Book 5
15
And, since AG. GH, HB are equal to one another, and DK, KL, LE are also equal to one another, therefore, as AG is to DK, so is GH to KL, and HB to LE. [V. 7]
15
Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [V. 12] therefore, as AG is to DK, so is AB to DE.
15
But AG is equal to C and DK to F; therefore, as C is to F, so is AB to DE.
15
Therefore etc. Q. E. D.
PROPOSITION 16.
16
If four magnitudes be proportional, they will also be proportional alternately.
16
Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D; I say that they will also be so alternately, that is, as A is to C, so is B to D.
16
For of A, B let equimultiples E, F be taken, and of C, D other, chance, equimultiples G, H.
16
Then, since E is the same multiple of A that F is of B, and parts have the same ratio as the same multiples of them, [V. 15] therefore, as A is to B, so is E to F.
16
But as A is to B, so is C to D; therefore also, as C is to D, so is E to F. [V. 11]
16
Again, since G, H are equimultiples of C, D, therefore, as C is to D, so is G to H. [V. 15]
16
But, as C is to D, so is E to F; therefore also, as E is to F, so is G to H. [V. 11]
16
But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less. [V. 14]
16
Therefore, if E is in excess of G, F is also in excess of H, if equal, equal, and if less, less.
