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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, so is D to E. If A be greater than C, D will be greater than F; if equal, equal; and if less, less.

a 8. 5.

b 13. 5.

d 10. 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 E is to F, as 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 A в с inversion, C is to B, as E to D: And E was shown to have to Fa greater ratio than C to B: therefore, E has to Fa greater ratio than c Cor. 13.5. E to De: but the magnitude to which the same has a greater ratio than it has to another, is the less of the twod: F therefore is less than D; that is, D is greater than F.

DEF

e 7. 5.

f 11. 5.

g 9. 5.

See N.

Secondly, Let A be equal to C; then will D be equal to F. Because A and C are equal, A is to B, as C is to B:

But A is to B, as E to F; and

Cis to B, as E to D; where

fore, E is to Fas E to Df; and
therefore D is equal to Fg.

Next, Let A be less than C;
then will D be less than F:
For, as was shown, C is to Bas
E to D; and, in like manner, B
is to A, as F to E; and Cis
greater than A; therefore Fis
greater than D, by case first;
consequently D is less than F.
Therefore, " if there be three,"
&c. Q. E. D.

ABCAB

C

DEF D E F

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 will have to the last of the first magnitudes the same ratio which the first of the others has to the last *.

• N. B. This proposition is usually cited by the words, "ex æquali," or " ex æquo."

137

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 Bas D to E; and as B is to C, so is E to F; then will A be to Cas D to F.

ABC DEF

For, of A and D take any equimultiples whatever G and H, and of B and E take any 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 G, H are equimultiples of A, D, and K, L equimultiples of B, E; as Gis to Kso 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, which, taken two and two, have the same ratio; if G be greater than M, H is

greater than N; if equal, equal;

GKM

HLN

a 4. 5.

and if less, less; and G, H are any equimultiples whatever b 20. 5. of A, D, and M, N are any equimultiples whatever of C, F:

Therefore, as A is to C, so is D to F.

c 5. def. 5.

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: then will A be to D, as E to H.

A. B. C. D.

E. F. G. Η.

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 Cis 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.

See N.

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 will have to the last of the first magnitudes the same ratio which the first of the others has to the last *.

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. Then A is to Cas D to F.

ABC

GHL

DEF

KMN

For, of A, B, D, take 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 magnitudes have the same ratio which their equimula 15. 5. tiples have a; as A is to B, so is Gto 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; b 11. 5. therefore G is to Has M to N b. And because as Bis to C so is D to E, and H, K are equimultiples of B, D, and L, M of C, E; as His to L, so is K to M: And it has been shown, that G is to Has 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

с 4. 5.

d 21. 5. is greater than N; if equal, equal; and if less, less d: and G, K are any equimultiples whatever of A, D; and L, Nany whatever of C, F; therefore, A is to C, as D to F.

*N. B. This proposition is usually cited by the words, "ex æquali in proportione perturbata;" or, "ex æquo inversely."

Next, Let there be four magnitudes, A, B, C, D, and other Book V.

four E, F, G, H, which, taken two and

A. B. C. D.

E. F. G. H.

two, in a cross order, have the same ratio, viz. A to B, as G to H; B to C, as F to G; and C to D, as E to F; then A is to D, as E to H.

Because A, B, C, are three magnitudes, and F, G, H, other three, which, taken two and two, in a cross order, have the same ratio; by the first case, A is to C, as F to H: But C is to D, as E is to F; 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. XXIV. THEOR.

If the first has to the second the same ratio which See N.

the third has to the fourth; and the fifth to the second, the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and sixth together have to the fourth.

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Let AB the first, have to C the second, the same ratio which DE the third has to F the fourth; and let BG the fifth have to C the second, the same ratio which EH the sixth has to F the fourth; then AG, the first and fifth together, will have to C the second, the same ratio which DH, the third and sixth together, has to F the fourth.

B

H

E

Because BG is to C, as EH to F; by inversion, C is to BG, as F to EH: And because, as AB is to C, so is DE to F; and as C to BG, so is F to EH; ex æquali, AB is to BG, as DE to EH: And because these magnitudes are proportionals, they will also be proportionals when taken jointly; as therefore AG is to GB, so is DH to HE; but as GB to C, so is HE to F. Therefore, ex æqualia, as AG is to C, so is DH to F. Wherefore, " if the first," &c. Q. E. D.

a 22. 5.

ACDFb 18.5.

COR. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth will be to the second, as

Book V. the excess of the third and sixth to the fourth: The demonstration of this is the same with that of the proposition, if division be used instead of composition.

COR. 2. The proposition holds true, of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest.

PROP. XXV. THEOR.

If four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together.

Let the four magnitudes AB, CD, E, F, be proportionals, viz. AB to CD, as E to F; and let AB be the greatest of a A & 14.5. them, and consequently F the least, then AB, together with F, are greater than CD, together with E.

с А. 5.

Take AG equal to E, and CH equal to F: Then, because as AB is to CD, so is E to F, and AG is equal to E, and CH equal to F; AB is to CD, as AG to CH. And because AB the whole, is B to the whole CD, as AG is to CH, the remainder GB will be to the remainder

G

D

H

b 19.5. HD, as the whole AB is to the wholeb CD: But AB is greater than CD, therefore GB is greater than HD: And because AG is equal to E, and CH to F; AG and F together are equal to CH and E together. If therefore to the unequal magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz. AG and F to GB, and CH and E to HD; AB and F together are greater than CD and E together: Therefore, " if four magnitudes," &c. Q. E. D.

ACEF

See N.

PROP. F. THEOR.

Ratios which are compounded of the same ratios, are the same with one another.

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