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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, Fother 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.

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Because A is greater than C, and B is any other magnitude,
A has to B a greater ratio than Chas to B: (v. 8.)
but as E to F, so is A to B; (hyp.)

therefore E has to Fa greater ratio than C to B: (v. 13.)
and because B is to C, as D to E; (hyp.)

by inversion, Cis to B, as E to D:

and E was shewn to have to F a greater ratio than Chas to B; therefore E has to Fa greater ratio than E has to D: (v. 13. Cor.) but the magnitude to which the same has a greater ratio than it has to another, is the less of the two: (v. 10.)

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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 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 "ex æquali,” or “ex æquo."

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First, let there be three magnitudes A, B, C, and as many others D, E, F, which taken two and two in order, have the same ratio, that is, such that A is to B, as D to E;

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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 Fany 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: (v. 4.)

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; therefore if G be greater than M, H is greater than N; and if equal, equal; and if less, less; (v. 20.) but G, H are any equimultiples whatever of A, D, and M, N are any equimultiples whatever of C, F; (constr.) therefore, as A is to C, so is D to F. (v. 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 A shall be to D, as E to H.

A.B.C.D

E. F.G.H

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 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.

PROPOSITION XXIII. THEOREM.

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 "ex æquali in proportione perturbatâ;" or ex æquo perturbato."

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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;

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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; (v. 15.)

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:
but as A is to B, so is E to F; (hyp.)
therefore as G is to H, so is M to N: (v. 11.)
and because as B is to C, so is D to E, (hyp.)
and that H, K are equimultiples of B, D, and L, M of C, E;
therefore as H is to L, so is K to M: (v. 4.)

and it has been shewn that G is to H, as M to N:

therefore, 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 is greater than N:

and if equal, equal; and if less, less: (v. 21.)

but G, K are any equimultiples whatever of A, D; (constr.)
and L, N any whatever of C, F;

therefore as A is to C, so is D to F. (v. def. 5.)

Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which taken two and 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 shall be to D, as E to H.

A.B.C.D

E.F.G.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 Cis 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.

PROPOSITION XXIV. THEOREM.

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

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, shall have to C the second, the same ratio which DH, the third and sixth together, has to F the fourth.

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Because BG is to C, as EH to F;

by inversion, Cis to BG, as F to EH: (V. B.) and because, as AB is to C, so is DE to F; (hyp.) and as C to BG, so is F to EH;

ex æquali, AB is to BG, as DE to EH: (v. 22.) and because these magnitudes are proportionals when taken separately, they are likewise proportionals when taken jointly; (v. 18.) therefore as AG is to GB, so is DH to HE: but as GB to C, so is HE to F: (hyp.)

therefore, ex æquali, as AG is to C, so is DH to F. (v. 22.)
Wherefore, if the first, &c. Q.E.D.

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

PROPOSITION XXV. THEOREM.

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 them, and consequently F the least. (v. 14. and A.)

Then AB together with Fshall be greater than CD together with E.

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Take AG equal to E, and CH equal to F.
Then because as AB is to CD, so is E to F,
and that AG is equal to E, and CH equal to F,
therefore AB is to CD, as AG to CH: (V. 11, and 7.)

and because AB the whole, is to the whole CD, as AG is to CH, likewise the remainder GB is to the remainder HD, as the whole AB is to the whole CD: (v. 19.)

but AB is greater than CD; (hyp.) therefore GB is greater than HD; (v. a.)

and because AG is equal to E, and CH to F;

AG and F together are equal to CH and E together: (1. ax. 2.) therefore if to the unequal magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz. to GB the two AG and F, and CH and E to HD;

AB and F together are greater than CD and E. (1. ax. 4.)
Therefore, if four magnitudes, &c. Q.E.D.

PROPOSITION F. THEOREM.

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

Let A be to B, as D to E; and B to C, as E to F. Then the ratio which is compounded of the ratios of A to B, and B to C, which, by the definition of compound ratio, is the ratio of A to C. shall be the same with the ratio of D to F, which, by the same definition, is compounded of the ratios of D to E, and E to F.

A.B.C
D.E.F

Because there are three magnitudes A, B, C, and three others D, E, F, which, taken two and two, in order, have the same ratio; ex æquali, A is to C, as D to F. (v. 22.)

Next, let A be to B, as E to F, and B to C, as D to E:

A.B.C
D.E.F

therefore, ex æquali in proportione perturbata, (v. 23.)
A is to C, as D to F;

that is, the ratio of A to C, which is compounded of the ratios of A to B, and B to C, is the same with the ratio of D to F, which is compounded of the ratios of D to E, and E to F.

And in like manner the proposition may be demonstrated, whatever be the number of ratios in either case.

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