shadow projected on it will not be uniform, as it is on the plane of the equinoctial. 1. 9 The rod EF, which projects the shodow, is called the Stile j also sometimes the Axis of the dial. The lines E 12, E 1, &c. which indicate the position of the shadow at the different hours, are called Hour Lines. The hour-lines are evidently the common section of the plane of the dial, and a plane passing through its axis and the sun. The point in which the axis of a dial meets its plane, which is also the common concourse of the hour-lines, is called its Centre. There are other technical terms belonging to this subject, but these we shall explain as we proceed. 10. The latitude of the place for which a dial is to be made, is an important element in their construction. This may be known by good maps, or it may be determined by astronomical observation.s. How to trace a Meridian Line on any Plane. 11. In constructing a dial, it is always necessary to determine the line in which the plane of the meridian meets the plane of the dial. If the plane of the dial is not horizontal, it will be convenient, in the first place, to trace a meridian line on a horizontal plane near it. A meridian may also be found by any three shadows of an upright pin or stile. Let OV (fig. 2) be the stile which stands at right angles to the plane in O, and ŎA, OA', O'A", its shadows at three different times of the day. Then, if AV, A'V, A′′V be joined, the angles AVO, A'VO, A'VO, are the sun's distances from the zenith at the times of noting the positions of the shadows; and these are known, because in the right-angled triangles AOV, A'OV, A′′OV, the sides about the right angles at O are known, from which the angles at V may be found. Let us now suppose that the sphere is projected stereographically on the horizontal plane AA'A", so that O is the centre of the primitive, the eye being in the nadir, then the lines AO, A′O, A"O, produced, will be the projections of azimuth circles; if the projections of the sun's places, in these circles, at the times of observation, be now found, a circle traced through them will evidently be the projection of the circle of declination, which the sun describes in the heavens that day; and the position of the meridian may now be found, because it will pass through the centre of that circle, and O, the centre of the horizon. Hence we derive the following construction. Make three right-angled triangles AOV, A'OV, A"′′OV (fig. 2), which have each VO VO, in fig. 3, the height of the stile; and bisect the angles at V, by the lines Va, Va', Va". Produce the shadows AO, AO, AO", so that Oa, Oa', Oa", of fig. 5, may be respectively equal to Oa, Oa, Oa", of fig. 4. Describe a circle through the points a, a, a", and from X its centre, draw a line through C; this will be in the direction of the meridian. For, by the principle of the stereographic projection of the sphere, if we take the horizonta' Q q plane A A' A", for the plane of projection; the lines Oa, Oa', Oa", will be the projections of circles passing through the zenith and the sun, at the times when the shadows have the positions OA, OA', OA"; and as by construction, Oa, Oa', Oa", are the tangents of half the zenith distances AVO, A'VO, A"VO, the points a, a, a", are the projected places of the sun; and the circle a, a', a", is the projection of the parallel it describes in the heavens on the day of observation, and OX, which passes through its centre, is the projection of the meridian. See Projection of the Sphere. 12. We may even find the latitude of the place of observation: for if P, the projection of the pole of the circle, be found, then OP will be the tangent of half the distance of the pole from the zenith (OV being taken as radius), that is, the tangent of half the complement of the latitude. 13. In this construction, no allowance is made for refraction, or change of declination. The zenith distances may, however, be corrected for refraction by the proper tables. And if the observation be made on the solstitial days, the error from change of declination will hardly be any thing. This method of tracing a meridian line was proposed by a very old author on dialling, named Mutio Oddi da Urbino, in a work called Gli Horologi Solari Nelle Superficie piane. 14. Another method of tracing a meridian line is, by observing when two stars which have the same right ascension, or whose right ascensions differ by 180°, come into the same vertical plane; for then they are both on the meridian. The observation may be made by means of a plane surface, kept in a vertical position by its own weight, or by any other suitable contrivance, and which is moveable about a vertical line. The pole star and the first of the tail of the Great Bear, are applicable to this purpose. In the beginning of 1911, their mean right ascensions were, Star, 191o 25' 3" 13 41 41 177 43 22 This difference, although not exactly 180 degrees, is yet sufficiently near; because when is on the meridian, the arc of 2° 16′ 38", by which the pole star has advanced in the small circle it describes, subtends an angle of about 4' only. The stars of Ophiuchus, and ẞ of the Dragon, are well adapted to the same purpose, the right ascensions and declinations are. of Ophiuchus, B of Dragon, 261° 32′ 26′′ 12° 42′ 29N". ... 261 32 33 52 26 47N. As these have almost the same right ascension, and differ 40° in their declination, they are very proper for determining the position of the meridian. 15. In whatever way a meridian is traced on the horizontal plane, it should be quite adjoining to the plane on which the dial is to he delineated; and it ought to be so placed, that a vertical plane passing along the meridian line, may cut the dial at the point where the axis Is to be fixed. To take a familiar example, we shall suppose that the dial plane is a vertical wall carefully smoothed and verified with a rule and plumb-line; and this being understood, it will be easy to suit the operation to any other plane. Let BC (fig. 4) be the meridian line on a horizontal table, and in. sq, two plumb-lines, which descend on the meridian line from a horizontal rod that has one end fixed in the wall, and the other supported on a stand. If the table admits of being pierced with two holes, the plumb-lines may with advantage pass through them, and the plummets hang suspended in vessels filled with water. They will thus be more steady, and more easily adjusted. The eye is now to be directed towards the wall, so that the visual ray may be in the plane of the plumb-lines; and then the line AK upon the wall, which they both appear to cover at once, will manifestly be the intersection of the plane of the meridian, and the plane of the dial; and, consequently, will be the twelve o'clock hour-line. A point A is now to be assumed, as the centre of the dial; and the axis AC must be fixed in the wall in such a position, that it may lie in the plane of the threads tn, sq, and make with a horizontal line AD, an angle equal to the latitude of the place, or with CR, a vertical line, an angle equal to the co-latitude; and then it will manifestly be parallel to the axis of the world. 16. The stile may have any shape that admits of its being firmly fastened to the dial; and, before it is fixed, it may be convenient to fasten a piece of wood to the wall, so that it may have a plane surface exactly in the plane of the meridian, as indicated by the plumb-lines, and a line traced on its surface in the position of the axis or edge of the stile; this board will serve to support the stile in its position, until it be fastened either with its plane in the plane of the meridian, or perpendicular to the plane of the dial; but it will look most symmetrical in this last position. In whatever way it is fixed, the edge which projects the shadow must be in the plane of the meridian, and parallel to the earth's axis. 17. When the position of the plane of a dial in respect of the earth's axis is known, the determination of the hour-lines is a geometrica! problem by no means difficult. As at every hour the sun is in one or other of twelve great circles of the sphere, which intersect at the poles of the heavens, and which make equal angles with one another, the general problem to be resolved is evidently this: Let there be twelve planes, which intersect in a straight line, and make equal angles with one another; and let these planes, indefinitely produced, meet another plane in any position whatever, to determine the lines in which they cut that plane. In resolving this problem, it will be convenient to begin with the more simple cases, and to reduce the others as much as possible to them. Equinoctial Dial. 18. This dial, seen obliquely in its proper position, is represented by the upper part of fig. 1. Its plane is parallel to the equator, and is the same as the plane of the equinoctial circle in the heavens; E is its centre, and EF its axis. As the hour circles in the heavens are perpendicular to the equinoctial circle, and divide it into twenty-four equal parts, the lines in which the plane of the dial cuts their planes, that is, the hour lines, will make twenty-four equal angles round the centre of the dial. It appears, then, that to delineate a dial of this kind, nothing more is necessary than to describe a circle on its plane, and to divide its circumference into twenty-four equal parts; and having drawn lines from the centre to the points of division, these will be the hour-lines against which the characters denoting the hours are to be written; if the axis be now fixed perpendicular to the plane, the dial will be constructed. In fixing this dial, the axis EF must be in the plane of the meridian, and must make with the horizontal meridian line, an angle equal to the latitude of the place, and then it will point to the pole of the heavens as it ought. 19. As the sun is one half of the year on the north side of the equinoctial, and the other half on the south side, it will be proper to trace hour-lines on both faces of the dial; and in north latitudes the hours will be shown on the upper face of the dial in summer, and on the lower face in winter ; but on the equinoctial days, neither face will be illuminated. The rays of the sun will always fall very obliquely on this dial in our latitudes, but to remedy this, a rim may be put round it, rising a little above the planes of its faces. The inside of the elevated part of the rim will be strongly illuminated by the sun's rays, and thus the hours will be more distinctly shown. 20. A dial of this construction, which admits of being adjusted to any latitude, is delineated at fig. 5. In this instrument, ABCD, and CDEF are two quadrangular pieces (which may be of ivory, wood, or metal), connected by means of a hinge C,D. An equinoctial dial is described on each side of ABCD, or on one of them, and in the centre I, a stile is placed at right angles to the planes of the dials. At G, in the middle of the piece EDCF, a magnetic needle is suspended. and covered with a plate of glass. At L, there is a quadrant fixe perpendicular to the plane of this piece, and divided into degrees. It passes through H, an aperture made to receive it in the upper piece. When the dial is to be used, it must be placed on a horizontal plane, so that the needle may be in the magnetic meridian. The upper piece must now be turned round the hinge, so that the planes of the two pieces may make with each other an angle equal to the latitude, as measured by the graduated quadrant. The hour of the day will then be shown by the axis I, on one or other of the two faces, except on the day of the equinox. Horizontal Dial. 21. A dial, traced on the horizontal plane, is called a Horizontal Dial. This is the most common and most useful of any, because it admits of being always illuminated when the sun shines. A dial of this kind is represented in perspective in fig. 6. The point C is the centre, and CK, which is directed to the pole of the heavens, and makes with the plane of the dial an angle equal to the latitude of the place, is the axis. To understand its nature and construction, let ABD be an equinoctial dial, whose axis EF is the prolongation of the axis of the horizontal dial; and let the planes of the two dials meet in the line PQ, and suppose the plane of the meridian to cut the plane of the horizontal dial in CM, and that of the equinoctial dial in EM; then the line PQ being the common intersection of the equinoctial and horizontal planes, which are perpendicular to the meridian, that line itself is perpendicular to the meridian. Let a plane passing through the sun's centre and the common axis of the dials, meet their planes in the lines EH, CH, these lines will manifestly be the positions of the shadows on the two dials at the same instant of time. 22. Now, at any given time, we know the angle HEM which the revolving shadow EH makes with the meridian line EM on the equinoctial dial, because it is the horary angle which the sun has to describe, or has described about the earth's axis, between the given time and noon, and which is always proportional to that time, reckoning 15 degrees of the angle to an hour. And in the triangle CEM, rightangled at E, we know the angle ECM, which is always equal to the latitude of the place for which the dial is to be constructed; and from these we must find the angle HCM, which the hour-line HC, on the horizontal dial, makes with CM, the meridian, or 12 o'clock line. Let us denote the horary angle HEM, which the sun describes between the given time and noon, by the letter E, and the angle HCM, which the hour-line on the horizontal dial makes with the meridian, by C, and let the angle ECM, the latitude of the place, be L; then, by plane trigonometry, in the two right-angled triangles, EMH, CMH, HM: ME:: tan. E: rad. and CH: HM :: rad. : tan. C; therefore, ex æquo inv. (see Geometry.) CM: ME: tan. E.: tan. C, : therefore, rad. : sin.L:: tan. E tan. C. Now, the first three terms of this proportion are known, therefore the last is also known; and we get this general formula for constr.cting a horizontal dial tan. C sin. L. tan. E. |