Book 9
PROPOSITIONS
PROPOSITION 1.
1
If two similar plane numbers by multiplying one another make some number, the product will be square.
1
Let A, B be two similar plane numbers, and let A by multiplying B make C; I say that C is square.
1
For let A by multiplying itself make D.
1
Therefore D is square.
1
Since then A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. [VII. 17]
1
And, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B. [VIII. 18]
1
But, if numbers fall between two numbers in continued proportion, as many as fall between them, so many also fall between those which have the same ratio; [VIII. 8] so that one mean proportional number falls between D, C also.
1
And D is square; therefore C is also square. [VIII. 22] Q. E. D.
PROPOSITION 2.
2
If two numbers by multiplying one another make a square number, they are similar plane numbers.
2
Let A, B be two numbers, and let A by multiplying B make the square number C; I say that A, B are similar plane numbers.
2
For let A by multiplying itself make D; therefore D is square.
2
Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. [VII. 17]
2
And, since D is square, and C is so also, therefore D, C are similar plane numbers.
2
Therefore one mean proportional number falls between D, C. [VIII. 18]
2
And, as D is to C, so is A to B; therefore one mean proportional number falls between A, B also. [VIII. 8]
2
But, if one mean proportional number fall between two numbers, they are similar plane numbers; [VIII. 20] therefore A, B are similar plane numbers. Q. E. D.
PROPOSITION 3.
3
If a cube number by multiplying itself make some number, the product will be cube.
3
For let the cube number A by multiplying itself make B; I say that B is cube.
3
For let C, the side of A, be taken, and let C by multiplying itself make D.
3
It is then manifest that C by multiplying D has made A.
