Elements of Algebra

CHAP. VI. OF ALGEBRA. 147 444. So that relations only differ according as their ratios are different; and there are as many different kinds of geometrical relations as we can conceive different ratios. The first kind is undoubtedly that in which the ratio becomes unity. This case happens when the two numbers are equal, as in $3:3 :: 4:4 :: a:a$; the ratio is here $1$, and for this reason we call it the relation of equality. Next follow those relations in which the ratio is another whole number. Thus, $4:2$ the ratio is $2$, and is called double ratio; $12:4$ the ratio is $3$, and is called triple ratio: $24:6$ the ratio is $4$, and is called quadruple ratio, &c. We may next consider those relations whose ratios are expressed by fractions; such as $12:9$, where the ratio is $\frac{4}{3}$, or $1\frac{1}{3}$; and $18:27$, where the ratio is $\frac{2}{3}$, &c. We may also distinguish those relations in which the consequent contains exactly twice, thrice, &c. the antecedent: such are the relations $6:12$, $5:15$, &c. the ratio of which some call subduple, subtriple, &c. ratios. Farther, we call that ratio rational which is an expressible number; the antecedent and consequent being integers, such as $11:7$, $8:15$, &c. and we call that an irrational or surd ratio, which can neither be exactly expressed by integers, nor by fractions, such as $\sqrt{5}:8$, or $4:\sqrt{3}$. 445. Let $a$ be the antecedent, $b$ the consequent, and $d$ the ratio. We know already, that $a$ and $b$ being given, we find $d = \frac{a}{b}$: if the consequent $b$ were given with the ratio, we should find the antecedent $a = bd$, because $bd$ divided by $b$ gives $d$: and lastly, when the antecedent $a$ is given, and the ratio $d$, we find the consequent $b = \frac{a}{d}$; for, dividing the antecedent $a$ by the consequent $\frac{a}{d}$, we obtain the quotient $d$, that is to say, the ratio. 446. Every relation $a:b$ remains the same, if we multiply or divide the antecedent and consequent by the same number, because the ratio is the same: thus, for example, let $d$ be the ratio of $a:b$, we have $d = \frac{a}{b}$; now the ratio of the relation $na:nb$ is also $\frac{na}{nb} = d$, and that of the relation $\frac{a}{n} : \frac{b}{n}$ is likewise $\frac{na}{nb} = d$. 447. When a ratio has been reduced to its lowest terms, L 2