Elements of Algebra

478 ADDITIONS. CHAP. I. 12. If we now multiply cross-ways the terms of the consecutive fractions, in the series $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, &c. we shall find $BA' - AB' = 1$, $CB' - BC' = AB' - BA'$, $DC' - CD' = BC' - CB'$, &c. whence we conclude, in general, that $BA' - AB' = 1$ $CB' - BC' = -1$ $DC' - CD' = 1$ $ED' - DE' = -1$, &c. This property is very remarkable, and leads to several important consequences. First, we see that the fractions $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, &c. must be already in their lowest terms; for if, for example, $C$ and $C'$ had any common divisor, the integer numbers $CB' - BC'$ would also be divisible by that same divisor, which cannot be, since $CB' - BC' = -1$. Next, if we put the preceding equations into this form, $\frac{B}{B'} - \frac{A}{A'} = \frac{1}{A'B'}$ $\frac{C}{C'} - \frac{B}{B'} = -\frac{1}{C'B'}$ $\frac{D}{D'} - \frac{C}{C'} = \frac{1}{C'D'}$ $\frac{E}{E'} - \frac{D}{D'} = -\frac{1}{D'E'}$, &c. it is easy to perceive, that the differences between the adjoining fractions of the series $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, are continually diminishing, so that this is necessarily converging. Now, I say, that the difference between two consecutive fractions is as small as it is possible for it to be; so that there can be no other fraction whatever between those two fractions, unless it have a denominator greater than the denominators of them. Let us take, for example, the two fractions $\frac{C}{C'}$ and $\frac{D}{D'}$, the difference of which is $\frac{1}{C'D'}$, and let us suppose, if possible,