transferring these divisions from B, as a center to the chord B C, we shall have the corresponding line of latitudes.

It is not necessary that these scales should all be projected to the same radius; but those which are used together, as the rhumbs and chords, the chords and longitudes, the sines, tangents, secants, and semitangents, and, lastly, the hours and latitudes, must be so constructed necessarily. In the accompanying diagram (plate 1, fig 2) we have laid down the hours and latitudes to a radius equal to the whole length of the scale, the other lines being laid down to the radius used in the foregoing construction.

The Line of Chords is used to set off an angle, or to measure an angle already laid down.

1st. To set off an angle, which shall contain D° from the point A, in the straight line. A B. Open the compasses to the extent of 60° upon the line of chords, which equals the radius to which this line has A been laid down (Euc. iv. prop.

B

15, Cor.), and, setting one foot upon A, with this extent describe an arc cutting A B in B; then, taking the extent of D from the same line of chords, set it off from в to c; and, joining A C, B A C is the angle required. Thus to set off an angle of 41°, having described the arc B c, as directed, with one foot of the compasses on B, and the extent of 41° on the line of chords, intersect B C in c, and join a c.

2nd. To measure the angle contained by the straight lines A B and A c already laid down. Open the compasses to the extent of 60° on the line of chords, as before, and with this radius describe the arc B C, cutting A B and A C, produced, if necessary, in the points в and c; then, extending the compasses from в to c, place one point of the compasses on the beginning, or zero point, of the line of chords, and the other point will extend to the number upon this line, indicating the degrees in the angle B A C. If, for instance, this point fall on the 41st division, or the first division beyond that marked 40 in the figure (plate 1, fig, 2), the angle B A C will contain 41°. The Line of Rhumbs is a scale of the chords of the angles of deviation from the meridian denoted by the several points and quarter points of the compass, enabling the navigator, without computation, to lay down or measure a ship's course "pon a chart. Thus, supposing the ship's course, to be

N.N.E. E. Through the point A, representing the ship's place upon the chart, draw the meridian A B, and with center A and distance equal to the extent of 60° upon the line of chords describe an arc cutting A B in B; then on the line of rhumbs take the extent to the third subdivision beyond the division marked 2, because N.N.E. is the second point of the compass from the north, and with one foot of the compasses on B describe an arc intersecting B C in C: join a c, and the angle B A C will represent the ship's course. On the other hand, if a

ship is to be sailed from the point A to a point on the line A c on a chart, draw the meridian A B, describe the arc B C with radius equal to chord of 60°, as before, and the extent from B to c, applied to the line of rhumbs, will give 2 pts. 3 qrs., denoting that the ship must be sailed by the compass N.N.E. E.

The Line of Longitudes shows the number of equatorial miles in a degree of longitude on the parallels of latitude indicated by the degrees on the corresponding points of the line of chords. Example.-A ship in latitude 60° N. sailing E. 79 miles, required the difference of longitude between the beginning and end of her course. Opposite 60 on the line of chords stands 30 on the line of longitudes, which is, therefore, the number of equatorial miles in a degree of longitude at that latitude. Hence, as 30: 79:: 60: 158 miles, the required difference of longitude.

The Lines of Sines, Secants, Tangents, and Semitangents are principally used for the several projections, or perspective representations, of the circles of the sphere, by means of which maps are constructed. Thus, the meridians and parallels of latitude being projected, the countries intended to be represented are traced out according to their respective situations and extent, the position of every point being determined by the intersection of its given meridian and parallel of latitude.

The plane upon which the circles are to be delineated is called the primitive, and the circumference of a circle, described with a radius, representing, upon the reduced scale of the drawing, the radius of the sphere, is called the circumference of the primitive. Lines, drawn from all the points of the circles to the eye, by their intersection with the primitive form the projection.

When the eye is supposed to be infinitely distant, so that the lines of vision are parallel to one another and perpendicular to the primitive, the projection is called orthographic. When the primitive is a tangent plane to the sphere, and the eye is supposed to be at the center of the sphere, the projection is called gnomonic. When the eye is supposed to be at the surface of the sphere, and the primitive to pass through the center, so as to have the eye in its pole, the projection is called stereographic.

The projection is further termed equatorial, meridional, or horizontal, according as the primitive coincides with, or is parallel to, the equator, or the meridian or horizon of any place.

Wol

30

CO

60

30

13010

30

120/150

30

GO

To delineate the Orthographic Projection of the Circles of the Terrestrial Sphere upon the Plane of the Meridian of any place. With a radius according to the contemplated scale of the projection, describe the circle wNES for the circumference of the primitive, and draw the vertical and horizontal diameters N s and w E, which will be the projections of a meridian perpendicular to the primitive, and of the equator, respectively. Take out from the line of sines the sines of the latitudes through which the parallels are to be drawn, and, reducing these sines to the radius of the primitive*, set off these reduced distances both ways from the center upon the line N s; and also both ways from the center upon the line w E, for the sines of the angles which the meridians, to be drawn at the same intervals as the parallels, make with the meridian N S. Through the divisions thus set off, upon the line N s draw straight lines parallel to w E, and such straight lines will be the projections of the several paral lels of latitude, which are to be numbered 0 to 90, from the equator to either pole for the latitudes. With distances from the center to the divisions set off upon w E as semi-minor axes, and the distance from c to N or s, equal to radius of primitive,

*If the proportional compasses be set in the proportion of the sine 90° on the line of sines to the radius of the primitive, one pair of points will give, reduced to this radius, the sines taken off by the other pair of points. The manner of taking from the sector a sine to any radius will be hereafter pointed

out.

as a common major axis, describe semi-ellipses*, and they will be the projections of the several meridians, which are to be numbered either way from the first meridian for the longitudes. In the figure the primitive coincides with the plane of the meridian of a place in 30° west longitude, or 150° east longitude, the sum of these two being 180°, as must always be the

case.

To delineate the Gnomonic Projection of the Circles of the Terrestrial Sphere upon a Plane parallel to the Equator.-In this case the meridians

will all be projected into straight lines, making the same angles one with another that their originals do on the surface of the sphere; the projection of the pole will be the center of the primitive, and the projections of the parallels of latitude will be circles described from the projection of the pole, as center, with distances equal to the tangents of the respective co

latitudes reduced to the radius of the primitive. The parallel of 45° will, therefore, coincide with the circumference of the primitive; the parallels of latitudes greater than 45° will lie within the primitive; and for latitudes less than 45° the parallels will fall without the primitive, the radii of their projec tions increasing as the latitude decreases, until the radius for projecting the equator becomes infinite. Describe, then, a

These semi-ellipses may be thus described. From any point p upon the straight edge of a piece of paper set off P C equal to the major axis, and P B equal to the minor axis: then move the paper W into various positions, but so that the point c may always be upon the line w E, and the point B upon the line N s, and the point p will, in every such position, coincide with a point in the required ellipse.

N

circle for the primitive; draw straight lines radiating from its center, and equally inclined one to another for the projections of equidistant meridians; and number them 0 to 180 both ways from the first meridian for the longitudes. With the tangents of the colatitudes, taken at intervals equal to the angle between two successive meridians, and reduced to the radius of the primitive, as distances, describe from the center of the primitive concentric circles; and number them 90 to 45 from the pole to the primitive for the latitudes, continuing the graduation beyond for the lower latitudes.

The gnomonic projection affords a good representation of the polar regions, but all places in latitudes lower than 60° appear greatly distorted. The gnomonic projection enlarges the representations of places at a distance from the center of projection beyond their proportionate true dimensions; and the orthographic, on the contrary, unduly contracts them; while both are adapted for representing best the countries at only a moderate distance from the center of projection.

To delineate the Stereographic Projection of the Circles of the Terrestrial Sphere upon the Horizon of any place.--With radius determined upon describe a circle for the primitive, and draw its vertical and horizontal diameters, N s and w E, which will be the projections of the meridian of the place and of the prime vertical respectively. From the center c set off upon the radius c s, produced, if necessary, the distance c 4, equal to the tangent of the latitude of the place reduced to the radius of the primitive; and with center A and distance a w or A E describe the circle w N E, which will be the projection of the meridian at right angles to N s, the meridian of the place; and, consequently, N will be the projection of the pole. Through a draw the right line a в at right angles to A C, and another line A D making any convenient angle with A B, and, setting off A B equal to the radius of the primitive, and A D equal to the sine of the colatitude, taken from the line of sines, join B D. Now take from the line of tangents the angles which the other meridians to be drawn are to make with the meridian w N E, or the complements of the angles which they are to make with N s, and set them off both ways from A upon the line ▲ D; through each of the divisions L, thus found, draw L o, parallels to B D, and we have at o the centers of the circles for describing the meridians*. With centers o and distances o N, describe the r cot. L

The distance A0=

cos.

where r represents the radius of the pri