CASE 3d.-Given both latitudes and the course, to find the distance and difference of longitude. A ship from the Lizard in N. lat. 49° 58', W. long: 5° 11' sails SWW until she come into N. lat. 43° 20', required the distance run, and the longitude come to. To secant of course, 4 points = 50° 37′ = 10.19706 CASE 4th.-Given both latitudes and distance, to find the course and difference of longitude. A ship from the Spurn in N. lat, 53° 41′, E. long. 0° 17', sails 220 miles in the NE quarter, and then finds by observation observation her latitude to be 56° 16', required the course As it is impossible on a plane surface to lay down with accuracy any considerable portion of the surface of a sphere, all maps and charts in which the meridians and parallels of latitude latitude are represented by straight lines, cutting one another at right angles and at equal intervals, must necessarily be erroneous. To remedy this inconvenience, various contrivances have been adopted, of which that possessing the greatest advantages is the method made known to the world by Gerard Mercalor, a native of the Netherlands, who in 1569, produced a chart in which the parallelism of the meridian was retained, but the parallels of latitude were placed at intervals increasing in magnitude as they receded from the equator towards the poles; thus by one error counterbalancing another, and presenting a chart which, although it contained no correct resemblance to any portion of the earth's surface, and particularly in high latitudes, yet admitted of a ship's course being laid down in a straight line on any point of the compass, whilst distances might be measured on it with the greatest accuracy. This contrivance of Mercator naturally attracted the notice of the learned; and in 1599 Edward Wright of Cambridge, published a treatise written many years before, in which he entered into the geometrical principles of the construction of such a map or chart, from which it became probable that Mercator had preeeded on a general notion of correcting one error by another, without rightly understanding the true grounds on which such corrections were to be founded. In speaking of parallel sailing, it was observed that any portion or the whole circumference of a parallel of latitude was in proportion to a corresponding portion, or the whole circumference of the equator in the proportion of the sine of the complement of the latitude of the given parallel to radius. Precisely analogous to this ratio is that of radius to the sine of the complement of the given latitude, or the secant of the latitude is to radius as the whole, or a portion of the equator to the whole, or a corresponding por tion of the given parallel. In the true Mercator's cart or map, therefore, where the meridians are all drawn parallel instead instead of converging to the poles, if we take from a scale of equal parts the sum of all the secants of the minutes, or other small equal portions contained in that latitude, and set it up. from the equator, we have the position of the parallel of the given latitude. This being done to every degree, or other convenient division comprehended within the limits of the proposed map or chart, one will be produced perfectly well adapted for performing operations in navigation; but in high latitudes the positions of places and countries will be distorted and erroneous in proportion to their increased distance from the equator. The meridians being all parallel right lines, but the parallels of latitude being placed at intervals constantly increasing in a given proportion, it is evident that the length of a degree of longitude at any given latitude will, on a Mercator's chart, bear to the length of a degree of latitude on the meridian at that parallel precisely the same proportion as they do in fact on the earth where the degrees of the meridian are (in a general sense) all equal, but where the meridians tending to meet at the poles, the degrees of longitude must continually decrease as they recede from the equator. (Vol. II. p. 19, &c.) Upon these principles tables have been formed containing the proportional magnitudes of each minute of latitude measured along a meridian from the equator to the poles : they are called tables of meridional parts, and are to be found in all complete treatises on Navigation. The meridians in the vicinity of the equator deviate so little from a parallel direction, that as far as latitude 6° the meridional parts or minutes do not sensibly differ from those on the globe,. but at that parallel the meridional parts are 361 instead of 360, the minutes in 6°; and in higher latitudes, where the degrees of longitude are much shorter than those of latitude. the meridional parts are greatly increased; thus at London in lat. 51° 31' where a degree of longitude contains 37. 34 minutes minutes of the equator, the meridional parts of a degree in minutes are 3618, instead of 3371, the minutes in 51° 31'. If as in plain sailing, a right angled triangle be constructed in which the hypothenuse represents the distance run by the ship on a given course, the perpendicular the difference of latitude, and the base the departure from the meridian of the place sailed from; if now the perpendicular be prolonged, and on it be laid off the meridional parts corresponding to the difference of latitude, and another base be drawn parallel to the departure, and meeting the distance or hypothenuse produced, this last base will represent the difference of longitude made good on the given course and distance; consequently the two triangles will be similar and their corresponding sides respectively proportional (Geom. Prop. 19 vol. i. p. 371): hence we have the means of solving the following cases in Mercator's sailing. CASE 1st.-Given the latitude and longitude sailed from, with the course and distance run by the ship, to find the place come to. If a ship from the Lizard in N. lat. 49° 58', and W. long. 5° 11' sail SW. by S. 560 miles, what is her place at the end of the course? By the rule given in plain sailing we have this proposition: As radius 10.00000 To co-sine of course, 3 points 33° 45' 560 |