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The length of the radius E'C' or EC, is just the cotangent of the latitude, if the radius CG = 1; for in triangle ECG, angle G is the complement of AGC, which is the latitude of the middle parallel C'P, and
GC: CE rad: tan G, or a:r=1:cotl,
where a CG, r = CE, and 7 = middle latitude; and hence, ra cot l.
Since the length of C'P is just the length of the middle parallel on the sphere APBD, whose radius CF is = cos l, or r'a cos l, if CF = r'; and the radius EC is ra cotl; therefore the number of degrees in the angle ME'N will be to the number of degrees of longitude in the map inversely as the radiir and r'. Or, if v, v', denote the number of degrees in angle ME'N, and in the longitude, then r: r' = v': v, or a cot l: a cos l = v' : v ;
v'v' sin l.
Thus, if 56°, and ' 15° 12', and a = 246, r = a cot l = 246 × ·6745 =165·9,
and v=v' sin l = 15°2 × ·829 = 12°·6 = 12° 36'.
533. This projection is used for large portions of the earth's surface, as for a hemisphere. The meridians and parallels of latitude are projected according to the methods explained in the problems of this projection, beginning with article 295.
I. When the projection is made on the plane of a meridian.
The meridians are described in the same manner as AGC in case 1, article 301. If, for instance, 12 meridians are to be described in the hemisphere ABCD, that is, at the distance of every 15°, for the meridian next to ADC, the angle FAE will be 15°; for the next it will be 30°; for the next 45°; for the next 60°; and so on. Or the meridians may be described thus:-Let the projection of a meridian, inclined to the primitive ACBD (fig. to art. 297), by an angle measured by the arc AF, be required. Join DF, and
through C, G, D, describe the circle CGD for the meridian required. When the meridian is to be inclined 15° to CAD, make AF = 15°; when the inclination is to be 30°, make AF = 30°; and so on.
The parallels of latitude are described thus:-Let CE or CM (first fig. to art. 298) be the distance of one of the parallels of latitude from the pole; then draw the tangent EL to meet the polar diameter CD produced, and EL is the radius, and L the centre of the projected parallel MNE. Or, join AE by a straight line, and it will cut CP in N; then, if a circle is described through MNE, it is the projection required. If the circle is 30° from the pole C, that is, if it is the parallel of latitude of 60°, make CE and CM each = 30°; for the latitude of 50°, make CE and CM each = 40°; and so on.
II. When the projection is to be made on some circle, as the horizon of a place, which cuts the meridian in the east and west points.
Let ACBD (fig. to art. 297) be the horizon of the place, AB the meridian of the place, and C,D the east and west points, and AF = the latitude. Join FD, and G will be the projected pole, for AG is the projection of the latitude (299.) The projection of any meridian, inclined to the meridian AGB of the place, is described in the same manner as the circle IFK (second fig. to art. 301), F being the pole, and BFD the meridian of the place. When the meridian is inclined 50° to BFD, make angle LFH = 50°; and similarly for any other meridian.
The parallels of latitude are in this case described as in case 3, article 298. If P is the pole, AB the meridian of the place, and B the north point, and D, C, the west and east points, then, to describe the parallel of latitude 30°, draw CP, and produce it to E; make EG and EF each = the complement of the latitude; draw CG and CF, cutting AB in I and H, and on IH describe the circle IKH for the required projection; and in a similar manner describe the projections of the other parallels of latitude.
534. There is a method of construction called the globular, which is not properly a projection, but it is useful, as it represents the magnitudes of different portions of the
earth very nearly in their proper proportions. In this method the radii of the polar and equatorial diameters AB, CD (fig. to art. 296), are divided into the same number of equal parts, into nine for instance, when the meridians and parallels are respectively 10° distant, and the meridians pass through the poles A, B, and the divisions of CD; also the four quadrants AC, CB, BD, DA, being divided into the same number of equal parts as the radii, the parallels for the northern hemisphere, for instance, pass through the corresponding divisions of the quadrants AC, AĎ, and radius AE; and similarly for the southern hemisphere.
If a point is taken for the projecting point in that equatorial diameter produced, which is perpendicular to CD, at three-fourths of the radius above the sphere, the projections of the meridians and parallels of latitude on the meridian ACBD would very nearly coincide with those of the globular
535. In the orthographic projection, the half meridians, as APB, are semi-ellipses, and the parallels of latitude are straight lines parallel to the equator CD, and passing through the equal divisions of the quadrants AC, AD. (See Geom. vol. II. Orthog. Proj. I. Cor. 3, and III.)
536. In Mercator's construction, the meridians and parallels of latitude are parallel straight lines; the former being equidistant for equal differences of longitude, but the distance between the latter, for equal differences of latitude, increases with the latitude. When these distances are very small, as 1', they are increased in the ratio of the cosine of latitude to radius, or of radius to the secant of latitude. (See article 514.)
Let e an elementary part of a terrestrial meridian, that is, a minute portion of latitude, as 1',
the enlargement of e on the projection, = the latitude, and R = the radius, then R: secle: e', or e' e sec l, when R=1; and if e = l', then e' in minutes = sec l.
Hence, if (l'), (2′), (3′),...denote the enlargement of l' in the latitudes of 1, 2, 3',...respectively, and if m1, M2,
m.,...denote the meridional parts for these latitudes respectively, then is
m1 = (1′), m2 = (1′) + (2′), m ̧ = (1′) + (2′) + (3′), &c. Or, m1 =(1′), m2 = m1 + (2′), m3 =m2 +(3′), &c.
At the latitude of 40° l', for example, if m' = the meridional parts for 40°, and m = the same for 40° 1′ or 241', then is m = m' + (241').
This method is not rigorously correct, for the enlargement of l' is made uniform at any particular latitude, whereas the minute portions of a minute that are farthest from the equator ought to be increased in a higher ratio than the other portions of it; but the difference is trifling, for in the latitude of 45° the error is only about 0'2; the meridional parts by the above, or Wright's method, being 3030-127, and its true value being 3029.939. If 1" were taken for the elementary part of the latitude, the error would be less, but the labour of calculation of the meridional parts would be much increased.
To construct a map by this projection.
1. When the map contains the equator.
Draw a straight line to represent the equator, and lay off on it from a convenient scale the number of degrees of longitude in the map. Through every 10th degree on this line draw perpendiculars to it for the meridians. Find the
meridional parts corresponding to the extreme latitude on the north, for instance; and as it is given in geographical miles, divide it by 60, and lay off the quotient from the same scale as the degrees of longitude on one of the meridians, and through its extremity draw a parallel to the equator for the extreme parallel of latitude. Find the meridional parts for the latitudes 10°, 20°, 30°,...and lay them off in the same manner, and draw the corresponding parallels.
2. When the map is limited by two parallels of latitude of the same name.
Draw a line to represent the lower parallel of latitude, in the same manner as the line representing the equator in the preceding case, and draw also the meridians in the same way; then find the meridional difference of latitude for the two extreme latitudes; divide it by 60, and lay off the quo
tient on a meridian as in the preceding case, and draw the extreme parallel as before. Find then the meridional difference of latitude corresponding to the lower latitude, and the latitude 10° higher, and, dividing it by 60, lay off the quotient from the lower parallel on a meridian, and it will reach the point through which the corresponding parallel passes; proceed in the same way for the parallels of 20, 30, 40,... degrees, and the map will be constructed (see 515.)
537. Dialling treats of the construction of sun-dials. A sun-dial is a surface, generally a plane, on which a system of lines is drawn, in such a manner that the coincidence of the shadow of a straight rod or edge with any of them, points out the hour of the day in apparent time.
538. The straight rod or edge is called the stile of the dial, and the system of lines are called hour lines.
When the stile is the edge of a plate, the latter is called a plate stile. The plane of the plate stile is generally placed perpendicularly to the plane of the dial, and its intersection with the plane of the dial is called the substile.
539. The inclination of the stile to the plane of the dial, that is, to the substile, is called the elevation of the stile.
The stile is always placed parallel to the earth's axis, and the hour lines are just the intersections with the surface of the dial of planes passing through the stile, which, with the plane of the meridian, are inclined to one another at an angle of 15° in succession. The earth is so minute an object compared with the distance of the sun, that the time shown on a dial at the earth's surface, is the same as would be indicated by the shadow of the earth's axis, were it a real axis, and the earth transparent, on a plane passing through the earth's centre parallel to the plane of the dial.
540. When the plane of the dial is horizontal, it is called a horizontal dial; when it is vertical, it is called a vertical or an erect dial; when the dial is vertical and perpendicular to the meridian, it is called a prime vertical dial; and