the angles, sides, or planes of the circumscribing figure touch all the angles of the figure within it.

LINES, ANGLES, AND FIGURES.

To divide a given right line into two equal parts.

From the extremities of the line as centres, and with any opening in the compasses, greater than half the given line, as a radius, describe arcs intersecting each other above, and below the given line. A line being drawn through these intersections will divide the given line into two equal parts.

An arc of a circle is bisected in the same manner.

To erect a perpendicular.

From the point A set off any length 4 times to C: from A as a centre with of those parts describe an arc at B, and from C with 5 of them cut the arc at B. Draw A B, which will be the perpendicular required. Any equimultiples of these numbers, 3, 4, 5, may be used for erecting a perpendicular. Plate 2, HEIGHTS AND DISTANCES, and PRACTICAL GEOMETRY, Fig.

To erect a perpendicular.

Set off on each side of the point A, any two equal distances, A D, A E. From D, and E as centres, and with any radius greater than half D E, describe two arcs intersecting each other in F. Through A, and F draw the line A F, and it will be the perpendicular required. Fig. 1.-Plate, PRACTICAL GEOMETRY.

To let fall a perpendicular.

From D as a centre, and with any radius, describe an arc intersecting the given line. From the points of intersection C, and E, with any radius greater than half, describe two arcs, cutting each other at F. Through D, and F draw a line, and D F will be the perpendicular required. Fig. 2.

To draw a line parallel to a given line.

From any point D in the given line with the radius D C, describe the arc c E, and from C with the same radius describe the arc D F. Take E C, and set it off from D to F. Through C, and F draw C F for the parallel required. Fig. 3.

To divide an angle into two equal parts.

From A,

From B as a centre with any radius describe an arc A C. and c with any radius describe arcs intersecting each other in D. Then draw B D, and it will bisect the angle. Fig. 4.

To divide a right angle into three equal parts.

From B as a centre with any radius describe the arc A C.

From A

with the radius AB cut the arc AC in D, and with the same radius

from c cut it in E. Then through the intersections D, and E draw the lines BD, BE, and they will trisect, or divide the angle into three equal parts. Fig. 5.

To find the centre of a circle.

Draw any chord A B, and bisect it by the perpendicular CD. Divide CD into two equal parts, and the point of bisection o will be the centre required. Fig. 6.

To describe an equilateral triangle.

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

To describe a square.

From the point B, draw BC perpendicular, and equal to A B. On A, and c, with the radius A B, describe arcs cutting each other in D. Draw the lines DA, DC, and the figure A B C D will be the square required. Fig. 8.

To inscribe a square in a circle.

Draw the diameters A B, C D perpendicular to each other. Then draw the lines A D, A C, B D, BC; and A B C D will be the square required. Fig. 9.

To inscribe an octagon in a circle.

Bisect any two arcs A C, 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 A B by the perpendicular C D. From A as a centre, and with A B as a radius, describe the arc B E, which divide into six equal parts; and from E as a centre describe the arcs 5 F, 4 G, 3 H, &c. Then from the intersection E as a centre, and with E A as a radius, describe the circle A I D B, which will contain A B six times. From F in like manner as a centre, and with FA as radius, describe the circle which will contain A B seven times; and so on for the other polygons. Fig. 10.

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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 A OB cutting the circumference in A, and B. Through D, and o draw DC. Then, join A B, A C, BC; and the figure A B C will be the triangle required. Fig. 11.