But LM was shown to be the same multiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of AB, CD.

Next, because HK is the same multiple of EB, that MN is of FD; and that KX is also the same multiple of EB, that NP is of FD; therefore HX is the same multiple (2. v.) of EB, that MP is of FD.

And because AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are equimultiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if less, less (5. Def. v.):

A C L

But if GH be greater than KX, by adding the common part HK to both, GK is greater than HX: wherefore also LN is greater than MP; and by taking away MN from both, LM is greater than NP: therefore, if GH be greater than KX, LM is greater than NP.

In like manner it may be demonstrated, that if GH be equal to KX, LM likewise is equal to NP; and if less, less:

And GH, LM are any equimultiples whatever of AE, CF, and KX, NP are any whatever of EB, FD: therefore (5 Def. v.), as AE is to EB, so is CF to FD. If, then, magnitudes, etc.

PROPOSITION XVIII.

Q. E. D.

THEOR. If magnitudes, taken separately, be proportionals, they shall also be proportionals when taken jointly; that is, if the first be to the second as the third to the fourth, the first and second together shall be to the second as the third and fourth together to the fourth. Let AE, EB, CF, FD be proportionals; that is, as AE to EB, so is CF to FD: they shall also be proportionals when taken jointly; that is, as AB to BE, so CD to DF.

Take of AB, BE, CD, DF any equimultiples whatever GH, HK, LM, MN: and again, of BE, DF take any whatever equimultiples KO, NP:

And because KO, NP are equimultiples of BE, DF, and that KH, NM are equimultiples likewise of BE, DF; if KO, the multiple of BE, be greater than KH, which is a multiple of the same BE, NP likewise, the multiple of DF, shall be greater than MN, the multiple of the same DF; and if KO be equal to KH, NP shall be equal to NM; and if less, less.

First, let KO be not greater than KH; therefore NP is not greater than NM:

And because GH, HK are equimultiples of AB, BE, and that AB is greater than BE, therefore GH is greater (3 Ax. v.) than HK; but KO is not greater than KH, wherefore GH is greater than KO.

In like manner it may be shown, that LM is greater than NP:

K

MI

N

C AGL

Therefore, it 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.

Next, let KO be greater than KH; therefore, as has been shown, NP is greater than NM:

And because the whole GH is the same multiple of the whole AB, that HK is of BE, the remainder GK is the same multiple of the remainder AE that GH is of AB (5. v.) ; which is the same that LM is of CD.

21

H

In like manner, because LM is the same multiple of CD, that MN is of DF, the remainder LN is the same multiple of the remainder CF, that the whole LM is of kthe whole CD (5. v.).

But it was shown 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.

C A CL

And because KO, NP are equimultiples of BE, DF, 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 them (6. v.).

First, let HO, MP be equal to BE, DF: and because AE is to EB, as CF to FD, and that GK, LN are equimultiples of AE, CF; GK shall be to EB, as LN to FD (Cor. 4. v.): but HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP. If, therefore, GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less (A. v.).

But let HO, MP be equimultiples of EB, FD:

And because (Hyp.) AE is to EB, as CF to FD, and that of AE, CF are taken equimultiples GK, LN; and of EB, o FD, the equimultiples HO, MP; if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less (5 Def. v.); which was likewise shown in the preceding case.

If therefore GH be greater then 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 shown, that if GH be equal to KO, LM is equal to NP; and if less, less.

F

A

And in the case in which KO is not greater than KH, it has been shown that GH is always greater than KO, с A and likewise LM than NP: but GH, LM are any equimultiples of AB, CD, and KO, NP are any whatever of BE, DF; therefore (5 Def. v.), as AB is to BE, so is CD to DF. If then magnitudes,

etc.

Q E. D.

PROPOSITION XIX.

THEOR. 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 to the whole.

Let the whole AB, be to the whole CD, as AE, a magnitude taken from AB, to CF, a magnitude taken from CD; the re

mainder EB shall be to the remainder FD, as the whole AB to the whole CD.

Because AB is to CD, as AE to CF (Hyp.); likewise, alternately (16. v.), BA is to AE, as DC to CF:

And because, if magnitudes taken jointly be proportionals, they are also proportionals (17. v.) when taken separately; therefore, as BE is to EA, so is DF to FC; and alternately, as BE is to DF, so is EA to FC:

But, as AE to CF, so, by the hypothesis, is AB to CD; therefore also BE, the remainder, shall be to the remainder DF (11. v.), as the whole AB to the whole CD. Wherefore, if the whole, etc. Q. E. 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 likewise is to the remainder, as the magnitude taken from the first to that taken from the other. The demonstration is contained in the preceding.

PROPOSITION 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 is to AE, as DC to CF.

Because AB is to BE, as CD to DF, by division (17. v.), AE is to EB, as CF to FD;

And by inversion (B. v.), BE is to EA, as DF to FC:

Wherefore, by composition (18. v.), BA is to AE, as DC is to CF. If, therefore, four, etc. Q. E. D.

PROPOSITION XX.

THEOR. If there be three magnitudes, and other three, which, taken two and two, have the same ratio; 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 (8. v.); therefore A has to B a greater ratio than C has to B:

But as D is to E, so is A to B; therefore (13. v.) 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 (B. v.) :

And D was shown to have to E a greater ratio than C to B; therefore D has to E a greater ratio than F to E (Cor. 13. v.). But the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two (10. v.); D is therefore 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 (7. v.):

But A is to B, as D to E; and C is to B, as F to E; wherefore D is to E, as F to E (11. v.); and therefore D is equal to F (9. v.):

Next, let A be less than C; D shall be less than F:

For C is greater than A; and, as was

shown in the first case, 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; and therefore D is less than F. Therefore, if there be three, etc. Q. E D.

PROPOSITION XXI.

THEOR. If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; 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.

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio (8. v.) that C has to B:

But as E to F, so is A to B; therefore (13. v.) E has to

F a 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 shown to have to F a greater ratio than C to B; therefore E has to F a greater ratio than E to A (Cor. 13. v.) D:

But the magnitude to which the same has a greater ratio D than it has to another, is the less of the two (10. v.); F therefore 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 (7. v.) to B, as C is to B: But A is to B, as E to F; and C is to B, as E to D; wherefore E is to F, as E to D (11. v.); and therefore D is equal to F (9. v.).

Next, let A be less than C: D shall be less than F. For C is greater than A; and, as was shown, C is to B, 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; and therefore, D is less than F. Therefore, if there be three, etc. Q. E. D.

10

PROPOSITION 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 of the others has to the last. 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.

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; as G is to K, so is (4. v.) 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; if G be greater than M, H is greater than N; and if equal, equal; and if less, less (20. v.):

And G, H are any equimultiples whatever of A, D (Constr.), and M, N are any equimultiples of C, F; therefore (Def.), as A is to C,

so is D to F.

G

C

E

F

K M

H L

N

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 E to F; and as B to C, so F to G; and as C to D, so G to H: A shall be to D, as E to H.

A. B. C. D. E. F. G. H.

Because A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio; by the foregoing case, A is to C, as E to G:

But C is to D, as G to H; wherefore again, by the first case, A is to D, as E to H; and so on, whatever be the number of magnitudes. Therefore, if there be any number, etc. Q. E. D.

PROPOSITION XXIII.

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

First, let there be three magnitudes A, B, C, and other three, D, E, F, which, taken two and two in a cross order, have the same ratio; that is, such, that A is to B, as E to F; and as B is to C, so is D to E: A is to C, as D to F.