*A^c 



CYCLOPEDIA: 



\r [J 



OR, A NEW 



UNIVERSAL DICTIONARY 



ARTS and SCIENCES. 



INCREMENTS. 



INCREMENTS, Method of, called by the French 

 Cah-iil (hs Differences Finies, is a branch of analyfis in- 

 vi'nted by- the learned Dr. Brook Tavlor, particularly ufefiil 

 in the fummation of feries, and applicable to feveral fubjeits 

 of mathematical invcftifration, where fcarcely any other me- 

 thod can be fucccfsfully employed. Mo)itucla obferves, 

 that had the human mind always purfued the path which 

 appears the moft natural, the theory of increrrcr.ts, or of 

 iinite difFerencrs, would have preceded that of fluxions, or 

 the differential calculus ; as it feems more natural for the 

 mind to be carried from the confideration of finite differ- 

 ences, to that of differences uidefinitely fmall, than that 

 the latter Ihould b? tlie precurfor of the former. Such, 

 however, was the faft ; for the liril diltinft notions of the 

 method of increments did not appear till the year 1 715, in 

 a work entitled " Methodus Incfementorum.'' .kc. by Dr. 

 Braok Taylor, in which both the direft and inverfe method 

 of increments are treated of in a very learned manner, and 

 ;m application of the fame to various interefHng problems ; 

 but the novelty of the fubjcft, and the concife mode cf ex- 

 preflion employed by its author, tojrcther with the very com- 

 plicated notation, rendered the work nearly unintelligible to 

 any man Icfs (killed in analyfis than the author hirafelf ; even 

 the enunciation of fome of the propofitions requires the 

 greateft poffible attention in order to comprehend their 

 jneaning ; but in other refpcCis, t!ie work bears (Irong and 

 evident marks of the lofty genius of its author, and contains, 

 in tlic fccond part, many very excellent applications of the 

 preceding theory to the folution of fomc of the moft inte- 

 refting and celebrated mathematical problems. Such, Iiow- 

 ever, bein£r the intricacy of the original work, it neceffarily 

 follows that it could only be read by tlie very firll rate mu- 

 VoL. XIX. 



thematicians ; and it was therefore fortunate that any of 

 them would condefccnd to illuftrate a fubjec^ in which they 

 could only aft a fecondary part ; fuch a perfon was however 

 found in M. Nichole, of the Royal Academy of Sciences, 

 who, having very early been in pofFefiion of a copy of Dr. 

 Taylor's work, and perceiving, at once, its general utility, 

 he undertook the taflc of illuftrating the principles upon 

 which it relied; and, by flmplifying the notation and opera- 

 tions, rendered it intelligible to readers of an inferior order: 

 his iirft paper on the fubjeft was publifhed in the Memoirs 

 of the Academy for 171 7, which was afterwards followed 

 by two others in 1723 and 1724. Dr. Taylor himfelf alfo, 

 in the Philofophical Tranfaftions, undertook an explanation 

 of certain parts of his work, and its farther application to 

 fome kinds of feries beyond thofe treated of in the original : 

 the fame was alfo done by M. Montmort, in the Tranfac- 

 tions for the years 1719 and 1720 ; which latter gentleman, 

 it feems, had conceived fomc idea of the theory before Dr. 

 Taylors work appeared ; and a trifling altercation took 

 place between tl»em as to the originality of fome of the no- 

 tions which was claimed by both parties. In 1763, Emer- 

 fon publilhcd his *■ Method of Increments ;" a work which., 

 at leaft, does this author as much credit as any that he ever 

 produced ; but the notation refembles, in a great meafure, 

 that of Dr. Taylor, which renders it now almoft obfolete. 

 Emerfon appears to have been extremely anxious to 

 bring the theory of increments to perfeftion ; and earneftly 

 urged thofe who were qualified for the tafli, to purfue the 

 paths he had opened to them. " I cannot,'' fays he, " pro- 

 mife that I ftiall have time and leifure hereafter to profe- 

 cute this fubjeft any farther. And as I have an earnefl de- 

 fire of advancing truth and improving fcience, let me here 



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