PROP. X. THEOR.

That magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the tmo: And that_magnitude to which the same has a greater ratio than it has unto another magnitude, is the lesser of the two.

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 than it has to A; B is less than A: For (7. Def. 5.) there is some multiple F of C, and some equimultiples E and D of B and A such, that Fis 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. 5.) less than A. That magnitude, therefore, &c. Q. E. D.

PROB. XI. THEOR.

Ratios that are the same to the same ratio, are the same to one another.

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

G be greater than L, H is greater than M, and K greater than Ñ; 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 toge ther; 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, 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: (5. Def. 5.) Let such be taken, and of C, E let G, H

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 equimrltiples; 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 great er 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 filth 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.

fore 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.)

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. But, as A is to B so is C to D; there- Therefore, if the first, &. Q. E. D.

ABCD ABCD ABCD

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.

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 num

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. There fore magnitudes, &c. Q. E. D

PROP. 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 of 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 as C is to D, so (11. 5.) is E to F: Again, because G, H are equimultiples of C, D, as C is to D, so is G to H; (15. 5.) but as C is to D, so is E to F. Wherefore, as E is to F, so is G to H. (11. 5.) But, when four magnitudes are proportionals, if the first be greater than the 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 AB; and G, H any whatever of C, D. Therefore A is to C, as B to D. (Def. 5.) If then four magnitudes, &c. Q. É. D.

PROP. XVII. THEOR.

If magnitudes, taken jointly, be proportionals, they shall also be propor. tionals 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 mul

tiple of AE, that HK is of EB, wherefore 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 mul

PROP. XVIII. THEOR.

If magnitudes taken separately be proportionals, they shall also be portionals when taken jointly, that is, if the first be to the second, as he third to the fourth, the first and second together shall be to the second, as the third and fourth together to the fourth.