Elements of Algebra

CHAP. I. ADDITIONS. 485 ferent intermediate fractions, all greater than $a$, but which will continually diminish, and will be such as to express the quantity $a$ more exactly than could be done by any other fraction greater than $a$, and expressed in simpler terms. Farther, if $\beta$ is also a number greater than unity, we may likewise place before the fractions $\frac{B}{B'}$ the fractions $\frac{A+1}{1}$, $\frac{2A+1}{2}$, $\frac{3A+1}{3}$, &c. as far as $\frac{\beta A+1}{\beta}$, that is $\frac{B}{B'}$, and these fractions will have the same properties as the other intermediate fractions. In this manner, we have these two complete series of fractions converging towards the quantity $a$. *Fractions increasing and less than $a$.* $\frac{A}{A'}$, $\frac{B+A}{B'+A'}$, $\frac{2B+A}{2B'+A'}$, $\frac{3B+A}{3B'+A'}$, &c. $\frac{\gamma B+A}{\gamma B'+A'}$, $\frac{C}{C'}$, $\frac{D+C}{D'+C'}$, $\frac{2D+C}{2D'+C'}$, $\frac{3D+C}{3D'+C'}$, &c. $\frac{\epsilon D+C}{\epsilon D'+C'}$, $\frac{E}{E'}$, $\frac{F+E}{F'+E'}$, $\frac{2F+E}{2F'+E'}$, $\frac{3F+E}{3F'+E'}$, &c. *Fractions decreasing and greater than $a$.* $\frac{A+1}{1}$, $\frac{2A+1}{2}$, $\frac{3A+1}{3}$, &c. $\frac{\beta A+1}{\beta}$, $\frac{B}{B'}$, $\frac{C+B}{C'+B'}$, $\frac{2C+B}{2C'+B'}$, $\frac{\delta C+B}{\delta C'+B'}$, &c. $\frac{D}{D'}$, $\frac{E+D}{E'+D'}$, $\frac{2E+D}{2E'+D'}$, $\frac{3E+D}{3E'+D'}$, &c. If the quantity $a$ be irrational, or transcendental, the two preceding series will go on to infinity, since the series of fractions $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, &c. which in future we shall call *principal fractions*, to distinguish them from the *intermediate fractions*, goes on of itself to infinity. (Art. 10.) But if the quantity $a$ be rational, and equal to any fraction, $\frac{v}{v'}$, we have seen in that article, that the series in question will terminate, and that the last fraction of that series will be