Elements of Algebra

CHAP. I. ADDITIONS. 477 shall then have $B > A$, but $c < B$, if it happens that the fraction $\frac{A}{B}$ has a different sign from $\gamma$; which is evident from the equation $\frac{C}{B} = \gamma + \frac{A}{B}$; because, in that case, $\gamma + \frac{A}{B}$ will be a number less than unity. Now, I say, in this case, we must have $D > B$; for since $\gamma = \pm 1$, we shall have (Art. 10), $c = \pm 1 + \frac{1}{d}$, and $c - \frac{1}{d} = \pm 1$; but as $c$ and $d$ are quantities greater than unity (Art. 3), it is evident, that this equation cannot subsist, unless $c$ and $d$ have the same signs; therefore, since $\gamma$ and $\delta$ are the approximate integer values of $c$ and $d$, these numbers $\gamma$ and $\delta$ must also have the same sign. Farther, the fraction $\frac{C}{B} = \gamma + \frac{A}{B}$ must have the same sign as $\gamma$, because $\gamma$ is an integer number, and $\frac{A}{B}$ a fraction less than unity; therefore $\frac{C}{B}$, and $\delta$, will be quantities of the same sign; consequently, $\frac{\delta C}{B}$ will be a positive quantity. Now, we have $\frac{D}{C} = \delta + \frac{B}{C}$; and hence, multiplying by $\frac{C}{B}$, we shall have $\frac{D}{B} = \frac{\delta C}{B} + 1$; so that $\frac{\delta C}{B}$ being a positive quantity, it is evident that $\frac{D}{B}$ will be greater than unity; and therefore $D > B$. Hence we see, that, if in the series $A, B, C$, &c. there be one term less than the preceding, the following will necessarily be greater; so that putting aside those less terms, the series will always go on increasing. Besides, if we choose, we may always avoid this inconvenience, either by taking the numbers $\alpha, \beta, \gamma$, &c. positive, or by taking them different from unity, which may always be done. The same reasonings apply to the series $A', B', C'$, &c. in which we have likewise $\frac{B'}{A'} = \beta$, $\frac{C'}{B'} = \gamma + \frac{A'}{B'}$, $\frac{D'}{C'} = \delta + \frac{B'}{C'}$, &c. whence we may form conclusions similar to the preceding.