Book V.

See Note.

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; a 5. def. 5. E shall also be greater than F; but E is

not greater than F, which is impossible; A
A therefore and B are not unequal; that is,
they are equal.

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 B
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

D

F

C

이

E

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 2: 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 Note. 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 a some equimultiples of A and B, and a 7. def. 5. 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 Fa 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 equi- A multiples of A and B, and D is greater than E; therefore A is b greater than B.

Next, Let C have a greater ratio to B than it has to A; B is less than A: for a 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 B because E and D are equimultiples of B and A, therefore B is b less than A. That magnitude, therefore, &c. Q. E. D.

D

F

이

E

b 4. Ax. 5.

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 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

Book V. 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. Again, because a 5. def. 5. 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 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 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 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 as E to F; and that G, H

K are equimultiples of A, C, E, and L, M, N equimultiples of Book V. 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. Where- a 5. 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 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 wholeb: for the same reason L, and L, M, N are any b 1.5. 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 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: 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, 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.de₤.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 D, and

S

Book V. of A and C, M and G are equimultiples: and of B and D, N and K are equimultiples; if M be greater than N, G is greater b 5. def. 5. than K; and if equal, equal; and if less, less; 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 c7.def. 5. B, than E has to Fe. Wherefore, if the first, &c. Q. E. D.

See N.

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

PROP. XIV. THEOR.

IF the first has to the second, the same 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 second B, the same 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 8. 5. A has to B a greater ratio than C to Ba; but, as A is to B, so

is C to D; therefore also C has to D a greater ratio than C has

b 13. 5. to Bb: but of two magnitudes, that to which the same has the c 10. 5. greater ratio is the lessere: wherefore D is less than B; that is, B is greater than D.

Secondly, If A be equal to C, B is equal to D: for A is to 495. B, as C, that is A, to D; B therefore is equal to Dd.

Thirdly, If A be less than C, B shall be less 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 less than D. Therefore, if the first, &c. Q. E. D.