Elements of Algebra

472 ADDITIONS. CHAP. I. value of the given quantity, we must always take $\alpha$, $\beta$, $\gamma$, \ldots the integer numbers which are nearest the quantities $a$, $b$, $c$, \ldots whether they be less, or greater than those quantities. Now, it is easy to perceive that if, for example, we do not take for $\alpha$ the integer number which is nearest to $a$, either above or below it, the following number $\beta$ will necessarily be equal to unity; in fact, the difference between $a$ and $\alpha$ will then be greater than $\frac{1}{2}$, consequently, we shall have $b = \frac{1}{a-\alpha}$ less than $2$; therefore $\beta$ must be equal to unity. So that whenever we find the denominators in a continued fraction equal to unity, this will be a proof that we have not taken the preceding denominators as near as we might have done; and, consequently, that the fraction may be simplified by increasing, or diminishing those denominators by unity, which may be done by the preceding formulæ, without the necessity of going through the whole calculation. 8. The method in Art. $4$ may also serve for reducing every irrational, or transcendental quantity to a continued fraction, provided it be expressed before in decimals; but as the value in decimals can only be approximate, by augmenting the last figure by unity, we procure two limits, between which the true value of the given quantity must lie; and, in order that we may not pass those limits, we must perform the same calculation with both the fractions in question, and then admit into the continued fraction those quotients only which shall equally result from both operations. Let it be proposed, for example, to express by a continued fraction the ratio of the circumference of the circle to the diameter. This ratio expressed in decimals is, by the calculation of Vieta, as $3,1415926535$ is to $1$; so that we have to reduce the fraction $\frac{3,1415926535}{10000000000}$ to a continued fraction by the method above explained. Now, if we take only the fraction $\frac{3,14159}{100000}$ we find the quotients $3, 7, 15, 1, \ldots$ and if we take the greater fraction $\frac{3,14160}{100000}$ we find the quotients $3, 7, 16, \ldots$ so that the third quotient remains doubtful;