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PROBLEM XL. To construct an horizontal dial
for any given latitude, by means of the terrestrial globe.
Elevate the pole to the latitude of the place, then bring the first meridian under the graduated edge of the strong brazen one, which will then be over the hour XII, or the equator.
As our globes have meridians drawn through every fifteen degrees of the equator, these meridians will represent the true circles of the sphere, and will intersect the horizon of the globe, in certain points on each side of the meridian. The distance of these points from the meridian must be carefully noted down upon a piece of paper, as will be seen in the example. The pupil need not, however, take out into his table the distances further than from XII to VI, which is just 90 degrees ; for the distances of XI, X, IX, VIII, VII, VI, in the forenoon, are the same from XII as, the distances of I, II, III, IV, V, 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.
No more hour-lines need be drawn than what answer to the sun's continuance above the horizon, on the longest day of the year, in the given latitude.
Example, Suppose the given place to be London, whose latitude is 51 deg. 30 min. north.
Elevate the north pole of the globe to 51.4 degrees
above the horizon; then will the axis of the globe have the same elevation above the broad paper circle, as the gnomon of the dial is to have above the plane thereof.
Turn the globe till the first meridian (which on English globes passes through London) is under the graduated side of the strong brazen meridian ; then observe and note the points where the hour circles intersect the horizon; and as on our globes the inner graduated circle, on the broad paper circle, begins from the two sixes, or east and west, we shall begin from thence, calling the hour VI OO we shall find the other hours interseeting the horizon at the following degrees;
V 18° 541 IV 36 24 III 57 II 65 41
I 78 9 which are the respective distances of the above hours from VI upon the plane of the horizon.
To transfer these, and the rest of the hours, upon an horizontal plane, draw the parallel right lines ac, and bd, plate 13, fig. 5, upon that plane, as far from each other as is equal to the intended thickness of the gnomon of the dial, and the space included between them, will be the meridian, or twelve o'clock line upon the dial; cross this meridian at right angles, by the line gh, which will be the six o'clock line ; then setting one foot of your compasses in the intersection a, describe the quadrant
with any convenient radius, or opening
of the compasses ; after this, set one foot of the compasses in the intersection b, as a centre, and with the same radius describe the quadrant fh; then divide each quadrant into 90 equal parts, or degrees, as in the figure.
Because the hour-lines are less distant from each other about noon, than in any other parts of the dial, it is best to have the centres of the quadrants at some distance from the centre of the dial-plane, in order to enlarge the hour-distances near XII; thus the centre of the plane is at A, but the centre of the quadrants is at a and b.
Lay a rule over 78°9', and the centre b, and draw there the hour-line of 1. Through b, and 65 41, gives the hour line of II. Through b, 51 57, that of III. Through the same centre, and 36 24, we obtain the hour-line of IV. And through it, and 18 54, that of V. And because the sun rises about four in the morning, continue the hour-lines of IV and V in the afternoon, through the centre b to the opposite side of the dial.
Now lay a rule successively to the centre a of the quadrant eg, and the like elevations or degrees of that quadrant 78 9, 65 41, 51 57, 36 24, 18 54, which will give the forenoon hours of XI, X, IX, VIII, and VII ; and because the sun does not set before VIII in the evening on the longest days, continue the hour-lines of VII and VIII in the afternoon, and all the hour-lines will be finished on this dial.
Lastly, through 51, degrees on either quadrant, and from its centre, draw the right line ag for the
axis of the gnomon agi, and from g let fall the per pendicular gi upon the meridian line ai, and there will be a triangle made, whose sides are ag, gi, and ia; if a plate similar to this triangle be made as thick as the distance between the lines ac and bd, and be set upright between them, touching at a and b, the line ag will, when it is truly set, be parallel to the axis of the world, and will cast a shadow on the hour of the day.
The trouble of dividing the two quadrants may be saved, by using a line of chords, which is always placed upon every scale belonging to a case of instruments.*
PROBLEM XLI. To delineate a direct south dial for
any given latitude, by the globe.
Let us suppose a south dial for the latitude of London.
Elevate the pole to the co-latitude of your place, 38° 30', and proceed in all respects as above taught for the horizontal dial, from VI in the morning to VI in the afternoon, only the hours must be reversed, as in plate 13, fig. 3; and the hypothenuse ag, fig. 4. of the gnomon agf, must make an angle with the dialplane equal to the co-latitude of the place.
As the sun can shine no longer than from VI in the morning to VI in the evening, there is no occasion for having more than twelve hours upon this dial.
* Or much more so by an appropriate set of dialling lines, placed on a 11.inch box scale, which is sold at our shop, io Holborn.
In solving this problem, we have considered our vertical south dial for the latitude of London, as an horizontal one for the complement of that latitude, or 38 deg. 30 min. all direct vertical dials may be thus reduced to horizontal ones in the same manner. The reason of this will be evident, if the globe be elevated to the latitude of London ; for, by fixing the quadrant of altitude to the zenith, and bringing it to intersect the horizon in the east point, it will point out the plane of the proposed dial.
This plane is at right angles to the meridian, and perpendicular to the horizon; and, it is clear, from the bare inspection of the globe, thus elevated, that its axis forms an angle with this plane, which is just the complement of that which it forms with the horizon, and is therefore just equal with the co-latitude of the place; and that, therefore, it is most simple to rectify the globe to that co-latitude.
The north vertical dial is the same with the south, only the style must point upwards, and that many of the hours, from its direction can be of no use.
PROBLEM XLII. To make an erect 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
dial declines' towards the east, (which we shall suppose in the present instance) count in the horizon the degrees of declination from the east point towards the north, and bring the lower