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Because BG is to C, as EH to F;

by inversion, C is 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 DĦto 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.

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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 F shall 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. Q.E.D.

PROPOSITION G. THEOREM.

If several ratios be the same to several ratios, each to each; the ratio which is compounded of ratios which are the same to the first ratios, each to each, shall be the same to the ratio compounded of ratios which are the same to 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 compounded of the ratios of K to L, and L to M, which are the same with the ratios of A to B, and C to D.

Again, 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 0, and O to P, which are the same with the ratios of E to F, and G to H: and it is to be shewn that the ratio of K to M, is the same with the ratio of N to P;

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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 so is) O to F: ex æquali, K is to M, as N to P. (v. 22.)

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PROPOSITION H. THEOREM.

If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios; and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is compounded of several of them; then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio compounded of these remaining ratios. 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 Eto M:

also, let the ratio of A to F, which is compounded of the first ratios, be the same with the ratio of G to M, which is compounded of the other ratios;

and besides, 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 same 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, shall be the same with the ratio of K to M, which is compounded of the remaining ratios of K to L, and Z to M of the other ratios.

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

Because, by the hypothesis, 4 is to D, as G to K,
by inversion, D is to A, as K to G; (V. B.)
and as A is to F, so is G to M; (hyp.)
therefore, ex æquali, D is to F, as K to M. (v. 22.)
If, therefore, a ratio which is, &c.

Q.E.D.

PROPOSITION K. THEOREM,

If there be any number of ratios, and any number of other ratios, such, that the ratio which is compounded of ratios which are the same to the first ratios, each to each, is the same to the ratio which is compounded of ratios which are the same, each to each, to the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the same to several of the first ratios, each to each, be the same to one of the last ratios, or to the ratio. which is compounded of ratios which are the same, each to each, to several of the last ratios; then the remaining ratio of the first, or, if there be more than one, the ratio which is compounded of ratios which are the same each to each, to the remaining ratios of the first, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio which is compounded of ratios which are the same each to each to these remaining ratios.

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, 0 to P, Q to R, be the other ratios :

and let A be to B, as S to T; and C to D, as 7'to V; and E to F, as V to X:

therefore, by the definition of compound ratio, the ratio of S to Xis compounded of the ratios of S to T, T to V, and to X, which are the same to 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 e; 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 a, a to b, b to c, and c to d, which are the same, each to each, to the ratios of G to H, K to L, M to N, 0 to P, and Q to R:

therefore, by the hypothesis, S is to X, as Y to d.

Also, let the ratio of A to B, that is, the ratio of S to T, which is one of the first ratios, be the same to the ratio of e to g, which is compounded of the ratios of e to f, and ƒ to g, which, by the hypothesis, are the same to the ratios of G to H, and K to Z, two of the other ratios;

and let the ratio of h to l be that which is compounded of the ratios of h to k, and k to 7, which are the same to the remaining first ratios, viz. of C to D, and E to F;

also, let the ratio of m to p, be that which is compounded of the ratios of m to n, n to o, and o to p, which are the same, each to each, to the remaining other ratios, viz. of M to N, O to P, and Q to R. Then the ratio of h to 7 shall be the same to the ratio of m to p; or h shall be to 7, as m to p..

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Because e is to f, as (G to H, that is, as) Y to Z;
and fis to g, as (K to L, that is, as) Z to a
therefore, ex æquali, e is to g, as Y to a: (v. 22.)
and by the hypothesis, A is to B, that is, S to T, as e to g;
wherefore S is to T, as Y to a; (v. 11.)
and by inversion, Tis to S, as a to Y: (V. B.)
but S is to X, as Y to D; (hyp.)

therefore, ex æquali, Tis to X, as a to d:

also, because h is to k, as (C to D, that is, as) Tto V; (hyp.)
and k is to l as (E to F, that is, as) V to X;
therefore, ex æquali, h is to l, as T to X:

in like manner, it may be demonstrated, that m is to p, as a to d;
and it has been shewn, that Tis to X, as a to d;

therefore h is to 7, as m to p. (v. 11.) Q. E.D.

The propositions & and K are usually, for the sake of brevity, expressed in the same terms with propositions F and H: and therefore it was proper to shew the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers.

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In the first four Books of the Elements are considered, only the absolute equality and inequality of Geometrical magnitudes. The Fifth Book contains an exposition of the principles whereby a more definite comparison may be instituted of the relation of magnitudes, besides their simple equality or inequality,

The doctrine of Proportion is one of the most important in the whole course of mathematical truths, and it appears probable that if the subject were read simultaneously in the Algebraical and Geometrical form, the investigations of the properties, under both aspects, would mutually assist each other, and both become equally comprehensible; also their distinct characters would be more easily perceived.

Def. I, II, In the first Four Books the word-part is used in the same sense as we find it in the ninth axiom, "The whole is greater than its part:"' where the word part means any portion whatever of any whole magnitude :. but in the Fifth Book, the word part is restricted to mean that portion of magnitude which is contained an exact number of times in the whole. For instance, if any straight line be taken two, three, four, or any number of times another straight line, by Euc. 1. 3; the less line is called a part, or rather a submúltiple of the greater line; and the greater, a multiple of the less line. The multiple is composed of a repetition of the same magnitude, and these definitions suppose that the multiple may be divided into its parts, any one of which is a measure of the multiple. And it is also obvious that when there are two magnitudes, one of which is a multiple of the other, the two magnitudes must be of the same kind, that is, they must be two lines, two angles, two surfaces, or two solids: thus, a triangle is doubled, trebled, &c., by doubling, trebling, &c. the base, and completing the figure. The same may be said of a parallelogram. Angles, arcs, and sectors of equal circles may be doubled, trebled, or any multiples found by Prop. xxvI-xxix, Book III.

Two magnitudes are said to be commensurable when a third magnitude of the same kind can be found which will measure both of them; and this third magnitude is called their common measure: and when it is the greatest magnitude which will measure both of them, it is called the greatest common measure of the two magnitudes : also when two magnitudes of the same kind have no common measure, they are said to be incommensurable. The same terms are also applied to numbers.

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Unity properly so called, may be assumed to represent that portion of every kind of magnitude which is taken as the measure, of all magnitudes of the same kind. The composition of units cannot produce Geometrical magnitude; three units are more in number than one unit, but still as different from magnitude as unity itself. Numbers may be considered as quantities, for we consider every thing that can be exactly measured, as a quantity.

Unity is a common measure of all rational numbers, and all numerical reasonings proceed upon the hypothesis that the unit is the same throughout the whole of any, particular process. Euclid has not fixed the magnitude of any unit of length, nor made reference to any unit of measure of lines, angles, surfaces, or volumes. Hence arises an essential difference between number and magnitude; unity, being invariable, measures all rational numbers; but though any quantity be assumed as the unit of magnitude, it is impossible to assert that this assumed unit will measure all other magnitudes of the same kind,

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All whole numbers therefore are commensurable; for unity is their common measure; also all rational fractions proper or improper, are commensurable;. for

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