meridians, and number them 0 to 180, both ways, from the first meridian, for the longitudes. For a parallel through any given latitude, take the difference of the complement of the given latitude and of the colatitude of the place from the line of semitangents, and, having reduced it to the radius of the primitive, set it off at r from c towards N for latitudes greater than the latitude of the place, and from c towards s for latitudes less than the lati take the sum of the complement of the given latitude and of the colatitude of the place from the line of semitangents, and set it off at s from c upon c N produced: then the circle described upon r s* as diameter will be the parallel required. Draw these parallels for intervals of latitude equal to the angles made by two successive meridians, and number them 90 to 0 from the pole N for the north latitudes, and again increasing from 0 on the other side of the equator for the south mitive, 7 the latitude of the place, and Z the angle at which the meridian is inclined to the meridian of the place. = * Diameter of parallel colatitude of place, and d = +rtan. (c + d) where tan. (c latitudes, if the place be in north latitude-or the converse, if the place be in south latitude. The practical application of the preceding methods of projection is usually confined to the representation of an entire hemisphere, or at least of a considerable portion of a sphere; but for laying down smaller portions of the sphere the method of development may be advantageously adopted. In this method the portion of the sphere to be represented is considered as coincident with a portion of a cone, touching the sphere in a circle which is the middle parallel of latitude of the country to be represented, and this portion of the cone when developed forms a portion of a sector of a circle. B To lay down the meridians and parallels of latitude for this development. 1. Take a straight line, B C A, for the middle meridian of the intended map, and divide it into equal parts, to represent degrees and minutes of latitude according to the scale determined upon for the map. 2. From one of these divisions, A, which is conveniently situated to form the center of the map, set off from A to c the cotangent of the middle latitude, reduced to a radius equal to 57.3 of the divisions previously marked off as degrees, or to 3438 of those marked off as minutes. 3. With c as a center and radius c A, describe the arc D A E for the middle parallel of latitude, and divide it into equal parts to represent degrees and minutes of longitude, the lengths of these parts having, to the lengths. of the parts previously set off on the meridian for degrees and minutes of latitude, the ratio cosine of middle latitude : radius. 4. With c as center, describe concentric arcs, through the divisions on CE, for the parallels of latitude; and draw straight lines, radiating from c, through the divisions on D A E for the meridians. In our figure the middle latitude is 55°; A B is equal to the length of 57.3°, or the radius of the sphere; 20 410 65 Ac is equal to the cotangent of 55, or the tangent of 35 reduced to this radius; and c, consequently, is the center for describing the parallels, and the radiating point for the meridians. In drawing a map of small extent, it is usual to make all the meridians and parallels of latitude straight lines; and to make the extreme parallels, and the meridian passing through the center of the map, proportional to their real magnitude. Another and more exact method is to make the meridian passing through the center of the map, and all the parallels. of latitude, straight lines, as in the last method. Then all the degrees on each of the parallels are made proportional to their magnitude, and the lines passing through the corresponding points of division on the parallels will represent the meridians. These will be curved lines, and not straight, as in the last method. This is usually called Flamstead's Projection, as it was first used by that astronomer in constructing his "Celestial Atlas;" and it is extremely useful in geographical maps for countries lying on both sides of the equator. A considerable improvement of this method, for countries of large extent, is to represent all the parallels of latitude by concentric circles, according to the principles of the conical development; and then to lay off the degrees on each parallel, proportional to their magnitude, and draw lines through the corresponding divisions of these parallels to represent the meridians. This delineation, perhaps, will give the different parts of a map of some extent in as nearly their due proportions as the nature of the case IX X XI XII I II 梁 * That is, the degrees on each parallel must have to a degree of latitude the ratio of radius: cosine of the latitude of the parallel. lines, a b, c d, as a double meridian line, at a distance apart equal to the thickness of the intended style, or gnomon Intersect them at right angles by another line, e f, called the six o'clock line. From the scale of latitudes take the latitude of the place with the compasses, and set that extent from c to e and from a to ƒ on the six o'clock line, and then, taking the whole of six hours between the parts of the compasses from the scale, with this extent set one foot in the point e, and with the other intersect the meridian line c d at d. Do the same from f to b, and draw the right lines e d and ƒ b, which are of the same lengths as the scale of hours. Place one foot of the compasses on the beginning of the scale, and, extending the other to any hour on the scale, lay these extents off from d to e for the afternoon hours, and from b to ƒ for the forenoon. In the same manner the quarters or minutes may be laid down, if required. The edge of a ruler being now placed on the point c, draw the first five afternoon hours from that point through the marks on the line d e, and continue the lines of 4 and 5 through the center c to the other side of the dial for the like hours of the morning. Lastly, lay a ruler on the point a, and draw the last five forenoon hours through the marks on the line ƒ b, continuing the hour lines of 7 and 8 through the center a to the other side of the dial, for the evening hours, and figure the hours to the respective lines. To make the Gnomon.-From the line of chords, always placed on the same dialling scale, take the extent of 60°, and describe from the center a the arc gn. Then with the extent of the latitude of the place, suppose London, 5110, taken from the same line of chords, set one foot in n, and cross the arc with the other at g. From the center at a draw the line a g for the axis of the gnomon agi, and from g let fall the perpendicular g i upon the horizontal meri dian line an, and there will be formed a triangle a g i. A plate or triangular frame similar to this triangle, and of the thickness of the interval of the parallel lines a c and b d, being now made and set upright between them, touching at a and b, its hypothenuse or axis a g will be parallel to the axis of the earth when the dial is fixed truly, and will cast its shadow on the hour of the day. re To construct an East or West Dial.-Draw the two meridian lines as before, and intersect it at right angles by another line, upon which set off, from the meridian lines, the tangents of 15°, 30°, 45°, &c., for every 15°, duced to a radius equal to the intended height of the style. The hour lines are to be drawn through the divisions thus marked, parallel to the meridian lines, and the meridian lines themselves are six o'clock hour lines. The gnomon is a plate in the form of a parallelogram, the breadth of which forms the height of the style or gnomon, and must be equal to the radius to which the tangents have been set off on the dial plate. It is set up between the meridian lines, perpendicular to the dial plate; and the dial is set up, so that the meridian lines, and consequently the edge of the gnomon, may be parallel to the earth's axis. As the sun only shines on the dial during half the day, if the dial fronts the east, it points out the time from sunrise to noon, or, if the dial fronts the west, from noon to night GUNTER'S LINES. These lines are graduated so as to form a scale of the logarithms of numbers, sines, and tangents; to which are sometimes added, for the use of the navigator, lines of the logarithms of the sine rhumbs and tangent rhumbs. They may be constructed as follows: : 1. To construct the Line of Logarithmic Numbers marked N C |