Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

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PROPOSITION 13.

If four numbers be proportional, they will also be proportional alternately.

Let the four numbers A, B, C, D be proportional, so that,

as A is to B, so is C to D; I say that they will also be proportional alternately, so that, as A is to C, so will B be to D.

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]

Therefore, alternately, whatever part or parts A is of C, the same part or the same parts is B of D also. [VII. 10]

Therefore, as A is to C, so is B to D. [VII. Def. 20] Q. E. D.

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PROPOSITION 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.

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.

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]