Kitap 8
21
For since N, O by multiplying E have made A, C, therefore, as N is to O, so is A to C, that is, E to F. [VII. 18]
21
But, as E is to F, so is H to L and K to M; therefore also, as H is to L, so is K to M and N to O.
21
And H, K, N are the sides of A, and O, L, M the sides of B.
21
Therefore A, B are similar solid numbers. Q. E. D.
PROPOSITION 22.
22
If three numbers be in continued proportion, and the first be square, the third will also be square.
22
Let A, B, C be three numbers in continued proportion, and let A the first be square; I say that C the third is also square.
22
For, since between A, C there is one mean proportional number, B, therefore A, C are similar plane numbers. [VIII. 20]
22
But A is square; therefore C is also square. Q. E. D.
PROPOSITION 23.
23
If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.
23
Let A, B, C, D be four numbers in continued proportion, and let A be cube; I say that D is also cube.
23
For, since between A, D there are two mean proportional numbers B, C, therefore A, D are similar solid numbers. [VIII. 21]
23
But A is cube; therefore D is also cube. Q. E. D.
PROPOSITION 24.
24
If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second will also be square.
24
For let the two numbers A, 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.
24
For, since C, D are square, C, D are similar plane numbers.
