a. 5. Def. 5.

Book V. H; and L, M of B, D; if G be greater than L, H is greater than M; and if equal, equal; and if lefs, lefs. Again, because C is to D, as E is to F, and H, K are taken equimultiples of C, E; and M, N of D, F; if H be greater than M, K is greater than N; and if equal, equal; and if lefs, lefs. but if G be greater than L, it has

been shewn that H is greater than M; and if equal, equal; and if lefs, lefs; therefore if G be greater than L, K is greater than N; and if equal, equal; and if lefs, lefs. and G, K, are any equimultiples whatever of A, E; and L, N any whatever of B, F. Therefore as A is to B, fo is E to F. Wherefore ratios that, &c. Q. E. D.

IF

F any number of magnitudes be proportionals, as one of the antecedents is to its confequent, fo fhall all the antecedents taken together be to all the confequents.

Let any number of magnitudes A, B, C, D, E, F, be propor tionals; that is, as A is to B, fo C to D, and E to F. as A is to B, fo fhall A, C, E together be to B, D, F together.

Take of A, C, E any equimultiples whatever G, H, K; and of

B, D, F any equimultiples whatever L, M, N. then because A is to
B, as C is to D, and as E to F; and that G, H, K are equimultiples

of A, C, E, and L, M, N equimultiples of B, D, F; if G be greater Book V. than L, H is greater than M, and K greater than N; and if equal, equal; arid if less, lefs 2. Wherefore if G be greater than L, then a. 5. Del. 5. G, H, K together are greater than L, M, N together; and if equal, equal; and if lefs, lefs. and G, and G, H, K together are any equimultiples of A, and A, C, E together, because if there be any number of magnitudes equimultiples of as many, each of each, whatever multiple one of them is of its part, the fame multiple is the whole of the whole . for the fame reafon L, and L, M, N b. i. si are any equimúltiples of B, and B, D, F. as therefore A is to B, fo are A, C, E together to B, D, F together. Wherefore if any number, &c. Q. E. D.

PROP. XIII. THEOR.

F the firft has to the fecond the fame ratio which the see Ni third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; the first shall also have to the fecond a greater ratio than the fifth has to the fixth.

Let A the firft have the fame ratio to B the fecond which C the third has to Ď the fourth, but C the third to D the fourth a greater ratio than E the fifth to F the fixth. alfo the first A shall have to the second B a greater ratio than the fifth E to the fixth F.

Because C has a greater ratio to D, than E to F, there are some equimultiples of C and E, and fome of D and F fuch, that the multiple of C is greater than the multiple of D, but the multiple of E is

not greater than the multiple of F a. let fuch be taken, and of C, E a. 7. Def. 5Jet G, H be equimultiples, and K, L equimultiples of D, F so that G be greater than K, but H not greater than L; and whatever multiple G is of C take M the fame multiple of A; and what multiple K is of D, take N the fame multiple of B. then because A is to B, as C to D, and of A and C, M and & are équimultiples, and of

b. 5.Def. 5.

Book V. B and D, N and K are equimultiples; if M be greater than N, G is greater than K; and if equa!, equal; and if lefs, lefs ; but G is greater than K, therefore M is greater than N. but H is not greater than L; and M, H are equimultiples of A, E; and N, L equimultiples of B, F. Therefore A has a greater ratio to B, than a. 7.Def. 5. E has to F 2. Wherefore if the firft, &c. Q. E. D.

See N.

a. 8. 5.

COR. And if the first has a greater ratio to the second, than the third has to the fourth; but the third the fame ratio to the fourth, which the fifth has to the fixth; it may be demonstrated in like manner that the first has a greater ratio to the second than the fifth has to the fixth.

TF the first has to the fecond the fame ratio, which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less.

Let the first A have to the fecond B the fame ratio, which the third C has to the fourth D; if A be greater than C, B is greater

than D.

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C to B a. but as A is to B, fo is C to

b. 13. 5.

C. 10. 5.

d. 9. 5.

A B C D A B C D A B C D D; therefore alfo C has to D a greater ratio than C has to Bb. but of two magnitudes, that to which the fame has the greater ratio is the leffer . wherefore D is less than B; that is, B is greater than D.

Secondly, If A be equal to C, B is equal D. for A is to B, as C, that is A, to D; B therefore is equal to Da.

Thirdly, If A be lefs than C, B fhall be lefs than D. for C is greater than A, and because C is to D, as A is to B, D is greater than B by the first case; wherefore B is lefs than D. Therefore if the first, &c. Q.E. D.

PROP. XV. THEOR.

AGNITUDES have the fame ratio to one another
which their equimultiples have.

Let AB be the fame multiple of C that DE is of F. C is to F, as AB to DE.

A

Because AB is the fame multiple of C that DE is of F, there are as many magnitudes in AB equal to C, as there are in DE equal to F. Let AB be divided into magnitudes, each equal to C, viz. AG, GH, HB; and DE into magnitudes, each equal to F, viz. DK, KL, LE. then the number of the first AG, GH, HB shall be equal to the number of the laft DK, KL, LE. and because AG, GH, HB are all equal, and that DK, KL, LE are alfo equal to one another; therefore AG is to DK, as GH to KL, and as HB to LE. and as one of the antecedents to its confequent, so are all the antecedents

Book V.

D

K

H

L

В С

E F

2. 7.5.

together to all the confequents together ; wherefore as AG is to b. 12. 5. DK, fo is AB to DE. but AG is equal to C, and DK to F. therefore as C is to F, fo is AB to DE. Therefore magnitudes, &c. Q. E. D.

PROP. XVI. THEOR.

F four magnitudes of the fame kind be proportionals, they shall also be proportionals when taken alternately.

Let the four magnitudes A, B, C, D be proportionals, viz. as A

whatever E and F; and of C and D take any equimultiples whatever

Book V. G and H. and because E is the fame multiple of A, that F is of B, and that magnitudes have the fame ratio to one another which their equimultiples have 2; therefore A is to B, as E is to F. but as A is to B, fo is C to D. wherefore as C is to D, fo b is E to F. again, because G, H

a. 15. 5.

b. II. 5.

H. But when four magnitudes are proportionals, if the first be greater than the third, the second shall be greater than the fourth; E. 14. 5 and if equal, equal; if lefs, lefs. Wherefore if E be greater than G, F likewife is greater than H; and if equal, equal; if lefs,

less. and E, F are any equimultiples whatever of A, B; and G, d. 5. Def. 5. H any whatever of C, D. Therefore A is to C, as B to Dd. If then four magnitudes, &c. Q. E. D.

IF

PROP. XVII. THEOR.

F magnitudes taken jointly be proportionals, they fhall also be proportionals when taken feparately, that is, if two magnitudes togetlter have to one of them, the fame ratio which two others have to one of these, the remaining one of the first two fhall have to the other, the fame ratio which the remaining one of the laft two has to the other of these.

Let AB, BE; CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, fo is CD to DF; they fhall also be proportionals taken separately, viz. as AE to EB, so CF to FD.

Take of AE, EB, CF, FD any equimultiples whatever GH,HK, LM, MN; and again, of EB, FD, take any equimultiples whatever KX, NP. and becaufe GH is the fame multiple of AE that HK is of EB, therefore GH is the fame multiple a of AE, that GK is of AB. but GH is the fame multiple of AE, that LM is of CF; wherefore GK is the fame multiple of AB, that LM is of CF. Again, be