PROBLEM XIX.

In a given square A B C D, to inscribe a regular

octagon.

1. Draw the diagonals A C, and B D, intersecting at e. 2. Upon the points A, B, C, D, as centres, with a radius e C, describe arcs hel, ken, meg, fei.

3. Join fn, ml, ki, hg, it will be the octagon required.

In a given circle to inscribe an equilateral triangle, an hexagon, or a dodecagon.

For the equilateral triangle.

1. Upon any point A, in the circumference with the radius A G, describe the arc B G F.

2. Draw B F, make B D equal to B F.

3. Join D F, and B D F will be the equilateral triangle required.

For the hexagon.

Carry the radius A G six times round the circumference, the figure A B C D E F will be the hexagon.

For the dodecagon.

Bisect the arc A B in h, and A h being carried twelve times round the circumference, will also form the dodecagon.

In a given circle to inscribe a square or an octagon.

1. Draw the diameters A C and B D, at right angles. 2. Join A B, B C, C D, D A, and A B C D will be the square.

For the octagon.

Bisect the arc A B in E, and A E being carried eight times round, will also form the octagon.

PROBLEM

PROBLEM. XXII.

In a given circle to inscribe a pentagon, or a

decagon.

For a pentagon.

1. Draw the diameters A C and B D, at right angles.

2. Bisect E C in f, upon f, with the distance of ƒ D describe the arc D g upon D, with the distance D g, describe the arc g H cutting the circle in H.

3. Join D H, and carry it round the circle five times, will form the pentagon.

For the decagon.

Bisect the arc D H in i, and D i being carried ten times round, will also form the decagon.

In a given circle to inscribe any regular polygon.

1. Draw the diameter A B, from E the centre, erect the perpendicular E F C, cutting the circle at F.

2. Divide E Finto four equal parts, and set three parts from F, to C.

3. Divide the diameter A B into as many equal parts as the polygon is required to have sides.

4. From C, through the second division in the diameter, draw C D.

5. Join A D, it will be the side of the polygon required.

Upon a given line A B, to describe an equilateral

triangle.

1. Upon the points A and B, with a radius equal to A B,

describe arcs, cutting each other at C.

2. Draw A C and B C, it will be the triangle required.