_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 |
+-------+--------+--------+---------+----------+---------+--------+
_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,
being vertical, will pass through the zenith, and, facing south, will
be at right angles to the meridian. Let QMa 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 XII o'clock line, and the line
EM, which is obviously horizontal, since M is the intersection of two
great circles ZM, QM, each at right angles to the vertical plane QZP,
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 a,
viz. ab, bc, &c.,--each quadrant aM, MQ, &c., containing 6,--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.
[Illustration: FIG. 4.]
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_;
or more simply,
tan AB = tan 15° cos _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.
In this ease the angles diminish as the latitudes increase, the
opposite result to that of the horizontal dial.
_Inclining, Reclining, &c., 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 ares 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.
[Illustration: FIG. 5.]
The mariner's compass may be employed as a first rough approximation.
It is well known that the needle of the compass, when free to move
horizontally, oscillates upon its pivot and settles in a direction
termed the magnetic meridian. This does not coincide with the true
north and south line, but the difference between them is generally
known with tolerable accuracy, and is called the variation of the
compass. The variation differs widely at different parts of the
surface of the earth, and is not stationary at any particular place,
though the change is slow; and there is even a small daily oscillation
which takes place about the mean position, but too small to need
notice here (see MAGNETISM, TERRESTRIAL).
With all these elements of uncertainty, it is obvious that the compass
can only give a rough approximation to the position of the meridian,
but it will serve to fix the style so that only a small further
alteration will be necessary when a more perfect determination has
been made.
[Illustration: FIG. 6.]