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Jf the chords of all the arcs from 0° to 90° of the quadrant EF be transferred to another straight line ef, a scale of chords will be formed, which is frequently wanted in making dials.
Construction of a Horizontal Dial by the Scales. 30. Let CM, C'M' be the meridian, and 6 C'Cl 6 the six o'clock hour-line (fig. 12.)
From the scale of latitudes take the extent from the beginning of the scale to the division corresponding to the latitude of the place for which the dial is to be made, and set it off from C to a, and from a to a'.
From the points a, a', place lines a l, a' W, each equal to the whole length of the scale of hours, to terminate at b and b' in CM, C'M', the meridian line.
Transfer the divisions of the scale of hours to the lines a b, d' W', numbering them as in the figure.
From the points C, C' draw the lines C1, C2, C3, &c. also a il, C 10, C 9, &c. and these will be the hour-lines of the dial.
The morning hours before VI, and evening hours after VI, are found as explained in the other constructions. And the stile is to be formed in all respects as has been described.
To demonstrate the truth of this construction, let the latitude foi which the dial is made be equal to the number of degrees in the arc Ep, (fig. 11.) Then, pq being drawn perpendicular to OF, and Eq drawn meeting the circle in r, and Dr joined, it is manifest, from the construction of fig. 11 and fig. 12, that the triangle DrE (fig. 11) is in all respects equal to the triangle aCb (fig. 12), so that Dr=Ca, TE=Cb, and DE=ab; and since, in fig. 13, rad. ; sin. Jat. :: EO : Oq :: Er : ID; therefore, in fig. 14, rad. : sin. lat. :: 6C : Ca.
Let H (fig. 12) be the point in which any one of the hour-lines (for example, that for IV in the afternoon) meets ab. In the six o'clock line, place CN equal to Cb; join 6N, and through H draw KHL parallel to CN, meeting the meridian in K, and the line 6N in L; and join CL. And because Nb and ab are similarly divided at L and H, and aH and Hb, in fig. 12, are respectively equal to a IV and IV b in fig. 11; therefore, Nb in fig. 12, and ab in fig. 11, are similarly divided at L and IV. Now the triangles NCO (fig. 12) and ubb (fig. 11) are manifestly similar ; therefore it is easy to see that the angle 1CL in fig. 12, must be equal to LOIV in fig. 11 ; and hence TCL in fig. 12, must be equal to the horary angle described by the sun between noon and IV in the afternoon.
Now LK=IK :HK :: tan. LCb : tan. HCu. But IK: HK : • 6C : Ca :: rad. : sin. lat. ; therefore, rad. : sin. lat. : : tan. hor, ang. : tan. HCb. Hence it follows that the angle which the hour-lino EC, or IVC, makes with the meridian, is, of the proper magnitude : and the same may be proved in like manner of all the others.
Construction of Horizontal Dials by a Globe. 31. The construction of a horizontal dial, and indeed of any dial whatever, as will appear farther on, may be very naturally deduced from the doctrine of the sphere. For, let aPBp (fig. 13) represent the earth, which we may suppose transparent, and let its equator be divided into 24 equal parts by meridian circles a, b, c, d, e, &c. ode of which is the geographical meridian of any given place, as Edioburgh, which we may suppose at the point a. If now the hour of 12 were marked at the equator, both upon that meridian and the opposite one, and all the rest of the hours in order on the other meridians, they will be the hour circles of Edinburgh, and the sun will move from one of them to another in an hour.
Now, if the sphere had an opake axis, terminating at the points Pp, the shadow of the axis, which is in the same plane with the sun and each meridian successively, would fall upon every particular meridian, and hour, when the sun came to the opposite meridian, and would therefore show the time at Edinburgh, and all other places on the same meridian. If the sphere were now cut through the middle, by a plane ABCD, in the rational horizon of Edinburgh, one half of the axis would be above the plane, and the other half below it; and if straight lines were drawn from the centre of the plane to those points where its circumference is cut by the hour circles of the sphere, those lines would be the hour circles of an horizontal dial for Edinburgh; for the shadow of the axis would fall upon each hour-line of the dial when it fell on the like hour-circle of the sphere.
32. It appears, then, that to construct a horizontal dial by the terrestrial globe, we must place the globe in such a position, that the arc of the brazen meridian between the pole and horizon may be equal to the latitude of the place, and that any one of the meridians on the globe may coincide with the brazen meridian ; and then the arcs of the horizon between its north point and its intersections with the 24 meridians on the globe will be the measures of the angles which the hour-lines on the dial must make with the meridian line.
33. From the same principles we may derive immediately the formula already investigated. For, let PHP be any hour circle which cuts the horizon in H, then in the right-angled spherical triangle PBH, there are given PB, one of its sides adjacent to the right angle B, equal to the latitude, and the angle HPB at the pole, which is equal to the hour angle from noon, to find HB, the arc of the horizon between the meridian and hour circle, passing through the sun, which arc is the measure of the angle at the centre of the dial contained by the meridian and hour-line corresponding to that hour circle.
By the principles of spherics (see Spherical Trigonometry), in any right-angled spherical triangle, radius is to the sine of either of the sides aboat the right angle, as the tangent of the adjacent angle to the tangent of the other side about the right angle ; that is, in the pre
vent caso, as radius to the sine of PB, the latitude; wo is the tangent of HPB, the horary angle in the heavens, to the tangent of HEB, the angle made by the hour-line and the meridian at the centre of the dial.
Vertical South or North Dials. 34. These dials are described upon vertical planes, facing directly to the south and north. They are represented in fig. 14 and fig. 15.
As the planes of these dials coincide with the prime vertical, that is, the great circle of the sphere which passes through the zenith and the east and west points of the horizon, their intersections with the meridian, or the XII o'clock hour-line, will be a vertical line. The theory of these dials might be investigated exactly in the same way as that of the horizontal dial, and particular rules formed for their construction ; but this is not necessary; for the geometrical constructions which have been investigated for a horizontal dial, may be made to apply to all dials whatever, by considering, that if a horizontal dial were transferred from the place for which it was made to any other place on the earth's surface, and fixed there in a position parallel to its original position, that is, with its plane parallel to the horizon of the place for which it was made, and its acis, as before, pointing to the pole of the heavens ; then, in its new position, it will indicate the hour of the day at its original position, precisely as it did before it was removed. This proposition, although not exactly, is almost exactly true, because of the great distance of the sun from the earth in comparison to the distance of one place on the earth from another.
35. From the above principle we may infer, that any plane dial whatever, at a given place, will be a horizontal dial for some place or other of the earth ; and, therefore, to construct a dial on a given plane, we have only to find what place of the earth has its horizon parallel to that plane, and then on the given plane to construct a horizontal dial for that place, and it will show the hour of the day there This, however, may not be the hour of the day at the place where we dial is intended to show time, but then it will differ from the true hour there always by the same given quantity, namely, by the difference of the longitudes of the two places reckoned in hours and minutes of time. For example, if it should be found that a certain plane at London was parallel to the horizon of St. Petersburg ; then a hori. zontal dial constructed on the plane for the latter place, would show the hour at St. Petersburg. But as the difference of the longitude between London and St. Petersburg is about 30 degrees, corresponding to two hours in time, the dial would indicate noon when it was only ten in the morning at London ; and it would show one o'clock wher the true time at London was eleven, and so on. However, the dial would be adapted to London if we wrote the character for the hous ten on the St. Petersburg meridian line, and that for eleven on the one o'clock hour-line, and so with the other hours.
36. The zenith of any place being in a line passing through the nlane perpendicular to its horizon, it is easy to see that two places