57. Proportion.—If from any two columns in the table of § 36 we remove the numbers or quantities in any two rows, we get a diagram such as that here shown. The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers. But the two pairs of compartments will correspond to a single pair of numbers, e.g. 2 and 6, in the standard series, so that, denoting them by M, N and P, Q respectively, M will be to N in the same ratio that P is to Q. This is expressed by saying that M is to N as P to Q, the relation being written M:N :: P:Q; the four quantities are then said to be in proportion or to be proportionals.
This is the most general expression of the relative magnitude of two quantities; i.e. the relation expressed by proportion includes the relations expressed by multiple, submultiple, fraction and ratio.
If M and N are respectively m and n times a unit, and P and Q are respectively p and q times a unit, then the quantities are in proportion if mq = np; and conversely.
IV. Laws of Arithmetic
58. Laws of Arithmetic.—The arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws:—
(i) Equalities and Inequalities.—The following are sometimes called Axioms (§ 29), but their truth should be proved, even if at an early stage it is assumed. The symbols “>” and “<” mean respectively “is greater than” and “is less than.” The numbers represented by a, b, c, x and m are all supposed to be positive.
(a) If a = b, and b = c. then a = c;
(b) If a = b, then a + x = b + x, and a − x = b − x;
(c) If a > b, then a + x > b + x, and a − x > b − x;