Make AB equal to 15, and AE equal to 5. At E erect the perpendicular ED, which make equal to 6; and join AD. With the radius AB, and D as a centre, describe an arc; and with B as a centre, and the radius AD, describe another arc, cutting the former in C. Draw the lines DC and BC, and the figure will be completed.

NOTE. The sum of all the interior angles of any quadrilate ral figure, is equal to four right angles.

PROBLEM XV.

Having the transverse and conjugate diameters given, to construct an ellipse.

Let the transverse diameter AB = 14, and the conjugate diameter CD = 8.

Draw the two diameters to bisect each other perpendicularly in the centre o. With the radius Ao, and the centre D, intersect AB in F and f: these two points will be the foci of the ellipse. Take any point m, in the transverse diameter, and with F and f as centres, and the radius Am, describe the arcs G, G,

Then with the same centres, and the radius Bm, describe arcs cutting the former in the points G, G, g, g; thus you will have four points in the circumference of the ellipse. Again, take a second point n, in the transverse diameter, and proceeding as before,

you will determine other four points. By the same method you may determine as many more as you please; through all of which, with a steady hand, draw the circumference of the ellipse. (See another method of construction on page 318.)

PROBLEM XVJ.

Upon a given line AB, to make a regular pentagon.

Make Bm perpendicular to AB, and equal to half of it. Draw Am, and produce it till mn be equal to Bm. With the radius Bn, and A and B as centres, describe arcs intersecting each other in o, which will be the centre of the circumscribing circle. From the point o, with the same radius, describe the circle ABCDE; and apply the line AB five times to the circumference, marking the angular points, which connect with right lines, and the figure will be completed.

PROBLEM XVII.

Upon a given line AB, to make a regular hexagon.

E

D

With A and B as centres, and the radius AB, describe arcs intersecting each other in o; and with o as a centre, and the same radius, describe the circle AB CDEF. Apply the line AB six times to the circumference, and it will form the hexagon required.

PROBLEM XVIII.

Upon a given line AB, to construct a regular octagon.

On the extremities of the given line AB, erect the indefinite perpendiculars AF and BE. Produce the line AB, both ways to m and n; and with the lines AH and BC, each equal to AB, bisect the angles mAF and nBE. Draw CD and HG parallel to AF or BE, and each equal to AB. With D and G as centres, and the radius AB, describe arcs intersecting A and BE, in the points F and E. Join CF, FE, and ED, and the figure will be completed.

PROBLEM XIX.

In a given triangle ABC, to inscribe a circle.

C

A

Bisect the angles A and B, with the lines Ao, Bo, and o will be the centre of the required circle; and its radius will be the nearest distance to any one of the sides; hence the circle may be described.

PROBLEM XX.

About a given triangle ABC, to circumscribe a circle ; or to describe the circumference of a circle through three given points A, B, C.

Bisect the sides AB, BC, with the perpendiculars mo and no; and o will be the centre of the circle, and its radius will be Ao, Bo, or Co.

PROBLEM XXI.

To make a triangle equal to a given trapezium ABCD.

D

C

A

Draw the diagonal B, and parallel to it draw CE, meeting AB produce in E. shall the triangle ADE be

ABCD.

Join the points DE; so equal to the trapezium

PROBLEM XXII.

To make a right angle by the line of chords on the plane scale.

Draw the unlimited line AB; then take in your compasses 60° from the line of chords, and with A as a centre, describe the arc ED. Take 90° from the same scale, and set off that extent from D to C. Draw the line AC; and CAD will be the angle required.

PROBLEM XXIII.

To make an acute angle that shall contain of degrees; suppose 35° 30'.

Draw the unlimited line AB; then take 60° in your compasses, and with A as a centre, describe the arc ED. Set off the angle 35o 30', from D to C. Draw the line AC; and CAD will be the angle required.

PROBLEM XXIV.

To make an obtuse angle that shall contain any number of degrees; suppose 128° 35'.