Elements of Algebra

484 ADDITIONS. CHAP. I. will be less than that of the same fraction and the fraction $\frac{B}{B'}$; now, we find $\frac{A}{A'} - \frac{B}{B'} = \frac{1}{A'B'}$ $\frac{B+A}{B'+A'} - \frac{B}{B'} = \frac{1}{(B'+A')B'}$ $\frac{2B+A}{2B'+A'} - \frac{B}{B'} = \frac{1}{(2B'+A')B'}$ $\frac{3B+A}{3B'+A'} - \frac{B}{B'} = \frac{1}{(3B'+A')B'}$ $\frac{C}{C'} - \frac{B}{B'} = \frac{1}{C'B'}$ Therefore, since these differences are also equal to unity divided by the product of the denominators, we may apply to them the reasoning of Article 12, to prove that no fraction, $\frac{m}{n}$, can fall between any one of the fractions $\frac{A}{A'}$, $\frac{B+A}{B'+A'}$, $\frac{2B+A}{2B'+A'}$, &c. and the fraction $\frac{B}{B'}$, if the denominator $n$ be less than that of the same fraction; whence it follows, that each of those fractions approximates towards the quantity $a$ nearer than any other fraction less than $a$, and having a less denominator; that is to say, expressed in simpler terms. 18. In the preceding Article, we have only considered the intermediate fractions between $\frac{A}{A'}$ and $\frac{C}{C'}$; but the same will be found true of the intermediate fractions between $\frac{C}{C'}$ and $\frac{E}{E'}$, between $\frac{E}{E'}$ and $\frac{G}{G'}$, &c. if $\epsilon$, $\eta$, &c. are numbers greater than unity. We may also apply what we have just said with respect to the first series $\frac{A}{A'}$, $\frac{C}{C'}$, &c. to the other series $\frac{B}{B'}$, $\frac{D}{D'}$, $\frac{F}{F'}$, &c. so that if the numbers, $\delta$, $\zeta$, are greater than unity, we may insert between the fractions $\frac{B}{B'}$ and $\frac{D}{D'}$, $\frac{D}{D'}$ and $\frac{F}{F'}$, &c. dif-