which have the same ratio, taken two and two, but in a cross order, viz. as A is to B, so is E to F, and as B is to C, so is D to E. If A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio (8. 5.) than C has to B: but as E to F, so is A to B; therefore (13. 5.) E has to F a greater ratio than C to B: and because B is to C, as D to E, by inversion, C is to B, as E to D: and E was shown to have to F a greater ratio than C to B; therefore E has to F a greater ratio than E to D (Cor. 13. 5.); but the magnitude to which the same has a greater ratio than it has to another, is the lesser of the two (10. 5.); F therefore is less than D, that is, D'is greater than F.

A

D

B

E F

Secondly, Let A be equal to C, D shall be equal to F. Because A and C are equal, A is (7. 5.) to B, as C is to B: but A is to B, as E to F; and C is to B,

as E to D; wherefore E is to F, as E to D (11. 5.) and therefore D is equal to F (9. 5.).

Next, Let A be less than C; D shall be less than F; for C is greater than A, and, as was shown, C D is to B, as E to D; and in like manner B is to A, as F to E; therefore F is greater than D, by case first; and therefore D is less than F. Therefore, if there be

three, &c. Q. E. D

E

PROP. XXII. THEOR.

с

A

F

DEF

If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words "ex æquali,"

or "ex æquo.

First, Let there be three magnitudes A, B, C, and as many

* See Note.

1

others D, E, F, which, taken two and two, have the same ratio, that is, such that A is to B, as D to E; and as B is to C, so is E to F; A shall be to C, as D to F.

B

K

D E F

M H L N

Take of A and D any equimultiples whatever G and H; and of B and E any equimultiples whatever K and L; and of C and Fany whatever M and N: then, because A is to B, as D to E, and that G, H are equimultiples of A, D, and K, L equimultiples of B, E; as G is to K, so is (4. 5.) H G to L. For the same reason, K is to M, as L to N and because there are three magnitudes G, K, M, and other three H, L, N, which, two and two, have the same ratio: if G be greater than M, H is greater than N; and if equal, equal; and if less, less (20. 5.); and G, H are any equimultiples whatever of A, D, and M,

N are any equimultiples whatever of C, F. 5.), as A is to C, so is D to F.

Therefore, (5. def.

Next, Let there be four magnitudes, A, B, C, D, and other four, E, F, G, H, which, two and two, have the

same ratio, viz. as A is to B, so is E to F, and A. B. C. D. as B to C, so F to G; and as C to D, so G to E. F. G. H. H: A shall be to D, as E to H.

Because A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio; by the foregoing case, A is to C, as E to G. But C is to D, as G is to H; wherefore again, by the first case, A is to D, as E to H: and so on, whatever be the number of magnitudes. Therefore, if there be any number, &c. Q. E. D.

PROP. XXIII. THEOR.

Ir there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words "ex æquali in proportione perturbata;" or, "ex æquo pertur bato."*

First, Let there be three magnitudes A, B, C, and other three D, E, F, which, taken two and two in a cross order, have the same ratio, that is, such that A is to B, as E to F; and as B is to C, so is D to E: A is to C, as D to F.

A

G

B

D

F

H

L

K

MIN

Take of A, B, D any equimultiples whatever G, H, K; and of C, E, F any equimultiples whatever L, M, N; and because G, H are equimultiples of A, B, and that magnitudes have the same ratio which their equimultiples have (15. 5.); as A is to B, so is G to H. And for the same reason, as E is to F, so is M to N; but as A is to B, so is E to F; as therefore G is to H, so is M to N (11. 5.). And because as B is to C, so is D to E, and that H, K are equimultiples of B, D, and L, M, of C, E; as H is to L, so is (4. 5.) K to M: and it has been shown, that G is to H, as M to N; then, because there are three magnitudes G, H, L, and other three K, M, N, which have the same ratio taken two and two in a cross order; if G be greater than L,

K is greater than N; and if equal, equal; and if less, less (21. 5.); and G, K are any equimultiples whatever of A, D; and L, N'any whatever of C, F; as, therefore, A is to C, so is D to F.

See Note.

A. B. C. D.

Next, Let there be four magnitudes, A, B, C, D, and other four E, F, G, H, which, taken two and two in a cross order, have the same ratio, viz. A to B, as G to H; B to C, as F to G; and C to D, as E to F: A is to D, as E to H.

1

E. F. G. H.

Because A, B, C are three magnitudes, and F, G, H other three, which, taken two and two in a cross order, have the same ratio; by the first case, A is to C, as F to H: but C is to D, as E is to F; wherefore again, by the first case, A is to D, as E to H: and so on, whatever be the number of magnitudes. Therefore, if there be any number, &c. Q. E. D.

PROP. XXIV. THEOR.

Ir the first has to the second the same ratio which the third has to the fourth; and the fifth to the second, the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and sixth together have to the fourth.*

Let AB the first, have to C the second, the same ratio which DE the third, has to F the fourth; and let BG the fifth, have to C the second, the same ratio which EH the sixth, has to F the fourth: AG, the first and fifth together, shall have to C the second, the same ratio which DH, the third and sixth together, has to F the fourth.

Because BG is to C, as EH to F; by inversion, C is to BG, as F to EH: and because as AB is to C, so is DE to F; and as C to BG, so F to EH ; ex æquali (22. 5.), AB is to BG, as DE to EH: and because these magnitudes are proportionals, they shall likewise be proportionals when taken

jointly (18. 5.) as, therefore, AG is to GB, so is DH to HE; but as GB to C, so is HE to F. Therefore, ex æquali (22. 5.), as AG is to C, so is DH to F. Wherefore, if the first, &c. Q. E. D.

* See Note.

COR. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second, as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition.

COR. 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest.

PROP. XXV. THEOR.

IF four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together.

1

Let the four magnitudes AB, CD, E, F, be proportionals; viz. AB to CD, as E to F; and let AB be the greatest of them, and consequently F the least (A. & 14. 15.). AB, together with F, are greater than CD, together with E.

B

G

Take AG equal to E, and CH equal to F: then, because as AB is to CD, so is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG to CH. And because AB the whole is to the whole CD, as AG is to CH, likewise the remainder GB shall be to the remainder HD, as the whole AB is to the whole (19. 5.) CD: but AB is greater than CD, therefore (A. 5.) GB is greater than HD and because AG is equal to E, and CH to F; AG and F together are equal to CH and E together. If, therefore, to the unequal

D

H

magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz. to GB the two AG and F, and CH and E to HD; AB and F together are greater than CD and E. Therefore, if four magnitudes, &c. Q. E. D.