+ Hyp.

5 Def. 5.

† Constr.

+ Hyp.

*5 Def. 5. ↑ Constr.

See N.

* 7 Def. 5.

multiple of A is greater than the multiple of C, but the mul-
tiple of B is not greater than that of C. Let these multiples
be taken; and let D, E be the equimultiples of
A, B, and F the multiple of C, such, that D may
be greater than F, but 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, and that D is greater than
F, therefore E is also greater than F: but E is
not greater than F; which is impossible:

*

B

therefore A 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 shall be equal to B.

For, if they are not equal, one of them must be 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, but not greater than D: and 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 *, therefore F the multiple of the third is 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.

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 lesser of the

two.

Let A have to C a greater ratio than B has to C: A shall be 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, 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 the equimultiples of A, B, and F the multiple of C, such, that D is greater than F, but E is not greater than F; therefore D

B

is greater than E: and, because D and E are equimultiples

of A and B, and that D is greater than E, therefore A is✶ ✶ 4 Ax. 5. greater than B.

Next, let C have a greater ratio to B than it has to A: B shall be less than A.

For* there is some multiple F of C, and some equimultiples 7 Def. 5. E and D of B and A, such, that F is greater than E, but not greater than D; therefore E is less than D: and because E and D are equimultiples of B and A, and that E is less than

D, therefore B is less than A. Therefore, that magnitude, 4 Ax. 5. &c. Q. E. D.

PROPOSITION 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 shall be 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 equi- G multiples of A, C, and L,

H

-K

A

C

E

and if less, less *. Again, because C is to D, as E is to F, and 5 Def. 5. 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 shewn 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. Wherefore, 5 Def. 5. ratios that, &c. 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,

Ꮶ ;

and

of B, D, F, any equimultiples whatever L, M, N: then, because A is to B, as C is to

is greater than M, and K greater than N; and if equal, equal; * 5 Def. 5. and if less, less*: wherefore if G be greater than L, then G,

⚫ 1.5.

† 5 Def. 5.

H, K together, are greater than L, M, N together; and if equal, equal; and if less, less: but 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 L, M, N are any equimultiples of B, and B, D, F: therefore as A is to B †, so are A, C, E together, to B, D, F together. Wherefore, if any number, &c. Q. E. D.

See N.

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 a greater ratio to D the fourth, than E the fifth has to F the sixth: also the first A shall have to the second B, a greater ratio than the fifth E has to the sixth F.

D and F, such, that the multiple of

is greater than the mul

tiple of D, but the multiple of E is not greater than the mul

tiple of F let these be taken, and let G, H be equimulti- ⚫7 Def. 5. ples of C, E, and K, L equimultiples of D, F, such, that G may 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 Hyp. 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: 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 B, than E has to F. Wherefore, if the first, &c. 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.

5 Def. 5.

+ Constr. + Constr.

⚫7 Def. 5.

E.J.

THEOR.-If the first have the same ratio to the second which See N.
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 the same ratio to the second B, which the third C has to the fourth D: if A be greater than C, B shall be greater than D.

⚫ 8. 5.

+ Hyp.

3

⚫ 13. 5.

ABC D ABC D Á BC D

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C has to B*: but, as A is to Bt, so is C to D; therefore also C has to D a greater ratio than C has to B*: but of two magnitudes, that to which the same has the greater ratio is the lesser : therefore D is less than B; that is, B is greater than D. Secondly, if A be equal to C, B shall be equal to D. For A is to B, as C, that is, A to D; therefore B is equal to D*. 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, therefore D is greater than B, by the first case; that is, B is less than D. Therefore, if the first, &c.

Q. E. D.

* 10. 5.

* 9.5.

• 7.5.

* 12. 5.

. 15. 5.

+ Hyp.

11. 5.

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 shall be to F, as AB to DE.

G

H

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, is 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: but as one of the antecedents to its consequent *, so are all the antecedents together to all the consequents together; 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, &c. Q. E. D.

PROPOSITION XVI.

BC EF

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

Let A, B, C, D be four magnitudes of the same kind, which are proportionals; viz. as A to B, so C to D: they shall also be proportionals when taken alternately; that is, A shall be 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 mag-
nitudes have the same ratio to one
another which their equimultiples
have, therefore A is to B, as E is to F:

E

A

C

B

D

F

but as A is to B so t is C to D; wherefore as C is to D, so is

E to F: again, because G, H are equimultiples of C, D, there