AE to EB, so let CF be to FD: they shall also be proportionals when taken jointly; that is, as AB to BE, so shall CD be to DF.

Take of AB, BE, CD, DF any equimultiples whatever GH, HK, LM, MN; and again, of BE, DF, take any equimultiples whatever KO, NP: and because KO, NP are equimultiples of BE, DF, and that KH, NM are likewise equimultiples of BE, DF; therefore if KO, the multiple of BE, be greater than KH, which is a multiple of the same BE, then NP, the multiple of DF, is also greater than NM, the multiple of the same DF; and if KO be equal to KH, NP is equal to NM; and if less, less.

H

OF

M

P

N

First, let KO be not greater than KH; therefore NP is not greater than N M :and because GH, HK, are equimultiples of AB, BE, and that AB is greater than BE, 3 Ax. 5. therefore GH is greater than HK; but KO is not greater than KH: therefore GH is greater than KO. In like manner it may be shewn, that LM is greater than NP. Therefore, if KO be not greater than KH, then GH, the multiple of AB, is always greater than KO, the multiple of BE; and likewise LM, the multiple of CD, is greater than NP, the multiple of DF.

* 5.5.

5.5.

O

H

P

M

N

Next, let KO be greater than KH; therefore, as has been shewn, NP is greater than NM: and because the whole GH is the same multiple of the whole AB, that HK is of BE, therefore the remainder GK is the same multiple of the remainder AE* that GH is of AB; which is the same that LM is of CD. In like manner, because LM is the same multiple of CD, that MN is of DF, therefore the remainder LN is the K same multiple of the remainder CF*, that the whole LM is of the whole CD: but it was shewn that LM is the same multiple of CD, that GK is of AE; therefore GK is the same multiple of AE, that LN is of CF; that is, GK, LN are equimultiples of AE, CF. And because KO, NP, are equimultiples of BE, DF, therefore if from KO, NP there be taken KH, NM, which are likewise equimultiples of BE, DF, the remainders HO, MP are either equal to BE, DF, or equimultiples of

6.5.

Hyp.

them*. First, let HO, MP be equal to BE, DF: then because+ AE is to EB, as CF to FD, and that GK, LN are equimultiples of AE, CF; therefore GK is to EB*, as LN to FD: but HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP: therefore if GK be greater than HO, LN is greater than* * A. 5. MP; and if equal, equal; and if less, less.

Cor. 4. 5.

But let HO, MP be equimultiples of EB, FD: then+ + Hyp. because AE is to EB, as CF to FD, and that of AE, CF are taken equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LN is greater than MP; and if equal, equal: and if less, less*; which was likewise shewn in the

O

⚫ 5 Def. 5.

P

+ 5 Ax. 1.

M

KB

N

† 4 Ax. 1.

preceding case. But if GH be greater than KO, taking KH from both, GK is greater than HO; wherefore also LN is greater than MP; and consequently adding NM to both, LM is greater than NP: therefore, 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 likewise LM greater than NP: but GH, LM are any equimultiples whatever of AB, CD+, and KO, NP are any whatever of BE, Constr. DF; therefore*, as AB is to BE, so is CD to DF. If ⚫ 5 Def. 5. then magnitudes, &c. Q. E. D.

PROP. XIX. THEOR.

If a whole magnitude be to a whole, as a magnitude taken See N. from the first is to a magnitude taken from the other;

the remainder shall be to the remainder as the whole 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 to the whole CD.

Because AB is to CD, as AE to CF; therefore alternately*, BA is to AE, as DC to CF: and because if magnitudes taken jointly be proportionals, they are also proportionals*, when taken separately; therefore, as BE is to EA, so is DF to FC; and alternately, as

16.5.

17. 5.

† 11.5.

* 17.5.

* B. 5. *18. 5.

See N.

8.5.

+ Hyp.

*13. 5.

+ B. 5.

BE is to DF, so is EA to FC: but as AE to
CF, so, by the hypothesis, is AB to CD;
therefore also BF the remainder is to the re-
mainder DF+, as the whole AB to the whole
CD. Wherefore, if the whole, &c. Q. E. D.

E

F

B D

COR. 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 likewise be to the remainder, as the magnitude taken from the first to that taken from the other. The demonstration is contained in the preceding.

PROP. E. THEOR.

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

Let AB be to BE, as CD to DF: then BA shall be to AE, as DC to CF.

Because AB is to BE, as CD to DF, therefore by division*, AE is to EB, as CF to FD; and by inversion*, BE is to EA, as DF to FC; wherefore, by composition*, BA is to AE, as DC is to CF. If therefore four, &c. Q. E. D.

E

BD

If there be three magnitudes, and other three, which, taken two and two, have the same ratio; 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 taken two and two have the same ratio, viz. as A is to B, so is D to E; and as B to C, so is E to F. If A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

Because A is greater than C, and B is any other magnitude, and that the greater has to the same magnitude a greater* ratio than the less has to it; therefore A has to B a greater ratio than C has to B: but as D is to Et, so is A to B; therefore* D has to E a greater ratio than C to B: and because B is to C, as E to F, by inversion†, C is to B, as F is to E:

A BC

DEF

5.

and D was shewn to have to E a greater ratio than C to B: therefore D has to E a greater * ratio than F to Cor. 13. E: but the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two; therefore D is greater than F.

Secondly, let A be equal to C; D shall be equal to F. Because A and C are equal to one another, A is to B, as C is to B*: but+ A is to B, as D to E; and + C is to B, as F to E; wherefore D is to E, as Fto E*; and therefore D is equal to F*.

Next, let A be less than C; D shall be less than F. For C is greater than A; and, as was shewn in the first case,

A BC
DEF

C is to B, as F to E, and in like manner, B is to A, as E to D; therefore F is greater than D, by the first case; that is, D is less than F. Therefore, if there be three, &c. Q. E. D.

PROP. XXI. THEOR.

10. 5.

* 7.5.
+ Hyp.
+ Hyp. &

B 5.

11. 5.

* 9.5.

If there be three magnitudes, and other three, which have See N. the same ratio taken two and two, but in a cross order; then if the first magnitude 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, viz. as A is to B, so is E to F, and as B is to C, so is D to E. If A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

:

A B C
DEF

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio * than C has to B: but as E to F, so is A to B: therefore* E has to Fa greater ratio than C to B and because + B is to C, as D to E, by inversion, C is to B, as E to D: and E was shewn to have to F a greater ratio than C has to B; therefore E has to F a greater ratio than E has to *D: but the magnitude to which the same has a greater ratio than it has to another, is the lesser* of the two: therefore F is less than D; that is, D is greater than F.

Secondly, let A be equal to C; D shall be equal to F. Because A and C are equal, A is * to B, as C is to *7. 5.

as E to D, and in like manner B is to A, as F to E; therefore F is greater than D, by case first; that is, D is less than F. Therefore, if there be three, &c.

See N.

4.5.

* 20. 5.

+ Constr.

*5 Def. 5.

PROP. XXII. THEOR.

If there be any number of magnitudes, and as many others, which taken two and two in order have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first has to the last of the others. N. B. This is usually cited by the words "ex æquali," or, "ex æquo."

First, let there be three magnitudes A, B, C, and as many others D, E, F, which taken two and two have the same ratio, that is, such that A is to B as D to E; and as B is to C, so is E to F: A shall be to C, as D to F.

ABC DEF
GKM HL N

Take of A and D any equimultiples whatever G and H; and of B and E any equimultiples whatever K and L; and of C and F any whatever M and N: then because A is to B, as D to E, and that G, H are equimultiples of A, D, and K, L equimultiples of B, E; therefore as G is to K, so is * H to L: for the same reason, K is to M as L to N: and because there are three magnitudes G,

K, M, and other three H, L, N, which two and two have the same ratio*; therefore if G be greater than M, H is greater than N; and if equal, equal; and if less, less: but G, H, are any equimultiples whatever of A, Dt, and M, N are any equimultiples whatever of C, F; therefore* as A is to C, so is D to F.

Next, let there be four magnitudes, A, B, C,
D, and other four E, F, G, H, which two and

two have the same ratio, viz. as A is to B, so is

A.B.C.D.

E.F.G.H