DIALS are instruments known to and constructed by the ancients, for the mea surement of time.
In constructing a sun-dial, the object is to find, by means of his shadow, the sun's distance at any time from the meri dian. When this distance is known, the hour is also known, provided we suppose the sun's apparent motion to be uniform, and that during the whole course of a day he moves in a circle parallel to the equator. Neither of these conditions is, in accurately fulfilled, but the er ror which this gives rise to is of small amount ; and it Is, moreover, sufficiently obvious that the use of a dial is not to indicate the hour with astronomical pre cision, but merely to give such an ap proximation as is necessary for the pur poses of civil life.
Dials are usually constructed on an immovable surface, and admit of an in finite number of different constructions, all depending on the nature of the sur face and its position with regard to the equator of the earth. The general prin ciples, however, are the same in all, and depend on the simplest elements of geometry and astronomy. The first part that claims attention is the style or glzo mon, or axis of the dial, which is usually a cylindrical rod, or the edge of a thin plate of metal. The style must be di rected perpendicularly to the terrestrial equator ; in which direction it may be considered, on account of the smallness of the earth's diameter in comparison of the distance of the sun, as coinciding with the axis of the diurnal rotation ; consequently the plane which passes through the style and its shadow on the surrounding surfaces, and which always passes through the centre of the sun, will be an hour plane, and turn with the sun as the sun turns round the style by the effect of the diurnal motion. All that remains to be done, in addition, is to dis cover, and describe, for the different hours of the day, the intersections of this variable hour plane with the surface on which the dial is to be constructed. On these intersections the shadow of the style will be projected every day at the same hour ; because at the same hour the sun must have returned to the same hour plane, although his distance from the equator may be different.
From these considerations it is mani fest that the whole theory of dialling is comprehended in the solution of this general problem :—" Twelve planes all intersecting each other in the same straight line, and making with each other equal angles of 15°, being given in posi tion; to find the intersections of those planes with any surface whatever, also given in form and position." The sur face which intersects the hour planes may be of any kind whatever, but for obvious reasons it is generally a plane ; and when its position with respect to the common intersection of the hour planes (which is the style of the dial) and to any one of those planes is given, the traces or intersections, which are in this case all straight lines, are the hour lines on the dial, and easily calculated by the ordinary rules of trigonometry or geo metry.
According to the position of the dial with respect to the horizon of the place, the dial is horizontal, vertical, or inclined. The most mon tion is the zontal Dial, or that in which the plane of the dial is parallel to the horizon, and ly makes with the style an gle equal to the latitude of the place. At the equator, this is the same as the polar dial ; but at all other places, the hour lines inter sect each other in the point in which the style intersects the plane of the dial, which point is called the centre, and the angles they make with one another, or with the xi' hour line, depend on the latitude.
After the horizontal dials, the construc tion most frequently employed is that in which the plane of the dial is vertical ; for example, when fixed on the wall of a house. In this case, the positions of the different hour lines depend on the lati tude of the place and on the aspect of the dial; that is to say, its position with re spect to the meridian. if the dial is per pendicular to the meridian, it is a south dial, or north, dial, according as it faces the south or north. (The vertical south dial is represented in the annexed figure.) When not perpendi cular to the meridian, the vertical dial is said to be declined. The formula for the hour lines of a south verti cal dial differs from that for a horizontal dial in no respect excepting that the sine of the latitude is changed into the co sine, the cause of which will be obvious when it is considered that the plane of the dial in passing from the horizontal to the south vertical direction preserves its inclination to the different hour planes unaltered ; while the angle which makes with the style, or the axis of the earth, is the complement of the angle it made with the same line in its former position. Let p, therefore, be the hour angle at the centre of the dial ; and put ting, as before, h. = the hour from noon, and 1 = the latitude, the formula for the south vertical dial is tan. p = tan. h cos. /; whence it follows that a horizontal dial constructed for any given latitude will be a south vertical dial for any place of which the latitude is the complement of the latitude of the former place,—a property which was discovered by the Arabians. The hour lines of the vertical north dial are found exactly in the same way as those of the south dial.