described on any one of these small surfaces is to the corresponding difference of latitude as radius is to the co-sine of the course, and as the course is the same on all these surfaces, it follows that the sum of all the distances described thereon is to the sum of the corresponding differences of latitude as radius is to the co-sine of the course; that is, the whole distance sailed on the globe is to the corresponding difference of latitude as radius is to the cosine of the course. In a similar manner it appears, that the distance described on the globe is to the sum of all the corresponding departures (or meridian distances) described on these different surfaces, as radius is to the sine of the course. So that the canons for calculating the whole difference of latitude and departure from the course and distance are the same, whether the earth be esteemed as an extended plane or a spherical surface, and the same is to be observed with respect to the other cases of Plane Sailing. * We shall, therefore, in all the calculations of sailing on the spherical surface of the earth, in which the course, distance, difference of latitude and departure occur, make use of the canons already taught in Plane Sailing, and shall construct the schemes exactly in the same manner. The only additional calculation in sailing on a spherical surface consists in determining the longitude from the departure: for in sailing on a plane, the departure and longitude are the same, but in sailing on a spherical surface, the whole departure (as was observed above) is equal to the sum of all the meridian distances made in sailing over the indefinite number of small surfaces, into which we have supposed the spherical surface to be divided, and the whole difference of longitude corresponding is equal to the sum of all the differences of longitude, deduced from each of these small meridian distances by Theorem IV. of Parallel Sailing. Several methods have been proposed for abridging the calculation of the difference of longitude from the departure, the most noted of which are those known by the names of Middle Latitude Sailing and Mercator's Sailing, the latter (which will be hereafter explained) is perfectly accurate, the former is only an approximation, but it is very much used in calculating short runs and days works, but in calculating large distances across distart parallels it is liable to error. The principle on which the calculations of Middle Latitude Sailing is founded, is this:-Instead of calculating the difference of longitude corresponding to the departure made on each of the small surfaces, into which we have supposed the sphere to be divided, and adding them together, the whole departure (or sum of the meridian distances) is calculated, and the longitude deduced therefrom by the rules of Parallel Sailing, using for the latitude the arithmetical mean between the latitude sailed from and that arrived to. On this supposition, we have the two first of the following theorems for calculating the departure from the difference of longitude or the difference of longitude from the departure, which are the same as Theo. III. and IV. of Parallel Sailing, except in writing departure for distance, and middle latitude for latitude: the other theorems are easily obtained by combining the two first with the common theorems of Plane Sailing; observing that the Middle Latitude is half the sum of the two latitudes if they are of the same name, or half their difference if of contrary names. THEOREM I. As radius is to the co-sine of the middle latitude, so is the difference of longitude to the departure. THEOREM II. As the co-sine of the middle latitude is to the radius, so is the departure to the difference of longitude. Now by case I. of Plane Sailing the radius is to the sine of the course, as the distance sailed is to the departure, and, if we combine this analogy with Theorem II. we shall have, Using (in estimating the difference of longitude corresponding to each of these small meridian distances) the latitude corresponding to the middle point of the surface on which these small meridian distances are respectively made. THEOREM III. As the co-sine of the middle latitude is to the sine of the course, so is the distance sailed to the difference of longitude. By Case II. of Plane Sailing, we have this analogy; as radius is to the tangent of the course, so is the difference of latitude to the departure; by combining this with Theorem II. we have THEOREM IV. As the co-sine of the middle latitude is to the tangent of the course, so is the difference of latitude to the difference if longitude. Whence we easily deduce the following, THEOREM V. As the difference of latitude is to the difference of longitude, so is the co-sine of the middle latitude to the tangent of the course. By means of the preceding theorems we have formed the following Table, which contains all the rules necessary for solving the various cases of Middle Latitude Sailing. 3 4 and Longitudes. Both Latitudes One Latitude, Both Latitudes Diff Lat. Rad Dep. Tang. Course. Dep. Diff. Lat. Diff. Long.:: Cos. Mid. Lat. : Tang. Course. Rad. Diff. Lat. Secant Course: Distance. {Sine Course: Depart. :: Rad. Distance. Course. Dif. Lat Rad.:: Dep. : Tang Course. Diff. Lat Rad. Dist.:: Co-sine Course: Diff. Lat. Course. Dist. Rad.:: Diff Lat. : Co-sine Course. Diff. Lat. Rad.: Dep: Co-tang. Course: Diff. Lat. Hence the other latitude and middle latitude are known. Course. Dist. Rad.; Dep. : Sine Course Dist. :: Co-sine Course: Diff. Lat. Hence we obtain the other latitude and middle latitude. Diff. Loog Co-sine Mid. Lat.; Dep.:: Rad. Diff. Long. We shall now proceed to illustrate these rules, by working an example in every case. CASE I. The latitudes and longitudes of two places given, to find their bearing and distance. Required the bearing and distance between Cape Cod light-house, in the latitude of 42° 5' N. longitude 70° 4' W. and the island of St. Mary, (one of the Western Islands) in the latitude of 36° 59′ N. and longitude 25° 10′ W. Draw the east and west line DC, with the chord of 60° describe the arch S about the centre D, to cut DC in Q; upon this arch, set off, from Q to S, the middle latitude 39° 32'; through D and S draw the line DB, which make equal to the difference of longitude 2694 miles; from B let fall upon DC the perpendicular BC, which continue towards A making AC equal to the difference of latitude 306 miles ;* join AD, and it is done. For by this method of construction, on the principles before explained, A will be the situation of Cape Cod, D the situation of St. Mary; CD will be the departure, which being measured will be found to be 2078 miles; the distance will be represented by AD, which being measured will be found to be 2099 miles, and the course from Cape Cod to St. Mary, will be represented by the angle CAD=81° 37′; therefore the course will be S. 81° 37′ E. or E. † S. nearly. NOTE. The course is put S. 81° 37' E. because St. Mary being in a less northern latitude than Cape Cod, is to the southward of it; it is also to the eastward of Cape Cod, because it is in a lesser western longitude. NOTE The log. of the departure above found 3,31760 is rather less than the log. of 2078= 3.31765; but in finding the course by the depar So is co-sine mid. lat. 39° 32′ ture, I have used the quantity found at the first To tang. of course 81° 37' BY GUNTER. 2 48572 3.43040 9.88720 13.31760 2.48572 10.83188 Extend from the radius or 90° to 50° 28' the complement of the middle latitude, on the line of sines; that extent will reach from the difference of longitude 2694, to the departure 2078, on the line of numbers. 2dly. Extend from the difference of latitude 306, to the departure 2078, If the place A be to the southward of D, the line AC should be set off upon the line CB, from C towards B. on the line of numbers; that extent will reach from radius or 45°, to the course 81° 57′ on the line of tangents. ; 3dly. Extend from the course 81° 37' to the radius 90° on the line of sines that extent will reach from the departure 2078 to the distance 2099 miles on the line of numbers. BY INSPECTION. RULE. Look for the middle latitude, as if it was a course in Plane Sailing, and the difference of longitude in the distance column, opposite to which, in the column of latitude, will stand the departure; having the difference of latitude and departure, the course and distance are found (as in case VI. Plane Sailing) by seeking in Tab. II. with the difference of latitude and departure, until they are found to agree in their respective columns; opposite to them will be found the distance in its column, and the course will be found at the top of that table, if the departure be less than the difference of latitude, otherwise at the bottom. Thus with one tenth of the difference of longitude 269.4 or 269, I enter Table II. and opposite to it, in the distance column of the Tables of 39° and 40°, I find 209.1 and 206.1 in the latitude column; now the middle latitude being nearly 3930, I take the mean of these, 207.6 for the departure, which being multiplied by 10, gives the whole departure 2076. Again, I enter Table I. with one tenth of the departure 207.6, and one tenth of the difference of latitude 30,6 and find that they agree nearly to a course of 74 points, and a distance of 210, which multiplied by 10, gives the sought distance 2100 miles nearly. CASE II. Both latitudes and departure from the meridian given, to find the course, distance, and difference of longitude. A ship in the latitude of 49° 57' N. and longitude of 15° 16′ W. sails southwesterly till her departure is 789 miles, and latitude in 39° 20′ N. Required the course, distance and longitude in? Latitude left 49° 57′ N. Latitude in 39 20 N. Diff. of lat. 10 37-637 miles. Sum of lats. 79 17 Middle lat. 44 38 BY PROJECTION. Draw the meridian ACD, on which take AC B equal to the difference of latitude 637 miles; draw CB perpendicular to AC, and make it equal to the departure 789 miles; about B as a centre describe an arch ab, on which set off the middle latitude 44° 38'; through B and b draw the line BD, meeting ACD in D; join AB and it is done; for AB will be the distance sailed, which being measured will be found 1014 miles; BD will be the difference of longitude 1109 miles, and the angle CAB will represent the course from the meridian 51° 5' 2.80414 As sine course 51° 5' 9.85225 Longitude sailed from 2.89708 Diff. Long 1109 miles 10.00000 Longitude in Diff Lat 9.89101 2 89708 10.00000 3.00607 150 16' W. 18 29 W. 33 45 W. BY GUNTER. 1st. The extent from the difference of latitude 637 to the departure 789, on the line of numbers, will reach from radius, or 45°, to the course 51° 5' on the line of tangents. 2dly. The extent from 51° 5' to radius, or 90°, on the line of sines, will reach from the departure 789, to the distance 1014 on the line of numbers. Sdly. The extent from the complement of middle latitude 45° 22′, to radius, or 90°, on the line of sines, will reach from the departure 789 to the difference of longitude 1109 on the line of numbers. BY INSPECTION. RULE. With the difference of latitude and departure, find the course and distance (as in Case VI. of Plane Sailing) by seeking in Tab. II. until the dif. ference of latitude and departure are found to correspond, against which in the distance column will be the distance; and if the departure be less than the difference of latitude, the course will be found at the top of that table, otherwise at the bottom. A Then take the middle latitude as a course, and find the departure in the latitude column, the number corresponding in the distance column will be the difference of longitude. In the present example, I take one tenth of the difference of latitude 637, and the departure 789; that is 63,7 and 78,9 the nearest numbers to these are 63,6 and 78,5, standing together over 51°, against the distance 101, which being multiplied by 10 gives 1010; whence the course by inspection is S. 51° W. and the distance 1010. Then I take one tenth of the departure, 78,9 and seek it in the column of latitude of 45° (which is the nearest to the middle latitude 44° 58'), the nearest number I find is 79.2, opposite which in the distance column stands 112, which being multiplied by 10 gives 1120 for the difference of longitude; this value differs a little from that found by logarithms, owing to the miles of middle latitude neglected, for if we were also to find the difference of longitude for the middle latitude 44° and proportion for the minutes, the result would come out nearly the same as by logarithms. CASE III. One latitude, course and distance given, to find the difference of latitude and difference of longitude. A ship in the latitude of 42° 30′ N. and longitude 58° 51′ W. sails S. E. by S. 591 miles. Required the Jatitude and longitude in? BY PROJECTION. Draw the meridian ADE (as in case I. Plane Sailing) upon A as a centre describe an arch with the chord of 60°, and upon it set off, from where it cuts AD, the course S. E. by S. or 3 points, through that point of the arch, and the point A, draw the line AC, which make equal to the distance 591 miles; from C let fall upon AD the perpendicular CD; then will CD be the departure 328 miles, and AD the difference of latitude 491 miles. Hence we obtain the latitude arrived at, and the middle latitude; draw the line CE making an angle with DC of 38° 24′ the middle latitude; and the distance CE will be the difference of longitude 419 miles, hence the longitude in is obtained. |