SOLUTION BY THE LINE OF LONGITUDE. Opposite to 46° the latitude on the line of chords, stands 41, the number of miles in a degree of longitude in that latitude. Therefore, 60: 41 :: 502 : 348 miles, the meridional distance or departure. BY THE TRAVERSE TABLE. With the colatitude = 44° as a course, and half the difference of longitude = 251 as distance, is found 174 departure, which doubled = 348, the meridional distance or departure. MIDDLE LATITUDE SAILING. MIDDLE LATITUDE SAILING is a method of finding a ship's place on the globe, by applying the principles of parallel sailing to a course made good on an oblique rhumb. When a ship sails on an oblique rhumb, that is, on a course between the meridian and parallel, she alters at the same time both her latitude and meridional distance. But the departure, found by plane sailing, will not be her meridional distance, either at the latitude sailed from or come to. For, at the greater latitude it will be too great, because the meridians converge toward the poles; and for a contrary reason, it will be too small at the less latitude. Whence it follows, that the departure is the true meridional distance, measured on a parallel, which lies between the two extreme parallels, namely, that sailed from and that come to. In middle latitude sailing, the departure is taken for the meridional distance, measured on the paral lel of the middle latitude; and, though these be strictly equal, yet the error, which arises from this method, is of no consequence in a day's run, except in very high latitudes. The middle latitude is found by taking half the sum of the two latitudes, if of the same name; or half their dif ference, if of contrary names; and the questions are solved by the help of the following proportions. 1. Cosine middle latitude : Radius :: Departure : Difference of longitude. This proportion is the inverse of the theorem of paral lel sailing. 2. Cosine middle latitude : Tangent of course :: Difference of latitude : Difference of longitude.* For examples in middle Latitude Sailing, see the examples in Mercator's Sailing. * This proportion is deduced from the foregoing, as is here demonstrated. MERCATOR'S SAILING. MERCATOR'S SAILING is the method of finding a ship's place on the globe by Mercator's chart, or equivalent tables. In Mercator's chart the meridians are drawn parallel to each other, as in the plane chart, and therefore the meridional distace is every where the same. But, in order to compensate for this error, parallels of latitude, which are equidistant on the globe, are not so on the chart, but are more distant the higher their latitudes. Or, in other words, the degrees of the meridian are not all equal, but increase in length the more remote they are from the equator. And this is so provided, that any very minute part of the artificial meridian bears the same proportion to a like part of the parallel of its latitude, as do the like parts of both on the globe itself.* This method of constructing a chart of the world, whose meridians and parallels are right lines, is hinted at in the writings of PTOLEMY. GERARD MERCATOR first published one of these charts, in 1556, with the theory of which he did not appear to be acquainted, for the parts of his meridian were not increased in the true ratio. In the year 1599 Mr. EDWARD WRIGHT published his Correction of Errors in Navigation, in which the theory is demonstrated, and the method of computation by a table of meridional parts explained. And Dr. HALLEY, in the Philosophical Transactions, first demonstrated, that the artificial meridian line is a scale of logarithmic tangents of half the colatitudes, beginning with radius. THEOREM I. Radius : Cosine of the latitude :: Secant of the latitude : Radius.* THEOREM II. The distance of any parallel of latitude from the equator, on Mercator's chart, is as the sum of the secants of all the arcs of latitude, beginning at 0, and increasing arithmetically, by an indefinitely small common difference, till the last arc be that of the given latitude.f :: AG= secant of lat. : AE= radius. Which was to be proved. † DEMONSTRATION. Radius cosine lat. :: part of the equator like part of the parallel. COR. The nautical meridian may be divided practically, by assuming arcs of latitude, whose common difference is known and determined; and the graduations will be more accurate the smaller the common difference. Thus, : And radius cosine lat. :: part of meridian like part of parallel, because the meridian and equator on the globe are equal. Again, radius: cosine lat. :: secant lat. : radius, by Theorem I. Therefore, secant lat. radius :: part of meridian : like part of parallel. And on Mercator's chart, secant lat. radius:: part of meridian lying in that latitude, (which part must consequently be indefinitely small): like part of parallel. Whence secant lat. radius = But radius is a constant quantity, and so likewise is the part of the parallel, because all the parallels of latitude are equal on Mercator's chart. Assume, therefore, any number of arcs of latitude increasing arithmetically from 0, by an indefinitely small common difference, and call the parts of Mercator's meridian, corresponding successively with the differences of latitude, by the letters a, b, c, &c. Then, Secant lat. 1. part of Mer. a = secant lat. 2. = secant lat. 3 &c. part of Mer. ¿ part of Mer. c' because they are respectively equal to the constant quantity radius part of parallel Whence sec. lat. 1: part of Mer. a :: secant lat. 2: part of Mer. 6 secant lat. 3: part of Mer. c, &c. And secant lat. 1 : part of Mer, a :: secant lat. 1+ secant lat. 2+ secant lat. 3, &c. : parts of Mer. a+b+c, &c. |