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Next, let there be four magnitudes, , B, C, D, and Book V. other tour E, F, G, H, which, taken tu'o and two, in a cross order, bave the same 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 maquitudes, 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 E to H; but C is to D, as E is to F; wherefore agiin, 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. TIIEOR. If the first has to the second the same ratio see 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; G! AG, the first and fifth together, shall

H . have to the second, the same ratio which DH, the third and sixth together, gi lias to F the fourth.

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 F to EH; ex æqualia, AB is to BG, as DE to EH: And

* 2%. 5. because these magnitudes are proportionals, they shall likewise be proportionals when taken jointly b; as therefore AG is to GB, so is DH 18. 5. 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.

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

Boor V. as the excess of the third and sixth to the fourth : The de.

monstration 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 aré greater than the other two together.

Let the four magnitudes AB, CD, E, F be proportion

als, viz. AB to CD, as E to F; and let AB be the greatest 2A. & 14.5. of them, and consequently F the least a. 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, so 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 B, to the whole CD, as AG is to CH, like a wise the remainder GB shall be to the

remainder HD, as the whole AB is to the - "19. 5. wholeb CD: But AB is greater than

| H1 * A. 5. 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 A C EX
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.

PROP. F. THEOR.
See N. RATios which are compounded of the same ra-

tios, are the same with one another.

Let A be to B, as D to E; and B to C, as E to F: The Book 1'. ratio which is compounded of the ratios of A to B, and B to C, which by the defini- | A. B. C. tion of compound ratio, is tlic ratio of A to D. E. F. C, is 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 æquali A is to C, as D to Fa.

- 22. 3. Next, let A be to B, as E to F, and B to C, as D to E; therefore, ex aquali in proportione perturbuta", A is to C, 23. 5. as D to F; that is, the ratio of A to C, which, is compounded of the ratios of A to B, and B | A. B. C. to C, is the same with the ratio of D to F, D. E. F. | which is compounded of the ratios of D to E, and E to F: Andin like manner the proposition may be demonstrated, whatever be the number of ratios in either case.

PROP. G. THEOR.

IF several ratios be the same with several ratios, See
each to each; the ratio which is 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 same with the other ratios,
each to each.

Let A be to B, as E to F; and C to D, as G toll: 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 de-
finition of compound ratio, is [A. B. C. D. K.L.M.!
compounded of the ra:ios of Kto E.F.G.H. N.O.P.I
L, and L to M, which are the
same with the ratios of A to B, and C to D: And as E to
F, so 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 (), 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

Buok V. H, and so is O to P:) Ex æqualia K is to M, as N to P. w Therefore, if several ratios, &c. Q. E. D.

a 22. 5.

PROP. H. THEOR.

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 rernaining 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; also, let the ratio of • Definition A to F, which is conipounded of a of com.. the first ratios, be the same with the RE F E pounded ratio of G to M, which is com- G. H. K. L. M. ,

pounded of the other ratios; And
besides, let the ratio of A to D,
which is compounder of the ratios of A to B, B to C, C to
D, be the same with the ratio of G to K, which is com-
pounded 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 inbB. 5. version", D is to A, as K to G; and as A is to F, so is G to “ 22. 5. M; thereforec, ex æquali, D is to F, as K to M. If there

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

ratio.

Boox V,

PROP. K. THEOR.

If there be any number of ratios, and any num - See N, ber of other ratios such, 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 same, each to each, with the last ratios; and if one of the first ratios, or the ratio which ie 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, each 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 same with the ratio come pounded of ratios which are the same 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 definition of compound ratio, the ratio of S to X is

h, k, 1.
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. compounded 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, so let Y be to Z; and K to L, as Z to a, M to N, as a to b, 0 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, Z to ag a to b, b to c, and c to d, which are the same, each to

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