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.
This has the same fault as some of the others—it is not a definition, but an inference. It occurs when the means in a proportion are equal, so that, in fact, there are four terms. As an illustration, let us take the numbers 4, 6, 9. Here the ratio of 4 : 6 is
