As many magnitudes, therefore, as there are in AB equal to E, fo many are there in AB and CD together, equal to E and F together.

And, confequently, whatever multiple AB is of E, the fame multiple will AB and CD together be of E and F to, gether. Q.E.D.

PROP. II. THEOREM.

If any number of magnitudes be multiples of the fame magnitude, and as many others be the fame multiples of another magnitude, each of each, the fum of all the former will be the fame multiple of the one, as the fum of all the latter is of the other.

Let any number of magnitudes AB, BE, be multiples of the fame magnitude C, and as many others DG, GH, the fame multiples of another F, each of each; then will the whole AE, be the fame multiple of C, as the whole DH, is of F.

For fince AB is the fame multiple of c that DG is of F (by Hyp.), there will be as many magnitudes in AB equal to c, as there are in DG equal to F.

And becaufe BE is the fame multiple of c that GH is of F (by Hyp.), there will be as many magnitudés in BE equal to C, as there are in CH equal to F.

As many magnitudes, therefore, as there are in the whole AE equal to c, fo many will there be in the whole DH equal to F.

The whole AE, therefore, is the fame multiple of c, as the whole DH is of F.

Q.E.D.

PROP. III. THEOREM.

If the first of four magnitudes be the fame multiple of the fecond as the third is of the fourth and if of the first and third there be ; taken equimultiples, thefe will also be equimultiples, the one of the fecònd, and the other of the fourth.

E

H

B

Let A the firft, be the fame multiple of в the second, as c the third, is of D the fourth; and let EF and GH be equimultiples of A and C ; then will EF be the fame multiple of B, that GH is of D.

For fince EF is the fame multiple of A that GH is of c (by Hyp.), there will be as many magnitudes in EF equal to A, as there are in GH equal to C.

Divide EF into the magnitudes EK, KF each equal to A (I. 35.); and GH into the magnitudes GL, LH, each equal to c.

Then

Then will the number of magnitudes EK, KF in the one, be equal to the number of magnitudes GL, LH in the

other.

And because A is the fame multiple of B that c is of D (by Hyp.), and EK is equal to A, and GL to C (by Conft.), EK will be the fame multiple of B, that GL is of D.

In like manner, fince KF is equal to A, and LH to C, KF will be the fame multiple of B, that LH is of D.

And fince EK, KF are each multiples of B, and GL, LH are each the fame multiples of D, the whole EF will be the fame multiple of B, as the whole GH is of D (V. 2.)

Q.E.D

PRO P. IV. THEOREM.

If the first of three magnitudes be greater than the second, and the third be any magnitude whatever, fome equimultiples of the first and second may be taken, and fome multiple of the third fuch, that the former fhall be greater than that of the third, but the latter not greater.

Let AB, BC be two unequal magnitudes, and D any other magnitude whatever; then there may be taken fome equimultiples of AB, BC, and fome multiple of D fuch,

that the multiple of AB fhall be greater than that of D, but the multiple of BC not greater.

For of BC, CA take any equimultiples GF, FE fuch, that they may be each greater than D; and of D take the multiples K and L fuch, that L may be that which is first greater than GF, and K that which is next less than L.

Then, because L is that multiple of D which is the first that becomes greater than GF, the next preceding multiple K will not be greater than GF; that is GF will not be less than K.

And, fince FE is the fame multiple of AC that GF is of BC (by Conft.), GF will also be the fame multiple of BC that EG is of AB (V. 1.)

The magnitudes EG and GF are, therefore, equimultiples of the magnitudes AB and BC, and L is a multiple of

D.

And, fince GF is not lefs than K, and EF is greater than D (by Conft.), the whole EG will be greater than K and D taken together.

But K and D, taken together, are equal to L (by Conft.); therefore EG will be greater than L, and FG not greater than L, as was to be shewn.

PROP.

PROP. V. THEOREM.

If four magnitudes be proportional, any equimultiples whatever of the antecedents will be proportional to any equimultiples whatever of the confequents.

any

Let A be to B as C is to D, and of A and C take equimultiples EK, FL; and of B and D any equimultiples GM, HN; then will EK be to GM, as FL is to HN.

For of EK and FL take any equimultiples whatever ÉP, FR; and of GM and HN any equimultiples whatever

GS, HT:

Then, fince EK is the fame multiple of A, that FL is of c (by Conft.), and of EK, FL have been taken the equimultiples EP, FR, EP will be the fame multiple of A, that FR is of C (V. 3.)

And, in the same manner, it may be fhewn, that Gs is the fame multiple of B, that HT is of D.

But A has the fame ratio to B that c has to D (by Hyp.); and of a and c have been taken the equimultiples EP, FR and of B and D the equimultiples GS, HT.

If, therefore, EP be greater than GS, FR will also be greater than HT; and if equal, equal; and if lefs, lefs (V. Def. 5.)