Book 5
PROPOSITION 14.
14
If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.
14
For let a first magnitude A have the same ratio to a second B as a third C has to a fourth D; and let A be greater than C; I say that B is also greater than D.
14
For, since A is greater than C, and B is another, chance, magnitude, therefore A has to B a greater ratio than C has to B. [V. 8]
14
But, as A is to B, so is C to D; therefore C has also to D a greater ratio than C has to B. [V. 13]
14
But that to which the same has a greater ratio is less; [V. 10] therefore D is less than B; so that B is greater than D.
14
Similarly we can prove that, if A be equal to C, B will also be equal to D; and, if A be less than C, B will also be less than D.
14
Therefore etc. Q. E. D.
PROPOSITION 15.
15
Parts have the same ratio as the same multiples of them taken in corresponding order.
15
For let AB be the same multiple of C that DE is of F; I say that, as C is to F, so is AB to DE.
15
For, since AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there also in DE equal to F.
15
Let AB be divided into the magnitudes AG, GH, HB equal to C, and DE into the magnitudes DK, KL, LE equal to F; then the multitude of the magnitudes AG, GH, HB will be equal to the multitude of the magnitudes DK, KL, LE.
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.
