Through the points of intersection draw the line m Cn, and it will divide A B into two equal parts.

2. To erect a perpendicular on a given point in a line. Let A B, fig. 2, be the given line and C the given point. On each side of point C measure off equal distances to a and b.

From the centres a and b, with any distance greater than a C or b C, describe two arcs intersecting each other in c.

Through C and e draw the line C c, and it will be per pendicular to the line A B.

3. To let fall a perpendicular from a given point. This is the 2d problem reversed, and one figure may serve for both.

From the given point D at any distance describe an arc, intersecting A B in a and b.

Proceed exactly as in problem 2d, only describing the arcs below the line A B, and the line D C E will be the perpendicular required.

PROJECTION OF MAPS.

I. To draw a map of the world on the globular projection. (See plate I. fig. 3.)

1st. Describe the circle N, E, S, W; and draw right lines cutting one another at right angles in the centre. N, S will represent the axis of the earth, and W, E the equator.

2d. Divide each quarter of these right lines into nine equal parts, proceeding from the centre to the circumfer ence; each division will represent ten degrees and may be numbered accordingly. Those on the axis will represent the latitude, and are to be numbered, from the equator towards the pole, 10, 20, &c. Those on the equator will represent the longitude, and are to be numbered so as to correspond with whatever point is fixed on as a first meridian. Figure 3 is supposed to be the western hemisphere, and London to be the first meridian and this, by

the usual method of projecting maps of the world, will fix the axis as io, from whence the numbers are continued westward, 120, 150, &c.

3d. Divide the four quarters of the outward circle into nine equal parts respectively, proceeding from the equator to the poles, and number them 10, 20, &c.

To draw the parallels of latitude The general rule is, to guide the compasses so that the lines may pass through the divisions in the outward edges and axis respectively, from 80 to 90. This however is attended with some difficulty, because the radius of every line being different, each requires a new centre. The centres can be easily found by the following process. Draw right lines from

the divisions on the axis to those on the circumference respectively, as a-a, fig. 3. Divide these right lines into two equal parts, and from the middle, b, let fall perpendiculars to a right line extending from the axis south or north, as b, b, c-the places where they respectively cut will be the centres, as c.

To draw the lines of longitude. Guide the compasses so that the lines may respectively pass through the divisions in the equator and the poles. The central points are found exactly in the same way as the latitude. See fig. 3,d, d.e, e, f.

II. To draw a map of the world on the polar projection.

(See plate I. fig. 4.)

In the polar projection the northern and southern hemispheres are projected on the plane of the equator, the poles being in the centre. It is but little used, as it exhibits the countries near the poles to the greatest advantage, while those near the Equator, which are of more importance, are much distorted.

It is extremely simple, and is executed by fixing one foot of the compasses in the poles, describing nine circles equidistant from each other to represent the parallels of latitude, the circles being ten degrees apart.

Divide the outer circles, or equator, into thirty-six equal parts, and draw lines from the pole to each point in the equator. These will represent the meridians of longitude.