Elements of Algebra

148 ELEMENTS SECT. III. it is easy to perceive and enunciate the relation. For example, when the ratio $\frac{a}{b}$ has been reduced to the fraction $\frac{p}{q}$, we say $a:b=p:q$, or $a:b::p:q$, which is read, $a$ is to $b$ as $p$ is to $q$. Thus, the ratio of $6:3$ being $\frac{6}{3}$, or $2$, we say $6:3::2:1$. We have likewise $18:12::3:2$, and $24:18::4:3$, and $30:45::2:3$, &c. But if the ratio cannot be abridged, the relation will not become more evident; for we do not simplify it by saying $9:7::9:7$. 448. On the other hand, we may sometimes change the relation of two very great numbers into one that shall be more simple and evident, by reducing both to their lowest terms. Thus, for example, we can say, $28844:14422::2:1$; or, $10566:7044::3:2$; or, $57600:25200::16:7$. 449. In order, therefore, to express any relation in the clearest manner, it is necessary to reduce it to the smallest possible numbers; which is easily done, by dividing the two terms of it by their greatest common divisor. Thus, to reduce the relation $57600:25200$ to that of $16:7$, we have only to perform the single operation of dividing the numbers $57600$ and $25200$ by $3600$, which is their greatest common divisor. 450. It is important, therefore, to know how to find the greatest common divisor of two given numbers; but this requires a Rule, which we shall explain in the following chapter. CHAP. VII. *Of the Greatest Common Divisor of two given Numbers.* 451. There are some numbers which have no other common divisor than unity; and when the numerator and denominator of a fraction are of this nature, it cannot be reduced to a more convenient form*. The two numbers $48$ and $35$, for example, have no common divisor, though each has its own divisors; for which reason, we cannot express the relation $48:35$ more simply, because the division of two numbers by $1$ does not diminish them. * In this case, the two numbers are said to be prime to each other. See Art. $66$.