Part I. Section III. Chapter 2. “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 w 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 = 2 - 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. 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.
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, of 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 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 same quantity; as in the numbers 4, 7, 10; and the decreasing progression is that in which the terms go on always diminishing by the same quantity, such as the numbers 9, 5, 1.
399 Let us suppose the numbers \(a\), \(b\), \(c\), to be in arithmetical progression; then \(a - b = b - c\), whence it follows, from the equality between the sum of the extremes and that of the means, that \(2b = a + c\); and if we subtract a from both, we have \(2b - a = c\).
400 So that when the first two terms \(a\), \(b\), of an arithmetical progression are given, the third is found by taking the first from twice the second. Let 1 and 3 be the first two terms of an arithmetical progression, the third will be 2·3 - 1 = 5; and these three numbers 1, 3, 5, give the proportion
1 - 3 = 3 - 5.
401 By following the same method, we may pursue the arithmetical progression as far as we please ; we have only to find the fourth term by means of the second and third, in the same manner as we determined the third by means of the first and second, and so on. Let \(a\) be the first term, and \(b\) the second, the third will be \(2b - a\), the fourth \(4b - 2a - b = 3b - 2a\), the fifth \(6b - 4a - 2b + a = 4b - 3a\), the sixth \(8b - 6a - 3b + 2a = 5b - 4a\), the seventh \(10b - 8a - 4b + 3a = 6b - 5a\), etc.
- Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
- Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.