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Book V. BA is to AE, as DC to CF: And becaufe, if mag.

b 17. 5.

nitudes, taken jointly, be proportionals, they are
alfo proportionals when taken feparately; there-
fore, as BE is to EA, fo is DF to FC; and alter-
nately, as BE is to DF, fo is EA to FC: But, as E
AE to CF, fo, by the hypothefis, 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.

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F

C

B D

COR. It 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 firft to that taken from the other: The demonftration is contained in the preceding.

IF

PROP. E. THEOR.

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

a 17. 5.

b B. 5.

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 divi-
fion, AE is to EB, as CF to FD; and by in- E
verfion, 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.

C

F

BD

See N.

PROP. XX. THEOR.

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

Let

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

a 8. 5.

DEF 13. 5.

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 A B C to B; therefore b D has to E a greater ratio than C to B: And because B is to C, as 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 C to B; therefore D has to E a 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 twod D is therefore greater than F.

Secondly, Let A be equal to C; D shall be equal to F: Be

caufe 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 to Ef; and therefore D is equal to F 8.

c Cor. 13.5.

d 10. 5.

e 7. 5.

f II. 5.

A B C

8 9.5.

АВ
A B C
DEF

Next, Let A be lefs than C; D fhall be lefs than F: For C is great-DEF

er than A, and, as was shown 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 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.

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Book V.

a 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.

Becaufe 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 E has to F a greater ratio than C to B: And because B is to C, as D to E, by inverfion, C is to B, as E to D: And E was shown to have to F a greater ratio than C to B; therec Cor. 13.5. fore E has to F a 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, Dis greater than F.

d 10. 5.

€ 7.5.

f 11. 5.

89.5.

e

A B C

DEF

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

Next, Let A be less than C;

D fhall be lefs than F: For C is A B C
greater than A, and, as was

hown, C is to B, as E to D, DEF

and in like manner B is to A,
as F to E; therefore F is great-
er than D, by cafe firft; and
therefore D is lefs than F.
Therefore, if there be three, &c.
Q. E. D.

A B

D DEF

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See N.

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."

66

First, Let there be three magnitudes A, B, C, and as ma- Book V. ny others 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.

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 A B C
B, E; as G is to K, fo is

H to

L: For the fame reafon, K is to G KM

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, His greater than N; and if equal, equal; and if lefs, lefs b: And G, H are any equimultiples whatever of A, D, and M, N are any e

DEF
HL N

a 4. 5.

b 20. 5.

quimultiples whatever of C, F: Therefore, as A is to C, foc 5. def. 5. 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 A. B. C. D. and as B to C, fo F to G; and as C to D, E. F. G. H. fo G to H: A fhall 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 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 fo on, whatever be the number of magnitudes. Therefore, if there be any number, &c. Q. E. D.

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Book V.

PROP. XXIII.

THE OR.

See N.

a 15. 5.

b II. 5.

€ 4. 5.

d 21. I.

Ioth

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 fhall have to the laft of the first magnitudes the fame ratio which the first of the others has to the laft. N. B. This is ufually cited by the words "ex aequali in proportione perturbata ;" or "ex aequo perturbate."

Firft, 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 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 be caufe G, H are equimultiples of

A, B, and that magnitudes have
the fame ratio which their equi-
multiples 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, fo is E to AB C
F; as therefore G is to H, fo is M GHL
to No. And because as B is to C,
fo is D to E, and that H, K are
equimultiples of B, D, and L,
M of C, E, as H is to L, fo is
CK to M: And it has been shown
that G is to H, as M to N: Then,
because there are three magni-
tudes 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 Ly

DEF KMN

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.

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