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When four magnitudes are proportionals it is usually expressed by saying, the first is to the second as the third is to the fourth.

7. When of the equimultiples of four magnitudes, taken as in the fifth definition, the multiple of the first is greater than the multiple of the second, but the multiple of the third is not greater than the multiple of the fourth, then the first is said to have to the second a greater ratio than the third has to the fourth; and the third is said to have to the fourth a less ratio than the first has to the second.

8. Analogy, or proportion, is the similitude of ratios. 9. Proportion consists in three terms at least.

10. When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.

[The second magnitude is said to be a mean proportional between the first and the third.]

11. When four magnitudes are continued proportionals, the first is said to have to the fourth, the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c. increasing the denomination still by unity, in any number of proportionals.

Definition of compound ratio. When there are any number of magnitudes of the same kind, the first is said to have to the last of them, the ratio which is compounded of the ratio which the first has to the second, and of the ratio which the second has to the 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 of the ratio of B to C, and of 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 that E has to F; and B to C the same ratio that G has to H; and C to D the same ratio that K has to L; 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 L.

In like manner, the same things being supposed, if M has to N the same ratio that A has to D; then, for the sake of shortness, M is said to have to N the ratio compounded of the ratios of E to F, G to H, and K to L.

12. In proportionals, the antecedent terms are said to be 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 the magnitude of proportionals, so that they continue still to be proportionals.

13. Permutando, or alternando, by permutation or alternately; when there are four proportionals, and it is inferred that the first is to the third, as the second is to the fourth. V. 16.

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

15. 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. V. 18.

16. 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. V. 17.

17. 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 is to its excess above the fourth. V. E.

18. Ex æquali distantia, or ex æquo, from equality of distance; when there is any number of magnitudes more than two, and as many others, such 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.

19. Ex æquali. This term is used simply by itself, when the first magnitude is to the second of the first rank, as the first is to the second of the other rank; and the second is to the third of the first rank, as the second is to the third of the other; and so on in order; and the inference is that mentioned in the preceding definition. V. 22.

20. Ex æquali in proportione perturbatâ seu inordinatâ, 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 the second is to the third of the first rank, as the last but two is to the last but one of the second rank; and the third is to the fourth of the first rank, as the last but three is to the last but two of the second rank; and so on in a cross order; and the inference is that mentioned in the eighteenth definition. V. 23.

AXIOMS.

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

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

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

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

PROPOSITION 1. THEOREM. If any number of magnitudes be equimultiples of as many, each of each; whatever multiple any one of them is of its part, the same multiple shall all the first magni

Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each: whatever multiple AB is of E, the same multiple shall AB and CD together, be of E and F together.

For, because AB is the same multiple of E, that CD is of F, as many magnitudes as there are in AB equal to E, so many are there in CD equal to F. Divide AB into the magnitudes AG, GB, each equal to E; and CD into the magnitudes CH, HD, each equal to F. Therefore the number of the magnitudes CH, HD, will be equal to the number of the magnitudes AG, GB.

And, because AG is equal to E, and CH equal to F, therefore AG and CH together are equal to E and F together ; and because GB is equal to E, and HD equal to F, therefore GB and HD together are equal to E and F together. [Axiom 2. Therefore as many magnitudes as there are in AB equal to E, so many are there in AB and CD together equal to E and F together. Therefore whatever multiple AB is of E, the same multiple is AB and CD together, of E and F together.

Wherefore, if any number of magnitudes &c. Q.E.D.

PROPOSITION 2. THEOREM. 'If the first 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; the first together with the fifth shall 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 of F the fourth, and let BG the fifth be the same multiple of the second, that EH the sixth is of F the fourth : AG, the first together with the fifth, shall be the same multiple of C the second, that DH, the third together with the sixth, is of F the fourth.

For, because AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there in DE equal to F. For the same reason, as many

A, magnitudes as there are in equal to C, so many are there in EH equal to F. Therefore as many magnitudes as there are in the whole AG equal to C, so many are there in the whole DH equal to F. Therefore AG is the same multiple of C that DH is of F.

Wherefore, if the first be the same multiple &c. Q.E.D.

COROLLARY. From this it is plain, that if any number of magnitudes AB, , GH be multiples of another C; and as many DE, EK, KL be the same multiples of F, each of each; then the whole of the first, namely, AH, is the same multiple of C, that the whole of the last, namely, H C F DL, is of F.

PROPOSITION 3. THEOREM.

If the first be the same multiple of the second that the third is of the fourth, and if of the first and the third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth.

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