← euclid / The Thirteen Books of Euclid’s Elements Volume II / Book VII / Propositions / VII.13 ← Prev Proposition · Commentary Next → [Folio 0319 | p. 313] 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. A B C 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] [VII. Def. 20]Therefore, as A is to C, so is B to D. Q. E. D. 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