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

nitudes, taken jointly, be proportionals, they are
B 17.5.

also proportionais b. when taken, separately; there-
fore, as BE is to DF, so is EA to FC; and alter- C

E.
nately, as BE is to EA, so is DF to FC: but, as
AE to CF, so, by the hypothesis, 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.

Cor. If the whole be to the whole, as a mag-
nitude taken from the first, is to a magnitude taken B D
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 preced-
ing.

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PROP. E. THEOR.

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 excess above the fourth.

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à 17. 5.
bB. 5.
c. 18. 5.

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

Because AB is to BE, as CD tó DF, by divi-
sione, AE is to EB, as CF to FD, and by inver. E- C
sion b, BE is to EA, as DF to FC. Wherefore, by FL
composition C, BA is to AE, as DC is to CF. *If,
therefore, four, &c. Q. E. D.

BD

PROP. XX. THEOR.

Sce Note.

If 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.

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, so is D to £; 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 ita ; therefore A has to B a greater ra

a 8.5. tio than C has to B; but as D is to E, so is A to B; thereforeb.D has to E a greater ratio A B

с

b 13. 5. ! than C to B; and because B is to C, as E to F,

D E 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 Ec; but the magnitude which has a greater

c Cor. 13.5. ratio than another to the same magnitude, is the greater of the twod: D is therefore greater than F.

d 10. 5. Secondly, Let A be equal to C; D shall be equal to F: because A and C are equal to one an. other, A is to B, as C is" to Be: but A is to B, as D to E; and C is

e 7.5. to B, as F to E: wherefore D is to E, as F to Ef; and therefore D is

f 11.5. equal to F8.

A

A Next, Let A be less than C;D shall be less than F: for C is great. D E F D E F er 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.

89.5

6

PROP. XXI. THEOR.

IF there be three magnitudes, and other three, which See Note: 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, loss.

T

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Baok 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 2 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; 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 inver

A B sion, C is to B, as E to D: and E was shown to

have to F a greater ratio than C to B ; there- D E F (Cor. 13. 5. fore E has to F a greater ratio than E to DC;

but the magnitude to which the same has a

greater ratio than it has to another, is the lesser d 10.5.

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. Be. e 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, f 11.5.

as E to Df; and therefore D is

equal to F 8 9.5.

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

A

С A B shown, C is to B, as E to D, and in like manner B is to A, D E F D E F as' F to E; therefore F is greater than D, by case first; and therefore D is less than F. Therefore, if there be three, &c. Q. E. D.

was

PROP. XXII. THEOR.

See Note.

IF there be any number of magnitudes, and as many 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 æquali," or "ex equo.

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, 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 equimultiples of B, A B C D E 1 E ;, as G is to K, so is a H to L. G K M HLN

a 4. 5, 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, D, and M, N are any equimultiples whatever of C, F. Therefore c, as A is to C, so is D c 5. def. 5. 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, and A. B.C. D. as B to C, so F to G; and as C to D, so G to E. F. G. H. 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 case, 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.

b 20 5.

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Book v.

PROP. XXIII. THEOR.

See Note.

IF 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 first magnitudes the same ratio which the first of the others has to the last. N. B. This is usu. ally cited by the words ex equali in proportione perturbata ," or, “ ex æquo 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 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 equimultiples havea; as A is to B, so is G to H. And, for the same reason, as E is to F, so is M to N; but as A is to B, so is E to F; A

D E as therefore G is to H so is M to G H L K M N 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 H is to L, so ise 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, I, 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, 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.

b 11. 5.

C 4.5.

d 21. 5.

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