tude taken from CD: the remainder EB shall be to the remainder FD, as the whole AB to the whole CD.

A

Because AB is to CD, as AE to CF: therefore alternately,* BA is to AE, as DC to CF : * 16. 5. and because if magnitudes taken jointly be proportionals, they are also proportionals,* when * 17. 5. taken separately; therefore, as BE is to EA, so is DF to FC; and alternately, as BE is to DF, so is EA to FC: but, as AE to CF, so, by the hypothesis, is E AB to CD; therefore also BF the remainder is to the remainder DF,† as the whole AB to the whole CD. Wherefore, if the whole, &c. Q. E. D.

C

BD

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 shall likewise be 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.

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 shall be to AE, as DC to CF.

Because AB is to BE, as CD to DF, therefore by division,* AE is to EB, as CF to FD; and by inversion,* 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.

PROP. XX. THEOR.

E

+11.5.

C

F

*17.5.

* B. 5.

* 18.5.

B D

If there be three magnitudes, and other three, which, taken two and two, have the same ratio ;

* 8.5.

*13. 5.

then 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, 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 B C therefore A has to B a greater ratio D E F Hyp. 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. 5. 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 *Cor.13. a greater* ratio than F to E: but the magnitude which has a greater ratio than another to the same magnitude, is the greater* of the two; therefore D is greater than F.

5.

*10.5.

*7.5.

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 B:* but + A

Hyp is to B, as D to E; and† C is + Hyp. & B. 5.

to B, as F to E; wherefore D A B C *11.5. is to E, as F to E; * and there- DEF fore D is equal to F.*

*9.5.

Next, let A be less than

C; D shall be less than F. For
C is 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;
that is, 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 same ratio taken two and two, but in a cross order; then 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.

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 B C

D-EF

*8.5. + Hyp. 13. 5.

+ Hyp.

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio* than C has to B: butt 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 Fa greater ratio than C has to B; therefore E has to F a greater ratio than E has to* D: but the magnitude to which the same *Cor.13. has a greater ratio than it has to another, is the lesser of the two: therefore F is less than D; *10. 5. that is, D is greater than F.

5.

Secondly, Let A be equal to C; D shall be equal to F. Because A and C are equal, A is * *7.5. to B, as C is to B: but A is to B,† as E to F; +Hyp. and C is to B, as E to D;

wherefore E is to F,* as E

*11.5.

to D; and therefore D is

A B C В С

*9.5.

equal* to F.

Next, let A be less than DEFDE

C: D shall be less than F.

For C is greater than A; and,

as was shown, C is to B, as

E to D, and in like manner B is to A, as F to
E; therefore F is greater than D, by case first;

P

*4.5.

that is, D is less than F. Therefore, if there be three, &c. Q.E.D.

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 has to the last of the others. N. B.
This is usually cited by the words
æquali," or ex æquo."

66

66

ex

First, let there be three magnitudes A, B, C, and as 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.

ABC DEF
GKM HLN

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, E; therefore as G is to K, so is 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, *20.5. which two and two have the same ratio; * therefore if G be greater than M, H is greater than N; and if equal, equal; and if less, less; but + Const. G, H are any equimultiples whatever of A, D,t and M, N are any equimultiples whatever of C, *5 Def. F; therefore,* as A is to C, so is D to F.

5.

A.B.C.D.

E.F.G.H

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 as B to C, so F to G; and as C to D, so G to H: A shall be to D, as E to H.

1

Because A, B, C are three magnitudes, and E, F, G other three, which taken two and two have the same ratio; therefore 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.

PROP. XXIII. THEOR.

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 has to the last of the others. N. B. This is usually cited by the words " æquali in proportione perturbatâ;" or ex æquo perturbato."

ex

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 shall be to C, as D to F.

:

ABC
GHL

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 have; therefore 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: butt as A is to B, so is E to F; therefore as G is to H,* so is M to N: and because † as B is to C, so is D to E, and that H, K are equimultiples of B, D,

15.5.

DEF

KM N

and L, M of C, E; therefore as H is to L, so is* *4. 5. K to M and it has been shown that G is to H,

: