Take the radius of the circle, and with it mark off six points on the circumference. Take two of these lengths of the radius and join their extreme points A and B, which will be the base. Now take this base as a radius and describe alternately two arcs cutting each other at C. Join A, C, and B, C, and a triangle is formed, whose sides being equal is termed an _equilateral triangle_. In order to ensure its being upright, erect a perpendicular at the centre, and let the two sides A, C, and B, C, meet that perpendicular where it intersects the circumferences. Or, begin the triangle at this point, and mark off two lengths of the radius, joining the extreme points as before; and do this at each side of the perpendicular; finally connecting the distant extremities of the two sides for a base. PROBLEM V. _To construct an upright square in a given circle._ Let fall a perpendicular, I, E, from the centre to the circumference, and with that as a radius and E as a centre, cut the circumference at A, B, C, and D, and join the points. The four-sided figure called a square is thus formed. [Illustration: _Fig. 5_] PROBLEM VI. _On a given right line_, A, B, _to construct a pentagon, or five-sided figure_. [Illustration: _Fig. 6_] Draw B, F, perpendicular and equal to the half of A, B. Produce A, F, to G, making F, G, equal to F, B. From the points A and B, with the radius B, G, describe arcs cutting each other at I. From I, with the radius I, B, describe a circle. Inscribe the successive chords A, E; E, D; D, C; C, B, which with the base A, B, completes the pentagon. If the circle be given, and a pentagon to be inscribed in it, the following is as simple as it is practical. From the centre erect a perpendicular, which shall meet the circumference at D. At each side of this point divide the circumference into five equal parts, and connect every two of them from D to E, from E to A, and from D to C, C to B. Now connect A and B and the pentagon is formed. PROBLEM VII. _On a given line_ A, B, _to construct a hexagon, or six-sided figure_. Take the length of the radius I, G, and lay it off from F to A, A to B, B to C, C to D, D to E, and E to F. [Illustration: _Fig. 7_] PROBLEM VIII. _To form an octagon, or eight-sided figure._ Refer back to _Fig. 5_. Draw the radius I, E, till it meets the circumference at E. Join the points E, A, and E, B. Repeat this at each of the four sides, and the octagon is formed. PROBLEM IX. _To form a decagon, or ten-sided figure._ Refer to _Fig. 6_, and proceed as in the preceding problem. PROBLEM X. _To construct a duo-decagon, or twelve-sided figure._ Refer to _Fig. 7_, and duplicate the chords, as already shown. We do not present 7, 9, or 11 sided figures, because they seldom or ever come into practice. Our object being to give what is useful and not overburden the memory unnecessarily.