3. From E as centre, with radius ED, describe an arc DF, cutting the diameter in F.

4. Draw the chord of the arc DF. This will be the length of a side of the pentagon.

5. With DF as radius starting from D, cut the circle in the points G, H, K, and L successively.

6. Join DG, GH, &c., by straight lines, and a regular pentagon will be inscribed within a given circle A.

Problem 64.

To inscribe a regular hexagon within a given circle A.

D

B

E

1. Draw any diameter BC.

2. With B and C as centres, and the radius of the circle AB, describe the arcs DAE and FAG.

3. Join BD, DF, &c., and the required regular hexagon is inscribed in the given circle A.

Problem 65.

To inscribe a regular heptagon within a given circle A.

1. Draw any radius AB, and from B, with BA as radius, describe an arc CAD cutting the circumference in points C and D.

B

M

2. Join CD by a straight line cutting AB in E. Then EC or ED will be the length of a side of the heptagon.

3. With EC or ED as radius, starting from B, cut the circle in the points F, G, . . . M successively.

4. Join BF, FG, &c., by straight lines, and a regular heptagon will be inscribed within a given circle A.

Problem 66.

To inscribe a regular octagon within a given circle A.

1. Draw any diameter BC, and bisect it by another diameter DE.

2. Bisect each of the four arcs by the diameters FK, HG.

F

E

3. Join the points BF, FD, &c., by straight lines, and the required regular octagon is inscribed in the given circle A.

Problem 67.

To inscribe a regular nonagon within a given circle 4.

1. Draw a diameter BC, and produce it one way indefinitely (say to the right), and bisect BC by another diameter DE.

2. From D, with radius DA, describe an arc cutting the arc DC in F.

3. From E, with radius EF, cut the produced diameter BC
in G.

4. From G, with radius GD, cut the diameter BC in H.
5. With BH, which is equal to a side of the nonagon, cut
the circle, starting from D, in the points K, L R.
Join these points by straight lines, and the required
regular nonagon is inscribed in the given circle A.

Problem 68.

To inscribe a regular decagon within a given circle A.

1. Draw any diameter BC, and a radius AD perpendicular to it from the centre of the circle (Pr. 2).

2. Bisect AD in E (Pr. 1), and join BE

3. From E, with EA as radius, describe an arc cutting BE

in F.

4. From B, with BF as radius, describe an arc cutting the

circumference in G.

5. Draw the straight line BG. It will be a side of the decagon.

6. With the straight line BG as radius, starting from G, cut the circle in the points H, K... P successively.

7. Join GH, HK, &c., by straight lines, and a regular decagon will be inscribed within a given circle A.

Problem 69.

To inscribe a regular un-decagon within a given circle A.

1. Draw two diameters BC and DE perpendicular to each other, and cutting each other in A.

S

C
M

BR

2. From E, with radius EA, describe an arc, cutting the quadrant EB in F.

3. From B, with the same radius, describe an arc, cutting the quadrant BD in G.

4. From F, with radius FG, describe an arc, cutting the radius AD in H.

5. Draw the straight line GH, it will be equal to a side of the un-decagon.

6. With the straight line GH as radius, starting from D, cut the circle in the points K, L, . . G successively.

7. Join DK, KL, &c., by straight lines, and a regular un-decagon will be inscribed within a given circle A.