Elements of Algebra

474 ADDITIONS. CHAP. I. $x = \alpha + \frac{1}{y}$; $x, y, z, \&c.$ representing here what was denoted in that article by $a, b, c, \&c.$ and substituting this value instead of $x$, we shall have, after removing the fractions, an equation of the same degree in $y$, which must have at least one positive, or negative root greater than unity. After seeking therefore for the approximate integer value of the root, and calling that value $\beta$, we shall then make $y = \beta + \frac{1}{z}$, which will give an equation in $z$, having likewise a root greater than unity, whose approximate integer value we must next seek, and so on. In this manner, the root required will be found expressed by the continued fraction $\alpha + \frac{1}{\beta + \frac{1}{\gamma + \frac{1}{\delta + \&c.}}}$ which will be terminated, if the root is commensurable; but will necessarily go on *ad infinitum*, if it be incommensurable. In the Memoirs just referred to, there will be found all the principles and details necessary to render this method and its application easy, and even different means of abridging many of the operations which it requires. I believe that I have scarcely left any thing farther to be said on this important subject. With regard to the roots of equations of the second degree, we shall afterwards give (Art. 33 et seq.) a particular and very simple method of changing them into continued fractions. 10. After having thus explained the genesis of continued fractions, we shall proceed to shew their application, and their principal properties. It is evident, that the more terms we take in a continued fraction, the nearer we approximate to the true value of the quantity which we have expressed by that fraction; so that if we successively stop at each term of the fraction, we shall have a series of quantities converging towards the given quantity. Thus, having reduced the value of $a$ to the continued fraction, $\alpha + \frac{1}{\beta + \frac{1}{\gamma + \frac{1}{\delta + \&c.}}}$ we shall have the quantities,