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Secondly, let A be equal to C; then D shall be equal to F. Because A and C are equal to one another, A is to B as C is to B (ƒ): but A is to B as D is to E; and C is to B as is to E; wherefore D is to E as F is to E (g); and therefore D is equal to F (h).

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Thirdly, let A be less than C; then 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 is to E, and in like manner B is to A, as E is to D; therefore F is greater than D, by the first case; and therefore D is less than F.

SCHOLIUM. The foregoing proposition may be algebraically expressed as follows:

THEOREM. If A, B, C be three magnitudes, and D, E, F three others, and if A: B : D: E, and B: C: E: F; then, if A be C, D is also > F; and if equal, equal; if less, less.

Because A: B :: D: E,

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C

F

therefore A: C:: D: F,

whence by the 5th definition it follows that if A is > C, D is > F; and if equal, equal; and if less, less.

PROPOSITION XXI.

THEOREM.-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; then. if A be greater than C, D shall be greater than F; and if equal, equal; and if less, less.

DEMONSTRATION. First, let A be greater than C; then D shall be greater than F: for because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C has to B (a): but as E is to F, so is A to B (b); therefore E has to F a greater ratio than C to B (c); and because B is to C, as D is to E (b), by inversion, C is to B, as E is to D and E was shown to have to F a greater ratio than C to B; therefore E has to Fa greater ratio than E to D (d); but the magnitude to which the same has a greater ratio than it has to another, is the lesser of the two (e); therefore F is less than D; that is, D is greater

than F.

:

Secondly, let A be equal to C; then D shall be equal to F. Because A and C are equal, A is to B, as C is to B (f): but A is to B, as E to F; and C is to B as E is to D; wherefore E is to F as E is to D (9); and therefore D is equal to F (h).

A

D

B C

E

F

(a) V. 8.
Hypoth.

(c) V. 13.
(d) V. 13, cor.
(e) V. 10.

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Thirdly, let A be less than C; then D shall be less than F. For C is greater than A, and, as was shown, C is to B, as E is to D, and in like manner, B is to A, as F is to E; therefore F is greater than D, by the first case; and therefore D is less than E.

SCHOLIUM.

This proposition may be algebraically expressed as follows:

THEOREM. If A, B, C be three magnitudes, and D, E
F three others, such that A: B:: E: F,
D: E; then if A be > C, D is also > F;
equal; if less, less.

Because A: B :: E: F,

and B: C ::
and if equal,

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whence by the 5th definition it follows that if A is > C, D is > F; and if equal, equal; if less, less.

PROPOSITION XXII.

THEOREM.-If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first has to the last of the first magnitudes the same ratio which the first has to the last of the others.

DEMONSTRATION. 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 is to E; and that B is to C, as E is to F; then A shall be to C, as D is 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

PROPOSITION XXI

Ter- there be the magutu, and other the same ratio taken two and two, but in a thf the first maritude be greater than the third, fill be greater than the sixth; and if equal, ec , less.

Let A, B, C be three magnitudes, and D, E, F here, which have the same ratio, taken two at tw but in a cross order, viz as A is to B, EF. and as B is to C, so is D to E; then Aereater than C, D shall be greater than F, and if equal, equal; and if less, less.

Dr. First, let A be greater than
C. then I shall be greater than F: for because
Amater than C. and Bis any other magnitude,
A to Ba greater ratio than C has to B (a):
bas Eis to F. so is A to B (6); therefore E has

A B

DF

a greater ratio than C to B (c); and because Estas D is to E (5), by inversion, C is to B, 2D and E was shown to have to Fa (a) V. rater rat than C to B; therefore E has to Fa athan E to D (d); but the magni... the same has a greater ratio than tas to arther, is the lesser of the two (e); (d) Fs less than D; that is, D is greater (e)

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Ser, let A be equal to C; then D shall be
ega F. Because A and Care equal, A is to
Bis to B): but A is to B, as E to F; and
Cat Bas Eis to D; wherefore E is to Fas E
Dander Dual F

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