Dial, an instrument for ascertaining the hour of the day by means of rays of light coming from the heavenly bodies. There are therefore solar, lunar, and astral dials. The sun dial only will be considered in this article. It is one of the oldest of human inventions, but its origin cannot be traced. The earliest historical mention of it is in the Old Testament, where we are told of the miracle wrought with the dial of Ahaz, king of Judah, at the instance of the prophet Isaiah, for a sign to Hezekiah, the son of Ahaz. The instrument used by the Chaldean historian and astronomer Berosus, early in the 3d century B. 0., is the most ancient of whose form we have any precise account. It was a hollow hemisphere, with its convexity turned toward the earth, and a button or small globule held in the spherical centre, which, by casting a shadow in the concavity, marked the succeeding hours from one rim to the opposite. It gave place to dials which required more mathematical knowledge for their construction; but a hemisphere properly adjusted will answer the purpose of a dial very well, and was used by several nations long after the commencement of our era.

Four have been discovered in modern times in Italy. One found at Tivoli in 1746 is supposed to have belonged to Cicero. Another was found in 1762 at Pompeii, and as it was adapted to the latitude of Memphis, it has been ascribed to the Egyptians, although no sun dial has ever been discovered among the ruins of Egypt, nor have any representations of it been found among their sculptures. It is believed, however, that they used some form of sun dial, notwithstanding the care they bestowed upon clepsydras, and their probable use of the pendulum. Perhaps the obelisks erected in honor of the sun were used as gnomons, and it has been suggested that the famous circle of Osy-mandyas might have been used to determine the azimuths of the heavenly bodies, and therefore the hours of the day. The tower of the winds at Athens, which from its architecture is judged to be of a somewhat later date than the time of Pericles, is an octagonal structure, and bears eight sun dials for the cardinal and intermediate points of the compass. They are described in Stuart's "Antiquities of Athens." Four others, known as the dials of Phoedrus, also found at Athens, are now in the British museum, and are described in Delam-bre's Histoire de l'astronomie ancienne.

Their construction shows that the Greeks used geometrical methods for vertical and also for declining dials. The first sun dial said to have been erected in Rome was by L. Papirius Cursor, who had taken it from the Samnites. About 30 years after another was placed near it by M. Valerius Messala, who brought it from Sicily in the second year of the first Punic war. It was made for the latitude of Catania, 4 1/2° south of Rome. The first dial constructed at Rome, and adapted to its latitude, is said to have been by the order of Q. Marcius Philippius, in 164 B. C. - The sun dial may have many forms, depending upon its position in regard to the sun, and upon the latitude in which it is used. The most common form at the present day is the horizontal dial. It consists of a horizontal plate upon which the hours are marked, and which supports a style or gnomon for casting the sun's shadow, having its edge parallel with the axis of the earth. Another form is that called the equinoctial dial, consisting of a staff or gnomon placed parallel with the axis of the earth, and passing perpendicularly through the centre of a circle divided into equal parts for marking the hours.

The slight deviation of the sun's apparent from the true time is not taken into account in the construction of dials, the correction being made after the observation is taken. Let fig. 1 represent the earth, N the north pole, A B the equator, and a, b, c, d, e, meridian lines; a, c, and e marking the quadrants. As the earth revolves on its axis with uniform motion once in 24 hours, each point moves through 15° every hour; therefore, if it is noon on the meridian a, in three hours after it will be noon on the meridian b, 45° from a, and in six hours the meridian c, marked 90°, will be brought vertically beneath the sun. All the rays of the sun which strike the earth are apparently parallel, because of his immense distance, which is about 12,000 times the earth's diameter. It will therefore be 6 o'clock in the morning on the meridian c when it is noon on the meridian a. Suppose a circle to be placed at the pole, in the plane of the horizon, and divided into degrees corresponding with those on the equator, and a staff which shall represent an extension of the earth's axis to pass through its centre; it follows that when the sun's path is north of the equator the staff will cast a shadow upon the circle, which will traverse it with uniform motion, passing through its 360° in 24 hours.

Such a circle and style would form an equinoctial dial, which, if placed on the meridian a at the equator, with its gnomon or style parallel with the earth's axis, would, while the sun's rays fall upon it, measure the time in the same way as at the pole; that is, the gnomon would traverse corresponding degrees at the same time. Instead of the shadow traversing the whole circumference of the dial, as it would at the pole, it can only traverse the lower half, between 6 in the morning and 6 in the evening. Place a similar dial D on the meridian a, between the equator and the north pole, also with its gnomon parallel to the earth's axis, and therefore inclined to the horizon with an angle equal to the latitude of the place, and the shadow of the gnomon will travel the face of the dial precisely as in the case of the circle C, with the difference that when the sun's path is north of the equator the shadow will be cast before 6 in the morning and after 6 in the evening, falling upon the northern face; and when the sun's path is south of the equator it will fall upon the south face, but not till after 6 in the morning nor until 6 in the evening.

In the horizontal dial the shadow does not traverse the hour circle with a uniform motion (except at the poles), but travels faster the further it recedes from the vertical position; so that the lines which mark the hours require to be further apart near the morning and evening than near noon. The hour lines may be determined by the following elementary method of plane trigonometry. Let ABCD, fig. 2, be a horizontal plane upon which the hour lines of a dial are to be described. Draw the meridian line E F, and from E erect E II parallel to the earth's axis, to represent the gnomon. Then, with a centre G on E H as an axis, describe the equinoctial circle I O F, its plane being parallel with the plane of the equator. Draw the meridian line I F, and also G M, G N, and G C, dividing the arc O F into equal segments. Draw E M, E N, and E C on the horizontal plane, meeting G M, G N, and G C in M, N, and C, and with a radius E F and a centre at E describe the arc F P, meeting E C in P; also with the same radius and centre describe the arc F K, meeting the line E H in K. The angle F G C is at the centre of the equinoctial arc, and has F C for its tangent, and the angle F E C is at the centre of the hour arc and also has F C for its tangent.

The corresponding equinoctial and hour arcs have therefore a common tangent. As G F is always perpendicular to E H, it follows that the radius of the equinoctial arc will always be the sine of the latitude arc F K. It will moreover be observed that the radii of the latitude and hour arcs will always be equal, and proportional to the radius of the equinoctial arc, as the hypothenuse of a right-angled triangle to one of its sides; and therefore that the common tangent F C will always measure a larger equinoctial than hour arc. All these relations being constant, we derive the following equations: rad. equinoc. arc = sin. lat. arc. tan. equinoc. arc = tan. hour arc.

Dial 060014

FlG. l.

Dial 060015

Fig. 2.

Multiplying equals by equal ratios, we have tan. hour arc x rad. equinoc. arc = sin. lat. arc x tan.equinoc. arc; and sin. lat. arc x tan. equinoc. arc. tan. hour arc = rad. equinoc. arc.

It will be observed that if the line E H were a rod, the circle I O F a material plane, and the angles at G each equal to 15°, the apparent motion of the sun would cause the rod's shadow to fall upon the lines G M, G N, and G C at the hours 1, 2, and 3 respectively, and also upon the lines E M, E N, and E C of the horizontal plane or hour arc. Let it be required to find the first hour angle on either side of the meridional line for a horizontal sun dial in the latitude of New York, which is 40° 42' 43". By logarithms:

sin. lat. arc 40° 42' 43" = ..............................

9.814419

tan.equinoc.are for 1 hour = 15 = ...........

9.428052

19.242461

rad.equinoe.are = .................

10.000000

tan. hour are, 9 54' 45" = .................

9.242461

The hour angles for 10 and 2 o'clock will be found by substituting 30° for 15° of the equinoctial circle, and for 9 and 3 o'clock by substituting 45°, and so on till 6 o'clock, when the angles will decrease in the same ratios in which they increased. A dial with its hour plane in a vertical position is called a vertical dial, and may be regarded as the complement of the horizontal, because the angle of inclination of the gnomon to the plane of the dial is the complement of that angle in the horizontal dial if taken to a latitude which is the complement of that for which it is intended as a vertical dial. Vertical dials have north and south faces, for the reason that the shadow of the sun can never fall upon the south face before 6 in the morning or after 6 in the evening; but when he rises before 6 in the morning his beams will fall upon the north face. This, being the counterpart of the south face, will show the hour. A horizontal dial and the south face of a vertical one are represented in fig. 3. Neither of these forms can be used with much accuracy in less than 20° of latitude, because as we approach the equator the gnomon becomes more and more parallel with the horizon, so that if its length remains the same the upper end will cast but a very small shadow during the hours near midday.

For instance, during the first hour before or after noon the sun in passing through 15° will cast a shadow on the horizontal plane but little more than one fourth the height of the gnomon. If this be 6 in., the edge of the shadow will be only 1 1/2 in. from the perpendicular. As the height of the upper end of the gnomon above the dial is to its length as the sine of an angle is to the radius, it follows that in lat. 30° the edge of the gnomon would require to be 12 in. in order to give it an elevation of 6 in. To retain this elevation in lat. 10°, the gnomon would have to be about 3 1/2 ft. long, and in lat. 5° about 7 ft.; which, for various reasons (one of which is that the oblique shadows would be too dim to be plainly discernible), makes its use impracticable. The nearer we approach the pole the more does the gnomon approach a perpendicular position, until at the pole it becomes an extension of the earth's axis; the hour angles, as marked upon the horizontal plane, becoming equal, as shown at E in fig. 1.

Dial 060016

Fig. 3.

- A glass cylinder having a rod for an axis and its surface marked with 24 equidistant lines, parallel with the axis, was used by Ferguson in the construction of dials, and is itself a form of equinoctial dial which may be used in any latitude by placing its axis parallel with the axis of the earth. It would thus be a modification of the dial of Berosus, if the hour lines were marked upon the latter as meridians, the shadow of the axis falling upon the parallel lines of the cylinder precisely as they would upon the meridians of the hemispheres. Fig. 4 represents the hemisphere of Berosus suspended in a graduated arc, by means of which its gnomon may be adjusted to the latitude of the place. It may also be turned upon its axis and held at any degree of inclination to the east or west, so that when the days are more than 12 hours long it will indicate the time before 6 in the morning and after 6 in the evening. The lines 5, 6, 7, 8, etc, are intended to represent meridians, and should meet at the poles A B. A magnetic needle and a pair of spirit levels facilitate its adjustment. Burt's solar compass (see Compass, Solar) contains a sun dial which is an elaboration of this plan. An hour arc is held in a plane having an inclination corresponding to the latitude of the place.

Over this arc an arm, attached to what is called the declination arc, is made to move until the sun's rays coincide with the axis of the arm, when the number of degrees traversed by the hour arc will indicate the sun's apparent time. This is the most correct sun dial that can be constructed.

Dial 060017

Fig. 4.