PROP. XVI. THEOR.

If four magnitudes all of the fame kind be propors tional, they will be alternately proportional.

Let the four magnitudes A, B, C, D all of the fame kind be proportional; viz. let A be to B, as c is to D: I fay they are also alternately proportional; that is, as A is tỏ c, fo is B to D.

For take any equimultiples E, F of A, B, and any other equimultiples G, H of C, D.

Then because E is the fame multiple of A, that F is of E, and the parts of magnitudes compared to one another

[by 15.5.] have the fame ratio, that their equimultiples have to one another; A will be to B, as E is to F. But as A is to B, fo is c to D: Wherefore [by ii.5.] as c is to D, so is E to F. Again, because G, H are equimultiples of c and D: c will be to D, as G is to H, But as c is to D, fo is E to F: Wherefore [by II. 5.] as E is to F, fo is G to H. But if four magnitudes be proportional, and the first be greater than the third; [by 14. EABF GCDH 5.] the fecond will be greater than the fourth: If the first bé

equal to the third, the second will be equal to the fourth: and if the first be less than the third, the fecond will be less than the fourth: If therefore E exceeds G; F will also exceed H; if E be equal to G, F will be equal to #, and if E be less than G, F will be less than H. But E, F are any equimultiples of A, B, and G, H any other equimultiples of C, D: Therefore [by 5. def. 5.] as A is to c, fa will B be to D.

If therefore four magnitudes all of the fame kind be proportional; they will be alfo alternately proportional: Which was to be demonftrated.

* Either lines, fuperficies, or folids. The thirteenth propofition of the feventh book answers to this in numbers.

PROP

PROP. XVII. THEOR.

If four magnitudes taken jointly be proportional; they will be alfo proportional when taken feparately*

Let the four magnitudes AB, BE, CD, DF taken jointly be proportional, and let A B be to B E, as CD is to D F: I fay they will be alfo proportional when taken feperately; that is, as A E is to E C, fo is CF to F D.

P

N

For take any equimultiples GH, HK, LM, MN of AE, EB, CF, FD, and any other equimultiples KX, NP of EB and FD. Then because GH is the fame multiple of A E, that HK is of EB; [by 1. 5.] G H will be the fame multiple of AE, that GK is of AB. But GH is the fame multiple of A E, that L M is of CF: Therefore G K will be the fame multiple of A B that L M is of C F. Again, because LM is the fame multiple of CF, that MN is of F D; LM will be the fame multiple of c F, that L N is of CD. But LM was the fame multiple of CF, that G K is of a B: Therefore G K is the fame multiple of A B, that L N is of CD: Wherefore G K, L N will be equimultiples of AB, C D. Again, because HK is the fame multiple of E B, that MN is of FD; but KX is the fame multiple of E B, that NP is of FD: Allo [by 2.5.] the compounded magnitude HX will be the fame multiple of E B, that MP is of F D. But fince as AB is to BE, fo is CD to DF, and there are taken any equimultiples G K, L N of AB, CD, and any other equimultiples H X, MP of E B, FD: Therefore [by 5. def. 5.] if G K exceeds H X ; L N will exceed MP: if GK be equal to HX, L N will be equal to M P ; and if GK be lefs than HX; LN

K

X

I

M

H

will be less than M P, Therefore G

let G K exceed H x; and taking

D

B

E

F

* Or thus: If two magnitudes (put or added) together have to one of them the fame ratio which two others taken together have to one of these last, the remaining one of the first two hall have to the other, the fame ratio which the remaining one of the laft two has to the other of them.

Q3

away

Book V away H K which is common, G H will exceed K x. But if GK exceeds HX; LN will exceed MP so that LN exceeds MP; and M N common to both being taken away, LM will exceed NP: wherefore if GH exceeds K X; LM will exceed N P. In like manner we demonstrate that if GH be equal to KX, LM is equal to NP; and if GH be lefs than KX; L M will alfo be less than N P. But & H, LM are any equimultiples of A E, C F, and K X, N P any other equimultiples of E B, FD: Therefore [by 5. def. 5.] as A E is to E B, fo will cr be to F D.

If therefore magnitudes when taken jointly are proportional; they will be alfo proportional when taken seperately. Which was to be demonstrated.

PROP. XVIII. THEOR.

If four magnitudes when taken feperately be propor→ tional, they will also be proportional taken jointly*. Let the four feperate magnitudes A E, E B, C F, F D be proportionals, viz. as A E is to E B, fo is C F, to F D: I fay they are also proportional when taken jointly; that is, as A B is to B E, fo is CD to F D.

G

F

D

E

B

For if it be not as AB is to B E, fo is CD to F D it will be as A B is to BD, fo is cp, to fome magnitude either leffer than F D, or greater than it.

Firft let it be to fome magnitude D G which is less than F D. Then because it is as AB is to B E, fo is CD to D G, these compounded magnitudes are proportional: Therefore [by 17. 5. they will be alfo proportional when they are divided: Wherefore it is as AE to EB, fo is CG to G D. But [by fuppofition] as AE is to E B, fo is CF to FD: Wherefore [by 11. 5.] as CG is to GD, fo is CF to A FD. But the firft magnitude CG is greater than the third CF: Therefore [by 14. 5.] the fecond G D will be greater than the fourth F D. But it is lefs too; which is abfurd. Therefore it is not as A B is to BE, fo is C D to a magnitude that is lefs than E D. After the fame manner we demonftrate that it cannot be as AB

C

Or thus: If the first of four magnitudes be to the fecoud as the third is to the fourth, the firft and fecond put together will be to the fecond, as the third and fourth added together is *o the fourth

is

is to B E, fo is CD to a magnitude greater than FD. Therefore it is as A B is to B E, fo C D to F d.

Wherefore if four magnitudes when taken feperately be proportional, they will also be proportional when they are taken conjointly. Which was to be demonftrated.

PROP. XIX. THEOR.

If one magnitude be to another, as a part taken away of the one is to a part taken away from the other; then shall the remaining part of the one magnitude be to the remaining part of the other, as one of the whole magnitudes is to the other whole one.

For let it be as one magnitude A B is to another c D, fa is the part A E of the one, to the part C F of the other: Then I say the remainder E B of the one is to the remain. der FD of the other, as the magnitude AB is to the magnitude C D.

B

For because it is as the whole A B is to the whole c D, fo is A E to CF; and alternately [by 16. 5.] as BA is to A E, fo is DC to CF. And because if magnitudes when taken joindy are proportionals, they will be [by 17.5.] proportio- D nal when taken feperately: Therefore as BE is to E A, fo will DF be to CF; and again, alternately, as B E is to DF, fo is EA to FC. But as A is to CF, fo [by suppofition] is AB to CD. Therefore [by 11. 5.] the remaining part E B will be to the remaining part F D, as the whole A B is to the whole C D.

F

Wherefore if one magnitude be to an- C other, as a part taken away from the one

E

A.

is to a part taken away from the other; then fhall the remaining part of one of the whole magnitudes be to the remaining part of the other, as one whole magnitude is to the other. Which was to be demonftrated.

Corollary. And because it is demonftrated [by 16.5.] as A B is to CD, fo is E B to F D : if it be alternately as

* This corallary is very corrupt; nor can it be corrected from any of the ancient copies: I have therefore altered the Q4

Book V. AB is to BE, fo is CD to D F, that is, conjoint mag nitudes proportional. But it has been demonftrated [by 16. and 19. 5.] that A B is to A E, as CD is to CF, which [by 17. def. 5.] is converse ratio. Therefore it is manifeft from hence if conjoint magnitudes be proportional, that they will also be proportional conversely.

PROP. XX. THEOR.

If there be three magnitudes and other three which taken two and two have the fame ratio. And if the first be greater than the third: then will the fourth be greater than the fixth: If the first be equal to the third, 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.

Let there be three magnitudes A, B, C, and D, E, F other three, which taken two and two are in the fame ratio, and let A be to B, as D is to E, and as B is to C, fo is E to F, and A is greater than I fay also that D is greater than F: If A be equal to C, D will be equal to F: and if A be less than C, D will be less than F.

For because A is greater than c, and B is any other magnitude, and [by 8. 5.] the greater of two magnitudes has a greater ratio to the fame magnitude than the leffer; the ratio of A to B, will be greater than the ratio of c to B. But as A is to B, fo is D to E, and [by inverfion] as c is to B, fo is F to E: Therefore D has a A B C D E F greater ratio to E, than F has to E. But of two magnitudes which have a ratio to the fame magnitude, that which has the greatest ratio is the greater magnitude [by 10. 5.]: Therefore D is greater than F : tranflation to preserve the fenfe. But the following is the most legitimate demonftration of converfe ratio. If A B be to BE, as CD is to D F, it will be [by dividing] as A E is to B E, fo is C F to D F, and [by inversion] as B E is to A E, fo is D F to CF; and [by compounding or addition] it will be as a B is to A fo is cp to CF, which is converfe ratio.

After