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DATA.

23. A: By: z

B: C::x:y
&c.

X: Y::b:c

Y: Z:: a:b

QUÆSITA.

A: Za: z.

§ 6. From a given Proportion, combined with Equimultiples, to prove Proportions involving new Ratios.

4.

a:b::c: d

A=ma, and C=mc

B=nb, and D=nd.

Cor. (1) do.

Cor. (2) do.

A: B:: C: D.

Ab: C: d.

a: B::c: D.

§ 7. From given Proportions, combined with Equations (or Inequalities), to prove Equations (or Inequalities).

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THE DEFINITIONS.

I.

A less magnitude is said to be a part of a greater when the less measures the greater; that is, 'when the less is contained a certain number of times exactly in the greater.'

II.

A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is 'when the greater contains the less a certain number of times exactly.'

Euclid always means, by a certain number of times,' 'more than once': but, in Algebra, ‘one' is considered as a number, so that a magnitude is a multiple of itself, and also a part of itself.

III.

Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity.'

This is too vague to be of any practical use. Euclid explains more clearly what he means by 'ratio,' when he comes to define 'identity of ratio' in Def. V. That Definition is applicable to incommensurable, as well as to commensurable, magnitudes whereas the Algebraical Definition of 'ratio,' and so of 'identity of ratio,' is applicable to commensurable magnitudes only.

IV.

Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

Euclid probably meant, by this Definition, to exclude infinite magnitudes.

V.

The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth: or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth: or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

The Algebraical Definition answering to this would be 'The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth, when the first is the same multiple, part, or fraction of the second which the third is of the fourth': but such a Definition would be quite superfluous, as it may be easily deduced from the Algebraical Definition of 'ratio.' This Definition is discussed in the Appendix.

VI.

Magnitudes which have the same ratio are called proportionals.

This Definition is also true in Algebra.

VII.

(To be omitted.)

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.

VIII.

Analogy, or proportion, is the similitude of

ratio.'

This might be expressed, equally well, as 'the equality of ratio,' or 'the identity of ratio.'

IX.

Proportion consists in three terms at least.

This is an Axiom.

X.

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.

XI.

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.

A.

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.

XII.

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

XIII.

Alternando. This word is used when there are four proportionals, and it is inferred that the first is to the third as the second to the fourth. (PROP. XVI.)

XIV.

Invertendo; when there are four proportionals, an it is inferred, that the second is to the first, as th fourth to the third. (PROP. B.)

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