128 PHILOSOPHICAL TRANSACTIONS. [aNN0 17I4. 



When Mr. Newton in his letter dated June 13, 1676, had explained his 

 method of Series, he added : " From these things it appears how much the 

 bounds of analysis are enlarged by such sort of infinite equations: for, by 

 their means, it extends itself, I had almost said, to all problems, excepting 

 the numeral ones of Diophantus, and the like; yet it does not become quite 

 universal, unless by some other methods of finding out infinite series. For, 

 there are some problems in which we cannot come to infinite series either by 

 division, or extraction of simple or affected roots. But how we are to proceed 

 in these cases, I have not leisure to show; nor to mention some other things 

 I have invented about the reduction of infinite Series into finite ones, where 

 the nature of the thing will bear it. For I forbore writing on these specula- 

 tions, now for almost these 5 years past, because I have long since been tired 

 of them." To this M. Leibnitz in his letter of August 27, 1676, answered: 

 " What you seem to say, that almost all difficulties (excepting Diophantus's 

 Problems) may be reduced to infinite Series, I cannot come into; for there 

 are several Problems so intricate and perplexed, as not to depend either on 

 equations, or quadratures : such are, among a great many others, the Problems 

 of the inverse method of tangents." And Mr. Newton in his letter of Oct. 24, 

 1676, replied : " When I said, that all Problems might be solved; I would be 

 understood to mean those especially, about which mathematicians have already 

 employed themselves, or at least such as can admit of mathematical reasoning. 

 For it is true, others may be devised, so involved with perplexed conditions, 

 that we cannot sufficiently comprehend them, and much less bear the fatigue 

 of such prodigious calculations, as they may require. However, that I may 

 not seem to exceed the bounds of modesty, I can resolve both the inverse 

 Problems of tangents, and others more difficult. And for this purpose I make 

 use of a twofold method; the one more concise, and the other more general. 

 At present I thought proper to express both in transposed letters, that I might 

 not be obliged, on account of others finding out the same thing, to alter my 

 design in some respects; thus, baccdcelOeffh, &c. that is, one method consists 

 in finding the fluent, or flowing quantity, from an equation involving it with 

 its fluxion ; the other method, only in assuming a series for any unknown 

 quantity, from which the rest may be commodiously deduced ; and in comparing 

 the homologous terms of the equation resulting to find out the terms of the 

 assumed Series." By these two letters of Mr. Newton's, it is certain, that he 

 had then, or rather upwards of 5 years before, found out the reduction of 

 Problems to fluxional equations, and converging Series ; and by M. Leibnitz's 

 answer to the first of those letters, it is as certain that he had not then found 

 out the reduction of Problems, either to different equations, or to converging 

 Series. 



