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Proposition 27

Similar solid numbers have to one another the ratio which a cubic number has to a cubic number.
Let A and B be similar solid numbers. I say that A has to B the ratio which cubic number has to cubic number.
Since A and B are similar solid numbers, therefore two mean proportional numbers C and D fall between A and B. | VIII.19
Take E, F, G, and H, the least numbers of those which have the same ratio with A, C, D, and B, and equal with them in multitude. | VII.33 or VIII.2
Therefore the extremes of them, E and H, are cubes. And E is to H as A is to B, therefore A also has to B the ratio which a cubic number has to a cubic number. | VIII.2,Cor.
Therefore, similar solid numbers have to one another the ratio which a cubic number has to a cubic number.
Q.E.D.

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(Forthcoming)

Book VIII Introduction - Proposition VIII.26 - Book IX Introduction.

© 1996
D.E.Joyce
Clark University