The earliest of all sun-dials of which we have any certain knowledge was the hemicycle, or hemisphere, of the Chaldean astronomer Berosus, who probably lived about 340 b.c. It consisted of a hollow hemisphere placed with its rim perfectly horizontal, and having a bead, or globule, fixed in any way at the centre. So long as the sun remained above the horizon the shadow of the bead would fall on the inside of the hemisphere, and the path of the shadow during the day would be approximately a circular arc. This arc, divided Into twelve equal parts, determined twelve equal intervals of time for that day. Now, supposing this were done at the time of the solstices and equinoxes, and on as many intermediate days as might be considered sufficient, and then curve lines drawn through the corresponding points of division of the different arcs, the shadow of the bead falling on one of these curve lines would mark a division of time for that day, and thus we should have a sun-dial which would divide each period of daylight into twelve equal parts.

These equal parts were called temporary hours; and, since the duration of daylight varies from day to day, the temporary hours of one day would differ from those of another; but this inequality would probably be disregarded at that time, and especially in countries where the variation between the longest summer day and the shortest winter day is much less than in our climates.

The dial of Berosus remained in use for centuries. The Arabians, as appears from the work of Albategnius, still followed the same construction about the year 900 a.d. Four of these dials have in modern times been found in Italy. One, discovered at Tivoli in 1746, is supposed to have belonged to Cicero, who, in one of his letters, says that he had sent a dial of this kind to his villa near Tusculum. The second and third were found in 1751— one at Castel-Nuovo, and the other at Rignano; and a fourth was found in 1762 at Pompeii. G. H. Martini, the author of a dissertation in German on the dials of the ancients, says that this dial was made for the latitude of Memphis; it may therefore be the work of Egyptians, perhaps constructed in the school of Alexandria.

It is curious that no sun-dial has been found among the antiquities of Egypt, and their sculptures give no indication of any having existed. It has, however, been supposed that the numerous obelisks found everywhere were erected in honour of the sun and employed as gnomons.

Herodotus has recorded that the Greeks derived from the Babylonians the use of the gnomon, but the great progress’ made by the Greeks in geometry enabled them in later times to construct dials of great complexity, some of which remain to us, and are proofs, not only of extensive knowledge, but also of great ingenuity.

Ptolemy’s Syntaxis treats of the construction of dials by means of his analemma, an instrument which solved a variety of astronomical problems. The constructions given by him were sufficient for regular dials, that is, horizontal dials, or vertical dials facing east, west, north, or south, and these are the only ones he treats of. It is certain, however, that the ancients were able to construct declining dials, as is shown by that most interesting monument of ancient gnomonics—the Tower of the Winds—which is still in existence at Athens. This is a regular octagon, on the faces of which the eight principal winds are represented, and over them eight different dials—four facing the cardinal points and the other four facing the intermediate directions. The date of the dials is long subsequent to that of the tower; for Vitruvius, who describes the tower in the sixth chapter of his first book, says nothing about the dials, and as he has described all the dials known in his time, we must believe that the dials of the tower did not then exist. The tower and its dials are described by Stuart in his Antiquities of Athens. The hours are still the temporary hours, or, as the Greeks called them, hectemoria.

As already stated, the learning and ingenuity of the Greeks enabled them to construct dials of various forms— among others, dials of suspension intended for travellers; but these are only spoken of and not explained; they may have been like our ring-dials.

The Romans were neither geometers nor astronomers, and the science of gnomonics did not flourish among them. The first sun-dial erected at Rome was in the year 290 b,c., and this Papirius Cursor had taken from the Samnites. A dial which Valerius Messala had brought from Catania, the latitude of which is five degrees less than that of Rome, was placedin the forum in the year 261 b.c. The first dial actually constructed at Rome was in the year 164 b.c., by order of Q. Marcius Philippus, but, as no other Roman has written on gnomonics, this was perhaps the work of a foreign artist. If, too, we remember that the dial found at Pompeii was made for the latitude of Memphis, and consequently less adapted to its position than that of Catania to Rome, we may infer that mathematical knowledge was not cultivated in Italy.

The Arabians were much more successful. They attached great importance to gnomonics, the principles of which they had learned from the Greeks, but they greatly simplified and diversified the Greek constructions. One of their writers, Abul-Hassan, who lived about the beginning of the 13 th century, taught them how to trace dials on cylindrical, conical, and other surfaces. He even introduced equal or equinoctial hours, but the idea was not supported, and the temporary hours alone continued in use.

Where or when the great and important step already conceived by Abul-Hassan, and perhaps by others, of reckoning by equal hours was generally adopted cannot now be determined. The history of gnomonics from the 13th to the beginning of the 16th century is almost a blank, and during that time the change took place. We can see, however, that the change would necessarily follow the introduction of clocks and other mechanical methods of measuring time; for, however imperfect these were, the hours they marked would be of the same length in summer and in winter, and the discrepancy between these equal hours and the temporary hours of the sun-dial would soon be too important to be overlooked. Now, we know that a balance clock was put up in the palace of Charles V. of France about the year 1370, and we may reasonably suppose that the new sun-dials came into general use during the 14th and 15th centuries.

Among the earliest of the modem writers on gnomonics must be named Sebastian Munster, a cordelier who published his HorologiographiaeΛ> Basel in 1531. He gives a number of correct rules, but without demonstrations. Among his inventions was a moon-dial, but this does not admit of much accuracy.

During the 17th century dialling was discussed at great length by all writers on astronomy. Clavius devotes a quarto volume of 800 pages entirely to the subject. This was published in 1612, and may be considered to contain all that was known at that time.

In the 18th century clocks and watches began to supersede sun-dials, and these have gradually fallen into disuse except as an additional ornament to a garden, or in remote country districts where the old dial on the church tower still serves as an occasional check on the modern clock by its side. The art of constructing dials may now be looked upon as little more than a mathematical recreation.

General Principles.— The diurnal and the annual motions of the earth are the elementary astronomical facts on which dialling is founded. That the earth turns upon its axis uniformly from west to east in 24 hours, and that it is carried round the sun in one year at a nearly uniform rate, is, we know, the correct way of expressing these facts. But the effect will be precisely the same, and it will suit our purpose better, and make our explanations easier, if we adopt the ideas of the ancients, of which our senses furnish apparent confirmation, and assume the earth to be fixed. Then, the sun and stars revolve round the earth’s axis uniformly from east to west once a day,—the sun lagging a little behind the stars, making its day some 4 minutes longer, so that at the end of the year it finds itself again in the same place, having made a complete revolution of the heavens relatively to the stars from west to east.

The fixed axis about which all these bodies revolve daily is a line through the earth’s centre; but the radius of the earth is so small, compared with the enormous distance of the sun, that, if we draw a parallel axis through any point of the earth’s surface, we may safely look on that as being the axis of the celestial motions. The error in the case of the sun would not, at its maximum, that is, at 6 A. Μ. and 6 p.M., exceed half a second of time, and at noon would vanish.

An axis so drawn is in the plane of the meridian, and points, as we know, to the pole,—its elevation being equal to the latitude of the place.

The diurnal motion of the stars is strictly uniform, and so would that of the sun be if the daily retardation of about 4 minutes, spoken of above, were always the same. But this is constantly altering, so that the time, as measured by the sun’s motion, and also consequently as measured by a sun-dial, does not move on at a strictly uniform pace. This irregularity, which is slight, would be of little consequence in the ordinary affairs of life, but clocks and watches being mechanical measures of time could not, except by extreme complication, be made to follow this irregularity, even if desirable, which is not the case.

The clock is constructed to mark uniform time in such wise that the length of the clock day shall be the average of all the solar days in the year. Four times a year the clock and the sun-dial agree exactly; but the sun-dial, now going a little slower, now a little faster, will be sometimes behind, sometimes before the clock—the greatest accumulated difference being about 16 minutes for a few days in November, but on the average much less. The four days on which the two agree are April 15, June 15, September 1, and December 24.

Clock-time is called mean time, that marked by the sundial is called apparent time, and the difference between them is the equation of time. It is given in most calendars and almanacs, frequently under the heading “clock slow,” “clock fast.” When the time by the sun-dial is known, the equation of time will at once enable us to obtain the corresponding clock time, or vice versa.

Atmospheric refraction introduces another error, by altering the apparent position of the sun; but the effect is too small to need consideration in the construction of an instrument which, with the best workmanship, does not after all admit of very great accuracy.

The general principles of dialling will now be readily understood. The problem before us is the following :— A rod, or style, as it is called, being firmly fixed in a direction parallel to the earth’s axis, we have to find how and where points or lines of reference must be traced on some fixed surface behind the style, so that when the shadow of the style falls on a certain one of these lines we may know that at that moment it is solar noon,—that is, that the plane through the style and through the sun then coincides with the meridian; again, that when the shadow reaches the next line of reference, it is 1 o’clock by solar time, or, which comes to the same thing, that the above plane through the style and through the sun has just turned through the twenty-fourth part of a complete revolution; and so on for the subsequent hours,—the hours before noon being indicated in a similar manner. The style and the surface on which these lines are traced together constitute the dial.

The position of an intended sun-dial having been selected —whether on church tower, south front of farm-stead, or garden wall—the surface must be prepared, if necessary, to receive the hour-lines.

The chief, and in fact the only practical difficulty will be the accurate fixing of the style, for on its accuracy the value of the instrument depends.

It must be in the meridian plane, and must make an angle with the horizon equal to the latitude of the place. The latter condition will offer no difficulty, but the exact determination of the meridian plane which passes through the point where the style is fixed to the surface is not so simple. We shall, further on, show how this may be done; and, in the meantime, we shall assume that we have found the true position, and have firmly fixed the style to the dial and secured it there by cross wires, or by other means. The style itself will be usually a strong metal wire whose thickness may vary with circumstances; and when we speak of the shadow cast by the style it must always be understood that the middle line of the thin band of shade is meant.

The point where the style meets the dial is called the centre of the dial. It is the centre from which all the hour-lines radiate.

The position of the xn o’clock line is the most important to determine accurately, since all the others are usually made to depend on this one. We cannot trace it correctly 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 xιι o’clock line will be found by the intersection 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 xιι o’clock line.

In one class of dials, namely, all the vertical ones, the XII 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 XII 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, <fcc., 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, II, hi, &c. o’clock lines was done by calculating the angle which each of these lines would make with the xιι 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 elaborate geometrical constructions, implying considerable mathematical knowledge, but also introducing increased chances of error. The chief source of error would lie in the uncertainty 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 position :—

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 neitheι vertical nor horizontal (these were further distinguished as reclining when leaning backwards from an observer, proclining 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 I>arallel to the axis of the earth. On the surface of the cylinder et equidistant generating lines be traced 15’ apart, one of them XU.. xn being in the meridian plane through AB, and the others ι...l, ∏...∏, &c., following in the order of the sun’s motion.

Then the shadow of the line AB will obviously fall on the line XII...XU at apparent noon, on the line ι...ι at one hour afternoon, 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, Aι, An, &c.

The construction of the dial consists in determining the angles made by Ai, Au, &c. with Axn; the line Axιι 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 mιde by the hour line with the xιι o’clock line in the two cases of a horizontal dial and of a vertical south dial Horizontal Dial.— Let PE7> (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, QMα, 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 xu o’clock line EA. Next divide the equatorial circle QMα into 24 equal parts ab, be, cd, &c. . . . of 15 % each, beginning from the meridian Pα, and through the various points of division and the poles draw the great circles P⅛p, Vcp, &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, π, in, &c., hour lines required; and the problem of the horizontal dial consists in calculating the angles which these lines make with the xιι o’clock line EA, whose position is known. The spherical triangles PAB, PAC, &c., 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, &c., are respectively 15°, 30°, &c., then tan. AB = tan. 15 β sin∙ latitude, tan. AC = tan. 30 β sin. latitude, kc.. &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 Hammerfest (Norway).

[table]

Thus the i o’clock hour line EB must make an angle on a Madras dial of only 3° 28' with the meridian EA, ll 0 51' on a London dial, 12° 31' at Edinburgh, and 14° 25' at Hammerfest. In the same way may be found the angles made by the other hour lines.

The calculations of these angles must extend throughout one quadrant from noon to vι o’clock, but need not be carried further, because all the other hour-lines can at once be deduced from these. —In the first place the dial is symmetrically divided by the meridian, and therefore two times equidistant from noon will have their hour lines equidistant from the meridian; thus the xι o’clock line and the I o’clock line must make the same angles with it, the xi o’clock the came as the II o’clock, and so on. And next, the 24 great circles, which were drawn to determine these lines, are in reality only 12; for clearly the great circle which gives i o’clock after midnight, and that which gives I o’clock after noon, are one and the same, and so also for the other hours. Therefore the hour lines between vι in the evening and VI the next morning are the prolongations of the remaining twelve.

Let us now remove the imaginary sphere with all its circles, and retain only the style EP and the plane HMA with the lines traced on it, and we shall have the horizontal dial.

On the longest day in London the sun rises a little after 4 o’clock, and sets a little before 8 o’clock; there is therefore no necessity for extending a London dial beyond those hours. At Edinburgh the limits will be a little longer, while at Hammerfest, which is within the Arctic circle, the whole circuit will be required.

Instead of a wire style it is often more convenient to use a metal plate from one quarter to half an inch in thickness. This plate, which is sometimes in the form of a right-angled triangle, must have an acute angle equal to the latitude of the place, and, when properly fixed in a vertical position on the dial, its two faces must coincide with the meridian plane, and the sloping edges formed by the thickness of the plate must point to the pole and form two parallel styles. Since there are two styles, there must be two dials, or rather two half dials, because a little consideration will show that, owing to the thickness of the plate, these styles will only one at a time cast a shadow. Thus the eastern edge will give the shadow for all hours before 6 o’clock in the morning. From 6 o’clock until noon the western edge will be used. At noon, it will change again to the eastern edge until 6 o’clock in the evening, and finally the western edge for the remaining hours of daylight.

The centres of the two dials will be at the points where the styles meet the dial face; but, in drawing the hour-lines, we must be careful to draw only those lines for which the corresponding style is able to give a shadow as explained above. The dial will thus have the appearance of a single dial plate, and there will be no confusion (see fig· 3).

The line of demarcation between the shadow and the light will be better defined than when a wire style is used; but the indications by this double dial will always be one minute too fast in the morning and one minute too slow in the afternoon. This is owing to the magnitude of the sun, whose angular breadth is half a degree. The well-defined shadows are given, not by the centre of the sun, as we should require them, but by the forward limb in the morning and by the backward one in the afternoon; and the sun takes just about a minute to advance through a space equal to its half-breadth.

Dials of this description are frequently met with in the country. Placed on an ornamental pedestal some 4 feet high, they form a pleasing and useful addition to a lawn or to a garden terrace. The dial plate is of metal as well as the vertical piece upon it, and they may be purchased ready for placing on the pedestal,—the dial with all the hour-lines traced on it, and the style-plate firmly fastened in its proper position, if not even cast in the same piece with the dial-plate.

When placing it on the pedestal care must be taken that the dial be perfectly horizontal and accurately oriented. The levelling will be done with a spirit-level, and the orientation will be best effected either in the forenoon or in the afternoon, by turning the dial-plate till the time given by the shadow (making the one minute correction mentioned above) agrees with a good watch whose error on solar time is known. It is, however, important to bear in mind that a dial, so built up beforehand, will have the angle at the base equal to the latitude of some selected place, such as London, and the hour-lines will be drawn in directions calculated for the same latitude. Such a dial can therefore not be used near Edinburgh or Glasgow, although it would, without appreciable error, be adapted to any place whose latitude did not differ more than 20 or 30 miles from that of London, and it would be safe to employ it in Essex, Kent, or Wiltshire.·

If a series of such dials were constructed, differing by 30 miles in latitude, then an intending purchaser could select one adapted to a place whose latitude was within 15 miles of his own, and the error of time would never exceed a small fraction of a minute. The following table will enable us to check the accuracy of the hour-lines and of the angle of the style,—all angles on the dial being readily measured with an ordinary protractor. It extends from 50° lat. to 59¾° lat., and therefore includes the whole of Great Britain and Ireland :— [table]

Vertical South Dial.— Let us take again our imaginary transparent sphere QZPA (fig. 4), whose axis PEp is parallel to the earth’s axis. Let Z be the zenith, and consequently, the great circle QZP the meridian. Through E, the centre of the sphere, draw a vertical plane facing south. This will cut the sphere in the great circle ZMA which, beinσ vertical, will pass through the zenith, and, facing south, will be at right angles to the meridian. Let QMα be the equatorial circle, obtained by drawing a plane through E at right angles to the axis PEp. The lower portion Ep of the axis will be the style, the vertical line EA in the meridian plane will be the xn o’clock line, and the horizontal line EM will be the vi o’clock line.. Now, as in the previous problem, divide the equatorial circle into 24 equal arcs of 15° each, beginning at α, viz., ab, be, &c.,—each quadrant aM, MQ, &c., containing six,—then through each point of division and through the axis Pp draw a plane cutting the sphere in 24 equidistant great circles. As the sun revolves round the axis the shadow of the axis will successively fall on these circles at intervals of one hour, and if these circles cross the vertical circle ZMA in the points A, B, C, &c., the shadow of the lower portion Ep of the axis will fall on the lines EA, EB, EC, &c., which will therefore be the required hour lines on the vertical dial, Ep being the style.

There is no necessity for going beyond the vi o’clock hour-line on each side of noon; for, in the winter months the sun sets earlier than 6 o’clock, and in the summer months it passes behind the plane of the dial before that time, and is no longer available.

It remains to show how the angles AEB, AEC, &c., may be calculated.

The spherical triangles pAB, pAC, &c., will give us a simple rule. These triangles are all right-angled at A, the side pA, equal to ZP, is the co-latitude of the place, that is, the difference between the latitude and 90°; and the successive angles ApB, ApC, &c. are 15°, 30°, &c., respectively. Then tan. AB=tan. 15° sin. co-latitude;

ør more simply, tan. AB=tan. 15 β ces. latitude, tan. AC—tan. 30° cos. latitude, &c., &c.

and the arcs AB, AC so found are the measure of the angles AEB, AEC, &c., required.

We shall, as examples, calculate the I o’clock hour angle AEB for each of the four places we had already taken in the horizontal dial. [table]

In this case the angles diminish as the latitudes increase, the opposite result to that of the horizontal dial.

Inclining, Redining, dec., Dials.— We shall not enter into the calculation of these cases. Our imaginary sphere being, as before supposed, constructed with its centre at the centre of the dial, and all the hour-circles traced upon it, the intersection of these hour-circles with the plane of the dial will determine the hour-lines just as in the previous cases; but the triangles will no longer be right-angled, and the simplicity of the calculation will be lost, the chances of error being greatly increased by the difficulty of drawing the dial-plane in its true position on the sphere, since that true position will have to be found from observations which can be only roughly performed.

In all these cases, and in cases where the dial surface is not a plane, and the hour-lines, consequently, are not straight lines, the only safe practical way is to mark rapidly on the dial a few points (one is sufficient when the dial face is plane) of the shadow at the moment when a good watch shows that the hour has arrived, and afterwards connect these points with the centre by a continuous line. Of course the style must have been accurately fixed in its true position before we begin.

Equatorial Dial.— The name equatorial dial is given to one whose plane is at right angles to the style, and therefore parallel to the equator. It is the simplest of all dials. A circle (fig. 5) divided into 24 equal arcs is placed at right angles to the style, and hour divisions are marked upon it. Then if care be taken that the style point accurately to the pole, and that the noon division coincide with the meridian plane, the shadow of the style will fall on the other divisions, each at its proper time. The divisions must be marked on both sides of the dial, because the sun will shine on opposite sides in the summer and in the winter months, changing at each equinox.

To find the Meridian Plane.— We have, so far, assumed the meridian plane to be accurately known; we shall proceed to describe some of the methods by which it may be found.

The ma