Book 8
26
For, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [VIII. 18]
26
Let it so fall, and let it be C; and let D, E, F, the least numbers of those which have the same ratio with A, C, B, be taken; [VII. 33 or VIII. 2] therefore the extremes of them D, F are square. [VIII. 2, Por.]
26
And since, as D is to F, so is A to B, and D, F are square, therefore A has to B the ratio which a square number has to a square number. Q. E. D.
PROPOSITION 27.
27
Similar solid numbers have to one another the ratio which a cube number has to a cube number.
27
Let A, B be similar solid numbers; I say that A has to B the ratio which a cube number has to a cube number.
27
For, since A, B are similar solid numbers, therefore two mean proportional numbers fall between A, B. [VIII. 19]
27
Let C, D so fall, and let E, F, G, H, the least numbers of those which have the same ratio with A, C, D, B, and equal with them in multitude, be taken; [VII. 33 or VIII. 2] therefore the extremes of them E, H are cube. [VIII. 2, Por.]
27
And, as E is to H, so is A to B; therefore A also has to B the ratio which a cube number has to a cube number. Q. E. D.
