T

PROP. XV. PRO B.

O infcribe an equilateral and equiangular hexagon in a given circle.

Let ABCDEF be the given circle; it is required to infcribe an equilateral and equiangular hexagon in it.

Find the centre Ġ of the circle ABCDEF, and draw the diameter AGD; and from D as a centre, at the distance DG, describe the circle EGCH, join EG, CG, and produce them to the points B, F; and join AB, BC, CD, DE, EF, FA : The hexagon ABCDEF is equilateral and equiangular.

F

A

B

G

Because G is the centre of the circle ABCDEF, GE is equal to GD: And because D is the centre of the circle EGCH, DE is equal to DG; wherefore GE is equal to ED, and the triangle EGD is equilateral; and therefore its three angles EGD, GDE, DEG are equal to one another, because the angles at the base of an ifofceles triangle are equal a; and the three angles of a triangle are equal to two right angles; therefore the angle EGD is the third part of two right angles: In the fame manner it may be demonftrated that the angle DGC is alfo the third part of two right angles: And because the ftraight line GC makes with EB the adjacent angles EGC, CGB equal c to two right angles; the remaining angle CGB is the third part of two right angles; therefore the angles E EGD, DGC, CGB, are equal to one another: And to thefe are equal d the vertical oppofite angles BGA, AGF, FGE: Therefore the fix angles EGD, DGC, CGB, BGA, AGF, FGE are equal to one another: But equal angles ftand upon equal e circumferences; therefore the fix circumferences AB, BC, CD, DE, EF, FA are equal to one another; And equal circumferences are fubtended by equal f ftraight lines; therefore the fix ftraight lines are equal to one another, and the hexagon ABCDEF is equilateral. It is all equiangulars for, fince the circumference AF is equal, ED to each of thefe add the circumference ABCD; therefore the whole circumference FABCD fhall be equal to the whole EDCBA:

D

H

And

And the angle FED ftands upon the circumference FABCD, Book IV. and the angle AFE upon EDCBA; therefore the angle AFE is equal to FED: In the fame manner it may be demonftrated that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED: Therefore the hexagon is equiangular; and it is equilateral, as was shown; and it is infcribed in the given circle ABCDEF. Which was to be done. COR. From this it is manifeft, that the fide of the hexagon is equal to the ftraight line from the centre, that is, to the semidiameter of the circle.

And if through the points A, B, C, D, E, F there be drawn ftraight lines touching the circle, an equilateral and equiangular hexagon fhall be described about it, which may be demonftrated from what has been said of the pentagon; and likewise a circle may be infcribed in a given equilateral and equiangular hexagon, and circumfcribed about it, by a method like to that ufed for the pentagon.

PROP. XVI. PRO B.

infcribe an equilateral and equiangular quinde. See N. cagon in a given circle.

A

Let ABCD be the given circle; it is required to infcribe an equilateral and equiangular quindecagon in the circle ABCD. Let AC be the fide of an equilateral triangle infcribed a in 3 2. 4. the circle, and AB the fide of an equilateral and equiangular pentagon infcribed b in the fame; therefore, of fuch equal parts b 11.4. as the whole circumference ABCDF contains fifteen, the circumference ABC, being the third part of the whole, contains five; and the circumference AB, which is the fifth part of the whole, contains three; therefore BC their difference B containes two of the fame parts: Bifect BC in E; therefore BE, EC are, each of them, the fifteenth part of the whole circumference ABCD: Therefore, if the ftraight lines BE,

E

F

D

c 30. 3.

EC be drawn, and ftraight lines equal to them be placed dd 1. around in the whole circle, an equilateral and equiangular quindecagon fhall be infcribed in it. Which was to be done.

Book IV.

And, in the fame manner as was done in the pentagon, if, through the points of divifion made by infcribing the quindecagon, ftraight lines be drawn touching the circle, an equilateral and equiangular quindecagon fhall be described about it: And likewife, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumfcribed about it.

THE

ELEMENT S

OF

EU CLI

L I D.

воок V.

DEFINITION S.

Book V.

A

I.

LESS magnitude is faid to be a part of a greater magnitude, when the lefs measures the greater, that is, ' when the less is contained a certain number of times exactly ' in the greater.'

II.

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

III.

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

IV.

Magnitudes are faid to have a ratio to one another, when the lefs can be multiplied fo as to exceed the other.

V.

The first of four magnitudes is faid to have the same ratio to the the fecond, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the fecond and fourth; if the multiple of the first be less than that of the second, the multiple of the third is alfo lefs 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 alfo equal to that of the fourth; H 4 or,

*Book V.

See N.

or, if the multiple of the first be greater than that of the se-
cond, the multiple of the third is also greater than that of
the fourth.
VI.

Magnitudes which have the fame ratio are called proportionals.
N. B. When four magnitudes are proportionals, it is
• ufually expreffed by faying, the firft is to the fecond, as the
third to the fourth.'

VII.

When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the firft is greater than that of the fecond, but the multiple of the third is not greater than the multiple of the fourth; then the first is faid to have to the fecond 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 lefs ratio than the firft has to the fecond.

VIII.

"Analogy, or proportion, is the fimilitude of ratios."

IX.

Proportion confifts in three terms at least.

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 fecond.

XI.

1

When four magnitudes are continual proportionals, the first is faid to have to the fourth the triplicate ratio of that which it has to the fecond, and fo on, quadruplicate, &c. increafing the denomination ftill by unity, in any number of propor

tionals.

Definition A, to wit, of compound ratio. When there are any number of magnitudes of the fame kind, the first is faid to have to the last of them the ratio compounded of the ratio which the first has to the fecond, and of the ratio which the fecond has to the third, and of the ratio which the third has to the fourth, and fo on unto the laft magnitude.

For example, if A, B, C, D be four magnitudes of the fame kind, the first A is faid to have to the laft 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 faid to be compounded of the ratios of A to B, B to C, and C to D: And