PROP. V. THEO. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other, the remainder shall be the same multiple of the remainder, that the whole is of the whole. As a particular arithmetical example, let us take the following: 25 is 5 times 5 5 is 5 times 1 Remainder 20 is 5 times 4 Algebraical Exposition. ma is the same multiple of a that mb is of b, .. ma- mb is m times b, for ma mb = m (a—b): à supposed to be greater than 6. PROP. VI. THEO. If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainder are either equal to these others, or equimultiples of them. when M' m' — 1. .. If two magnitudes be equimultiples, &c. For this case, as a particular arithmetical example, let us take From 40 take 3 times 4, the remainder is 28, and from 50 take 3 times 5, the remainder is 35; it is evident that From 40 take 9 times 4, and from 50 take 9 times 5, then the remainders are 4 and 5. Algebraically. ma is the same multiple of a that m b is of b; from m a take n a, the remainder is ma - na = (m n)a; from m b take n b, the remainder is nb = (m - n) b. m b Then it is evident that (mn) a is the same multiple of a, that (m — n)b is of b; and (m — n) a = a, and (m — n) b c = b, when m− n = 1. PROP. A. THEO. If the first of the four magnitudes has the same ratio to the second, which the third has to the fourth, then, if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less. Geometricians make use of the technical term " Invertando," by invertion, when there are four proportionals, and it is inferred, that the second is to the first as the fourth to the third. Let A B C D, then, by "invertando" it is inferred B: A:: D: C. |