DIALLING 155 on the dial until the style has been itself accurately fixed in its proper place, as will be explained hereafter. When that is done the xn o clock line will be found by the inter section of the dial surface with the vertical plane which contains the style ; and the most simple way of drawing it on the dial will be by suspending a plummet from some point of the style whence it may hang freely, and waiting until the shadows of both style and plumb line coincide on the dial. This single shadow will be the xn o clock line. In one class of dials, namely, all the vertical ones, the xn o clock line is simply the vertical line from the centre ; it can, therefore, at once be traced on the dial face by using a fine plumb line. The xn o clock line being traced, the easiest and most accurate method of tracing the other hour lines would at the present day when good watches are common, be by marking where the shadow of the style falls when 1, 2, 3, &c., hours have elapsed since noon, and the next morning by the same means the forenoon hour lines could be traced ; and in the same manner the hours might be subdivided into halves and quarters, or even into minutes. But formerly, when watches were not, the tracing of the I, n, in, &c. o clock lines was done by calculating the angle which each of these lines would make with the xn o clock line. Now, except in the simple cases of a horizontal dial or of a vertical dial facing a cardinal point, this would require long and intricate calculations, or elabor ate geometrical constructions, implying considerable mathe matical knowledge, but also introducing increased chances of error. The chief source of error would lie in the uncer tainty of the data ; for the position of the dial-plane would have to be found before the calculations began, that is, it would be necessary to know exactly by how many degrees it declined from the south towards the east or west, and by how many degrees it inclined from the vertical. The ancients, with the means at their disposal, could obtain these results only very roughly. Dials received different names according to their posi tion : Horizontal dials, when traced on a horizontal plane ; Vertical dials, when on a vertical plane facing one of the cardinal points ; Vertical declining dials, on a vertical plane not facing a cardinal point ; Inclining dials, when traced on planes neithei vertical nor horizontal (these were further distinguished as reclin ing when leaning backwards from an observer, prodining when leaning forwards) ; Equinoctial dials, when the plane is at right angles to the earth s axis, &c. &c. We shall limit ourselves to an investigation of the simplest and most usual of these cases, referring the reader, for further details, to the later works given at the end of this article. Dial Construction. A very correct view of the problem of dial construction may be obtained as follows : Conceive a transparent cylinder (fig. 1) having an axis AB parallel to the axis of the earth. On the surface of the cylinder let equidistant generating lines be traced 15 apart, one of them xii.. xii being in the meridian plane through AB, and the others I...T, n...n, &c., following in the order of the sun s motion. Then the shadow of the line AB will obviously fall on the line XII. ..xii at apparent noon, on the line I...T at one hour after noon, on II. ..II at two hours after noon, and so on. If now the cylinder be cut by any plane MN representing the plane on which the dial is to be traced, the shadow of AB will be intercepted by this plane, and fall on the lines Axil, Ai, Ail, &c. The construction of the dial consists in determining the angles made by Ai, An, &c. with Axn ; the line Axil itself, being in the vertical plane through AB, may be supposed known. For the purposes of actual calculation, perhaps a trans parent sphere will, with advantage, replace the cylinder, and we shall here apply it to calculate the angles rnide by the hour line with the xii o clock line in the two cases of a horizontal dial and of a vertical south dial Pig. 1. Horizontal Dial. Let PEp (fig. 2), the axis of the supposed transparent sphere, be directed towards the north and south poles of the heavens. Draw the two great circles, HMA, QMa, the former horizontal, the other perpendicular to the axis Pp, and therefore coinciding with the plane of the equator. Let EZ be vertical, then the circle QZP will be the meridian, and by its intersection A with the horizontal will determine the xii o clock line EA. Next divide the equatorial circle QMa into 24 equal parts ab, be, cd, &c. ... of 15 each, beginning from the meridian Pa, and through the various points of division and the poles draw the great circles Tbp, ~Pcp, &c. . . . These will exactly correspond to the equidistant generating lines on the cylinder in the previous construction, and the shadow of the style will fall on these circles after successive intervals of 1, 2, 3, &c. hours from noon. If they meet the horizontal in the points B, C, D, &c., then EB, EC, ED, &c. . . . will be the i, n, in, &c., hour lines required ; and the problem of the horizontal dial consists in calculating the angles which these lines make with the xn o clock line EA, whose position is known. The spherical triangles PAB, PAG, &P., enable us to do this readily. They are all right-angled at A, the side PA is the latitude of the place, and the angles APB, APC, &o., are respectively 15, 30, &c., then tan. AB = tan. 15" sin. latitude, tan. AC = tan. 30 sin. latitude, &c., &c. These determine the sides AB, AC, &c. that is, the angles AEB, AEC, &c., required. For examples, let us find the angles made by the I o clock line at the following places Madras, London, Edinburgh, and Hammer-
fest (Norway).