Elements of Algebra

294 ELEMENTS SECT. IV. $\frac{r}{p} = \frac{2q}{p} + 1$, or $r = 2q + p$; whence we infer that the double of each term, added to the preceding term, will give the succeeding one. If, therefore, we begin again with 0, 1, we shall have the series, $0, 1, 2, 5, 12, 29, 70, 169, 408, \&c.$ Whence it follows, that the value of $x$ will be expressed still more accurately by the following fractions: $x = \frac{1}{0}, \frac{2}{1}, \frac{5}{2}, \frac{12}{5}, \frac{29}{12}, \frac{70}{29}, \frac{169}{70}, \frac{408}{169}, \&c.$ which, consequently, will always approximate nearer and nearer the true value of $x = 1 + \sqrt{2}$; so that if we take unity from these fractions, the value of $\sqrt{2}$ will be expressed more and more exactly by the succeeding fractions: $\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \frac{99}{70}, \frac{239}{169}, \&c.$ For example, $\frac{99}{70}$ has for its square $\frac{9801}{4900}$, which differs only by $\frac{1}{4900}$ from the number 2. 795. This method is no less applicable to equations, which have a greater number of dimensions. If, for example, we have the equation of the third degree $x^3 = x^2 + 2x + 1$, we must make $x = \frac{q}{p}$, $x^2 = \frac{r}{p}$, and $x^3 = \frac{s}{p}$; we shall then have $s = r + 2q + p$; which shews how, by means of the three terms $p, q,$ and $r$, we are to determine the succeeding one, $s$; and, as the beginning is always arbitrary, we may form the following series: $0, 0, 1, 1, 3, 6, 13, 28, 60, 129, \&c.$ from which result the following fractions for the approximate values of $x$: $x = \frac{0}{0}, \frac{1}{0}, \frac{1}{1}, \frac{3}{1}, \frac{6}{3}, \frac{13}{6}, \frac{28}{13}, \frac{60}{28}, \frac{129}{60}, \&c.$ The first of these values would be very far from the truth; but if we substitute in the equation $\frac{60}{28}$, or $\frac{15}{7}$, instead of $x$, we obtain $\frac{3375}{343} = \frac{225}{49} + \frac{30}{7} + 1 = \frac{3388}{343}$, in which the error is only $\frac{13}{343}$. 796. It must be observed, however, that all equations are not of such a nature as to admit the application of this method; and, particularly, when the second term is wanting, it cannot be made use of. For example, let $x^2 = 2$; if we wished to make $x = \frac{q}{p}$, and $x^2 = \frac{r}{p}$, we should have