From the point B, with any radius, describe the arc A C. From A and C, with the same, or any other radius, make the intersection m. Draw the line Bm and it will

bisect the angle A B C, as required.

PROBLEM III.

To draw a line parallel to a given line A B.

CASE 1. When the parallel line is to pass through a given point C.

From any point m, in the line A B, with the radius m C, describe the arc C n. From the centre C, with the same radius, describe the arc m r. Take the distance Cn in the compasses, and apply it from m to r. Through C and r draw the line D E, and it will be the parallel required.

CASE 2. When the parallel line is to be at a given distance from A B.

From any two points m and n, in the line A B, with a radius equal to the given distance, describe the arcs r and o. Draw the line CD, to touch these arcs, without cutting them, and it will be the parallel required.

Note. This Problem may be more easily performed by means of a parallel ruler, which may also be used to advantage in several operations in Practical Geometry.

PROBLEM IV.

To erect a perpendicular from a given point C, in a given line A B.

On each side of the point C, take any two equal dis tances, Cm and Cn. From m and n, as centres, with any radius greater than C m or Cn, describe two arcs cutting each other in r. Draw the line Cr, and it will be the perpendicular required.

F

PROBLEM V.

From a given point C, to let fall a perpendicular upon a given line AB.

With C as a centre, and any radius a little exceeding the distance of the given line, describe an arc cutting AB in m and n. With the centres m and n, and the same or any other radius exceeding half their distance, describe arcs intersecting each other in r. Draw the line Cr; and CD will be the perpendicular required..

PROBLEM VI.

To find the centre of a given circle, or one already de

scribed.

C

B

D

Draw any chord AB, and bisect it perpendicularly with CD, which will be a diameter.

Bisect C D in the

'point o, which will be the centre required.

PROBLEM VII.

To make a triangle with three given lines, any two of which, taken together, must be greater than the third. (Euclid I. 22.)

Let the given lines be A B=12, A C=10, and B C 8.

From any scale of equal parts (which is to be understood as employed likewise in the following Problems) lay off the base A B. With the centre A, and radius A C, describe an arc. With the centre B, and radius BC, describe another arc, cutting the former in C. Draw the lines A C and B C, and the triangle will be completed.

Note. A trapezium may be constructed in the same manner; having the four sides and one of the diagonals.

PROBLEM VIII.

Having given the base, the perpendicular, and the place of the perpendicular upon the base, to construct a triangle.

Let the base A B=12, the perpendicular C D=6, and the distance A D=7.

Make A B equal to 12, and A D equal to 7. At D erect the perpendicular DC, which make equal to 6. Join A C and B C, and the figure will be completed.

Note. A trapezium may be constructed in a similar manner, by having one of the diagonals, the two perpendiculars let fall thereon from the opposite angles, and the places of these perpendiculars upon the diagonal; and a trapezoid may be constructed by drawing the two parallel sides perpendicularly to their base or given distance.

PROBLEM IX.

To describe a square whose side shall be equal to a given line.

Upon one extremity B, of the given line, erect the perpendicular BC, which make equal to A B. With A and C as centres, and the radius AB, describe arcs cutting each other in D. Join AD and CD and the square will be completed.

PROBLEM X.

To describe a rectangular parallelogram, whose length and breadth shall be equal to two given lines.

Let the length A B=12, and the breadth B C=6,

6.

At B erect the perpendicular B C, which make equal to With A as a centre, and the radius B C, describe an