Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/176

158 158 DIALLING middle of one end of the table. With H for centre, describe any number of concentric arcs of circles, AB, CD, EF, Ac. A bead P, kept in its place by friction, is threaded on the plummet line at some convenient height above H. Every thing being thus prepared, let us follow the shadow of the bead P as it moves along the surface of the table during the day. It will be found to describe a curve ACE .... FDB, approaching the point H as the sun advances towards noon, and receding from it afterwards. (The curve is a conic section an hyperbola in these regions.) At the moment when it crosses the arc AB, mark the point A ; AP is then the direction of the sun, and, as AH is horizontal, the angle PAH is the altitude of the sun. In the afternoon mark the point B where it crosses the same arc ; then the angle PBH is the altitude. But the right- angled triangles PHA, PHB are obviously equal ; and the sun has therefore the same altitudes at those two instants, the oue before, the other after noon. It follows that, if the sun has not changed its declination during the interval, the two positions will be symmetrically placed one on each side of the meridian. Therefore, drawing the chord AB, and bisecting it in M, HM will be the meridian line. Each of the other concentric arcs, CD, EF, &c., will furnish its meridian line. Of course these should all coincide, but if not, the mean of the positions thus found must be taken. The proviso mentioned above, that the sun has not changed its declination, is scarcely ever realized ; but the change is slight, and may be neglected, except per haps about the time of the equinoxes, at the end of March and at the end of September. Throughout the remainder of the year the change of declination is so slow that we may safely neglect it. The most favourable times are at the end of June and at the end of December, when the sun s declination is almost stationary. If the line HM be produced both ways to the edges of the table, then the two points on the ground vertically below those on the edges may be found by a plummet, and, if permanent marks be made there, the meridian plane, which is the vertical plane passing through these two points, will have its position perfectly secured. To place the Style of a Dial in its True Position. Before giving any other method of finding the meridian plane, we shall complete the construction of the dial, by showing how the style may now be accurately placed in its true position. The angle which the style makes with a hanging plumb-line, being the co-latitude of the place, is known, and the north and south direction is also roughly given by the mariner s compass. The style may therefore be already adjusted approximately correctly, indeed, as to its inclination but probably requiring a little horizontal motion east or west. Suspend a fine plumb-line from some point of the style, then the style will be properly adjusted if, at the very instant of noon, its shadow falls exactly on the plumb-line, or, which is the same thing, if both shadows coincide on the dial. This instant of noon will be given very simply by the meridian plane, whose position we have secured by the two permanent marks on the ground. Stretch a cord from the one mark to the other. This will not generally be horizontal, but the cord will be wholly in the meridian plane, and that is the only necessary condition. Next, suspend a plummet over the mark which is nearer to the sun, and, when the shadow of the plumb-line falls on the stretched cord, it is noon. A signal from the observer there to the observer at the dial enables thb latter to adjust the style as directed above. Other Methods of finding the Meridian Plane. We have dwelt at some length on these practical operations because they are simple and tolerably accurate, and because they want neither watch, nor sextant, nor telescope nothing more, in fact, than the careful observation of shadow lines. The polar star may also be employed for finding the meridian plane without other apparatus than plumb-lines. This star is now only about 1 21 from the pole ; if there fore a plumb-line be suspended at a few feet from the observer, and if he shift his position till the star is exactly hidden by the line, then the plane through his eye and the plumb-line will never be far from the meridian plane. Twice in the course of the 24 hours the planes would be strictly coincident. This would be when the star crosses the meridian above the pole, and again when it crosses it below. If we wished to employ the method of determin ing the meridian, the times of the stars crossing would have to be calculated from the data in the Nautical Almanac, and a watch would be necessary to know when the instant arrived. The watch need not, however, be very accurate, because the motion of the star is so slow that an error of ten minutes in the time would not give an error of one- eighth of a degree in the azimuth. The following accidental circumstance enables us to dis pense with both calculation and watch. The right ascen sion of the star rj Ursce Majoris, that star in the tail of the Great Bear which is farthest from the &quot; pointers,&quot; happens to differ by a little more than 12 hours from the right ascension of the polar star. The great circle which joins the two stars passes therefore close to the pole. When the polar star, at a distance of about 1J from the pole, is crossing the meridian above the pole, the star rj Ursce Majoris, whose polar distance is about 40&quot;, has not yet reached the meridian below the pole. When 17 Ursce Majoris reaches the meridian, which will be within half an hour later, the polar star will have left the meridian ; but its slow motion will have carried it only a very little distance away. Now at some instant between these two times much nearer the latter than the former the great circle joining the two stars will be exactly vertical ; and at this instant, which the observer determines by seeing that the plumb-line hides the two stars simultaneously, neither of the stars is strictly in the meridian ; but the deviation from it is so small that it may be neglected, and the plane through the eye and the plumb-line taken for meridian plane. In all these cases it will be convenient, instead of fixing the plane by means of the eye and one fixed plummet, to have a second plummet at a short distance in front of the eye ; this second plummet, being suspended so as to allow of lateral shifting, must be moved so as always to ba