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Proposition 24

If two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square.
Let the two numbers A and B have to one another the ratio which the square number C has to the square number D, and let A be square. I say that B is also square.
| Since C and D are square, C and D are similar plane numbers. Therefore one mean proportional number falls between C and D. | VIII.18
And C is to D as A is to B, therefore one mean proportional number falls between A and B also. And A is square, therefore B is also square. | VIII.18 VIII.22
Therefore, if two numbers have to one another the ratio which a square number has to a square number, and the first is square, then the second is also a square.
Q.E.D.

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(Forthcoming)

Book VIII Introduction - Proposition VIII.23 - Proposition VIII.25.

© 1996
D.E.Joyce
Clark University