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Book V.

See N.

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AGNITUDES which have the fame ratio to the fame magnitude are equal to one another; and those 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 others let A be the greater; then, by what was fhewn 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 mul tiple 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 D is greater than F;

2.5. Def. 5. E fhall also be greater than F*; but E is A
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 B
equal to B. for if they are not, one of
them is greater than the other; let A be

the greater, therefore, as was fhewn in

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Prop. 8th, there is fome multiple F of C, and fome equimultiples E and D of B and A fuch, that F is greater than E, and not greater than D. but 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 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.

Book V.

PROP. X. THEOR.

THAT magnitude which has a greater ratio than see N. another 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 fome multiple of a.7.Def.5. 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 equimultiples 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 equimultipies of A and B, and D is greater

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A

than it has to A; B is lefs than A. for B 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

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b. 4. Az. J.

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than D. E therefore is lefs than D; and because E and D are equimultiples of B and A, therefore B is less than A. That magnitude therefore, &c. Q. E. D.

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PROP. XI. THEOR.

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

Let A be to B, as Cis 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 of A, C are taken equimultiples G,

Book V. H; 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, because C is to a. 5. Def. 5. 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

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been fhewn that H is greater than M; and if equal, equal; and if less, 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.

PROP. XII. THEOR.

IF 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 propor tionals; 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

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B, D, F any equimultiples whatever L, M, N. then because A is to
B, as C is to D, and as 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 Book V. than L, H is greater than M, and K greater than N; and if equal, n equal; and if less, lefs. Wherefore if G be greater than L, then a. 5. Def. 5. G, H, K together are greater than L, M, N together; and if equal, equal; and if lefs, lefs. 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 b. for the fame reafon L, and L, M, N b. 1. s are any 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.

PRO P. XIII. THEOR.

IF the firft has to the fecond the fame ratio which See N.

the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; the first shall alfo have to the fecond a greater ratio than the fifth has to the fixth.

Let A the firft 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 fhalf 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 mul tiple of C is greater than the multiple of D, but the multiple of E is

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not greater than the multiple of F. let fuch be taken, and of C, E a.7 Def.?%

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 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 A and C, M and G are equimultiples, and of

Book V. 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 lefs, lefs ; but h. 5. Def. 5. 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 a. 7-Def. 5. E has to F. Wherefore if the firft, &c. Q. E. D.

Sce N.

a. 8. 5.

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

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F the firft has to the fecond the fame ratio, which the third has to the fourth; then, if the firft be greater than the third, the fecond shall be greater than the fourth; and if equal, equal; and if less, less.

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 is C to

2

3

A B C D

A B C D

A B C D

b. 13. 3. 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.

e. 10. 5.

d. 9. 5.

Secondly, If A be equal to C, B is equal D. for A is to B, as C, that is A, to D; B therefore is equal to D.

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

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