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Stereograpale Horizontal rejection
CONSTRUCTION OF MAPS Arrowsmiths Projection
of A MAP of the WORLD, on the Plane of a Meridian
PRACTICAL CONSTRUCTION OF MAPS.
Definitions. Prob. 1. To construct a Map of the World, on the plane of a 'meridian, according to the globular projection of the sphere.- (See fg. 1 and 2, plate I)
1. The globular projection of the sphere, is that in which equal spaces on the surface of the globe are represented by equal spaces of the projected map, as nearly as a spherical surface can be represented on a plane.
2. The plane of a meridian is the plane of one of the great circles of the sphere passing through both poles, and crossing the equator at right angles.
3. A hemispherical map of the world, is a representation of the en. tire surface of the globe of Earth, projected on the plane of one of its great circles.
1. To draw the meridians, or circles of longitude,
2 From C as a centre, with any radius, CA or CB, according to the size of your paper, describe the circle ANBS.
3. This circle is then the plane of your projection.
4. Divide the four radii, CA, CN, CB, CS, each into nine equal parts.
5. Now, to draw the meridian 80° west of Greenwich, we have the two poles 909 90°, and the point 80° in the equator, or diameter AB.
6. From N as a centre, and with NC as a radius, describe the arc ZCZ; also from S, with the same radius, describe the arc XCX.
7. Then remove the compasses to the point 80 on the equator, and describe the arcs 1,1, and 2,2.
8. Through the intersections, as at 1 and 2, draw lines from 2,2, through the points 1,1, till they intersect the diameter BA produced in D.
9. Then will D be the centre from which, with the radius D 80°, OF DN, or DS, the meridian of 80° west longitude from Greenwich must be described.
Nute.-The same radius will draw the meridiau expressing 140° W. L.; and, in the other bemisphere, the corresponding meridians of east longitude.
• A B represents the equator, and NS the aris meridian.
10. The meridian of 50° is drawn in the same manner as that of 80°, except that the point 50 on the equator (AB) is the centre from which, with the radius CB, the intersections are made at a,a, and 6,6, on the arcs decribed from N and S. The point E, where the lines la, ba, meet on the equator, is the centre for the meridian of 50° W. long. The same radius serves for the other three meridians 30° within the circle of projection.
Note.-In this manner are all the other meridians for both hemispheres to be drawn; as may be seen in fig. 7th, Plate II.
II. To draw the parallels of latitude.-Fig. 2, plate I.
1. The same construction remaining, and the radii CN, CS, divided each into nine equal parts.
2. Divide the circumference ANBS into thirty-six equal parts ; so shall each quadrant AN, BN ; AS, BS, be divided into nine equal parts, which, being again subdivided into ten equal parts each, will give us 360 equal parts or degrees.
3. To draw, now, the parallel of 30° north latitude, set one foot of the compasses in the point 30 on the axis meridian NS, and with any radius describe the circle KTL.
4. Set, again, one foot of the compasses in the points 30, 30, in the circumference, and intersect the circle KTL in the points s, s, n, n.
5. Through nn and ss draw the straight lines nn, ss, meeting each other in the point R of the axis SN, produced to R.
6. with R as a centre, and R30 as a radius, describe now the parallel of 30° north latitude.
Note.-The same radius is, of course, used for 30° south latitude.
7. After the same process the parallel of 60° north latitude is drawn, as may be easily conceived by inspecting the figure.
Note.-And in this way proceed with all the other parallels in both hemispheres ; as is evident froin fig. 7th, Plate II.
8. This is the rationale of construction. That which respects the minutiæ of filling up the map requires only the attention of the delineator, in strictly observing the proper situations of objects or lines, as far as they respectively bear on the practice of map-making.
Prob. 2. To project a map of the world on the plane of a meridian, according to the stereographic projection of the sphere.—(fig. 3, plate I.)
I. To draw the circles of latitude.
1. Describe, from any centre C, the circle ENQS, which will represent one half of the earth's surface.
2. Draw the diameters EQ and NS, intersecting each other at right angles.
3. EQ will be the equator, and NS the axis meridian.
4. Divide the circumference into 360 equal parts; numbered 10, 20, 30, &c. as on the figure.
9. From E to 140 draw the line E 140; bisect the portion a 140
p v, and from v raise the perpendicular vr, which produce till it meet NS produced in x. The point x will now be the centre from which to describe the parallel za 140; or, which is the same thing, the 50th degree of south latitude. | Note.—The same radius will serve for the 50th parallel of north latitude ; and after the same manner all the other parallels in both hemispheres are drawn, as they are fully shown in fig. 8, plate II.
II. To draw the circles of longitude.
1. The unequal divisions of the equator, as indicated by the numbers 10, 20, 30, &c. on the radius CQ, and which are obvious from No. 1, fig. 11, plate III.
2. The points 20, 40, 60, 80, will now be centres on which the circles of longitude Syn are to be drawn ; but, for the remaining circles, produce the diameter EQ, and from N, through every tenth degree in the quadrant NQ, draw lines cutting that diameter produced, and the points of intersection will give the centres for the remaining circles of longitude : it is necessary, however, to observe to the young geographer, that each centre is twenty degrees distant from the preceding one.
3. For the circles of longitude in the other semi-hemisphere, the centres may be formed by setting off the proper distances on the diameter EQ, produced the contrary way.
Prob. 3. To draw a map of the earth on the plane of a meridian, according to the globular projection of the sphere.
Note.—Though Problem 1 exhibited this projection, we are induced to give it by another process; since, by this variety, more skill will be acquired in the practice of mathematical geography.
1. To draw the circles of latitude. Fig. 4, plate I.
1. Describe the circles ENQS; draw the diameters EQ and NS at right angles; the former will represent the equator, and the latter the axis meridian.
2. Divide the quadrant QS into nine equal parts, 10, 20, 30, &c. From each of these divisions draw lines, as Ef 20, Eg 30, Ed 60, &c.
3. Divide into two equal parts the portions f20, g30, 260, &c. and from c, the point of division, let fall the perpendiculars cF, cG, and «D, produced till they cut the polar diameter extended indefinitely.
4. The points F, G, D, will be centres, from which the circles of latitude xf20, 2G30, zd60, are to be described, and which will be the true representation of the parallels 20, 30, and 60 degrees of south latitude.
5. In the same manner, draw the paralels for every tenth or fifth degree in that semi-hemisphere.
6. To obtain those in the northern hemisphere, set off on the line SN produced, in the opposite direction, the distances which served as centres of the southern parallels : and thus the northern ones may be described for every tenth or fifth degree of latitude. II. To draw the circles of longitude. Fig. 4, plate I.
1. Divide the quadrant EN into equal parts, 10, 20, 30, &c. Divide also the quadrant SQ into two equal parts at s, of 450 each, and let fall the perpendicular ss from the point s.
2. Set off, on the line NS produced, SR equal to ss (see fig. 10, plate III.), then, lines drawn from R to 10, 20, 30, &c. in the quadrant WS, will divide the radius in the points 10, 20, 30, &c. through which the circles of longitude are to be drawn (in fig. 4, plate I), after the following manner :
3. To find the centre of the circle S 30 N, join the points S 30, and N 30; divide the two lines into two equal parts in o, and let fall the perpendiculars o 30; and the point t, where they meet, is the centre of a circle of 30 degrees of longitude. The other centres may be found exactly in the same manner.
Or the centres may be found, mechanically, and very readily, from the following TRIGONOMETRICAL Table of Radu:
1. Let the radius of the circle be divided into 100 equal parts by a scale ; then the meridian or circle of
10° 20° 30° 40° 50°
of those parts set off 42 from C towards Q, 61.5 (added to the distance 88 between C and the se133 veral points 10, 20, 30, 208
&c. in the radius EC. 464
2. Thus the radius of the circle of 10° of longitude is equal to the distance between 10 in the line EC, and 10 in the line QC; the radius of that of 50°=the distance between 50 and 50; that of 80° between 80 in the line EC and a given point in EQ, produced ; which, taken from C, will be = 342 parts, of which radius is = 100.
Prob. 4. To project a map of the earth stereographically, according to the horizontal projection of the sphere, and answering to the latitude of London. See fig. 5, plate I.
1. To draw the meridians.
1. With any radius, SC or SD, describe the circle CPD, which divide as usual into 360°, and draw CD, PS, at right angles to each other; then will PS be the first meridian, or the north and south azimuth.
Note. Asimulh, or verlical circles, are great circles of the sphere, intersecting each other in the zenith and dadir, and cutting the horizon at right angles.
2 On the quadrant DP set off DE=511 = the latitude of London, and draw parallel to CD, WE, the east and west azimuth of the place. Bisect WE in the point Z, which will be London, or the place of projection. The letters E, S, W, N, represent the four cardinal points, bearing due east and west, north and south, from London (Z). as the centre of the projection. And P, the pole of the meridional projection, is also the pole of the horizontal projection.