Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/30

18 this kind are products; quotients, roots, rectangles, ſquares, cubes, ſquare and cubic ſides, and the like. Theſe quantities I here conſider as variable and indetermined, and increaſing or decreaſing as it were by a perpetual motion or flux; and I underſtand their momentaneous increments or decrements by the name of Moments; ſo that the increments may be eſteem'd as added, or affirmative moments; and the decrements as ſubducted, or negative ones. But take care not to look upon finite particles as ſuch. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the juſt naſcent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their firſt proportion as naſcent. It will be the ſame thing, if, inſtead of moments, we uſe either the Velocities of the increments and decrements (which may alſo be called the motions, mutations, and fluxions of quantities) or any finite quantities proportional to thoſe velocities. The coefficient of any generating ſide is the quantity which ariſes by applying the Genitum to that ſide.

Wherefore the ſenſe of the Lemma is, that if the moments of any quantities A, B, C, &c. increaſing or decreaſing by a perpetual flux, or the velocities of the mutations which are proportional to them, be called a, b, c, &c. the moment or mutation of the generated rectangle AB will be aB + bA; the moment of the generated content ABC will be aBC + bAC + cAB: and the moments of the generated powers, A2, A3, A4, A½, A3/2, A⅓, A⅔, A-1, A-2, A-½ will be 2aA, 3aA2, 4aA3, ½aA-½, 3/2aA½, ⅓aA-⅔, ⅔aA-⅓, -aA-2, -2aA-3, -½aA- 3/2 reſpectively. And in general, that the moment of any power A$$^\frac{n}{m}$$ will be $$\scriptstyle \frac{n}{m}$$aA$$^\frac{n-m}{m}$$. Alſo that the moment