B. For the same construction being made, D may in like manner be shown to be equal to E: therefore, if F be greater than D, it is likewise greater than E; if equal, equal; and if less, less. But F is any multiple whatever of C, and D, E are any like multiples whatever of A, B: therefore (V. def. 5.) C: A :: C: B. Therefore equal magnitudes, &c.

PROP. VIII. THEOR.

OF 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 to the greater.

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 Fig. 1. has a greater ratio to BC than to

AB.

:

E

Fig. 2.

E

E

F

L K

D

Fig. 3.

B

If the magnitude which is not the greater of the two AC, CB, be F 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 Figs. 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 each of the cases, take H the double of 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.

:

:

Then, because Lis 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 (V. 1.) of CB, that EG is of AB; wherefore EG and FG are like multiples 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 like multiples of AB, BC, and L is a multiple of D; therefore (V. def. 7.) AB has to D a greater ratio than BC has to D.

Also D has to BC a greater ratio than it has to AB. For, the same construction being made, it may be shown, in like manner, that L is greater than FG, but that it is not greater than EG : and L is a multiple of D; and FG, EG are like multiples of CB, AB; therefore (V. def. 7.) D has a greater ratio to CB than to AB: wherefore, of unequal magnitudes, &c.

PROP. IX. THEOR.

MAGNITUDES which have the same ratio to the same magnitude are equal: and those to which the same magnitude has the same ratio are equal.

D

E

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 proposition, there are some like multiples of A and B, and some multiple of C, such that the multiple of A is greater than the mul- a tiple of C, but the multiple of B is not greater than B that of C. Let such multiples be taken, and let D, E be the multiples 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 like multiples D, E are taken, and of C a multiple F is taken; and that D is greater than F; E shall also (V. def. 5.) be greater than F; but E is not greater than F, which is impossible; A, therefore, and B are not unequal, that is, they are equal.

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 proposition 8th, there is some multiple F of C, and some like multiples E and D, of B and A, such that F is greater than E, and not greater than D; but because C: B:: C: 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 (V. def. 5.) than D, the multiple of the fourth but F is not greater than D, which is impossible. Therefore A is equal to B: wherefore magnitudes which, &c.

:

PROP. X. THEOR.

THAT magnitude which has a greater ratio than another has to the same magnitude, is the greater of the two and that magnitude, to which the same has a greater ratio than it has to another magnitude, is the less of the two.

Let A have to C a greater ratio than B has to C; A is greater

D

than B. For, since A has a greater ratio to C, than B has to C, there are (V. def. 7.) some like multiples 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 be taken, and let D, E be like multiples of A, B, and F a multiple of C, such, that D is greater than F, but E is not greater A than F. Therefore D is greater than E: and because D and E are like multiples of A and B, and D is greater than E; therefore (V. ax. 4.) A is greater than B.

Next, let C have a greater ratio to B than it has to A; B is less than A. For (V. def. 7.) there is some multiple F of C, and some like multiples

B

F

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 like multiples of B and A, therefore (V. ax. 4.) B is less than A. That magnitude, therefore, &c.

PROP. XI. THEOR.

RATIOS that are equal to the same ratio, are equal to one another.

G

A

B

K

E

F

Let A: B::C: D, and C: D:: E:F; then A: B:: E: F. Take of A, C, E, any like multiples whatever, G, H, K; and of B, D, F, any like multiples whatever L, M, N. Therefore, (V. def. 5.) since A: B:: C: D, and G, H are taken like multiples of A, C, and L, M of B, D; if G be greater than L, H is greater than M; if equal, equal; and if less, less. Again, because C: D:: E: F, and H, K are taken like multiples of C, E; and M, N, of D, F; if H be greater than M, K is greater than N; if equal, equal; and if less, less. But, if G be greater than L, it has been shown that H is greater than M; if equal, equal; and if less, less therefore if G be greater than L, K is greater than N; if equal, equal; and if less, less and G, K are any like multiples whatever of A, E; and L, N any whatever of B, F. Therefore (V. def. 5.) A:B::E: F: wherefore, ratios, &c.

L

M

PROP. XII. THEOR.

N

IF If any number of magnitudes be proportionals, as any one of the antecedents is to its consequent, so are all the antecedents taken together to all the consequents.

Let any number of magnitudes A, B, C, D, E, F be propor

tionals; that is, as A: B:: C: D::F: F: then as A is to B, so are A, C, E together to B, D, F together.

G

H

K

A

C

E

B

D

F

M

N

L

Take of A, C, E any like multiples whatever G, H, K; and of B, D, F any like multiples whatever L, M, N: then (V. def. 5.) because A is to B, as C is to D, and as E to F, and that G, H, K are like multiples of A, C, E; and L, M, N like multiples of B, D, F; if G be greater than L, H is greater than M, and K greater than N; if equal, equal; and if less, less. Therefore, if G be greater than L, then G, H, K together are greater than L, M, N together; if equal, equal; and if less, less. And G, and G, H, K together are any like multiples of A, and A, C, E together; because if there be any number of magnitudes like multiples of as many, each of each, whatever multiple one of them is of its part, the same multiple (V. 1.) is the whole of the whole. For the same reason L, and L, M, N are any like multiples of B, and B, D, F. As therefore (V. def. 5.) A is to B, so are A, C, E together, to B, D, F together: wherefore, if any number, &c.

PROP. XIII. THEOR.

IF the first have 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 will 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: A has also to B a greater ratio than E to F.

M

G

H

A

C

E

B

Because C has a greater ratio to D, than E to F, there are (V. def. 7.) some like multiples of C and E, and some of D and F such, that the multiple of C is greater than that of D, but the multiple of E is not greater than that of F. Let such be taken, and of C, E let G, H be like multiples, and K, L like multiples 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 same multiple of A; and whatever multiple K is of D, take N the same multiple of B. Then, (V. def. 5.) because A: B:: C: D, and of A and C, M and G are like multiples: and of B and D,

N and K are like multiples; if M be greater than N, G is greater than K; if equal, equal; and if less, less; but G is greater than K, therefore M is greater than N. But H is not greater than L; and M, H are like multiples of A, E; and N, L like multiples of B, F: therefore (V. def. 7.) A has a greater ratio to B, than E has to F. Wherefore, if the first, &c.

Cor. If the first have 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.

IF the first have to the second the same ratio which the third has to the fourth; then if the first be greater than the third, the second is greater than the fourth; if equal, equal; and if less, less.

:

Let A B C 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

2

3

A B C D

A B C D A B C D

has (V. 8.) to B a greater ratio than C to B: but A: B:: C: D; therefore also (V. 13.) C has to D a greater ratio than C has to B. But (V. 10.) of two magnitudes, that to which the same has the greater ratio is the less: wherefore D is less than B.

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 (V. 9.) is equal to D.

Thirdly, if A be less than C, B shall be less than D. For C is greater than A; and because C: D::A: B, D is greater than B, by the first case. Therefore, if the first, &c.

PROP. XV. THEOR.

MAGNITUDES have the same ratio to one another that their like multiples have.

Let AB be the same multiple of C, that DE is of F: Cis to F, as AB to DE.