of the first Ratio are not of the same Kind with the Terms of the latter. Therefore instead of that, it may not be improper to add this Demonstration following : If four Magnitudes are proportional, they will be so conversely : For let A B be to B E, as CD a DF. And then dividing it is, as AE is to BE, so is CF to DF: And this inversely is, as BE is to AE, so is DF to CF; which by compounding becomes, as AB is to AE, fo is CD to CF ; which by the 17th Definition is converse Ratio ; By S. Cunn.

PROPOSITION XX.

THE OR EM.
If there be three Magnitudes, and others equal to

them in Number, which being taken two and two
in each Order, are in the same Ratio. And if the
first Magnitude be greater than the third, then the
fourth will be greater than the sixth: But if the
first be equal to the third, then the fourth will be
equal to the sixth; and if the first be less than the
third, the fourth will be less than the sixtb.

ET A, B, C, be three Magnitudes,

in Number, taken two and two in each Order, are in the same Proportion, viz. let A be to B, as D is to E, and B to C, as E to F; and let the first Magnitude A be greater

than the third C. I say the fourth D is also greater than the sixth F. And if Á B C A be equal to C, D is equal to F. But if A be less than C, D is less than F.

For because A is greater than C, and B is any other Magnitude; and fince a greater Magnitude hath * a greater Proportion

*8 of this. to the fame Magnitude than a leffer hath, A will have a greater Proportion to B, than C to B. But as A is to B, so is D to D E F E; and inversly, as C is to B, so is F to E. Therefore allo D will have a greater Proportion to E, than F has to E. But of Magnitudes having Proportion to the same Magnitude, that which has the greater

Proportion

* 10 of tbis. Proportion is * the greater Magnitude. Therefor

is greater than F. In the same manner we dem strate, if A be equal to C, then D will be also e to F, and if A be less than C, then D will be than F. Therefore, if there be three Magnitudes, others equal to them in Number: which being taken

and two in each Order, are in the same Ratio. If firft Magnitude be greater than the third, then the fou will be greater than the sixth : But if the first becq to the third, then the fourth will be equal to the fixt and if the first be less than the third, the fourth will less than the fixth; which was to be demonstrated.

THEOREM.
If there be three Magnitudes, and others equal t

them in number, which taken two and two, ar
in the same Proportion, and the Proportion bu
perturbate ; if the first Magnitude be greater than
ibe 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 sixth; if less, less.
L

ET three Magnitudes, A, B, C, be proportional;

and others D, E, F, equal to them in Number, Let their Analogy likewise be perturbate, viz. as A is to B, fo is E to F; and as B is to C, fo is D to E; if the first Magnitude A be greater than the third C. I say, the fourth D is also greater than the sixth F. And if A be equal to C, then D is equal to F; but if A be less than C, then is less than F.

A в с For fince A is greater thanC, and B is * 8 of ibis. some other Magnitude, A will have *

greater Proportion to B, than C has to B.
But as A is to B, so is E to F; and in-
versly, as C is to B, fo is E to D. Where-
fore also E shall have a greater Proportion
to F than E to D. But that Magnitude to

which the fame Magnitude has a greater D E F + 10 of ibis. Proportion, is f the lefser Magnitude.

Therefore

а

Therefore F is less than D; and so D shall be greater
than F. After the same manner we demonstrate, if
A be equal to C, D will be also equal to F; and
if A be less than C, D will also be less 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 first 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
fixtb; if less, less; which was to be demonstrated.
PROPOSITION XXII.

THEORE M.
If there be any Number of Magnitudes, and others

equal to them in Number, which taken two and
two, are in the same Proportion ; then they shall
be in the same Proportion by Equality.
ET there be
any

, C, and others D, E, F, equal to them in Number, which taken two and two, are in the same Proportion, that is, as A is to B, fo is D to E, and as B is to C,

I say, they are also proportional by Equality, viz. as A is to C, lo is

so is E to F.

D to F.

For let G, H, be Equimultiples of A,D; and K, L, any Equimultiples of B, E, and likewise M, N, any Equi- A B C D E F multiples of C, É. Then because A is to B, as D is to E, GKM HLN and G, H, are Equimultiples of A, D, and K, L, Equimultiples of B, E; it Thall be * as G is to K, fo is H to L. For the same Reason also it will be, as K is to M, fo is L to N.

And since there are three Magnitudes, G, K, M, and others H, L, N, equal to them in Num

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