Therefore F is lefs than D; and fo D fhall be greater than F. After the fame manner we demonftrate, if A be equal to C, D will be alfo equal to F; and if A be lefs than C, D will also be lefs than F. If, therefore, there are three Magnitudes, and others equal to them in Number, which taken two and two, are in the fame Proportion, and the Proportion be perturbate; if the firft Magnitude be greater than the third, then the fourth will be greater than the fixth; but if the first be equal to the third, then is the fourth equal to the fixth; if lefs, lefs; which was to be demonftrated.

PROPOSITION XXII.

THEOREM.

If there be any Number of Magnitudes, and others equal to them in Number, which taken two and two, are in the fame Proportion; then they shall be in the fame Proportion by Equality.

ET there be any number of Magnitudes A, B, C, and others D, E, F, equal to them in Number, which taken two and two, are in the fame Proportion, that is, as A is to B, fo is D to E, and as B is to C, fo is E to F. I fay, they

are also proportional by Equality, viz. as A is to C, fo is D to F.

For let G, H, be Equimultiples of A,D; and K, L, any Equimultiples of B, E; and

likewife M, N, any Equi- A B C D E F
multiples of C, F. Then be-

caufe A is to B, as D is to E, GK MHL N
and G, H, are Equimultiples
of A, D, and K, L, Equi-
multiples of B, E; it fhall
be* as G is to K, fo is H
to L. For the fame Reason
alfo it will be, as K is to M,
fo is L to N. And fince
there are three Magnitudes,

G, K, M, and others H, L, N, equal to them in Num

4 of this,

ber, which being taken two and two in each Order, 20 of this. are in the fame Proportion. If G exceeds M, * H will exceed N; if G be equal to M, then H fhall be equal to N; and if G be lefs than M, H fhall be lefs than N. But G, H, are Equimultiples of A, D, and M, N, any other Equimultiples of C and F. Whence as A is to C, fo fhall + D be to F. Therefore, if there be any Number of Magnitudes, and others equal to them in Number, which taken two and two, are in the fame Proportion, then they shall be in the fame Proportion by Equality; which was to be demonstrated.

+ Def. 5. of this.

PROPOSITION XXIII.

THE ORE M.

If there be three Magnitudes, and others equal to them in Number, which, taken two and two, are in the fame Proportion; and if their Analogy be perturbate, then fhall they be alfo in the Jame Proportion by Equality.

LE

and o

ET there be three Mags
nitudes A, B, C,
thers equal to them in Num-
ber D, E, F, which, taken
two and two, are in the fame
Proportion, and their Analo-
gy be perturbate, that is, as A
is to B, fo is E to F; and as
B is to C, fo is D to E. I fay,
as A is to C, fo is D to F.

For let G, H, L, be Equi-
multiples of A, B, D, and
K, M, N, any Equimulti-
ples of C, E, F.

Then becaufe G, H, are
Equimultiples of A and B,

and fince Parts have the fame

Proportion as their like Multiples when taken corre* 15 of this. fpondently, it fhall be as A is to B, fo is G to H; and by the fame Reafon, as E is to F, fo is M to N. † 11 of this. But A is to B as E is to F. Therefore, † as G is to H, fo is M to N. Again, becaufe B is to C, as D

is to E, and H, L are Equimultiples of B and D, as likewife K, M any Equimultiples of C, E; it fhall be as H to K, fo is L to M. But it has been alfo proved, that as G is to H, fo is M to N. Therefore, because three Magnitudes, G, H, K, and others, L, M, N, equal to them, in Number, which taken two and two are in the fame Proportion, and their Analogy is perturbate; then if G exceeds K, alfo L* will exceed * 21 of this. N; and if G be equal to K, then L will be equal to N; and if G be less than K, L will likewife be lefs: than N. But G, L are Equimultiples of A, D; and K, N Equimultiples of C, F. Therefore, as A is to C, fo fhall D be to F. Wherefore, if there be three Magnitudes, and others equal to them in Number, which taken two and two are in the fame Proportion; and if their Analogy be perturbate, then shall they be alfo in the fame Proportion by Equality; which was to be demonftrated.

PROPOSITION XXIV.

THEORE M.

If the first Magnitude has the fame Proportion to
the fecond, as the third to the fourth; and if the
fifth has the fame Proportion to the fecond, as the
fixth bas to the fourth, then fhall the first, com-
1 pounded with the fifth,
have the fame Proportion
to the Jecond, as the third
compounded with the fixth
bas to the fourth.

G

H