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AGNITUDES which have the same ratio to
the same magnitude are equal to one another; and those to which the fame 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 shewn in the preceding Propofition, there are some equimultiples of A and B, and some multiple of C such, 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 such multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple 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;
D 2. s. Def. s. E shall also be greater than F*; but E is A
not greater than F, which is impossible.
E them is greater than the other ; let A be the greater, therefore, as was shewn in Prop. 8th, there is some multiple F of C, and some equimultiples E and D of B and A fach, 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 zhe multiple of the second; F the multiple of the third is greater than D the multiple of the fourth a. but F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore magnitudes which, &c. Q.E.D.
THAT magnitude which has
PROP. X. THEOR.
another has unto the fame magnitude is the
Let A have to C a greater ratio than B has to C; A is greater
PRO P. XI. THEOR.
ATIOS that are the same to the same ratio, are
the same to one another.
Let A be to B, as Cis to D; and as C to D, fo let E be to F;
Take of A, C, E any equimultiples whatever G, H, K, and of
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 less, lessé. Again, because C is to 2. 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, 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 a. Wherefore ratios that, &c.
Q. E. D.
PRO P. XII. THEOR.
of the antecedents is to its confequent, fo 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, 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 because A is to
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, equal; and if lefs, less. Wherefore if G be greater than L, then a.s. Def. s. 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 b. for the same reason L, and L, M, N b. 1. are any equimultiples of B, and B, D, F. as thereföre 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. F the first has to the second the same 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 sixth i 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 fixth. 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
L not greater than the multiple of F. let such be taken, and of C, E a.7 Defa 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 mul. tiple K is of D, take N the fame multiple of B. then because A is to B, as C to D, and of A andC, M and G are equimultiples, and of
Book v. B and D, N and K are equimultiples ; if M be greater than N,
WG is greater than K; and if equal, equal; and if lefs, less b; but 1. 5. Def. s. G is greater than K, therefore M is greater than N. bat 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 2.7-Def. 5. E has to F:. 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.
PRO P. XIV. THEOR.
the third has to the fourth ; then, if the first be greater than the third, the second fhall be greater than the fourth; and if equal, equal ; and if less, less.
Let the first 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, fo is C to
2. 8. si
e. 10. 5.
A B C D A B C D A B C D b. 13. 5. D; therefore also C has to Da greater ratio than C has to Bb. but of
two magnitudes, that to which the same 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 D. for A is to B, as C, that is A, to D; B therefore is equal to D d.
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. Therefore if the first, &c. Q. E. D.
d. 9. 5.