Elements of Algebra

130 ELEMENTS SECT. III. 395. In order to demonstrate this principal property, let $a-b=p-q$; then if we add to both $b+q$, we have $a+q=b+p$; that is, the sum of the first and fourth terms is equal to the sum of the second and third: and, inversely, if four numbers, $a, b, p, q$, are such, that the sum of the second and third is equal to the sum of the first and fourth; that is, if $b+p=a+q$, we conclude, without a possibility of mistake, that those numbers are in arithmetical proportion, and that $a-b=p-q$; for, since $a+q=b+p$, if we subtract from both sides $b+q$, we obtain $a-b=p-q$. Thus, the numbers 18, 13, 15, 10, being such, that the sum of the means ($13+15=28$) is equal to the sum of the extremes ($18+10=28$), it is certain that they also form an arithmetical proportion; and, consequently, that $18-13=15-10$. 396. It is easy, by means of this property, to resolve the following question. The first three terms of an arithmetical proportion being given, to find the fourth? Let $a, b, p$, be the first three terms, and let us express the fourth by $q$, which it is required to determine, then $a+q=b+p$; by subtracting $a$ from both sides, we obtain $q=b+p-a$. Thus, the fourth term is found by adding together the second and third, and subtracting the first from that sum. Suppose, for example, that 19, 28, 13, are the three first given terms, the sum of the second and third is 41; and taking from it the first, which is 19, there remains 22 for the fourth term sought, and the arithmetical proportion will be represented by $19-28=13-22$, or by $28-19=22-13$, or, lastly, by $28-22=19-13$. 397. When, in an arithmetical proportion, the second term is equal to the third, we have only three numbers; the property of which is this, that the first, minus the second, is equal to the second, minus the third; or that the difference between the first and second number is equal to the difference between the second and third. The three numbers 19, 15, 11, are of this kind, since $19-15=15-11$. 398. Three such numbers are said to form a continued arithmetical proportion, which is sometimes written thus, $19:15:11$. Such proportions are also called *arithmetical progressions*, particularly if a greater number of terms follow each other according to the same law. An arithmetical progression may be either *increasing*, or *decreasing*. The former distinction is applied when the terms go on *increasing*; that is to say, when the second exceeds the first, and the third exceeds the second by the