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Book V.

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Next, Let there be four magnitudes, A, B, C, D, and o-
ther four E, F, G, H, which, taken two and
two in a cross order, have the same ratio, viz. A. B. C. D.
A to B, as G to H; B to C, as F to G; and E. F. G. H.
C 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
lame 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 magni-
tudes. Therefore, if there be any number, &c. Q. E. D.

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

third has to the fourth; and the fifth to the second, the fame 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 fixth together have to the fourth,

Let AB the first, have to the second, the fame ratio which
DE the third, has to F the fourth; and let BG the fifth, hayo
to C the second, the same ratio which EH
: the fixth, has to F the fourth : AG, the G
first and fifth together, shall have to C the

H
second, the same ratio which DH, the
third and fixth together, has to f the
fourth.

B
Because BG is to C, as EH to F; by in.

E
version, C is to BG, as F to EH; And be-
cause, as AB is to C, so is DE to F; and
as C to BG, fo F to EH; ex aequali", AB

a 22. 5a
is to BG, as DE to EH: And because
these magnitudes are proportionals, they
shall likewise be proportionals when taken AC DF
jointly; as therefore AG is to GB, fo is
DH to HE; but as GB to C, so is HE to F. Therefore, ex
aequalia, as AG is to C, fo is DH to F. Wherefore, if the
first, &c. Q. E. D.

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

the

b 18. 39

K 3

Book V. the excess of the third and Gxth to the fourth : The demonftra

tion of this is the same with that of the propoßtion, if divifion 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 fame ratio that the correfponding one of the second rank has to a fourth magnitude ; as is manifeft.

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IF four magnitudes of the same kind are proportionals,

the greatest and laest of them together are greater than the other two together.

15.

I.et 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 consequently F the least'. AB, together with F, are greater

than CD, together with E.

Take AG equal to E, and CH equal to F: Then, because as AB is to CD, lo is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG to CH.

B
And because AB the whole, is to the
whole CD, as AG is to CH, likewise the G D

remainder GB shall be to the remainder
b 19. 5. HD, as the whole AB is to the wholeb HH

CD: But AB is greater than CD, therea A. S.

forec GB is greater than HD: And be.
cause 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

A C E F 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.

PRO P. F. THE O R.

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

Let 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

A. B. C. of compound ratio, is the ratio of A to C, is

D. E. F. 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.

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 aequali, A is to C, as D to Fa.

Next, Let A be to B, as E to F, and B to C, as D) to to E; therefore, ex aequali in proportione perturbata b, A is to C, as D to F; that is, the ratio of A

b 23. s.

A. B. C. to C, which is compounded of the ratios of A to

D, E. F. 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.

a 22. S.

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IF several ratios be the same with several ratios, each to See N.

each ; the ratio which is compounded of ratios which are the same with the first ratios, each to each, is the faine 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. K. L. M. ed of the ratios of K to L, and

E. F. G. H. N. 0. P. L to M, which are the same with the ratios of A to B, and C to D: And as E to F, fo let N be to 0); and as G to H, so let O be to P; then the ratio of N to P is compounded of the ratios of N to O, and 0 to P, which are the same with the ratios of E to F, and G to H: And it is to be shown that the ratio of K to M, is the same 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 0; and as L to M, so is (C to D, and so is G to H,

and

K 4

Book V. and so is) O to P: Ex aequali', K is to M, as N to P. Therew fore, if several ratios, &c. Q. E. D.

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See N.

IF a ratio compounded of several ratios be the same

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 same with one of the last ratios, or with the ratio compounded of any of the last ; then the ratio compounded of the remaining ratios of the first, or the remaining ratio of the first, if but one remain, is the same with the ratio compounded of those remaining of the last, or with the remaining ratio of the last.

Let the first ratios be those 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: Allo, let the ratio of A to a Definition F, which is compounded of the first of com

ratios be the same with the ratio of G A. B. C. D. E. F. pounded ratio.

to M, which is compounded of the G. H K L. M.
other ratios : And besides, let the ra-
tio of A to D, which is compounded of the ratio 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 first 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 same 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 hypothesis, A is to D, as G to K, by inB. s.

verbion , D is to A, as K to G; and as A is tó F, so is to ( 22. s. M; therefore, ex aequali, D is to F, as K to M. If therefore

a ratio which is, &c. Q E. D.

PROP.

Book V.

PRO P. K.

THEOR.

If there be any number of ratios, and any number of sce N.

other ratios fuch, that the ratio compounded of ratios which are the same with the first ratios, each to each, is the same 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 same with several of the first ratios, each to each, be the same with one of the last ratios, or with the ratio compounded of ratios which are the same, cach to each, with several of the last ratios : Then the ratio compounded of ratios which are the same 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 those 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 first 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 des finition of compound ratio, the ratio of S to X is compounded

h, k, l.
A, B; C, D, E, F.

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

e, f, g. m, n, o, p.

of the ratios of S to T, T to V, and V to X, which are the same with the ratios of A to B, C to D, E to F, each to each : Also, as G to H, to let y be to Z; and K to L, as Z to a; M to N, as a to b, O to P, as b to c; and Q to R, as c to d: Therefore, by the same definition, the ratio of Y to d is compounded of the ratios of Y to Z, 2 to a, a to b, b to c, and

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