at the perpendicular. If then the hypothenuse of any rightangled triangle whatever be found in the column of distances, in the traverse table; the perpendicular will be opposite in the latitude column, and the base in the departure column; the angle at the perpendicular being at the top or bottom of the page. Ex. 1. Given the hypothenuse 24, and the angle at the perpendicular 54°; to find the base and perpendicular by inspection. Opposite 24 in the distance column, and over 54° will be found the base 19.54 in the departure column, and the perpendicular 13.94 in the latitude column. 2. Given the angle at the perpendicular 3704, and the base 46; to find the hypothenuse and perpendicular. Under 37°4, look for 46 in the departure column; and opposite this will be found the perpendicular 60.5 in the latitude column, and the hypothenuse 76 in the distance column. 3. Given the perpendicular 36, and the base 30.21; to find the hypothenuse and angles. Look in the columns of latitude and departure, till the numbers 36 and 30.21 are found opposite each other; these will give the hypothenuse 47, and the angle at the perpendicular 40o. SECTION II. PARALLEL AND MIDDLE LATITUDE SAILING. 52. BY the methods of calculation in plane sailing, a ship's course, distance, departure, and difference of latitude are found. There is one other particular which it is very important to determine, the difference of longitude. The departure gives the distance between two meridians in miles. But the situations of places on the earth, are known from their latitudes and longitudes; and these are measured in degrees. The lines of longitude, as they are drawn on the E globe, are farthest from each other at the equator, and gradually converge towards the poles. A ship, in making a hundred miles of departure, may change her latitude in one case 2 degrees, in another 10, and in another 20. It is important, then, to be able to convert departure into difference of longitude; that is, to determine how many degrees of longitude answer to any given number of miles, on any parallel of latitude. This is easily done by the following THEOREM. 53. As the cosine of latitude, So is the departure, By this is to be understood, that the cosine of the latitude is to radius; as the distance between two meridians measured on the given parallel, to the distance between the same meridians measured on the equator. Let P (Fig. 21.) be the pole of the earth, A a point at the equator, La place whose latitude is given, and LO a line perpendicular to PC. Then CL or CA is a semi-diameter of the earth, which may be assumed as the radius of the tables; PL is the complement of the latitude, and OL the sine of PL, that is, the cosine of the latitude. If the whole be now supposed to revolve about PC as an axis, the radius CA will describe the equator, and OL the given parallel of latitude. The circumferences of these circles are as their semi-diameters OL and CA, (Sup. Euc. 8. 1.) And this is the ratio which any portion of one circumference has to a like portion of the other. Therefore OL is to CA, that is, the cosine of latitude is to radius, as the distance between two meridians measured on the given parallel, to the distance between the same meridians measured on the equator. Cor. 1. Like portions of different parallels of latitude are to each other, as the cosines of the latitudes. Cor. 2. A degree of longitude is commonly measured on the equator. But if it be considered as measured on a parallel of latitude, the length of the degree will be as the cosine of the latitude. The following table contains the length of a degree of longi- D.L.) Miles. D.L. Miles. D.L., Miles. D.L. Miles. D.L. Miles. (D.L. Miles. 1557.95 3051.96|| 45|42.43 60 30.00 75 15.53 900.00 The length of a degree of longitude in different parallels 54. The sailing of a ship on a parallel of latitude* is call- 55. The Geometrical Construction is very simple. Make For Cos C: BC :: R: CD (Trig. 121.) * See Note C. 56. The parts of the triangle may be found hy inspection in the traverse table. (Art. 51.) The angle opposite the departure is D the complement of the latitude, and the difference of longitude is the hypothenuse CD. If then the departure be found in the departure column under or over the given number of degrees in the co-latitude, the difference of longitude will be opposite in the distance column. Example I. A ship leaving a port in Lat. 38° N. Lon. 16° E. sails west on a parallel of latitude 117 miles in 24 hours. What is her longitude at the end of this time? Cos 35°: Rad :: 117: 148=2° 28' the difference of longitude. This subtracted from 16o leaves 13° 31' the longitude required. Example II. What is the distance of two places in Lat. 46° N. if the longitude of the one is 2° 13' W. and that of the other 1 17' Ε.? As the two places are on opposite sides of the first meridian, the difference of longitude is 2° 13'+1° 17′ = 3° 30', or 210 minutes. Then Rad: Cos 46°:: 210: 145.88 miles, the departure, or the distance between the two places. Example III. A ship having sailed on a parallel of latitude 138 miles, finds her difference of longitude 4° 3' or 243 minutes. What is her latitude ? Diff. Lon. 243: Dep. 138 :: Rad: Cos Lat. 55° 23'. Example IV. On what part of the earth are the degrees of longitude half as long as at the equator? Ans. In latitude 60. MIDDLE LATITUDE SAILING. 57. BY the method just explained, is calculated the difference of longitude of a ship sailing on a parallel of latitude. But instances of this mode of sailing are comparatively few. It is necessary then to be able to calculate the longitude when the course is oblique. If a ship sail from A to C, (Fig. 18.) the departure is equal to om+sn+tC. But the sum of these small lines is less than BC, and greater than AD. (Art. 40.) The departure, then, is the meridian distance measured not on the parallel from which the ship sailed, nor on that upon which she has arrived, but upon one which is between the two. If the exact situation of this intermediate parallel could be determined, by a process sufficiently simple for common practice, the difference of longitude would be easily obtained. The parallel usually taken for this purpose, is an arithmetical mean between the two extreme latitudes. This is called the Middle Latitude. The meridian distance on this parallel is not exactly equal to the departure. But for small distances, the errour is not material, except in high latitudes. The middle latitude is equal to half the sum of the two extreme latitudes, if they are both north or both south; but to half their difference, if one is north and the other south. 58. In middle latitude sailing, all the calculations are made in the same way as in plane sailing, excepting the proportions in which the difference of longitude is one of the terms. The departure is derived from the difference of longitude, and the difference of longitude from the departure, in the same manner as in parallel sailing, (Arts. 53, 54.) only substituting in the theorem the term middle latitude for latitude. THEOREM I. As the cosine of middle latitude, So is the departure, To the difference of longitude. 59. The learner will be very much assisted in stating the proportions, by keeping the geometrical construction steadily in his mind. In Fig. 20 we have the lines and angles in plane sailing, and in Fig. 22, those in parallel sailing. By |