PROPOSITION II.

THEORE M.

If the first be the fame Multiple of the fecond, as the third is of the fourth; and if the fifth be the fame Multiple of the fecond, as the fixth is of the fourth, then fball the first, added to the fifth, be the fame Multiple of the fecond, as the third, added to the fixth, is of the fourth.

L'

Α

D

ET the firft A B be the fame Multiple of the fecond C, as the third D E is of the fourth F; and let the fifth BG be the fame Multiple of the fecond C, as the fixth EH is of the fourth F. I fay, the first added to the fifth, viz. A G, is the fame Multiple of the fecond C, as the third added to the fixth, viz. DH, is of the fourth F.

B+

E

HF

For, because AB is the fame Multipile of C, as DE is of F; there are as many Magnitudes equal to C in A B, as there are Magnitudes equal to Fin D E. And, for the fame Reason, there are as many Magnitudes equal to C in B G, as there are Magnitudes equal to Fin E H. Therefore there are as many Magnitudes equal to C, in the whole A G, as there are Magnitudes equal to F in D H. Wherefore A G is the fame Multiple of C, as DH is of F. And fo the firft, added to the fifth, A G, is the fame Multiple of the fecond C, as the third, added to the fixth, D H, is of the fourth F. Therefore, if the first be the fame Multiple of the fecond, as the third is of the fourth; and if the fifth be the fame Multiple of the fecond, as the fixth is of the fourth; then fhall the firft, added to the fifth, be the fame Multiple of the fecond, as the third, added to the fixth, is of the fourth; which was to be demonftrated.

PRO

PROPOSITION III.

THEOREM.

If the first be the fame Multiple of the fecond, as the third is of the fourth, and there be taken Equ multiples of the first and third; then will the Magnitudes fo taken be Equimultiples of the fecond and fourth.

LET

ET the firft A be the fame Multiple of the fecond
B, as the third C is of the fourth D; and let E F,
GH, be Equimultiples of A
and C. I fay, EF, is the fame
Multiple of B as C H is of D.

F

H

For, because EF is the fame Multiple of A, as G H is of C, there are as many Magnitudes K+ equal to A in E F, as there are Magnitudes equal to C in G H. Now divide EF into the Magnitudes E K, K F, each equal to A, and G H into the Magnitudes GL, L H, each equal

L+

EAB G CD

to C. Then the Number of the Magnitudes E K, K F, will be equal to the Number of the Magnitudes GL, LH. And because A is the fame Multiple of B, as C is of D, and E K is equal to A, and GL to C; EK will be the fame Multiple of B, as GL is of D. For the fame Reason K F fhall be the fame Multiple of B, as L H is of D. Therefore because the first E K is the fame Multiple of the second B, as the third GL is of the fourth D, and K F the fifth is the fame Multiple of B, the fecond, that L H, the fixth, is of D the fourth: Therefore the first added to the fifth, EF, fhall be the fame Multiple of the fecond B, as 2 of this. the third added to the fixth, GH, is of the fourth D. If, therefore, the first be the fame Multiple of the fecond, as the third is of the fourth, and there be taken Equimultiples of the firft and third; then will the Magnitudes fo taken, be Equimultiples of the fecond and fourth; which was to be demonftrated.

PRO

PROPOSITION IV.
THEOREM.

If the first have the fame Proportion to the second,
as the third to the fourth; then also shall the
Equimultiples of the first and third have the fame
Proportion to the Equimultiples of the fecond and
fourth, according to any Multiplication whatfo-
ever, if they be fo taken as to answer each other.

LET the firft A have the fame Proportion to the

fecond B, as the third C hath to the fourth D; and let E and F, the Equimultiples of A and C, be any how taken; as alfo G and H, the Equimultiples of B and D. I fay, E is to G as F is to H.

For take K and L, any Equimultiples of E and F; and alfo M and N, any, of G and H.

Then, because E is the fame Multiple of A, as F is of C, and K and L are taken Equimultiples of E *3 of this. and F; therefore K will be the fame Multiple of A, as L is of C.

this.

For the fame Reafon, M is the fame KEABGM
Multiple of B, as N is of D.

And

fince A is to B, as C is to D, and LFCDHN K and L are Equimultiples of A and C; and alfo M and N Equimultiples of B and D; if K exceeds + Def. 5 of M, then + L will exceed N; if K be equal to M, L will be equal to N; and if K is lefs than M, then L will be less than N: But K and Lare Equimultiples of E and F, alfo M and N are Equimultiples of G and H. Therefore, as E is to G, fo fhall F be to H. Wherefore, if the first have the fame Proportion to the fecond, as the third to the fourth; then alfo fhall the Equi

Def. 5.

multiples

multiples of the first and third have the fame Proportion to the Equimultiples of the fecond and fourth, according to any Multiplication whatfoever, if they be fo taken as to anfwer each other; which was to be demonftrated.

Because it is demonftrated, if K exceeds M, then L will exceed N; and if K be equal to M, L will be equal to N; and if K be less than M, L will be lefs than N: It is manifeft, likewise, if M exceeds K, that N fhall exceed L; if equal, equal; but if lefs, lefs. And therefore, as G is to E, fo is * H to F. Def. 5.

Coroll. From hence it is manifeft, if four Magnitudes be proportional, that they will be alfo inverfely proportional.

PROPOSITION V.

THEOREM.

If one Magnitude be the fame Multiple of another Magnitude, as a Part taken from the one is of a Part taken from the other; then the Refidue of the one shall be the fame Multiple of the Refidue of the other, as the Whole is of the Whole.

LET the Magnitude A B be the fame Multiple of
the Magnitude CD, as the Part taken away A E,
is of the Part taken away CF. I fay,
that the Refidue E B is the fame Mul-
tiple of the Refidue FD, as the Whole
AB is of the whole CD.

For, let E B be fuch a Multiple of
CG, as A E is of CF.

Then, becaufe A E is the fame Multiple of CF, as E B is of CG, A E will be the fame Multiple of C F, as A B is of F G. But A E and A B are put Equimultiples of CF and CD: Therefore

B

E+1

G

+ F

AD

3 of this.

A B is the fame Multiple of G F, as of CD; and fo GF is equal to CD. Now let CF, which is + Ax. 2. of common, be taken away; then the Refidue G C is this.

equal

equal to the Refidue D F. And then, because A E is the fame Multiple of C F, as E B is of CG, and C G is equal to DF; AE fhall be the fame Multiple of CF, as E B is of F D. 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 CD; and fo the Refidue E B is the fame Multiple of the Refidue FD, as the Whole A B is of the Whole C D. Wherefore, if one Magnitude be the fame Multiple of another Magnitude, as a Part taken from the one is of a Part taken from the other; then the Refidue of the one fhall be the fame Multiple of the Refidue of the other, as the Whole is of the Whole; which was to be demonftrated.

PROPOSITION VI.
THEOREM.

If two Magnitudes be Equimultiples of two Mag-
nitudes, and fome Magnitudes Equimultiples of
the fame, be taken away; then the Refidues
are either equal to thofe Magnitudes, or elfe
Equimultiples of them.

LET two Magnitudes A B C D, be Equimultiples

of two Magnitudes E, F; and let the Magnitudes A G, CH, Equimultiples of the fame E, F, be taken from A B, CD; I fay the Refidues G B, HD, are either equal to E, F, or are Equimultiples of

them.

A

K

For, firft, Let G B be equal to E. I fay, HD is alfo equal to F. For let CK be equal to F. Then, because AG is the fame Multiple of E, as CH is of F; and G B is equal to E; and CK to F; A B will be the fame Multiple of E, as K H is of F. But A B and C D are put Equimultiples of E and F. Therefore K H is the G++ H fame Multiple of F, as CD is of F.

+ C

B

EF

And because K H and CD are
Equimultiples of F; KH will be equal to C D. Take

away