PROBLEM XIII.

Having the Base and the Two Perpendiculars given, to construct a

Trapezoid.

and the trapezoid will be completed.

PROBLEM XIV.

B

Having the Four Sides given, to construct a Quadrilateral Figure which

has one Right Angle.

Let the sides AB=7, let the angle at B be a right angle.

Draw the line AB equal to 7, and erect the perpendicular BC, equal to 4 chains. With C as a centre, and the radius CD, describe an arc; and with A as a centre, and the radius DA, describe another arc, cutting the for

BC= 4, CD = 6, and DA = 3 chains; and

B

mer in D. Draw the lines CD and DA, and the figure will be completed.

CD

PROBLEM XV.

Having the Transverse and Conjugate Diameters given, to

construct an Ellipsis.

Let the transverse diameter AB=7, and the conjugate diameter

=

4 chains.

the centre o.

Draw the two diameters to bisect each other perpendicularly in With the radius Ao, and the centre C or D, intersect AB in F and f. These points will be the foci of the ellipse. Take any point m, in the transverse diameter, and with F and ƒ as centres

and the radius Am, describe the arcs G, G and g, g. Then with the same centres and the radius Bm, describe arcs cutting the former in the

points G, G, g and g: thus will you have four points in the circumference of the ellipse. After this, 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, you must draw the circumference of the ellipse.

NOTE. An ellipse may also be constructed as follows:-Having found the foci F and f, as before, take a thread equal in length to the transverse diameter AB, and fasten its ends, with two pins, in the points F and ƒ; then stretch the thread to its greatest extent; and by moving a pencil round, within the thread, keeping it always tight, you will trace out the curve of the ellipse.

The principle upon which this construction is founded may be seen in Prob. X. Part VI.

PROBLEM XVI.

To reduce a given Trapezium ABCD to a Triangle of equal area.

D

A

C

B

E

Draw the diagonal DB, and parallel to it draw CE, meeting AB produced in E. Draw DE; the triangle ADE is equal to the trapezium

ABCD.

NOTE. This and the following problem may be applied in finding the areas of trapeziums and irregular polygons, by first reducing them to triangles.

PROBLEM XVII.

To reduce an Irregular Polygon ABCDE, of five sides,
to a Triangle of equal area.

Extend the side gonals CE and CA.

AE both ways at pleasure, and draw the dia-
Parallel to these diagonals draw the lines DF,

B

D

A

E

and BG; then draw CF and CG and GCF will be the triangle required.

NOTE. Any irregular polygon of more than five sides may be brought to a triangle of equal area by reducing it successively to a figure with one side less, until you bring it to a figure of three sides. Thus the trapezium ABCF or GCDE is equal to the polygon ABCDE as well as the triangle GCF.

PROBLEM XVIII.

To raise a Perpendicular from any Point D, in a given Line AB, by a scale of equal parts.

tion; but 3, 4, and 5 are the least whole numbers that will make a right-angled triangle.

PROBLEM XIX.

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 E 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.

C

Ꭺ

B

PROBLEM XX.

To make an Acute Angle equal to any Number of Degrees,
suppose 33° 30'.

Draw the unlimited line AB; then take 60° in your compasses, and with A as a centre describe the arc ED. the angle 33° 30′

Then set off from D to C.

E

Draw the line AC;

To make an Obtuse Angle equal to any Number of Degrees,
suppose 125° 30'.

PROBLEM XXII.

To find the Number of Degrees contained in any given Angle BAC.

With the chord of 60°, and A as a centre, describe the arc mn. Take the distance mn in your compasses, and apply it to the line of chords and it will show the number of degrees required.

B

NOTE.-Angles may be more expeditiously laid down or measured by means of a semicircle of brass called a Protractor, the arc of which is divided into 180°.

PROBLEM XXIII.

To lay down a Line making a given Angle with the Meridian,
or North and South Line.

EXAMPLES.

1. Let it be required to lay down a line that ranges N.E., making an angle of 45° with the meridian line. (See The Compass, Part II.).

B

N

E

S

Draw the meridian line AN; and with the sweep of 60°, taken from the line of chords in your compasses, and A as a centre, describe the arc BC.

Set off the given angle 45° from в to C; draw the line AC, and it will range N.E., as was required.

NOTE. If the line had ranged N.W., the angle must have been set off on the other side of the meridian AN; and AD would have been the direction of the line.