To describe an Equilateral Triangle.

From the points A, B, as centres, and with AB as radius, describe arcs intersecting each other in C. Draw CA, CB, and the figure ABC will be the Triangle required. Fig. 7.

AB.

To describe a Square.

From the point в draw BC perpendicular, and equal to
On A and with the radius AB, describe arcs cutting
Draw the lines DA, DC and the figure ABCD

each other in D.

C,

will be the Square required. Fig. 8.

To inscribe a Square in a Circle.

Draw the diameters AB, CD perpendicular to each other. Then draw the lines AD, AC, BD, BC; and ABCD will be the Square required. Fig. 9.

To inscribe an Octagon in a Circle.

Bisect any two arcs AC, BC of the Square ABCD in G and E. Through the points G and E and the centre o draw lines, which produce to F and H.

Join AF, FD, DH, &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 AB by the perpendicular CD. From A as a centre, and with AB as a radius, describe the arc BE, which divide into six equal parts; and from E as a centre describe the arcs 5 F, 4G, 3 H, &c. Then from the intersection E as a centre, and with EA as a radius describe the circle AIDB, which will contain AB six times. From F in like manner as a centre, and with FA as radius, describe the circle AKLB, which will contain AB 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 DO of the given circle, describe an arc AOB cutting the circumference in A and B. Through D and o draw DC. Then join AB, AC, BC; and the figure ABC will be the Triangle required. Fig. 11.

G G

To inscribe a Hexagon in a Circle.

Bisect the arcs AC, BC in E and F, and join AD, DB, BF, &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 AD of the hexagon in G, and AG being carried twelve times round the circumference will form the

Dodecagon. Fig. 11.

To inscribe a Pentagon, Hexagon, or Decagon in a Circle.

Draw the diameter AB, and make the radius DC perpendicular to AB. Bisect DB in E. From E as a centre, and with EC as radius, describe an arc, cutting AD in F. Join CF which will be the side of the Pentagon, CD that of the Hexagon, and DF that of the Decagon. Fig. 12.

To find the Angles at the centre, and circumference of a Regular Polygon.

Divide 360 by the number of sides of the given Polygon, and the quotient will be the Angle at the centre; and this angle being subtracted from 180, the difference will be the Angle at the circumference required.

Table, showing the Angles at the Centre, and Circumference. No. of Sides. Angles at Centre. Angles at Circumference.

Names.

To inscribe any regular Polygon in a Circle.

From the centre C draw the radii CA, CB making an angle equal to that at the centre of the proposed Polygon, as contained in the preceding Table. Then the distance AB