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round, we determine the bearing and distance, at once, of all places from London, or any other place (Z), in the centre.

Scholia. 1. Fig. 7th, plate II., represents the entire globular projection of the sphere.

2. Fig. 8, plate II., represents the complete stereographic projection of the sphere.

3. If, in No. 1, fig. 7th, we take from the whole projection the rectangular figure ABCD, we shall have the skeleton of the Map of Asia.

4. And if, in No. 2, fig. 7th, we take from the hemisphere the parallelogram EFGH, it will be the skeleton of the Map of Africa.

5. Also, if, in No. 1, fig. 8th, we take the rectangular figure IKLM, it will be the skeleton of the Map of North America.

6. Finally, if, in No. 2, fig. 8th, we take the parallelogram NOPQ, it will be the skeleton of the Map of South America.

Note.-Projected on a large scale, these skeletons will afford the most ample exercises for the display of juvenile genius and taste in the subsequent execution of filling up, shading, lettering, and colouring.

Prob. 6. To construct a map of the world, on the plane of a meridian, according to the orthographic projection of the sphere. (Fig. 10, plate III.)

1. To describe the meridians, which, in this projection, are ellipses. -Provided they be described through every tenth degree on the equator, the distance of each successive meridian, from the centre of the map, is found by means of the parallels drawn through the corresponding divisions of the circumference, as shown in No. 1, fig. 10. If these elliptical meridians are drawn with a pair of elliptical compasses, through every fifteenth degree of the equator from the centre of the map, they will appear as in fig. 9, plate II.

2. To draw the parallels of latitude, which are straight lines.If they are at 10° separate from each other, lines drawn parallel to the equator, through every successive tenth degree of the circumference, will indicate the parallels of latitude. But, if they be drawn through every fifteenth degree of latitude, they will appear as in fig. 9, plate II.

Tote. This projection is chiefly useful for astronomical purposes, and then only when you represent a sign of the Zodiac. But it is obvious that twelve such projections would furnish the means of depicting the twelve signs; and this is no inconsiderable advantage to those who study "Squire's Grammar of Astronomy."

Prob. 7. To project, on the plane of the equator, a map of the world, according to the globular projection of the sphere. (Fig. 10, plate III.)

Obs. In fig. 10, No. 1, the point R is distant from N equal to the line ss or the sine of 45° in fig. 4, plate 1. This point R is the place of the eye. whence the spectator, supposing the sphere pellucid, views the entire hemi sphere WSE. (No. 1, fig. 10, plate III.) The lines which pass from 10, 20' 30, &c. in WS to the eye at R, cross WP obliquely in the points 10, 20, 30' &c. Thus, the equal division, or nearly so, of the radius WP is obtained; and this is, in fact, the principle of the globular projection of the sphere.

I. To draw the meridians; which, in this map, are straight lines, radiating from P, the pole.

Divide the circumference into 36 or 360 equal parts, and to each

of these equal divisions draw straight lines, as seen in No. 2; and the meridians for one hemisphere, projected on the plane of the equator, are now laid down.

II. To draw the parallels of latitude; which, in this map, are concentric circles, described, with their respective radii, from P the com

mon centre.

From the Observation with which we have prefaced this construction, we know that WP, the radius (No. 1), is divided into nine equal parts. Then, with the respective radii, P 10, P 20, P 30, &c. of No. 1, describe the concentric circles, as shown in No. 2, and one hemisphere is completed, so far as respects the projection of the meridians and parallels.

Note.-Countries, sca coasts, towns or places, mountains, rivers, &c. are now to be laid down from tables of Latitude and Longitude.

Prob. 8. To construct a map of the world, on the plane of the equator, according to the stereographic projection of the sphere. (See fig. 2, plate III.

Obs. As, in this projection, the eye of the observer is placed on the surface of the sphere, and in either pole, as at N (No. 1), the straight lines drawn from the equal divisions of the quadrant WS to the point N, cut the radius WP into unequal divisions, we derive, at once, the principle of the stereographic projection of the sphere, in which equal spaces on the surface of the earth are represented by unequal spaces on the projection; the space W 10 being double of P 80; and, consequently, if P 80 be 1, W 10 will be 2; and any quadrilateral continued between W 10 and 10° of latitude, on a projection on the plane of the meridian, will be four times the size of a quadrilateral comprehended by P 80 and 10° of latitude, because the square of 2 is 4, and the square of 1 is 1.

I. To project the meridians; which, in this, as in the last Problem, are straight lines.

The directions given in that Problem apply perfectly to this. The process is the same in both, and needs no further illustration.

II. To draw the parallels of latitude, which, as in the former projection, are concentric circles.

Take P as the common centre, and with the radii P 10, P 20, P 30, &c. of No. 1, respectively, describe the concentric circles, which shall represent the parallels, as shown in No. 2.

Note. The note subjoined to the last problem is to be fully observed in the execution of this projection.

Prob. 9. To project a map, on the plane of the equator, according to the orthographic projection of the sphere. (See fig. 12, plate III.)

Obs. 1. As the eye of the observer is supposed, in this projection, to be situated at an infinite distance from the surface of the sphere, all the lines which are drawn on it are straight lines. On this principle, the meridians would appear, to an eye so situated, as in No. 1, fig. 12, plate III.; and the parallels would also be straight lines, as represented in fig. 9, plate II.

2. But, on the plane of a meridian, a map constructed according to this projection has its meridians drawn elliptical, while its parallels are straight lines. (See fig. 9, plate II.)

3. Whereas, on the plane of the equator, the same laws are observed as in

the two last projections, and the meridians are radiating straight linee; while the parallels are concentric circles, described from the common pole.

I. To draw the meridians.

Proceed as in the two last problems for their meridians.
II. To draw the parallels of latitude.

Through the points 80, 70, 60, 50, 40, 30, 20, and 10, of the semicircle NWS (fig. 12, No. 1), draw straight lines parallel to NPS, and the divisions 80, 70, &c. on PW, are the divisions which indicate the law of the projection, and the radii for the concentric parallels, which are respectively drawn on No. 2

Scholium. On reviewing these projections, the globular (fig. 10) has decidedly the advantage of presenting equal spaces of latitude throughout its suc cessive geographic quadrilaterals; the stereographic (fig. 11) presents unequal spaces, diminishing towards the pole, but allowing us more space than the globular for those countries situated near the equator; and, in this respect, answering better than the other the conditions of the projection. The ortho. graphic is the reverse of this last, as it allows to the polar regions more space than either of the other two; but then the countries contiguous to the equator are abandoned to a greater error in respect of latitude than even the polar regions in the stereographic projection. For geographical purposes, the globular is preferable; for astronomical uses, the stereographic merits atten tion, when the signs of the Zodiac, or stars within the tropics, are to be laid down, as seen from either pole; and the orthographic suits best the delineation of the arctic or antarctic constellations

Prob. 10. To project a map of Asia, according to the globular projection. (See fig. 13, plate IV.)

1. Having drawn any indefinite line AX, and assumed a distance Ak for 10° of latitude, set off this distance nine times from A toward X. The point 9, or 90, will be the pole.

2. With the distance Ak set off AB=AD, because the degrees of longitude on the equator correspond with those of latitude on a meridian of the sphere.

3. At the point a, or 70° of latitude, set off ab, and ad, each equal to 20 52; the number of geographical miles corresponding to 70°.

4. Through the points Bb, Dd, draw the oblique lines Bb, Dd, which, by the laws of decreasing longitude is constructed, terminate in the point X. This point (X) is, therefore, the common centre for all the parallels of latitude; and it is 30 degrees beyond the pole P, or at the same distance north of the parallel of 60 degrees, that the equator is south of it.

5. On the equator, EAQ, set off the portion AB, or AD, as often as may be necessary to answer the conditions of the projection; and from each of these points, 50, 60, 70, &c. draw lines to the point X, and they will indicate the meridians, which are all straight lines.

6. Set, now, one foot of the compasses in the point X, and with the other describe the successive concentrics, 10, 20, 30, &c. for the parallels of latitude.

7. Through the point A draw ML, at right angles to AX; raise the two perpendiculars MO, LW, and draw OW, completing the parallelogram OMLW.

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Orthographic Projection of a Map of the World
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