a circle. Produce AO and BO to meet the circle in C

and D, and bisect the angles DOA and COB (8) by the diameter EOF. Bisect each angle COE, EOB, AOF, FOD, by the diameters

IOK and HOG. Join all

the points where these diameters meet the circumference by the lines CI, IE, EH, HB, &c., and the required decagon is completed, having each side equal to АВ.

.

The ratio of A B to BO is very nearly as 1 1.618.

44. To construct a regular hexagon, or figure of six sides, the length of a side being given.-Let A B (fig. 37) be the given side. On A B

(26) construct an equilateral triangle A OB; and from O as a centre, with OA as radius, describe a circle. Produce A O and BO to meet the circle in D and E; and from A and B as centres, with A O as radius, draw arcs cutting the circle in C and F.

Join BC, CD, DE, EF,

Fig. 37.

FA, and the hexagon is completed, each of these lines being equal to A B.

It is evident that A O is equal to A B.

45. To construct a regular octagon, or figure of eight

sides, the length of a side being given.-Let A B (fig. 38) be

scribe a circle cutting A D in the point D. Join ED and A D. Draw DOK parallel to A B intersecting AC in the point O. Through O draw FOG parallel to AD. From O as a centre, with OA for radius, describe a circle passing through A, E and D. Draw the diameter EOH and join all the points where the diameters meet the circle by the lines DG, GC, CH, &c., and the required octagon is completed, each side being equal to A B.

The ratio of A B to A O is nearly as 1:1-307.

46. To construct a dodecagon, or figure of twelve sides, the length of a side being given.-Let AB (fig. 39) be the given side. Erect the line BD (3) at right angles to A B, and from B as a centre, with B A for radius, describe a circle cutting BD in the point D. Divide the right angle DBA into twelve equal parts (4), and take DBE equal to two of those parts. Bisect A B in C, and draw the line CO perpendicular to AB meeting BE in O. From O as a centre, with O A or O B as radius, describe a

circle through B and A. circle in K and L.

Produce A O and BO to meet the From A and K as centres, with

OA for radius, describe arcs cutting the circle in P, F, H and M. Draw GON at right angles to A K, and bisect

the angle AOP by the line QOI. Join all the points B, F, G, H, &c., upon the circle, and the dodecagon will be completed, having each side equal to A B.

The ratio of A B to AO is nearly as 1 : 1.932.

In the same manner we can construct a regular polygon of any given number of sides, provided that given number is some product of 3 into a power of 2. The right angle ABD must be divided (9) into the same number of parts as there are to be sides to the figure, and the angle DBE taken equal to two of those parts; then the point O where BE meets the bisecting perpendicular CO will be the centre of the circumscribing circle. Then from A and B as centres, with A B for radius, describe arcs cutting the circle in

Q and F, and so proceed round the circle until all the points where the polygon touches the circle are found.

In this way a quindecagon or fifteen-sided figure can be constructed; for (10) we can divide the right angle ABD (fig. 39) into fifteen equal parts. Take DBE equal to two of those parts and proceed in the construction the same way as for the dodecagon.

For a figure of sixteen equal sides, we must divide the right angle ABD into eight equal parts (8) and take DBE equal to one of them.

For a figure of twenty equal parts we must first divide the angle ABD into five equal parts (10) and bisect each of those parts, by which the angle will be divided into ten equal parts; and DBE being taken equal to one of those parts the figure can be constructed in the manner above described.

CHAPTER II.

THE CIRCLE.

Fig. 40.

47. DEFINITIONS.-A curve is a line made by the motion of a point the direction of which is constantly changing. A circle is a curve produced by the motion of a point about a fixed point from which it is always at the same distance the fixed point about which the motion takes place is called the centre of the circle, and the curved line thus produced is called the circumference. If a straight line or rod O A (fig. 40) is made to revolve about one of its extremities O, its other extremity A will trace out a circle, and the length OA is called the radius of circle; the line AOB, which is double the radius, is called the diameter of the circle. If a line AT is drawn at right angles (3) to AO), at the outer end of any radius, AT is called the tangent to

the circle, which it touches without cutting. Any line as CD which is drawn across the circle but does not pass through the centre is called a chord. The part of the circle CED cut off by the chord is called an arc, and the space included between the arc and chord is called a segment. If we draw the lines OC, OD, from the centre to the ends of the