Elements of Algebra

CHAP. I. ADDITIONS. 487 This problem will be easily resolved by the theory which we have explained. We shall begin by reducing the fraction proposed to a continued fraction after the method of Art. 4, observing to take all the approximate values less than the real ones, in order that the numbers $β, γ, δ, \&c.$ may be all positive; then, by the assistance of the numbers found, $a, β, γ, \&c.$ we form, according to the formulæ of Art. 10, the fractions $\frac{A}{A'}, \frac{B}{B'}, \frac{C}{C'}, \&c.$ the last of which will necessarily be the same as the fraction proposed: because in that case the continued fraction terminates. Those fractions will alternately be less and greater than the given fraction, and will be successively expressed in greater terms; and farther, they will be such, that each of those fractions will be nearer the given fraction than any other fraction can be, which is expressed in terms less simple. So that by these means we shall have all the fractions, that will satisfy the conditions of the problem, expressed in lower terms than the fraction proposed. If we wish to consider separately the fractions which are less, and those which are greater, than the given fraction, we may insert between the above fractions as many *intermediate* fractions as we can, and form from them two series of converging fractions, the one all less, and the other all greater than the fraction proposed (Art. 16, 17, and 18); each of which series will have separately the same properties, as the series of principal fractions $\frac{A}{A'}, \frac{B}{B'}, \frac{C}{C'}, \&c.$ for the fractions in each series will be successively expressed in greater terms, and each of them will approximate nearer to the value of the fraction proposed, than could be done by any other fraction whether less, or greater, than the given fraction, but expressed in simpler terms. It may also happen, that one of the *intermediate* fractions of one series does not approximate towards the given fraction so nearly, as one of the fractions of the other series, although expressed in terms less simple than the former; for this reason, it is not proper to employ *intermediate* fractions, except when we wish to have the fractions sought either all less, or all greater, than the given fraction. 20. *Example* 1. According to M. de la Caille, the solar year is $365^d. 5^h. 48'. 49''$, and, consequently, longer by $5^h. 48'. 49''$ than the common year of $365^d$. If this difference