equator, which set off on the meridian. These distances are found in a table of meridional parts to each latitude, where it is to be noted, that if the equator be graduated into minutes, the meridional parts are to be taken as in the table; but if it be divided only into degrees, the meridional parts must previously be reduced to equatorial de grees by dividing by 60. THEOREM III. The proper difference of latitude : The meridional difference of latitude on Mercator's chart, :: The departure The difference of longitude.* *DEMONSTRATION. COR.. Let there be taken an indefinitely small part of an oblique rhumb on the globe; then, radius : cosine lat. :: difference of longitude : departure. And secant, lat. : radius :: diff. long. departure." Or, departure : radius :: diff. long. : sec. lat. Now, any definite part of the rhumb may be conceived to be composed of a number of such indefinitely small parts, crossing a like number of parallels of latitude. Therefore, the sum of all the departures, or the whole departure answering to the definite part of the rhumb : the like mul, tiple of radius :: the sum of all the differences of longitude corresponding with the departures, or the whole difference of longitude made : the aggregate of the secants of all the latitudes. That is to say, the departure: difference of longitude :: the multiple of radius answering to the number of parallels of lat crossed : the aggregate of all the secants of those latitudes. Again, let the same indefinitely small part of the same oblique rhumb be taken, and, by Theorem II. secant lat. : radius : indefinitely small part of Mercator's meridian in that latitude, or the meridional difference of latitude : like part of the parallel, or proper difference of latitude, because the parallels on the chart are equal to the meridians on the globe. COR. Hence it follows, that the rhumbs are represent ed on Mercator's chart by right lines.* THEOREM Or, proper difference of latitude radius :: meridional dif. ference of latitude : secant of latitude. And if the same definite part of the rhumb be taken as before, the number of indefinitely small parts, which may be conceived to make the whole difference of latitude, will also be equal to the number of parallels crossed. Therefore, by the argument already used, the proper difference of latitude Mer. diff. lat. :: the multiple of radius answering to the number of parallels crossed : the aggregate of all the secants of those latitudes. difference of But it has been proved, that the departure: longitude :: the multiple of radius answering to the number of parallels crossed the aggregate of all the secants of those Tatitudes. Whence, proper difference of latitude :: Mer. diff. lat. departure difference of longitude. Which was to be proved. B * For, if AB repre- k sent the proper difference of latitude, and BC the departure made by running on the rhumb AC; and if Ab be the meridional difference of latitude, and be the parallel of latitude, at which a ship has arrived; in Mercator's chart, the line AC prolonged will cut the parallel in c, and be will be the difference of longitude. For the tri angles THEOREM IV. The meridional difference of latitude The difference of longitude :: Radius : The tangent of the course.* EXAMPLES. angles ABC, Abc, are similar; being both right-angled, and having a common angle at A. Whence AB, or the proper dif ference of latitude : Ab, or the mer. diff. lat. :: BC, or the departure bc, or the difference of longitude, which agrees with the theorem. And in like manner, every departure whatever, which can be made by sailing on the rhumb AC, will have the corresponding difference of longitude terminated in the same right line continued. It is, therefore, easily inferred, that lines on Mercator's chart, which cross the meridians at the same latitudes, as the rhumbs do the meridians on the globe, are right lines. That is to say, the rhumbs are represented, on Mercator's chart, by right lines. Which was to be shewn. DEMONSTRATION. For, proper difference of latitude :: meridional difference of latitude :: departure : difference of longitude. Theorem III. Or, proper difference of latitude : departure :: merid ional difference of latitude : difference of longitude. Now, proper difference of latitude : departure :: radius tangent course; according to plane sailing. Therefore, meridional difference of latitude difference of EXAMPLES. 1. It is required to find the course and distance from the land's end to the island of Bermudas. Solution by Middle Latitude Sailing. Land's end in lat. 50°06′N. and long. 6oc′ W. 81 26 Diff. 18 46 or 1126m. S. 58 48 or 3528m. W. Sum Sum 40 43 is middle lat. 49 17 comiddle lat. Construction. Make an angle CAE of as many degrees and minutes as the middle latitude, viz. 40° 43'.From the angular point A set off AE=3528m. the difference of longitule, and from its outer extremity E let fall the A perpendicular EC. AC will be the departure or meridion al distance. From C on CE, prolonged if necessary, set off CD= 1126 miles the difference of latitude, and join AD. The angle ADC, which is opposite to the departure, will be the course, and AD will be the distance required. The The triangle AEC is constructed on the principles of parallel sailing, already explained, and the triangle ADC is constructed on the principles of plane sailing; difference of latitude and departure being given, to find the rest. It is obvious, that the other method of constructing questions in parallel sailing may be applied instead of that, which is here used; but it is presumed to be unnecessary to enter into a detail on that head. Computation, To find the departure by inverting the first proportion. To find the course by the second proportion. Difference of latitude 1126 ar. co. 6'94846 Difference of long. 3528 3'54753 |