Book 5
18
Therefore, as AB is to BE, so is not CD to a less magnitude than FD.
18
Similarly we can prove that neither is it in that ratio to a greater; it is therefore in that ratio to FD itself.
18
Therefore etc. Q. E. D.
PROPOSITION 19
19
If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.
19
For, as the whole AB is to the whole CD, so let the part AE subtracted be to the part CF subtracted; I say that the remainder EB will also be to the remainder FD as the whole AB to the whole CD.
19
For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [V. 16]
19
And, since the magnitudes are proportional componendo, they will also be proportional separando, [V. 17] that is, as BE is to EA, so is DF to CF, and, alternately, as BE is to DF, so is EA to FC. [V. 16]
19
But, as AE is to CF, so by hypothesis is the whole AB to the whole CD.
19
Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD. [V. 11]
19
Therefore etc. [
19
Porism. From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.
19
] Q. E. D.
PROPOSITION 20.
20
If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and, if less, less.
20
Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, 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; and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and, if less, less.
20
For, since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, [V. 8] therefore A has to B a greater ratio than C has to B.
