3. A ship from lat. 28° 32' N., has run the following courses, viz., 1st, N.W. by N., 20 miles; 2d, S.W., 40 miles; 3d, N. E. by E., 60 miles; 4th, S. E. 55 miles; 5th, W. by S., 41 miles; 6th, E. N. E., 66 miles. Required her present latitude, the distance made good, and the direct course from the place left. The direct course is due east, and distance 70.2 miles, the ship being in the same latitude at the end as at the beginning of the traverse. 4. A ship from lat. 41° 12′ N., sails S.W by W. 21 miles; S.W. S. 31 miles; W.S.W. S., 16 miles; S. E., 18 miles; S.W. W., 14 miles; and W. N., 30 miles; required the latitude of the place arrived at, and the direct course and distance. Lat. 40° 5' N.; course S. 52° 49' W.; distance 111.7 miles. 5. A ship runs the following courses, viz. 1st, S. E., 40 miles; 2d, N. E., 28 miles; 3d, S.W. by W., 52 miles; 4th, N.W. by W., 30 miles; 5th, S. S. E., 36 miles; 6th, S. E. by E., 58 miles; required the direct course and distance made good. Direct course S. 25° 42′ E., or S. S. E. E. nearly; distance 95.69 miles. These examples will sufficiently illustrate the principles of plane sailing, in which, course, distance, difference of latitude, and departure, are the only quantities which enter into the problem, two of them being always given. The determination of the difference of longitude made on any course, which is the distance between the meridians measured on the equator, cannot be effected by these principles, for this element is not the same as if the meridians were all parallel to each other, as is the case with the other elements. The finding of the difference of longitude is the easiest when the ship sails due east or due west, that is, upon a parallel of latitude; this is called PARALLEL SAILING. 99. The theory of parallel sailing is comprehended in the following proposition, which admits of a variety of other applications. The arc of a great circle comprehended between two of its secondaries is to the arc of a parallel small circle, comprehended between the same secondaries, as radius unity is to the cosine of the distance between the great circle and its parallel, measured on one of the secondaries. (See Spherical Geom., Prob. 2, Cor. 6.) Applied to the case under consideration the above proposition would run as follows, viz: The cosine of the latitude of the parallel is to the distance run as the radius to the difference of longitude. This may be demonstrated as follows: Let IQH represent the equator, and BDA any parallel of latitude; cr will be the radius of the equator, and Cв the radius of the parallel. Let BD be the distance sailed, then the difference of longitude will be measured by the arc IQ of the equator, and since similar arcs are to each as the radii of the circles to which they belong, we have H A CB: CI: dist. BD: diff. long. 1Q B D I But CB is the cosine of the latitude IB to the radius c1, and as cosine and radius are proportional in different circles, CB CI: cos lat. : R The first two terms of these proportions being the same, the last are proportional, and we have : cos lat. Rad.:: distance: diff. long. (1) Corollary hence if the distance between any two meridians, measured on a parallel in latitude L be D, and the distance of the same meridians, measured on a parallel, in latitude L' be D', we shall have (Spher. Geom., Prop. II., Cor. 6), COS LD cos L': D' for both the ratios of (2) will be equal to R: diff. long. (2) By referring to proportion (1) it will be seen that if any one of the legs of a right-angled triangle represent the distance run on any parallel, and the adjacent acute angle be equal to the degrees of lat. of that parallel, then the hypothenuse will represent the difference of longitude, since this hypothenuse will be determined by that proportion. The right angled triangle used in plane sailing may therefore be employed here, changing the names of its elements, viz., course into latitude, difference of latitude into distance, and distance into difference of longitude. And a traverse table computed to degrees and fractions of a degree instead of points and quarter points, may be employed to solve problems in parallel sailing. Formula (1) above may be expressed by the following rule. Divide the distance sailed by the cosine of the latitude, and the quotient will be the difference of longitude. EXAMPLES. 1. A ship from latitude 53° 56' N., longitude 10° 18′ E., has sailed due west, 236 miles required her present longitude. By the rule 2. If a ship sail E. 126 miles, from the North Cape, in lat. 70° 10' N., and then due N., till she reaches lat. 73° 26' N.; how far must she sail W. to reach the meridian of the North Cape? Here the ship sails on two parallels of latitude, first on the parallel of 71° 10′, and then on the parallel of 73° 26', and makes the same difference of longitude on each parallel. Hence by the corollary, 3. A ship in latitude 32° N. sails due east, till her difference of longitude is 384 miles; required the distance run. 325.6 miles. 4. If two ships in latitude 44° 30' N., distant from each other 216 miles, should both sail directly south till their distance is 256 miles, what latitude would they arrive at? 32° 17' N. 5. Two ships in the parallel of 47° 54' N., have 9° 35' difference of longitude, and they both sail directly south, a distance of 836 miles: required their distance from each other at the parallel left, and at that reached. 385.5 miles, and 496.9 miles. MIDDLE LATITUDE SAILING. 100. Having seen how the longitude which a ship makes when sailing on a parallel of latitude may be determined, we come now to examine the more general problem, viz., to find the longitude a ship makes when sailing upon any oblique rhumb. There are two methods of solving this problem, the one by what is called middle latitude sailing, and the other by Mercator's sailing. The first of these methods is confined in its application, and is moreover somewhat inaccurate even where applicable; the second is perfectly general, and rigorously true; but still there are cases in which it is advisable to employ the method of middle latitude sailing, in preference to that of Mercator's sailing; it is, therefore, proper that middle latitude sailing should be explained, especially since, by means of a correction to be hereafter noticed, the usual inaccuracy of this method may be rectified. very inaccurate when the course is small, and the distance run great; for it is plain that the middle latitude distance between the extreme meridians will be much greater than the departure, if the track A B cuts the successive meridians at a very small angle. The principle approaches nearer to accuracy as the angle A of the course increases, because then as but little advance is made in latitude, the several component departures lie more in the immediate vicinity of the middle latitude parallel. But since in very high latitudes, a small advance in latitude makes a considerable difference in meridional distance, this principle is not to be recommended in such latitudes if much accuracy is required. By means, however, of a small table of corrections, constructed by Mr. WORKMAN, the imperfections of the middle latitude method may be removed, and the result of it rendered in all cases accurate. This table we have given at the end of the present volume. The rules for middle latitude sailing may be thus deduced. dop diflong. dist. B It has been seen at (Art. 97), that the difference of latitude, departure, and distance sailed on any oblique rhumb, may be Á all accurately represented by the sides AB', B'B, ab, of a right angled plane triangle. Now, by the present hypothesis, the departure B'в is equal to the middle latitude distance between the meridians of the places sailed from, and arrived at, so that the difference of longitude of the two places of the ship is the same as if it had sailed the distance B'в on the middle latitude parallel; the determination of this difference of longitude is, therefore, reduced to a case of parallel sailing; and since, as we have seen (p. 215), the formula for parallel sailing is a proportion which expresses the relation between the elements of a right angled plane triangle in which the base is the dist. sailed, the angle at the base the lat., and the hypothenuse the diff. of long., let B'BA' be this triangle, in which, according to the theory of mid. lat. sailing, the departure B'в takes the place of the dist. sailed. From these triangles, the two partial ones of which are right angled, and the total one not, we have the following theorems, viz., in the triangle A'B'B, that is, COS A'BB': BB': radius: BA' A 1. Cos. mid. lat. : departure :: radius: diff. of long. In the triangle A'BA, which is not right angled, that is, sin A': AB sin A: A'B; II. Cos mid. lat. : distance: sin course: diff. long. In the triangle ABB', we have the proportion (Art. 41), R: tan A: AB' :: BB' comparing this with the first proportion above, observing that the extremes of this are the means of that, we have that is, AB': A'B COS A'BB': tan A; ш. Diff. lat.: diff. long. :: cos mid. lat. : tan course. |