Book V. BA is to AE, as DC to CF. and because if magni-
tudes taken jointly be proportionals, they are alfo pro-
b. 17.5. portionals b when taken feparately; therefore as BE
is to EA, fo is DF to FC; and alternately, as BE is
to DF, fo is EA to FC. but as AE to CF, fo, by the E-
Hypothefis, is AB to CD; therefore alfo BE the re-
mainder shall be to the remainder DF, as the whole
AB to the whole CD. Wherefore if the whole, &c.
Q. E. D.

A

F

C

B D

COR. If the whole be to the whole, as a magnitude taken from the firft is to a magnitude taken from the other; the remainder likewife is to the remainder, as the magnitude taken from the first to that taken from the other. the Demonftration is contained in the preceeding.

PROP. E. THEOR.

F four magnitudes be proportionals, they are alfo proportionals by converfion, that is, the firft is to its excefs above the fécond, as the third to its excefs above the fourth.

A

a. 17. 5.

b. B. s.

c. 18. 5.

Let AB be to BE, as CD to DF; then BA is to
AE, as DC to CF.

Because AB is to BE, as CD to DF, by division *, E
AE is to EB, as CF to FD; and by inversion ↳, BE
is to EA, as DF to FC. Wherefore, by Compofition,
BA is to AE, as DC is to CF. If therefore four, &c.
Q. E. D.

F

C

Sce N.

B D

IF

PROP. XX. THEOR.

F there be three magnitudes, and other three, which taken two and two have the fame ratio; if the first be greater than the third, the fourth fhall be greater than the fixth; and if equal, equal; and if lefs, less.

Let

Let A, B, C be three magnitudes, and D, E, F other three, Book V. which taken two and two have the fame ratio. viz. as A is to B, fo is D to E; and as B to C, fo is E to F. If A be greater than C, D fhall be greater than F; and if equal, equal; and if lefs, less,

Because A is greater than C, and B is any other magnitude,and that the greater has to the fame magnitude a greater ratio than the less has to it; therefore A has to B a greater ratio than C has to B. but as D is to E, fo is A to B; therefore b D has to E A B a greater ratio than C to B. and becaufe B is to C, D E as E to F, by inverfion, C is to B, as F is to E; and D was shewn to have to E a greater ratio than C to B; therefore D has to E a greater ratio than F to E. but the magnitude which has a greater ratio

a. 8. S

C b. 13. 5.

F

c. Cor. 13:5.

than another to the fame magnitude, is the greater of the two. D d. 1o. 5. is therefore greater than F.

Secondly, Let A be equal to C; D fhall be equal to F. because

A and C are equal to one another, A

is to B, as C is to B. but A is to

B, as D to E; and C is to B, as F
to E; wherefore D is to E, as F to
Ef; and therefore D is equal to F %.
Next, Let A be less than C; D A B C
shall be less than F. for C is greater DEF

than A, and, as was fhewn in the first
cafe, C is to B, as F to E, and in like
manner B is to A, as E to D; therefore
F is greater than D, by the firft cafe;
and therefore D is lefs than F. There-
fore if there be three, &c. Q. E. D.

PROP. XXI. THEOR.

c. 7. 5.

f. 11. 5.

g. 9.5.

АВС
DEF

IF there be three magnitudes, and other three, which

have the fame ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth fhall be greater than the fixth; and if equal, equal; and if lefs, lefs.

Let

Book V.

8. s.

b. 13. s.

Let A, B, C be three magnitudes, and D, E, F other three, which have the fame ratio taken two and two, but in a cross order, viz. as A is to B, fo is E to F, and as B is to C, fo is D to E. If A be greater than C, D fhall be greater than F; and if equal, equal; and if lefs, lefs.

a

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C has to B. but as E to F, fo is A to B; therefore b E has to F a greater ratio than C to B. and be

caufe B is to C, as D to E, by inverfion, C is to A B C B, as E to D. and E was fhewn to have to Fa D E F greater ratio than C to B; therefore E has to Fa

c.Cor.13.5. greater ratio than E to D.

but the magnitude to

which the fame has a greater ratio than it has to

J. 10. s. another, is the leffer of the two. F therefore is

e. 7. 5.

lefs than D; that is, D is greater than F.

e

Secondly, Let A be equal to C; D shall be equal to F. Because A and C are equal, A is to B, as C is to B. but A is to B, as E to F; and C is to B, as E to D;

f. 11. 5. wherefore E is to F, as E to Df; 9. 5. and therefore D is equal to F §.

See N.

Next, Let A be less than C;

А В С

D shall be less than F. for C is AB Ċ
greater than A, and, as was fhewn,

C is to B, as E to D, and in like DEF
manner B is to A, as F to E; there-

fore F is greater than D, by cafe
firft; and therefore D is lefs than
F. Therefore if there be three,
&c. Q. E. D.

PROP. XXII. THEOR.

D

IF there be any number of magnitudes, and as many

others, which taken two and two in order have the

mag

fame ratio; the first shall have to the last of the first nitudes the fame ratio which the first of the others has to the laft. N. B. This is ufually cited by the words "ex "aequali, or, ex aequo."

First,

First, Let there be three magnitudes A, B, C, and as many others Book V. D, E, F, which taken two and two have the fame ratio, that is fuch that A is to B, as D to E; and as B is to C, fo is E to F. A fhall be to C, as D to F.

a

A B C
G KM

DEF

HLN 2.45

Take of A and D any equimultiples whatever G and H; and of B and E any equimultiples whatever K and L; and of C and F any whatever M and N. then because A is to B, as D to E, and that G, H are equimultiples of A, D, and K, L equimultiples of B, E; as G is to K, fo is H to L. for the fame reason K is to M, as L to N. and because there are three magnitudes G, K, M, and other three H, L, N, which two and two have the fame ratio; if G be greater than M, H is greater than N; and if equal, equal; and if

lefs, lefs b. and G, H are any equi

multiples whatever of A, D, and M, N are any equimultiples whatever of C, F. therefore as A is to C, fo is D to F.

C

Next, Let there be four magnitudes A, B, C, D, and other four

E, F, G, H, which two and two have the fame ratio, viz. as A is to B, fo is E to F; and as B to C, fo F to G; and as C to D, fo G to H. fhall be to D, as E to H.

A

A. B. C. D.

E. F. G. H.

Because A, B, C are three magnitudes, and E, F, G other three, which taken two and two have the fame ratio; by the foregoing cafe, A is to C, as E to G. but C is to D, as G is to H; wherefore again, by the first case, A is to D, as E to H. and fo on, whatever be the number of magnitudes, Therefore if there be any number, &c. Q. E. D.

b. 20. 5.

c. 5. Def. §.

PROP.

Book V.

See N.

IF

PROP. XXIII. THEOR.

F there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the fame ratio; the first shall have to the last of the firft magnitudes the fame ratio which the firft of the others has to the last. N. B. This is ufually cited by the words "ex aequali in proportione perturbata, or, ex aequo per"turbate."

First, Let there be three magnitudes A, B, C, and other three D, E, F, which taken two and two in a crofs order have the fame ratio, that is such that A is to B, as E to F; and as B is to C, fo is D to E. A is to C, as D to F.

Take of A, B, D any equimultiples whatever G, H, K; and of C, E, F any equimultiples whatever L, M, N. and because G, H are equimultiples of A, B, and that magnitudes have the fame ratio

a. 15. 5. which their equimultiples have a ;
as A is to B, fo is G to H. and for

the fame reason, as E is to F, fo is M
to N. but as A is to B, fo is E to

ARE REN

F; as therefore G is H, fo is M to AB C DEF b. 11. 5. Nb. and becaufe as B is to C, fo is G H L KMN

D to E, and that H, K are equi-
multiples of B, D, and L, M of C,
E; as H is to L, fo is K to M.
and it has been fhewn that G is to
H, as M to N. then because there
are three magnitudes G, H, L, and
other three K, M, N which have
the fame ratio taken two and two

in a cross order; if G be greater

d. a1. s. than L, K is greater than N; and if equal, equal; and if lefs, lefs".

and G, K are any equimultiples whatever of A, D; and L, N any whatever of C, F; as therefore A is to C, fo is D to F.

Next,