viding lines, will cut each other in D, then D is the center of the circle.

2nd. From D let fall a perpendicular to either of the sides, as at F, then DF is the radius with which to describe the circle from the point D.

PROBLEM 14, Figure 6.

On the Given Line AB to Construct an Equilateral Triangle, the Line AB to become One of its Sides.

1st. With a radius equal to the given line, from the points A and B, draw two arcs intersecting each other in C.

2nd. From draw CA and CB to complete the figure.

PLATE 12.

CONSTRUCTION OF POLYGONS.

PROBLEM 15, FIGURE 1.

On a Line AB to Construct a Square whose Side shall be Equal to the Given Line.

1st. With the length AB for a radius, from the points A and B, describe two arcs cutting each other in C.

2nd. Bisect the arc CA or CB in D.

3rd. From C, with a radius equal to CD, cut the arc BE in E, and the arc AF in F

4th. Draw AE, EF and FB, which complete the square.

PROBLEM 16, FIGURE 2.

In the Given Square GHKJ to Inscribe an Octagon.

1st. Draw the diagonals GK and HJ intersecting each other in P. 2nd. With a radius equal to half the diagonal from the corners GHK and J, draw arcs cutting the sides of the square in 0, 0, 0, etc. 3rd. Draw the lines 0,0,0,0, etc., and they will complete the octagon.

NOTE. This mode is used by workmen when they desire to make a piece of wood for a roller, or any other purpose. It is first made square and the diagonals drawn across the end; the distance of one-half the diagonal is then set off, as from G to R in the diagram, and a gauge set from H to R, which, run on all the corners, gives the lines for reducing the square to an octagon; the corners are again taken off, and finally finished with a tool appropriate to the purpose. The center of each face of the octagon gives a line in the circumference of the circle, running the whole length of the piece; and, as there are eight of those lines equidistant from each other, the further steps in the process are very simple.

PROBLEM 17, FIGURE 3.

In a Given Circle to Inscribe an Equilateral Triangle, a Hexagon, and

Dodecagon.

1st. For the triangle: With the radius of the given circle from any point in the circumference, as at A, describe an arc cutting the circle in B and C.

2nd. Draw the right line BC, and, with a radius equal to BC from the points B and C, cut the circle in D.

3rd. Draw DB and DC, which complete the triangle.

4th. For the hexagon: Take the radius of the given circle and carry it round on the circumference six times; it will give the points ABEDFC; through them draw the sides of the hexagon. The radius of a circle is always equal to the side of an hexagon inscribed.

5th. For the dodecagon: Bisect the arcs between the points found for the hexagon, which will give the points for inscribing the dodecagon.

PROBLEM 18, FIGURE 4.

In a Given Circle to Inscribe a Square and an Octagon.

1st. Draw a diameter AB, and bisect it with a perpendicular by Problem 1, giving the points CD.

2nd. From the points ACBD draw the right lines forming the sides. of the square required.

3rd. For the octagon: Bisect the sides of the squåre, and draw perpendiculars to the circle, or bisect the arcs between the points ACBD, which will give the other angular points of the required octagon.

PROBLEM 19, FIGURE 5.

On the Given Line OP to Construct a Pentagon, OP being the Length of the Side.

1st. With the length of the line OP from O, describe the semicircle PQ, meeting the line PO extended in Q.

2nd. Divide the semicircle into five equal parts, and from O draw lines through the divisions 1, 2, and 3.

3rd. With the length of the given side from P cut 01 in S, from Scut O 2 in R, and from Q cut O 2 in R; connect the points OQRSP by right lines, and the pentagon will be complete.