But G, and G, H, K together, are any equimultiples whatever of A, and A, C, E together (by Conft.); and L, and L, M, N together, are any equimultiples whatever of B, and B, D, F together; whence, as A is to в, fo is A, C and E together, to B, D and F together (V.Def. 5:)

PROP. XIII. THEOREM.

Q. E. D.

Equimultiples of any two magnitudes have the fame ratio as the magnitudes themfelves.

Let CD be the fame multiple of A that EF is of B; then will CD have the fame ratio to EF that A has to B.

For, fince CD is the fame multiple of A that EF is of B? there are as many magnitudes in CD equal to A, as there are in EF equal to B.

Let CD be divided into the magnitudes CG, GH, HD each equal to A (I. 25.); and EF into the magnitudes EK, KL, LF, each equal to в.

Then, the number of magnitudes CG, GH, HD in the one, will be equal to the number of magnitudes FK, KL LF in the other.

And, because DH, HG, GC are all equal to each other, as are alfo FL, LK, KE, DH will be to FL AS HG to LK, and as GC to KE (V.9.)

And, fince any antecedent is to its confequent, as all the antecedents are to all the confequents (V. 12.), FL will be to DH, as FE is to DC.

But DH is equal to A (by Const.), and FL is equal to B ; therefore B will be to A, as FE to DC; and, inversely, DC to FE as A to B. Q. E. D.

PRO P. XIV. THEOREM.

If four magnitudes of the fame kind be proportional, and the first be greater than the third, the fecond will also be greater than the fourth, and if equal, equals and if lefs, lefs.

Let A be to B as c is to D; then if A be greater than C, B will also be greater than D; and if equal, equal; and if lefs, lefs.

First, let a be
be greater

tian D.

than c; then в will be greater

B

Tor, of A, C take the equimultiples E, G, and of в the multole F fuch, that E may be greater than F, but G not greate (V.4.); and make н the fame multiple of p that F is of:

Then, because A is to B as C is to D (by Hyp.), and E, G are Quimultiples of A, C, and F, H of B, D (by Conf.), F beng greater than F, G will alfo be greater than H (V. D. 5.)

And, fince, by conftruction, F is not less than G, and G has been proved to be greater than H, F will likewife be greater than H.

But F and H are equimultiples of B and D (by Conft.); therefore, fince F is greater than H, B will alfo be greater than D.

Secondly, let A be equal to c; then will в be equal

to D.

For, A is to B as c is to D (by Hyp.); or, fince a is equal to C, A is to B as A is to D; therefore B is equal to D (V. 10.)

Thirdly, let A be lefs than c; then will в be less

than D.

For, c is to D as a is to B, by the propofition; therefore c being greater than A, D will also be greater than B, by the firft cafe. Q. E. D.

PROP. XV. THEOREM.

If four magnitudes of the fame kind be proportional, they will be proportional alfo when taken alternately.

Let A be to в as c is to D; then, alfo, alternately, A will be to c as в is to D.

For, of A and B take any equimultiples whatever E and G; and of C and D any equimultiples whatever F and H : Then,

L

4

Then, fince E is the fame multiple of A that G is of B, and that magnitudes have the fame ratio as their equimultiples (V. 13.), A is to в as E is to G.

But A is to B as c is to D, by the propofition; whence c is to D as E is to G (V. 11.)

In like manner, because F is the fame multiple of c that H is of D, C will be to D as F is to H (V. 13.)

But c has been fhewn to be to D as E is to G; confequently, E will be to G as F is to H (V. II.)

Since, therefore, E has the fame ratio to G that F has to H, if E be greater than F, G will also be greater than H; and if equal, equal; and if less, less (V. Def. 5.)

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

Q. E. D.

PROP. XVI. THEOREM.

If four magnitudes be proportional, the fum of the first and fecond, will be to the firft or fecond, as the fum of the third and fourth, is to the third or fourth.

Let AE be to EB as CF is to FD; then will AB be to Be,

or AE, as CD is to DF, or CF.

For, fince AE is to EB as CF is to FD (by Hyp.); therefore, alternately, AE will be to CF as EB to FD (V. 15.)

And, fince the antecedent is to its confequent as all the antecedents are to all the confequents (V. 12.), AE will be to CF as AB is to CD.

But ratios which are the fame to the fame ratio are the fame to each other (V. 11.); whence AB will be to CD as EB is to FD; and, alternately, AB to EB as CD to DF (V. 15.)

Again, fince AE has been fhewn to be to CF as ab is to CD, therefore, by alternation, AE will be to AB as CF is to CD (V. 15.)

But quantities which are directly proportional are also proportional when taken inversely; whence AB will be to AE as CD is to CF (V.7.)

PROP. XVII. THEOREM.

Q. E. D.

If four magnitudes be proportional, the difference of the first and fecond, will be to the first or fecond, as the difference of the third and fourth, is to the third or fourth.

Let AB be to BE as CD is to DF; then will AE be to

AB, or EB, as CF is to CD, or FD.

For,