, the art of drawing sun, moon, and star-dials on any sort of surface, whether plane or curved.

Dialling is wholly founded on the first motion of the heavenly bodies, and chiefly the sun; or rather on the diurnal rotation of the earth: so that the elements of spherics, and spherical trigonometry, should be understood, before a person advances to the doctrine of dialling.

The principles of dialling may be aptly deduced from, and illustrated by, the phenomena of a hollow or transparent sphere, as of glass. Thus, suppose aPcp to re- present the earth as transparent; and its equator as divided into 24 equal parts by so many meridian semicircles a, b, c, d, e, &c, one of which is the geographical meridian of any given place, as London, which it is supposed is at the point a; and if the hour of 12 were marked at the equator, both upon that meridian and the opposite one, and all the rest of the hours in order on the other meridians, those meridians would be the hour circles of London: because, as the sun appears to move round the earth, which is in the centre of the visible heavens, in 24 hours, he will pass from one meridian to another in an hour. Then, if the sphere had an opake axis, as PEp, terminating in the poles P and p, the shadow of the axis, which is in the same plane with the sun and with each meridian, would fall upon every particular meridian and hour, when the sun came to the plane of the opposite meridian, and would consequently shew the time at London, and at all other places on the same meridian. If this sphere were cut through the middle by a solid plane ABCD in the rational horizon of London, one half of the axis EP would be above the plane, and the other half below it; and if straight lines were drawn from the centre of the plane to those points where its circumference is cut by the hour circles of the sphere, those lines would be the hour lines of an horizontal dial for London; for the shadow of the axis would fall upon each particular hour line of the dial, when it fell upon the like hour circle of the sphere.

If the plane which cuts the sphere be upright, as AFCG, touching the given place, for ex. London, at F, and directly facing the meridian of London, it will then become the plane of an erect direct south dial; and if right lines be drawn from its centre E, to those points of its circumference where the hour circles of the sphere cut it, these will be the hour lines of a vertical or direct south dial for London, to which the hours are to be set in the figure, contrary to those on an horizontal dial; and the lower half Ep of the axis will cast a shadow on the hour of the day in this dial, at the same time that it would fall upon the like hour circle of the sphere, if the dial plane was not in the way.

If the plane, still facing the meridian, be made to incline, or recline, any number of degrees, the hour circles of the sphere will still cut the edge of the plane in those points to which the hour lines must be drawn straight from the centre; and the axis of the sphere will cast a shadow on these lines at the respective hours. The like will still hold, if the plane be made to decline by any number of degrees from the meridian towards the east or west; provided the declination be less than 90 degrees, or the reclination be less than the co-latitude of the place; and the axis of the sphere will be the gnomon: otherwise, the axis will have no elevation above the plane of the dial, and cannot be a gnomon.

Thus it appears that the plane of every dial represents the plane of some great circle on the earth, and the gnomon the earth's axis; the vertex of a right gnomon the centre of the earth or visible heavens; and the plane of the dial is just as far from this centre as from the vertex of this stile. The earth itself, compared with its distance from the sun, is considered as a point; and therefore, if a small sphere of glass be placed upon any part of the earth's surface, so that its axis be parallel to the axis of the earth, and the sphere have such lines upon it, and such planes within it, as above described; it will shew the hours of the day as truly as if it were placed at the earth's centre, and the shell of the earth were as transparent as glass. Ferguson, lect. 10.

The principal writers on Dials, and Dialling, are the following: Vitruvius, in his Architecture, cap. 4 | and 7, lib. 9: Sebastian Munster, his Horolographia: John Dryander de Horologiorum varia Compositione: Conrade Gesner's Pandectæ: Andrew Schoner's Gnomonicæ: Fred. Commandine de Horologiorum Descriptione: Joan. Bapt. Benedictus de Gnomonum Umbrarumque Solarium Usu: Joannes Georgius Schomberg, Exegesis Fundamentorum Gnomonicorum: Solomon de Caus, Traité des Horologes Solaires: Joan. Bapt. Trolta, Praxis Horologiorum: Desargues, Maniere Universelle pour poser l'Essieu & placer les Heures & autres Choses aux Cadrans Solaires: Ath. Kircher, Ars magna Lucis & Umbræ: Hallum, Explicatio Horologii in Horto Regio Londini: Tractatus Horologiorum Joannis Mark: Clavius, Gnomonices de Horologiis; in which he demonstrates both the theory and the operations after the rigid manner of the ancient mathematicians: Dechales, Ozanam, and Schottus, gave much easier treatises on this subject; as did also Wolfius in his Elementa: M. Picard gave a new method of making large dials, by calculating the hour lines; and M. De la Hire, in his Dialling, printed in 1683, gave a geometrical method of drawing hour lines from certain points, determined by observation. Everhard Walper, in 1625, published his Dialling, in which he lays down a method of drawing the primary dials on a very easy foundation; and the same foundation is also described at length by Sebastian Munster, in his Rudimenta Mathematica, published in 1651. In 1672, Sturmius published a new edition of Walper's Dialling, with the addition of a whole second part, concerning inclining and declining dials, &c. In 1708, the same work, with Sturmius's additions, was re-published, with the addition of a 4th part, containing Picard's and De la Hire's methods of drawing large dials, which makes much the best and fullest book on the subject. Peterson, Michael, and Muller, have each written on Dialling, in the German language: Coetfius, in his Horologiographia Plana, printed in 1689: Gauppen, in his Gnomonica Mechanica: Leybourn, in his Dialling: Bion, in his Use of Mathematical Instruments: Wells, in his Art of Shadows. There is also a treatise by M. Deparceux, 1740. Mr. Ferguson has also written on the same subject in his Lectures on Mechanics; besides Emerson, in his Dialling; and Mr. W. Jones, in his Instrumental Dialling.

Universal Dialling Cylinder, is represented in fig. 2, plate vii, where ABCD is a glass cylindrical tube, closed at both ends with brass plates, in the centres of which a wire or axis EFG is fixed. The tube is either fixed to an horizontal board H, so that its axis may make an angle with the board equal to that which the earth's axis makes with the horizon of any given place, and be parallel to the axis of the world; or it may be made to move on a joint, and elevated for any particular latitude. The twenty-four hour lines are drawn with a diamond on the outside of the glass, equidistant from each other, and parallel to the axis. The XII next B stands for midnight, and the XII next the board H for noon. When the axis of this instrument is elevated according to the latitude, and the board set level, with the line HN in the plane of the meridian, and the end towards the north; the axis EFG will serve as a gnomon or stile, and cast a shadow on the hour of the day among the pa- rallel hour lines, when the sun shines on the instrument. As the plate AD at the top is parallel to the equator, and the axis EFG perpendicular to it, right lines drawn from the centre to the extremities of the parallels will be the hour lines of an equinoctial dial, and the axis will be the stile. An horizontal plate ef put down into the tube, with lines drawn from the centre to the several parallels, cutting its edge, will be an horizontal dial for the given latitude; and a vertical plate gc, fronting the meridian, and touching the tube with its edge, with lines drawn from its centre to the parallels, will be a vertical south dial: the axis of the instrument serving in both cases for the stile of the dial: and if a plate be placed within the tube, so as to decline, incline, or recline, by any given number of degrees, and lines be drawn, as above, a declining, inclining, or reclining dial will be formed for the given latitude. If the axis with the several plates sixed to it be drawn out of the tube, and set up in sunshine in the same position as they were in the tube, AD will be an equinoctial dial, ef an horizontal dial, and ge a vertical south dial; and the time of the day will be shewn by the axis EFG. If the cylinder were wood, instead of glass, and the parallel lines drawn upon it in the same manner, it would serve to facilitate the operation of making these several dials. The upper plate with lines drawn to the several intersections of the parallels, which appears obliquely in fig. 2, would be an equinoctial dial as in fig. 3, and the axis perpendicular to it be its stile. An horizontal dial for the latitude of the elevation of the axis might be made, by drawing out the axis and cutting the cylinder, as at efgh, parallel to the horizontal board H; the section would be elliptic as in fig. 4. A circle might be described on the centre, and lines drawn to the divisions of the ellipse would be the hour lines; and the wire put in its place again, as E, would be the stile. If this cylinder were cut by a plane perpendicular to the horizontal board H, or to the line SHN, beginning at g, the plane of the section would be elliptical as in fig. 5, and lines drawn to the points of intersection of the parallels on its edge would be the hour lines of a vertical direct south dial, which might be made of any shape, either circular or square, and F the axis of the cylinder would be its stile. Thus also inclining, declining, or reclining dials might be easily constructed, for any given latitude. Ferguson, ubi supra.

Dialling Globe, is an instrument made of brass, or wood, with a plane fitted to the horizon, and an index; particularly contrived to draw all sorts of dials, and to give a clear exhibition of the principles of that art.

Dialling Lines, or Scales, are graduated lines, placed on rules or the edges of quadrants, and other instruments, to expedite the construction of dials. The principal of these lines are, 1. A scale of six hours, which is only a double tangent, or two lines of tangents each of 45 degrees, joined together in the middle, and equal to the whole line of fines, with the declination set against the meridian altitudes in the latitude of London, suppose, or any place for which it is made: the radius of which line of sines is equal to the dialling scale of six hours. 2. A line of latitudes, which is sitted to the hour scale, and is made by this | canon: as the radius is to the chord of 90 degrees; so are the tangents of each respective degree of the line of latitudes, to the tangents of other arches: and then the natural sines of those arches are the numbers, which, taken from a diagonal scale of equal parts, will graduate the divisions of the line of latitude to any radius. The line of hours and latitudes is generally for pricking down all dials with centres. For the method of constructing these scales, see Scale.

Dialling Sphere, is an instrument made of brass, with several semicircles sliding over one another, on a moving horizon, to demonstrate the nature of the doctrine of spherical triangles, and to give a true idea of the drawing of dials on all manner of planes.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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