to B. Conceive the path of the ship to be divided into portions Ab, bc, cd, &c., so small that each may differ insensibly from a straight line, and draw meridians through these several divisions, as also the parallels of latitude bb', cc', dd', &c.; we shall thus have a series of triangles described on the surface of the globe, but so small that each may be considered as a plane triangle. These triangles are all similar, for the angles at b', c', d', &c., are right-angles, and the ship's path cuts all the meridians at equal angles; hence (Geom., Prop. 18, B. 4,) Ab: Ab':: bc: bc':: cd: cd', &c. therefore, (Geom., Prop. 6, B. 2,) Β ́ B ab: ab' : : ab + bc + cd + &c., : ab' + bc' + cd' + &c. But ab+bc + cd + &c., is the whole distance sailed, and Ab'+bc' + cd + &c. AB', is the difference of latitude between A and B ; consequently, if a right angled triangle a BB', similar to the small triangle abb', be constructed, that is, one in which the angle a is equal to the course, and if the hypothenuse A B represent the distance sailed, the side A B' will represent the difference of latitude. Moreover the other side BB', or that opposite to the course, will represent the sum b'b + c'c + d'd + &c. of all the minute departures which the ship makes from the successive meridians which it crosses; for as the tri angle ABB', in this last diagram, is similar to the small triangle Abb', in the former, we have Ab bb': AB: BB' but in the first figure we have to B. Ab bb': bc ec' :: cd: dd', &c. (1); .. ab: bb' :: ab+bc+cd+&c. : bb'+cc'+ dd' +&c. ... (2); consequently, since the three first terms of (1) are respectively equal to those of (2), the fourth term BB', of (1), must be equal to the fourth term, bb' + cc' + dd' + of (2), &c. This last quantity is called the departure of the ship in sailing from a It follows, therefore, that the distance sailed, the difference of latitude made, and the departure, are correctly represented by the hypothenuse and sides of a right angled plane triangle, in which the angle opposite the departure is the course, so that when any two of these four things are given, the others may be found simply by the resolution of a right angled plane triangle; as far, therefore, as these particulars are concerned, the results are the same as if the ship were sailing on a plane surface, the meridians being parallel straight lines, and the parallels of latitude cutting them at right angles; and hence that part of Navigation in which only distance sailed, departure, difference of latitude, and course are considered, is called Plane sailing. EXAMPLES. 1. A ship from latitude 47° 30' N. has sailed S. W. by S. 98 miles. What latitude is she in, and what departure has she made? Let c be the place sailed from, св the meridian, the angle c 3 points 33° 45', and CA 98 miles, the distance sailed; then CB will be the difference of latitude, and BA the departure. Hence by the formulæ for the solution of right angled triangles, As rad. 10. : Distance 98 1.991226 : Dist. 1.991226 : : cos.course 33045' 9.919846 :: sin. course 9.744739 : Diff. of lat. 81.48 1.911072 : Departure 54.45 1.735965 . Latitude left 47.0 30' N. Diff. of lat.=81.48 minutes=1 22 S. Dep.=54.45 miles W. Latitude in 46 8 N. 2. A ship sails for 24 hours on a direct course, from lat. 38° 32' N., till she arrives at lat. 36° 56' N.; the course is between the S.and E., and the rate 54 miles an hour. Required the course, distance, and departure. Lat. left 38° 32' N. 24 x 5% = 132 miles, the distance. Lat. in 36 56 N. Diff. 1 36 = 96 miles As Dist. 132 2.120574 As rad. 10. 10. 2.120574 :: Diff. lat. 96 1.982271) :: sin. course 9.836477 : cos. course 43°20'9.861697 : Dep. 90.58 1.957051 . : Rad. : Dist. . Hence the course is S. 43° 20' E., and the departure 90.58 miles E. 3. A ship sails from lat. 3° 52' S. to lat. 4° 30' N., the course being N. W. by W. 1 W.; required the distance and departure. Distance, 1065 miles; Departure, 938.9 miles W. 4. Two ports lie under the same meridian, one in latitude 52° 30' N., and the other in latitude 47° 10' N. A ship from the southernmost sails due east, at the rate of 9 miles an hour, and two days after meets a sloop which had sailed from the northernmost port; required the sloop's direct course and distance run. Course S. 53° 28' E., or S. E. E.; distance run 537.6 miles. 5. If a ship from lat. 48° 27' S. sail S. W. by W. 7 miles an hour, in what time will she arrive at the parallel of 500 S? In 23,914 hours. 6. If after a ship has sailed from lat. 40° 21' N. to lat. 460 18' N., she be found 216 miles to the eastward of the port left; required her course and distance sailed. Course N. 31° 11' E., distance 417.3 miles. Traverse Sailing. 98. When a ship, in going from one place to another, sails on different courses, it is called traverse sailing ; and the determination of the single course and distance from the one place to the other is called working or compounding the traverse. To effect this, it is obviously merely necessary to find the difference of latitude, and departure, due to each distinct course, to take the aggregate of these for the whole difference of latitude and departure, and from these to find, as in last article, the single course and distance. It is usual in thus compounding courses to form a table consisting of six columns, called a traverse table, and in the first column to register the several component courses, and against them, in the second column, the proper distances; the next two columns, marked N. and s., ar to receive the several differences of latitude, whether N. or S., due to each course, and distance, and the two remaining columns marked E. and W. are to receive, in like manner, the corresponding eastings and westings, that is, the departures. When these several particulars are all inserted, the columns are added up, and the difference of the results of the N. and S. columns will be the required difference of latitude, and the difference of the results of the E. and W. columns will be the corresponding departure. (See page 134.) The columns appropriated to the differences of latitude and departures are usually filled up from a table already computed to every quarter point of the compass, and to all distances from one mile up to 100 or 120; so that, by entering this table with any given course and distance, the proper difference of latitude and departure is found by inspection. Most books on navigation and also surveying, contain a second and more enlarged traverse table, being computed to every course from a quarter of a degree up to forty-five degrees. This latter a table we have not thought it necessary to insert in our collection, but the former we have given. (Table IV.) This table shows, by inspection, the difference of latitude and departure due to any proposed course and distance. The course is found at the top or bottom of the page, and the distance at the left or right of the half page. If the distance sailed be more than 120 miles, it will exceed the limits of the table; but the difference of latitude and departure may still be determined from it by this simple operation : divide the given distance by any number that will give a quotient not exceeding 120; enter the table with this quotient, and multiply the corresponding diff. of lat. and dep. by the assumed divisor, and there will result the diff. of lat. and dep. due to the proposed distance. Or take any numbers whose sum is equal the given distance, the sum of their differences of lat. and dep. will be the lat. and dep. of the given distance. These rules depend upon the principle that for the same course the differences of latitude and departure are proportional to the distance run; which will be evident if we recollect that dist., ; diff. of lat. and dep. form a right angled triangle, and that two right angled triangles are similar when an acute angle of one is equal to an acute angle in the other. But there is another mode of finding the direct course and distance, much practised by seamen, viz., by construction. To facilitate this construction the mariner's scale is employed, which is a two foot flat rule exhibiting several seales on each side, by help of which and a pair of compasses the usual problems in sailing may be all solved. One of these scales is a scale of chords, commonly called a scale of rhumbs, being confined to every quarter point of the compass; and another is a more enlarged scale of chords, being to every single degree. Both these scales are constructed in reference to the same common radius, so that the chords on the scale of rhumbs belong to that circle whose radius equals the chord of 60° on the scale of chords; and the method of laying down a traverse from these scales, and one of equal parts, and of thence measuring the equivalent single course, and distance |