890. 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.