Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is E to F.
But as A is to B, so is C to D; therefore also, as C is to D, so is E to F. [V. 11]
Again, since G, H are equimultiples of C, D, therefore, as C is to D, so is G to H. [V. 15] But, as C is to D, so is E to F; therefore also, as E is to F, so is G to H. [V. 11]
But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth;
if equal, equal; and if less, less. [V. 14]
Therefore, if E is in excess of G, F is also in excess of H, if equal, equal, and if less, less.
Now E, F are equimultiples of A, B, and G, H other, chance, equimultiples of C, D; therefore, as A is to C, so is B to D. [V. Def. 5]
卽戊與己。若甲與乙也。本篇十五
庚與辛。若丙與丁也。
夫甲與乙。若丙與丁。而戊與己。亦若甲與乙。
卽戊與己。亦若丙與丁矣。依顯庚與辛。若丙與丁。卽戊與己。亦若庚與辛也。本篇十一
次三試之。若戊大於庚則己亦大於辛也。
若等、亦等。若小、亦小。
任作幾許倍。恆如是也。本篇十四
則倍一甲之戊。倍三乙之己。與倍二丙之庚。倍四丁之辛。其等、大、小、必同類也。而甲與丙。若乙與丁矣。
Complete text
Book I