PROPOSITION XVII.

THEOREM.

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

LET

ET the compounded Magnitudes A B, BE, CD, DF, be proportional; that is, let A B be to BE, as CD is to D F. I fay, thefe Magnitudes divided are proportional; viz. is A E is to E B, fo is C F to FD. For let GH, H K, L M, and MN, be Equimultiples of A E EB, CF, and FD; and K X, and N P, any Equimultiples of E B and FD.

K

H

X

Ꮐ

I

P

B

N+ 1 of this.]

D

E+FM+

A

Because G H is the fame Multiple of A E, as HK is of EB; therefore GH* is the fame Multiple of AE as GK is of A B. But GH is the fame Multiple of A E, as L M is of C F. Wherefore GK is the fame Multiple of A B, as LM is of CF. Again, becaufe L M is the fame Multiple of CF, as MN is of FD, L M will be the fame Multiple of CF, as L N is of CD. Therefore GK is the fame Multiple of A B, as L N is of CD. And fo G K and L N will be Equimultiples of AB and CD. Again, becaufe H K is the fame Multiple of EB, as MN is of FD; as likewife K X the fame Multiple of E B, as N P is of FD; the compounded Magnitude HX is + alfo the fame Multiple of E B, as † 2 of sbis. MP is of F D. Wherefore, fince it is, as A B is to BE, fo is CD to DF; and G K and L N are Equimultiples of A B and CD; and alfo H X and M P any Equimultiples of E B and FD; if G K exceeds HX, then NL will exceed MP; and if G K be ‡ Def. 5. equal to H X, then LN will be equal to M P; if lefs, lefs. Now let G K exceed HX; then if HK, which is common, be taken away, GH fhall exceed K X. But when G K exceeds H X, then L N exceeds M P; therefore L N does exceed M P. If M N, which is

common,

common, be taken away, then L M will exceed N P. And fo, if G H exceed K X, then L M will exceed NP. In like manner we demonftrate, if GH be equal to K X, that L M will be equal to N P ; and if lefs, lefs. But GH and L M are Equimultiples of A É and C F and K X and N P are any Equimul*Def. 5. tiples of E B and F D. Whence,* as A E is to E B, fo is CF to F D. 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, fhall be proportional.

LET the divided proportional Magnitudes be A E,

EB, CF, FD; that is, as A E is to F B, fo is
CF to FD. I fay, they are alfo propor-
tional when compounded; viz. as A B is
to B E, fo is CD to D F.

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

Firft, Let it be to a leffer, viz. to GD.
Then, because A B is to B E, as CD is to
DG, compounded Magnitudes are pro-
* 17 of this, portional; and confequently* they will

A

C

E+F+

G+

B D

be proportional when divided. Therefore A E is to E B, as C G is to GD. But (by the Hyp.) as A E is to EB, fo is CF to FD. Wherefore, alfo, as +11 of this CG is to GD, fo + is CF to F D. But the firft CG is greater than the third CF; therefore the fecond 14 of this. D G fhall be greater than the fourth D F. But it is lefs, which is abfurd. Therefore A B is not to BE, as CD is to D G. We demonftrate in the fame manner, that A B to B E is not as CD to a greater than D F. Therefore A B to B E muft 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 demonstrated.

PRO

PROPOSITIO N XIX.
THEORE M.

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.

L

ET the Whole A B be to the Whole CD, as the Part taken away A E, is to the Part taken away CF. Ifay, the Refidue E B is to the Refidue F D, as the Whole A B is to the Whole C D.

Α

† 17 of ibi.

For, because the whole A B is to the Whole CD, 16 of this. as A E is to CF; it shall be alternately, as A B is to AE, fo is CD to CF. Then, becaufe compounded Magnitudes, being proportional, will be + alfo proportional when divided; as BE is to EA, fo is DF to FC: And again, it will be by Alternation, as BE is to DF, fo is É A to FC. But as E A is to E+ C FC, fo (by the Hyp.) is A B to CD. And therefore t the Refidue E B fhall be to the Refidue FD, as the Whole A B to the Whole C D. Wherefore, 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; which was to be demonftrated.

F+11 of this.

B D

Coroll. If four Magnitudes be proportional, they will be likewife converfly proportional. For let A B be to BE, as CD to DF; then (by Alternation,) it fhall be, as A B is to CD, fo is BE to DF. Wherefore, fince the Whole A B is to the Whole CD, as the Part taken away BE is to the Part taken away DF; the Refidue A E to the Refidue CF fhall be as the Whole A B to the Whole C D. And again, (by Inverfion and Alternation) as A B is is to A E, fo is CD to C F. Which is by converse Ratio.

The Demonftration of converfe Ratio, laid down in this Corollary, is only particular. For Alternation (which is used herein) cannot be applied but when the four proportional Magnitudes are all of the fame Kind, as

will

will appear from the 4th and 7th Definitions of this Book. But converfe Ratio may be used when the Terms of the first Ratio are not of the fame Kind with the Terms of the lat

ter.

Therefore, inftead of that, it may not be improper to add this Demonflration following: If four Magnitudes are proportional, they will be fo converfly: For, let A B be to * 17 of this. BE, as CD to D F. And then dividing, it is, * as + Cor. 4. of AE is to BE, fo is CF to DF: And this inverfly, t as BE is to AE, fo is DF to CF; which by compound18 of tbis. ing becomes, tas AB is to A E, fo is CD to CF; which by the 7th Definition, is converfe Ratio: By S. Cunn.

this.

PROPOSITION
THE ORE M.

XX.

If there be three Magnitudes, and others equal to them in Number, which, being taken two and two in each Order, are in the jame Ratio; and 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 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.

ETA, B, C, be three Magnitudes, and

LE

D, E, F, others equal to them in Num-
ber, which being 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
DD is alfo greater than the fixth F. And if
A be equal to C, D is equal to F. But if
A be lefs than C, D is lefs than F.

is

ABC

For, because A is greater than C, and B any other Magnitude; and fince a greater 8 of this. Magnitude hath greater Proportion to the fame Magnitude than a leffer hath; A will have a greater Proportion to B, than C hath to B. But as A is to B, fo is D to E+; therefore D hath a greater Proportion to E, than C hath to B. Now inverfly, as C is to B, fo is F to E. Therefore alfo D will have a greater Propor

† By Hyp.

DEF

tion to E, than F has to E. But of Magnitudes have ing Proportion to the fame Magnitude, that which has the greater Proportion is the greater Magnitude. † 10 of this. Therefore D is greater than F. In the fame manner we demonstrate, if A be equal to C, then D will be alfo equal to F; and if A be less than C, then D will be less than F. Therefore, 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; 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; which was to be demonstrated. PROPOSITION XXI.

THE ORE M.

If there be three Magnitudes, and others equal to them in Number, whic, thaken 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.

L

ET three Magnitudes, A, B, C, be proportional; and others, D, E, F, equal to them in Number. Let their Analogy likewife 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 D is less than F.

For, fince A is greater than C, and B is 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; whence E has a greater Proportion to F, than Chath to B Now 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

ABC

DEF

of this.

bears