Elements of Algebra

126 ELEMENTS SECT. III. which gives $1$ for the product, as the nature of the thing required. 377. If we multiply the series which we found for the value of $\frac{1}{(a+b)^2}$ by $a+b$ only, the product ought to answer to the fraction $\frac{1}{a+b}$, or be equal to the series already found, namely, $\frac{1}{a} - \frac{b}{a^2} + \frac{b^2}{a^3} - \frac{b^3}{a^4} + \frac{b^4}{a^5}$, &c. and this the actual multiplication will confirm. $\frac{1}{a^2} - \frac{2b}{a^3} + \frac{3b^2}{a^4} - \frac{4b^3}{a^5} + \frac{5b^4}{a^6}$, &c. $a+b$ $\frac{1}{a} - \frac{2b}{a^2} + \frac{3b^2}{a^3} - \frac{4b^3}{a^4} + \frac{5b^4}{a^5}$, &c. $+ \frac{b}{a^2} - \frac{2b^2}{a^3} + \frac{3b^3}{a^4} - \frac{4b^4}{a^5}$, &c. $\frac{1}{a} - \frac{b}{a^2} + \frac{b^2}{a^3} - \frac{b^3}{a^4} + \frac{b^4}{a^5}$, &c. as required. SECTION III. Of Ratios and Proportions. CHAP. I. Of Arithmetical Ratio, or of the Difference between two Numbers. 378. Two quantities are either equal to one another, or they are not. In the latter case, where one is greater than the other, we may consider their inequality under two different points of view: we may ask, how much one of the quantities is greater than the other? Or we may ask, how many times the one is greater than the other? The