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 Flamsteed'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 * 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, ef, 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 f 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 a gi, and from g let fall the perpendicular g i upon the horizontal meri dian line a n, 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. right angles by another line, upon which set off, from the meridian lines, the tangents of 15°, 30°, 45°, &c., for every 15°, reduced 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 - Having fixed upon a convenient length for the entire scale, which must be exactly equal to the length of twenty of the primary divisions of the diagonal or vernier scale, or of the beam compasses (p. 48), by which it is to be divided, bisect it, and figure it 1 at the commencement on the left hand, 1 again in the middle, and 10 at the end. The half line, then, is taken for unity, or the logarithm of 10, and, consequently, the whole line represents 2, or the logarithm of 100. The lengths corresponding to the three first figures of the logarithms of 2, 3, &c., up to 9, as found in the common table of logs., may now be taken off from the diagonal scale, or the length corresponding to four or even five figures may be estimated upon a vernier scale, or upon the beam compasses, if the scale be not less than twenty inches in length. These lengths are to be set off from the 1 at the commencement of the line for the logarithms of 2, 3, &c., to 9, and again from the 1 at the middle of the line for the logarithms of 20, 30, &c., to 90 The divisions thus formed are to be subdivided by setting off, in the same manner, the three, four, or five first figures of the logarithms of 11, 12, 13, &c., to 19; of 2·1, 22, 2-3, &c., to 2.9, and so on, each of the primary divisions being thus subdivided into ten; and these again are to be subdivided each into ten, or five, or two, as the length of the secondary divisions may admit, by setting off the logarithms of 1∙11, 1·12, 1·13, &c.; or of 1·12, 1·14, &c.; or of 1 15, 1·25, &c.; and the scale is completed. 9. To construct the Line of Logarithmic Sines marked S.The whole length of the scale is taken as the logarithm of the radius, and, since this extent upon the line of numbers represents 2, or the logarithm of 100, it follows that the lines of sines, tangents, &c., are to be scales of the logarithms of the sines, tangents, &c., to radius 100, of which the logarithm is 2. whereas the logarithmic tables of sines, tangents, &c., are set down to a radius, of which the logarithm is 10. By taking 8, then, from each of the tabulated values of the logarithmic sines, tangents, &c., we should obtain the logarithmic sines, tangents, &c., to radius 100, and the three, four, or five first figures of these reduced values are to be set off, from the left hand towards the right, by one of the scales, or by the beam compasses, as explained in the construction of the line of numbers; 1st, for every 10 degrees, then for every degree, and then for every half degree, every 10 minutes, and every 5 minutes, as far as the length of the several primary divisions will admit. The line is then numbered 1, 2, 3, &c., at every degree to 10, and afterwards 20, 30, 40, &c., at every ten degrees to 90, which stands at the extreme right, since sine 90° equals radius. The tabulated logarithmic sine of 34′ 23′′, being 8.0000669, will coincide, or nearly so, with the zero point upon our scale, and consequently angles smaller than this cannot be taken off from the sines. This remark applies equally to the line of tangents, the tabulated logarithmic tangent of 34' 23" being 8.0000886. By taking the extents backwards from right to left, and reckoning them as forward distances, the line of sines becomes a line of cosecants *, giving us, in fact, the excesses of the logarithmic cosecants above the logarithmic radius; and, by taking the complements of the required angles, the line of sines becomes a line of cosines when measured forwards from left to right, and a line of secants when measured backwards from right to left. 3. To construct the Line of Logarithmic Tangents marked T. -8 being taken from each of the tabulated values of the logarithmic tangents up to 45°, the extents corresponding to these values are to be set off upon the scale, and numbered from left to right, in a similar manner to that in which the logarithmic sines were set off and numbered upon the line of logarithmic sines. The logarithmic tangent of 45° extends to the extreme right of the scale, coinciding in extent with the sine of 90°, since tangent 45° equals radius, and the logarithmic tangents of the angles from 45° to 90'are measured backwards from the extreme right to the complement of the angle required, these extents giving us, in fact, the excesses of the logarithmic tangents sought above the logarithmic radius t * Cosecant = sin. and sec. = COS. = - log. rad. = and log. secant 2 log. rad. or, log. secant tan. of compt. log. tan. of compt.; = log. rad. - log. tan. of compt. |