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.