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 urd. Now this fundamental Point one would think hould be very clearly made out, conidering how much is built upon it, and that its Influence extends throughout the whole Analyis. But let the Reader judge. This is given for Demontration. Suppoe the Product or Rectangle AB increaed 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 leer 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 oon as the Sides A and B are increaed 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 ubduct 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.