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.
[Illustration: FIG. 7.]
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.
[Illustration: FIG. 8.]
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.
To complete the construction,--with the centres F and G, and radii FA
and GA, draw the two arcs AY and AZ which will limit the hour-lines;
for in an observation the bead will always be found between them. The
forenoon and afternoon hours may then be marked as indicated in the
figure. The dial does not of itself discriminate between forenoon and
afternoon; but extraneous circumstances, as, for instance, whether the
sun is rising or falling, will settle that point, except when close to
noon, where it will always be uncertain.
To _rectify_ the dial (using the old expression, which means to
prepare the dial for an observation),--open the small door, by turning
it about its hinge, till it stands well out in front. Next, set the
thread in the line FG opposite the day of the month, and stretching it
over the point A, slide the bead P along till it exactly coincide
with A.
To find the hour of the day,--hold the dial in a vertical position in
such a way that its plane may pass through the sun. The verticality is
ensured by seeing that the bead rests against the card without
pressing. Now gradually tilt the dial (without altering its vertical
plane), until the central line of sunshine, passing through the open
slit of the door, just falls along the sun-line. The hour-line against
which the bead P then rests indicates the time.
[Illustration: FIG. 9.]
The _sun-line_ drawn above has always, so far as we know, been used as
a _shadow-line_. The upper edge of the rectangular door was the
prolongation of the line, and, the door being opened, the dial was
gradually tilted until the shadow cast by the upper edge exactly
coincided with it. But this shadow tilts the card one-quarter of a
degree more than the sun-line, because it is given by that portion of
the sun which just appears above the edge, that is, by the upper limb
of the sun, which is one-quarter of a degree higher than the centre.
Now, even at some distance from noon, the sun will sometimes take a
considerable time to rise one-quarter of a degree, and by so much time
will the indication of the dial be in error.
The central line of light which comes through the open slit will be
free from this error, because it is given by light from the centre of
the sun.
The card-dial deserves to be looked upon as something more than a mere
toy. Its ingenuity and scientific accuracy give it an educational
value which is not to be measured by the roughness of the results
obtained.
The theory of this instrument is as follows:--Let H (fig. 9) be the
point of suspension of the plummet at the time of observation, so that
the angle DAH is the north declination of the sun,--P, the bead,
resting against the hour-line VX. Join CX, then the angle ACX is the
hour-angle from noon given by the bead, and we have to prove that this
hour-angle is the correct one corresponding to a north latitude DAC, a
north declination DAH and an altitude equal to the angle which the
_sun-line_, or its parallel AC, makes with the horizontal. The angle
PHQ will be equal to the altitude, if HQ be drawn parallel to DC, for
the pair of lines HQ, HP will be respectively at right angles to the
sun-line and the horizontal.
Draw PQ and HM parallel to AC, and let them meet DCE in M and N
respectively.
Let HP and its equal HA be represented by a. Then the following values
will be readily deduced from the figure:--