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

Á
nitudes, taken jointly, be proportionals, they are
b 17. s.

also proportionalso when taken separately; there-
fore, as BE is to EA, so is DF to FC; and alter- C
nately, as BE is to DF, so is £ A to FC : But, as E
AE to CF, so, by the hypothesis, is AB to CD;

F
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. It the whole be to the whole, as a inagnitude taken from the first, is to a magnitude taken

BD 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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IF
F four magnitudes be proportionals, they are also pro-

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

1

Let AB be to BE, as CD to DF; then BA is to

A
AE, as DC to CF.
Because AB is to BE, as CD to DF, by divi.

C
Gon", AE is to EB, as CF to FD ; and by in. E
version, BE is to EA, as DF to FC. Wherefore,
by composition", BA is to AE, as DC is to CF:

F
If, therefore, four, &c. Q. E. D.

a 17. S b B. 5. C 18. s.

B D

PRO P. XX.

THE O R.

See N.

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 sixth ; and if equal, equal ; and if less, less.

Let

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

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

a 8. 5. has to it; therefore A has to B a greater ratio than C has to B : But as D is to Ē, fo is A Á B C to B; therefore D has to E a greater ratio than C to B; And because B is to C, as E to F,

D E F b 13. 5. 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 Eo: But the magnitude which has a greater

c Cor. 13. S. ratio than another to the same 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 an

e 7. 5. other, 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

fil. S. E, as F to Ef; and therefore D is equal to F.

A B C
Next, Let A be less than C;D

A B Ć shall be less than F: For C is great

D E F

D E F er than A, and, as was shown in the firit 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.

8 9. 5.

PRO P. XXI.

THE O R.

31

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 equal, qual; and if less, less.

Let

E

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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, fo is D to E. If A be greater than C, D shall be greater than F; and if equal, equal; and if lefs, lefs.

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

than C has to B : But as E to F, so is A to B; b 13. 5.

therefored E has to F a greater ratio than C to
B: And because B is to C, as D to E, by inver- A B C
fion, C is to B, as E to D: And £ was shown to D E F

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 Do;

but the magnitude to which the same has a
greater ratio than it has to another, is the leffer
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. Be. €7. 5. cause 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; and therefore D is & 9. S.

equal to F 8.

Next, Let A be less than C D fhall be less than F: For C is A B CÀ A B C greater than A, and, as was shown, C is to B, as E to D, D E F D E F 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 less than F. Therefore, if there be three, &c. Q. E. D.

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See N.

IF.
F there be any number of magnitudes, and as mang

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 ex aequo."

First,

First, Let there be three magnitudes A, B, C, and as ma- Book V. e ny others D, E, F, which, taken two and cwo, 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 to F.
Take of A and D any equimultiples whatever G and H

i
and of B and E any equimultiples
whatever K and Land of Cand
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 D E Ė
B, E; as G is to K, fo is H to

a 4. 5.
L: For the fame reason, K is to Ģ ĶM HLN
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, e-
qual; and if less, less 6: And G,
H are any equimultiples whatever
of A, D, and M, N are any e-
quimultiples whatever of C, F: Therefore, as A is to C, loc 5. def. 5o
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 F A. B. C. D.
and as B to C, fo Fco G; and as C to D, E. F. G. H.
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 café, 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. There-
fore, if there be any number, &c. Q. E. D.

b 20. S.

JA

K 2

P POP

Book V,

PRO P. XXIII.

.

THE O R.

See N.

F there be any number of magnitudes, and as many

others, which, taken two and two, in a cross order, have the farne 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 in proportione perturbata ;or “ex aequo perturbate."

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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 fame 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 be caufe G, H are equimultiples of A, B, and that magnitudes have the fame ratio which their equimultiples have; as A is to B, so is G to H: And for the same reason, as E is to F, fo is M to N: But as A is to B, fo is E to ABC Ο E F F; as therefore G is to H, so is M G H L KMN to Nb. 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, so is

K to 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, K is greater than N ; and if equal, equal; and if less, less *; 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.

C 4. 5.

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Next,

d 21. I.

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