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m A Again, let B

m C

be somewhat more than n, then also so is

somewhat

D

more than n, and therefore m A is > n B, and m C is > n D. Now, collecting these results we have

If m A be < n B, then m C is < n D,

if m An B, then m C =
= n

n D,

or if m A is n B, then m C is > nŊ.

which being compared with Euclid's definition, as symbolically expressed at page 18, will be found to be identical.

That the definition of proportion here given by Euclid was only meant to be applied to geometrical quantities, is evident from the fact that he has given another for proportional numbers in the seventh book; but it should be observed that all his conclusions may be generalized so as to apply with equal truth, in the case of numbers, by the substitution of the word quantity" for "magnitude."

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m A
B

=

n, is than n, or is

The perfection of Euclid's method is, that one demonstration suffices either when than n, whereas, with all other methods, when rigorous proof is requisite, they require two demon

strations to each proposition, one when

m A
B

=

n, and another when

m A

is

> or < n; and this latter case has usually to be proved from the former by a "reductio ad absurdum."

It should be observed that, in any proportion, the first and second terms must be of the same kind, and the third and fourth of the same kind, but the two pairs may differ; thus, the first and second magnitudes may be two lines or angles, while the third and fourth are surfaces or solids.

7. When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the first is greater than that 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 magnitude has to the fourth; and, on the contrary, 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 equality of ratios.

In this definition the term "equality" has been substituted for "similitude," the word employed by Euclid. The whole definition might have been omitted, as being unnecessary.

9. Proportion consists in three terms at least.

This is rather an inference than a definition. Three quantities may form a proportion when the middle term is both the consequent of the first ratio and the antecedent of the second; thus, when A: B: B: C. In such a case B is termed a mean proportional between A and C. When a series of quantities are such that each middle term is the consequent of that which precedes it, and the antecedent of that which follows it, or when, in other words, every term bears an equal ratio to that which follows it, such a series is said to be in continued proportion. In any proportion the first and last terms are called the extremes, and all the others the mean terms.

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.

11. When four magnitudes are continual 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.

12. When there are any number of magnitudes of the same kind, the first is said to have to the last of them the ratio 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 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 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 which A has to D; then, for shortness sake, M is said to have to N, the ratio compounded of the ratios of É to F, G to H, and K to L.

Arithmetically ratios are compounded by multiplying together all the antecedents of the separate ratio for a new antecedent, and all the consequents together for a new consequent. Thus the ratio 120 : 960 is compounded of the ratios 3 : 6, 5: 10, and 8: 16, for 3 × 5 × 8 = 120, and 6 x 10 x 16 = 960.

A duplicate ratio that which is compounded of two equal ratios, as of A: B, B: C; a triplicate ratio is compounded of three equal ratios, as of A: B, BC, C: D; a quadruplicate ratio, is compounded of four equal ratios; a quintuplicate of five equal ratios, and so on.

Thus, if A, B, C, be in continued proportion, then

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that is, AC :: A: B3, or A is to C in the duplicate ratio of A to B. Again, if A, B, C, D, be in continued proportion,

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or A is to D in the triplicate ratio of A to B, and so on with any number of quantities in continued proportion.

13. In proportionals, the antecedent terms are called homologous to one another, as also the antecedents 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.

14. 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 shown in the 16th proposition of this book.

15. 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. Proposition B, book 5.

16. 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. Proposition XVIII., book 5.

17. 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. Proposition XVII., book 5.

18. 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. Proposition E, book 5.

19. Ex æquali (sc. distantiâ), 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."

20. 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 Proposition XXII., book 5.

21. Ex æquali, in proportione; perturbatâ, seu inordinata, from equality, in perturbate or disorderly proportion (Prop. 4, Lib. II. Archimedis de sphæra et cylindro); 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 19th definition. It is demonstrated in Proposition XXIII., book 5. The following table will serve to illustrate and explain the foregoing seven last definitions.

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The terms subduplicate, subtriplicate, and sesquiplicate ratios being frequently employed in astronomy should be defined.

If three quantities be in continued proportion, the first is said to have to the second the subduplicate ratio of that which the first has to the third. Thus, if A, B, C, are in continued proportion, A is said to have to B the subduplicate ratio of that which A has to C, and may be expressed algebraically A: B :: A: C1.

If four quantities be in continued proportion, the first is said to have to the second the subtriplicate ratio of that which the first has to the fourth. Thus, if A, B, C, D, are in continued proportion, A is said to have to B the subtriplicate ratio of that which A has to D, and may be expressed algebraically, A : B :: A3 : D3.

A sesquiplicate ratio is the ratio compounded of the simple ratio and the subduplicate, and may be expressed algebraically, A : B :: A: c.

AXIOMS.

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

Or if equals be multiplied by the same, the products are equal.

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

Or if equals be divided by the same, the quotients are equal.

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

4. Of two magnitudes that one of which a multiple is greater than the same multiple of the other, is the greater.

In the following propositions lines are employed by Euclid to represent proportional magnitudes, but it should be understood that any similar magnitudes might have been employed, such as plane figures, solid bodies, or angles.

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