Draw B D.

Bisect A B and B D, and from the intersection O of the bisecting lines, with radius O A or O B, describe a circle.

From A and B set off the length A B around the circle -viz., E F G H.

Draw D E, E F, F G, G H, and H A, which will complete the heptagon.

To construct a pentagon on this principle, divide the quadrant into five parts, and set off one beyond C.

For a hexagon, divide the quadrant into six, and set off two beyond C.

For an octagon, divide into eight, and set off four beyond E, &c.

To inscribe a regular Heptagon in a given circle. (Fig. 63.)

F

Fig. 63.

From any point, as A, with the radius of the circle, describe arcs cutting the circumference in B and C.

Draw the line B C and the radius A O, which will bisect B C in D.

Set off the length D B around the circle, and join the points A E F G H I J, and the heptagon will be completed.

It will be seen that B C is one side of the equilateral triangle, which could be inscribed in the circle, and thus as D B is half of B C, half of the side of the inscribed equilateral triangle gives the side of a regular heptagon, which can be inscribed in the same circle.

To construct a regular Octagon on the given line A B. (Fig. 64.)

Produce A B on each side.

Erect perpendiculars at A and B.

From A and B, with radius A B, describe the quadrants CD and E F.

Bisect these quadrants, then A G and B H will be two more sides of the octagon.

At H and G draw perpendiculars, GI and H K, equal to A B.

Draw G H and I K.

Make the perpendiculars A and B equal to G H or I K-viz., A L and B M.

Draw IL, L M, and M K, which will complete the octagon.

To inscribe an Octagon in the square A B C D. (Fig. 65.)

Draw diagonals, A D and C B, intersecting each other in O.

From A B C and D, with radius equal to A O, describe

quadrants cutting the sides of the square in E F G H IJ KL.

Join these points, and an octagon will be inscribed in the square.

To inscribe an Octagon in a given Circle. (Fig. 66.)

ポ

Fig. 66.

B

Draw the diameter A B, and bisect it by C D.

Bisect the quadrants A C, C B, A D, and B D, in the points E F G H.

Draw lines connecting all the eight points, which will complete the required octagon.

As all other polygons may be constructed on the prin

ciples already shown, it will be unnecessary to give further examples of them.

To inscribe an Equilateral Triangle in a regular Pentagon, A B C D E. (Fig. 67.)

Fig. 67.

К

From E, with any radius, describe a semicircle, F G.

From F and G, with the same radius, describe arcs cutting the semicircle in H and I.

(The radius with which a semicircle is struck, divides it into three equal parts).

From E draw a line through H and I, cutting the sides of the pentagon in J K.

Draw J K, which will complete the equilateral triangle in the pentagon.

E