Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/175

157 DIALLING 157 tlie meridian plane will be the xn o clock line, and the hori zontal line EM 1 will be the vi o clock line. Now, as in the pre vious problem, divide the equatorial circle into 24 equal arcs of 15 each, beginning at a, viz., ab, be, &c., each quadrant aM, MQ, &c., containing six, then through each point of division and through the axis Pp draw a plane cutting the sphere in 24 equidistant great circles. As the sun revolves round the axis the shadow of the axis will successively fall on these circles at intervals of one hour, and if these circles cross the vertical circle ZMA in the points A, B, C, &c., the shadow of the lower portion Ep of the axis will fall on the lines EA, EB, EC, &c., which will therefore be the required hour lines on the vertical dial, Ep being the style. There is no necessity for going beyond the VI o clock hour-line on each side of noon ; for, in the winter months the sun sets earlier than 6 o clock, and in the summer months it passes behind the plane of the dial before that time, and is no longer available. It remains to show how the angles AEB, AEC, &c., may be calculated. The spherical triangles pAB, pAC, &c., will give us a simple rule. These triangles are all right-angled at A, the side pA, equal to ZP, is the co-latitude of the place, that is, the differ ence between the latitude and 90 ; and the successive angles ApB, ApC, &c. are 15, 30, &c., respectively. Then tan. AB = tan. 15 sin. co-latitude; or more simply, tan. AB = tan. 15 cos. latitude, tan. AC = tan. 30 cos. latitude, &c., &c. and the arcs AB, AC so found are the measure of the angles AEB, AEO, &c., required. &quot;We shall, as examples, calculate the I o clock hour angle AEB for each of the four places we had already taken in the horizontal dial. Madras (13 4 N. lat.) Log. tan. 15 9-42805 Log. cos. 13 4 9-98861 Log. tan. 14 38 9-41666 Edinburgh (55 57 N. lat.) Log. tan. 15 9-42805 Log. cos. 55 57 9-74812 London (51 30 N. lat.) Log. tan. 15 9-42805 Log. cos. 51 30 9-79415 Log. tan. 9 28 . 9 22220 Hammerfest (73 40 X. lat.) Log. tan. 15 9 42805 Log. cos. 73 40 9-44905 Log. tan. 8 32 . 9 17617 Log. tan. 4 19 8-87710 In this case the angles diminish as the latitudes increase, the opposite result to that of the horizontal dial. Inclining, Reclining, &c., Dials. We shall not enter into the calculation of these cases. Our imaginary sphere being, as before supposed, constructed with its centre at the centre of the dial, and all the hour-circles traced upon it, the intersection of these hour-circles with the plane of the 1 EM is obviously horizontal, since M is the intersection of two great circles ZM, QM, each at right angles to the vertical plane QZP. dial will determine the hour-lines just as in the previous cases ; but the triangles will no longer be right-angled, and the simplicity of the calculation will be lost, the chances of error being greatly increased by the difficulty of drawing the dial-plane in its true position on the sphere, since that true position will have to be found from obser vations which can be only roughly performed. In all these cases, and iu cases where the dial surface is not a plane, and the hour-lines, consequently, are not straight lines, the only safe practical way is to mark rapidly on the dial a few points (one is sufficient when the dial face is plane) of the shadow at the moment when a good watch shows that the hour has arrived, and afterwards connect these points with the centre by a continuous line. Of course the style must have been, accurately fixed in its true position before we begin. Equatorial Died. The name equatorial dial is given to one whose plane is at right angles to the style, and therefore parallel to the equator. It is the simplest of all dials. A circle (fig. 5) divided into 24 equal arcs is placed at right angles to the style, and hour divisions are marked upon it. Then if care be taken that the style point accurately to the pole, and that the noon division coincide with the meridian plane, the shadow of the style will fall on the other divisions, each at its proper time. The divisions must be marked on both sides of the dial, because the sun will shine on opposite sides in the summer and in the winter months, changing at each equinox. To find the Meridian Plane. We have, so far, assumed the meridian plane to be accurately known ; we shall pro ceed to describe some of the methods by which it may be found. The mariner s compass may be employed as a first rough approximation. It is well known that the needle of the compass, when free to move horizontally, oscillates upon its pivot and settles in a direction termed the magnetic meridian. This does not coincide with the true north and south line, but the difference between them is generally known with tolerable accuracy, and is called the variation of the compass. The variation differs widely at different parts of the surface of the earth, being now about 20 W. in London, 7 W. in New York, and 17* E. in San Francisco. Nor is the variation at any place stationary, though the change is slow. We said that now the variation in London is about 20 W. ; in 1837 it was about 24 W. ; and there is even a small daily oscillation which takes place about the mean position, but too small to need notice here. With all these elements of uncertainty, it is obvious that the compass can only give a rough approximation to the position of the meridian, but it will serve to fix the style so that only a small further alteration will be necessary when a more perfect determination has been made. A very simple practical method is the following : Place a table (fig. 6), or other plane surface, in such a position that it may receive the sun s rays both in the morning and in the afternoon. Then carefully level the surface by means of a spirit-level. This must be done very accurately, and the table in that position made perfectly secure, so that there be no danger of its shifting during the day. Next, suspend a plummet SH from a point S, which must be rigidly fixed. The extremity H, where the plum met just meets the surface, should be somewhere near the