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