Book V.

a. 16. 5. b. 17. 5.

a. 17. 5.

b. B. 5.

c. 18. 5.

b

Because AB is to CD, as AE to CF; like-
wife, alternately BA is to AE, as DC to CF.
and because if magnitudes taken jointly be pro-
portionals, they are also proportionals when
taken feparately; therefore as BE is to EA, fo E
is DF to FC; and alternately, as BE is to DF,
fo is EA to FC. but as AE to CF, fo, by the
Hypothesis, is AB to CD; therefore alfo BE the
remainder fhall be to the remainder DF, as the
whole AB to the whole CD. Wherefore if the
whole, &c. Q. E. 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 Demonstration is contained in the preceding.

PROP. E, THEOR.

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

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

Becaufe AB is to BE, as CD to DF, by E
divifion, AE is to EB, as CF to FD; and
by inverfion, 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.

A

C

F

B D

Sce N.

PROP. XX. THE O R.

IF 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 shall be greater than the fixth; and if equal, equal; and if lefs, lefs.

137

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, lefs.

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 lefs has to it"; therefore A has to B a greater ratio than C has A B to B. but as D is to E, fo is A to B ; therefore b D has to E a greater ratio than C to B. and be- DE caufe B is to C, as E to F, by inverfion, C is to B, as F is to E; and D was fhewn to have to Ea

a. 8. 5.

C

b. 13.5

F

greater ratio than C to B; therefore D has to Ea
greater ratio than F to E. but the magnitude
which has a greater ratio than another to the fame magnitude, is
the greater of the two. D is therefore greater than F.
Secondly, Let A be equal to C; D shall 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 lefs than C; D A B C fhall be less than F. for C is great- DEF

er than A, and, as was fhewn in

the firft 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 first cafe; and therefore

c.Cor. 13.5.

d. 10. S.

c. 7. 5.

f. 11. 5.

8.9.5.

A B C
DEF

D is less than F. Therefore if there be three, &c. Q. E. D.

PROP. XXI. THEOR.

IF there be three magnitudes, and other three, which

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

Book V.

2. 8. 5.

b. 13. 5.

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 becaufe B is to C, as D to E, by inverfion, C is to B, as E to D. and E was fhewn to have to Fa 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 another, is the leffer of the two . F therefore is lefs than D; that is, D is greater than F.

d. 10. 5.

c. 7. S.

f. 11. 5. 8.9.5.

Sce N.

A B C
DEF

Secondly, Let A be equal to C; D fhall 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; wherefore E is to F,as E to Df; and therefore D is equal to F.

Next, Let A be less than C; D fhall be lefs than F. for Cis greater than A, and, as was fhewn, C is to B, as E to D, and in like manner B is to A, as F to E; therefore F is greater than D, by cafe firft; and therefore D is lefs than F.

A B C
DEF

ABC
DEF

'Therefore if there be three, &c. Q. E. D.

IF

F there be any number of magnitudes, and as many others, which taken two and two in order have the fame ratio; the firft fhall have to the laft of the firft magnitudes the fame ratio which the firft of the others has to the laft. N. B. This is ufually cited by the words "ex aequali, or, ex aequo."

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

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 Gis

to K, fo is H to L. for the fame reafon 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;

A B C

GKM

DEF
HLN

a. 4. 5.

and if lefs, lefs b. and G, H are any equimultiples whatever of A, b. 20 5. D, and M, N are any equimultiples whatever of C, F. therefore c. 5. Def. s. as A is to C, fo is D to F.

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. A fhall be to D, as E to H.

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 cafe, A is to D, as E to H. and so on, whatever be the number of magnitudes. Therefore if there be any number, &c. Q. E. D.

Book V.

See N.

8, 15. 5.

b. 11. 5.

C. 4. 5.

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 firft fhall have to the laft of the firit 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 perturbate."

66

Firft, 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 fuch 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 which
their equimultiples have ; as A is
to B, fo is G to H. and for the fame
reafon, as E is to F, fo is M to N.

but as A is to B, so is E to F; as A B C
therefore G is to H, fo is M to N b.

and because as B is to C, fo is D to G H L

E, and that H, K are equimultiples
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 than L, K is greater

D E F

K MN

d. 23. 5. than N; and if equal, equal; and if lefs, lefs d. 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.