ple of A, that E is of B, and that A is equal to B; D is (1. Ax. 5.) equal to E: therefore, if D be greater than F, E is greater than F: and if equal, equal; if less, less: and D, E are any equimultiples of A, B, and F is any multiple of C. Therefore (5. def. 5.) as A is to C, so is B to C.

Likewise C has the same ratio to A, that it has to B: for having made the same construction, D may in like manner be shown equal to E: therefore, if F be greater than D, it is likewise greater than E; and if equal, equal; if less, less: and F is any multiple whatever of C, and D, E are any equimultiples whatever of A, B. Therefore, C is to A, as C is to B (5. def. 5.). Therefore equal magnitudes, &c. Q. E. D.

PROP. VIII. THEOR.

Or unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less, than it has to the greater.*

Fig. 1.

E

Let AB, BC be unequal magnitudes, of which AB is the greater, and let D be any magnitude whatever: AB has a greater ratio to D than BC to D; and D has a greater ratio to BC than to AB.

A

F

B

L

K

H D

If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG, the doubles of AC, CB, as in Fig. 1. But, if that which is not the greater of the two AC, CB be less than D (as in Fig. 2. and 3.) this magnitude can be multiplied, so as to become greater than D, whether it be AC, or CB. Let it be multiplied, until it become greater than D; and let the other be multiplied as often; and let EF be the multiple thus taken of AC, and FG the same multiple of CB; therefore EF and FG are each of them greater than D: and in every one of the cases, take H the double D, K, its triple, and so on, till the multiple of D be that which first becomes greater than FG: let L be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L.

* See Note.

Then, because L is the multiple of D, which is the first that becomes greater than FG; the next preceding multiple K is not greater than FG; that is, FG is not less than K: and since EF is the same multiple of AC, that FG is of CB; FG is the same multiple of CB, that EG is of AB (1. 5.): wherefore EG and FG are equimultiples of AB

and CB: and it was shown, that FG was not less than K, and, by the construction, EF is greater than D; therefore the whole EG is greater than K and D together; but, K together with D, is equal to L; therefore EG is greater than L; but FG is not greater than L; and EG, FG are equimultiples of AB, BC, and L is a multilple of D; there

fore (7. def. 5.) AB has to D a greater ra tio than BC has to D. Also D has to BC a greater ratio than it has to AB, for, having made the same construction, it may be shown, in like manner, that L is greater than FG, but that it is not

Fig. 2.

E

F

A

Fig. 3.

E

A

greater than EG: and L is a multiple of D; and FG, EG are equimultiples of CB, AB; therefore D has to CB a greater ratio (7. def. 5.) than it has to AB. Wherefore, of unequal magnitudes, &c. Q. E. D.

PROP. IX. THEOR.

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 be not equal, one of them is greater than the other; let A be the greater; then, by what was shown in the preceding

* See Note.

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 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; E shall also be greater than F (5. def. 5.): but E is not greater than F, which is impossible; A therefore and B are not unequal; that is, they are equal.

A

D

이

E

B

Next, let C have the same ratio to each of the magnitudes A and B; A is equal to B: for if they be not, one of them is greater than the other; let A be the greater; therefore, as was shown in Prop. 8th, there is some 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 multiple 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 fourth (5. def. 5.) : but F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c. Q. E. D.

PROP. X. THEOR.

THAT magnitude which has a greater ratio than another 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 lesser 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 (7. def. 5.) 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 it let them

* See Note.

be taken, and let D, E be equimultiples of A, B, and F a multiple of C such, 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 equimultiples of A and B, and D is greater than E; therefore A is (4. Ax. 5.) greater than B.

Next, let C have a greater ratio to B that it has to A; B is less than A: for 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 it is not greater than D: E therefore B is less than D; and because E and D are equimultiples of B and A, therefore B is (4. Ax. 5.) less than A. That magnitude therefore, &c. Q. E. D.

PROP. XI. THEOR.

RATIOS that are the same to the same ratio, are the same 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 Bas 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, since A is to B, as C to D, and G, H are taken equimultiples of 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 less, less (5. def. 5.). Again, because C is to D, as E 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 been shown 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 (5. def. 5.). Wherefore ratios that, &c. Q. E. D.

PROP. XII. THEOR.

Ir 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, so is C to D, and E to F: as A is to B, so shall A, C, E together be to B, D, F together.

G

Take of A, C, E any equimultiples whatever G, H, K; and of B,

A

B

L

D, F any equimultiples whatever, L, M, N: then because A is to B, as C is to D, and 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 than L, H is greater than M, and K greater than N; and if equal, equal; and if less, less (5. def. 5.). Wherefore, 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 same multiple is the whole of the whole (1. 5.): for the same reason L, and L, M, N are any equimultiples 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.

PROP. XIII. THEOR.

Ir the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; 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 sixth: 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 equimultiples of C and F, and some of D and F, such, than the multiple of C is greater than the multiple of D, but the multiple

* See Note.