Parallel Sailing. 16. The entire theory of parallel sailing is comprehended in the following proposition, viz. The cosine of the latitude of the parallel, is to the distance run, as radius to the difference of longitude. Let IQH represent the equator, and FDN any parallel of latitude: then, CI will be the radius of the equator, and EF the radius of the parallel. Suppose FD to be the distance sailed, then the difference of longitude will be measured. by IQ, the arc intercepted on the equator. Then, since similar arcs are to each other as their radii (Bk. V. Prop. xi. Cor.), we have, EF CI :: dist. FD diff. long. IQ. But EF is the sine of PF, or cosine of FI, the latitude, and CI is the radius of the sphere: hence, cos. lat. : R :: distance diff. longitude. Corollary. If we denote by D the distance between any two meridians, measured on the parallel whose latitude is L; and by D' the distance between the same meridians measured on the parallel whose latitude is L', the arcs will be similar, and we shall have (Bk. V. Prop. xi. Cor.), that is, cos. L : D :: cos. L' : D', cos. L : Cos. L' :: D: D'. Hence, when the longitude made on different parallels is the same, the distances sailed are proportional to the cosines of the parallels of latitude. By referring to Th. V. page 45, we see that in any right angled triangle R or : cos, angle at base :: hyp. : base, and by comparing this with the proportion, G to the degress of latitude of that parallel, then the hypothenuse will represent the difference of longitude. It follows therefore, that any problem in parallel sailing, may be solved as a simple case of plane sailing. For, if we regard the latitude as the course, the distance run as the base, the difference of longitude will be the hypothenuse of the corresponding right angled triangle. 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 sails E. 126 miles, from the North Cape, in lat. 71° 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 710 10', and then on the parallel of 73° 26', and makes the same difference of longitude on each parallel. Hence, by the corollary, As cos. lat. 71° 10′ arith. comp. 0.491044 3. A ship in latitude 32° N. sails due E. till her difference of longitude is 384 miles: required the distance run. Ans. 325.6 miles. 4. If two ships in latitude 44° 30′ N., distant from each other 216 miles, should both sail directly S. till their distance is 256 miles, what latitude would they arrive at ? Ans. 32° 17' S. 5. Two ships in the parallel of 47° 54' N., have 9° 35′ difference of longitude, and they both sail directly S., a distance of 836 miles required their distance from each other at the parallel Middle Latitude Sailing. 17. 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 which 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. Middle latitude sailing proceeds on the supposition that the departure or sum of all the meridional distances, b'b, c'c, d'd, &c. from 0 to T, is equal to the distance M'M of the meridians of O and T, measured on the middle parallel of latitude between 0 and T. M M The middle latitude is half the sum of the two extreme latitudes, if they are both of the same name, and to half their difference if they are of contrary names. This supposition becomes very inaccurate when the course is small, and the distance run great; for it is plain that the middle latitude distance will receive a much greater accession than the departure, if the track OT cuts the successive meridians at a very small angle. The principal approaches nearer to accuracy as the angle O 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 parallel M'M. But still, in very high latitudes, a small advance in latitude makes a considerable difference in meridional distance; hence, this principle is not to be used in such latitudes, if much accuracy is required. By means, however, of a small table of corrections, constructed by Mr. Wakeman, the imperfections of the middle lat itude method may be removed, and the results of it rendered in all cases accurate. This table we have given at the end of this work. Τ dif long. dep T The rules for middle latitude sailing may be thus deduced. We have seen, in the first case of plane sailing, that if a ship sails on an oblique rhumb from O to T, that the hypothenuse OT will represent the distance; OT the difference of latitude, and TT, the departure. Now, by the present hypothesis, the departure TT is equal to the middle parallel of latitude between the meridians of the places sailed from and arrived at so that the difference of longitude of these two places is the same as if the ship had sailed the distance TT on the middle parallel of latitude. The determination of the difference of longitude is, therefore, reduced to the case of parallel sailing: for, TT now representing the distance on the parallel, if the angle TTO be made equal to the latitude of that parallel, we shall have, by the last case, the difference of longitude represented by the hypothenuse OʻT. We therefore have the following theorem: I. In the triangle OTT that is, cos. O'TT' : TT :: R: TO'; cos. mid. lat. dist. : departure :: R diff. longitude. II. In the triangle OʻTO sin. O' : OT :: sin O : O'T; that is, since sin. O'=cos. O TT' cos. mid. lat. : distance :: sin. course : diff. longitude. III. In the triangle OTT', we have R: tangent O :: ΟΤ' : TT; comparing this with the first proportion, and observing that the extremes of this are the means of that, we have that is, OT: O'T :: cos. O'TT' : tangt. 0; diff. lat. diff. long. :: : cos. mid. lat. : tangt. course. These three propositions comprise the theory of middle latitude sailing; and when to the middle latitude sailing, the proper correction, taken from Mr. Workman's table, is applied, these theorems will be rendered accurate. In the table of pages 93 and 94, the middle latitude is to be found in the first column to the left. Then, along in the hori the proper correction to be added to the middle latitude to obtain the latitude in which the meridian distance is accurately equal to the departure. Thus, if the middle latitude be 37°, and the difference of latitude 18°, the correction will be found on page 94, and is equal to 0° 40′. EXAMPLES. 1. A ship, in latitude 51° 18′ N., longitude 22° 6′ W., is bound to a place in the S. E. quarter, 1024 miles distant, and in lat. 37° Ñ.: what is her direct course and distance, as also the difference of longitude between the two places? In this operation the middle latitude has not been corrected, so that the difference of longitude here determined is not without error. To find the proper correction, look for the given middle latitude, viz. 44° 9', in the table of corrections, the nearest to which we find to be 45°; against this and under 14° diff. of lat. we find 27', and also under 15° we find 31', the difference between the two being 4'; hence, corresponding to 14° 18′ the correction will be about 28', Hence, the corrected middle latitude is 44° 37', therefore, Cos. corrected mid. lat. : tangt. course :: diff. lat. : diff. long. 44° 37′ ar. comp. 0.147629 785.3 9.813899 3.933487 2.895015 therefore, the error in the former result is about 6 miles. 2. A ship sails in the N. W. quarter, 248 miles, till her departure is 135 miles, and her difference of longitude 310 miles: required her course, the latitude left, and the latitude come to. Course N. 32° 59' W; Ans. Lat. left 62° 27' N.; lat. in 65° 55′ N. 3. A ship, from latitude 37° N., longitude 9° 2′ W., having sailed between the N. and W., 1027 miles, reckons that she has made 564 miles of departure: what was her direct course, and the latitude and longitude reached? |