If Magnitudes compounded, are proportional; they
Shall also be proportional when divided.

ET the compounded Magnitudes AB, BE, CD, DF, be proportional, that is, let AB be to BE as CD is to DF. I fay, thefe Magnitudes divided are proportional, viz. as AE is to EB, fo is CF to FD. For let GH, HK, LM, MN, be Equimultiples of AE, EB, CF, FD, and KX, NP, any Equimultiples of EB, FD.

X

P

K

F

G

A C L

H

as L M is of CF. Again, because L M is the fame Multiple of CF as MN is of FD, LM will be the *the fame Multiple of CF, as LN is of CD, Therefore GK is the fame. Multiple of AB, as LN is of CD. And fo GK, LN will be Equimultiples of AB, CD. Again, because HK is the fame Multiple of EB, as MN is of FD; as likewife KX the fame Multiple of EB, as NP is of FD, the taf this. compounded Magnitude HX is + alfo the fame Multiple of EB, as MP is of F D. Wherefore, fince it is as AB is to BE, fo is CD to DF; and GK, LN are Equimultiples of AB, CD; and alfo HX, MP any Equimultiples of EB, FD: If G K exceeds HX, then LN will exceed MP: and if GK be equal to HX, then LN will be equal to MP; if lefs, lefs. Now let G K exceed HX; then if HK, which is common, be taken away, GH fhall exceed KX, But when GK exceeds HX, then LN exceeds MP; therefore LN does exceed MP. If MN, which is

Def. 5.

common,

common, be taken away, then LM will exceed NP. And fo if GH exceeds K X, then L M will exceed NP. In like manner we demonftrate, if GH be equal to KX, that L M will be equal to NP; and if lefs, lefs. But GH, LM, are Equimultiples of AE, CF; and KX, NP, are any Equimultiples of EB, FD. Whence, as AE is to EB, fo CF to Def. 5. FD. Therefore, if Magnitudes compounded, are proportional; they shall also be proportional when divided; which was to be demonftrated.

PROPOSITION XVIII.

THEOREM.

If Magnitudes divided be proportional, the fame alfo being compounded, shall be proportional.

L

ET the divided proportional Magnitudes be A E, AE, EB, CF, FD; that is, as AE is to EB, fo is CF to FD. I fay, they are alfo proportional when compounded, viz. as AB is to BE, fo is CD to DF.

For if AB be not to BE, as CD is to DF, AB fhall be to BE as CD is to a Magnitude, either greater or less E than FD.

A

1 B

C

F

G

D

*17 of this.

Firft, let it be to a leffer, viz. to GD.: Then because AB is to BE as CD is to DG, compounded Magnitudes are proportional; and confequently* they will be proportional when divided. Therefore A E is to EB as CG is to GD. But (by the Hyp.) as AE is to EB, fo is CF to FD. Wherefore alfo as CG is to GD, fotis CF to FD. But the firft CG is greater + 11 of this. than the third CF; therefore the fecond DG fhall be

greater than the fourth DF. But it is lefs, which is † 14 of this. abfurd. Therefore AB is not to BE, as CD is to DG. We demonftrate in the fame manner, that AB to BE, is not as CD to a greater than DF. Therefore AB to BE, must neceffarily be as CD is to DF. And fo if Magnitudes divided be proportional, they will also be proportional when compounded; which was to be de monstrated.

PRO

PROPOSITION XIX.

THEOREM.

If the whole be to the whole, as a Part taken away is to a Part taken away; then fhall the Refidue be to the Refidue, as the Whole is to the Whole.

ET the whole AB be to the whole CD, as the Part taken away AE is to the Part_taken away CF. I fay, the Refidue E B is to the Refidue FD, as the whole AB is to the whole CD.

For because the whole AB is to the whole CD, *16 of this. as AE is to CF; it shall be alternately as AB is to AE, fo is CD to CF. Then because compounded Magnitudes, being proportional, will be +17 of this. + alfo proportional when divided.

,

As

E

A

C

F

B D

BE is to EA, fo is DF to FC: And
again, it will be by Alternation, as BE to
DF, fo is EA to FC. But as EA to FC,
fo (by the Hyp.) is AB to CD. And
therefore the Refidue EB, fhall be to the
Refidue FD, as the whole AB to the whole
CD. Wherefore, if the Whole be to the
Whole, as a Part taken away is to a Part
taken away; then shall the Refidue be to the Refidue, as the
whole is to the whole; which was to be demonftrated.
Coroll. If four Magnitudes be proportional, they will
be likewise converfely proportional. For let AB
be to BE, as CD to DF; then (by Alternation)
it fhall be as AB is tó CD, fo is BE to DF.
Wherefore fince the whole AB is to the whole
CD, as the Part taken away BE is to the Part
taken away DF; the Refidue AE to the Refidue
CF, fhall be as the whole A B to the whole CD.
And again (by Inverfion and Alternation) as AB is to
AE, fo is CD to CF. Which is by converse Ratio.
The Demonftration of converfe Ratio, laid down in
this Corollary, is only particular. For Alternation
(which is ufed herein) cannot be applied but when the
four propertional Magnitudes are all of the fame Kind,
as will appear from the 4th and 17th Definitions of this
Book. But converfe Ratio may be used when the Terms

of

of the firft Ratio are not of the fame Kind with the Terms of the latter. Therefore inftead of that, it may not be improper to add this Demonftration following: If four Magnitudes are proportional, they will be fo converfely: For let AB be to BE, as CD to DF. And then dividing it is, as AE is to BE, fo is CF to DF: And this inversely is, as BE is to AE, fo is DF to CF; which by compounding becomes, as AB is to AE, fo is CD to CF; which by the 17th Definition is converfe Ratio; By S. Cunn.

PROPOSITION XX.

THEOREM.

If there be three Magnitudes, and others equal to them in Number, which being taken two and two in each Order, are in the fame Ratio. And 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 the fourth will be equal to the fixth; and if the first be less than the third, the fourth will be less than the fixth.

B

ET A, B, C, be three Magnitudes, and D, E, F, others equal to them in Number, taken two and two in each Order, are in the fame Proportion, viz. let A be to B, as D is to E, and B to C, as E to F; and let the firft Magnitude A be greater than the third C. I fay the fourth Dis also greater than the fixth F. And if A be equal to C, D is equal to F. But if A be less than C, D is lefs than F. For because A is greater than C, and B is any other Magnitude; and fince a greater Magnitude hath a greater Proportion 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, fo is D to D E F E; and inverfly, as C is to B, fo is F to E.

*

Therefore alfo D will have a greater Proportion to E, than F has to E. But of Magnitudes having Proportion to the fame Magnitude, that which has the greater

Proportion

#8 of this.

* 10 of this. Proportion is * the greater Magnitude. Therefor is greater than F. In the fame manner we dem ftrate, if A be equal to C, then D will be alfo 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 fame Ratio. If firft Magnitude be greater than the third, then the fou will be greater than the fixth: But if the first be eq to the third, then the fourth will be equal to the fixt and if the first be less than the third, the fourth will lefs than the fixth; which was to be demonstrated.

PROPOSITION XXI.

THEOREM.

If there be three Magnitudes, and others equal t them in number, which taken two and two, ar in the fame Proportion, and the Proportion b 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, less.

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 fay, the fourth D is also greater than the fixth F. And if A be equal to C, then D is equal to F; but if A be less than C, then Dis less than F.

For fince A is greater thanC, and B is of ibis. fome other Magnitude, A will have a greater Proportion to B, than C has to B. But as A is to B, fo is E to F; and inverfly, as C is to B, fo is E to D. Wherefore alfo E fhall have a greater Proportion to F than E to D. But that Magnitude to which the fame Magnitude has a greater + 10 of this. Proportion, is † the leffer Magnitude,

DE F

Therefore