Bisect any two arcs A C, B C of the square A B C D in G, and E. Through the points G, and E, and the centre O draw lines, which produce to F, and H. Join A F, F D, D H, &c. and they will form the octagon required. [Fig. 9.]
On a line to describe all the several polygons, from the hexagon to the dodecagon.
Bisect A B by the perpendicular C D. From A as a centre, and with A B as a radius, describe the arc B E, which divide into six equal parts; and from E as a centre describe the arcs 5 F, 4 G, 3 H, &c. Then from the intersection E as a centre, and with E A as a radius, describe the circle A I D B, which will contain A B six times. From F in like manner as a centre, and with F A as radius, describe the circle A K L B, which will contain A B seven times; and so on for the other polygons. [Fig. 10.]
To inscribe in a circle an equilateral triangle.
From any point D in the circumference as a centre, and with the radius D O of the given circle, describe an arc A O B cutting the circumference in A, and B. Through D, and O draw D C. Then, join A B, A C, B C; and the figure A B C will be the triangle required. [Fig. 11.]
To inscribe a hexagon in a circle.
Bisect the arcs A C, B C in E, and F, and join A D, D B, B F, &c., which will form the hexagon. Or carry the radius six times round the circumference, and the hexagon will be obtained. [Fig. 11.]
To inscribe a dodecagon in a circle.
Bisect the arc A D of the hexagon in G, and A G being carried twelve times round the circumference, will form the dodecagon. [Fig. 11.]
To inscribe a pentagon, hexagon, or decagon, in a circle.