Book 8
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.
24
Therefore one mean proportional number falls between C, D. [VIII. 18]
24
And, as C is to D, so is A to B; therefore one mean proportional number falls between A, B also. [VIII. 8]
24
And A is square; therefore B is also square. [VIII. 22] Q. E. D.
PROPOSITION 25.
25
If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.
25
For let the two numbers A, B have to one another the ratio which the cube number C has to the cube number D, and let A be cube; I say that B is also cube.
25
For, since C, D are cube, C, D are similar solid numbers.
25
Therefore two mean proportional numbers fall between C, D. [VIII. 19]
25
And, as many numbers as fall between C, D in continued proportion, so many will also fall between those which have the same ratio with them; [VIII. 8] so that two mean proportional numbers fall between A, B also.
25
Let E, F so fall.
25
Since, then, the four numbers A, E, F, B are in continued proportion, and A is cube, therefore B is also cube. [VIII. 23] Q. E. D.
PROPOSITION 26.
26
Similar plane numbers have to one another the ratio which a square number has to a square number.
26
Let A, B be similar plane numbers; I say that A has to B the ratio which a square number has to a square number.
