CASE 2. When the given point C is nearly opposite to the end of the given line.

From C draw the line CM to meet AB, in any point M. Bisect the line CM in the point N; and with the centre N, and radius CN or MN, describe an arc, cutting AB in D. Draw the line CD, and it will be the perpendicular required.

NOTE. Perpendiculars may be more easily raised or let fall by means of a square, or of a feather-edged scale or ruler, with a line across it perpendicularly to its edges.

PROBLEM VI.

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

C

B

D

Draw any chord AB; and bisect it perpendicularly with CD, which will be a diameter. Bisect CD 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 AB = 12, AC = 10, and BC= 8.

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

Make AB equal to 12, and AD equal to 7. At D erect the perpendicular DC, which make ual to 6. Join AC and BC, and the figure will be completed.

C

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.

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 AB. 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 AB = 12, and the breadth BC = 6.

At B erect the perpendicular BC, which make equal to 6. With A as a centre, and the radius BC, describe an arc; and with C as a centre, and the radius AB, describe another arc, cutting the former in D. Draw the lines AD and CD, and the rectangle will be completed.

PROBLEM XI.

Upon a given right line to construct a regular rhombus. Let the given line AB = 8.

With A and B as

Draw the line AB equal to 8. centres, and the radius AB, describe arcs cutting each other in D; then with B and D as centres, and the same radius, make the intersection C. Draw the lines AD, DC, and BC, and the rhombus will be completed.

PROBLEM XII.

To construct an irregular rhombus, having given its side and perpendicular height.

Let the side 8, and the perpendicular = 6.

Draw AB equal to 8; at A erect the perpendicular AE, which make equal to 6; and draw EC parallel to AB. With the radius AB, and A as a centre, make the intersection D; and with the same radius, and B as a centre, make the intersection C. Jon AD, DC, and CB, and the figure will be completed.

PROBLEM XIII.

Having any two right lines given, to construct a regular rhomboid.

Let the given lines be AB = 12, and BC = 6.

Draw the line AB equal to 12. Take in your compasses the line BC, and lay it from A to E. With A and E as centres, and the radius AE, make the intersection D. Then with B as a centre, and the same radius, describe an arc; and with D as a centre, and the radius AB, describe another arc, cutting the former in C. Draw the lines AD, DC, and BČ, and the rhomboid will be completed.

PROBLEM XIV.

Having given the base, the perpendicular, and the place of the perpendicular upon the base, to construct an irregular rhomboid.

Let the base AB = 15, the perpendicular ED = 6, and the distance AE = 5.