+10 of this. bears a greater Proportion, + is the leffer Magnitude. Therefore F is lefs than D; and fo D fhall be greater than F. After the fame manner we demonftrate, if A be equal to C, D will also be equal to F; and if A be less than C, D will be alfo less than F. If, therefore, there are three Magnitudes, and others equal to them in Number, which, taken two and two, are in the fame Proportion, and the Proportion be perturbate; if the first Magnitude be greater than the third, then the fourth will be greater than the fixth; but if the first be equal to the third, then is the fourth equal to the fixth; if lefs, lefs; which was to be demonftrated.

PROPOSITION XXII.

THEOREM.

If there be any Number of Magnitudes and others equal to them in Number, which, taken two and two, are in the fame Proportion; then they fhall be in the fame Proportion by Equality. LET there be any Number of Magnitudes, A, B,

and C; and others D, E, and F, equal to them in Number, which, taken two and two, are in the fame Proportion; that is, as A is to B, fo is D to E; and as B is to C, fo is E to F. I fay, they

are also proportional by Equality,
viz. as A is to C, fo is D to F.

For let G and H be any Equimultiples of A and D; and K and L any Equimultiples of B and E; and likewife M and N, any Equimultiples of C and F. Then because A is to B, as D is to E; and G and H are Equimultiples of A and D; and K and L Equimultiples of B and *4 of this. E, it fhall be, as G is to K, fo is H to L. For the fame Reafon, alfo, it will be as K is to M, fo is L to N. And fince there are three Magnitudes G, K, and M, and others H, L, and N, equal to them in Number, which, being taken two and two, in each

*

ABCDEF

GKM HLN

Order,

Order, are in the fame Proportion; therefore if G exceeds M, * H will exceed N; if G be equal to M, * 20 of this then H fhall be equal to N ; and if G be lefs than M, H fhall be lefs than N. But G and H are Equimultiples of A and D; and M and N any other Equimultiples of C and F. Whence as A is to C, fo fhall Dt be to F. Therefore, if there be any Number of + Def. 5. of Magnitudes, and others equal to them in Number, which, taken two and two, are in the fame Proportion; then they fhall be in the fame Proportion by Equality; which was to be demonftrated.

PROPOSITION XXIII.

PROBLEM.

If there be three Magnitudes; and others equal to them in Number, which, taken two and two, are in the fame Proportion; and if their Analogy be perturbate, then shall they be alfo in the fame Proportion by Equality.

L E T there be three Magnitudes

A, B, and C ; and others equal to them in Number, D, E, and F, which, taken two and two, ate in the fame Proportion, and their Analogy be perturbate; that is, as A is to B, fo is E to F; and as B is to C, fo is D to E. I fay, as A is to C, fo is D to F.

A B C D E F

For, let G, H, and L, be Equi- GHKL MN multiples of A, B, and D; and K, M, and N, any Equimultiples of

C, E, and F.

Then, becaufe G and H are Equimultiples of A and B, and fince Parts have the fame Proportion as their like Multiples, when taken corre

bis.

fpondently, it fhall be *, as A is to B, fo is G to H: and, by the fame Reafon, as E is to F, fo is M to N. 15 of this. But A is to B, as E to F. Therefore, as G is to H, fo is M to N. Again, becaufe B is to C as D is 11 of this. to E; and H and L are Equimultipies of B and D ; L

aş

as likewife K and M any Equimultiples of C and E it fhall be as H is to K, fo is L to M. But it has been alfo proved, that as G is to H, fo is M to N Therefore, because three Magnitudes G, H, and K, and others, L, M, and N, equal to them in Number, which, taken two and two, are in the fame Proportion, and their Analogy is perturbate; then if G ex21 of this ceeds K, alfo L* will exceed N; and if G be equal to K, then L will be equal to N; and if G be lefs than K, L will likewife be less than N. But G and L are Equimultiples of A and D; and K and N Equimultiples of C and F. Therefore, as A is to C, fo fhall D be to F. Wherefore, if there be three Magnitudes, and others equal to them in Number, which, taken two and two, are in the fame Proportion; and if their Analogy be perturbate, then fhall they be alfo in the fame Proportion by Equality; which was to be demonftrated.

G

If the first Magnitude bas the fame Proportion to
the fecond, as the third to the fourth; and if
the fifth bas the fame Proportion to the fecond,
as the fixth bas to the fourth; then shall the
firft compounded with the fifth,
have the fame Proportion to
the fecond, as the third, com-
pounded with the fixth, has to
the fourth.

LET the firft Magnitude AB have

the fame Proportion to the fecond C, as the third DE has to the fourth F. Let alfo the fifth B G have the fame Proportion to the fecond C, as the fixth E H has to the fourth F. I fay, A G, the firft, compounded with the fifth, has the fame Proportion to the fecond C, as DH, the third, compounded with the fixth, has to the fourth F.

B+

AC

H

E+

D

For,

For, because B G is to C, as E H is to F; it thall be (inverfly), as C is to B G, fo is F to E H. Then, fince A B is to C, as DE is to F ; and as C is to BG, fo is F to EH; it fhall be, * by Equality, as * 22 of this. A B is to GB, fo is D E to E H. And because Magnitudes, being divided, are proportional, they fhall also

be + proportional when compounded. Therefore, as † 18 of this. AG is to BG, fo is D H to HE: But as GB is to Hyp. C, fo alfo is HE to F. Wherefore, by Equality*, it fhall be, as A G is to C, fo is D H to F. Therefore, if the firft Magnitude has the fame Proportion to the fe cond, as the third to the fourth; and if the fifth has the fame Proportion to the fecond, as the fixth has to the fourth; then hall the firft, compounded with the fifth, have the fame Proportion to the fecond, as the third compounded with the fixth has to the fourth; which was to be demonftrated.

PROPOSITION XXV.

PROBLEM.

If four Magnitudes be proportional; the greatest, and the least of them, will be greater than the other two.

LET four Magnitudes, A B, CD, E, F, be proportional, whereof A B is to C D, as E is to F; let A B be the greatest of them, and

F the leaft. I fay, A B and F, are B greater than CD and E.

For, let A G be equal to E, and CH to F. Then, because A B is to CD as E is to F; and fince AG and CH are each equal to E and F; it fhall be as A B is to D C, fo is AG to C H. And because the Whole AB is to the Whole CD, as the Part taken away A G is to the Part taken away CH; it fhall alfo be, as the Refidue G B to the Refidue HD, fo is the Whole A B to the Whole C D. But A Bis greater than CD; therefore, alfo,

G+

D

H+

29 of this.

A

CEF

1

G B shall be greater than H D. And fince A G is equal to E, and C H to F; A G and F will be equal to CH and E. But if equal Things are added to unequal Things, the Wholes fhall be unequal. Therefore GB, HD, being unequal, for G B is the greater, if A G and F are added to GB; and CH and E to HD; then AB and F will neceffarily be greater than CD and E. Wherefore, if four Magnitudes be proportional; the greatest and the leaft of them will be greater than the other two; which was to be demonftrated.

The END of the FIFTH BOOK.

EUCLID's