away CH, which is common; then the Refidue KC is equal to the Refidue HD. But K C is equal to F. Therefore HD is equal to F; and fo G B fhall be equal to E, and HD to F.

In like manner we demonftrate, if G B was â 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.

PROPOSITION VII.
THE ORE M.

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

DA

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 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.

K

EBC F

I fay,

* Def. 5.

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 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 lefs than E. But F is a Multiple of C; and D and E, any other Equimultiples of A and B ; 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.

THEOREM.

The greater of any two unequal Magnitudes has 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.

LE

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

F

Because A B is greater than C, make B E equal to C, that is, let A B exceed C by A E; then AE, multiplied fome Number of Times, will be greater than D. Now let A E be multiplied until it exceeds D, and let that Multiple of A E greater than D be F Ġ.

Make

A

G+E+

GH the fame Multiple of K HB
E B, and K of C, as FG is

of A E. Alfo, affume L double
to D, M triple, and fo on, un-
til fuch a Multiple of D is had,
as is the neareft greater than K;
let this be N, and let M be a
Multiple of D, the neareft lefs
than N.

Now,

because N is
N is the

nearest Multiple of D greater N M L

than

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 A E, as G H is of EB; FG fhall be of this. the fame Multiple of A E, as FH is of A B; but F G is the fame Multiple of A E, as K is of C: Wherefore F H is the fame Multiple of AB as K is of C; that is, F H and K are Equimultiples of A B and C. Again, because G H is the fame Multiple of E B, as K is of C, and E B is equal to C; GH fhall be + equal † Ar. 1. to K. But K is not less than M: Therefore G H fhall not be less than M. But F G is greater than D: Therefore the whole F H 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 F H 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 Multiple of D; therefore A B will have a greater Ratio Def. 7. to D, than C has to D. I fay, moreover, that D has a greater Ratio to C than it has to A B: For the fame Conftruction remaining, we demonftrate, as before, that N exceeds K, but not FH. And N is a Multiple of D, and F H and K are Equimultiples of A B and C. Therefore D has a greater Proportion to C, than D hath to A. B. Wherefore, the greater of any two unequal Magnitudes has a greater Proportion to fome third Magnitude than the lefs has; and the third Magnitude bath a greater Proportion to the leffer of the two Magnitudes than it has to the greater; which was to be demonftrated.

PROPOSITION IX.

THE ORE M.

Magnitudes which have the fame Proportion to one
and the fame Magnitude, are equal to one ano-
ther; and if a Magnitude bas the fame Proper-
tion to other Magnitudes, thefe Magnitudes are
equal to one another.

ET the Magnitudes A and B have the fame Pro-
ortion to C. I fav, A is equal to B.

K 2

For,

For, if it was not, A and B would not have the 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 demonstrated.

PROPOSITION X.

THEORE M.

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

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

For, if it be not greater, it will either be equal or lefs. But A is not equal to B, because

*7 of this. then both A and B would have * the

A

fame Proportion to the Magnitude C; but they have not. Therefore A is not equal to B: Neither is it lefs than t8 of this. B; for then A would have + a less Proportion to C, than B would have; but it hath not a lefs Proportion. Therefore A is not less than B. But it has been proved likewife not to be equal to it: Therefore A fhall be greater than P. Again, let C have a greater Proportion to B than to A. I fay, B is lefs than A.

+

B

For,

the *7,of this.

For, if it be not lefs, it is greater or equal. Now, B is not equal to A, for then C would have 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 Propor- † 8 of tbis. tion 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 less 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 demonstrated.

PROPOSITION XI.
PROBLEM.

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

LETA 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.

G.

A.

B.

L

For, take G, H, and K, Equimultiples of A, C, and

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