THEOREM XII. In two ranks of proportionals, if the corresponding terms be multiplied together the products will be proportionals. If a:b=c:d, and ef=g:h; then ae : bf = cg :dh. 8 For 08 , jf dh' that is, ae : bf = cg : dh. This is called compounding the proportions. The proposition is true if applied to any number of proportions. e a с ae = and = ', therefore, IN THEOREM XIII. If four quantities be proportionals, the like powers or roots of these quantities will be proportionals. If a :6=c:d, then a" : 7" =(: d". a : 6 : If three quantities, a, b, c, be in continued proportion, that is, if b:c, then a : c= a : 6%. 6 For 6 ; multiply by a a ū IV. INCOMMENSURABLE QUANTITIES. These theorems are applicable to incommensurable quantities. In the definition of Proportion, it is supposed that one quantity is some determinate multiple, part, or parts of another : or that the fraction formed by taking one of the quantities as a numerator and the other as a denominator, is a determinate fraction. This will be the case whenever the two quantities have any common measure whatever. Let x be a common measure of a and b, and let a = mx, b = nx, a But, if the quantities are incommensurable, the value of n cannot be tor are whole numbers. Yet a fraction of this kind may be found which will express its value to any required degree of accuracy—that is, a a = nx ; fraction which may approach the limit b as nearly in value as we please. Suppose x to be a measure of b, and let b also let a be greater than mx, but less than (m + 1)x; then is greater than m + I but less than , or, the difference between is less a m m a 6 n n I I than ; and as x is diminished, since nx = 6, n is increased and n n m diminished; therefore, by diminishing x, the difference between n a and 7 may be made less than any that can be assigned. In other a words, the incommensurable ratio g is the limit which a varying com may approach as nearly in value as we please, n but never reach. a If c and d as well as a and 6 be incommensurable, and if, when ever the magnitudes m and n are increased, then is equal to á For, if they are not equal, they must have some assignable difference ; m + I and because each of them lies between and this difference I is less than ; but since n may, by the supposition, be increased n may be diminished without limit, that is, it may be n a come less than any assignable magnitude ; therefore, and a have d a с no assignable difference; that is, 6 d Hence, all the propositions respecting proportionals are true of the four magnitudes a, b, c, d, when incommensurable. C 1 V. EUCLID'S DEFINITION OF PROPORTION. It will be useful to compare the definition of proportion which has been given in this chapter with that which is given in the Fifth Book of Euclid. The latter definition may be stated thus : Four quantities are pro portionals when, if any equimultiples be taken of the first and third, and, also, any equimultiples of the second and fourth, the multiple of the third is greater than, equal to, or less than the multiple of the fourth, according as the multiple of the first is greater than, equal to, or less than the multiple of the second. We will first show that the property involved in this definition follows from the algebraical definition. For, suppose a : 6 :: 0 :d, then therefore, pa _ pc d' qb Hence, pc is greater than, equal to, or less than qd, according as pa is greater than, equal to, or less than gb. Next, we may deduce the algebraical definition of proportion from Euclid's. Let a, b, c, d be four quantities, such that pc is greater than, equal to, or less than qd, according as pa is greater than, equal to, or less than qb, then shall à First, suppose c and d are commensurable ; then we can take p and q such that pc = qd; hence, by P hypothesis, pa = qb. pa pc Thus = I = gb and era qd" a C = qd' a = a If, however, a, b, c, d be incommensurable, the above equalities cannot be obtained ; but we can always make pa approach as near as we please to qb by giving proper values to p and q : i. e., we can make pa differ from qb by a quantity less than b, or make pa lie between qb and (q + 1). Then, also, will pc lie between qd and (2 + 1)d; i e., both . and lie between and 6 d a and 9 may be increased without limit; therefore, 7 đ' or a, b, c, d are proportionals according to the algebraical definition. It will be seen that Euclid's definition of proportion includes both commensurable and incommensurable quantities. Also, с |