PROPOSITION 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 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 (8. v.), there are some equimultiples of A and B, and some multiple of C, such that A 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 (Hyp.) 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 B greater than F (5 Def. v.): but E is not greater (Constr.) than F; which is impossible; A therefore and B are not unequal; that is, they are equal.

E

Next, let C have the same 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 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 (Hyp.), as C is to A, and that F the multiple of the first is greater than E the multiple of the second (5 Def. v.); F the multiple of the third is greater than D, the multiple of the fourth: but F is not (Constr.) greater than D, which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, etc. Q. E. D.

PROPOSITION 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 less 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. v.) 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 Dis 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. v.) greater

than B.

Next, let C have a greater ratio to B than it has to A;

B

E

VOL. II.

H

B is less than A: for (7 Def. v.) 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 (4 Ax. v.) less than A. That magnitude, therefore, etc.

PROPOSITION XI.

Q. E. D.

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 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. v.).

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 less, less (5 Def. v.).

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. v.). Wherefore, ratios that, etc. Q. E. D.

PROPOSITION 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 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

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. v.):

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. v.): for the same reason L, and L, M, N are any equimultiples of B, and B, D, F: as therefore A is to B (5 Def. v.) so are A, C, E together to B, D, F together. Wherefore, if any number, etc.

Q. E. D.

PROPOSITION 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, 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 (7 Def. v.): let such be taken, and of C, E, let G, H be equimultiples, and K, L equimultiples of D, F, such

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 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 than K; and if equal, equal; and if less, less (5 Def. v.):

But G is greater (Constr.) than K; therefore M is greater than N: but H is not greater (Constr.) 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 E has to F (7 Def. v.). Wherefore, if the first, etc.

Q. E. D.

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.

PROPOSITION XIV.

THEOR. If the first has the same ratio to the second 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 (Hyp.), and B is any other magnitude, A has to B a greater ratio than C to B (8. v.):

But, as A is to B, so is C to D; therefore also C has to D a greater ratio than C has to B (13. v.):

But of two magnitudes, that to which the same has the greater ratio is the less (10. v.). 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 B, as C,

that is, A to D; B therefore is equal to D (9. v.).

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, etc. Q. E. D.

PROPOSITION XV.

THEOR. Magnitudes have the same ratio to one another which their equimultiples have.

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

a

Because AB is the same 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 last, DK, KL, LE:

And because AG, GH, HB are all equal, and that DK, KL, LE are also equal to one another; therefore AG is to DK as GH to KL, and as HB to LE (7. v.):

H

K

B C E F

And as one of the antecedents to its consequent, so are all the antecedents together to all the consequents together (12. v.); wherefore, as AG is to DK, so is AB to DE: but AG is equal to C, and DK to F; therefore, as C is to F, so is AB to DE. Therefore, magnitudes, etc. Q. E. D.

PROPOSITION XVI.

THEOR. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately.

Let the four magnitudes A, B, C, D be proportionals, viz. as A to B, so C to D: they shall also be proportionals when taken alternately; that is, A is to C, as B to D.

Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H:

And because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another which their equimultiples have (15. v.); therefore A is to B, as E is to F:

But as A is to B, so (Hyp.) is C to D; wherefore as C is to D, so (11. v.) is E to F:

Again, because G, H are equimultiples of C, D, as C is to D, so is G to H (15. v.): but as C is to D, so is E to F. fore, as E is to F, so is (11. v.).

Where

A

G to H в

D

H

But when four magnitudes are F proportionals, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; if less, less (14. v.). Wherefore, if E be greater than G, F likewise is greater than H; and if equal, equal; if less, less: and E, F are any (Constr.) equimultiples whatever of A, B; and G, H any whatever of C, D: therefore A is to C, as B to D (5 Def. v.). If, then, four magnitudes, etc. Q. E. D.

PROPOSITION XVII.

THEOR. If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last 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, so is CD to DF: they shall 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 because GH is the same multiple of AE, that HK is of EB, wherefore GH is the same multiple (1. v.) of AE, that GK is of AB:

But GH is the same multiple of AE, that LM is of CF; wherefore GK is the same multiple of AB, that LM is of CF.

Again, because LM is the same multiple of CF, that MN is of FD; therefore LM is the same multiple (1. v.) of CF, that LN is of CD: