646 PHILOSOPHICAL TRANSACTIONS, [anNO I74'2-3. 



tions are found to agree nearly with those which have been deduced from other 

 theories, and from astronomical observations. 



A fluid being supposed to gravitate towards two given centres with equal and 

 invariable forces, it is shown, that the figure of the fluid must be that of an 

 oblong spheroid, and that those two centres must be the foci of the generating 

 ellipse. The nature of the figure is also shown, when the fluid gravitates to- 

 wards several centres, or when it revolves on its axis; but these are mentioned 

 briefly, because such theories are of little or no use for discovering the figures 

 of the planets. 



In the 12th chapter, the author proceeds to consider the more concise me- 

 thods, by which the fluxions of quantities are usually determined, and to de- 

 duce general theorems more immediately applicable to the resolution of geome- 

 trical and philosophical problems, In the method of infinitesimals, the ele- 

 ment, by which any quantity increases or decreases, is supposed to become in- 

 finitely small, and is generally expressed by two or more terms, some of which 

 become infinitely less than the rest, and therefore being neglected as of no im- 

 portance, the remaining terms form, what is called the difi^erence of the quan- 

 tity proposed. The terms that are neglected in this manner, are the very same 

 which arise in consequence of the acceleration or retardation of the generating 

 motion, during the infinitely small time in which the element is generated; 

 and therefore these differences are in the same ratio to each other as the gene- 

 rating motions or fluxions. Hence the conclusions in this method are accu- 

 rately true, without even an infinitely small error, and agree with those that are 

 deduced by the method of fluxions. 



It is usual in this method to consider a curve as a polygon of an infinite 

 number of sides, which, being produced, give the tangents of the curve, and, 

 by their inclination to each other, measure its curvature. But it is necessary in 

 some cases, if we would avoid error, to resolve the element of the curve into 

 several infinitely small parts, or even sometimes into infinitesimals of the second 

 order; and errors that might otherwise arise in its application, may, with due 

 care, be corrected by a proper use of this method itself, of which some in- 

 stances are given. If we were to suppose, for example, the least arc that can 

 be described by a pendulum to coincide with its chord, the time of the vibra- 

 tion derived from this supposition will be found erroneous; but by resolving 

 that arc into more and more infinitely small parts, we approach to the true time 

 in which it is described. By supposing the tangent of the curve to be the pro- 

 duction of the rectilineal element of the curve, the subtense of the angle of 

 contact is found equal to the second dift'erence or fluxion of the ordinate; but 



