PROPOSITION 2.
If two numbers by multiplying one another make a square number, they are similar plane numbers.
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.
For let
A by multiplying itself make
D; therefore
D is square.
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]
And, since
D is square, and
C is so also, therefore
D,
C are similar plane numbers.
Therefore one mean proportional number falls between
D,
C. [
VIII. 18]
And, as
D is to
C, so is
A to
B; therefore one mean proportional number falls between
A,
B also. [
VIII. 8]
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.
If a cube number by multiplying itself make some number, the product will be cube.
For let the cube number
A by multiplying itself make
B; I say that
B is cube.