7. Description of the Sinical Quadrant.
This instrument, Fig. 13, which is chiefly of use in navigation, consists of several concentric quadrantal arches, divided into eight equal parts by radii, with straight lines crossing one another at right angles, and parallel to the rectangular radii.
Any one of the arches, as BC, is used to represent the horizon, or meridian, though it may represent a quadrant of any great circle of a sphere. If BC is taken as a quadrant of the horizon, either of the sides as AB, may represent the meridian ; and the other sides as AC will represent a parallel, or a line of east and west ; while all the other lines parallel to AB will be also meridians, and all those parallel to AC east and west lines, or pa rallels.
The eight equal parts into which the limb is divided by the radii contain 11° 15', and represent the eight parts of the compass on the quarter of the horizon. The arch BC is divided into 90'; and by means of dia gonal lines each degree is subdivided into twelve parts, or five minutes each. A thread is fixed to the centre, and divides the horizon, by being laid over any degree of the quadrant.
If this quadrant is taken to represent a quarter of the meridian, one of its sides AB may be taken for the common radius of the meridian and the equator, and then the other AC will be half the axis of the world. The degrees of the circumference BC will represent degrees of latitude, and the lines parallel to the side AB assumed from every point of latitude to the axis AC, will be the radii of the parallels of latitude, and likewise the cosines of these latitudes.
If it is now required to find the degrees of longitude contained in 83° of the lesser leagues in the parallel of 48°. Lay the thread over 48° of latitude, on the cir cumference, and count the 83 leagues from A upon AB. These will terminate at II, allowing fur every small in terval four leagues, and the interval between the broad lines twenty leagues. By then tracing out the parallel JIG from the part H to the thread, the part AG of the thread will show, that 125 greater, or equinoctial leagues, make 6° 15'; allowing 20 leagues to a degree, and 3' for one league; and, con•equently, that 83 lesser leagues Ali, which make the difference of longitude of the course, and are equal to the radius of the parallel GI, make 6° 15' of the above-mentioned parallel.
When the ship sails on an oblique course, this course, besides the north and south greater leagues, gives lesser leagues easterly and westerly, to be converted into de grees of longitude of the equator. But as these leagues are made neither on the parallel of departure, nor on that of arrit al, but in all the intermediate ones, a mean proportional parallel between them must be found. For this purpose, the quadrant has a scale of cross la titudes, so that we have only to take with the compasses the middle point between the parallels of which we want the mean, and the middle point will be the mean paral lel required.
The chief use of the sinical quadrant, is to form tri angles upon it similar to those made by a ship's course with tne meridians and parallels; the sides of these tri angles being measured by the equal intervals between the concentric quadrants, and the lines of N. and S. E. and W. Every fifth of the lines and arcs is distinguish ed by a broader line, so that if each interval is made to stand for a league, there will he five between two adja cent broad lines ; or if each interval represent four leagues, there will be twenty leagues, or a sea degree, between two adjacent broad lines.
If we suppose a ship to have sailed 150 leagues north east, one-fourth north, which is the third point, and makes an angle of 38° 45' with the north part of the me ridian, then making A the place of departure, reckon by means of the concentric arch along the point the ship sailed on, as AD, 150 leagues from A to D, then is the point D the point of the plane to which the ship has ar rived. Let DE be the parallel to the side AC, and we shall then have a right angled triangle A ED, similar to that of the ship's course, difference of longitude and la titude. The side AE gives 125 leagues fur the difference of latitude northwards, which makes 6° 15', reckoning twenty leagues to a degree; and the side DE gives eighty-three lesser leagues, answering to the parallels which, when reduced as above shown, will give the dif ference of longitude.