sbct.i.] DISSERTATION SECOND. 7 



small alteration, the law for the series of terms which 

 expresses the root of any binomial quantity whatsoever. 

 Thus he was put in possession of another valuable dis- 

 covery, the Binomial Theorem, and at the same time 

 perceived that this last was in reality, in the order of 

 things, placed before the other, and afforded a much easier 

 access to such quadratures than the method of interpola- 

 tion, which, though the first road, appeared now, neither 

 to be the easiest nor the most direct. 



It is but rarely that we can lay hold with certainty of the 

 thread by which genius has been guided in its first dis- 

 coveries. Here we are proceeding on the authority of 

 the author himself, for in a letter to Oldenburgh, 1 Se- 

 cretary of the Royal Society of London, he has entered 

 into considerable detail on this subject, adding (so ready 

 are the steps of invention to be forgotten), that the facts 

 would have entirely escaped his memory, if he had not 

 been reminded of them by some notes which he had 

 made at the time, and which he had accidentally fallen on. 

 The whole of the letter just referred to, is one of the most 

 valuable documents to be found in the history of inven- 

 tion. 



In all this, however, nothing occurs from which it can be 

 inferred that the method of fluxions had yet occurred to the 

 inventor. His discovery consisted in the method of reduc- 

 ing the value of y, the ordinate of a curve, into an infinite 

 series of the integer powers of x the abscissa, by division, 

 or the extraction of roots, that is, by the Binomial Theorem : 

 after which, the part of the area belonging to each term 

 could be assigned by the arithmetic of infinites, or other 

 methods already known. He has assured us himself, how- 

 ever, that the great principle of the new geometry was 



1 Commercium Epislolicum, Art. 55. 



