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: A is equal to B: for if they be 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: 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 shall also be greater than F (5. def. 5): but E is not greater than F, which is impossible; A therefore and B are not unequal; that is, they are equal.

A

B

D

이

E

F

Next, let C have the same ratio to each of the magnitudes A and B ; A is equal to B: for if they be not, one of them is greater than the other; let A be the greater; therefore, as was shown in Prop. 8th, there is some multiple F of C, and some equimultiples E 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 that F, the multiple of the first, is greater than E, the multiple of the second; F the multiple of the third, is greater than D, the multiple of the fourth (5. def. 5.): but F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c. Q. E. D.

• See Note.

PROP. X. THEOR.

THAT magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two: and that magnitude, to which the same has a greater ratio than it has unto another magnitude is the lesser of the two.*

Let A have to C a greater ratio than B has to C: A is greater than B: for, because A has a greater ratio to C, than B has to C, there are (7. def. 5.) 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 it: 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 E and, because D and E are equimultiples of A and B, and D is greater than E; therefore A is (4. Ax. 5.) greater than B.

A

Next, Let C have a greater ratio to B than it has to A; B is 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 it is not greater than D: E therefore is less than D; and B because E and D are equimultiples of B and A, therefore B is (4. Ax. 5.) less than A. That magnitude, therefore, &c. Q. E. D.

PROP. XI. THEOR.

D

E

F

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 as C to D, so 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, since A is to B, as C to D, and G, H are taken equimultiples of

* See Note.

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, less (5. def. 5.). Again, because C is to D, as E 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 less, less:

but if G be greater than L, it has been shown that H is greater than M; and if equal, equal; and if less, less: therefore, if G be greater than L, K is greater than N; and if equal, equal; and if less, less: and G K are any equimultiples whatever of A, E; and L, N any whatever of B, F: therefore as A is to B, so is E to F (5. def. 5.). Wherefore, ratios that, &c. Q. E. D.

IF

PROP. XII. THEOR.

If any number of magnitudes be proportionals as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents.

Let any number of magnitudes A, B, C, D, E, F be proportionals; that is, as A is to B, so is C to D, and E to F: as A is to B, so shall 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 E to F; and that G, H, K are equi

multiples of A, C, E, and L, M, N equimultiples 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 (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 (1. 5.) for the same reason L, and L, M, N are any equimultiples of B, and B, D, F: 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 which 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: also the first A shall 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, than the multiple of C is greater than the multiple of D, but

the multiple of E is not greater than the multiple of F (7. def. 5.); let 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 D,

*. See Note.

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, less (5. def. 5.). Again, because C is to D, as E 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 less, less:

:

but if G be greater than L, it has been shown that H is greater than M; and if equal, equal; and if less, less therefore, if G be greater than L, K is greater than N; and if equal, equal; and if less, less: and G K are any equimultiples whatever of A, E; and L, N any whatever of B, F: therefore as A is to B, so is E to F (5. def. 5.). 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 shall all the antecedents taken together be to all the consequents.

Let any number of magnitudes A, B, C, D, E, F be proportionals; that is, as A is to B, so is C to D, and E to F: as A is to B, so shall 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 E to F; and that G, H, K are equi