Page:Newton's Principia (1846).djvu/269

A, together with $\scriptstyle {\frac {1}{A}}$ drawn into a, will be the moment of 1, that is, nothing. Therefore the moment of $\scriptstyle {\frac {1}{A}}$, or of A^-1, is $\scriptstyle {\frac {-a}{A^{2}}}$. And generally since $\scriptstyle {\frac {1}{A^{n}}}$ into Aⁿ is 1, the moment of $\scriptstyle {\frac {1}{A^{n}}}$ drawn into Aⁿ together with $\scriptstyle {\frac {1}{A^{n}}}$ into naA^n-1 will be nothing. And, therefore, the moment of $\scriptstyle {\frac {1}{A^{n}}}$ or A^-n will be $\scriptstyle -{\frac {na}{A^{n+1}}}$.   Q.E.D.

Case 5. And since A^½ into A^½ is A, the moment of A^½ drawn into 2A^½ will be a (by Case 3); and, therefore, the moment of A^½ will be $\scriptstyle {\frac {a}{2A{\frac {1}{2}}}}$ or ½aA-½. And, generally, putting $\scriptstyle A^{\frac {m}{n}}$ equal to B, then A^m will be equal to Bⁿ, and therefore maA^m-1 equal to nbB^n-1, and maA^-1 equal to nbB^-1, or $\scriptstyle nbA^{-{\frac {m}{n}}}$; and therefore $\scriptstyle {\frac {m}{n}}aA^{\frac {m-n}{n}}$ is equal to b, that is, equal to the moment of $\scriptstyle A^{\frac {m}{n}}$.   Q.E.D.

Case 6. Therefore the moment of any generated quantity A^mBⁿ is the moment of A^m drawn into Bⁿ, together with the moment of Bⁿ drawn into A^m, that is, maA^m-1 Bⁿ + nbB^n-1 A^m; and that whether the indices m and n of the powers be whole numbers or fractions, affirmative or negative. And the reasoning is the same for contents under more powers.   Q.E.D.

Cor. 1. Hence in quantities continually proportional, if one term is given, the moments of the rest of the terms will be as the same terms multiplied by the number of intervals between them nd the given term. Let A, B, C, D, E, F, be continually proportional; then if the term C is given, the moments of the rest of the terms will be among themselves as -2A, -B, D, 2E, 3F.

Cor. 2. And if in four proportionals the two means are given, the moments of the extremes will be as those extremes. The same is to be understood of the sides of any given rectangle.

Cor. 3. And if the sum or difference of two squares is given, the moments of the sides will be reciprocally as the sides.

SCHOLIUM.

In a letter of mine to Mr. J. Collins, dated December 10, 1672, having described a method of tangents, which I suspected to be the same with Slusius's method, which at that time was not made public, I subjoined these words: This is one particular, or rather a Corollary, of a general me-