SECTION VIII.
ON THE PROPORTION OF NUMBERS.
170. When two numbers are named in any problem, it is usually necessary, in some way or other, to compare the two; that is, by considering the two together, to establish some connexion between them, which may be useful in future operations. The first method which suggests itself, and the most simple, is to observe which is the greater, and by how much it differs from the other. The connexion thus established between two numbers may also hold good of two other numbers; for example, 8 differs from 19 by 11, and 100 differs from 111 by the same number. In this point of view, 8 stands to 19 in the same situation in which 100 stands to 111, the first of both couples differing in the same degree from the second. The four numbers thus noticed, viz.:
8, 19, 100, 111,
are said to be in arithmetical[26] proportion. When four numbers are thus placed, the first and last are called the extremes, and the second and third the means. It is obvious that 111 + 8 = 100 + 19, that is, the sum of the extremes is equal to the sum of the means. And this is not accidental, arising from the particular numbers we have taken, but must be the case in every arithmetical proportion; for in 111 + 8, by (35), any diminution of 111 will not affect the sum, provided a corresponding increase be given to 8; and, by the definition just given, one mean is as much less than 111 as the other is greater than 8.
171. A set or series of numbers is said to be in continued arithmetical proportion, or in arithmetical progression, when the difference between every two succeeding terms of the series is the same. This is the case in the following series:
| 1, | 2, | 3, | 4, | 5, | &c. |
| 3, | 6, | 9, | 12, | 15, | &c. |
| 1½, | 2, | 2½, | 3, | 3½, | &c. |
The difference between two succeeding terms is called the common difference. In the three series just given, the common differences are, 1, 3, and ½.
172. If a certain number of terms of any arithmetical series be taken, the sum of the first and last terms is the same as that of any other two terms, provided one is as distant from the beginning of the series as the other is from the end. For example, let there be 7 terms, and let them be,
a b c d e f g.
Then, since, by the nature of the series, b is as much above a as f is below g (170), a + g = b + f. Again, since c is as much above b as e is below f (170), b + f = c + e. But a + g = b + f; therefore a + g = c + e, and so on. Again, twice the middle term, or the term equally distant from the beginning and the end (which exists only when the number of terms is odd), is equal to the sum of the first and last terms; for since c is as much below d as e is above it, we have c + e = d + d = 2d. But c + e = a + g; therefore, a + g = 2d. This will give a short rule for finding the sum of any number of terms of an arithmetical series. Let there be 7, viz. those just given. Since a + g, b + f, and c + e, are the same, their sum is three times (a + g), which with d, the middle term, or half a + g, is three times and a half (a + g), or the sum of the first and last terms multiplied by (3½), or ⁷/₂, or half the number of terms. If there had been an even number of terms, for example, six, viz. a, b, c, d, e, and f, we know now that a + f, b + e, and c + d, are the same, whence the sum is three times (a + f), or the sum of the first and last terms multiplied by half the number of terms, as before. The rule, then, is: To sum any number of terms of an arithmetical progression, multiply the sum of the first and last terms by half the number of terms. For example, what are 99 terms of the series 1, 2, 3, &c.? The 99th term is 99, and the sum is