Chapter VIII: Part 8

Ptolemy's _Almagest_ 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 gnomics--the Tower of the Winds 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 hours are still the temporary hours or, as the Greeks called them, _hectemoria_.

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 Messalla had brought from Catania, the latitude of which is five degrees less than that of Rome, was placed in 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, Abu'l Hassan, who lived about the beginning of the 13th 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 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:--

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

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).

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.

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.

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.

A very simple practical method is the following:--

Place a table (fig. 6), or other plane surface, in such a position
that it may receive the sun's rays both in the morning and in the
afternoon. Then carefully level the surface by means of a
spirit-level. This must be done very accurately, and the table in that
position made perfectly secure, so that there be no danger of its
shifting during the day.

Next, suspend a plummet SH from a point S, which must be rigidly
fixed. The extremity H, where the plummet just meets the surface,
should be somewhere near the middle of one end of the table. With H
for centre, describe any number of concentric arcs of circles, AB, CD,
EF, &c.

A bead P, kept in its place by friction, is threaded on the plummet
line at some convenient height above H.

Everything being thus prepared, let us follow the shadow of the bead P
as it moves along the surface of the table during the day. It will be
found to describe a curve ACE ... FDB, approaching the point H as the
sun advances towards noon, and receding from it afterwards. (The curve
is a conic section--an hyperbola in these regions.) At the moment when
it crosses the arc AB, mark the point A; AP is then the direction of
the sun, and, as AH is horizontal, the angle PAH is the altitude of
the sun. In the afternoon mark the point B where it crosses the same
arc; then the angle PBH is the altitude. But the right-angled
triangles PHA, PHB are obviously equal; and the sun has therefore the
same altitudes at those two instants, the one before, the other after
noon. It follows that, _if the sun has not changed its declination_
during the interval, the two positions will be symmetrically placed
one on each side of the meridian. Therefore, drawing the chord AB, and
bisecting it in M, HM will be the meridian line.

Each of the other concentric arcs, CD, EF, &c., will furnish its
meridian line. Of course these should all coincide, but if not, the
mean of the positions thus found must be taken.

The proviso mentioned above, that the sun has not changed its
declination, is scarcely ever realized; but the change is slight, and
may be neglected, except perhaps about the time of the equinoxes, at
the end of March and at the end of September. Throughout the remainder
of the year the change of declination is so slow that we may safely
neglect it. The most favourable times are at the end of June and at
the end of December, when the sun's declination is almost stationary.
If the line HM be produced both ways to the edges of the table, then
the two points on the ground vertically below those on the edges may
be found by a plummet, and, if permanent marks be made there, the
meridian plane, which is the vertical plane passing through these two
points, will have its position perfectly secured.

_To place the Style of a Dial in its True Position._--Before giving
any other method of finding the meridian plane, we shall complete the
construction of the dial, by showing how the style may now be
accurately placed in its true position. The angle which the style
makes with a hanging plumb-line, being the co-latitude of the place,
is known, and the north and south direction is also roughly given by
the mariner's compass. The style may therefore be already adjusted
approximately--correctly, indeed, as to its inclination--but probably
requiring a little horizontal motion east or west. Suspend a fine
plumb-line from some point of the style, then the style will be
properly adjusted if, at the very instant of noon, its shadow falls
exactly on the plumb-line,--or, which is the same thing, if both
shadows coincide on the dial.

This instant of noon will be given very simply, by the meridian plane,
whose position we have secured by the two permanent marks on the
ground. Stretch a cord from the one mark to the other. This will not
generally be horizontal, but the cord will be wholly in the meridian
plane, and that is the only necessary condition. Next, suspend a
plummet over the mark which is nearer to the sun, and, when the shadow
of the plumb-line falls on the stretched cord, it is noon. A signal
from the observer there to the observer at the dial enables the latter
to adjust the style as directed above.

_Other Methods of finding the Meridian Plane._--We have dwelt at some
length on these practical operations because they are simple and
tolerably accurate, and because they want neither watch, nor sextant,
nor telescope--nothing more, in fact, than the careful observation of
shadow lines.

The Pole star, or _Ursae Minoris_, may also be employed for finding
the meridian plane without other apparatus than plumb-lines. This star
is now only about 1° 14' from the pole; if therefore a plumb-line be
suspended at a few feet from the observer, and if he shift his
position till the star is exactly hidden by the line, then the plane
through his eye and the plumb-line will never be far from the meridian
plane. Twice in the course of the twenty-four hours the planes would
be strictly coincident. This would be when the star crosses the
meridian above the pole, and again when it crosses it below. If we
wished to employ the method of determining the meridian, the times of
the stars crossing would have to be calculated from the data in the
_Nautical Almanac_, and a watch would be necessary to know when the
instant arrived. The watch need not, however, be very accurate,
because the motion of the star is so slow that an error of ten minutes
in the time would not give an error of one-eighth of a degree in the
azimuth.

The following accidental circumstance enables us to dispense with both
calculation and watch. The right ascension of the star [eta] _Ursae
Majoris_, that star in the tail of the Great Bear which is farthest
from the "pointers," happens to differ by a little more than 12 hours
from the right ascension of the Pole star. The great circle which
joins the two stars passes therefore close to the pole. When the Pole
star, at a distance of about 1° 14' from the pole, is crossing the
meridian above the pole, the star [eta] _Ursae Majoris_, whose polar
distance is about 40°, has not yet reached the meridian below the
pole.

When [eta] _Ursae Majoris_ reaches the meridian, which will be within
half an hour later, the Pole star will have left the meridian; but its
slow motion will have carried it only a very little distance away. Now
at some instant between these two times--much nearer the latter than
the former--the great circle joining the two stars will be exactly
vertical; and at this instant, which the observer determines by seeing
that the plumb-line hides the two stars simultaneously, neither of the
stars is strictly in the meridian; but the deviation from it is so
small that it may be neglected, and the plane through the eye and the
plumb-line taken for meridian plane.

In all these cases it will be convenient, instead of fixing the plane
by means of the eye and one fixed plummet, to have a second plummet at
a short distance in front of the eye; this second plummet, being
suspended so as to allow of lateral shifting, must be moved so as
always to be between the eye and the fixed plummet. The meridian plane
will be secured by placing two permanent marks on the ground, one
under each plummet.

This method, by means of the two stars, is only available for the
upper transit of _Polaris_; for, at the lower transit, the other star
[eta] _Ursae Majoris_ would pass close to or beyond the zenith, and
the observation could not be made. Also the stars will not be visible
when the upper transit takes place in the daytime, so that one-half of
the year is lost to this method.

Neither could it be employed in lower latitudes than 40° N., for there
the star would be below the horizon at its lower transit;--we may even
say not lower than 45° N., for the star must be at least 5° above the
horizon before it becomes distinctly visible.

There are other pairs of stars which could be similarly employed, but
none so convenient as these two, on account of _Polaris_ with its very
slow motion being one of the pair.

_To place the Style in its True Position without previous
Determination of the Meridian Plane._--The various methods given above
for finding the meridian plane have for ultimate object the
determination of the plane, not on its own account, but as an element
for fixing the instant of noon, whereby the style may be properly
placed.

We shall dispense, therefore, with all this preliminary work if we
determine noon by astronomical observation. For this we shall want a
good watch, or pocket chronometer, and a sextant or other instrument
for taking altitudes. The local time at any moment may be determined
in a variety of ways by observation of the celestial bodies. The
simplest and most practically useful methods will be found described
and investigated in any work on astronomy.

For our present purpose a single altitude of the sun taken in the
forenoon will be most suitable. At some time in the morning, when the
sun is high enough to be free from the mists and uncertain refractions
of the horizon--but to ensure accuracy, while the rate of increase of
the altitude is still tolerably rapid, and, therefore, not later than
10 o'clock--take an altitude of the sun, an assistant, at the same
moment, marking the time shown by the watch. The altitude so observed
being properly corrected for refraction, parallax, &c., will, together
with the latitude of the place, and the sun's declination, taken from
the _Nautical Almanac_, enable us to calculate the time. This will be
the solar or apparent time, that is, the very time we require.
Comparing the time so found with the time shown by the watch, we see
at once by how much the watch is fast or slow of solar time; we know,
therefore, exactly what time the watch must mark when solar noon
arrives, and waiting for that instant we can fix the style in its
proper position as explained before.

We can dispense with the sextant and with all calculation and
observation if, by means of the pocket chronometer, we bring the time
from some observatory where the work is done; and, allowing for the
change of longitude, and also for the equation of time, if the time we
have brought is clock time, we shall have the exact instant of solar
noon as in the previous case.

In former times the fancy of dialists seems to have run riot in
devising elaborate surfaces on which the dial was to be traced.
Sometimes the shadow was received on a cone, sometimes on a cylinder,
or on a sphere, or on a combination of these. A universal dial was
constructed of a figure in the shape of a cross; another universal
dial showed the hours by a globe and by several gnomons. These
universal dials required adjusting before use, and for this a
mariner's compass and a spirit-level were necessary. But it would be
tedious and useless to enumerate the various forms designed, and, as a
rule, the more complex the less accurate.

Another class of useless dials consisted of those with variable
centres. They were drawn on fixed horizontal planes, and each day the
style had to be shifted to a new position. Instead of hour-_lines_
they had hour-_points_; and the style, instead of being parallel to
the axis of the earth, might make any chosen angle with the horizon.
There was no practical advantage in their use, but rather the reverse;
and they can only be considered as furnishing material for new
mathematical problems.

_Portable Dials._--The dials so far described have been fixed dials,
for even the fanciful ones to which reference was just now made were
to be fixed before using. There were, however, other dials, made
generally of a small size, so as to be carried in the pocket; and
these, so long as the sun shone, roughly answered the purpose of a
watch.

The description of the portable dial has generally been mixed up with
that of the fixed dial, as if it had been merely a special case, and
the same principle had been the basis of both; whereas there are
essential points of difference between them, besides those which are
at once apparent.

In the fixed dial the result depends on the _uniform_ angular motion
of the sun round the fixed style; and a small error in the assumed
position of the sun, whether due to the imperfection of the
instrument, or to some small neglected correction, has only a trifling
effect on the time. This is owing to the angular displacement of the
sun being so rapid--a quarter of a degree every minute--that for the
ordinary affairs of life greater accuracy is not required, as a
displacement of a quarter of a degree, or at any rate of one degree,
can be readily seen by nearly every person. But with a portable dial
this is no longer the case. The uniform angular motion is not now
available, because we have no determined fixed plane to which we may
refer it. In the new position, to which the observer has gone, the
zenith is the only point of the heavens he can at once practically
find; and the basis for the determination of the time is the
constantly but _very irregularly_ varying zenith distance of the sun.

At sea the observation of the altitude of a celestial body is the only
method available for finding local time; but the perfection which has
been attained in the construction of the sextant enables the sailor to
reckon on an accuracy of seconds. Certain precautions have, however,
to be taken. The observations must not be made within a couple of
hours of noon, on account of the slow rate of change at that time, nor
too near the horizon, on account of the uncertain refractions there;
and the same restrictions must be observed in using a portable dial.

To compare roughly the accuracy of the fixed and the portable dials,
let us take a mean position in Great Britain, say 54° lat., and a mean
declination when the sun is in the equator. It will rise at 6 o'clock,
and at noon have an altitude of 36°,--that is, the portable dial will
indicate an average change of one-tenth of a degree in each minute, or
two and a half times slower than the fixed dial. The vertical motion
of the sun increases, however, nearer the horizon, but even there it
will be only one-eighth of a degree each minute, or half the rate of
the fixed dial, which goes on at nearly the same speed throughout the
day.

Portable dials are also much more restricted in the range of latitude
for which they are available, and they should not be used more than 4
or 5 m. north or south of the place for which they were constructed.

We shall briefly describe two portable dials which were in actual use.

_Dial on a Cylinder._--A hollow cylinder of metal (fig. 7), 4 or 5 in.
high, and about an inch in diameter, has a lid which admits of
tolerably easy rotation. A hole in the lid receives the style shaped
somewhat like a bayonet; and the straight part of the style, which, on
account of the two bends, is lower than the lid, projects horizontally
out from the cylinder to a distance of 1 or 1½ in. When not in use the
style would be taken out and placed inside the cylinder.

A horizontal circle is traced on the cylinder opposite the projecting
style, and this circle is divided into 36 approximately equidistant
intervals.[2] These intervals represent spaces of time, and to each
division is assigned a date, so that each month has three dates marked
as follows:-January 10, 20, 31; February 10, 20, 28; March 10, 20, 31;
April 10, 20, 30, and so on,--always the 10th, the 20th, and the last
day of each month.

Through each point of division a vertical line parallel to the axis of
the cylinder is drawn from top to bottom. Now it will be readily
understood that if, upon one of these days, the lid be turned, so as
to bring the style exactly opposite the date, and if the dial be then
placed on a horizontal table so as to receive sunlight, and turned
round bodily until the shadow of the style falls exactly on the
vertical line below it, the shadow will terminate at some definite
point of this line, the position of which point will depend on the
length of the style--that is, the distance of its end from the surface
of the cylinder--and on the altitude of the sun at that instant.
Suppose that the observations are continued all day, the cylinder
being very gradually turned so that the style may always face the sun,
and suppose that marks are made on the vertical line to show the
extremity of the shadow at each exact hour from sun-rise to
sun-set-these times being taken from a good fixed sun-dial,--then it is
obvious that the next year, on the _same date_, the sun's declination
being about the same, and the observer in about the same latitude, the
marks made the previous year will serve to tell the time all that day.

What we have said above was merely to make the principle of the
instrument clear, for it is evident that this mode of marking, which
would require a whole year's sunshine and hourly observation, cannot
be the method employed.

The positions of the marks are, in fact, obtained by calculation.
Corresponding to a given date, the declination of the sun is taken
from the almanac, and this, together with the latitude of the place
and the length of the style, will constitute the necessary data for
computing the length of the shadow, that is, the distance of the mark
below the style for each successive hour.

We have assumed above that the declination of the sun is the same at
the same date in different years. This is not quite correct, but, if
the dates be taken for the second year after leap year, the results
will be sufficiently approximate.

When all the hour-marks have been placed opposite to their respective
dates, then a continuous curve, joining the corresponding hour-points,
will serve to find the time for a day intermediate to those set down,
the lid being turned till the style occupy a proper position between
the two divisions. The horizontality of the surface on which the
instrument rests is a very necessary condition, especially in summer,
when, the shadow of the style being long, the extreme end will shift
rapidly for a small deviation from the vertical, and render the
reading uncertain. The dial can also be used by holding it up by a
small ring in the top of the lid, and probably the vertically is
better ensured in that way.

_Portable Dial on a Card._--This neat and very ingenious dial is
attributed by Ozanam to a Jesuit Father, De Saint Rigaud, and probably
dates from the early part of the 17th century. Ozanam says that it was
sometimes called the _capuchin_, from some fancied resemblance to a
cowl thrown back.

_Construction._--Draw a straight line ACB parallel to the top of the
card (fig. 8) and another DCE at right angles to it; with C as
centre, and any convenient radius CA, describe the semicircle AEB
below the horizontal. Divide the whole arc AEB into 12 equal parts at
the points r, s, t, &c., and through these points draw perpendiculars
to the diameter ACB; these lines will be the hour-lines, viz. the line
through r will be the XI ... I line, the line through s the X ... II
line, and so on; the hour-line of noon will be the point A itself; by
subdivision of the small arcs Ar, rs, st, &c., we may draw the
hour-lines corresponding to halves and quarters, but this only where
it can be done without confusion.

Draw ASD making with AC an angle equal to the latitude of the place,
and let it meet EC in D, through which point draw FDG at right angles
to AD.

With centre A, and any convenient radius AS, describe an arc of circle
RST, and graduate this arc by marking degree divisions on it,
extending from 0° at S to 23½° on each side at R and T. Next determine
the points on the straight line FDG where radii drawn from A to the
degree divisions on the arc would cross it, and carefully mark these
crossings.

The divisions of RST are to correspond to the sun's declination, south
declinations on RS and north declinations on ST. In the other
hemisphere of the earth this would be reversed; the north declinations
would be on the upper half.

Now, taking a second year after leap year (because the declinations of
that year are about the mean of each set of four years), find the days
of the month when the sun has these different declinations, and place
these dates, or so many of them as can be shown without confusion,
opposite the corresponding marks on FDG. Draw the _sun-line_ at the
top of the card parallel to the line ACB; and, near the extremity, to
the right, draw any small figure intended to form, as it were, a door
of which a b shall be the hinge. Care must be taken that this hinge is
exactly at right angles to the _sun-line_. Make a fine open slit c d
right through the card and extending from the hinge to a short
distance on the door,--the centre line of this slit coinciding
accurately with the _sun-line_. Now, cut the door completely through
the card; except, of course, along the hinge, which, when the card is
thick, should be partly cut through at the back, to facilitate the
opening. Cut the card right through along the line FDG, and pass a
thread carrying a little plummet W and a _very_ small bead P; the bead
having sufficient friction with the thread to retain any position when
acted on only by its own weight, but sliding easily along the thread
when moved by the hand. At the back of the card the thread terminates
in a knot to hinder it from being drawn through; or better, because
giving more friction and a better hold, it passes through the centre
of a small disk of card--a fraction of an inch in diameter--and, by a
knot, is made fast at the back of the disk.