|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 ratio, 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.|
To illustrate this proposition, consider the two similar plane numbers a = 18 and b = 8, as illustrated in the Guide to VII.Def.21. According to VIII.18, there is a mean proportional between them, namely, 12. And the square of the mean proportional is their product, ab = 144.
Let a and b be the given similar plane numbers. Then there is a mean proportional between them (VIII.18). And, since a:b = a2:ab, therefore there is also a mean proportional between a2 and ab (VIII.1). But since a2 is a square, therefore ab is also a square (VIII.22). Thus, the product of the original similar plane numbers is a square.
|Next proposition: IX.2||
© 1996, 2002