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Proposition 5

Plane numbers have to one another the ratio compounded of the ratios of their sides.
Let A and B be plane numbers, and let the numbers C and D be the sides of A, and E and F the sides of B. I say that A has to B the ratio compounded of the ratios of the sides.
| The ratios being given which C has to E and D to F, take the least numbers G, H, and K that are continuously in the ratios C, E, D, and F, so that C is to E as G is to H, and D is to F as H is to K. Multiply D by E to make L. | VIII.4
Now, since D multiplied by C makes A, and multiplied by E makes L, therefore C is to E as A is to L. But C is to E as G is to H, therefore G is to H as A is to L. | VII.17
Again, since E multiplied by D makes L, and further multiplied by F makes B, therefore D is to F as L is to B. But D is to F as H is to K, therefore H is to K as L is to B. | VII.17
But it was also proved that, H as G is to H as A is to L, therefore, ex aequali, L as G is to K as A is to B. | VII.14
But G has to K the ratio compounded of the ratios of the sides, therefore A also has to B the ratio compounded of the ratios of the sides.
Therefore, plane numbers have to one another the ratio compounded of the ratios of their sides.
Q.E.D.

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(Forthcoming)

Book VIII Introduction - Proposition VIII.4 - Proposition VIII.6.

© 1996
D.E.Joyce
Clark University