First, let E be greater than G, then G is less than E. Because A is to B, as C is to D (Hyp.), and of A and C, the first and third, G and H are equimultiples; and of B and D, the second and fourth, E and F are equimultiples. But G is less than E. Therefore H is less than F (V. Def. 5), that is F is greater than H. Wherefore, if E be greater than G, F is greater than H. In like manner, if E be equal to G, F may be shown to be equal to H; and if less, less. But E and F, are any equimultiples whatever of B and D (Const.); and G and H any whatever of A and C. Therefore, as B is to A, so is D to C (V. Def. 5). Therefore, if four magnitudes, &c. Q. E. D.

If the first be the same multiple of the second, or the same part of it, that the third is to the fourth; the first is to the second, as the third is to the fourth.

Let the first A, be the same multiple of the second B, that the third C, is of the fourth D. Then A is to B as C is to D.

A
E

B
G-

F

D

H

Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H.

Because A is the same multiple of B that C is of D (Hyp.), and E the same multiple of A, that F is of C (Const.). Therefore E is the same multiple of B, that F is of D (V. 3), that is, E and F are equimultiples of B and D. But G and H are equimultiples of B and D (Const.). Therefore if E be a greater multiple of B than G is of B, F is a greater multiple of D than H is of D; that is, if E be greater than G, F is greater than H. In like manner, if E be equal to G, or less than G, it may be shown that F is equal to H, or less than H. But E and F are any equimultiples whatever of A and C (Const.); and G and H any equimultiples whatever of B and D. Therefore A is to B, as C is to D (V. Def. 5).

Next, let the first A be the same part of the second B, that the third C is of the fourth D. A is to B, as C is to D.

Because A is the same part of B

that C is of D. Therefore B is the A- B same multiple of A that D is of C.

D

Wherefore, by the preceding case, B is to A, as D is to C. Therefore inversely, A is to B, as C is to D (V. B.). Wherefore, if the first be the same multiple, &c. Q. E. D.

PROP. D. THEOREM.

If the first be to the second as the third is to the fourth, and if the first be a multiple, or a part of the second; the third is the same multiple, or the same part of the fourth.

Let A be to B as C is to D.

In the first place, let A the first be a multiple of B the second. The third C is the same multiple of the fourth D.

A

B--
E-

D-

F

Take E equal to A, and whatever multiple A or E is of B, make F the same multiple of D.

Because A is to B, as C is to D (Hyp.); and of B the second, and D the fourth, equimultiples have been taken, E and F. Therefore A is to E, as C is to F (V. 4, Cor.). But A is equal to E (Const.). Therefore C is equal to F (V. A). But F is the same multiple of D that A is of B (Const.). Therefore C is the same multiple of Ď that A is of B.

Next, let A the first, be a part of B the second. The third C is the same part of the fourth D.

Because A is to B, as C is to D (Hyp.). Inversely, B is to A, as D to C. (V. B). But A is a part of B (Hyp.). Therefore B is a multiple

B

C-- D

of A. Wherefore, by the preceding case, D is the same multiple of C; that is, C is the same part of D that A is of B. Therefore, if the first, &c. Q. E. D.

This proposition is the converse of the preceding one.

Equal magnitudes have the same ratio to a magnitude of the same kind: and conversely, a magnitude has the same ratio to equal magnitudes of the same kind. Let A and B be equal magnitudes, and C any other of the same kind. First, the magnitudes A and B have each the same ratio to C.

Take of A and B any equimultiples whatever D and E, and of C any multiple whatever F.

Because D is the same multiple

of A, that E is of B (Const.), and A is equal to B (Hyp.). Therefore D is equal to E (V. Ax. 1). Wherefore, if D be greater than F, E is greater than F; if equal, equal; and if less, less. But D and E are any equimultiples of A and B (Const.), and F is any multiple of C. Therefore, as A is to C, so is B to C (V. Def. 5).

Secondly, the magnitude C has the same ratio to A that it has to B. For the same construction being made, it may be shown as above, that D is equal to E. Therefore if F be greater than D, F is likewise greater than E; if equal, equal; and if less, less. But F is any multiple whatever of C, and D and E are any equimultiples whatever of A and B. Therefore C is to A as C is to B (V. Def. 5). Therefore, equal magnitudes, &c. Q. E. D.

Of unequal magnitudes, the greater has a greater ratio to a magnitude of the same kind, than the less has: and conversely, a magnitude has a greater ratio to the less of unequal magnitudes of the same kind, than it has to the greater. Let AB, BC be two unequal magnitudes, of which AB is the greater, and let D be any other magnitude of the same kind.

First, the greater A B has a greater ratio to D, than B C has to D.

If the magnitude which is not the greater of the two A C and CB, be not less than D (as in fig. 1), take EF and F G, the doubles of A C and CB. But, if that which is not the greater of the two A C and CB, be less than D (as in figs. 2 and 3), take equimultiples of A C and C B,—viz., EF and F G, each greater than D. In all the cases, take H the double of D, K its triple, and so on, till L the multiple of D be found which first becomes greater than FG; and K the multiple of D which is next less than L, or the next preceding which is not greater than FG: that is, F G is not less than K.

Because EF is the same multiple of A C, that FG is of CB (Const.). Therefore F G is the same multiple of CB that EGis of AB (V. 1), that is, EG and F G are equimultiples of AB and C B. But FG is not less than K, and (Const.) E F is greater than D. Therefore the whole EG is greater than K and D together. But K together with D is equal to L (Const.). Therefore E G is greater than L. But F G is not greater than L (Const.); and EG and FG were proved to be equimultiples of A B and B C; and L is a multiple of D (Const.). Therefore A B has to D a greater ratio than B C has to D (V. Def. 7).

Secondly, D has to the less BC a greater ratio than it has to AB.

For, the same construction being made, it may be shown, as above, that I is greater than FG, but not greater than EG; and L is a multiple of D (Const.). And FG and EG were proved to be equimultiples of CB and A B. Therefore D has to CB a greater ratio than it has to AB (V. Def. 7). Wherefore, of unequal magnitudes, &c. Q. E. D.

PROP. IX. THEOREM

Magnitudes which have the same ratio to a magnitude of the same kind, are equal to one another: and conversely, magnitudes to which a magnitude of the same kind has the same ratio, are equal to one another.

Let the magnitudes A and B have each the same ratio to a magnitude C of the same kind. The magnitude A is equal to the magnitude B.

A

For if A be not equal to B, one of them must be greater than the other. Let A be the greater. As in the preceding proposition, take D and E, equimultiples of A and B, and Fa multiple of C, such that D

B

D

C

F

E

may be greater than F, but E not greater than F. Because A is to C as B is to C (Hyp.), and of A and B, are taken equimultiples D and E, and of C is taken a multiple F. But D is greater than F (Const.). Therefore E is also greater than F (V. Def. 5). But E is not greater than F (Const.); which is impossible. Therefore A and B are not unequal; that is, they are equal.

Next, let the magnitude C have the same ratio to each of the magnitudes A and B. The magnitude A is equal to the magnitude B.

For, if A be not equal to B, one of them must be greater than the other; let A be the greater. As in the preceding proposition, take E and D equimultiples of B and A, and F a multiple of C, such that F may be greater than E, but not greater than D.

Because C is to B, as C is to A (Hyp.), and F the multiple of the first, is greater than E the multiple of the second. Therefore F, the multiple of the third, is greater than D the multiple of the fourth (V. Def. 5). But F is not greater than D (Hyp.); which is impossible. Therefore A and B are not unequal; that is, they are equal. Wherefore magnitudes which, &c. Q. E. Ú.

Of two magnitudes, that which has the greater ratio to another magnitude of the same kind, is the greater of the two; and that magnitude to which another magnitude of the same kind has the greater ratio, is the less of the two.

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

As, in the preceding proposition, take D and E equimultiples of A and B, and F a multiple of C, such, that D is greater than F, but E not greater than F (V. Def. 7).

Because D and E are equimultiples of A and

B, and D is greater

than E (Const.). There

A.

B

fore A is greater than B (V. Ax. 4).

D

F

E

Next, let C have a greater ratio to B than to A. B is less than A. Take a multiple F of C, and equimultiples of E and D, of B and A, such that F is greater than E, but not greater than D (V. Def. 7).

Because E and D are equimultiples of B and A, and E is less than D (Const.). Therefore B is less than A (V. Ax. 4). Therefore, of two magnitudes, &c. Q. E. D.

PROP. XI, THEOREM.

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

If A is to B as C is to D; and C is to D, as E is to F. A is to B, as E is to F.

Take of A, C, and E, any eqimultiples whatever, G, H, and K: and of B, D, and F any equimultiples whatever, L, M, and N.

B

L

H

K

E

F

M

N

Because A is to B as Cis to D, and G and H are taken equimultiples of A and C; and L and M, of B and D. If G be greater than GL, H is greater than M; if equal, equal; and if less, less (V. Def. 5). Again, because C is to D, as É is to F, and H and K are taken equimultiples of C and E; and M and N, of D and 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. But G and K are any equimultiples whatever of A and E; and L and N any whatever of B and F. Therefore, A is to B as E is to F (V. Def. 5). Wherefore, ratios that, &c. Q. E. D.

PROP. XII. THEOREM.

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

A

B

L

H

K

E

D

F

M

N

Let any number of magnitudes A, B, C, I E, and F, be proportionals; that is, as A is to B, so is C to D; and as C is to D, so is E to F. A is to B, as A, C, and E together, is to B, D, and F together. Take of A, C, and E any Gequimultiples whatever G, H, and K; and of B, D, and F any equimultiples whatever, L, M, and N. Because A is to B, as C is to D; and C is to D as E is to F; and that G, H, and K are equimultiples of A, C, and E; and L, M, and N; equimultiples of B, D, and F. Therefore, if G be greater than L, H is greater than M, and K greater than N; if equal, equal; and if less, less (V. Def. 5). Wherefore if G be greater than L, then G, H, and K together, are greater than L, M, and N together; if equal, equal; and if less, less. But if there be any number of magnitudes equimultiples of as many others, each of each, whatever multiple one of them is of its part, the same multiple is the whole of the whole (V. 1). Therefore G, and G, H, and K together, are any equimultiples of A, and A, C, and E together. For the same reason L, and L, M, and N together, are any equimultiples of B, and B, D, and F together. Therefore A is to B, as A, C, and E together, are to B, D, and F together (V. Def. 5). Wherefore, if any number, &c. Q. E. D.

PROP. XIII. THEOREM.

If 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 has also 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