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.

Next, let C have a greater ratio to B that 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 B is less than D; and 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.

A

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 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.

Ir 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 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. 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.

Ir 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 F, and some of D and F, such, than the multiple of C is greater than the multiple of D, but the multiple

* See Note.

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, 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. 5.): 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 (7. def. 5.). 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.

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 has to B a greater ratio than C to B (8. 5.); but as A is to B, so is C to

A B

D A B C D

A B D; therefore also C has to D a greater ratio than C has to B (13. 5.): but of two magnitudes, that to which the same has the greater ratio is the lesser (10. 5.): 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. 5.).

* See Note.

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.

PROP. 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.

G

A

K

D

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. 5.) and as one of the antecedents to its consequent, so are all the antecedents together to all the consequents together (12. 5.); 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.

H

PROP. XVI. THEOR.

B

CE F

Ir 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. 5.); therefore A is to B, as E is to F: but as A is to B so is C to D: wherefore

5.). But, when four magnitudes are 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. 5.). 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 equimultiples whatever of A, B; and G, H any whatever of C, D. Therefore A is to C, as B to D (5. def. 5.). If then four magnitudes, &c. Q. E. D.

PROP. XVII. THEOR.

Ir 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.

X

P

Take of AE, EB, CF, FD, any equimultiples whatever of 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, therefore GH is the same multiple (1. 5.) 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. 5.) of CF, that LN is of CD: but LM was shown to be the same multiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of CD: that is, GK, LN are equimultiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; and that KX is also the same multiple of EB, that NP is of FD; therefore HX is the same multiple (2. 5.) of EB, that MP is of FD. And because AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are equimultiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if less, less (5. def. 5.): but if GH be greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore also LN is greater than MP; and by taking away MN from both, LM is greater than NP: therefore, if GH be greater

• See Notc.

K

N

H

M

B D

E

F