PROPOSITION II.

THEOREM. If the first be the fame Multiple of the second, as

the third is of the fourth ; and if the fifth be the same Multiple of the second, as the fixth is of the fourth; then all the first, added to the fifth, be the same Multiple of the second, as the third, added to the fixth, is of tbe fourtb.

E T the firft A B be the fame Multiple of the se

cond C, as the third D E is of the fourth F; and let the fifth BG be the same Multiple of the second C, as the sixth

Α.

D
EH is of the fourth F. I say,
the first added to the fifth, viz.
A G, is the same Multiple of the

B +

E second C, as the third added to

+ the fixth, viz. DH, is of the fourth

G C С НЕ For, because A B is the same Multipile of C, as D E is of F; there are as many Magnitudes equal to C in A B, as there are Magnitudes equal to F in D E. And, for the same Realon, 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 , DH. Wherefore A G is the same Multiple of C, as DH is of F. And so the first, added to the fifth, A G, is the same Multiple of the second C, as the third, added to the sixth, DH, is of the fourth F. Therefore, if the first be the same Multiple of the second, as the third is of the fourth ; and if the fifth be the same Multiple of the second, as the fixth is of the fourth; then Mall the first, added to the fifih, be the same Multiple of the second, as tbe third, added to the sixth, is of the fourth ; which was to be demonstrated.

PRO

PROPOSITION III. .

THEOREM.
If the first be the same Multiple of the second, as

the third is of ibe fourth, and there be taken
Equ multiples of the first and third ; then will
the Magnitudes so taken be Equimultiples of
the second and fourib.

L

ET the first A be the same Multiple of the second

B, as the third C is of the fourth D; and let E F, GH, be Equimultiples of A

F and C. I say, EF, is the same

H Multiple of B as C H is of D.

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

EAB G CD nitudes GL, LH, each equal to C. Then the Number of the Magnitudes E K, K F, will be equal to the Number of the Magnitudes GL, L H. And because A is the same Multiple of B, as C is of D, and E K is equal to A, and G L to C; E K will be the same Muliiple of B, as GL is of D. For the same Reason K F shall be the same Multiple of B, as L H is of D. Therefore because the first E K is the same Multiple of the second B, as the third GL is of the fourth D, and K F the fifth is the same Multiple of B, the second, that LH, the fixth, is of D the fourth : Therefore the first added to the fifth, EF, shall be * the same Multiple of the second B, as * 2 of ebis. the third added to the sixth, G H, is of the fourth D. If, therefore, the first be the same Multiple of the second, as the third is of the fourth, and there be taken Equimultiples of the first and third i then will the Magnitudes to taken, be Equimultiples of the second and fourth ; which was to be demonstrated,

PRO

to H.

PROPOSITION IV.

THEOREM.
If the first have the same Proportion to the second,

as the third to the fourtb; then also fall ibe
Equimultiples of the first and third bave the same
Proportion to the Equimultiples of the second and
fourth, according to any Multiplication whatso-

ever, if they be to taken as to answer each other. LET the firft A have the same Proportion to the

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

For take K and L, any Equimul-
tiples of E and F; and also M and
N, any, of G and H.

Then, because E is the same
Multiple of A, as F is of C, and K

and L are taken Equimultiples of E * 3 of tbis. and F; therefore K will be * the

same Multiple of A, as L is of C.
For the fame Reason, M is the fame KE A BGM
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 also M and N Equi-

multiples of B and D; if K exceeds Defo 5 of M, then + L will exceed N; if K

be equal to M, L will be equal to
N; and if K is less than M, then L
will be less than N: But K and
L are Equimultiples of E and F,
allo M and N are Equimulti-

ples of G and H. Therefore, as I Dif.5.

E is to G, lo shall I F be to H.
Wherefore, if the first have the same
Proportion to the second, as the third
to the fourth; then also shall the Equi-

multiples

this.

multiples of the forft and third have the same Proportion to the Equimultiples of the second and fourth, according to any Multiplication whatsoever, if they be so taken as to answer each other ; which was to be demonstrated.

Because it is demonstrated, 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 less than N: It is manifeft, likewise, if M exceeds K, that N fall exceed L; if equal, equal ; but if less, less. And therefore, as G is to E, so is * H to F.

Dof. 5.

Coroll. From hence it is manifeft, if four Magnitudes

be proportional, that they will be also inverfely proportional.

P R O POSITION V.

THEOREM.
If one Magnitude be the same Multiple of another

Magnitude, as a Part taken from the one is of
a Part taken from the other; then the Re.
fidue of the one shall be the fame Multiple of
the Residue of the other, as the Whole is of
tke Whole.

E+1

LET the Magnitude A B be the same Multiple of

the Magnitude CD, as the Part taken away A E, is of the Part taken away CF. I say, that the Residue E B is the same Mul. B tiple of the Refidue FD, as the whole A B is of the whole CD.

1 For, let E B be such a Multiple of

С
CG, as A E is of CF.
Then, because A E is the same Mul-

+ F
tiple of C F, as E B is of CG, A E will
be * the same Multiple of CF, as A B is
of F G. But A E and A B are put Equi AD

* 3 of this. multiples of C F and CD: Therefore A B is the fame Multiple of GF, as of CD; and so GF is + equal to CD. Now let CF, which is + Ax. 2.cf common, be taken away, then the Residue G C is ibis.

equal

equal to the Refidue DF. And then, because A E is
the same Multiple of CF, as E B is of CG, and CG
is equal to DF; A E fhall be the same Multiple of
CF, as E B is of F D. But A E is put the same Mul-
tiple of CF, as AB is of CD. Therefore E B is
the same Multiple of F D, as A B is of CD; and so
the Residue E B is the fame Multiple of the Residue
FD, as the Whole A B is of the Whole CD.
Wherefore, if one Magnitude be the same Multiple of
another Magnitude, as a Part taken from the one is of a
Part taken from the other ; then the Residue of the one
mall be the same Multiple of the Refidue of the other, as
the Whole is of the Whole ; which was to be demon-
ftrated.

PROPOSITION VI.

THEORE M.
If two Magnitudes be Equimultiples of two Mag-

nitudes, and some Magnitudes Equimultiples of
the same, be taken away; then the Residues
are eitber equal to those Magnitudes, or else

Equimultiples of them.
LET two Magnitudes A B C D, be Equimultiples

of two Magnitudes E, F; and let the Magni-
tudes A G, CH, Equimulciples of the same E, F, be
taken from A B, CD; I say the Refidues GB, HD,
are either equal to E, F, or are Equimultiples of
them.

For, first, Let G B be equal to E. I say, HD is
also equal to F. For let C K be
equal to F. Then, because A G is А
the same Multiple of E, as CH is K
of F; and G B is equal to E; and
CK to F; A B will be * the same +C
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
same Multiple of F, as C D is of
E.

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

away

* 1 of tbis.

B)