Therefore if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less.

[V. A.

Again, suppose that HO and MP are equimultiples of EB and FD.

Then, because AE is to EB as CF is to FD; [Hypothesis. and that GK and LN are equimultiples of AE and CF, and HO and MP are equimultiples of EB and FD; therefore if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less; [V. Definition 5. which was likewise shewn on the preceding supposition.

H

K

B

AF

P

M

N

But if GH be greater than KO, then by taking the common magnitude KH from both, GK is greater than HO; therefore also LN is greater than MP;

and, by adding the common magnitude NM to both, LM is greater than NP.

Thus if GH be greater than KO, LM is greater than NP.

In like manner it may be shewn, that if GH be equal to KO, LM is equal to NP; and if less, less.

And in the case in which KO is not greater than KH, it has been shewn that GH is always greater than KO, and also LM greater than NP.

But GH and LM are any equimultiples whatever of AB and CD, and KO and NP are any equimultiples whatever of BE and DF, [Construction.

therefore AB is to BE as CD is to DF.

Wherefore, if magnitudes &c. Q.E.D.

[V. Definition 5.

If a whole magnitude be to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall be to the remainder as the whole is to the whole.

Let the whole AB be to the whole CD as AE, a magnitude taken from AB, is to CF, a magnitude taken from CD: the remainder EB shall be to the remainder FD as the whole AB is to the whole CD.

For, because AB is to CD as AE is to CF, [Hypothesis. therefore, alternately, AB is to AE as CD is to CF.

[V. 16. And if magnitudes taken jointly be proportionals, they are also proportionals when taken separately;

[V. 17. therefore EB is to AE as FD is to CF; therefore, alternately, EB is to FD as AE is to CF.

[V. 16. But AE is to CF as AB is to CD; [Hyp. therefore EB is to FD as AB is to CD. [V.11.

Wherefore, if a whole &c. Q.E.D.

B

F

COROLLARY. If the whole be to the whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall be to the remainder as the magnitude taken from the first is to the magnitude taken from the other. The demonstration is contained in the preceding.

PROPOSITION E. THEOREM.

If four magnitudes be proportionals, they shall also be proportionals by conversion; that is, the first shall be to its excess above the second as the third is to its excess above the fourth.

Let AB be to BE as CD is to DF: AB shall be to AE as CD is to CF

For, because AB is to BE as CD is

to DF;

[Hypothesis.

therefore, by division, AE is to EB as CF is to FD;

[V. 17.

E

F

[V. B.

and, by inversion, EB is to AE as FD
is to ČF.
Therefore, by composition, AB is to AE
as CD is to CF.

[V. 18.

B D

Wherefore, if four magnitudes &c. Q.E.D.

PROPOSITION 20. THEOREM.

If there be three magnitudes, and other three, which have the same ratio, taken two and two, then, if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio taken two and two; that is, let A be to B as D is to E, and let B be to C as E is to F: if A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

First, let A be greater than C: D

shall be greater than F.

For, because A is greater than C, and B is any other magnitude,

therefore A has to B a greater ratio than Chas to B.

But A is to B as D is to E;

[V. 8.

[Hypothesis.

therefore D has to E a greater ratio than

C has to B.

[V. 13. And because B is to C as E is to F, [Hyp. therefore, by inversion, C is to B as F is to E. [V. B. And it was shewn that D has to E a greater ratio than C has to B;

therefore D has to E a greater ratio than F has to E;

therefore D is greater than F.

[V. 13, Cor.

[V. 10.

Secondly, let A be equal to C: D shall

be equal to F.

For, because A is equal to C, and B is any other magnitude,

therefore A is to B as C is to B.

But A is to B as D is to E,

and Cis to B as Fis to E,

[V. 7. [Hypothesis.

D E F

[Hyp. V. B.

[V. 11.

[V.9.

therefore D is to E as F is to E;

and therefore D is equal to F.

Lastly, let A be less than C: D shall

be less than F.

For Cis greater than A;

and, as was shewn in the first case, C is to B as Fis to E;

and, in the same manner, B is to A as Ę is to D;

therefore, by the first case, F is greater than D;

that is, D is less than F.

Wherefore, if there be three &c. Q.E.D.

D

PROPOSITION 21. THEOREM.

If there be three magnitudes, and other three, which have the same ratio, taken two and two, but in a cross order, then if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio, taken two and two, but in a cross order; that is, let A be to B as E is to F, and let B be to C as D is to E: if A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

First, let A be greater than C: D shall be greater than F.

and C is to B as E is to D; [Hyp. V. B. therefore E is to Fas E is to D; [V. 11. and therefore D is equal to F. [V.9.

Lastly, let A be less than C: D shall be less than F.

For Cis greater than A;

and, as was shewn in the first case, C is to B as E is to D;

and, in the same manner, B is to A as F is to E;

therefore, by the first case, Fis greater than D;

that is, D is less than F.

Q

DE

Wherefore, if there be three &c.

Q.E.D.