K

A

C

Take away CH which is com-
mon; 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.

GH

B DE F

In like, Manner we demonftrate if G B was a Multiple of E, that HD is the 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 elfé Equimultiples of them; which was to be demonftrated.

PROPOSITION VII.

PROBLEM.

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

L

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 Propor-
tion 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 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 Pleafure. 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 D, E are Equimul tiples of A, B; and F is any Multiple of C. There* Def. 5. fore it will be as A is to C, fo is B to C.

*

E B C F

I say, moreover, that C has thefame Proportion to

A

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 alfo 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 fhall* C be to B: Where- * Def. s
fore equal Magnitudes have the fame Proportion to
the fame Magnitude, and the fame Magnitude to
equal ones; which was to be demonftrated.

PROPOSITION VIII.
THEOREM.

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

LET 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 greater Proportion to C, than it has to AB.

G

F

A

E

KHB C

Because AB is greater than C, make BE equal to C, that is, let AB exceed C by AE; then AE multiplied fome Number of Times, will be greater than D. Now let AE be multiplied until it exceeds D, and let that Multiple of AE, greater than D,be FG, Maké GH the fame Multiple of EB, and K of C, as FG is of AE. Also, affume L double to D, P 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 neareft lefs than N.

Now because N is the

nearest Multiple of D greater N M P

Ꭰ

than

Ax. 1.

than K, M will not be greater than K, that is, K will not be less than M. And fince FG is the fame Multiple of AE; as GH is of EB; FG fhall be * 1 of this. * the fame Multiple of AE; as FH is of AB; but FG is the fame Multiple of AE, às K is of C; wherefore FH is the fame Multiple of AB, as K is of C; that is, FH, K, are Equimultiples of AB and C. Again, because GH is the fame Multiple of EB, as K is of C, and EB is equal to C; GH fhall bet equal to K. But K is not less than M. Therefore GH fhall 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 neareft leffer than N: Wherefore FH is greater than N. And fo fince F H 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 a greater Ratio to D, than C has to D. I fay, moreover, that D has a greater Ratio to C, than it has to AB; for the fame Conftruction remaining, we demonftrate, as before, 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 a greater Proportion to C, than D hath to B. Wherefore the greater of any two unequal Magnitudes, has a greater Proportion to fome third Magnitude, than the lefs has; and that third Magnitude hath a greater Proportion to the leffer of the two Magnitudes, than it has to the greater.

PROPOSITION IX.

THEOREM.

Magnitudes which have the fame Proportion to one and the fame Magnitude, are equal to one another; and if a Magnitude has the fame Proportion to other Magnitudes, thefe Magnitudes are equal to one another.

IET the Magnitudes A and B have the fame Pro

portion to C. I fay, A is equal to B.

For

For if it was not, A and B would not have the* 8 of this, fame Proportion to the fame Magnitude C; but they have. Therefore A

is equal to B.

Again, let C have the fame Proportion to A as to B. I fay, A is equal to B.

B.

A

For if it be not, C will not have the fame Proportion to A as to B; but it hath: therefore A is neceffarily equal to B. Therefore, Magnitudes that have the fame Proportion to one and the fame Magnitude, are equal to one another; and if a Magnitude has the fame Proportion to other Magnitudes, thefe Magnitudes are equal to one another; which was to be demonftrated.

PROPOSITION X.

THEOREM.

Of Magnitudes having Proportion to the fame Magnitude, that which has the greater Proportion, is the greater Magnitude: And the Magnitude to which the fame bears a greater Proportion, is the leffer Magnitude.

ET A have a greater Proportion to C, than B
has to C. I fay, A is greater than B.

7 of this

A

C+8 of this,

B

For if it be not greater, it will either be equal or lefs. But A is not equal to B, because then both A and B would have the fame Proportion to the fame Magnitude C; but they have not. Therefore A is not equal to B: Neither is it lefs than B; for then A would have † a lefs Proportion to C than B would have; but it hath not a lefs Proportion: Therefore A is not lefs than B. But it has been proved likewife not to be equal to it: Therefore A fhall be greater than B. Again, let C have a greater Proportion to B than to A. I fay, B is lefs than A.

For if it be not lefs, it is greater, or equal. Now *7 of this. B is not equal to A; for then C would have * the fame Proportion to A as to B; but this it has not. Therefore A is not equal to B; neither is B greater than A; for if it was, C would have a lefs Proportion to B than to A; but it has not: Therefore B is not greater than A. But it has also been proved not to be equal to it. Wherefore B fhall be lefs than A. Therefore of Magnitudes having Proportion to the fame Magnitude, that which has the greater Proportion, is the greater Magnitude: And that Magnitude to which the fame bears a greater Proportion, is the leffer Magnitude; which was to be demonftrated.

PROPOSITION XI.

THEOREM.

Proportions that are one and the fame to any third, are alfo the fame to one another.

L

G.

A

B

L.

ET A be to B as C is to D, and C to D as E
to F. I fay, A is to B as E is to F.
For take G, H, K, Equimultiples of A, C, E; and

this.

L, M, N, other Equimultiples of B, D, F. Then because A is to B as C is to D, and there are taken G, H, the Equimultiples of A and C, and L, M any 5 Def of Equimultiples of B, D; if G exceeds L, * then H will exceed M; and if G be equal to L, H will be equal to M; and if lefs, leffer. Again, because as C is to D, fo is E to F; and H and K are taken Equimultiples of C and E; as likewife M, N, any Equimultiples of D, F; if H exceeds M*, then K will exceed N; and if H be equal to M, K will be equal to N; and if lefs, leffer. But if H exceeds M, G will also exceed L; if equal, equal; and if lefs, lefs. Wherefore if G exceeds L, K will alfo exceed N; and if

G