of wchwhich he had written to him some years before p. 45. But M when Mr Oldenburg received theym they proved to be the last of the last a harmony of series, one of wchwhich was the last of the eight series wchwhich Mr Oldenburgh had sent him the year before & the others were different & served only to make up the harmony being found by subducting the negative terms from the preceding affirmative ones in these & such like progressions

1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} &c = 1

1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} &c = 1

. The first progression gives this series

\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} &c = 1

. The second gives this

\frac{2}{3} + \frac{2}{15} + \frac{2}{35} + \frac{2}{63} + \frac{2}{99} &c = 1

And a quarter of the first series & half of yethe second wthwith the summ of ho & the series sent him by Grego Oldenburg compose the harmony. p. 61. And tho the series sent him by Mr Oldenburg was sent by Mr Gregory to Mr Collins in the beginning of the year 1671 & a copy of Mr Gregory's letter was sent to Mr Leibnitz in summer 1676 whereby he could not but know that the series was Gregorys (p 47 yet he published it in the Acta Leipsica as his own without letting the world know that he received it from Oldenburg & Collins & that Collings had it from Gregory A.C. 1671 who found it by M Newtons method in yethe beginning of yethe year 1671 or before winter between the years in the beginning of the year 1671 or before. In yethe year 1674 he In the year 1674 I he pretended to have two series for the circle but what they were & whence he had them is uncertain seing he did not know them to be any of the eight series sent him the next year by Mr Oldenburg, nor A yeare or two after the publishing of Mr Newtons Principia Philosophiæ Mr Leibnitz had be having been long from home in a At the same time he compared anoth And in his Tentamen de motuum cœlestium causis, (another treatise written at the same time) he [wherin he prætended to have found the principall Propositions in in Mr Newtons book said book relating to the Pl motions of the Planets] mathematicians say that in writing his Tentamen de mot.motuum cœlest.cœlestium causis he forgot to reexamin the calculati process of finding the 8 his 19th Proposition wchwhich is the chief of Mr Newtons Propositions relating to the motions of yethe Planets & that & that the Proposit he could no man could find theat Proposition by such an erroneous process as he has set down. They do not much blame him for his errors (those Errors are easily committed in calculations & but They blame him not for committing them but for pretending to find Mr Newtons Proposition by them., But had forgot to tell his telet us know us which of the Propositions he communicated to yethe Academy at Paris & to acknowledge that he left Paris before he found understood the differential method & publishing & publishing them hastily with a designe to make that Proposition his own. They blame him not for committing an error some errors in his first essay, but for adapting an erroneous calculation to that Mr Newtons Propopsition with intention to make it his own: but for adapting a calculation to another man's Proposition with a designe to make himself the first inventor, tho the by the errors of the calculation it appears that he did not invent it. And that Mr & yet allowing that Mr Newton by himself attained to principles like the differential method [& that Mr Keil is a young man unacquainted with things done before his time] by wchwhich allowance he seems to mean that although Mr Newton had the method of fluencts in the year 1676 when he wrote his two Letters of 13 Iune & 24 Octob. & had not at that time heard of the Differential method, yet Mr Leibnits had the differential method is the older., being known to Mr Leibnitz above nine years before 127150 That yethe R. Society is satisfied in the fidelity of their Committee in printing no papers but what such as appeared to them to be genuine & in omitting none wchwhich they they thought me in their opinion made for Mr Leibnitz & particularly they are satisfied that the three Letters to Mr Collins in Iu&ly dated in & Aug. 1669 found in the hand of Dr Barrow & the Analysis copy of the Analysis referred to in those Letters were & found in the hand of Mr Collins are genuine, & that the Letter of Mr Iames Gregory to Mr Collins dated 15 Feb. 167

\frac{0}{1}

& found in the hand of the author & copied in the hand of Mr Collins, i genuine & the Letter of Mr Oldenburg to Mr Leibnitz dated 15 Apr 1675 & found in english in the hand writing of Mr Collins & in Latin in one of the letter Books of the R. Society (ibeing translated into Latin by Mr Oldenburgh & the Answer of Mr Oldenburg M Leibnitz dated from Paris 20 May 1675 are all of the genuine & the Letter of Mr Leibnitz to Mr Oldenburgh dated from Paris 12 May 1676 are & found in the archives among th in the hand writing of Mr Leibnitz among the ancient papers left by Mr Oldenburgh in the custody of the R. Socitety wthwith notes on the back side in the hand of Mr Oldenburg & the Letter of Mr Newton to Mr Collins 10 Decem 1672 found in the hand of the author among the Papers of Mr Collins with a copy thereof in the hand of Mr Collins, are all of them genuine. & the Collections of the papers of Mr Iames Gregory sent to Mr Leibnitz at Paris 1676 & found in the hand of Mr Collins among are all of them genuine. And of since Mr Leibnitz proposes to publish a new Commercium And we see no reason to doubt of the sincerity of any of the Letters published by Dr Wallis. And of since Mr Leibnitz proposes to print a further Commercium Epistolicum and f if for giving credit to the Letters in his custody he pleases to take attested copies of such letters as he intends to publish & send the Originalls to any of yethe forreign ministers here, or to or any other friend to be examined before us by a Committee before a Committee of the R. S. by such as know the hands of Mr Oldenburgh, Mr Collins, Mr Newton, Mr Gregory, Dr Wallis, Dr Barrow &c, & to let attested copies to be taken of them here: the Originals shall be returned wthwith attestestedattested copies of any Letters in orour custody which his friends shall think material to be published by him: by wchwhich means those Letters wchwhich he shall publish will become authentick & of indisputable credit & authority,. wchwhich Whereas A letter from Mr Leibnitz wchwhich being produced wherin he represents his inclination to publish a further Commercium Epistolicum wchwhich may for serve for an history in Literatur & desires copies of such Letters as are in the custody of the R. Society our hands to be published with those in his own custody: if for giving credit to those in his own custody preventing disputes about the credit of the Originals he pleases to take (after he has taken attested copies of thoseem in his own custody in his own custody), to send yoryour Originals to any foreign Minister or other friend in England to be eximeined before a Committee of the R. S. by such as know the hands of Mr Oldenburge, Mr Collins, Mr Gregory, Dr Wallis, &c & to let attested copies be taken of them here: the Originalls shall be returned, with attested copies taken of any letters in orour custody wchwhich his friends shall think material to be published desire, the [Letters already printed being first examined by the Originals before his frothe Committee in the presence of his friends] Originalls of all the Letters & MSS in orour custody printed or to be printed being for also examined before his friends where any doubt sho 120151 4 Slusius in yethe November 1672 gave notice to Mr O of his new method of tangents to Mr O & communicated it Mr Oldenburg to him in Ianuary following & founded it th on three Lemmas the first of wchwhich was this. Differentia duarum dDignitatum ejusdem gradus applicata ad Differentiam Laterum dat partes singulares gradus inferioris ex binomio laterum.

\frac{y^{3} - x^{3}}{y - x} = y y + y x + x x

In general In general lLet

d y

be yethe infinitely small difference of y & x & n the index of the dignity & the difference of

y^{n} - x^{n}

will be

n - 1

n o y^{n - 1}

n d y y^{n - 1}

: 5 Mr Newton being at that time desired by Mr Oldenburg to communicate his Method of Tangents, sent it to him in a letter dated 10 Decem 1672. & subjoyned Hoc est unum particulare vel Corollare [It proved to be yethe same wthwith that of Slusius. & And the same method was found also communicated by Hudden & But to Schooten in yethe year 1659. But Mr Newton found in a different deduced it from a Method much more general. For when he had described it he subjoyns] And subjoyned Hoc est unum particulare vel Corollarium potius Methodi Generalis . . . . . . . surdis sunt immunes. Mr Newtons method of Tangents proved to be the same wthwith that of Slusius. And it appearses since that the same method was communicated to Mr by Hudde to Schooten in 166059. But their methods were not so general as that of Mr Newton.s. 6 12 Mr Oldenburg 15 Apr 1675 sent to Mr Leibnitz several series one of wchwhich was that of Mr Gregory for finding the arch of a circle whose tangant is given. And Mr Leibnitz by his Letter of 20th May following owned the receipt of yethe Letter. A Towards the end of the year Mr Gregodry died & Mr Collins the next year at the request of Mr Leibnitz & some others sent colledcted extrats out of Mr L Gregories Letters & Papers & sent them to Mr Oldenburg to be sent to Mr Leibnitz at Paris & returned back after they had been perused there. They were sent a little before the 11th of August 1676 as Mr Collins mentioned in a letter of that date written to Mr the brother of the said Gregory. And Mr Tschurnhause being then at Paris in his Letter to Mr Old. dated 1 Sept 1676, has this reference to the Collection. Similia porro, saith he, quæ in hac re præstitit eximius Geometra Gregorius memoranda certe sunt, si quidem optimæ famæipsius consulturi sunt qui ipsius relicta MSS luci publicæ ut exponantur operam navabunt. In this Collection was Mr a 6 Dr Wallis in his Arithmetick published A. 1657, Chap. 33 Prop. 68, teaches tought how to reduce the fraction

\frac{A}{1 - R}

by perpet. divis. into the series

A + A R + A R^{2} + A R^{3} + A R^{4} + &c

. 7 The Lor Vicount Brounker squared the Hyperbola by this series

\frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + \frac{1}{7 \times 8} + \frac{1}{9 \times 10} + &c

that is, by this

1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + \frac{1}{9} - \frac{1}{10} + &c

joyning every two terms into one. This was published in the Philosophical Transactions in April 1668. 8 Mercator by the Division of Dr Wallis demonstrated yethe Quadratures of Vicount Brounker after another manner & published it the same year about August or September. Mr Ne 9 The next year in August Dr Barrow communicated to Mr Collings the Analysis per Æquationes numero terminorum infinitas wchwhich is the first Tract isn this Collection of Mr Iones. And Mr Collins thenceforward for some years together communicated several things out of it to his friends, [& one of the series for the Circle he communicated to Mr Iame Greg] 10 Mr Iames Gregory having by this a series sent to him by Mr Collins found Mr Newtons method of Series of Series sent back in to Mr Collins in yethe begining of the year 1671 several other series, & gave Mr Collins leave to communicate them freely. One of those series was for the arc of a circle whose tangttangent was given vizt 11 Mr Leibnitz was at London in the years 1671 1672 & 1673 till March & then went to Paris, & in 1674 pr wrote to Mr Oldenburg (15 Iuly) that he had a wonderful Theoreme wchwhich gave him the area of a circle or of any sector thereof in a series of rational numbers produced continually in infinitūum & 26 October t explained himself by representin that by by one & the same method he could find either the whole circumference or any arch whose sine was given tho the proportion of the arc to yethe whole circumference was not given known. If the proportion was known it this method or ThWonderful Theorem gave him the whole circumference: if not it gave him at least the Arc. And in his Letter of May 12 1676 he desired Mr Oldenburg to help him to the demonstration of the Theorem, that is to Mr Newtons method of finding it 129)152 * Vpon reading this Letter before the R. Society, Mr Newton represented that these last words reflected upon the Committee of the R. S. as if they had printed things partially, & that as he did not print the Commercium Epistolicum himself but left it to the Committee to print what they thought fit to select out of the Archives f of the R. S. least he should seem to make himself a witness in his own cause, so he did not think Mr Leibnitz a proper person to print a Commercium in his own cause. But if he had any ancient Letters wchwhich he had received from England & desired pleased to send the Originals to any friend in England who might produce them to be examined before the R. S. by t be e by such as knew the hands , & to have let attested copies be taken of them: they might then be the Originals should be returned & the Letters might then be printed either in the Transactions or in Germany as Mr Committee to themselves in publishing the Commercium Epistolicum, represented further that he could have sp supplied them Committee wthwith some ancient Letters of moment then in his own custody but forbore to do it & least he should seem to make himself a witness in his own cause. And to prove the truth of what he said, he produced two original Letters written to himself the one written by Mr Leibnitz from Hanover

\frac{7}{17}

Mar. 1693, the other by Dr Wallis from Oxford 10 Apr. 1695 directing which directo to the a Passage in the first volum Preface to the first Volume of his works where the DrDoctor gives his op (Homo vetus & intelligentissimus) gives his opinion very f fully in this matter. And these two Letters, after they had been examined before the before the R. S. by persons who knew the hands were read to before them Society, & left in their Archives & thence copied by Mr Newton in the Appendix to his Remarks printed above pag 100 & 105. NB Vpon the reading of this Letter to the R. Society Mr Newton represented that he was so far from printing the Commercium Epistolicum himself that he did not so much as deliver to the Committee of the Society the Letters wchwhich he had in his own custody. &And to prove this produced ta letter of Mr Leibnitz dated & another of Dr Wallis dated both wchwhich after the hands had been examined were deposited in the Archives of the Society. Here instead of proving his accusation agtagainst Dr Keill wchwhich he was bound to do in justice, he tells the R. Society that they are imprudent & unjust unless they will post tot documenta vitæ approve all approve of his candor & allow him to be a witness in his own cause against Dr Kell the Doctor And yet in that is to say the Royal Society are imprudent & unjust unless they will allow him to a witness in his own cause against Dr Keill. Here again he reflects upon the justi prudence & justice of the R. S. telling them that: they are imprudent & injust if they suffer his candor to be questioned while he himself is questioning the candor of others. They are unjust unless they allow him (post totot documenta vitæ) to be a witness in his own cause, contrary to the laws of all nations. 130)153 7 Mr Leibnitz indeed was the first who published the invention of first differences. but he published nothing but what he might deduce from Mr Newtons Letters. For in Mr Newtons Letter to Mr Collins dated 10 Decem 1672, a copy of wchwhich was sent to him by Mr Old. among the extracts of Gregories Letters 26 Iune 1676, Mr Newton described a method of Tangents wchwhich he conjectured to be the same wthwith the methods of Gregory & Slusius, & said that it was a branch or Corrollary of of a general Method of solving the abstruser Problems without any troblesome calculus & without stopping at surds. And the method of tangents be described by the following example. Pone CB applicatam ad AB in quovis angulo dato terminarej ad quamvis Curvam CA, & dicatur AB x & BC y, habitudoque inter x et y exprimatur per quamlibet æquationem puta

x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} = 0

, qua ipsa determinatur Curva. Regula ducendi tangentem hæc est. Multiplica æquationis terminos per quamlibet progressionem Arithmeticam juxta dimensiones y, puta

\begin{array}{r} x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} \\ 010023 \end{array}

; ut et juxta dimensiones x, puta

\begin{array}{r} x^{3} - 2 x x y + b x x - b b x + b y y - y^{3} \\ 322100 \end{array}

Prius productum erit Numerator, et posterius divisum per x denominator Fractioneis quæ exprimet longitudinem [subtangentis] BD ad cujus extremitatem D ducenda est Tangens CD. Est ergo longitudo

BD = \frac{−2 x x y + 2 b y y}{3 x x - 4 x y + 2 b x - b b}

. This far Mr Newton. And this is as much as to say that as all the terms of the æquation multiplied by the indices of the dignities of the ordinate y absiaabscissa x & divided by the Ordinate y abscissa x are to all the terms of the æquation multiplied by the indices of the dignities of the abscissa x ordinate y & divided by the abscissa x ordinate y, so is the Ordinate xy=BC to the subtangent BD, & (by the method of tangents of Dr Barrow) so is dy to dx. And by multiplying the extremes & means &you have the equation

3 x x d x - 4 x y d x + 2 b x d x - b b d x = −2 x x d y + 2 b y d y

. [Or

3 x x d x - 4 x y d x + 2 b x d x - b b d x - 2 x x d y + 2 b y d y

.] And this is as much as to say that if you multiply all the every termsterm of the Æquation by the indiceexs of the dignities of x the in that term & divide it by

\frac{d x}{x}

& multiply it by dx & & then multiply all the every terms by the indices of the dignitieys of y therin & by dy & divid put the summ of all the terms produced will equall to O you will have an Equation involving the differences of x & y vizt. And is the solution of the first Proposition of Book of Quadratures when there are but two indeterminate quantities. And when there are more, the same operation applied to them solution all gives the solution. [And if the equation involve a surd, let theis equation be (for instance

x^{3} - 2 x x y + b x x - b b x + a \sqrt{x x - b y} = 0

. Put

a \sqrt{x x - b y} = b z

& the equation will become] And this is the solution of the Proposition Data æquatione fluentes quotcunque quantitates involvente invenire fluxiones. wchwhich is the first Proposition of yethe Book of Quadratures. wchwhich was set down in Mr Newtons Letter of 24 Iune 1676 & published in his Book of Principles. And therefore Mr Newton understood the solution of thatis Proposition in the year 1672, & Mr Leibnitz had sufficient light into it from the copy of the said Letter in the year 1676, besides the light wchwhich he received into it from Mr Newtons Letters of Iune 13 & Octob. 24. 1676. And what he published in the Acta Eruditorum for in the year 1684 amounts to nothing more then the solution of thatis Proposition. I cited Dr Barrow's method of Tangents because Mr Leibnitz procured the Doctors Lectures when he was in London the second time, & carried them with him into Germany; & in his Letter of 21 Iune 1677 deduced from thence the differential method of Tangents from the m of Slusius as Gregory had done before. For Gregory tells us this in his Letter to Collins 5 Sept. 1670, a coppy of wchwhich was sent to Mr Leibnitz in the Exceprpta of Gregories Letters 26 Iune 1676. 10 In the end of the year 16 1715 Mr Leibnitz in the Postscript of a Letter to Abbe Conti wrote thus: They that have writ against me have made no difficulty to attack my candour by interpretations forced & ungrounded: they shall not have the pleasure to see me answer to the slender to reasons of a sort of men who use me so ill &c. If this was the case he should have shewn by some instances wherein they had misinterpreted the originall Letters. The nNotes upon the Letters are of no authority what but what they derive from the Letters. And the Reader is judge of that. Neither the Letters nor the Notes upon them have hitherto been answered & therefore they are now published as unsanswerable 11 Mr Newton in his Answer in his Answer to the said Postscript dated 26 Feb. 171

\frac{5}{6}

wr pressed Mr Leibnitz in these words. Mr Leibnitz has hitherto refused to answer, knowing well that it is impossible to answer to matters of fact. For a pretence to his silence he adlledges at present that he has not seen the Book, & that he has not leasure to examin it, but has del desired a famous Mathematician to do it &c. And Mr Leibnitz in his Answer to Mr Newtons Letter, 9 Apr. 1716 repr still excused himself from answering saying: For writing an answering from point to point, in to the L work published against me, it would require a work as big at least as that one must enter into a great detail of the quantity of minutes pr wchwhich passed 30 or 40 years ago, wheofwherof I remberremember but little; I must search my old Letters, many of wchwhich are lost, besides that I for the most part I kept no minutes of my own, & the rest are buried in a great heap of papers wchwhich I cannot search out without a great deal of time & patience. But I have no leasure to do it, being charged at present with business of quite another nature. Thus far Mr Leibnitz. He had time to keep a large correspondence by Letters & enter into Metaphysical disputes: but had no time to shew thre or flour any any defect or flaw in flaws in the Epistles or Notes upon them Commercium Epistolicum. However, upon his death wchwhich happened in November following his friends gave out quod Commercio Epistolicuom Anglaorum aliud quoddam suum idemque amplius opponere decreverat as you may see in his Elogium printed in the Acta Eruditorum for Iuly 1717. 4B. In the flying Paper above mentioned Mr Leibnitz wrote: Certe aut miram ejus [Newtoni] oblivionem esse oportet aut magnam contra conscientiæ testimonium conscientiæ iniquitatem, si indulgentiam accusationem (ut ex indulgentia testimonium probat colligas) probat, qua quidam ejus asseclæ etiam seriem, quæ arcus circularis magnitudinem ex tangente exhibet, a Gregorio hausisse Lei. . . . um volunt. And Mr Newton in his Letter to Abbe Conti dated 26 Feb 1617

\frac{5}{6}

to be sent communicated to him, pressed him about this matter in these words. In his Letter of 20 May 1675 he acknowedgedacknowledged — — — dated 15 Feb. 1671 (your self being one of them) These Letters were compared with the print Baron Kilmanseg being present. And Mr Leibniz in his Letter to Madam Kilmansegg 19 Apr 1716 endeavoured to excuse himself in these words Cependant Eo men in mentioning this series endeavoured to excuse himself by saying that Mr Newton himself applauded it. It was found afterwards that one named Mr Gregory had found the just same series with him But that was what he understood late. Thus he allowed the Letters wchwhich is all that I here contend for) & that at length he understood He was therefore aat length satisfied by themse Letters that the series Gregory had found the same series. Abbe Conti & several other Gentlemen (amongst whom was Count Kilmansegg) having viewed the Originals & compared them wthwith the printed copies: Mr Leibnitz in a letter to the Countess did not deny what think fit to question what had been approved, but endeavoured to excused himself as well as he could, if as if he had not read the Letters of Mr Oldenburg when he first reccived them. So then the credit of the Letters in the Commercium is still safe. 154 8 Mr Leibnitz in a Letter to Mr Chamberlain dated from Vienna 25 Aug. 1714 wrote that it was being probable that there were still some Letters that regarded him among those of Mr Oldenburg & Mr Collins, wchwhich were not yet published, he desired that the R. S. would give order to communicate them to him. For, said he, when I return to Hannover, I may also publish a Commercium epistolicum which may serve for Histoire Litteraire. I shall be disposed t no less to publish the Letters wchwhich may be alledged against me then those that favour me & leave the jugment to the publick. But upon reading this to the Society, it was represented that these words reflected upon their Committee & that as they had not referred it to him but to a Committee of other persons to Collect & publish the ancient Letters so he had left it was not right th & they so Mr Leibnitz was not to be made a juge in his one cause. That If he had any ancient letters relating to this matter they ought to be examined & attested allowed by other people before they were printed. then That they might send him attested copies of any Letters, but were not to part with the Originals, & that he himself had Originalsl letters relating to this matter, but did not produce then to the Committee because he would not go about to make himself a witness in his own cause. And to prove this he produced two old Letters one of Mr Leibnitz dated

\frac{7}{17}

March 1693 & another of Dr Wallis written dated 10 April 1695, both written to himself. And the Letters being examined he t before the R. Society by those who knew the hands, were laid up in their Archives. 9 In the end of the year 1715 & beginning of the next year Mr Leibnitz — – – – – different from himself. By this citation he is a witness that Mr Iohn Bernoulli was not the author. By printing the Letter in Latin wthwith this citation Mr Leibnitz is a witness that Bernoully was not the author. And Mr Bernoulli by affirming per omnia humanitatis sacra that he wrote no such Letter is a third witness. And the style & contents of the whole flying paper points at Mr Leibnitz for the author thereof. For the word illaudibiles is peculiar to Mr Leibnitz. No body but Mr Leibnitz could know what passed between him & Mr Huygens at Paris 38 years before. Mr Leibnitz g in yethe year 1684 published the elements of the method by Addition Substraction Multiplication & Division & called this the Algorithm of his method. And the Mathematician objects that ♁♁ that he did not beleive ytthat Mr Newton when from the begi when he wrote his Principles & before, did so much as dream of his calculus of fluxions & fluents or the reduction thereof to general Analytical operations ad instar Algorithmi vel Regularum Arithemticarum. The word Algorithmus in this sence is peculiar to Mr Leibnitz this sense And Mr Bernoulli & the Argument goes beyond what Bernoulli ever affirmed. And if Mr BernoullBernoulli had writ been the author of the Letter of 7th Ione it would have been act of plagiary in Mr Leibnitz to leave out the sentence in wchwhich Bernoulli is cited by the author. Mr Leibnitz by correcting has confessed it to be his own. 5 In this flying Paper it is represented that when the ancient Epistles published in the Commercium Epistolicum were written Mr Newton did not so much asdream of the Method of fluxions because there are no prickt Letters in all the Commercium; tno not in theis Principia Philosophia. And is not this as trifling as if one should say that when Mr Newton wrote his Introduction to his Book of Quadratures he did not so much as in wchwhich he explains the method of Fluxions in words at length & illustrated his explanation wthwith exples examples in solving of Problems, he did not so much as dream of the method of Fluxions, because there are no prickt letters in all that Introduction. And when he wrote the second Lemma of this second Book of Principles & added a Scholium in which he said that the foundation of the Method of fluxions was conteined in that Lemma & that Lemma & that (h in his Letter of Octob correspondence wthwith Mr Oldenbur Leibnitz in the year 1676 he had couched it in this sentence [; Data æquatione fluentes quotcunque quantitates involvente fluxiones invenire et vice versa: he did not so much as dreame of the method of fluxions when he wrote those things because ther are no prict letters in what he then wrote. When he wrote his Analysis per series wchwhich Dr Barrow in the year 16969 sent to Mr Collins he used the symbol

\frac{a a}{64 x}

in the very same sense in wchwhich Mr Leibnitz afterwards used the symbol

\int \frac{a a}{64 x}

, & the symbols & ov, ox, oy, oz in the very same sense in wchwhich Mr Leibnitz afterwards used the symbols dx, dz & therefore in the year 1664 did at least dream of the method wchwhich Mr Leibnitz afterwards called summatory & differential. 131155 I said in my Letter of 24 Octob 1676 that I wrote a Tract five years before concerning the method of Series & another method together, & said then that the other method proceeded without sticking at surds, & was founded on this Proposition: Data æquatione fluentes quotcunque quantitates involvente, Fluxiones invenire; & vice versa. And the two parts of this Proposition include imply the direct & inverse methods of fluxions, called by Mr Leibnitz the differential & summatory methods. And in that Letter I there said further that the method there spoken of, extended to Problems about maxima & minima, & about tangents directly & inversely, & Quadratures & to some other Problems; & gave me the general Series there set down for Quadratures & other Series of like nature. And in my Letter of 10 Decem. 1672 (a copy of wchwhich was sent by Mr Oldenburge to Mr Leibnitz & came to his hands) I mentioned some other sorts of Problems, & particularyparticularly those of finding the curvatures of Curves, & the tangents of Mechanical Curves. And in my Letter of 13 Iune 1676 I said that my Analysis (composed of the method of Series & other methods vizt the methods of fluxions & arbitrary series) extended to almost all sorts of Problems except perhaps some numeral ones like those of Diophantus. And Mr Leibnitz in his Letter of sAug. 27. 1676 replyed that he did not beleive that my method could be so general, or extend to inverse Problemes of tangents & many others: tho he had reduced one inverse Probleme of Tangents (that of Beaune) to a Quadrature. All those things are old. And if all this, & the method of extracting fluents out of Equations involving fluxions & that of arbitrary series set down in my Letter of Octob. 24 1676 as known to me some years before, & the Analysis per series numero terminorum infinitas, & with my Letter to Mr Collins Novem 8 1676 published by Mr Iones, & the consent of the Book of Quadratures with all the rest, & with my Letter to Mr Collins Novem 8 1676 published by Mr Iones,, be not sufficient security: no man must hereafter venture to tCommunicate any thing in writing, before he has secured it to himself by printing publishing it in print. And if after such a concession as Mr Leibnitz made in these his Letters now recited & printed seventeen years ago, men may be allowed to go back upon any pretence whatever: even printing will be no security after witnesses are dead. Out of the Book of Mr Nicolas Fatio de Duillier, intituled Investigatio Geometrica Solidi rotundi in quod minima fiat resistentia, & published in the year 1699. Quæret forsan Cl. Leibnitius unde mihi cognitus sit iste Calculus quos utor. Ejus equidem fundamenta, ac plerasque Regulas proprio Marte, Anno 16987 circa Mensem Aprilem et sequentes, alijsque deinceps annis, inveni; quo tempore neminem eo calculi genere præter meipsum uti putabam. Nec mihi minus cognitus foret si nondum natus esset Leibnitius. Alijs igitur glorietur discipulis me certe non potest. Quod satis patebit si olim Literæ quæ inter Leibnitium et me Clarissimum Hugenium meque intercesserunt, publici juris fiant. Newtonum tamen primum ac pluribus annis vetustissimum hujus cCalculi Inventorem, ipsa rerum evidentia coactus agnosco: aqu a quo utrum quicquam mutuatus sit Leibnitius secundus ejus inventor, malo eorum quam meum sit judicium, quibus visæ fuerint Newtoni Literæ, alijque ejusdem manuscripti cCodices. Neque modestioris Newtoni silentium, aut prona Leibnitij sedulitas inventionem hujus Calculi sibi passim tribuentis ullis imponet, qui ea pertractarint quæ ipse evolvi Instrumenta. NB. Mr Fatio wrote this not as a Iudge but as a Witnesse. He related what he had seen, & his testimony is the stronger because it was against himself, & he understood the Methods of us all wchwhich he wrote. Part of a Letter of Dr Wallis to Mr Leibnitz Iuly 30. 1697 Optaverim item ut tibi vacet tuum cCalculum Differentialem & Newtono suam Fluxionum Fluxionum Methodum, justo ordine exponere; ut quid sit utrique commune et quid intersit discriminis, et utramque distinctius intelligamus. NB The doctor here wishes that the common method as deue to me & the improvements as due to Mr Leibnitz were stated between us. ButAnd Mr Leibnitz returned no answer to this but under the colour of having improved the method went on to call it his own, & at length when my witnesses were dead, & he thought himself strong enough in disciples fell foul upon Dr Keill for saying nothing more then what Dr Wallis had said before without being complained of. A&nd ain his Letter of 29 Decem 1711 began to claimed a right to the whole method & pressed that I would declare my opinion in this matter; that is that I would retract what I had published in the Introduction to the book of Quadaratures & what Dr Wallis had published in the Preface to the two first Volumes of his Works, & what Mr Leibnitz himself had in those days conceded to the Doctor. The next year Mr Leibnitz acknowledged that I was the first that had proved by a specimen made publick, that I had that part of the Method whereby the Solidum minimæ resistentiæ, the Curva linea celerrimi descensus & the Curva cCatenaria & velaria were invented. And after all this I when I wrote the Introduction to the Book of Quadratures I ( wchwhich (to the best of my memory) was in the year 16703 or 1704) I did not suspect that I should be called to account for saying there that I invented the Method of fluxions by degrees in the year 1665 & 1666; the seven or eight years after be required to give my opinion in this matter, in order to make me retract, & & be put to all this trouble in defending my self. The next year Mr Leibnitz in his Answer to Mr Fatio Out of the Answer of Mr Leibnitz published in the Acta Lipsiensia for May 1700 Certe cum Elementa – – – – satis intelli. — prodiere. NB Mr Leibnitz in his Letter of 27 Iune By my Letter of 24 Octob 1676 & his Answer 121 Iune 1677 it is evident that when he published the elements NB Let the first part of this sentence be compared with my Letters of 13 10 Decem 1672, 13Iune 1676 & & & 24 Octob 1676 both wchwhich came to his hands & with his own Letter of 21 Iune 1677: & let the latter part thereof be compared with his Letter of

\frac{7}{17}

March 1693 printed above, & with his concession that I was the first that who had proved by a specimen made publick, that I had [ that part of the Math infinitesimall Method Calculus by wchwhich the Solidum rotundum minimæ resistentiæ] ,& the cCurva cellerrimi descensus, were found & such like Pr invented. And aAfter all this, when I wrote the Introduction NB. The first part of this Answer should have been made to Dr Wallis had it been consonant to our Letters of 10 Decem. 17672, 24 Octob. 1676 & 21 Iune 1677. The second part admits of no retraction upon a pretence that I had deceived hism. The third part relates only to names & symbols. In the seems to relate to the place above mentioned in the Preface to the first & second Volumes of the Doctors worth as recital concerning the likeness of the methods. In the same Answer he acknowledgesd that I was the first who had proved by a specimen made publick that I had that part of the infinitesimal method calculus by wchwhich the soli Solidum rotundum minimæ resistentiæ & the Curva Celerrimi descenses &c were invented. Mr Fatio returned an Answer to Mr Leibnitz but the Editors of the Acta would not print it, & what it was I do not know. — & then challenged Mr Fatio to solve the Probleme of Mr Bernoulli for finding a Series of Curves which should cut another given series at right angles. Which makes it probable that M. Bernoulli hath ever since reserved the solution of that Problem for a challeng. Consessus Arbitrorum dDelectorum a Societate Regia, ad quam D. Leibnitius provocaverat ad Calcem hujusce Commercij Epistolicis annexas. Ea est ut sequitur, occasione nata ex Literis D. Leibnitij Vindicatione hujus Inventionis suo proprio Autori a D. Ioanhanne Keill nunc Professore apud Oxonienses & [Literis D. Leibnitij postulantis ut a Regia Societate remedium postulantis petentis, nempe ut ipsis Iudicibus, Keilius] publice sententiam suam] et provocatione D. Leibnitij ad judicium R. Societatis. 132)156 2 Mr Newton In his my Letter of 24 Octob 1676 I represented that five years before that time that is in the year 1671 he I wrote a tract Book of this Method & the method of series together but for the sake of quiet desisted from my designe of publishing it, & The True Book is still in being, & conteins a very plain description of those two methods. The first Proposition of the Book of Relatione quantitatum fluentium inter se data fluxionum relationem determinare. When Mr F Leibnitz fir In the year 1684 Mr Fatio Craig desired me to explain to him the elements of the differential calculus then newly printed in the Acta Leip Eruditorum. I did so & told him that it was my method a little altered in another dress as would appear if the Letters wchwhich past between me & Mr Leibnitz seven or eight years before should be published & Mr Craig is still alive & remembers this. In Dr Wallis in the Preface to the two first Volumes of his works printed in spring 1695 wrote that in those Letters I had explained to Mr Leibnitz the Method, found by me 10 years before or above that is in the year 1666 or before. Mr Fatio in the year 1669 wrote that I In the AcctAccount of these two Volumes published in the Acta Eruditorum the next year notice was taken of this passage was taken notice of & not denied contradicted. Dr Wallis in a letter dated 1 Decem. 1696 Dr Wallis gave notice of this passage to Mr Leibnitz, & in the Letters wchwhich followed between them it was not contradicted. Mr Fatio in the year 1699 Dr Wallis in the 3d Volume of his workes published the Letters wchwhich had passed between me & Mr Leibnitz in the years 1676 & 1677. And Mr Fatio the same year (viz 1699) published that I was the oldest inventor by many years & confirmed his opinion by what he had seen in my manuscripts, & Mr Leibnitz in his answer did not contradict him And all this was done without any encouragement from me.. In the year 17054 I published (in the Indtroduction to my book of Quadratures) I published that I found the method of fluxions by degrees in the years 16565 & 1666. And For this had been long allowed in England & was not yet contradicted abroad. For this was not yet disputed. But the next year the Leibnitians began to contradict it. But Dr Wallis being now dead, the Leibnitians in the Acta Eruditorum for Ianuary 1705 in giving an Account of this book wrote the Introduction to this book published the Passage above mentioned, the meaning of wchwhich is to tell thes Reader & there by which they began to tell the world was told that I did not invented the method so early nor write a book upon it in the year 1671 nor mentioned it in my Letters of 13 Iune & 24 Octob 1676 was the first inventor but had & 10 Decem 1672 nor always used fluxions for the Leibnitians differences even as Faber had substituted motions for the method of Cavallerius. With this Accusation the dispute began, & Mr Leibnitz has made this accusation his own by saying in his Letter of 29 Decem. 1711 against Dr Frustra ad Exemplum Actorum Lipsiensium [Keilius] provocat ut sua dicta excuset; in illis enim circa hanc rem quicquam cuiquāam detractum non reperio sed suum potius passim suum cuiquue tributum. The Leibnitians therefore in giving the lye to me, & Dr Wallis & Dr Keil Mr Fatio, & in beginning now to contradictinging the tradition wchwhich had hitherto remained in England, & endeavouring to transfer the method right first of invention from me to Mr Leibnits are the aggressors, & ought to prove what they affirm. In his Letters of 4 March & 29 Decem. 16 1711 he de appealed to me & declined disputing wthwith Dr Keill & desired that I would give my opinion in the matter, & that is that I would retract. For he knew what had been published by Dr Wallis Dr Keil Mr Fatio & me,. aAnd And now he tells you compares this dispute to a combate In the same Letter he knew that I refused to contend wthwith Dr Keill pretending because the Dr was not authorised by me, & desired that I would give my opinion in this matter, that is, that I would either retract what condemn Dr Keill & retract what Dr Wallis Mr Fatio & I had published some years before or enter into the lists with him as he expresses himself in his last letter.. And by this attaque he [has given Dr Wallis Mr Fatio & me the lye & charged me wthwith plagiary & ought to prove the accusation upon pain of being deemed guilty of calumny] is the aggressor the 134)158 And the Editors of the Acta Eruditorum still insist upon this concession of Mr Newton. And yet Mr Newton in the Scholium wh never acknowledged anything more then that Mr Leibnitz had the differential Method when he wrote his Letter dated Iune 21. 1677, & the till Series that he had a Method for finding the converging Series wh by the transmutation of figures when a year or two before [And Dr Wallis had read Mr Newtons Letters & his Scholium upon the 2d Lemma of the second book of his Principles wherein he acknowledged these things & therefore as he knew what Mr Newton had written so he did not omitted not what he knew. But Mr Leibnitz & his friends have endeavoured to make the world Germans beleive that Mr Leibnitz gra Newton acknowledged what he never did acknowledge.] How he then came by the Method differential Method Dr Walls has told them truly, & as for the pretended Method of converging Series it is of little or no use, I do not know that any other use has been made of it then to colour over the pretences of Mr Leibnitz to the Series of Mr Gregory. If he means that h Mr Newton ever acknowledged that Mr Leibnitz had the Differential method before the year 1677, he is in a mistake Dr Wallis menant that Mr Leib Newton explained the method of fluxions to Mr Leibnitz Dr Wallis affirmed that Mr Newton explained to Mr Leibnits in the year 1676, Mr Newton never allowed that Mr Leinitz had the Differential method before the year 1677. And Mr Leibnitz himself in the Acta Eruditorum for the year April 1691 pag 178 acknowledged that he found it after he returned home from Paris to enter upon other business. And as for the pretended method of cinfinite series by means of a transmutation of figures it is of little or no use. I know do not know that ill any other use has been made of it then to colour over the pretences of Mr Leibnitz to the Series of Mr Gregory. Mr Leibnitz in his Answer to Mr Fatio printed in the Acta Eruditorum for the year 1700 pag. 203, wrote thus. Spee [Newtonus] scit unus omnium optime, satisque indicavit publice cum sua Mathematica Naturæ Principia publicaret, Anno 1687 nova quædam inventa Geometrica quæ ipsi communia mecum fuere NEVTRVM LVCI AB ALTERO ACCETPÆ, sed meditationibus quemque suis debere, & a me decennio ante [i.e. anno 1677] exposita fuisse. But iIn the Book here referred unto Mr Newton did not acknowledge that Mr Leibnitz found this Method without receiving light from his the three above mentioned Letters, & if he had acknowledged it, yet second Inventors have no right: & Dro Wallis told him the contrary without being confuted or contradicted; & if he had inven found it without without apart the assistance of Mr Newton, yet second inventors have no right. Mr Leibnitz in his aforesaid Answer to Mr Fatio published in the Acta Eruditorūum for May 1700, writes this also. Certe cum elementa — — — — satis intellexi. Here he again acknowledged that the Book of Principles gave him great light into Mr Newton Method: now he & yet he now denyes that this book conteins any othing of that Method in it. Here he pr denyes pretended that by that book he received the first light into Mr Newton's method, before that book came abroad he knew nothing more of Mr Newtons inventions then that he had a certain method of Tangents, & that by that book he received the first light into Mr Newtons method of fluxions. But in his Letter of Iune 21 1677 he acknowledged that Mr Newtons method extend also to Quadratures & was like his own. His words are: Arbitror quæ — — — differentialem. Mr Newton had in his three Letters above mentioned (copies of wchwhich he MrLeibnitz had received from Mr Oldenburgh) explained his represented his method so general as to reach to by the help of Equations both finite & infinite to determin maxima & minima, tangents, areas, leng solid contents, & centers of gravities of figures, lenghts & curvities of Curves lines & curvilinear figures even without taking away radicals & to extend to the Curves usually called Mechanical, & to inverse Problems of tangents & others more difficult, & to almost all Problemes except perhaps some numeral ones like those of Diophantus. &And had repr Mr Leibnitz in his Letter of 27 Aug. 1677 represented that he could not beleive that Mr Newtons method was so general. Mr Newton had said in his Letter of Decem 10 167 the first of the three Letters that had set set down his method of Tangents deduced from this general method & Mr illustrated wthwith an example, & said that thatis Method of Tangents was but a branch or corollary of his general method, & that he took the method of Tangents of Slusius to be of the same kind; & thereupon Mr Leibnitz in his return from Paris through England & Holland into Germany was considering how to improve the method of Tangents Tangents of Slusius & extend it to all sorts of Problems, as we have shewed above out of his Letters. ② Mr Newton in the last of the said three Letters added that his Method gave series Theoremes for Quadratures by series which brake off & became finite when the Quadrature might be expressed in a finite equation, & set down the first Series or Theorem wthwith several examples. ① Mr Newton in his second Letter shewed how to resolve any dignity of a Binomium into a series, & thereby its well known that if the second term of the D Binomium be indefinitely small, the second term of the series will be the first term first difference of the Dignity, & the following terms will be proportional to the following Differences. ② Mr Newton also in the last — — examples. And after Mr Newton had made so large an explanation of his method, & Mr Leibnitz had got some light into it, & in his th in exp in his Letter of Iune 21 explained how the method wchwhich he was fallen into answered to the discription wchwhich Mr Newton had given of his Method in drawing Tangents, giving the Method of Slusius, as a Corollary & faciliating proceeding without taking away fractions & surds & faciliating Quadratures: Its wonde to tell the Germans that when he first published the Method of S his Differential he knew nothing more of Mr Newtons performances Inventions then that it he had a certain method of Tangents is very wonderfull extraordinary, & wants an explanation. When Mr Leibnitz first published his Differential Method – – – – – some years before the year 1676. It was not enough to mention a methodus similis without saying – – – – candor & justice. The Editors of the Acta Eruditorum have represented that Mr Ne