Book V.

See N.

M

AGNITUDES which have the fame ratio to the fame magnitude are equal to one another; and thofe to which the fame magnitude has the fame ratio are equal to one another.

Let A, B have each of them the fame 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 propofition, there are fome equimultiples of A and B, and fome multiple of C fuch, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let fuch multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C, fo that D may be greater than F, and E not greater than F: But, 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 Dis greater than F; E fhall alfo be greata 5. def, 5, er than F; but E is not greater than F, which is impoffible; A therefore and B are not unequal; that is, they are equal. Next, Let C have the fame 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 B
greater; therefore, as was fhown in Prop.
8th, there is fome multiple F of C, and
fome equimultiples E and D of B and A

c

D

F

E

fuch, that F is greater than E, and not greater than D; but becaufe C is to B, as C is to A, and that F, the multiple of the firft, is greater than E, the multiple of the fecond; F the multiple of the third, is greater than D, the multiple of the fourth * ; But F is not greater than D, which is impoffible. Therefore A is equal to B. Wherefore magnitudes which, &c. Q. E. D,

PROP,

Book V.

THA

PROP. X. THEOR.

HAT magnitude which has a greater ratio than an- see N. other has unto the fame magnitude is the greater of the two: And that magnitude to which the fame has a greater ratio than it has unto another magnitude is the leffer 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 fome equimultiples of A and B, and 27. Def. 5. fome multiple of C fuch, 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 equi multiples of A, B, and F a multiple of C fuch, 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 equi multiples of A and B, and D is greater than E; therefore A is greater than B.

Next, Let C have a greater ratio to B than it has to A; B is lefs than A: For there is fome multiple F of C, and fome equimultiples E and D of B and A fuch, that F is greater than E, but is not greater than D: E therefore is lefs than D; and becaufe E and D are equimultiples of B and A, therefore B is lefs than A. magnitude, therefore, &c. Q. E. D.

That

B

D

F

b 4. Ax. 5.

E

Ꭱ

ATIOS that are the fame to the fame ratio, are the
fame to one another.

Let A be to B, as C is to D; and as C to D, fo 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, fince A is to B, as C to D, and G, H are taken equimultiples of

1

Book V,

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 lefs, lefs. Again, be a 5. def. 5. caufe 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 lefs, lefs: But, if G

be greater than L, it has been shown that H is greater than M; and if equal, equal; and if lefs, lefs; therefore, if G be greater than L, K is greater than N; and if equal, equal; and if lefs, lefs And G, K, are any equimultiples whatever of A, E; and L, N any whatever of B, F: Therefore, as A is to B, fo is E to F. Wherefore ratios that, &c. Q. E. D.

F any number of magnitudes be proportionals, as one of the antecedents is to its confequent, fo fhall all the antecedents taken together be to all the confequents.

Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, as A is to B, fo C to D, and E to F: As A is to B, fo fhall 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, becaufe A is to B, as C is to D, and as E to F; and that G, H,

K

K are equimultiples of A, C, E, and L, M, N equimultiples of Book V. 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 lefs, lefs. Where- a 5. def. S. fore, 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 fame multiple is the whole of the whole: For the fame reafon L, and L, M, N are any b 1. 5. equimultiples of B, and B, D, F: As therefore A is to B, fo are A, C, E together to B, D, F together. Wherefore, if any number, &c. Q. E. D.

It

F the firft has to the fecond the fame ratio which the see N third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; the firft fhall allo have to the fecond a greater ratio than the fifth has to the fixth.

Let A the first, have the fame ratio to B the fecond, 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 fixth: Alfo the firft A fhall have to the second B a greater ratio than the fifth E to the fixth F.

Because C has a greater ratio to D, than E to F, there are fome equimultiples of C and E, and fome of D and F fuch, 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: Let a 7. def. 5. fuch be taken, and of C, E let G, H be equimultiples, and K, L equimultiples of D, F, fo that G be greater than K, but H not greater than L; and whatever multiple G is of C, take M the fame multiple of A; and what multiple K is of D, take N the fame multiple of B: then, because A is to B, as C to D, and

of

Book V.

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 5. def. 5. than K; and if equal, equal; and if lefs, lefs b; 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 a 7. def. 5. to B, than E has to F. Wherefore, if the first, &c. Q. E. D.

See N.

a 8. 5.

COR. And if the first has a greater ratio to the fecond, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the fixth; it may be demonstrated in like manner, that the first has a greater ratio to the second, than the fifth has to the fixth.

IF

the first has to the fecond, the fame ratio which the third has to the fourth; then, if the first be greater than the third, the fecond fhall be greater than the fourth; and if equal, equal; and if lefs, lefs.

Let the firft A, have to the fecond B, the fame 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: But, as A is to B, fo

b 13. 5.

C IO. 5.

d 9. 5.

A B CD

AB CD

A B C D is C to D; therefore alfo C has to D a greater ratio than C has to Bb: But of two magnitudes, that to which the fame has the greater ratio is the leffer: Wherefore D is lefs 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 Dd.

Thirdly, If A be lefs than C, B fhall be less than D: For C is greater than A, and because C is to D, as A is to B, Dis greater than B, by the first cafe; wherefore B is lefs than D. Therefore, if the firft, &c. Q. E. D.

PROP.