Elements of Algebra

146 ELEMENTS SECT. III. CHAP. VI. *Of Geometrical Ratio.* 440. The Geometrical ratio of two numbers is found by resolving the question, *How many times is one of those numbers greater than the other?* This is done by dividing the one by the other; and the quotient will express the ratio required. 441. We have here three things to consider; 1st, the first of the two given numbers, which is called the antecedent; 2dly, the other number, which is called the consequent; 3dly, the ratio of the two numbers, or the quotient arising from the division of the antecedent by the consequent. For example, if the relation of the numbers $18$ and $12$ be required, $18$ is the antecedent, $12$ is the consequent, and the ratio will be $\frac{18}{12} = 1\frac{1}{2}$; whence we see that the antecedent contains the consequent once and a half. 442. It is usual to represent geometrical relation by two points, placed one above the other, between the antecedent and the consequent. Thus, $a:b$ means the geometrical relation of these two numbers, or the ratio of $a$ to $b$. We have already remarked that this sign is employed to represent division*, and for this reason we make use of it here; because, in order to know the ratio, we must divide $a$ by $b$; the relation expressed by this sign being read simply, $a$ is to $b$. 443. Relation therefore is expressed by a fraction, whose numerator is the antecedent, and whose denominator is the consequent; but perspicuity requires that this fraction should be always reduced to its lowest terms: which is done, as we have already shewn, by dividing both the numerator and denominator by their greatest common divisor. Thus, the fraction $\frac{18}{12}$ becomes $\frac{3}{2}$, by dividing both terms by $6$. The algebraists of the sixteenth and seventeenth centuries paid great attention to these different kinds of numbers and their mutual connexion, and they discovered in them a variety of curious properties; but as their utility is not great, they are now seldom introduced into the systems of mathematics. F. T. * It will be observed that we have made use of the symbol $\div$ for division, as is now usually done in books on this subject.