Elements of Algebra

CHAP. II. OF ALGEBRA. 129 CHAP. II. *Of Arithmetical Proportion.* 390. When two arithmetical ratios, or relations, are equal, this equality is called an arithmetical proportion. Thus, when $a - b = d$, and $p - q = d$, so that the difference is the same between the numbers $p$ and $q$, as between the numbers $a$ and $b$, we say that these four numbers form an arithmetical proportion; which we write thus, $a - b = p - q$, expressing clearly by this, that the difference between $a$ and $b$ is equal to the difference between $p$ and $q$. 391. An arithmetical proportion consists therefore of four terms, which must be such, that if we subtract the second from the first, the remainder is the same as when we subtract the fourth from the third; thus, the four numbers 12, 7, 9, 4, form an arithmetical proportion, because $12 - 7 = 9 - 4$. 392. When we have an arithmetical proportion, as $a - b = p - q$, we may make the second and third terms change places, writing $a - p = b - q$: and this equality will be no less true; for, since $a - b = p - q$, add $b$ to both sides, and we have $a = b + p - q$: then subtract $p$ from both sides, and we have $a - p = b - q$. In the same manner, as $12 - 7 = 9 - 4$, so also $12 - 9 = 7 - 4$*. 393. We may in every arithmetical proportion put the second term also in the place of the first, if we make the same transposition of the third and fourth; that is, if $a - b = p - q$, we have also $b - a = q - p$; for $b - a$ is the negative of $a - b$, and $q - p$ is also the negative of $p - q$; and thus, since $12 - 7 = 9 - 4$, we have also, $7 - 12 = 4 - 9$. 394. But the most interesting property of every arithmetical proportion is this, that the sum of the second and third term is always equal to the sum of the first and fourth. X This property, which we must particularly consider, is expressed also by saying that the sum of the means is equal to the sum of the extremes. Thus, since $12 - 7 = 9 - 4$, we have $7 + 9 = 12 + 4$; the sum being in both cases 16. * To indicate that those numbers form such a proportion, some authors write them thus: $12 . 7 :: 9 . 4$. K