VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. 121 



tells you, p. 19, that by the same method, tangents may be drawn to mecha- 

 nical curves. And in his letter of Dec. 10, 1672, he adds, that problems about 

 the curvature of curves, geometrical or mechanical, are resolved by the same 

 method. Whence it is manifest, that he had then extended the method to the 

 second and third moments. For when the areas of curves are considered as 

 fluents, as is usual in this analysis, the ordinates express the first fluxions, the 

 tangents are given by the second fluxions, and the curvatures by the third. And 

 even in this Analysis, p. 16, where Mr. Newton says, momentum est superficies 

 cum de solidis, et linea cum de superficiebus, et punctum cum de lineis agitur, 

 it is all one as if he had said, that when solids are considered as fluents, their 

 moments are superficies, and the moments of those moments, or second 

 moments, are lines, and the moments of those moments, or third moments, 

 are points, in the sense of Cavallerius. And in his Principia Philosophiae, 

 where he frequently considers lines as fluents described by points, whose velo- 

 cities increase or decrease, the velocities are the first fluxions, and their increase 

 the second. And the problem, data asquatione fluentes quantitates involvente 

 fluxiones invenire et vice versa, extends to all the fluxions, as is manifest by the 

 examples of its solution, published by Dr. Wallis, tom. 2, p. 39 1, 392, 396. 

 And in lib. 2, Princip. prop. 14, he calls the second difference the difference of 

 moments. 



Now the better to know what kind of calculation Mr. Newton used, in or 

 before the year 1669, when he wrote this compendium of his Analysis, I will 

 here set down his demonstration of the first rule above-mentioned. " Let the 

 base AB, of any curve ab^, fig. 3, be = x, and the ordinate bd = y, and the 

 area abd = z, as before; likewise let Bj3 = 0, bk = v, and the rectangle BpHK 

 (ov) = the space bjS^d ; therefore, aj3 will be = x -\- 0, and A^fi = z •{• ov. 

 These things being premised, from assuming at pleasure the relation between x 



and z, 1/ is sought for as follows: let the equation ^jc^ = z, or -pc^ = zz, be 

 taken at pleasure : then substituting a? -|- o or Aj3 for x, and z -|- oi; or a^(^ for z, 

 you have -*- into a^ -\- 3x^o + 3a?o^ -j- 0^ = (from the nature of the curve) z^ -\- 

 2zov -|- oV: and taking away equals, viz. ^^ and zz, and dividing the rest by 0, 

 there remains -f- into 3x'^ -f- 3xo -\- 0^ = 2zy -{• ov^. Now if we suppose, that 

 Bj3 decreases in infinitum and vanishes, or that is nothing, v and y will be 

 equal, and the terms multiplied by will vanish; consequently there will remain 

 ^ X 3xx = 2zv, or ^xx (= zy) = ^^x^y, or a?*(= -) = y. Therefore, e contra, 



* 3 n — — 



if x^ = y, ^x^ will be = z, or universally, if — ~ X ax " =z z\ or putting 



VOL. VI. R 



