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equal to the latitude of the place for which the dial is made.

26. If the pole of the globe be elevated to the co-latitude of the given place, and any meridian be brought to the north point of the horizon, the rest of the meridians will cut the horizon in the respective distances of all the hours from XII, for a direct south dial, the gnomon of which must form an angle with the plane of the dial equal to the co-latitude of the place; and the hours on this dial must be placed in a direction contrary to that in which they stand on the horizontal dial.

27. But if the globe have more than twentyfour meridian semicircles upon it, we must take the following method for making horizontal and south dials:-Elevate the pole to the latitude of the place, and turn the globe until any particular meridian (suppose the first) comes to the north point of the horizon, and the opposite meridian will cut the horizon in the south. Then set the hour index to the uppermost XII on its circle, and turn the globe westward until 15° of the equator pass under the brazen meridian, and the hour index will be at I, for the sun moves 15° every hour), and the first meridian will cut the horizon in the number of degrees from the north point that I is distant from XII. Turn on until other 15° of the equator pass under the brazen meridian, and the hour index will then be at II, and the first meridian will cut the horizon in the number of degrees that II is distant from XII: and so, by making 15° of the equator pass under the brazen meridian for every hour, the first meridian of the globe will cut the horizon in the distances of all the hours from XII to VI, which is just 90°; and then the distances of X1, X, IX, VIII, VII, and VI, in the forenoon will be the same from XII, as the distance of I, II, III, IV, V, and VI, in the afternoon: and these hour lines continued through the centre, will give the opposite hour lines on the other half of the dial.

28. To make a horizontal dial for the latitude of London, which is 51° 30′ north, elevate the north pole of the globe 51° 30′ above the north point of the horizon; and then turn the globe, until the first meridian (which, on the British terrestrial globe, is that of London), cuts the north point of the horizon, and set the hour index to XII at noon. Then turning the globe westward until the index points successively to I, II, III, IV, V, and VI, in the afternoon, or until 15°, 30°, 45°, 60°, 75°, and 90° of the equator pass under the brazen meridian, the first meridian of the globe will cut the horizon in the following numbers of degrees from the north towards the east, viz. 113, 241, 38, 53, 71, and 90; which are the respective distances of the above hours from XII upon the plane of the horizon. To transfer these, and the rest of the hours, to a horizontal plane, draw the parallel right lines a c, and db, fig. 3, upon that plane, as far from each other as is equal to the intended thickness of the gnomon or stile of the dial, and the space included between them will be the meridian or twelve o'clock line on the dial. Cross this meridian at right angles with the six o'clock line, kh, and setting one foot of the compasses in

the intersection a, as a centre, describe the quadrant ke with any convenient radius or opening of the compasses; then setting one foot in the intersection b, as a centre, with the same radius describe the quadrant fh, and divide each quadrant into ninety equal parts or degrees, as in the figure.

29. As the hour lines are less distant from each other about noon than in any other part of the dial, it is best to have the centres of these quadrants at a little distance from the centre of the dial plane, on the side opposite to XII, in order to enlarge the hour distances thereabouts, under the same angles on the plane. Thus the centre of the plane is at C, but the centres of the quadrants are at a and b. Lay a ruler over the point b, and (keeping it there for the centre of all the afternoon hours in the quadrant fh), draw the hour line of I through 11° 30' in the quadrant; the hour line of II through 24° 30′; of III through 38° 5′; IV through 53° 30′; and V through 71° 4: and, because the sun rises about four in the morning on the longest days, at London, continue the hour lines of IV and V in the afternoon through the centre b to the opposite side of the dial.

30. the other quadrant is now to be divided, but it is very obvious that the same minute process need not be gone through in doing so, as the divisions already laid down may be readily transferred to the quadrant e k; as the labor of dividing both may be much shortened by working from a scale, having a line of chords upon it, as will be shown presently.

31. If a plate similar to this triangle be made as thick as the distance between the lines ac and bd, and set upright between them, touching at a and b, its hypothenuse a g will be parallel to the axis of the world, when the dial is truly set; and will cast a shadow on the hour of the day.

32. To make an erect direct south dial, fig. 4, elevate the pole to the co-latitude of the place, and proceed in all respects as above for the horizontal dial, and from VI in the morning to VI in the afternoon; only the hours must be reversed, as in the figure; and the hypothenuse a g, of the gnomon a gh, must make an angle with the dial-plane equal to the co-latitude of the place. As the sun can shine no longer on this dial than from six in the morning until six in the evening, there is no occasion for having any more than twelve hours upon it.

33. To make a direct dial, declining from the south towards the east or west, elevate the pole to the latitude of the place, and screw the quadrant of altitude to the zenith. Then, if the dial decline towards the E. (which we shall suppose it does), count in the horizon the degrees of declination, from the E. point towards the N. and bring the lower end of the quadrant to that degree of declination at which the reckoning ends. Then bring any particular meridian of the globe (suppose the first) directly under the graduated edge of the upper part of the brazen meridian, and set the hour to XII at noon. Then, keeping the quadrant of altitude at the degree of declination in the horizon, turn the globe eastward on its axis, and observe the degrees cut by the

first meridian in the quadrant of altitude (counted from the zenith), as the hour circle comes to XI, X, IX, &c., in the forenoon, or as 15, 30, 45, &c. degrees of the equator pass under the brazen meridian at these hours respectively; and the degrees then cut in the quadrant by the first meridian, are the respective distances of the forenoon hours from XII on the plane of the dial.

34. Then, for the afternoon hours, turn the quadrant of altitude round the zenith until it comes to the degree in the horizon opposite to that where it was placed before; namely, as far from the W. point of the horizon towards the S. as it was set at first from the E. point towards the N.; and turn the globe westward on its axis, until the first meridian comes to the brazen meridian again, and the hour index to XII; then, continue to turn the globe westward, and as the index points to the afternoon hours, I, II, III, &c., or as 15°, 30°, 45°, &c., of the equator pass under the brazen meridian, the first meridian will cut the quadrant of altitude in the respective number of degrees from the zenith that each of these hours is from XII on the dial. And when the first meridian goes off the quadrant at the horizon in the forenoon, the hour index shows the time when the sun will come upon this dia., and when it goes off the quadrant in the afternoon, the index will point to the time when the sun goes off the dial. Having thus found all the hour distances from XII, lay them down upon the dial plane, either by dividing a semicircle into two quadrants of 90° each (beginning at the hour line of XII), or by the line of chords, as above directed.

35. In all declining dials, the line on which the stile or gnomon stands (commonly called the substile line) makes an angle with the twelve o'clock line, and falls among the forenoon hour lines, if the dial declines towards the E; and among the afternoon hour lines, when the dial declines towards the W. that is, to the left hand from the twelve o'clock line in the former case, and to the right hand from it in the latter.

drant to the opposite degree in the horizon, namely, as far from the W. towards the N. and then proceed in all respects as above.

38. Thus when our declining dial is finished, we have four dials, viz. 1. A north dial declining eastward by the same number of degrees; 2. A north dial declining the same number west; 3. A south dial, declining east; and, 4. A south dial declining west; only placing the proper number of hours, and the stile or gnomon respectively, upon each plane. For, in the S. W. plane, the substilar line falls among the afternoon hours; and in the S. E. of the same declination, among the forenoon hours, at equal distances from XII. And so all the morning hours on the W. decliner, will be like the afternoon hours on the E. decliner; and the S. W. decliner, the N. E. decliner, by only extending the hour lines, stile and substile, quite through the centre: the axis of the stile (or edge that casts the shadow on the hour of the day), being in all dials whatever, parallel to the axis of the world, and consequently pointing towards the north pole of the heaven in north latitudes, and towards the south pole, in south latitudes.

METHOD OF CONSTRUCTING DIALLING LINES.

39. Describe, with any opening of the compasses, as EA, fig. 5, according to the intended length of the scale, the circle A D C B, and cross it at right angles by the diameters C E A and D E B; divide the quadrant A B first into 9 equal parts, and then each part into 10; so shall the quadrant be divided into 90 equal parts or degrees. Draw the right line AFB for the chord of this quadrant; and, setting one foot of the compasses in the point A, extend the other to the several divisions of the quadrant, and transfer these divisions to the line AFB by the arcs 10, 20, 30, &c., and this will be a line of chords, divided into 90 unequal parts.

40. Divide the quadrant CD into 90 equal parts, and from each point of division draw right lines, as i, k, l, &c., to the line C E; all perpendicular to that line, and parallel to DE, which will divide E C into a line of sines; and although these are seldom put among the dialling lines on a scale, yet they assist in drawing the line of latitudes. For if a ruler be laid upon the point D, and over each division in the line of sines, it will divide the quadrant CB into 90 unequal parts, as Ba Bb, &c., shown by the right lines 12 a, 20 b, 30 c, &c., drawn along the edge of the ruler. If the right line B C be drawn, subtending this quadrant and the nearest distances, Ba, Bb, Bc, &c., be taken in the compasses from B, and set upon this line in the same manner as directed for the line of chords, it will make a line of latitudes BC, equal in length to the line of chords A B, and an equal number of divisions, but very unequal as to their lengths.

36. To find the distance of the substile from the twelve o'clock line, if the dial declines from the S. towards the E. count the degrees of the declination in the horizon from the E. point toward the N. and bring the lower end of the quadrant of altitude to that degree of declination where the reckoning ends; then, turn the globe until the first meridian cuts the horizon in the like number of degrees, counted from the S. point toward the E. and the quadrant and the first meridian will then cross one another at right angles; and the number of degrees of the quadrant, which are intercepted between the meridian and the zenith, is equal to the distance of the substile line from the twelve o'clock line; and the number of degrees of the first meridian, which are intercepted between the quadrant and the N. pole, is equal to the elevation of the stile above the plane of the dial. 41. Draw the right line D G A, subtending the 37. If the dial declines westward from the S., quadrant DA; and parallel to it, draw the right count that declination from the E. point of the line rs, touching the quadrant D B at the nume horizon towards the S. and bring the quadrant of ral figure 3. Divide this quadrant into six altitude to the degree in the horizon at which the equal parts, as 1, 2, 3, &c., and through these reckoning ends; both for finding the forenoon points of division draw right lines from the centre hours, and distance of the substile from the meri-E to the liners, which will divide it at the points dian: and for the afternoon hours, bring the qua- where the six hours are to be placed, as in the

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