Coroll. From hence it is manifeft, that the Side of the Hexagon is equal to the Semidiameter of the Cirele. And if we draw, thro' the Points A, B, C, D, E, F, Tangents to the Circle, an equilateral and equiangular Hexagon, will be defcribed 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.

LE

ET ABCD be a Circle given. It is required to defcribe an equilateral and equiangular Quindecagon in the fame.

Let AC be the Side of an equilateral Triangle infcribed in the Circle A B C D, and A B the Side of a Pentagon. Now, if the whole Circumference of the Circle ABCD be divided into fifteen equal Parts, the Circumference A B C, one Third of the Whole, fhall be five of the faid fifteen equal Parts; and the Circumference A B, one Fifth of the Whole, will be three of the faid Parts: Wherefore the remaining Circumference BC will be two of the faid Parts. if BC be bifected in the Point E, then BE, or EC, will be one fifteenth Part of the whole Circumference ABCD. And fo, if B E, E C, be joined, and either EC, or E B, be continually applied in the Circle, there fhall be an equilateral and equiangular Quindecagon defcribed in the Circle A B CD; which was to be done.

And

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

EUCLI D's

ELEMENTS,

BOOK V.

DEFINITION S.

1. A Part is a Magnitude of a Magnitude, the Lefs of the Greater, when the Leffer mea

fures 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 whatfoever, are either both together, greater, equal, or lefs, than the Equimultiples of the fecond and fourth, if those be taken that anfwer each other.

That

That is, if there be four Magnitudes, and you take any Equimultiples of the first and third, and also any Equimultiples of the fecond and fourth; and if the Multiple of the firft be greater than the Multiple of the fecond; and alfo 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 fecond; and also the Multiple of the third equal to the Multiple of the fourth; or, laftly, if the Multiple of the first be less than the Multiple of the fecond; and also that of the third less than that of the fourth, and these Things happen according to every Multiplication whatfoever: Then the four Magnitudes are in the fame Ratio; the first to the fecond, as the third to the fourth.

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

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

Numbers are proportional, when the firft is either the fame Multiple of the fecond, as the third is of the fourth, or elfe 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 ufual received Definition of proportional Numbers; but this much eafiér flows from that, than that from this; which may be thus demonstrated:

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 must have according to the fifth Definition; and let the firft be a Multiple of the fecond: I fay, the third is alfo the fame Multiple of the fourth. For Example: Let A be equal to 5B: Then C fhall be equal to 5D. Take any Num

1 4

ABC: D

2A, 10B, 2C, 10D

Number, for Example, 2, by which let 5 be multiplied, and the Product will be 10: And let 2A, 2C, be Equimultiples of the first and third Magnitudes A and C: Alfo, let 10B and roD be Equimultiples of the fecond and fourth Magnitudes B and D. Then (by Def. 5.) if 2A be equal to 10B, 2C fhall be equal to 10D. But fince A (from the Hypothefts) is five Times B, 24 fhall be equal to 10B; and fo 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 fome Part of B; then B will be a Multiple of A: And fo (by Cafe 1.) D will be the fame Multiple of C; and accordingly C fhall 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 fay, 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. 1 fay, C is alfo equal to D. For, becaufe A is equal B, each of them being multiplied by 4, then 4A will be equal to 5B. And fo, if the Equimultiples of the first A B C D and third, viz. 4A, 4C, be af

fumed; as alfo the Equímulti

ples of the fecond and fourth, 4A, 5B, 4C, 5D viz. 5B, 5D; and (by the Definition) if 4A is equal to 5B; then 4C is equal to 5D. But 4A has been proved equal to 5B, and fo 4C fhall be equal to 5D, and C equal to D. W. W. D.

n

And univerfally, if A be equal to-B, C will be

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