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third, and of the ratio which the third has to the fourth, and so on unto the last magnitude.

For example, if a, b, c, d be four magnitudes of the same kind, the first a is said to have to the last d the ratio compounded of the ratio of a to b, and the ratio of b to c, and the ratio of c to d; or, the ratio of a to d is said to be compounded of the ratios of a to b, b to c, and c to d:

And if a has to b the same ratio which e has to f; and b to c the same ratio that g has to h; and c to d the same that k has to 1; then by this definition, a is said to have to d the ratio compounded of ratios which are the same with the ratios of e to f, g to h, and k to l. And the same thing is to be understood when it is more briefly expressed by saying, a has to d the ratio compounded of the ratios of e to f, g to h, and k to 1.

In like manner, the same things being supposed, if m has to n the same ratio which a has to d; then, for shortness sake, m is said to have to n the ratio compounded of the ratios of e to f, g to h, and k to 1.

XII. In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another.

Geometers make use of the following technical words to signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals.

XIII. Permutando, or alternando, by permutation, or alternately. This word is used when there are four proportionals, and it is inferred that the first has the same ratio to the third which the second has to the fourth; or that the first is to the third as the second to the fourth: as is shewn in the 16th Prop. of this 5th Book.

XIV. Invertendo, by inversion; when there are four proportionals, and it is inferred that the second is to the first as the fourth to the third. Prop. B. Book 5.

XV. Componendo, by composition; when there are four proportionals, and it is inferred that the first, together with the second, is to the second as the third, together with the fourth, is to the fourth. 18th Prop. Book 5.

XVI. Dividendo, by division; when there are four proportionals, and it is inferred that the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth. 17th Prop. Book 5.

XVII. Convertendo, by conversion; when there are four proportionals, and it is inferred that the first is to its excess above the second as the third to its excess above the fourth. Prop. E. Book 5.

XVIII. Ex æquali (sc. distantia), or ex æquo, from equality of distance; when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred that the first is to the last of the first rank of magnitudes as the first is to the last of the others. Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken, two and two.

XIX. Ex æquali, from equality. This term is used simply by itself, when the first magnitude is to the second of the first rank as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order. And the inference is as mentioned in the preceding definition, whence this is called "ordinate proportion." It is demonstrated in 22d Prop. Book 5.

XX. Ex æquali in proportione perturbata seu inordinata, from equality in perturbate or disorderly proportion.* This term is used when the first magnitude is to the second of the first rank as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank, and so on in a cross order: and the inference is as in the 18th definition. It is demonstrated in the 23d Prop. of Book 5.

AXIOMS.

I. Equimultiples of the same or of equal magnitudes are equal to one another.

II. Those magnitudes of which the same or equal magnitudes are equimultiples are equal to one another.

III. A multiple of a greater magnitude is greater than the same multiple of a less.

IV. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

PROPOSITION I.-THEOREM.

If any number of magnitudes be equimultiples of as many others, each of each; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the others.

LET any number of magnitudes a b, c d be equimultiples of as many others e, f, each of each: whatsoever multiple ab is of e, the same multiple shall a b and cd together be of e and fto- a gether.

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Because ab is the same multiple of e that cd is of f, as many magnitudes as are in a b equal to e, so many are there g in cd equal to f. Divide a b into magnitudes equal to e, viz. ag, gb; and cd into ch, hd equal each of them to f: the number therefore of the magnitudes ch, hd, shall be equal to the number of the others ag, gb; and because ag is equal to e, and ch to f, therefore a g and ch together are equal to (i. 2 ax.) e and f together. For the same reason, because gb is equal to e. and hd to f; gb and hd together are equal to e and f together. Wherefore as many magni- h tudes as are in ab equal to e, so many are there in ab, cd together equal to e and f together. Therefore, whatsoever multiple a b is of e, the same multiple is a b and c d together d of e and f together.

Therefore, if any magnitudes, how many soever, he equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other for the same demonstration holds in any number of magnitudes, which was here applied to two. Q. E. D.

* Archimedes de Sphæra et Cylindro, Prop. 4. lib. 2.

PROPOSITION II.-THEOREM.

If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth. LET ab the first, be the same multiple of c the second, that de the third is off the fourth; and bg the fifth, the same multiple of c the second, that eh the sixth is of f the fourth: then is a g, the first, together with the fifth, the same multiple of c the second, that dh, the third, together with the sixth, is of f the fourth.

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Because ab is the same multiple of c, that de is off; there are as many magnitudes in ab equal to c, as there are in de equal to f. In like manner, as many as there are in bg equal to c, so many are there in eh equal to f: as many, then, as are in the whole ag equal to c, so many are there in the whole dh equal to f; therefore a g is the same multiple of c, that dh is off; that is, ag the first and fifth together, is the same multiple of the second c, that dh the third and sixth together is of the fourth f. If, therefore, the first be the same multiple, &c. Q. E. D.

COR. From this it is plain, that if any number of magnitudes ab, bg, gh, be multiples of another c; and as many de, ek, kl be the same multiples of f, each of each; the whole of the first, viz. ah, is the same multiple of c, that the whole of the last, viz. dl, is of f

PROPOSITION III.-THEOREM.

If the first be the same multiple of the second, which the third is of the fourth; and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth. LET a the first, be the same multiple of b the second, that c the third is of d the fourth; and of a, c, let the equimultiples e f g h be taken: then ef is the same multiple of b, that g h is of d.

Because ef is the same multiple of a, that gh is of c, there are as

many magnitudes in ef equal to a, as

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are in gh equal to c Let ef be di- f vided into the magnitudes e k, kf, each equal to a, and gh into gl, 1h, each equal to c the number therefore of the magnitudes ek, kf, shall be equal to the number of the others g1, 1h: and because a is the same multiple of b, that c is of d, and that ek is equal to a, and gl to c; therefore ek is the same multiple of b, that gl is of d. For the same reason, kf is the same multiple of b, that 1h is of d; and so, if there be more parts in ef, gh, equal to a, c because, therefore, the first ek is the same multiple of the second b, which the third gl is of the fourth d, and that the fifth kf is the same multiple of the second b, which the sixth 1h is of the fourth d; ef the first together with the fifth, is the same multiple (ii. 5) of the second b, which gh the third together with the sixth, is of the fourth d. If, therefore, the first, &c. Q. E. D.

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PROPOSITION IV. THEOREM.

If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth, viz. the equimultiple of the first shall have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth.

LET a the first have to b the second, the same ratio which the third c has to the fourth d; and of a and c let there be taken any equimultiples whatever e, f; and of b and d any equimultiples whatever g, h: then e has the same ratio to g, which f has to h

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Take of e and f any equimultiples whatever k, l, and of g, h, any equimultiples whatever m, n: then, because e is the same multiple of a that f is of c and of e and f have been taken equimultiples k, 1; therefore k is the same multiple of a, that 1 is of c (v. 3). For the same reason, m is the same multiple of b, that n is of d and because, as a is to b, so is c to d (hypoth.), and of a and c have been taken certain equimultiples k,l: and of b and d have been taken certain equimultiples m, n; if therefore k be greater than m, 1 is greater than n and if equal, equal; if less, less (v. def. 5). And k, 1 are any equimultiples whatever of e, f; and m, n any whatever of g h: as therefore e is to g, so is (v. def. 5) f to h. Therefore, if the first, &c. Q. E. D.

COR. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth. And in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth.

Let a the first have to b the second, the same ration which the third c has to the fourth d, and of a and c let e and f be any equimultiples whatever; then e is to b, as f to d.

Take of e, f any equimultiples whatever k, l, and of b, d any equimultiples whatever gh; then it may be demonstrated, as before, that k is the same multiple of a, that 1 is of c. And because a is to b, as c is to d, and of a and c certain equimultiples have been taken, viz. k and 1; and of b and d certain equimultiples g, h; therefore, if k be greater than g, 1 is greater than h; and if equal, equal; if less, less (v. def. 5). And K1 are any equimultiples of e, f, and g, h any whatever of b, d; as therefore e is to b, so is f to d. And in the same way the other case is demonstrated.

PROPOSITION V.—THEOREM.

If one magnitude be the same multiple of another which a magnitude taken from the first is of a magnitude taken from the other, the remainder shall be the same multiple of the remainder that the whole is of the whole.

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LET the magnitude ab be the same multiple of cd, that ae taken from the first, is of cf taken from the other; the remainder eb shall be the same multiple of the remainder fd, that the whole ab is of the whole cd.

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the same multiple of fd that ae is of cf: ag, therefore ae is (v. 1) the same multiple of cf, that eg is of cd. But ae, by the hypothesis, is the same multiple of cf, that ab is of cd: therefore eg is the same multiple of cd that ab is of cd; wherefore eg is equal to a b (v. ax. 1). Take from them the common magnitude a e; the remainder ag is equal to the remainder eb. Wherefore, since ae is the same multiple of cf, that ag is of fd, and that ag is equal to eb; therefore ae is the same multiple of cf, that eb is of fd. But ae is the same multiple of of that ab is of cd; therefore eb is the same multiple of fd, that a b Therefore, if one magnitude, &c. Q. E. D.

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