Elements of Algebra

CHAP. I. ADDITIONS. 481 $(B'c + A') < (B'(\gamma + 1) + A') < (C' + B')$, also $(C'd + B') < (C'(\delta + 1) + B') < (D' + c')$, &c. so that the errors in taking the fractions $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, &c. for the value of $a$, would be respectively less than $\frac{1}{A'B'}$, $\frac{1}{B'C'}$, $\frac{1}{C'D'}$, &c. but greater than $\frac{1}{A'(B' + A')}$, $\frac{1}{B'(C' + B')}$, $\frac{1}{C'(D' + C')}$, &c. which shews how small those errors are, and how they go on diminishing from one fraction to another. But farther, since the fractions $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, &c. are alternately less and greater than the quantity $a$, it is evident, that the value of that quantity will always be found between any two consecutive fractions. Now, we have already seen (Art. 12), that it is impossible to find, between two such fractions, any other fraction whatever, which has a denominator less than one of the denominators of those two fractions; whence we may conclude, that each of the fractions in question, express the quantity $a$ more exactly than any other fraction can, whose denominator is less than that of the succeeding fraction; that is to say, the fraction $\frac{C}{C'}$, for example, will express the value of $a$ more exactly than any other fraction $\frac{m}{n}$, in which $n$ would be less than $D'$. 15. If the approximate values $\alpha$, $\beta$, $\gamma$, &c. are all, or partly, greater than the real values, then some of those numbers will necessarily be negative (Art. 3), which will also render negative some terms of the series $A$, $B$, $C$, &c. $A'$, $B'$, $C'$, &c. consequently, the differences between the fractions $\frac{A}{A'}$, $\frac{B}{B'}$, $\frac{C}{C'}$, &c. and the quantity $a$, will no longer be alternately positive and negative, as in the case of the preceding articles: so that those fractions will no longer have the advantage of giving the limits in *plus* and *minus* of the quantity $a$; an advantage which appears to me of very great importance, and which must therefore in practice make us always prefer those continued fractions, in which the denominators are all positive. Hence, in what follows, we shall only attempt an investigation of fractions of this kind. II