PROBLEM XVII.

To describe the segment of a circle, by means of two rules, to any length A B, and perpendicular height C D, in the middle of A B, without making use of the centre.

It will be most convenient for practice to make the rules CE and C F each equal to A B, as room is sometimes wanted. »

1. Place the rules to the height at C, bring the edges close to A and B, tack them together at C, and fix a rod across them to keep them tight.

2. Put in pins at A and B, then move your rules round these pins, hold a pencil to the angular point at C, it will describe the segment required.

FIG. II. By means of a triangle, let A B be the length of the segment, and C D the perpendicular height in the middle.

1. Through the points D and B, draw D B.

2. Draw D E parallel to A B, for conveniency, make D E equal to D B, and join E B.

3. Make a triangle E, D, B, put in pins at the points A, D, B, then move your triangle round the points D and B, and the` angular point will describe half the segment; the other half will be described in the same manner, which will complete the whole segment, as was required.

FIG. III. Another method by means of points; let AR be the length, and C D bisecting A B perpendicular, the height. 1. Through D, draw G H parallel to A B.

2. Draw D B, the half chord.

3. From B, make B H perpendicular to D B, cutting G H in H, make D G equal to D H.

4. Draw A F and B E, each perpendicular to A B, cutting GII in F and E.

5. Divide D G, DH; CA, C B; and A F, BE; each into a like number of equal parts, as five.

6. Draw the cross lines 4 4, 3 3, 22, 1 1, &c.

7. From the division on A F, and B E, draw lines to D, cutting the other cross lines at d, e, f, g, &c.

8. Put pins in these points, bend a slip round them, and draw the curve by it, it will be the segment required.

FIG. IV. Another method, by points nearly true, when the segment is very flat, let A B be the length, and C D bisecting A B, the perpendicular height.

1. Draw A E, and B F, perpendicular to A B, each equal

to C D.

2. Divide C B, and C A, each into the same number of equal parts, as 5.

3. From the points 4, 3, 2, 1, &c. on A B, draw the perpendicular 4 4, 3 3, 2 2, 1 1, &c. to A B.

4. Divide A F, and B E, into five equal parts each.

5. Draw lines from the points 1, 2, 3, 4, 5, at each end, to D, and complete the segment in the same manner as FIG. III.

PROBLEM XVIII.

In a given triangle A, B, C, to inscribe a circle. 1. Bisect any two angles A and C, with the lines A D, and C D.

2. From D, the point of intersection, let fall the perpendicular D E, it will be the radius of the circle required.