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Proposition 2

If two numbers multiplied by one another make a square number, then they are similar plane numbers.
Let A and B be two numbers, and let A multiplied by B make the square number C. I say that A and B are similar plane numbers. Multiply A by itself to make D. Then D is square.
| Now, since A multiplied by itself makes D, and multiplied by B makes C, therefore A is to B as D is to C. | VII.17
And, since D is square, and C is so also, therefore D and C are similar plane numbers.
Therefore one mean proportional number falls between D and C. And D is to C as A is to B, therefore one mean proportional number falls between A and B also. | VIII.18 VIII.8
But, if one mean proportional number falls between two numbers, then they are similar plane numbers, therefore A and B are similar plane numbers. | VIII.20
Therefore, if two numbers multiplied by one another make a square number, then they are similar plane numbers.
Q.E.D.

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(Forthcoming)

Book IX Introduction - Proposition IX.1 - Proposition IX.3.

© 1996
D.E.Joyce
Clark University