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EUCLI D's ELEMENTS.

BOOK V.

DEFINITIONS.

PART, is a Magnitude of a Magnitude,
the less of the greater, when the leffer
measures the greater.

II. But a Multiple is a Magnitude of a
Magnitude, the greater of the leffer,

when the leffer measures the greater, III. Ratio, is a certain mutual Habitude of the Magnitudes of the same kind, according to Quantity. IV. Magnitudes are said to have Proportion to each other, which being multiplied can exceed one another, V. Magnitudes are said to be in the same Ratio, the first to the second, and the third to the fourth, when the Equimultiples of the first and third, compared with the Equimultiples of the second and fourth, ac cording to any Multiplication whatsoever, are either both together greater, equal, or less, than the Equiz multiples

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multiples of the second and fourth, if those be taken that answer each other.

That is, if there be four Magnitudes, and you take any Equimultiples of the first and third, and also any Equimultiples of the second and fourth. And if the Multiple of the first be greater than the Multiple of the second, and also the Multiple of the third greater than the Multiple of the fourth: Or, if the Multiple of the first be equal to the Multiple of the second; and also the Multiple of the third equal to the Multiple of the fourth: Or, lastly, if the Multiple of the first be less. than the Multiple of the second; and also that of the third less than that of the fourth; and these Things happen according to every Multiplication whatsoever; then the four Magnitudes are in the same Ratio, the first to the second, as the third to the fourth.

VI. Magnitudes that have the same Proportion, are called Proportionals.

Expounders usually lay down here that Definition which Euclid has given for Numbers only, in his seventh Book, viz. That

Magnitudes are said to be Proportionals, when the first is the same Equimultiple of the second, as the third is of the fourth, or the same Part, or Parts.

But this Definition appertains only to Numbers and Commenfurable Quantities; and so since it is not universal, Euclid did well to reject it in this Element, which treats of the Properties of all Proportionals; and to substitute another general one, agreeing to all Kinds of Magnitudes. In the mean time, Expounders very much endeavour to demonstrate the Definition here laid down by Euclid, by the ufual received Definition of Proportional Numbers; but this much eafier flows from that, than that from this; which may be thus demonftrated:

First, Let A, B, C, D, be four Magnitudes which are in the fame Ratio, according to the Conditions that Magnitudes in the same Ratio must have, laid down 'in the fifth Definition. And let the first be a Multiple of the second. I say, the third is also the fame

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same Multiple of the fourth. For Example: Let A be equal to 5 B. Then C shall be equal to 5 D. Take any Number. For Example, 2, by which let sbe

multiplied, and the Product will be 10: And let 2A, 2C, be E- A: B:: C: D quimultiples of the first and 2A, 10B, 2C, 10D third Magnitudes A and C:

Also, let 10B and 10D be Equimultiples of the second and fourth Magnitudes B and D. Then (by Def. 5.) if 2 A be equal to 10B, 2C shall be equal to IOD. But since A (from the Hypothesis) is five Times B, 2A shall be equal to 10B; and so 2C equal to 10D, and C equal to 5D; that is, C will be five Times D. W. W. D.

Secondly, Let A be any Part of B; then C will be the fame Part of D. For because A is to B, as Cis to D, and fince A is some Part of B; then B will be a Multiple of A: And so (by Cafe 1.) D will be the fame Multiple of C, and accordingly C shall be the fame Part of the Magnitude D, as A is of B. W.W.D.

Thirdly, Let A be equal to any Number of whatfoever Parts of B. I say, C is equal to the fame Number of the like Parts of D. For Example: Let A be a Fourth Part of five Times B; that is, let A be equal to & B. I say, C is also equal to D. For because A is equal to & B, each of them being multiplied by 4, then 4A will be equal to 5 B. And fo if the Equimultiples of the first and third, viz. 4A, 4C be af- A: B:: C: D. sum'd; as also the Equimultiples 4A, 5B, 4C, 5D. of the second and fourth, viz.

5.B, 5D, and (by the Definition) if 4 A is equal to 5B; then 4C is equal to 5 D. But 4 A has been prov'd equal to 5 B, and so 4C shall be equal to 5D, and Cequal to D. W. W. D.

And universally, If A be equal to "B, C will be

n

equal to D. For let A and C

m

be multiplied by m, and Band A: B:: C: D

D byn. And because A is equal

toB; m A shall be equal to

m

mA, B, mC, nD

"B; wherefore (by Def. 5) mC will be equal to nD

and C equal to D. W. W. D.

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