Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/25

Fluxions of all other Products and Powers; be the Coefficients or the Indexes what they will, integers or fractions, rational or ſurd. Now this fundamental Point one would think ſhould be very clearly made out, conſidering how much is built upon it, and that its Influence extends throughout the whole Analyſis. But let the Reader judge. This is given for Demonſtration.^[1] Suppoſe the Product or Rectangle AB increaſed by continual Motion: and that the momentaneous Increments of the Sides A and B are a and b. When the Sides A and B were deficient, or leſſer by one half of their Moments, the Rectangle was $\textstyle {{\overline {A-{\frac {1}{2}}a}}\times {\overline {B-{\frac {1}{2}}b}}}$ i.e. $\textstyle {AB-{\frac {1}{2}}aB-{\frac {1}{2}}bA+{\frac {1}{4}}ab}$. And as ſoon as the Sides A and B are increaſed by the other two halves of their Moments, the Rectangle becomes $\textstyle {{\overline {A+{\frac {1}{2}}a}}\times {\overline {B+{\frac {1}{2}}b}}}$ or $\textstyle {AB+{\frac {1}{2}}aB+{\frac {1}{2}}bA+{\frac {1}{4}}ab}$. From the latter Rectangle ſubduct the former, and the remaining difference will be aB + bA. Therefore the Increment of the Rectangle generated by the intire Increments a and b is aB + bA.

Q.E.D.

1. Naturalis Philosophiæ principia mathematica, I. 2. lem. 2.