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Coroll. From hence it is manifeft, that the Side of the Hexagon is equal to the Semidiameter of the Circle. And if we draw thro' the Points A, B, C, D, E, F, Tangents to the Circle, an equilateral and equiangular Hexagon will be described about the Circle, as is manifeft from what has been faid concerning the Pentagon. And fo likewife may a Circle be infcribed and circumfcribed about a given Hexagon; which was to be done.

PROPOSITION XVI.

PROBLEM.

To defcribe an equilateral and equiangular Quindecagon in a given Circle.

LET ABCD be a Circle given. It is required

to describe an equilateral and equiangular Quindecagon in the fame.

Let AC be the Side of an equilateral Triangle infcribed in the Circle ABCD, and AB the Side of a Pentagon. Now if the whole Circumference of the Circle ABCD be divided into fifteen equal Parts, the Circumference ABC, one third of the whole, fhall be five of the faid fifteen equal Parts; and the Circumference AB, one fifth of the whole will be three of the faid Parts. Wherefore the remaining. Circumference BC, will be two of the faid Parts. And if BC be bifected in the Point E, BE, or EC, will be one fifteenth Part of the whole Circumference ABCD. And fo if BE, EC, be joined, and either EC, or EB, be continually applied in the Circle, there Thall be an equilateral and equiangular Quindecagon defcribed in the Circle ABCD; which was to be done.

If, according to what has been faid of the Pentagon, Right Lines are drawn thro' the Divifions of the Circle touching the fame, there will be described about the Circle an equilateral and equiangular Quindecagon. And, moreover, a Circle may be infcribed, or circumfcribed, about a given equilateral and equiangular Quindecagon.

EUCLID's

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EUCLID's

ELEMENTS.

BOOK V.

DEFINITIONS.

A

PART, is a Magnitude of a Magnitude, the lefs of the greater, when the leffer meafures 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 Magnitudes of the fame kind, according to Quantity. IV. Magnitudes are faid to have Proportion to each other, which being multiplied can exceed one another.

V. Magnitudes are faid to be in the fame Ratio, the first to the fecond, and the third to the fourth, when the Equimultiples of the first and third, compared with the Equimultiples of the fecond and fourth, according to any Multiplication whatsoever, are either both together greater, equal, or lefs, than the Equimultiples of the fecond and fourth, if those be taken that anfwer each other.

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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, laftly, if the Multiple of the firft be less than the Multiple of the fecond; and alfo that of the third lefs than that of the fourth, and these Things happen according to every Multiplication whatsoever; then the four Magnitudes are in the fame Ratio, the first to the second, as the third to the fourth.

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

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

Magnitudes are faid to be Proportionals, when the firft is the fame Equimultiple of the fecond, as the third is of the fourth, or the fame Part, or Parts.

But this Definition appertains only to Numbers and commenfurable Quantities; and fo fince it is not univerfal, Euclid did well to reject it in this Element, which treats of the Properties of all Proportionals; and to fubftitute another general one, agreeing to all Kinds of Magnitudes. In the mean time, Expounders very much endeavour to demonftrate the Definition here laid down by Euclid, by the usual received Definition of proportional Numbers; but this much easier flows from that, than that from this; which may thus demonftated:

be

First, Let A, B, C, D, be four Magnitudes which are in the fame Ratio, according to the Conditions that Magnitudes in the fame Ratio muft have laid 6 down in the fifth Definition. And let the first be a Multiple of the fecond. I fay, the third is alfo the fame Multiple of the fourth. For Example: Let A

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