130 PHILOSOPHICAL TRANSACTIONS. [aNNO 1714. 



with Mr. Newton's letter of June 13, 1676. And Mr. Newton having de- 

 scribed, in these two letters, that he had a very general analysis, consisting 

 partly of the method of converging series, partly of another method, by which 

 he applied those series to the solution of almost all problems (except perhaps 

 some numeral ones like those of Diophantus) and found the tangents, areas, 

 lengths, solid contents, centres of gravity, and curvitures of curves, and 

 curvilinear figures geometrical or mechanical, without sticking at surds; and 

 that the method of tangents of Slusius was but a branch or corollary of this 

 other method : Mr. Leibnitz, on his returning home through Holland, was 

 meditating on the improvement of the method of Slusius. For in a letter to 

 Mr. Oldenburg, dated from Amsterdam, Nov. -la, i67^j he wrote thus: 

 ** The method of tangents, published by Slusius, does not reach so far but that 

 jsomething farther might be done in that kind, which would be of very great 

 use in all sorts of problems, even in my method (without extractions) of re- 

 ducing equations to series ; to wit, a certain short table of tangents might be 

 calculated, and continued so far, till its progression appear, so that any one 

 might continue it as far as he pleased, without any calculation." This was the 

 improvement of the method of Slusius into a general method, which Mr. 

 Leibnitz was then thinking on, and by his words, " Something further might 

 be done in that kind, which would be of very great use in all sorts of problems," 

 it seems to be the only improvement which he had then in his mind, for extend- 

 ing the method to all sorts of problems. The improvement by the differential 

 calculus was not yet in his mind, but must be referred to the next year. 



Mr. Newton, in his next letter dated Oct. 24, 1676, mentioned the analysis 

 communicated by Dr. Barrow to Mr. Collins, in the year 1669, and also an- 

 other tract written in 1671, about converging series, and about the other 

 method by which tangents were drawn after the method of Slusius, and maxima 

 and minima were determined, and the quadrature of curves was made more 

 easy, and this without sticking at radicals, and by which series were invented 

 which brake off, and gave the quadrature of curves in finite equations, when it 

 was possible. And the foundation of these operations he comprehended in this 

 sentence, expressed enigmatically as above, having given an equation involv- 

 ing any number of fluent quantities, to find the fluxions, and vice versa. 

 Which puts it past all dispute that he had invented the method of fluxions be- 

 fore that time. And if other things in that letter be considered, it will appear 

 that he had then brought it to great perfection, and made it exceedingly general ; 

 the propositions in his book of quadratures, and the methods of converging 

 series, and of drawing a curve line through any number of given points, being 

 then known to him. For when the method of fluxions proceeds not in finite 



