PROPOSITION IV.

THEOREM.

If the first have the fame Proportion to the fecond, as the third to the fourth; then also shall the Equimultiples of the firft and third have the fame Proportion to the Equimultiples of the second and fourth, according to any Multiplication whatsoever, 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 Equimul-
tiples of E and F; and alfo M and
N, any, of G and H.

Then, becaufe 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.

And

LFCDHN

For the fame Reason, M is the fame KEABG M Multiple of B, as N is of D. fince A is to B, as C is to D, and 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 L are 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

1 Def. 5.

multiples

multiples of the firft and third have the fame Proportion to the Equimultiples of the fecond and fourth, according to any Multiplication whatsoever, if they be fo taken as to answer 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 lefs than M, L will be lefs than N: It is manifeft, likewife, 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 also inversely 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 fhall 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 C F. I fay, that the Refidue E B is the fame Multiple of the Refidue FD, as the Whole A B is of the Whole C D.

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

B

G

E+ 1.

+F

* I of this.

AD

and fo

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 GF. But A E and A B are put Equimultiples of CF and CD: Therefore AB is the fame Multiple of GF, as of CD; GF is equal to CD. Now, let CF, which is * Ax. 2 of common, be taken away; then the Refidue GC is this.

equal

equal to the Refidue DF. And then, because AE is the fame Multiple of C F, as E B is of CG, and CG is equal to DF; AE fhall be the fame Multiple of CF, as E B is of FD. But A E is put the fame Multiple of CF, 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 Residue FD, as the Whole AB is of the Whole CD. 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 Magnitudes, and fome Magnitudes Equimultiples of the fame, be taken away; then the Refidues are either equal to thofe Magnitudes, or elfe Equimultiples of them.

LE

ET two Magnitudes A B, CD, be Equimultiples of two Magnitudes E, F; and let the Magnitudes A G, CH, Equimultiples of the fame, E, F, be taken from AB, 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 elfo equal to F. For let CK be equal to F. Then, because A G is the fame Multiple of E, as CH is of F ; and G B is equal to E; and * 1 of this, CK to F; AB will be the fame Multiple of E, as K H is of F. But AB and CD are put Equimultiples of E and F. Therefore K H is the G++H fame Multiple of F, as CD is of F.

+C

BD EF

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

away

away C H, which is common; then the Refidue KC is equal to the Refidue HD. But KC is equal to F. Therefore HD is equal to F; and fo GB fhall be equal to E, and HD to F.

In like manner we demonftrate, if G B was a Multiple of E, that HD is a like Multiple of F. Therefore, if two Magnitudes be Equimultiples of two Magnitudes, and fome Magnitudes, Equimultiples of

the fame, be taken away; then the Refidues are either equal to thofe Magnitudes, or elfe Equimultiples of them ; which was to be demonftrated.

Equal Magnitudes have the fame Proportion to
the fame Magnitude; and one and the fame
Magnitude bave the fame Proportion to equal
Magnitudes.

LE

ET A, B, be equal Magnitudes, and let C be any other Magnitude. I fay, A and B have the fame Proportion to C; and likewife

C has the fame Proportion to A as to
B.

For take D, and E, Equimultiples of A and B ; and let F be any other Multiple of C.

Now, becaufe D is the fame Multiple of A, as E is of B, and A is equal to B, D fhall be alfo equal to E; but F is a Magnitude taken at Pleasure. Therefore if D exceeds F, then E will exceed F; if D be equal to F, E will be equal to F; and if lefs, lefs. But

DA

EBC F

D and E are Equimultiples of A and B; and F is any other Multiple of C. Therefore it will be as A is* Def. 5. to C, fo is B to C.

I fay, moreover, that C has the fame Proportion to A as to B. For the fame Conftruction remaining, we prove, in like manner, that D is equal to E. Therefore, if F exceeds D, it will also exceed E; if it be equal to D, it will be equal to E; and if it be less than D, it will be lefs than E. But F is a Multiple of C; and D and E, any other Equimultiples of A and B ; Def. 5. therefore, as C is to A, fo fhall C be to B. Wherefore, equal Magnitudes have the fame Proportion to the fame Magnitude, and the fame Magnitude to equal ones ; which was to be demonftrated.

PROPOSITION VIII.

THEORE M.

The greater of any two unequal Magnitudes bas a greater Proportion to fome third Magnitude, than the lefs bas; and that third Magnitude bath a greater Proportion to the leffer of the two Magnitudes, than it has to the greater.

L

ET AB and C be two unequal Magnitudes, whereof AB is the greater; and let D be any third Magnitude. I fay, AB has a greater Proportion to D, than C has to D; and D has a greates Proportion to C, than it has to AB.

F

Because A B is greater than C, make BE equal to C, that is, let AB exceed C by AE; then A E, multiplied fome Number of Times, will be greater than D. Now let

A

G+E+

AE be multiplied until it ex-
ceeds D, and let that Multiple
of AE greater than D be F G.
Make GH the fame Multiple K H
of E B, and K of C, as FG
is of AE. Alfo, aflume L
double to D, M triple, and fo
on, until fuch a Multiple of
D is had, as is the nearest
greater than K; let this be N,
and let M be a Multiple of D,
the nearest less than N.

Now, becaufe N is the
nearest Multiple of D greater N M

than