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2. A lighthouse was observed to bear from a ship NNE., and after sailing 15 miles on a WNW. course, its bearing was found to be NE. b E.; required its distance from the last place of the ship. Ans. 27 miles.
3. A cape was observed to bear E. 6 S. from a ship, and after sailing NE. 18 miles, its bearing was SE. 6 E.; what was the distance of the cape from the second place of the ship. Ans. 39 miles.
525. After taking the departure, the next important problem is to find the bearing and distance of the port bound for, which is solved by article 518. After performing a day's sailing, as nearly as possible in the proper direction, the place of the ship is then to be determined by article 521; and then again, if necessary, the bearing and distance of the intended port; and these problems are to be successively repeated during the voyage. The place of the ship, determined in this manner, is said to be the place by account; or its latitude and longitude are said to be the latitude and longitude by account. As the place of a ship, determined in this manner, cannot be depended upon on a long voyage, on account of the errors occasioned by unknown currents, storms, and the unavoidably imperfect means of measuring the courses and distances, it becomes necessary to employ the principles of practical astronomy to determine the latitude and longitude with greater accuracy. This method of determining the various elements in navigation, is called nautical astronomy.
526. By the principles of nautical astronomy, the time at the ship's place, the variation of the compass, the latitude and longitude, and various other elements used in navigation, can be determined. As the complete solutions of these problems have already been given in the problems in practical astronomy, excepting the circumstances peculiar to navigation, by which the solutions are in some cases modified, it will be necessary here merely to add the methods of calculating the effect of these circumstances.
527. PROBLEM I.-To find the variation of the compass.
Find the azimuth or amplitude of some celestial object by the methods formerly given in the articles 466, 473; and find also its bearing per compass, and the difference between the azimuth and bearing will give the variation of the compass.
1. The azimuth of the sun was found to be S. 48° 54' E. when its bearing was S. 77° 1′ E.; what was the variation of the compass? Ans. 2 points E.
2. The amplitude of a star was found to be E. 10° 15′ N. when its bearing was S. 84° 12′ E.; what was the declination of the needle? Ans. 16° 3′ W. be N. 68° 10′ E. the variation of Ans. 11° 55' E.
3. The azimuth of a star was found to when its bearing was NE. 6 E.; what was the compass?
528. PROBLEM II. Having given two altitudes of a celestial body, the ship having sailed for several hours during the interval, to reduce the first altitude to the place at which the second was taken.
'Find the angle of inclination between the ship's course and the bearing of the body at the first place of observation, or its supplement when greater than a right angle; then
The radius is to the cosine of this angle, as the distance run to the correction in minutes; which is to be applied by addition or subtraction to the first altitude, according as the inclination is less or greater than a right angle."
If d the distance, i = the inclination, c = the correction, a' the first altitude, and a = the first altitude reduced to the second place, then
Rad: cos id: c, and a = a' ± c.
It is evident that c can be found by inspection of the table of difference of latitude and departure, by considering i as the course, and d the distance; then c will be found in the latitude column.
1. The altitude of a star when east of the meridian was observed to be 20° 40', and its bearing at the time was
SE. b S.; and after sailing 40 miles W. 6 S., its altitude was again observed when it was west of the meridian; what would have been its altitude at the time of the first observation, if it had been taken at the place of the second observation? Ans. Here i=3+7=10 pts. = 112° 30', d = 40, c = 15′·3, and a = 20° 24′·7.
2. The sun's altitude was observed to be 30° 41'5, and its bearing was SE. 6 E.; and after sailing 48 miles E. b S., its altitude was again taken; required the sun's altitude at the latter place of the ship at the time of the first observation. Ans. Here i 2 pts. = 22° 30′, d = 48, c = 44'3, and a = 31° 25'8.
The principle of the rule may be proved thus:-Let S be the zenith of the place of the first observation, and S' that of the second, B and B' the positions of the body at these two instants of time, SS' the intermediate distance sailed by the ship in minutes of space, HRn' and H'Rn the horizons of S and S'; then BH and B'H' are the two altitudes. Now, to find the altitude Bh of the body when at B, supposing the altitude to be taken then at S'; produce BS to m, and from S' draw the perpendicular mS' from S' on Sm; then, since SS', and consequently mS, is a small distance, mn may be considered as differing insensibly from S'h, at least for ordinary nautical purposes; but S'h is a quadrant, as also SH; hence mn = SH nearly, therefore mS nH nearly. Therefore Bh, the altitude of B, when taken at S', which is nearlyBn, is less than BH by mS. Now, angle BSS' is evidently i, SS'd, mS= c, and radius: cos i=d: c, which is the rule. If the ship had sailed from S to S", instead of S', it could in the same manner be shown, by drawing S'm' perpendicularly to SH, that Sm' would require to be added to the altitude of B, taken at S, in order to obtain its altitude at the same time if it were taken at S".
CONSTRUCTION OF MAPS AND CHARTS.
529. Maps and charts are representations of portions, or of the whole, of the surface of the earth, with meridians and parallels of latitude at some convenient distance from each other, as at 5 or 10 degrees. The principal kinds of construction are the plane construction, the method of conical projection, the stereographic projection, and Mercator's projection.
530. In the first method of plane construction, the meridians are parallel straight lines, as are also the parallels of latitude. It is used for a very small portion of the earth's surface, extending only a few degrees in c length and breadth, as for a portion of a kingdom.
The breadth from north to south, AC, is the number of degrees of lati- E tude, each of which is equal to 60 geographical or 69.02 imperial miles ; and the length from east to west, AB, is just the length of the number of degrees A of longitude contained in it, estimated on the parallel EF of middle latitude by the proportion in article 507.
Let L', L= the lengths of a degree of longitude at the equator, and at the middle latitude,
the middle latitude;
then rad: cos / L': L, and LL' cos l, if R = 1; and since L'60 geographic miles,
L 60 cos l.
If l = 56°, then L = 60 × ·5592 = 33·552 miles.
531. In the second method of plane construction, the parallels of latitude are parallel straight lines, and the meridians are converging straight lines. This method is used for projecting larger portions of the earth's surface, as for a kingdom.
The breadth of the map in this case is AB, which is equal to the length of the degrees in the latitude contained in the map, as in the preceding method. The parallels of latitude are perpendicular to AB, which is the middle meridian. The
lengths of the degrees of longitude in the parallel CD are the lengths for the latitude of B found as those in EF in the preceding method; and the lengths for the parallel EF are also those corresponding to the latitude of A, and the corresponding divisions of EF and CD are joined by straight lines, which are the meridians.
532. The method of conical projection, or conical development, is used for still larger portions of the earth's surface, as for a continent. This method is thus derived :-A conic surface is supposed to touch the earth's surface along the parallel of the middle latitude, and the former surface is supposed to coincide with that of the earth for a few degrees of latitude on both sides of the middle parallel.
Let APB be the earth, C the middle latitude, and CE a tangent, meeting c the axis DP produced; then E is the vertex of the cone, CE its slant side, A and ED its axis. When this conic surface is developed on a plane, it will be represented by E'MN, in which the points E', C', correspond to E and C in the above figure. The breadth of the map MQ, therefore, is just the length of the number of degrees of latitude contained in the map, as in those of the last two articles; and the length of the middle parallel C'P is just the same length as on the earth, which is computed exactly in the same manner as the length of the middle parallel EF in article 530; and the meridians E'S, E'T, &c., are drawn through points of division on C'P from the vertex E', which is the centre of the parallels of latitude.