i88 HISTORY OF SCIENCE. 



kinds of algebraical series, and the discovery of facile methods of find- 

 ing the terms of such series. Finally, he was led -to the discovery of 

 the celebrated Binomial Theorem, which gave a method of expressing 

 any root of a quantity consisting of two terms, by means of a series 

 composed of those terms combined according to a given law. This 

 rule was immediately applicable to numberless cases of ordinary mathe- 

 matical analyses, and especially to problems involving the areas of 

 curves. Newton did not publish these discoveries to the world, but 

 showed to Dr. Barrow and other friends, some years before 1669, a 

 MS. treatise, in which series were employed and general methods 

 clearly indicated. In the year 1673, Leibnitz, the distinguished ma- 

 thematician, historian, and metaphysician, visited England, where he 

 made the acquaintance of several of the most eminent mathematicians. 

 Some years afterwards Newton, at the request of Oldenburg, the secre- 

 tary of the Royal Society, wrote an account of his method of determining 

 the areas of curved figures, together with a description of his invention 

 of fluxions, concealed under an anagram. This was, by Oldenburg, 

 sent to Leibnitz, and in 1677 the latter returned to Oldenburg a short 

 account of an equally general method of calculation which he had 

 himself invented, and which he called the Differential Calculus. He 

 gave in his communication the notation and the principal rules for this 

 calculus, but the first published account of it did not appear until 1684. 

 Though Newton had invented fluxions previous to his temporarily 

 leaving Cambridge in 1666, it was only in the first edition of the "Prin- 

 cipia" which appeared in 1687, that he for the first time made public 

 the fundamental principle of the fluxionary calculus, without giving its 

 notation. Leibnitz, in his first account of his differential calculus, de- 

 scribed the notation he had invented, and pointed out its application 

 to drawing tangents to curves, and to determining maxima and minima. 

 He makes a reference to a similar calculus of Newton, and does not 

 claim for himself the merit of having been the first inventor of the 

 general method. Newton, in a note in his " Principia" says : " In a 

 correspondence which took place, about ten years ago, between that 

 very skilful geometer G. G. Leibnitz and myself, I announced to him 

 that I possessed a method of determining maxima and minima, of 

 drawing tangents, and of performing similar operations, which was 

 applicable to rational and irrational quantities, and I concealed the 

 same in transposed letters involving this sentence (data equatione quin- 

 teunque fluentes quantitates in volvente fluxiones invenire et vice versa). 

 That illustrious man replied that he also had fallen on a method of 

 the same kind, and he communicated to me his method, which scarcely 

 differed from mine except in the notation and in the idea of the gene- 

 ration of the quantities." These statements seem to show that each of 

 these eminent men gave the other, at that time, credit for having in- 

 dependently arrived at the same discovery as himself. A few years 

 later, however, there arose between the two distinguished philosophers 



