D: And in every one of the cases, take H the double of D, Book V. K its triple, and so 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 less than L.

E

Then, because L is the multiple of D, which is the first that becomes greater than FG, the next preceding multiple K is not greater than FG, that is, FG is not less than K: And since EF is the same multiple of AC, that FG is of CB; FG is the same multiple of CB, that EG is of AB; wherefore a 1. 5. EG and FG are equimultiples of AB and CB; and it was shown, that FG was not less than K, and, by the construction, EF is greater than D; therefore, the whole EG is greater than K and D together: But K, together with D, is equal to L; therefore EG is greater than L: but FG is not greater than F L; and EG, FG are equimultiples of AB, BC, and L is a multiple of D; therefore, AB has to D a greater ratio than BC has to D.

E

F

A

A

C

GB

b 7. def. 5.

LK HD

G B

Also, D has to BC a greater L K D

ratio than it has to AB: For,

having made the same construc

tion, it may be shown, in like manner, that L is greater than FG, but that it is not greater than EG; and L is a multiple of D; and FG, EG are equi

b

multiples 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.

PROP. IX. THEOR.

Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another.

Let A, B, have each of them the same ratio to C; then will A be 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 shown in the preceding proposition, there are some equimultiples of A and B, and some multiple of C such, 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 such multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C, so that D may be greater than F, and E not greater than F: Then, 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; D being greater than A a 5. def. 5. F, E will also be greater than Fa; but E is not greater than F, which is impossible; A therefore and B are not unequal; that is, they are equal.

D

E

F

Next, Let C have the same ratio to each of the magnitudes A and B; then will A be equal to B: For, if they are not, one of them is greater than the other; let A be the greater; therefore, as was shown in Prop. 8, there is some multiple F, of C, and some equimultiples É and D, of B and A, such that F is greater than E, and not greater than D; but because C is to B as C is to A, and F the multiple of the first, is greater than E, the multiple of the second; F, the multiple of the third, is also greater than D, the multiple of the fourth But F is not greater than D, which is impossible. Therefore, A is equal to B. Wherefore, "magnitudes which," &c. Q. E. D.

Book V.

PROP. X. THEOR.

That magnitude, which has a greater ratio than See N. another has to the same magnitude, is the greater of the two: And that magnitude, to which the same has a greater ratio than it has to another magnitude, is the less of the two.

Let A have to C a greater ratio than B has to C; then is A greater than B: For, because A has a greater ratio to C, than B has to C, there are some equimultiples of A and B, a 7. def. 5. and some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than the multiple of C. Let them be taken, and let D, E, be equimultiples of A, B, and F a multiple of C such, that D is greater than F, but E is not greater than F: Therefore D is greater than Ě: And because D and E are equimultiples of A and B, and D is greater than A E; therefore A is greater than B.

D

E

b 4. Ax. 5.

F

Next, Let C have a greater ratio to B than it has to A; then is B less than A: For there is some multiple F of C, and some equimultiples E and D of B and A such, that F is greater than E, but is not greater than D: E therefore is less than D; and because E and D are equimultiples of B and A, therefore, B is less than A. "That magnitude, there"fore," &c. Q. E. D.

PROP. XI. THEOR.

Ratios that are the same to the same ratio, are the same to one another.

Let A be to B as C is to D; and C to D as E is to F; then will A be to B, as E to F.

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

Book V. equimultiples of 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 less, a 5. def. 5. less. Again, because C is to D, as E is to F, and H, K are 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 less, less: But if G be greater than L, it has been shown that H is also greater than M; and if equal, equal; and if less, less; therefore, if G be greater than L, K is also greater than N; and if equal, equal; and if less, less: And since G, K are any equimultiples whatever of A, E; and L, N, are any equimultiples whatever of B, F: Therefore, as A is to B, so is E to Fa. 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 consequent, so are all the antecedents, taken together, to all the consequents.

Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, let A be to B, as C to D, and as E is to F: Then, as A is to B, so will A, C, E together, be to B, D, F, together.

For, of A, C, E take 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 G, H,

K are equimultiples of A, C, E, and L, M, N equimultiples Book V. of B, D, F; if G be greater than L, H is greater than M, and K greater than N; and if equal, equal; and if less, less a. a 5. def. 5. Wherefore, if G be greater than L, then G, H, K together are greater than L, M, N together and if equal, equal; and if less, less. 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 same multiple is the whole of the whole: For the same reason, L, and b 1. 5. L, M, N together are any equimultiples of B, and B, D, F together: As therefore A is to B, so are A, C, E together to B, D, F together. Wherefore, " if any number," &c. Q. E. D.

PROP. XIII. THEOR.

If the first has to the second the same ratio that See N. the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first shall also have to the second a greater ratio than the fifth has to the sixth.

Let A the first, have the same ratio to B the second, 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 sixth: Then will the first A have to the second B a greater ratio than the fifth E to the sixth F.

Because C has a greater ratio to D, than E to F, there are some equimultiples of C and E, and some 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 Fa: Let a 7. def. 5. such be taken, and of C, E let 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 same multiple of A; and whatever multiple K is of D, take N the same multiple of B: Then, because A is to B as C to