Elements of Algebra

496 ADDITIONS. CHAP. II. the same principles, it will be proper to treat of them in a direct manner, without supposing any thing of what has been before demonstrated: by which means we shall have the satisfaction of seeing how necessarily these subjects lead to the theory of Continued Fractions. Besides, this theory will be rendered much more evident, and receive from it a greater degree of perfection. 23. Problem 1. A positive quantity $a$, whether rational or not, being given, to find two integer positive numbers, $p$ and $q$, prime to each other; such, that $p - aq$ (abstracting from the sign), may be less than it would be, if we assigned to $p$ and $q$ any less values whatever. In order to resolve this problem directly, we shall begin by supposing that we have already found values of $p$ and $q$, which have the requisite conditions; wherefore, assuming for $r$ and $s$, any integer positive numbers less than $p$ and $q$, the value of $p - aq$ must be less than that of $r - as$, abstracting from the signs of these two quantities; that is to say, taking them both positive: now, if the numbers $r$ and $s$ be such, that $ps - qr = \pm 1$, (the upper sign applying when $p - aq$ is a positive number, and the under, when $p - aq$ is a negative number) we may conclude, in general, that the value of the expression $y - az$ will always be greater (abstracting from the sign) than that of $p - aq$, as long as we give to $z$ and $y$ only integer values, less than those of $p$ and $q$, we may hence draw the following conclusion. First, it is evident, that we may suppose, in general, $y = pt + ru$, and $z = qt + su$, $t$ and $u$ being two unknown quantities. Now, by the resolution of these equations, we have $t = \frac{sy - rz}{ps - qr}$, and $u = \frac{qy - pz}{qr - ps}$; then, since $ps - qr = \pm 1$, $t = \pm (sy - rz)$, and $u = \pm (qy - pz)$; it is evident, that $t$ and $u$ will always be integer numbers, since $p, q, r, s, y,$ and $z$ are supposed to be integers. Therefore, since $t$ and $u$ are integer numbers, and $p, q, r, s$ integer positive numbers, it is evident, in order that the values of $y$ and $z$ may be less than those of $p$ and $q$, that the numbers $t$ and $u$ must necessarily have different signs. Now, I say, that the value of $r - as$ will also have a different sign from that of $p - aq$; for, making $p - aq = P$, and $r - as = R$, we shall have $\frac{p}{q} = a + \frac{P}{q}$, $\frac{r}{s} = a + \frac{R}{s}$; but the equation, $ps - qr = \pm 1$, gives $\frac{p}{q} - \frac{r}{s} = \pm \frac{1}{qs}$;