D: And in every one of the cafes take H the double of D, K its triple, and fo on, till the multiple of D be that which first becomes greater than FG: Let L be that multiple of D which is first greater than FG, and K the multiple of D which is next lefs than L.

E

F

Fig. 2.

Fig. 3.

E

Book V.

Then, because L is the multiple of D which is the first that becomes greater than FG, the next preceeding multiple K is not greater than FG; that is, FG is not lefs than K: And fince EF is the fame multiple of AC, that FG is of CB; FG is the fame multiple of CB, that EG is of AB; wherefore EG and a 1.5. FG are equimultiples of AB and CB; And it was fhewn, that FG was not less than K, and, by the conftruction, EF is greater than D; therefore the whole EG is greater than Kand D together: But K together with D is equal to L; therefore EG is greater than L; but FG is not greater than L; and EG, FG are equimultiples of AB, BC, and L is a multiple of D; therefore b AB has to Da greater ratio than BC has to D.

Alfo D has to BC a greater ratio than it has to AB: For, having made the fame conftruction, it may be fhewn, in like manner, that L is greater than

A

ct

G B

F

A

L KHD

G B

b 7. def. 5.

LKD

FG, but that it is not greater than EG: And L is a multiple of D; and FG, EG are equimultiples of CB, AB; therefore D has to CB a greater ratio than it has to AB. Wherefore, of unequal magnitudes, &c. Q. E. D.

Book V.

See N.

M

AGNITUDES which have the fame ratio to the fame magnitude are equal to one another; and thofe to which the fame magnitude has the fame ratio are equal to one another.

D

Let A, B have each of them the fame ratio to C; A is equal to B: For, if they are not equal, one of them is greater than the other; let A be the greater; then, by what was fhewn in the preceeding propofition, there are fome equimultiples of A and B, and fome multiple of C fuch, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let fuch multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C, fo that D may be greater than F, and E not greater than F: But, because A is to C, as B is to C, and of A, B are taken equimultiples D, E, and of C is taken a multiple F; and that D is greater than F; E fhall also be greater a 5. def. 5. than F; but E is not greater than F, which is impoffible; A therefore and B are not unequal; that is, they are equal. Next, Let C have the fame ratio to each of the magnitudes A and B; A is equal to B: For, if they are not, one of them is greater than the other; let A be the greater; therefore, as was fhewn in Prop. 8th, there is fome multiple F of C, and fome equimultiples E and D of B and A fuch, that F is greater than E, and not greater than D; but becaufe C is to B, as C is to A, and that the multiple of the first is greater than E the multiple of the fecond; Fthe multiple of the third is greater than D the multiple of the fourth: But Fin not greater than D, which is impoffible. Therefore A is equal to B. Wherefore magnitudes which, &c. Q. E. D.

B

E

F

PROP.

Book V.

TH

PROP. X. THEO R.

HAT magnitude which has a greater ratio than an- See N. other has unto the fame magnitude is the greater of the two: And that magnitude to which the fame has a greater ratio than it has unto another magnitude is the leffer of the two.

2

Let A have to Ca greater ratio than B has to C; A is greater than B: For, becaufe A has a greater ratio to C, than B has to C, there are fome equimultiples of A and B, and a 7. Def, 5. fome multiple of C fuch, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it: Let them be taken, and let D, E be equimultiples of A, B, and F a multiple of C fuch, that D is greater than F, but E is not greater than F: Therefore D is greater than E; And, because D and E are equimultiples of A and B, and D is greater than E; therefore A is greater than B.

b

Next, Let C have a greater ratio to B than it has to A; B is lefs than A: Fora there is some multiple F of C, and fome equimultiples E and D of B and A fuch that F is greater than E, but is not greater than D: E therefore is lefs than D; and because E and D are equimultiples of B and A, therefore B is lefs than A. That magnitude, therefore, &c. Q. E. D.

R

D

E

b 4. Ax. 5.

F

ATIOS that are the fame to the fame ratio, are the
fame to one another.

Let A be to B as C is to D; and as C to D, fo let E be to F; A is to B, as E to F.

Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F any equimultiples whatever L, M, N. Therefore, fince A is to B, as C to D, and G, H are taken equimultiples of

Book V.

a 5. def. 5.

A, C, 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, be caufe 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, Kis greater than N; and if equal equal; and if lefs, less: But, if G

be greater than L, it has been fhewn, 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 less, lefs: And G, K, are any equimultiples whatever of A, E; and L, Nany whatever of B, F: Therefore, as A is to B, fo is E to F. Wherefore ratios that, &c. Q. E. D.

PROP. XII. THEOR.

IF 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 equimpltiples whatever G, H, K;

and of B, D, F any equimultiples whatever L, M, N: Then, becaufe A is to B, as C is to D, and as E to F; and that G, H,

K

K are equimultiples of A, C, E, and L, M, N equimultiples of Book V. B, D, F; if G be greater than L, His greater than M, and K greater than N; and if equal, equal; and if lefs, lefs. Where- a §. def. 5., fore, if G be greater than L, then 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 are any ь 1. 5. equimultiples 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.

IF

[F the firft has to the fecond the fame ratio which the See N. third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; the first shall alfo have to the fecond a greater ratio than the fifth has to the fixth.

Let A the first have the fame ratio to B the fecond which C the third has to D the fourth, but C the third to D the fourth, a greater ratio than E the fifth to F the fixth: Alfo the firft A fhall have to the fecond B a greater ratio than the fifth E to the fixth F.

Becaufe C has a greater ratio to D, than E to F, there are fome equimultiples of C and E, and fome of D and F such, 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: Let a 7. def. §. fuch be taken, and of C, E let G, H be equimultiples, and K, L equimultiples of D, F, fo 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