In practice, both the construction and calculation above indicated are superseded by the use of the table of difference of latitude and departure, which is given in treatises on navigation, and is called a traverse table. The numbers in the table are nothing more than the computed values of the sides of right-angled triangles; the hypothenuse, or the distance, and the adjacent angle, or the course, being given. Thus, by referring to such a table, the courses and distances being used as arguments, the numbers in the columns N. S. E. NV. above, might have been found sufficiently near the truth. And, conversely, seeking In the table the difference of latitude ( =76) and the departure (= 96), the corresponding distance ( =122) would be seen in its proper column, and the angle or course at the bottom of the page.
The logarithmic or Gunter's scale [SCALE] W58 formerly, for the sake of expedition, much used in the resolution both of plain and 'spherical triangles for the purposes of navigation. If, it were re quired by that instrument to find the values of cg and qd in the triangle cqd, the following proportions might be worked by taking in the compasses the distance from 90° to 60' 34' on the line of sines, and applying that distance on the line of numbers from towards zero ; the other foot of the compasses would fall on 44, which is the value of gd; again by taking the dis tance from to 26' on the line of sines, and applying it on the line of numbers, from 50.5, as before, the other foot of the compasses would fall on 25, which is the value of cq. But it is evident that when the angle is small, or nearly a right angle, the instrument must be very inaccurate.
Should a ship, on any part of the earth's surface, sail for a short time in a direction either due east or due west, so that during that time it might be considered, without sensible error, as sailing on the circum ference of a parallel of latitude, the determination of its place is obtained by a different process. Thus, the earth being supposed to be a sphere, the length, in miles, of any arc of the equator between the meridian circles passing through its extremities is to the length, in miles, of the arc between the same meridians on any given parallel of latitude as radius is to the cosine of the latitude of the parallel.
Therefore, when the number of geographical miles passed over on any parallel of latitude is known by the log (all due corrections being supposed to be made), the difference of longitude corresponding to that distauce may be found at once by the above proportion. Evidently also, if any three whatever of the terms are given, the fourth can be found ; and thus every variation of the case may be resolved. This is called parallel sailing.
But the tables of difference of latitude and departure may be ren dered available for finding the required term if we consider the latitude of the parallel on which the ship is sailing to represent what is called the course in those tables ; the distance in miles on the parallel as the difference of latitude, and the difference of longitude in geo graphical miles as the distance in the tables; and then, by inspection as before, the required term may be found.
The third method of operating, which is called middle latitude sailing, has been defined under LONGITUDE AND LATITUDE,METIIODS Or FINDING, and we have here only to point out its application. Let A E be a portion of the rhumb-line which a ship describes while her motion continues to coincide with the directiou of one point of the compass, that is to say, while it makes a constant angle with the meridians of her successive places. Let this curve be divided into any parts, A B, B C, &e., of small extent, so that each part may, without sensible error, be considered ae a straight line: and imagine both meridians and parallels of latitude to be drawn through A, B, C, fie. Then the several triangles B e b, c B c, &c., being considered as plane triangles, if the con stant angle n A b, c B c, &c., be represented by A, we shall have—