Part 17
Where or when the great and important step already conceived by Abu'l 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 modern writers on gnomonics was SEBASTIAN MÜNSTER (q.v.), who published his _Horologiographia_ at Basel in 1531. He gives a number of correct rules, but without demonstrations. Among his inventions was a moon-dial,[1] but this does not admit of much accuracy.
During the 17th century dialling was discussed at great length by many 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 twenty-four hours, and that it is carried round the sun in one year at a nearly uniform rate, is 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 four 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.M. 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 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 four 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.
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 sixteen 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 sun-dial 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 farmstead 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. At present we shall assume that the style has been fixed in its true position. The style itself will be usually a stout metal wire, 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 XII 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. When that is done the XII 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 XII 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, &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 did not exist, the tracing of the I, II, III, &c., o'clock lines was done by calculating the angle which each of these lines would make with the XII 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 neither 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.
_Dial Construction._--A very correct view of the problem of dial construction may be obtained as follows:--
[Illustration: FIG. 1.]
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 ... I, II ... II, &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 ... I 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 AXII AI, AII, &c.
The construction of the dial consists in determining the angles made by AI, AII, &c. with AXII; the line AXII itself, being in the vertical plane through AB, may be supposed known.
For the purposes of actual calculation, perhaps a transparent sphere will, with advantage, replace the cylinder, and we shall here apply it to calculate the angles made by the hour-line with the XII o'clock line in the two cases of a horizontal dial and of a vertical south dial.
_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
[Illustration: FIG. 2.]
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 circle will determine the XII o'clock line EA. Next divide the equatorial circle QMa into 24 equal parts ab, bc, cd, &c. ... of 15° each, beginning from the meridian Pa, and through the various points of division and the poles draw the great circles Pbp, 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 circle in the points B, C, D, &c., then EB, EC, ED, &c. ... will be the I, II, III, &c., hour-lines required; and the problem of the horizontal dial consists in calculating the angles which these lines make with the XII 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° 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.
The I o'clock hour-line EB must make an angle with the meridian EA of 11° 51' on a London dial, of 12° 31' at Edinburgh, of 11° 23' at Paris, 12° 0' at Berlin, 9° 55' at New York and 9° 19' at San Francisco. 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 VI 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 XI o'clock line and the I o'clock line must make the same angles with it, the X o'clock the same 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 VI 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 before 4 o'clock, and sets a little after 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).
[Illustration: 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. 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 m. 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 m. in latitude, then an intending purchaser could select one adapted to a place whose latitude was within 15 m. 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 59½° lat., and therefore includes the whole of Great Britain and Ireland:--
+-------+--------+--------+---------+----------+---------+--------+ | LAT. |XI. A.M.| X. A.M.| IX. A.M.|VIII. A.M.|VII. A.M.|VI. A.M.| | | I. P.M.|II. P.M.|III. P.M.|IIII. P.M.| V. P.M.|VI. P.M.| +-------+--------+--------+---------+----------+---------+--------+ | 50° 0'| 11° 36'| 23° 51'| 37° 27'| 53° 0' | 70° 43'| 90° 0'| | 50 30 | 11 41 | 24 1 | 37 39 | 53 12 | 70 51 | 90 0 | | 51 0 | 11 46 | 24 10 | 37 51 | 53 23 | 70 59 | 90 0 | | 51 30 | 11 51 | 24 19 | 38 3 | 53 35 | 71 6 | 90 0 | | 52 0 | 11 55 | 24 28 | 38 14 | 53 46 | 71 13 | 90 0 | | 52 30 | 12 0 | 24 37 | 38 25 | 53 57 | 71 20 | 90 0 | | 53 0 | 12 5 | 24 45 | 38 37 | 54 8 | 71 27 | 90 0 | | 53 30 | 12 9 | 24 54 | 38 48 | 54 19 | 71 34 | 90 0 | | 54 0 | 12 14 | 25 2 | 38 58 | 54 29 | 71 40 | 90 0 | | 54 30 | 12 18 | 25 10 | 39 9 | 54 39 | 71 47 | 90 0 | | 55 0 | 12 23 | 25 19 | 39 19 | 54 49 | 71 53 | 90 0 | | 55 30 | 12 27 | 25 27 | 39 30 | 54 59 | 71 59 | 90 0 | | 56 0 | 12 31 | 25 35 | 39 40 | 55 9 | 72 5 | 90 0 | | 56 30 | 12 36 | 25 43 | 39 50 | 55 18 | 72 11 | 90 0 | | 57 0 | 12 40 | 25 50 | 39 59 | 55 27 | 72 17 | 90 0 | | 57 30 | 12 44 | 25 58 | 40 9 | 55 36 | 72 22 | 90 0 | | 58 0 | 12 48 | 26 5 | 40 18 | 55 45 | 72 28 | 90 0 | | 58 30 | 12 52 | 26 13 | 40 27 | 55 54 | 72 33 | 90 0 | | 59 0 | 12 56 | 26 20 | 40 36 | 56 2 | 72 39 | 90 0 | | 59 30 | 13 0 | 26 27 | 40 45 | 56 11 | 72 44 | 90 0 | +-------+--------+--------+---------+----------+---------+--------+