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b 17. 5.

Book V: lya, BA is to AE, as DC is to CF; and because

A
if magnitudes, taken jointly, be proportionals,
a 16. 5.

they are also proportionals when taken sepa-
rately; therefore, as BE is to EA, so is DF to

с
FC, and alternately, as BE is to DF, so is EA to E-
FC: But, as AE to CF, so by the hypothesis, is

F
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.
Cor. If the whole be to the whole, as a mag-

B D nitude 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.

PROP. E. THEOR.

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If 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 ex-
cess above the fourth.
Let AB be to BE, as CD to DF; then BA

А
is to AE, as DC to CF.
Because AB is to BE, as CD to DF, by di-

С 17.5. visiona, AE is to EB, as CF to FD ; and by in- E B. 5. version , BE is to EA, as DF to FC. Where

F * 18. 5. fore, by composition, BA is to AE, as DC is

to CF: If, therefore, four, &c. Q. E. D.

B D

PROP. XX. THEOR.

See N. Ir 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.

A 8. 3.

13, 5.

as

Let A, B, C, be three magnitudes, and D, E, Fother three, Book V. which, taken two and two, have the same ratio, viz, as A is to B, so is D to E; and as B to C, so is E to F. If A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

Because A is greater than C, and B is any other magnitude, and that the greater has to the same magnitude a greater ratio than the less has to it a; therefore A has to B a greater ratio than C has to B: But as D is to E, so

A B с is A to B; therefore b D has to E a greater ratio than C to B; and because B is to C, D E to F, by inversion, C is to B, as F is to E; and D was shown to have to E a greater ratio than Cto B; therefore D has to E a greater

c Cor. 13.5. ratio than F to E. But the magnitude which has a greater ratio than another to the same magnitude, is the greater of the twod; 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 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 tc Ef;

f 11. 5. and therefore D is equal to Fs. A B C Next, Let A be less than C;

В с D shall be less than F: For Cis D E F

D E F greater than · A, and, as was shown in the first case, 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 10. 5.

© 7. 5.

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PROP. XXI. THEOR.

If there be three magnitudes, and other three, See X.
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 equal, equal; and if less, less.

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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 shall be greater than F; and if equal, equal; and if less, less.

Because A is greater than C, and B is any * 8. 5. other magnitude, A has to B a greater ratio a

than C has to B: But as E to F, so is A to B : 13. 5. therefore b E has to F a greater ratio than C to B: And because B is to C, as D to E, by

A B C inversion, C is to B, as E to D: And E was shown to have to Fa greater ratio than C to D E F

B; therefore E has to f a greater ratio than Cor. 13. 5. E to DC; but the magnitude to which the same

has a greater ratio than it has to another, is d 10. 5. the lesser of the twod: 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. e 7. 5. Because 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 $11. 5. is to F, as E to Df; and 8 9. 5. therefore D is equal to F&.

Next, Let A be less than C;
D shall be less than F: for C
is greater than A, and, as was

A B C Α. Β
shown, C is to B, 'as E to D,
and in like manner B is to A, E F

E F
as F to E; therefore Fis great-
er than D, by case first; and
therefore, D is less than F.
Therefore, if there be three,
&c. Q.E.D.

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PROP. XXII. THEOR.

See N. If there be any number of magnitudes, and as many

other, 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 æquali," or "er sequo.

First, Let there be three magnitudes A, B, C, and as Book V. many 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, so 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 equimultiples of A, D, and K, L A B C D E F equimultiples of B, E; as G is to K, so isa H to L: For the C K M H L N - 4.5. 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, lessb; and G, H are any equin ultiples whatever of A, D, and M, N are any equimultiples whatever of C, F: Therefore, as A is to C, . 5 Def. 5. so 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 same ratio, viz. as A is to B, so is E to A. B. C. D. F; and as B to C, so F to G; and as C to E. F. G. H. 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 samne ratio; by the foregoing case, A is to C, as E to G: But Cis 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 nuinber, &c. Q.E.D.

h 20. 5.

Book v.

PROP. XXIII. THEOR.

See n. If there be any number of magnitudes, and as

many others, which, taken two and two, in a cross order, bave 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 æquali in proportione perturbata;" or“ ex æquo perturbato."

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so is

st, 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 their
* 15. 5. equimultiples havea : as A is to

B, so is G to H: And for the
same reason, as E is to F,

A B C D E F
M to N: But as A is to B, so

is E to F; as therefore G is to G H L K M N + 11. 5. H, so is M to Nb. And because

as B is to C, so is D to E, and
that H, K, are equimultiples

of B, D, and L, M of C,E; as
< 4. 5. H is to L, so iscKto M. And

it has been shown 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, Kiş 21. 5. greater than N: and if equal, equal; and if less, lessd ; 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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