BOOK V.

DEFINITION S.

1. A less magnitude is faid to be a part of a greater, when the lefs is contained a certain number of times in the greater.

2. A greater magnitude is faid to be a multiple of a lefs, when the greater is equal to a certain number of

times the less.

3. Ratio is a certain mutual relation of two magnitudes of the fame kind, which arifes from confidering the quantity of each.

4. When four magnitudes are compared together, the firft and third are called the antecedents, and the second and fourth the confequents.

5. Four magnitudes are faid to be proportional, when any equimultiples whatever of the antecedents, are, each of them, either equal to, greater, or less, than any equimultiples whatever of their confequents.

6. Inverse ratio is, when the confequents are made the antecedents, and the antecedents the confequents.

7. Alternate ratio is, when antecedent is compared with antecedent, and confequent with confequent.

8. Compounded ratio is, when each antecedent and its confequent, taken as one quantity, is compared, either with the confequents, or the antecedents.

9. Divided ratio is, when the difference of each antecedent and its confequent, is compared, either with the confequents, or the antecedents.

PROP. I. THEOREM.

If any number of magnitudes be equimultiples of as many others, each of each; whatever multiple any one of them is of its part, the fame multiple will all the former be of

Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each; then whatever multiple AB is of E, the fame multiple will AB and CD together, be of E and F together.

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

Divide AB into magnitudes equal to E (I. 35.), which let be AG, GB; and cp into magnitudes equal to F, which let be CH, HD.

Then the number of magnitudes CH, HD, in the one, will be equal to the number of magnitudes AG, GB, in

the other.

And because AG is equal to E, and CH to F (by Const.), AG and CH, taken together, will be equal to E and F taken together.

For the fame reafon, becaufe GB is equal to E, and HD to F, GB and HD taken together, will be equal to E and F taken together.

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 toQ. E. D.

gether,

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 because BE is the fame multiple of c that GH is of F (by Hyp.), there will be as many magnitudes in BE equal to c, as there are in GH 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 firft and third there be taken equimultiples, these will also be equimultiples, the one of the fecond, and the other of the fourth.

Let A the firft, be the fame multiple of в the fecond, 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 E, 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.

B

And because A is the fame multiple of в 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.

PROP. IV. THEOREM.

If the first of three magnitudes be greater than the fecond, and the third be any magnitude whatever,, fome equimultiples of the first and fecond may be taken, and fome multiple of the third such, 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 some multiple of D such,