Book 7
13
For since, as A is to B, so is C to D, therefore, whatever part or parts A is of B, the same part or the same parts is C of D also. [VII. Def. 20]
13
Therefore, alternately, whatever part or parts A is of C, the same part or the same parts is B of D also. [VII. 10]
13
Therefore, as A is to C, so is B to D. [VII. Def. 20] Q. E. D.
PROPOSITION 14.
14
If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.
14
Let there be as many numbers as we please A, B, C, and others equal to them in multitude D, E, F, which taken two and two are in the same ratio, so that, as A is to B, so is D to E, and, as B is to C, so is E to F; I say that, ex aequali, as A is to C, so also is D to F.
14
For, since, as A is to B, so is D to E, therefore, alternately, as A is to D, so is B to E. [VII. 13]
14
Again, since, as B is to C, so is E to F, therefore, alternately, as B is to E, so is C to F. [VII. 13]
14
But, as B is to E, so is A to D; therefore also, as A is to D, so is C to F.
14
Therefore, alternately, as A is to C, so is D to F. [id.]
PROPOSITION 15.
15
If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.
15
For let the unit A measure any number BC, and let another number D measure any other number EF the same number of times; I say that, alternately also, the unit A measures the number D the same number of times that BC measures EF.
15
For, since the unit A measures the number BC the same number of times that D measures EF, therefore, as many units as there are in BC, so many numbers equal to D are there in EF also.
15
Let BC be divided into the units in it, BG, GH, HC, and EF into the numbers EK, KL, LF equal to D.
15
Thus the multitude of BG, GH, HC will be equal to the multitude of EK, KL, LF.
