Elements of Algebra

CHAP. V. OF ALGEBRA. 89 the quotient is expressed, as we have already observed, by a fraction: thus, if we have to divide 1 by $1-a$, we obtain the fraction $\frac{1}{1-a}$. This, however, does not prevent us from attempting the division according to the rules that have been given, nor from continuing it as far as we please; and we shall not fail thus to find the true quotient, though under different forms. 290. To prove this, let us actually divide the dividend 1 by the divisor $1-a$, thus: $1-a)1 \quad * \quad (1+\frac{a}{1-a}$ $\underline{1-a}$ remainder $a$ or, $1-a)1 \quad * \quad (1+a+\frac{a^2}{1-a}$ $\underline{1-a}$ $a$ $\underline{a-a^2}$ remainder $a^2$ To find a greater number of forms, we have only to continue dividing the remainder $a^2$ by $1-a$; $1-a)a^2 \quad * \quad (a^2+\frac{a^3}{1-a}$ $\underline{a^2-a^3}$ $a^3$ covered by Mercator, about the middle of the last century; and soon after, Newton discovered those which are derived from the extraction of roots, and which are treated of in Chapter XII. of this section. This theory has gradually received improvements from several other distinguished mathematicians. The works of James Bernoulli, and the second part of the "Differential Calculus" of Euler, are the books in which the fullest information is to be obtained on these subjects. There is likewise in the Memoirs of Berlin for 1768, a new method by M. de la Grange for resolving, by means of infinite series, all literal equations of any dimensions whatever. F. T.