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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 fame ratio, viz.

A to B, as G to H;

B to C, as F to G; and

A is to D, as E to H.

A. B. C. D.
E. F. G. H.

C to D, as E to F:
Becaufe 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 first cafe, A is to C, as F to H: But C is
to D, as E is to F; wherefore again, by the first cafe, A is to
D, as E to H And fo on, whatever be the number of magni-
tudes. Therefore, if there be any number, &c. Q. E. D.

Book V.

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IF

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[F the firft has to the fecond the fame ratio which the See N. third has to the fourth; and the fifth to the fecond, 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 firft, have to C the fecond, the fame ratio which DE the third, has to F the fourth; and let BG the fifth, have to C the fecond, 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

Because BG is to C, as EH to F; by inverfion, C is to BG, as F to EH; And becaufe, 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 thefe magnitudes are proportionals, they fhall likewise 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. aequali, as AG is to C, fo is DH to F. firft, &c. Q. E. D.

E

H

ACDF

Therefore, ex
Wherefore, if the

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

K 3

the

a 22. 5

b 18. 5

Book V. the excefs of the third and fixth to the fourth: The demonftration of this is the fame with that of the propofition, if divifion be used instead 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 second magnitude the fame ratio that the correfponding one of the fecond rank has to a fourth magnitude; as is manifeft.

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I four magnitudes of the fame kind are proportionals, the greatest and laeft 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 leaft. AB, together with F, are greater than CD, together with E.

15.

B

H

D

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 remainder GB fhall be to the remainder b 19. 5. HD, as the whole AB is to the whole b CD: But AB is greater than CD, therefore GB is 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 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.

a A. 5.

ACEF

Sce N.

R

PROP. F. THEOR.

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

Let

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A. B. C.

151

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 others 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 I) to to E
therefore, ex aequali in proportione perturbata
b, 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. 5.

b. 23. 5.

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IF feveral ratios be the fame with several ratios, each to see N.
each; the ratio which is compounded of ratios which
are the fame with the firft ratios, each to each, is the
fame with the ratio compounded of ratios which are the
fame with the other ratios, each to each.

Then the ra

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:
tio 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 fhown 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

1

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

a 22. 5.

See N.

of com

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 first, 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 first, or the remaining ratio of the first, if but one remain, is the fame with the ratio compounded of those remaining of the laft, or with the remaining ratio of the laft.

A. B. C. D. E. F.
G. H. K. L. M.

Let the first 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 those 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 first ratios be the fame with the ratio of G pounded ratio. to M, which is compounded of 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.

b B. 5.

€ 22. 5.

Because, by the hypothefis, A is to D, as G to K, by invertion, 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.

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Book V.

PROP. K. THEOR.

IF there be any number of ratios, and any number of See N. other ratios fuch, that the ratio 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, each to each, with the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the fame with several of the first ratios, each to each, be the fame with one of the last ratios, or with the ratio compounded of ratios which are the fame, each to each, with feveral of the laft ratios: Then the ratio compounded of ratios which are the fame with the remaining ratios of the first, each to each, or the remaining ratio of the first, if but one remain; is the fame with the ratio compounded of ratios which are the fame with thofe remaining of the last, each to each, or with the remaining ratio of the last.

Let the ratios of A to B, C to D, E to F be the firft ratios; and the ratios of G to H, K to L, M to N, O to P, Q to R, be the other ratios: And let A be to B, as S to T; and C to D, as T to V; and E to F, as V to X: Therefore, by the definition of compound ratio, the ratio of S to X is compounded

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G, H; K, L, M, N, O, P, Q, R. Y, Z, a, b, c, d.

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which are the
each to each :
as Z to a; M

of the ratios of S to T, T to V, and V to X,
fame with the ratios of A to B, C to D, E to F,
Alfo, as G to H, fo let Y be to Z; and K to L,
to N, as a to b, O to P, as b to c; and Q to R, as c to d:
Therefore, by the fame definition, the ratio of Y to d is com-
pounded of the ratios of Y to Z, Z to a, a to b, b to c, and

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