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Book V.

PROP. XXIII.

THEOR.

See N.

2 15. 5.

b 11. 5.

C 4.5.

b 23. I.

IF

there be any number of magnitudes, and as many others, which taken two and two, in a cross order, have the fame ratio; the first shall have to the last of the first magnitudes the fame ratio which the firft of the others has to the laft. N. B. This is ufually cited by the words "ex aequali in proportione perturbata ;" or "ex aequo 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 fame ratio, that is, fuch that A is to B, as E to F; and as B is to C, fo is D to E: A is to C, as D to F.

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 fame ratio which their equi-
multiples have; as A is to B,
fo is G to H: And for the fame
reafon, as E is to F, fo is M to
N: But as A is to B, fo is E to AB C
F; as therefore G is to H, fo is M GHL

DEF KMN

to Nb: And becaufe as B is to C,
fo is D to E, and that H, K are
equimultiples of B, D, and L,
M of C, E; as H is to L, fo is
CK to M: And it has been thewn
that G is to H, as M to N: Then,
because there are three magni-
tudes G, H, L, and other three
K, M, N which have the fame
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 lefs, lefs d;
and G, K are any equimultiples whatever of A, D; and L, N
any whatever of C, F; as, therefore, A is to C, fo is D to F.

Next, Let there be four magnitudes, A, B, C, D, and o- Book. V. ther four E, F, G, H, which, taken two and

two in a cross order, have the fame ratio, viz. A. B. C. D. A to B, as G to H; B to C, as F to G; and C | E. F. G. H. to D, as E to F: 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 fame ratio; by the firft cafe, A is to C, as F to H: But C is to D, as E is to F; wherefore again, by the firft cafe, A is to D as E to H: And fo on, whatever be the number of magnitudes. Therefore, if there be any number, &c. Q. E. D.

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IF the firft has to the fecond the fame ratio which the Sec N. third has to the fourth; and the fifth to the second the fame ratio which the fixth has to the fourth; the first and fifth together fhall have to the fecond, the fame ratio which the third and fixth together have to the fourth.

Let AB the first have to C the second the fame ratio which DE the third, has to F the fourth; and let BG the fifth have to C the second the fame ratio which EH

the fixth has to F the fourth: AG, the G first and fifth together, fhall have to C the fecond the fame ratio, which DH, the third and fixth together, has to F the fourth.

B

E

H

Because BG is to C, as EH to F; by inverfion, Cis to BG, as F to EH: And because. as AB is to C, fo is DE to F; and as C to BG, fo F to EH; ex aequali, AB is to BG, as DE to EH: And because these magnitudes are proportionals, they fhall likewife be proportionals when taken jointly ; as therefore AG is to GB, fo is DH to HE; but as GB to C, fo is HE to F. Therefore, ex aequali, as AG is to C, so is DH to F. Wherefore, if the first, &c. Q. E. D.

a 22. 5.

ACDF 13.

COR. 1. If the fame hypothefis be made as in the propofition, the excess of the first and fifth fhall be to the fecond, as

K 3

the

b 18.5.

Book V.

the excess of the third and fixth to the fourth: The demonftra tion of this is the fame with that of the propofition, if divifion be ufed inftead of compofition.

COR. 2. The propofition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the fecond magnitude the fame ratio that the correfponding one of the fecond rank has to a fourth magnitude; as is manifeft.

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F four magnitudes of the fame kind are proportionals, the greatest and last 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, a A. & 14. and confequently F the least. AB together with F are greater than CD together with E.

IS.

b 19.5.

A. S.

B

D

H

Take AG equal to E, and CH equal to F: Then, because as AB is to CD, fo is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG to CH. And because AB the whole is to the whole CD, as AG is to CH, likewife the re- G mainder GB fhall be to the remainder HD, as the whole AB is to the whole b CD: But AB is greater than CD, therefore GB it greater than HD: And becaufe 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 ACEF 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. Therefore, if four magnitudes, &c. Q. E. D.

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A which are compounded of the fame ratios, are the home with one another.

Let

a

A. B. C.

Let A be to B, as D to E; and B to C, as E to F: The ra- Book V. tio 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, is the fame with the ratio of D to F, which, by the fame definition, is compounded of the ratios of D to E, and E to F.

D. E. F.

Because there are three magnitudes A, B, C, and three o thers D, E, F which taken two and two in order, have the fame ratio; ex aequali, A is to C as D to Fa.

A. B. C.

D. E. F.

Next, Let A be to B, as E to F, and B to C, as D to E;
therefore, ex aequali in proportione perturbata
, 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 fame 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 propofition may be
demonftrated, whatever be the number of ratios in either cafe.

a 22. S.

b 23.5.

It

PROP. G. THEOR.

[F feveral ratios be the fame with feveral ratios, each see N. to each; the ratio which is compounded of ratios which are the fame with the first ratios, each to each, is the fame with the ratio compounded of ratios which are the fame with the other ratios, each to each.

Let A be to B, as E to F; and C to D, as G to H: And let A be to B, as K to L; and C to D, as L to M: Then the ratio of K to M, by the definition

of compound ratio, is compound- A. B. C: D.

E. F. G. H.

K. L. M.

N. O. P.

ed of the ratios of K to L, and
L to M, which are the fame with
the ratios of A to B, and C to D:
And as E to F, fo let N be to O; and as G to H, fo let O be to
P; then the ratio of N to P is compounded of the ratios of N to
O, and O to P, which are the fame with the ratios of E to F,
and G to H: And it is to be fhewn that the ratio of K to M,
is the fame with the ratio of N to P, or that K is to M, as N to P.
Because K is to L, as (A to B, that is, as E to F, that is, as)
N to O; and as L to M, fo is (C to D, and fo is G to H,
K 4
and

Book V.

a 22. 5.

and fo is) O to P: Ex aequali, K is to M, as N to P. Therefore, if feveral ratios, &c. Q. E. D.

Soc N.

IF

PROP. H. THEOR.

a ratio compounded of feveral ratios be the fame with a ratio compounded of any other ratios, and if one of the first ratios, or a ratio compounded of any of the firft, be the fame with one of the laft ratios, or with the ratio compounded of any of the laft; then the ratio compounded of the remaining ratios of the firft, or the remaining ratio of the first, if but one remain, is the fame with the ratio compounded of thofe remaining of the laft, or with the remaining ratio of the last.

Let the firft ratios be thofe of A to B, B to C, C to D, D to E, and E to F; and let the other ratios be thofe of G to H, H to K, K to L, and L to M: Alfo let the ratio of A to a Definition F, which is compounded of the

of com

pounded ratio.

b B. 5.

C 52. 2.

firft ratios be the fame with the ratio A. B. C. D. E. F. of G to M, which is compounded off G. H. K. L. M. the other ratios: And befides, let the

ratio of A to D, which is compounded of the ratios of A to B, B to C, C to D, be the fame with the ratio of G to K, which is compounded of the ratios of G to H, and H to K: Then the ratio compounded of the remaining firft ratios, to wit, of the ratios of D to E, and E to F, which compounded ratio is the ratio of D to F, is the fame with the ratio of K to M, which is compounded of the remaining ratios of K to L, and L to M of the other ratios.

Because, by the hypothefis, A is to D, as G to K, by inverfion, D is to A, as K to G; and as A is to F, fo is G to M; therefore, ex aequali, D is to F, as K to M. If therefore a ratio which is, &c. Q. E. D.

PROP.

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