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Book V.

PROP. IX.

THEOR.

See N.

MAGNITUDES which have the same ratio to the same

magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another.

Let A, B have each of them the same 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 sewn in the preceeding Proposition, there are some equimultiples of A and B, and some multiple of C such, 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 such multiples be taken, and let D, E, be the equimultiples of A, B, and F the multíple of C so 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; E shall also be

D 1. 5. Def. 5. greater than Fa; but ! is not greater than F, A

which is impossible. A therefore and B are
not unequal; that is, they are equal.

F
Next, Let C have the same ratio to each c
of the magnitudes A and B; A is equal to B.
for if they are not, one of them is greater

B than the other; let A be the greater, there

E fore, as was shewn in Prop. 8th, there is fome multiple F of C, and some equimultiples E and D of B and A such, 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 mulciple of the first is greater than E the multiple of the second; F the multiple of the third is greater than D the multiple of the fourtha. but F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore magnitudes which, &c. Q. E. D.

PROP.

Book V.

PROP. X. THEOR. THAT magnitude which has a greater ratio than ano- See N.

ther has unto the same magnitude is the greater of the two. and that magnitude to which the same 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 some equimultiples of A and B, and some multiple of Ca. 7. Def. s. such, 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 Fa multiple of C such, that D

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 b

F greater

b. 4. As. s than B. Next, Let C have a greater ratio to B than

B it has to A; B is less than A. for a 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 is not greater than D. E therefore is less than D; and because E and D are equimultiples of Band A, therefore B is bless than A. That magnitude therefore, &c. l. E. D.

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RATIOS that are the fame to the same ratio, are the

fame 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 fince A is to B, as C to D, and of A, C are taken equimultiples G,

H; and

Book V. H; and L, M of B, D; if G be greater than L, H is greater than MM; and if equal, equal; and if lefs, lessa. Again, because Cis to a. s. Def. s. 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

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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. Wherefore ratios that, &c. Q. E. D.

PRO P. XII. THE O R.

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.

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, 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

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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

of A, C, E, and L, M, N equimultiples of B, D, F; if Ġ be greater Book V. than L, H is greater than M, and K greater than N; and if equal, m equal; and if less less. Wherefore if G be greater than L, then G, a. s. Def.sk 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

wholeb. for the same reason L, and L, M, N are any equimultiples b. 1. so I 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.

See

PROP: XIII. THEOR. F ihe first has to the second the same ratio which the Seč N.

third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first thall also have to the second a gteater ratio than the fifth has to the sixth.

Let A the first have the same ratio to B the second which the third has to D the fourth, but the third to the fourth a greater ratio than E the fifth to F the sixth. also the first A Mall 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 M G.

H A

C

E B

D

K

Ń

L not greater than the multiple of Fa. let such be taken, and of C, E a. 7. Def. s. 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 miltiple 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

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 less, less b; but G is b. s. Def.s.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 equimul

tiples of B, F. Therefore A has a greater ratio to B, than E has to #. 7. Def. 5. Fa. Wherefore if the first, &c. Q. E. D.

Cor. And if the first has 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.

Scc N.

IF

F the first has to the second the fame 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 firsi 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*. but as A is to B, so is C to D;

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A B C D A B C D AB CD 6.13.5. therefore also C has to D a greater ratio than C has to Bb. but of

two magnitudes, that to which the same has the greater ratio is the 6. 10. s. lesser. wherefore D is less than B; that is, B is greater than D.

Secondly, If A be equal to C, B is equal D. for A is to B, as & 9.5. C, that is A, to D; B therefore is equal to Dd.

Thirdly, If A be less than C, B shall be less than D. for Cis 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 forA, &c. Q.E. D.

PROP.

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