Elements of Algebra

CHAP. I. OF ALGEBRA. 127 results which constitute the answers to these two questions are both called relations, or ratios. We usually call the former an arithmetical ratio, and the latter a geometrical ratio, without however these denominations having any connexion with the subject itself. The adoption of these expressions is entirely arbitrary. 379. It is evident, that the quantities of which we speak must be of one and the same kind; otherwise we could not determine any thing with regard to their equality, or inequality: for it would be absurd to ask if two pounds and three ells are equal quantities. So that in what follows, quantities of the same kind only are to be considered; and as they may always be expressed by numbers, it is of numbers only that we shall treat, as was mentioned at the beginning. 380. When of two given numbers, therefore, it is required how much the one is greater than the other, the answer to this question determines the arithmetical ratio of the two numbers; but since this answer consists in giving the difference of the two numbers, it follows, that an arithmetical ratio is nothing but the difference between two numbers; and as this appears to be a better expression, we shall reserve the words ratio and relation to express geometrical ratios. 381. As the difference between two numbers is found by subtracting the less from the greater, nothing can be easier than resolving the question how much one is greater than the other: so that when the numbers are equal, the difference being nothing, if it be required how much one of the numbers is greater than the other, we answer, by nothing; for example, $6$ being equal to $2 \times 3$, the difference between $6$ and $2 \times 3$ is $0$. 382. But when the two numbers are not equal, as $5$ and $3$, and it is required how much $5$ is greater than $3$, the answer is, $2$; which is obtained by subtracting $3$ from $5$. Likewise $15$ is greater than $5$ by $10$; and $20$ exceeds $8$ by $12$. 383. We have therefore three things to consider on this subject; 1st. the greater of the two numbers; 2d. the less; and 3d. the difference: and these three quantities are so connected together, that any two of the three being given, we may always determine the third. Let the greater number be $a$, the less $b$, and the difference $d$; then $d$ will be found by subtracting $b$ from $a$, so that $d = a - b$; whence we see how to find $d$, when $a$ and $b$ are given.