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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 110; or, which is the sanie thing, the 50th degree of south latitude. 1. 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 num. bers 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 projectiov, 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, d60, &c. and from c, the point of division, let fall the perpendiculars cF, G, 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.
3. In the same manner, draw the parallels 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 latitudę. 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 111.), then, lines drawn from R to 10, 20, 30, &c. in the quadrant WS, will divide the radi 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 r, where they meet, is the centre of a circle of 30 degrees of longitude. The other centres may be found exactly in the saine manner.
Or the centres may be found, mechanically, and very readily, from the following TRIGONOMETRICAL Table of Radii :
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.- Azimuth, or rertical circles, are great circles of the sphere, intersecting each other in the zenith and nadir, 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.
3. Describe now the meridians of the meridional projection CPD, observing to allow them to pass through the pole P, beyond the primitive circle, and to louch the horizontal projection in the segment WNE.
4. To describe the parallels of latiluae, lay a ruler upon W, and move it to every degree, or every tenth degree of the meridian PD continued, marking where it cuts the meridian NS, as through these points the parallels on this side of the pole must all pass. But, as they nave not a common centre, the points through which they have to pass on the other side of the pole are found by moving the ruler along every tenth degree of the meridian PC continued ; for wherever the ruler intersects the meridian, NS will be the opposite points through which the parallels are to pass.
5. Having now got the diameters of the parallels, we have only to bisect each of them, and with one half, as a radius, describe the correspondent parallel. In fine, the projection may now be completed, as shown in fig. No. 2.
Prob. 5. By the globular projection of the sphere, to construct a map with azimuth lines, to show the l'earing and distance of all places within the map, from London, or any other given place in the centre. (See fig. 6, plate I.)
Ohs. It will at once be perceived that, in this partial projection, the longitude and latitude of places are neglected, because the map is restricted to the bearing and distance only of places from a station in the centre.
1. Having described a circle of any convenient radius, we cross it with two diameters ; of which NS represents the meridian of the face assumed as the centre, or the north and south line ; and WE, he east and west line, may be considered the parallel of latitude passing over the place. The intersection of the diameters, as at Z, indicates the place in the centre.
2. Divide each quadrant into 90°, as shown in the exterior circle of the figure ; and the whole inner circle into 32 equal parts, to indicate the points of the mariner's compass. (See the Table of Angles, &c. page 589). The lines radiating from 2 are the bearing lines. The three concentric circles, described from the coinmon centre 2, may be assumed as one degree each ; and the scale will then contain 180 geographical, or 208, English, miles.
3. Suppose, now, we place London in the centre; then, by the help of the Tables of Latitude and Longitude, in the sequel of this Atlas, we may transfer, into this projection, all places within 180 miles of London.
Note,-1. The numbers 1, 2, 3 degrees, on the scale from the centre, are arbitrary, and may be reckoned 10, 20, 30 degrees; in which case our projection would embrace 1860 geographical, or 2083 English, miles. Or, if the radius, or scale, be divided into 4, 5, 6, or any given number of equal parts, each of those parts may represent 4, 5, 6, or 40, 50, 60, degrees.
2. The scale in this projection must be considered a moveable slip of Bris. tol board, graduated according to the radius of the projection, and rivetted on the map by means of a neat button. Its use is obvious ; for, by moving it