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Proposition 1

If two similar plane numbers multiplied by one another make some number, then the product is square.
Let A and B be two similar plane numbers, and let A multiplied by B make C. I say that C is square. Multiply A by itself to make D. Then D is square.
| Since then 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 A and B are similar plane numbers, therefore one mean proportional number falls between A and B. | VIII.18
Since as many number fall in continued proportion between those which have the same ration, therefore one mean proportional number falls between D and C also. | VIII.8
And D is square, therefore C is also square. | VIII.22
Therefore, if two similar plane numbers multiplied by one another make some number, then the product is square.
Q.E.D.

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This proposition is used in X.29.

Book IX Introduction - Proposition IX.2.

© 1996
D.E.Joyce
Clark University