Elements of Algebra

CHAP. I. ADDITIONS. 467 we may also seek for the integer number, which shall be nearest the value of $b$; and this number being called $β$, we shall again have $b - β$ equal to a fraction less than unity; and, consequently, $\frac{1}{b-β}$ will be equal to a quantity greater than unity, which we may represent by $c$; so that, to assign the value of $c$, we have only to seek, in the same manner, for the integer number nearest to $c$, which being represented by $γ$, we shall have $c - γ$ equal to a quantity less than unity; and, consequently, $\frac{1}{c-γ}$ will be equal to a quantity, $d$, greater than unity, and so on. From which it is evident, that we may gradually exhaust the value of $a$, and that in the simplest and readiest manner; since we only employ integer numbers, each of which approximates, as nearly as possible, to the value sought. Now, since $\frac{1}{a-α} = b$, we have $a - α = \frac{1}{b}$, and $a = α + \frac{1}{b}$; likewise, since $\frac{1}{b-β}=c$, we have $b=β+\frac{1}{c}$; and, since $\frac{1}{c-γ} = d$, we have, in the same manner, $c = γ + \frac{1}{d}$, &c.; so that by successively substituting these values, we shall have $a \left\{ \begin{array}{l} =α+\frac{1}{b}, \\ \\ =α+\frac{1}{β+\frac{1}{c}}, \\ \\ =α+\frac{1}{β+\frac{1}{γ+\frac{1}{d}}}; \end{array} \right.$ and, in general, $a=α+\frac{1}{β+\frac{1}{γ+\frac{1}{δ+\text{&c.}}}}$; It is proper to remark here, that the numbers $α, β, γ$, &c. which represent, as we have shewn, the approximate integer values of the quantities $a, b, c$, &c. may be taken each in two different ways; since we may with equal propriety take, for the approximate integer value of a given quantity, either of the two integer numbers between which that quan- HH 2