Now, it is evident that if is greater than unity, is greater than unity; but if is greater than unity, ma is greater than nb; and if is greater than unity, mc is greater than nd. In like manner, if ma be equal to nb, mc is equal to nd; and if less, less. The foregoing is an easy proof of the converse of the theorem which is contained in Euclid’s celebrated Fifth Definition. Next, to prove Euclid’s theorem—that if, according as the multiple of the first of four magnitudes is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth; the ratio of the first to the second is equal to the ratio of the third to the fourth. Dem.—Let, a, b, c, d be the four magnitudes. First suppose that a and b are commensurable, then it is evident that we can take multiples na, mb, such that na = mb. Hence, by hypothesis, nc = md. Thus, therefore = . Next, suppose a and b are incommensurable. Then, as in a recent note, we can find two numbers m and n, such that is greater than unity, but less than unity. Hence lies between and . Now, since by hypothesis, when is greater than unity, is greater than unity; and when is less than unity, is less than unity. Hence, since lies between and , lies between the same quantities. Therefore the difference between and is less than ; and since n may be as large as we please, the difference is nothing; therefore vii. When of the multiples of four magnitudes (taken as in Def. v.) the multiple of the first is greater than that of the second, but the multiple of the third not greater than that of the fourth, the first has to the second a greater ratio than the third has to the fourth. This, instead of being a definition, is a theorem. We have altered the last clause from that given in Simson’s Euclid, which runs thus:—“The first is said to have to the second a greater ratio than the third has to the fourth.” This is misleading, as it implies that it is, by convention, that the first ratio is greater than the second, whereas, in fact, such is not the case; for it follows from the hypothesis that the first ratio is greater than the second; and if it did not, it could not be made so by definition. We have made a similar change in the enunciation of the Fifth Definition. Let a, b, c, d be the four magnitudes, and m and n the multiples taken, it is required to prove, that if ma be greater than nb, but mc not greater than nd, that the ratio a : b is greater than the ratio c : d. Dem.—Since ma is greater than nb, but mc not greater than nd, it is evident that is greater than ; therefore is greater than ; that is, the ratio a : b is greater than the ratio c : d. ix. Proportion consists of three terms at least.