Elements of Algebra

CHAP. I. ADDITIONS. 473 whence we see, that, in order to extend the continued fraction only beyond three terms, we must adopt a value of the circumference, which has more than six figures. If we take the value given by Ludolph to thirty-five decimal places, which is 3,14159, 26535, 89793, 23846, 26433, 83279, 50288; and if we work on with this fraction, as it is, and also with its last figure 8 increased by unity, we shall find the following series of quotients, 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 1; so that we shall have $\frac{Circumference}{Diameter} = 3 + \frac{1}{7} + \frac{1}{15} + \frac{1}{1} + \frac{1}{292} + \frac{1}{1} + \ldots$ And as there are here denominators equal to unity, we may simplify the fraction, by introducing negative terms, according to the formulæ of Art. 7, and shall find $\frac{Circumference}{Diameter} = 3 + \frac{1}{7} + \frac{1}{16} - \frac{1}{294} - \frac{1}{3} - \frac{1}{3} + \ldots$ $\frac{Circumference}{Diameter} = 3 + \frac{1}{7 + \frac{1}{16 + \frac{1}{-294 + \frac{1}{-3 + \frac{1}{-3 + \ldots}}}}}$ 9. We have elsewhere shewn how the theory of continued fractions may be applied to the numerical resolution of equations, for which other methods are imperfect and insufficient*. The whole difficulty consists in finding in any equation the nearest integer value, either above, or below the root sought; and for this I first gave some general rules, by which we may not only perceive how many real roots, positive or negative, equal or unequal, the proposed equation contains, but also easily find the limits of each of those roots, and even the limits of the real quantities which compose the imaginary roots. Supposing, therefore, that $x$ is the unknown quantity of the equation proposed, we seek first for the integer number which is nearest to the root sought, and calling that number $\alpha$, we have only, as in Art. 3, to make * See the Memoirs of the Academy of Berlin, for the years 1767 and 1768; and Le Gendre's Essai sur la Theorie des Nombres, page 133, first edition.