Again, let the firft B, be the fame part of the second A, as the third D, is of the fourth c; then will в have to A the fame ratio that D has to c.

B

For A is the fame multiple of в that c is of D (by Hyp.); therefore A will have to в the fame ratio that e has to D (V. 8.)

And, fince A is to B as c is to D, therefore, also, inversely, B is to A as D is to c (V. 7.)

Q. E. D.

PROP. IX. THEOREM.

Equal magnitudes have the fame ratio to the fame magnitude, and the fame has the fame ratio to equal magnitudes.

Let A and B be equal magnitudes, and c any other magnitude whatever; then A will have to c the fame ratio that в has to c.

B

For of A and B take any equimultiples whatever D and E; and of c any multiple whatever F:

Then, because D is the fame multiple of A that E is of B, and A is equal to B, D will also be equal to E.

And, fince D and E are equal to each other, if D be greater than F, E will also be greater than F; and if equal, equal; and if lefs, less.

But D and E are any equimultiples whatever of A and B, and F is any multiple whatever of c; therefore A is to c as B is to c (V. Def. 5.)

Again, let A and B be equal magnitudes, and c any, other magnitude whatever; then c has to A the fame ratio that it has to B.

For, having made the fame conftruction as before, D may, in like manner, be fhewn to be equal to E:

And, fince D is equal to E, if F be greater than D, it will also be greater than E; or if equal, equal; or if lefs, lefs.

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.)

PROP. X. THEOREM.

Q. E. D.

Magnitudes which have the fame ratio to the fame magnitude are equal to each other; and those to which the fame magnitude has the fame ratio are equal to each other.

Let A have to c the fame ratio that в has to c; then

will A be equal to B.

For, if they be not equal, one of them must be greater than the other.

Let A be the greater; and of A and B take the equimultiples D and E, and of c the multiple F fuch, that D may be greater than F, and E not greater than F (V. 4.)

Then, fince A is to c as B is to C, and D and E are equimultiples of A and B, and F is a multiple of c, d being greater than F, E will also be greater than F (V. Def. 5.)

But, by conftruction, E is not greater than F; whence it is greater and not greater at the fame time, which is abfurd.

The magnitude A is, therefore, not greater than ; and in the fame manner it may be fhewn that it is not lefs; confequently they are equal to each other.

Again, let c have to A the fame ratio that it has to B ; then will A be equal to B.

For, fince c is to A as C is to B, therefore, also, inverfely, A will be to c as B is to c (V. 7.)

But magnitudes which have the fame ratio to the fame magnitude have been fhewn to be equal to each other; therefore A is equal to E.

Q. E. D.

PROP. XI. THEOREM.

Ratios which are the fame to the fame

ratio, are the fame to each other.

Let A be to B as c is to D, and c to D as E is to F; then will A be to в as E is to F.

For, of A, C and E take any equimultiples whatever G, H and K; and of B, D and F any equimultiples whatever L, M and N:

Then, fince A is to B as c is to D (by Hyp.), and G and H are equimultiples of A and c, and L and M of в and D, if G be greater than L, H will be greater than M; and if equal, equal; and if lefs, lefs (V. Def. 5.)

And, because c is to D as E is to F (by Hyp.), and н and K are equimultiples of c and E, and M and N of D and F ; if H be greater than M, K will be greater than N ; and if equal, equal; and if less, less (V. Def. 5.)

But if G be greater than L, it has been fhewn that H will also be greater than M; and if equal, equal; and if less, lefs; whence, if G be greater than L, K will also be greater than N; and if equal, equal; and if lefs, lefs.

And fince G and K are any equimultiples whatever of A and E, and L and N are any equimultiples whatever of E and F, A will have to в the fame ratio that E has to F (V. Def. 5.)

Q. E. D.

PROP. XII. THEOREM.

If any number of magnitudes be propor tional, either of the antecedents will be to its confequent, as the fum of all the antecedents is to the fum of all the confequents.

Let A be to E as c is to D, and as E is to F; then will A be to B, as A, C and E together, are to B, D and F together.

For, of A, C and E take any equimultiples whatever G, H and K; and of B, D and F any equimultiples whatever L, M and N :

Then, fince A is to E, as c is to D (by Hyp.), and G, H are equimultiples of A, C, and L, M of B, D, if G be greater than L, H will be greater than M, and if equal, equal; and if lefs, lefs (V. Def. 5.)

And because A is alfo to в as E is to F (by Hyp.), and G, K are equimultiples of A, E, and L, N of B, F, if a be greater than L, K will be greater than N; and if equal, equal; and if lefs, lefs (V. Def. 5.)

ན

From hence it follows, that if G be greater than L, G, H and K together, will be greater than L, M and N together; and if equal, equal; and if lefs, lefs.