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Book V.

Because AB is to CD, as AE to CF ; like

wile, alternately · BA is to AE, as DC to CF. A 2. 16. 5.

and because if magnitudes taken jointly be prob. 17. 5. portionals, they are also proportionals 6 when

С taken separately; therefore as BE is to EA, fo E is DF to FC; and alternately, as BE is to DF, F fo is EA to FC. but as AE to CF, fo, by the Hypothefis, is AB to CD; therefore also BE the remainder shall be to the remainder DF, as the whole AB to the whole CD. Wherefore if the whole, &c. Q. E. D.

B D Cor. If the whole be to the whole, as a magnitude taken from the first is to a magnitude taken from the other, the remainder likewise is to the remainder, as the magnitude taken from the first to that taken from the other. the Demonstration is contained in the preceding

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PROP. E THEOR.
IF
F four magnitudes be proportionals, they are also

proportionals by conversion, that is, the first is to its excess above the second, as the third to its excess above the fourth.

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

С
Because all is to BE, as CD to DF, by E
division, AE is to EB, as CF to FD; and

F
by inversion b, BE is to EA, as DF to FC.
Wherefore, by Composition , BA is to AÉ,
as DC is to CF. If therefore four, &c,
Q. E. D.

B D

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PRO P. XX. THE O R.
IT
F there be three magnitudes, and other three, which

taken two and two have the same ratio; if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

b. 13. 5.

Let A, B, C be three magnitudes, and D, E, F other three, Book V. which taken two and two have the same 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 shall 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 less has to it";

a. 8. 3. therefore A has to B a greater ratio than C has A B C 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. D E F cause B is to C, as E to F, by inversion, 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

c.Cor.13.5. which has a greater ratio than another to the same magnitude, is the greater of the two d. D is therefore greater than F.

Secondly, Let A be equal to C; D shall be equal to F. because A and Care equal to one another, A is to B, as Cis to Be, 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 FB, Next, Let A be less than C;D

A B C fhall be less than F. for C is great- D E F

A B C er than A, and, as was shewn in

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

d. 1o. S.

c. 7.5.

f. 11.5.

8. 9. s.

PRO P. XXI. THEO R.

If there be three magnitudes, and other three, which

have the same ratio taken two and two, but in a cross order ; if the first magnitude be greater than the third, the fourth shall be greater than the sixth ; and if cqual, equal ; and if lefs, less.

Book v. Let A, B, C be three magnitudes, and D, E, F other three,

which have the same ratio taken two and two, but in a cross order,
viz. as A is to B, so is E to F, and as B is to C, so
is D to E. If A be greater than C, D Mall be great-
er than F; and if equal, equal ; and if less, less.

Because A is greater than C, and B is any other 2. 8. s. magnitude, A has to B, a greater ratio a than C b. 13. 5.

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- A B C
caufe B is to C, as D to E, by inversion, C is to

DEF
B, as E to D. and E was shewn to have to F a

greater ratio 'than C to B; therefore E has to Fa c.Cor.13.5. greater ratio than E to Do. but the magnitude to

which the fame has a greater ratio than it has to d. 10. 5.

another, is the lesser of the two d. F therefore is
less than D; that is, D is greater than F.

Secondly, Let A be equal to C; D shall be equal to F. Because c. 7. S. A and C are equal, A ise 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; 8. 9. 5.

and therefore D is equal to F3,

Next, Let A be less than C; D Mall be less than F. for Cis A B C A B C greater than A, and, as was

D E F D E F shewn, 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 first; and therefore D is lels than F. "Therefore if there be three, &c. Q. E. D.

f. 11. s.

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Sce N.

F there be

any

number of others, which taken two and two in order have the same ratio ; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words ex aequali, or, cx aequo."

a. 4. $.

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 same 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 shall 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 equi- A B C D E F multiples of A, D, and K, L

G K M H LN equimultiples of B, E, as G is to K, fo is · H to L. for the same 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 same ratio ; if G be greater than M, H is greater than N; and if equal, equal ; and if less, less b. and G, H are any equimultiples whatever of A, b. 20 s. D, and M, N are any equimultiples whatever of C, F. therefore c.s. 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 A.B. C.D.
two have the same ratio, viz. as A is to B, so is E.F. G. H.
E to F; and as B to C, fo F to G; and as C to
D, so G to H. A shall be to D, as E to H.

Because A, B, C are three magnitudes, and E, F, G other three, which taken two and two have the same ratio ; by the foregoing cale, 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 so on, whatever be the number of magnitudes. Therefore if there be any number, &c. Q. E. D.

Book V.

PROP. XXIII. THEO R.

Sec N.

IF
F there be any number of magnitudes, and as many

others, which, taken two and two, in a cross order, have the same ratio ; the first shall have to the last of the firit magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words ex aequali in proportione perturbata,

or, ex aequo perturbate."

First, Let there be three magnitudes A, B, C, and other three D, E, F, which taken two and two in a cross order have the same ratio, that is such that A is to B, as E to F; and as B is to C, so 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 same ratio which , 15. S. their equimultiples have; as A is

to B, fo is G to H. and for the fame
reason, as E is to F, so is M to N.

but as A is to B, so is E to F; as À B C D E F b. 11. S.

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 K MN
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 mewn 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 same ratio
taken two and two in a cross order;

if G be greater than L, K is greater 25. 5. than N; and if equal, equal; and if less, less 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, so is D to F.

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