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when its plane coincides with the meridian, it may be called a meridian dial.
When the plane of the dial is perpendicular to that of the meridian, but makes an oblique angle with the horizon, it is called an inclining dial; when the dial is vertical, but inclined at an oblique angle to the meridian, it is called a declining dial.
541. A dial that is inclined at an oblique angle to each of the three planes-that of the horizon, of the meridian, and of the prime vertical-may be called an oblique dial.
Dials may be constructed by means of a terrestrial globe, by dialling scales,* by stereographic projection, and by the principles of spherical trigonometry; the method explained
here is the last method.
542. PROBLEM I.-To construct a horizontal dial.
Let SNT be the plane of the dial, extended to cut the celestial sphere, P the pole, PNS the plane of the meridian, and CPT the plane of an hour circle, and CP the direction of the stile; then PN is the latitude, and TCT' the hour line corresponding to the meridian,
which gives the hours of the same name in the forenoon and afternoon, as for instance five o'clock in the afternoon and morning; also SN is the hour line of twelve.
Let PN, the latitude of the place,
hangle TPN, the hour angle in degrees,
and t = NT, the distance in degrees of the hour line
then in the triangle PNT, having the right angle N,
sin l: tant;
Rad sin / cot h⚫ tan t, and cot h: rad
rad = 1, tan t =
sin l cot h
sin l⚫ tan h.
For any given latitude = 7, the above proportion will give the values of t, when h 15°, 30°, 45°, &c.
*See Fergusson's Lectures, and "Uranographie" by Francœur.
EXERCISE. Find the angular distances of the hour lines in succession from the hour line of noon, for the latitude of Edinburgh or 55° 57', for a horizontal dial.
Ans. For 1h P.M. or 11h A.M., t = 12° 31′.
t = 25 34.
t = 39 39.
t = 55
543. PROBLEM II.-To construct a prime vertical dial.
Let TNZ be the plane of the dial produced to cut the celestial sphere, P the pole of an opposite name from the latitude, PZN the plane of the meridian, and CPT the plane of an hour circle, and CP the direction of the stile; then PN is the co-latitude, and TCT' the hour line corresponding to that meridian, which gives the hours of the same name at T and T' in the forenoon and afternoon.
Using the same notation as in the preceding problem, excepting c for the co-latitude PN, instead of l, it appears from the triangle PNT that
R⚫sin ccot h⚫ tan t, or cot h: rad = cos l: tant;
rad 1, tant=
cos / cot h
cos ltan h.
EXERCISE. Find the angular distances of the hour lines in succession from the hour line of noon for a place in latitude 39°, for a vertical dial.
Ans. For 1h P.M. or 11h A.M., t = 11° 50′.
544. The inclination of the plane of an oblique dial to the horizon, or plane of a horizontal dial, is called its inclination; and its inclination to the prime vertical is called
its declination; and the inclination of the planes of two meridians passing through the hour line of noon and the substile, is called the declining plane's difference of longitude; also the inclination of the hour line of noon on the corresponding horizontal dial, with the intersection of the declining plane and the horizontal plane, may be called the declining plane's deviation; and the angle contained by a horizontal line through the centre of the dial perpendicular to this intersection, with the hour line of noon on the horizontal dial, may be called the direction of the inclination.
545. In a horizontal and vertical dial, the elevation of the stile is respectively equal to the latitude and co-latitude of the place; for (last two figures) CP' is the direction of the substile, and angle PCN is in the former, and = c in the latter figure.
Hence, if a horizontal dial, constructed for a given place, is carried to any other place on the same meridian, and placed in a plane parallel to the horizon of the former placethat is, parallel to its first position-it will be an inclining dial for the latter place; and its inclination at this place will be equal to the difference of the latitudes of the two places. Also, the elevation of the stile of an inclining dial at any place, is equal to the sum or difference of the latitude and inclination. Hence, also, to construct an inclining dial at a given place, find the latitude of the place to whose horizon the plane of the inclined dial is parallel, and construct it as a horizontal dial for this place, and it will be the required dial; the latitude of this latter place will be the sum or difference of the latitude of the given place and the inclination.
546. At the equator the stile and substile of a horizontal dial coincide, and all the hour lines coincide with that of noon; and therefore it would be necessary to fix the stile above the plane of the dial, and parallel to the north and south line, and then the hour lines would all be parallel to this line, and distant from it by the tangent of the corresponding hour circle's inclination to the meridian, the height of the stile being radius. Let s the height of the stile above the dial, h—the same element as in article 542.
and t perpendicular distance of the corresponding hour line from that of noon; then is ts tan h.
The stile of a prime vertical dial at the equator would evidently be perpendicular to its plane, and the hour line of noon would be a vertical line through its centre; also, all the hour lines would make an angle of 15° with each other in succession.
547. A meridian dial at any place is just a horizontal dial at the equator, at a place differing by 90° of longitude from the given place; but the north and south line on the latter would be the six o'clock line on the former; the one o'clock line on the former would be the seven o'clock line on the latter; and so on. The dial may therefore be constructed in a manner similar to the horizontal dial of the last article, and it may be observed that the construction of meridian dials is independent of the latitude.
548. If the plane of the dial has an inclination to the east or west without any declination, that is, if a horizontal dial turn about the axis of the prime vertical, then, in the triangle PNT (fig. to art. 542), the latitude PN and angle P remain the same; but angle N, instead of being = 90°, will be 90° the inclination; and NT can be calculated by article 356 or 364. Similarly, if a prime vertical dial turn round the axis of the horizon, then, in the triangle PNT, the co-latitude PN, and the hour angle P, are unaltered; but the angle N, instead of being 90°, = is = 90° the declination; and NT can be computed as above (art. 543.)
549. As in article 545, a horizontal dial was supposed to revolve over some angle about the axis of the meridian, so that the latitude for which it would be a horizontal dial was altered without any change in the longitude; so, if a horizontal dial at any place be made to revolve over some angle about its stile as an axis, the latitude of the place for which it would then be a horizontal dial would remain the same, but the longitude would be altered just to the extent of the declining plane's difference of longitude. If this difference is one hour east, then the twelve o'clock line on the dial in its original position would be the eleven o'clock line in its new position. If, however, the difference of longitude is
not an exact number of hours, the twelve o'clock line of the original dial would not be an hour line on the dial in its new position, but would represent an instant of time, differing from noon by the difference of longitude.
550. To determine the hour lines of the oblique dial explained in the preceding article, when the difference of longitude is east, the hour angle P (fig. to art. 542) would require to be increased by the difference of longitude, and the computation of the corresponding hour angle would then be performed exactly as in that article.
Thus, let SEN represent the horizon of the given place, and RN'E the new position of the plane of the dial, Z being the zenith of the given place, and Z' that of the oblique dial; P the pole; Z'PN' a meridian through Z', R then it is perpendicular to RN'E, and the inclination of the two meridians SPN, Z'PN', is the difference of longitude of the oblique dial, and also PNPN'. Let PT'T be an hour circle, then PNT corresponds to triangle PNT in the figure of article 542, in which
PN1, NPT=h, and NT= t;
PN', N'PT'h', and N'T't';
then is l'l, and h'=h+d, if d = difference of longitude. Therefore, in triangle PN'T' (as in 542),
Rad sin cot h' tan t', or cot h': rad sin 7: tan ť; from which is found, which is the distance from N' (the noon of the dial in its first position) of the hour line corresponding to the hour angle h.
551. The hour lines for any oblique dial can be found in a manner similar to that given in the preceding article. For this purpose it is merely necessary to find the latitude and longitude of the place for which the oblique dial would be horizontal, supposing the inclination and its direction to be given; which can easily be accomplished as follows:
Let SEN (preceding figure) be a horizontal dial at the given place, and RN'E any oblique dial, the other parts of the figure representing the parts explained in the preceding