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

Find the centre G of the circle ABCDEF, and draw the diameter AGD; and from Das 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.

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; and. Cor. 5. 1. the three angles of a triangle are equal b to two right angles; b 32. 1. therefore the angle EGD is the third part of two right angles:

In the same manner it may be de-
monftrated that the angle DGC is
alfo the third part of two right
angles: and because the straight
line GC makes with EB the ad-
jacent angles EGC, CGB equal c
to two right angles; the remain-
ing angle CGB is the third part
of two right angles; therefore E
the angles EGD, DGC, CGB, are
equal to one another: and to these

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

H

angles stand upon equal e circumferences; therefore the fix e 26. 3. circumferences AB, BC, CD, DE, EF, FA are equal to one another: and equal circumferences are subtended by equal f f 29.3. straight lines; therefore the fix straight lines are equal to one another, and the hexagon ABCDEF is equilateral. It is also equiangular; for, fince the circumference AF is equal to ED, to each of these add the circumference ABCD; therefore the whole circumference FABCD shall be equal to the whole EDCBA: and the angle FED stands upon the circumference

Book IV. cumference FABCD, and the angle AFE upon EDCBA; therefore the angle AFE is equal to FED: in the same manner it may be demonstrated 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 inscribed in the given circle ABCDEF. Which was to be done.

Cor. From this it is manifest, that the fide of the hexagon is equal to the straight 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 straight lines touching the circle, an equilateral and equiangular hexagon shall be described about it, which may be demonstrated from what has been faid of the pentagon; and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon.

a 2.4.

PROP. XVI. PROB.

TO inscribe an equilateral and equiangular quindecagon in a given circle.

Let ABCD be the given circle; it is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD.

Let AC be the fide of an equilateral triangle inscribed a in the circle, and AB the fide of an

equilateral and equiangular pen

b tr. 4. tagon inscribed b in the same; therefore, of such equal parts 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

B

E

C

A

F

D

30. 3. BC their difference contains two of the same parts: bisect

BC

!

BC in E; therefore BE, EC are, each of them, the fifteenth Book IV. part of the whole circumference ABCD: therefore if the straight lines BE, EC be drawn, and straight lines equal to them be placed d around in the whole circle, an equilateral d1.4. and equiangular quindecagon shall be inscribed in it. Which was to be done.

And in the same manner as was done in the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon shall be described about it: And likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it.

ELE

ELEMENTS

OF

GEOMETRY.

BOOK V.

DEFINITIONS.

I.

A

Less magnitude is faid

to be a part of a greater magni- Book V.

tude, 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.

III.

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

IV.

Magnitudes are said to be of the same kind, when the less can be multiplied so as to exceed the greater; and it is only fuch magnitudes that are faid to have a ratio to one ano