.b 17.5.

Book V. BA is to AE, as DC to CF: and because, if magnitudes, taken jointly, be proportionals, they are also proportionais b. when taken, separately; therefore, as BE is to DF, so is EA to FC; and alternately, 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.

E.

A

B

F

C

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.

å 17. 5.

b B. 5. c 18. 5.

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.

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-
siona, AE is to EB, as CF to FD, and by inver- E-
sion, BE is to EA, as DF to FC. Wherefore, by
composition, BA is to AE, as DC is to CF. If,
therefore, four, &c. Q. E. D.

C

FL

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 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.

A

a 8. 5.

B

b 13.5.1

E

F

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; therefore A has to B a greater ratio than C has to B; but as D is to E, so is A to B; therefore 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 Ec; 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.

D

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

c Cor. 13.5.

d 10. 5.

cause 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 to B, as F to E: wherefore D is to E, as F to Ef; and therefore D is equal to Fg.

A

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

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, less.

T

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.

a 8.5.

b 13. 5.

Because A is greater than C, and B is any 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; therefore E has to F a greater ratio than C to B: and because B is to C, as D to E, by inversion, C is to B, as E to D and E was shown to have to F a greater ratio than C to B; there

& 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 of the twod; F therefore is less than D; that is, D is greater than F.

a 10. 5.

e 7. 5.

f 11. 5. 89.5.

Secondly, Let A be equal to C; D shall be equal to F. Be-
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
equal to F3.

Next, Let A be less than C;
D shall be less than F: for C is

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

E

A

F

D

BE

F

See Note.

PROP. XXII. THEOR.

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 æquo."

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,

GKMHL N

a 4. 5,

D, and K, L equimultiples of B, A B C D E F
E; as G is to K, so is a 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 great-
er 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 equimul-

b 20 5.

tiples whatever of C, F. Therefore, as A is to C, so is Dc 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.

Book V.

PROP. XXIII. THEOR.

See Note.

a 15. 5.

b 11. 5.

c 4. 5.

d 21. 5.

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 usually cited by the words "ex æquali in proportione perturbata" or, " ex æquo 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.

G

B

C

H

L

K M N

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 as therefore G is to 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 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, 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, so is D to F,

: