1 Def. 5. shall † F be to H. Wherefore, if the first have the

same Proportion to the second, as the third to the fourth; then also, Mall the Equimultiples of the firf 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 te answer each; which was to be demonstrated.

Because it is demonstrated, if K exceeds M, then L will exceed N; and if it be equal to it, it will be equal; and if less, lefser. It is manifest likewise, if M exceeds K, that N shall exceed L; if equal,

equal; but if lefs, less. And therefore as G is to E, * Def: 5. fo is * H to F.,

Coroll. From hence it is manifest, if four Magnitudes

be proportional, that they will be also inversely proportional.

PROPOSITION V.

THE O RE M.
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 shall be the same Multiple of the Residuc
of the other, as the whole is of the whale.
ET the Magnitude AB be the fame Multiple

of the Magnitude CD, as the Part taken away
AE is of the Part taken away CF. I
say that the Residue EB is the same B
Multiple of the Residue FD, as the

G whole AB is of the whole CD.

For let EB be such a Multiple of
CG as AE is of CF.

E
Then because AE is the same Mul-

F
tiple of CF, as E B is of CG, AE
of this will be * the same Multiple of CF,

as AB is of GF. But AE and AB A D are put Equimultiples of CF and CD.

Therefore AB is the fame Multiple of GF as of +2 exism CD; and fo G F is t equal to CD. Now let CF, which is common, bę takon away, and the Residue

GC

of Ibis.

GC is equal to the Residue DF. And then becaufe AE is the same Multiple of CF, as EB is of CG, and CG is equal to DF; AE shall be the fame Multiple of CF, as EB is of FD. But AE is put the fame Multiple of CF as AB is of CD. Therefore EB is the fame Multiple of FD, as AB is of CD: and so the Residue EB is the same Multiple of the Residue. FD, as the whole AB 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 shall be the same Multiple of the Residue of the other, as the whole is of the whole ; which was to be demonstrated.

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 Refidues are
either equal to those Magnitudes, or else Equi-
multiples of them.
ET two Magnitudes AB, CD, be Equimulti-

ples of two Magnitudes E, F, and let the Magnitudes AG, CH, Equimultiples of the fame E, F, be taken from AB, CD. I say, the Residues GB, HD, are either equal to E, F, or are Equimultiples of them.

For first, Let GB be equal to E. I say, HD is also equal to F. For let CK be equal to F. Then because AG A is the same Multiple of E, as CH

K is of F; and GB is equal to E; and CK to F; AB will be * the

с

* 1 of this. fame Multiple of E, as KH is of F. But A B and CD are put

G

H Equimultiples of E and F.

1
Therefore KH is the same Mul-
tiple of F, as CD is of F. B D E F

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

Take away CH which is com-
mon; then the Residue KC is

K
equal to the Residue HD. But

A
KC is equal to F. Therefore

с
HD is equal to F; and fo GB
shall be equal to E, and HD to
F.

G | H
In like, Manner we demon-
ftrate if G B was a Multiple of
E, that HD is the like Multiple
of F. Therefore, if two Mag-
nitudes be Equimultiples of two B D E F
Magnitudes, and some Magni-
tudes Equimultiples of the fame be taken away; then the
Residues are either equal to those Magnitudes, or else
Equimultiples of them; which was to be demonstrated?

PROPOSITION VII.

PROBLEM.
Equal Magnitudes have the same Proportion to tbe

Tame Magnitude ; and one and the same Magni-
tude has the same Proportion to equal Magnitudes.
ET A, B, be equal Magnitudes, and let C be

any other Magnitude. I say, A and B have the same Proportion to C; and likewise Chas the same Proportion to A as to B.

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

D A
Now because D is the same
Multiple of A, as E is of B, and
A is equal to B, D shall be also
equal to E; but F is a Magnitude
taken at Pleasure. Therefore if
D exceeds F, then E will exceed

E B C F
F; if D be equal to F, E will be
equal to F; and if less, less. But D, E are Equimut
tiples of A, B; and F is any Multiple of C. There-
fore it will be * as A is to C, so is B to C.
I say, moreover, that has the same Proportion to

А

Déf. 5o

A as to B. For the same Construction 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 less than E. But if F is Multiple of C; and D, E, any other Equimultiples of A, B. Therefore as C is to A, fo shall * C be to B: Where- * Def. go fore equal Magnitudes have the same Proportion to the same Magnitude, and the fame Magnitude to équal ones; which was to be demonstrated.

PROPOSITION VIII.

THEOREM.
The greater of any two unequal Magnitudes, has

a greater Proportion to some third Magnitude,
than the less has; and that third Magnitude
bath a greater Proportion to the lesser of the two
Magnitudes, than it has to the greater.
ET AB and C be two unequal Magnitudes,

is let
third Magnitude. I say, AB has a greater Proportion
to D, than C has to D; and D has a greater Propor-
tion to C, than it has to AB.

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

G AE be multiplied until it ex

A ceeds D, and let that Multiple of AE, greater than D,be FG.

E
Maké ĞH the same Multiple
of EB, and K of C, as FG K H B
is of AE. Also, assume L
double to D, P triple, and so
on, until such 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 because N is the
Dearest Multiple of D greater N M PL D

than

thän K, M will not be greater than K, that is, ķ will not be less than M. And since FG is the same

Multiple of AE; às ĠH is of EB; FG shall be * 1 of this. * the fame Multiple of AE; as FH is of AB, but

FG is the same Multiple of A E, as K is of C; wherefore FH is the same Multiple of AB, as K is of C; that is, FH, K, are Equimultiples of AB and C. Again, because GH is the same Multiple of

EB, as K is of C, and EB is equal to C; GH shall if Ax. I. be f equal to K. But K is not less than M.

Therefore GH shall not be less than M; but FG is greater than D. Therefore the whole FH will be greater than M and D; but M and D together, are equal to N; because M is a Multiple of D, the nearest lesser than N: Wherefore FH is greater than N. And so finĉe FH exceeds N, and K does not, and FH and

K are Equimultiples of A B and C, and N is another Def. 7. Multiple of D; therefore A B will have I a greater

Ratio to D, than C has to D. I say, moreover, that D has a greater Ratio to C, than it has to AB; for the fame Construction remaining, we demonstrate, as béfore, that N exceeds K, but not FH. And N is a Multiple of D, and FH, K, are Equimultiples of AB and C. Therefore D has I a greater Proportion to C, than D hath to B. Wherefore the greater of any two unequal Magnitudes, has a greater Proportion to some third Magnitude, than the lefs has; and that third Magnitude hath a greater Proportion to the lefser of the tws Magnitudes, than it has to the greater.

PROPOSITION IX. .

THEOREM.
Magnitudes which have the same Proportion to one

and the fame Magnitude, are equal to one an-
other; and if a Magnitude has the same Pro-
portion to other Magnitudes, these Magnitudes
are equal to one another.

fame Proportion to Č. I say, A is equal to B.

For