176. When we use the term part of a number or fraction in the remainder of this section, we mean, one of the various sets of equal parts into which it may be divided, either the half, the third, the fourth, &c.: the term multiple has been already explained (102). By the term multiple-part of a number we mean, the abbreviation of the words multiple of a part. Thus, 1, 2, 3, 4, and 6, are parts of 12; ½ is also a part of 12, being contained in it 24 times; 12, 24, 36, &c., are multiples of 12; and 8, 9, ⁵/₂, &c. are multiple parts of 12, being multiples of some of its parts. And when multiple parts generally are spoken of, the parts themselves are supposed to be included, on the same principle that 12 is counted among the multiples of 12, the multiplier being 1. The multiples themselves are also included in this term; for 24 is also 48 halves, and is therefore among the multiple parts of 12. Each part is also in various ways a multiple-part; for one-fourth is two-eighths, and three-twelfths, &c.

177. Every number or fraction is a multiple-part of every other number or fraction. If, for example, we ask what part 12 is of 7, we see that on dividing 7 into 7 parts, and repeating one of these parts 12 times, we obtain 12; or, on dividing 7 into 14 parts, each of which is one-half, and repeating one of these parts 24 times, we obtain 24 halves, or 12. Hence, 12 is ¹²/₇, or ²⁴/₁₄, or ³⁶/₂₁ of 7; and so on. Generally, when a and b are two whole numbers, a/b expresses the multiple-part which a is of b, and b/a that which b is of a. Again, suppose it required to determine what multiple-part (2⅐) is of (3⅕), or ¹⁵/₇ of ¹⁶/₅. These fractions, reduced to a common denominator, are ⁷⁵/₃₅ and ¹¹²/₃₅, of which the second, divided into 112 parts, gives ¹/₃₅, which repeated 75 times gives ⁷⁵/₃₅, the first. Hence, the multiple-part which the first is of the second is ⁷⁵/₁₁₂, which being obtained by the rule given in (121), shews that a/b, or a divided by b, according to the notion of division there given, expresses the multiple-part which a is of b in every case.

178. When the first of four numbers is the same multiple-part of the second which the third is of the fourth, the four are said to be geometrically[27] proportional, or simply proportional. This is a word in common use; and it remains to shew that our mathematical definition of it, just given, is, in fact, the common notion attached to it. For example, suppose a picture is copied on a smaller scale, so that a line of two inches long in the original is represented by a line of one inch and a half in the copy; we say that the copy is not correct unless all the parts of the original are reduced in the same proportion, namely, that of 2 to (1½). Since, on dividing two inches into 4 parts, and taking 3 of them, we get (1½), the same must be done with all the lines in the original, that is, the length of any line in the copy must be three parts out of four of its length in the original. Again, interest being at 5 per cent, that is, £5 being given for the use of £100, a similar proportion of every other sum would be given; the interest of £70, for example, would be just such a part of £70 as £5 is of £100.

Since, then, the part which a is of b is expressed by the fraction a/b, or any other fraction which is equivalent to it, and that which c is of d by c/d, it follows, that when a, b, c, and d, are proportional, a/b = c/d. This equation will be the foundation of all our reasoning on proportional quantities; and in considering proportionals, it is necessary to observe not only the quantities themselves, but also the order in which they come. Thus, a, b, c, and d, being proportionals, that is, a being the same multiple-part of b which c is of d, it does not follow that a, d, b, and c are proportionals, that is, that a is the same multiple-part of d which b is of c. It is plain that a is greater than, equal to, or less than b, according as c is greater than, equal to, or less than d.

179. Four numbers, a, b, c, and d, being proportional in the order written, a and d are called the extremes, and b and c the means, of the proportion. For convenience, we will call the two extremes, or the two means, similar terms, and an extreme and a mean, dissimilar terms. Thus, a and d are similar, and so are b and c; while a and b, a and c, d and b, d and c, are dissimilar. It is customary to express the proportion by placing dots between the numbers, thus:

a : b ∷ c : d

180. Equal numbers will still remain equal when they have been increased, diminished, multiplied, or divided, by equal quantities. This amounts to saying that if

It is also evident, that a + p-p, a -p + p, ap/p, and a/p × p, are all equal to a.

181. The product of the extremes is equal to the product of the means. Let a/b = c/d, and multiply these equal numbers by the product bd. Then,