VOL. LXXVI.] PHILOSOPHICAL TRANSACTIONS. 63 



In the 3d part Dr. W. remarks that the first algebraists divided quantities, and 

 extracted their roots, no further than the quantities themselves: they did not per- 

 ceive the utility of proceeding any further, otherwise the operation would have 

 been the same continued. Mr. Gregory St. Vincent, and Mr. Mercator divided, 

 and Sir Isaac Newton divided and extracted the roots of quantities, in which only 

 one unknown quantity x is contained, by the operations then used by arithme- 

 ticians, into series ascending or descending, according to the dimensions of x in 

 infinitum. They clearly saw the utility of it in finding the fluents of fluxions, 

 as Dr. Wallis and others some little time before had found the fluent of the 



^ m 



fluxion ax^'x; or, which is the same, the area of a curve whose ordinate is ax~ 

 and abscissa is x. M. Leibnitz asked from Mr. Newton the cases in which the 

 above-mentioned serieses would converge; for it would be altogether useless when 

 they diverge, and of little use when they converge slowly. 



To this question an answer, Dr. W. believes, was first given in the Medita- 

 tiones, viz. reduce the function to its lowest terms; and also in such a manner 

 that the quantities contained in the numerator and denominator may have no 

 denominator: make the denominator a = O, and every distinct irrational quan- 

 tity contained in it = O; and also every distinct irrational quantity h contained 

 in the numerator = O; then, let a be the least root, affirmative or negative, 

 (but not = O) of the above-mentioned resulting equations, the ascending series 

 will always converge, if the value of x is contained between a. and — a; but if ^ 

 be greater than a or — «, the above-mentioned series will not converge. If the 

 above-mentioned series, s, be multiplied into x, and its fluent found; then will 

 the series denoting the fluent contained between two values a and ^, of the quan- 

 tity X, converge, when a and b are both contained between a and — a : the 

 fluent always converges faster than the series s, the unknown quantity x having 

 the same values in both. The infinite series a^ + mdJ^^x -|- m . ^-^ a'"~V -j- 

 &c. = (a -f- a?) will always converge when a is greater than x, and diverge 

 when less; and consequently its convergency does not depend on the index m, 

 unless when a; = + a: and in the Meditationes Analyticae are given the cases in 

 which it converges or diverges when ip a = ar; and also if the series af -f- 

 maoif^'^ -f- &c. =z {x •{- aY descends according to the dimensions of x, when it 

 converges or diverges. 



Sir Isaac Newton, in the binomial theorem, reduced the power or root of a 

 binomial into a series proceeding according to the dimensions of the terms con- 

 tained in the binomial. M. de Moivre reduced the power or root of a multino- 

 mial into a like series; but in all cases, except the most simple, we must still 

 recur to the common division, extraction of roots, &c. Messrs. Euler, Mac- 

 laurin, and other mathematicians, finding tha the serieses before-mentioned 



