A

E

many will there be in CD equal to F: Divide A B into parts equal to E, viz. AG, G B, and CD into parts equal to F, viz. C H, HD: Then the multitude of the parts c H, HD of the one is equal to the multitude of the parts AG, GB of G the other. And because A G is equal to E, and C H to F; AG, C H will be equal to E, F. By the fame reafon G B is equal to B E, and HD to F: Therefore GB, HD are equal to E, F: Wherefore there are as C many parts in A B equal to E, as there are in A B, C D equal to E, F: Therefore a B, CD will be the fame multiple of E, F, as H.

AB is of E.

Therefore if there be how many foever magnitudes equimultiples of as many other DI

magnitudes, each of each; the fame multiple one magnitude is of one, all shall be of all. Which was to be demonftrated.

The fifth and fixth propofitions of the feventh book of Euclid answer to this in numbers. And what is here pronounced of multiple ratio only, is afterwards demonstrated univerfally at prop. 12. of any proportionals rational, or irrational whatsoever. And this propofition was firft put down to prove others, which were neceflary towards the demonstration of the twelfth propofition,

PROP. II. THEOR.

If the first of four magnitudes be the fame multiple of the fecond, as the third is of the fourth, and there be a fifth the jame multiple of the fecond, as a fixth magnitude is of th fourth; then the first and fifth taken together will be the fame multiple of the fecond, as the third and fixth are of the fourth.

For let the first of four magnitudes A B be the fame multiple of the fecond c, as the third D E is of the fourth F. And let a fifth magnitude B G be the fame multiple of the second C, as a fixth EH is of the fourth : I fay, AG the first

P 3

and

D

A

B

G

E

and fifth taken together will be the fame multiple of the fecond c, as D H the third and fixth taken together is of the fourth F.

For because A B is the fame multiple of c, as D E is of F, F there will be as many magni

tudes in AB equal to C, as there are in D E equal to F. And by the fame reafon as many as are in B G equal to c, fo many there are in E H equal to F. Therefore as many as there are in the whole A G, equal to C, fo many there will be in the whole D H equal to F: Therefore D H is the fame multiple of F, as AG is of c: Wherefore AG the first and fifth taken together, is the fame multiple of the second c, as DH the third and fixth, is of the fourth F.

H

Wherefore, if the first of four magnitudes be the fame multiple of the fecond, as the third is of the fourth, and à fifth magnitude be the fame multiple of the fecond, as a fixth is of the fourth, then the first and fifth taken together will be the fame multiple of the fecond, as the third and fixth is of the fourth.

This propofition fuprofes the fifth and fixth magnitudes to be equimultiples of the fecond and fourth. But the propofition would hold the fame, if the fifth were only equal to the fecond, and the fixth to the fourth.

PROP. III. THEOR.

If the first of four magnitudes be the fame multiple of the fecond as the third is of the fourib, and any equimultiples of the first and third be taken; then, by equality, each of the affumed equimultiples will be an equimultiple of each, the one of the fecond, and the other of the fourth magnitude .

For let the first of four magnitudes A be the fame multiple of the fecond B, as the third c is of the fourth D; and

take

take E F, G H equimultiples of A, c: I fay, E F is the fame multiple of B, as G H is of D.

c;

F

K

HT

L

For because EF is the fame multiple of A, as GH is of c; there will be as many magnitudes in G H equal to c as there are in EF equal to A. Divide EF into the magnitudes E K, K F equal to A, and G H into the magnitudes GL, LH equal to then will the multitude of EK, KF be equal to the multitude of GL, L H. And because A is the fame multiple of B, as c is of D, and EK is equal to A, and GL equal to C; E K will be the fame multiple of B, as GL is of D. By the fame reafon K F is the fame multiple of B, as L H is of D. Then because the first E K is the fame multiple of the fecond B, as the third G L is of the fourth D; and the fifth K F is the fame multiple of the fecond B, as the fixth LH is of the fourth D: the magnitude E F made up of the first and fifth will [by 2. 5.] be the fame multiple of the fecond в, as GH the third and fixth is of the fourth D.

E

I I

ΙΙ

A B GC D

If therefore the first of four magnitudes be the fame multiple of the second, as the third is of the fourth, and any equimultiples of the firft and third be taken; then, by equality, each of the affumed magnitudes will be an equimultiple of each, the one of the fecond, and the other of the fourth magnitude. Which was to be demonftrated.

This is demonftrated nniverfally at prop. 22. it being here only propofed in multiple ratio; and that for the fake of de-. monftrating the fourth propofition.

PROP. IV. THEOR. If the first of four magnitudes has the fame ratio to the fecond, as the third bas to the fourth; then will any equimultiples of the first and third bave the fame ratio to any equimultiples of the fecond and fourth, when compared to one another.

For let the first magnitude ▲ have the same ratio to the fecond B, as the third c has to the fourth D. Take E, ►

any equimult ples of A, C, and G, H, any other equimultiples of B, D: I fay, E will be to G, as F is to H.

For let K, L be equimultiples of E, F, and M, N equimultiples of G, H.

Then because E is the fame multiple of A, as F is of c, and K, L are equimultiples of E, F; [by 3.5.] K will be the fame multiple of A, as L is of c. By the fame reason м will be the fame multiple of B, as N is of D.

KEAB G M And because as A is to B, so is

c to D, and K, L, are taken equimultiples of A, C, and M, N other equimultiples of B, D: If κ exceeds м 1 [by 5. def. 5.] will exceed N; if κ be equal to M, L will be equal to N; if K be lefs than M, L will be less than N and K, L are equimultiples of E, F: But M, N any other equimultiples of G, H: Therefore [by 5. def. 5.]'as E is to G, fo will F be to H.

I I LFCDII N

Wherefore if the firft of four magnitudes has the fame ratio to the fecond, as the third has to the fourth; then will any equimu'tiples of the first and third have the fame ratio to any equimultiples of the second and fourth, when compared to one another. Which was to be demonftrated.

Corollary. Therefore because it has been demonstrated if k exceeds M, L will alfo exceed N; if the one be equal to the one, the other will be equal to the other; if lefs, lefs it is manifeft if м exceeds K, N will exceed L ; if M be equal to K, N will be equal to L; if lefs, lefs: Therefore as G is to E, fo is H to F. Wherefore from hence it is manifest, that if four magnitudes be proportionals, they will be proportionals inversely.

PRQ P.

PROP. V. THEOR.

If one magnitude be the fame multiple of another, as a part taken away from the one is of a part taken away from the other; then shall the part remaining of the one be the fame multiple of the part remaining of the other, as one of the whole magnitudes is of the other ".

For let the magnitude A B be the fame multiple of the magnitude CD, as the part A E taken away of the one is of the part CF taken away of the other: I fay, the part E B remaining of the one, will be the fame multiple of the part F D remaining of the other, as the whole magnitude AB is of the whole magnitude c D.

E

C

F

For make EB the fame multiple of C G, as A E is of cr. Then because [by 1. 5.] AE is the fame multiple of C F as A B is of GF, and Ą ɛ A is the fame multiple of c F, as A B is of CD; AB is the fame multiple of GF, CD: Wherefore GF is equal to CD. Take away C F, which is common, from both Then the remainder G C is equal to the remainder DF. And because A E is the fame multiple of CF, as E B is of GC, and C B is equal to D F ; A E will be the fame multiple of CF, as EB is of FD. But A E is put the fame multiple of c F, as A B is of CD: Therefore E B is the fame multiple of F D, as A B is of of CD: Wherefore the remainder E B will be the fame multiple of the remainder F D, as the whole A B is of the whole c D.

B

D

Therefore if one magnitude be the fame multiple of another, as a part taken away from the one is of a part taken away from the other; then fhall the part remaining of the one be the fame multiple of the part remaining of the other, as one of the whole magnitudes is of the other. Which was to be demonstrated.

This propofition, propofed here only in multiple proportion, is univerfally demonftrated at prop. 19. and the feventh and eighth propofitions of the feventh book answer to it in numbers.

PROP.