The study of the motion of a material system is much assisted by the
use of a series of diagrams representing the configuration,
displacement and acceleration of the parts of the system.
_Diagram of Configuration._--In considering a material system it is
often convenient to suppose that we have a record of its position at
any given instant in the form of a diagram of configuration. The
position of any particle of the system is defined by drawing a
straight line or vector from the origin, or point of reference, to the
given particle. The position of the particle with respect to the
origin is determined by the magnitude and direction of this vector. If
in the diagram we draw from the origin (which need not be the same
point of space as the origin for the material system) a vector equal
and parallel to the vector which determines the position of the
particle, the end of this vector will indicate the position of the
particle in the diagram of configuration. If this is done for all the
particles we shall have a system of points in the diagram of
configuration, each of which corresponds to a particle of the material
system, and the relative positions of any pair of these points will be
the same as the relative positions of the material particles which
correspond to them.
We have hitherto spoken of two origins or points from which the
vectors are supposed to be drawn--one for the material system, the
other for the diagram. These points, however, and the vectors drawn
from them, may now be omitted, so that we have on the one hand the
material system and on the other a set of points, each point
corresponding to a particle of the system, and the whole representing
the configuration of the system at a given instant.
This is called a diagram of configuration.
_Diagram of Displacement._--Let us next consider two diagrams of
configuration of the same system, corresponding to two different
instants. We call the first the initial configuration and the second
the final configuration, and the passage from the one configuration to
the other we call the displacement of the system. We do not at present
consider the length of time during which the displacement was
effected, nor the intermediate stages through which it passed, but
only the final result--a change of configuration. To study this change
we construct a diagram of displacement.
Let A, B, C be the points in the initial diagram of configuration, and
A', B', C' be the corresponding points in the final diagram of
configuration. From o, the origin of the diagram of displacement, draw
a vector oa equal and parallel to AA', ob equal and parallel to BB',
oc to CC', and so on. The points a, b, c, &c., will be such that the
vector ab indicates the displacement of B relative to A, and so on.
The diagram containing the points a, b, c, &c., is therefore called
the diagram of displacement.
In constructing the diagram of displacement we have hitherto assumed
that we know the absolute displacements of the points of the system.
For we are required to draw a line equal and parallel to AA', which we
cannot do unless we know the absolute final position of A, with
respect to its initial position. In this diagram of displacement there
is therefore, besides the points a, b, c, &c., an _origin_, o, which
represents a point absolutely fixed in space. This is necessary
because the two configurations do not exist at the same time; and
therefore to express their relative position we require to know a
point which remains the same at the beginning and end of the time.
But we may construct the diagram in another way which does not assume
a knowledge of absolute displacement or of a point fixed in space.
Assuming any point and calling it a, draw ak parallel and equal to BA
in the initial configuration, and from k draw kb parallel and equal to
A'B' in the final configuration. It is easy to see that the position
of the point b relative to a will be the same by this construction as
by the former construction, only we must observe that in this second
construction we use only vectors such as AB, A'B', which represent the
relative position of points both of which exist simultaneously,
instead of vectors such as AA', BB', which express the position of a
point at one instant relative to its position at a former instant, and
which therefore cannot be determined by observation, because the two
ends of the vector do not exist simultaneously.
It appears therefore that the diagram of displacements, when drawn by
the first construction, includes an origin o, which indicates that we
have assumed a knowledge of absolute displacements. But no such point
occurs in the second construction, because we use such vectors only as
we can actually observe. Hence the diagram of displacements _without
an origin_ represents neither more nor less than all we can ever know
about the displacement of the material system.
_Diagram of Velocity._--If the relative velocities of the points of
the system are constant, then the diagram of displacement
corresponding to an interval of a unit of time between the initial and
the final configuration is called a diagram of relative velocity. If
the relative velocities are not constant, we suppose another system in
which the velocities are equal to the velocities of the given system
at the given instant and continue constant for a unit of time. The
diagram of displacements for this imaginary system is the required
diagram of relative velocities of the actual system at the given
instant. It is easy to see that the diagram gives the velocity of any
one point relative to any other, but cannot give the absolute velocity
of any of them.
_Diagram of Acceleration._--By the same process by which we formed the
diagram of displacements from the two diagrams of initial and final
configuration, we may form a diagram of changes of relative velocity
from the two diagrams of initial and final velocities. This diagram
may be called that of total accelerations in a finite interval of
time. And by the same process by which we deduced the diagram of
velocities from that of displacements we may deduce the diagram of
rates of acceleration from that of total acceleration.
We have mentioned this system of diagrams in elementary kinematics
because they are found to be of use especially when we have to deal
with material systems containing a great number of parts, as in the
kinetic theory of gases. The diagram of configuration then appears as
a region of space swarming with points representing molecules, and the
only way in which we can investigate it is by considering the number
of such points in unit of volume in different parts of that region,
and calling this the _density_ of the gas.
In like manner the diagram of velocities appears as a region
containing points equal in number but distributed in a different
manner, and the number of points in any given portion of the region
expresses the number of molecules whose velocities lie within given
limits. We may speak of this as the velocity-density.
_Diagrams of Stress._--Graphical methods are peculiarly applicable to
statical questions, because the state of the system is constant, so
that we do not need to construct a series of diagrams corresponding to
the successive states of the system. The most useful of these
applications, collectively termed Graphic Statics, relates to the
equilibrium of plane framed structures familiarly represented in
bridges and roof-trusses. Two diagrams are used, one called the
diagram of the frame and the other called the diagram of stress. The
structure itself consists of a number of separable pieces or links
jointed together at their extremities. In practice these joints have
friction, or may be made purposely stiff, so that the force acting at
the extremity of a piece may not pass exactly through the axis of the
joint; but as it is unsafe to make the stability of the structure
depend in any degree upon the stiffness of joints, we assume in our
calculations that all the joints are perfectly smooth, and therefore
that the force acting on the end of any link passes through the axis
of the joint.
The axes of the joints of the structure are represented by points in
the diagram of the frame. The link which connects two joints in the
actual structure may be of any shape, but in the diagram of the frame
it is represented by a straight line joining the points representing
the two joints. If no force acts on the link except the two forces
acting through the centres of the joints, these two forces must be
equal and opposite, and their direction must coincide with the
straight line joining the centres of the joints. If the force acting
on either extremity of the link is directed towards the other
extremity, the stress on the link is called pressure and the link is
called a "strut." If it is directed away from the other extremity, the
stress on the link is called tension and the link is called a "tie."
In this case, therefore, the only stress acting in a link is a
pressure or a tension in the direction of the straight line which
represents it in the diagram of the frame, and all that we have to do
is to find the magnitude of this stress. In the actual structure
gravity acts on every part of the link, but in the diagram we
substitute for the actual weight of the different parts of the link
two weights which have the same resultant acting at the extremities of
the link.
We may now treat the diagram of the frame as composed of links without
weight, but loaded at each joint with a weight made up of portions of
the weights of all the links which meet in that joint. If any link has
more than two joints we may substitute for it in the diagram an
imaginary stiff frame, consisting of links, each of which has only two
joints. The diagram of the frame is now reduced to a system of points,
certain pairs of which are joined by straight lines, and each point is
in general acted on by a weight or other force acting between it and
some point external to the system. To complete the diagram we may
represent these external forces as links, that is to say, straight
lines joining the points of the frame to points external to the frame.
Thus each weight may be represented by a link joining the point of
application of the weight with the centre of the earth.
But we can always construct an imaginary frame having its joints in
the lines of action of these external forces, and this frame, together
with the real frame and the links representing external forces, which
join points in the one frame to points in the other frame, make up
together a complete self-strained system in equilibrium, consisting of
points connected by links acting by pressure or tension. We may in
this way reduce any real structure to the case of a system of points
with attractive or repulsive forces acting between certain pairs of
these points, and keeping them in equilibrium. The direction of each
of these forces is sufficiently indicated by that of the line joining
the points, so that we have only to determine its magnitude. We might
do this by calculation, and then write down on each link the pressure
or the tension which acts in it.
We should in this way obtain a mixed diagram in which the stresses are
represented graphically as regards direction and position, but
symbolically as regards magnitude. But we know that a force may be
represented in a purely graphical manner by a straight line in the
direction of the force containing as many units of length as there are
units of force in the force. The end of this line is marked with an
arrow head to show in which direction the force acts. According to
this method each force is drawn in its proper position in the diagram
of configuration of the frame. Such a diagram might be useful as a
record of the result of calculation of the magnitude of the forces,
but it would be of no use in enabling us to test the correctness of
the calculation.
But we have a graphical method of testing the equilibrium of any set
of forces acting at a point. We draw in series a set of lines parallel
and proportional to these forces. If these lines form a closed polygon
the forces are in equilibrium. (See MECHANICS.) We might in this way
form a series of polygons of forces, one for each joint of the frame.
But in so doing we give up the principle of drawing the line
representing a force from the point of application of the force, for
all the sides of the polygon cannot pass through the same point, as
the forces do. We also represent every stress twice over, for it
appears as a side of both the polygons corresponding to the two joints
between which it acts. But if we can arrange the polygons in such a
way that the sides of any two polygons which represent the same stress
coincide with each other, we may form a diagram in which every stress
is represented in direction and magnitude, though not in position, by
a single line which is the common boundary of the two polygons which
represent the joints at the extremities of the corresponding piece of
the frame.
We have thus obtained a pure diagram of stress in which no attempt is
made to represent the configuration of the material system, and in
which every force is not only represented in direction and magnitude
by a straight line, but the equilibrium of the forces at any joint is
manifest by inspection, for we have only to examine whether the
corresponding polygon is closed or not.
The relations between the diagram of the frame and the diagram of
stress are as follows:--To every link in the frame corresponds a
straight line in the diagram of stress which represents in magnitude
and direction the stress acting in that link; and to every joint of
the frame corresponds a closed polygon in the diagram, and the forces
acting at that joint are represented by the sides of the polygon taken
in a certain cyclical order, the cyclical order of the sides of the
two adjacent polygons being such that their common side is traced in
opposite directions in going round the two polygons.
The direction in which any side of a polygon is traced is the
direction of the force acting on that joint of the frame which
corresponds to the polygon, and due to that link of the frame which
corresponds to the side. This determines whether the stress of the
link is a pressure or a tension. If we know whether the stress of any
one link is a pressure or a tension, this determines the cyclical
order of the sides of the two polygons corresponding to the ends of
the links, and therefore the cyclical order of all the polygons, and
the nature of the stress in every link of the frame.
_Reciprocal Diagrams._--When to every point of concourse of the lines
in the diagram of stress corresponds a closed polygon in the skeleton
of the frame, the two diagrams are said to be reciprocal.
The first extensions of the method of diagrams of forces to other
cases than that of the funicular polygon were given by Rankine in his
_Applied Mechanics_ (1857). The method was independently applied to a
large number of cases by W. P. Taylor, a practical draughtsman in the
office of J. B. Cochrane, and by Professor Clerk Maxwell in his
lectures in King's College, London. In the _Phil. Mag._ for 1864 the
latter pointed out the reciprocal properties of the two diagrams, and
in a paper on "Reciprocal Figures, Frames and Diagrams of Forces,"
_Trans. R.S. Edin._ vol. xxvi., 1870, he showed the relation of the
method to Airy's function of stress and to other mathematical methods.
Professor Fleeming Jenkin has given a number of applications of the
method to practice (_Trans. R.S. Edin._ vol. xxv.).
L. Cremona (_Le Figure reciproche nella statica grafica_, 1872)
deduced the construction of reciprocal figures from the theory of the
two components of a wrench as developed by Möbius. Karl Culmann, in
his _Graphische Statik_ (1st ed. 1864-1866, 2nd ed. 1875), made great
use of diagrams of forces, some of which, however, are not
reciprocal. Maurice Levy in his _Statique graphique_ (1874) has
treated the whole subject in an elementary but copious manner, and R.
H. Bow, in his _The Economics of Construction in Relation to Framed
Structures_ (1873), materially simplified the process of drawing a
diagram of stress reciprocal to a given frame acted on by a system of
equilibrating external forces.
[Illustration: FIG. 1.--Diagram of Configuration.]
Instead of lettering the joints of the frame, as is usually done, or
the links of the frame, as was the custom of Clerk Maxwell, Bow places
a letter in each of the polygonal areas enclosed by the links of the
frame, and also in each of the divisions of surrounding space as
separated by the lines of action of the external forces. When one link
of the frame crosses another, the point of apparent intersection of
the links is treated as if it were a real joint, and the stresses of
each of the intersecting links are represented twice in the diagram of
stress, as the opposite sides of the parallelogram which corresponds
to the point of intersection.
This method is followed in the lettering of the diagram of
configuration (fig. 1), and the diagram of stress (fig. 2) of the
linkwork which Professor Sylvester has called a quadruplane.
In fig. 1 the real joints are distinguished from the places where one
link appears to cross another by the little circles O, P, Q, R, S, T,
V. The four links RSTV form a "contraparallelogram" in which RS = TV
and RV = ST. The triangles ROS, RPV, TQS are similar to each other. A
fourth triangle (TNV), not drawn in the figure, would complete the
quadruplane. The four points O, P, N, Q form a parallelogram whose
angle POQ is constant and equal to [pi] - SOR. The product of the
distances OP and OQ is constant. The linkwork may be fixed at O. If
any figure is traced by P, Q will trace the inverse figure, but turned
round O through the constant angle POQ. In the diagram forces Pp, Qq
are balanced by the force Co at the fixed point. The forces Pp and Qq
are necessarily inversely as OP and OQ, and make equal angles with
those lines.
[Illustration: Fig. 2.--Diagram of Stress.]
Every closed area formed by the links or the external forces in the
diagram of configuration is marked by a letter which corresponds to a
point of concourse of lines in the diagram of stress. The stress in
the link which is the common boundary of two areas is represented in
the diagram of stress by the line joining the points corresponding to
those areas. When a link is divided into two or more parts by lines
crossing it, the stress in each part is represented by a different
line for each part, but as the stress is the same throughout the link
these lines are all equal and parallel. Thus in the figure the stress
in RV is represented by the four equal and parallel lines HI, FG, DE
and AB. If two areas have no part of their boundary in common the
letters corresponding to them in the diagram of stress are not joined
by a straight line. If, however, a straight line were drawn between
them, it would represent in direction and magnitude the resultant of
all the stresses in the links which are cut by any line, straight or
curved, joining the two areas. For instance the areas F and C in fig.
1 have no common boundary, and the points F and C in fig. 2 are not
joined by a straight line. But every path from the area F to the area
C in fig. 1 passes through a series of other areas, and each passage
from one area into a contiguous area corresponds to a line drawn in
the diagram of stress. Hence the whole path from F to C in fig. 1
corresponds to a path formed of lines in fig. 2 and extending from F
to C, and the resultant of all the stresses in the links cut by the
path is represented by FC in fig. 2.
Many examples of stress diagrams are given in the article on BRIDGES
(q.v.).
_Automatic Description of Diagrams._
There are many other kinds of diagrams in which the two co-ordinates
of a point in a plane are employed to indicate the simultaneous values
of two related quantities. If a sheet of paper is made to move, say
horizontally, with a constant known velocity, while a tracing point is
made to move in a vertical straight line, the height varying as the
value of any given physical quantity, the point will trace out a curve
on the paper from which the value of that quantity at any given time
may be determined. This principle is applied to the automatic
registration of phenomena of all kinds, from those of meteorology and
terrestrial magnetism to the velocity of cannon-shot, the vibrations
of sounding bodies, the motions of animals, voluntary and involuntary,
and the currents in electric telegraphs.
In Watt's indicator for steam engines the paper does not move with a
constant velocity, but its displacement is proportional to that of the
piston of the engine, while that of the tracing point is proportional
to the pressure of the steam. Hence the co-ordinates of a point of the
curve traced on the diagram represent the volume and the pressure of
the steam in the cylinder. The indicator-diagram not only supplies a
record of the pressure of the steam at each stage of the stroke of the
engine, but indicates the work done by the steam in each stroke by the
area enclosed by the curve traced on the diagram. (J. C. M.)
DIAL and DIALLING. Dialling, sometimes called gnomonics, is a branch of
applied mathematics which treats of the construction of sun-dials, that
is, of those instruments, either fixed or portable, which determine the
divisions of the day (Lat. _dies_) by the motion of the shadow of some
object on which the sun's rays fall. It must have been one of the
earliest applications of a knowledge of the apparent motion of the sun;
though for a long time men would probably be satisfied with the division
into morning and afternoon as marked by sun-rise, sun-set and the
greatest elevation.
_History._--The earliest mention of a sun-dial is found in Isaiah
xxxviii. 8: "Behold, I will bring again the shadow of the degrees which
is gone down in the _sun-dial_ of Ahaz ten degrees backward." The date
of this would be about 700 years before the Christian era, but we know
nothing of the character or construction of the instrument. The earliest
of all sun-dials of which we have any certain knowledge was the
hemicycle, or hemisphere, of the Chaldaean astronomer Berossus, who
probably lived about 300 B.C. It consisted of a hollow hemisphere placed
with its rim perfectly horizontal, and having a bead, or globule, fixed
in any way at the centre. So long as the sun remained above the horizon
the shadow of the bead would fall on the inside of the hemisphere, and
the path of the shadow during the day would be approximately a circular
arc. This arc, divided into twelve equal parts, determined twelve equal
intervals of time for that day. Now, supposing this were done at the
time of the solstices and equinoxes, and on as many intermediate days as
might be considered sufficient, and then curve lines drawn through the
corresponding points of division of the different arcs, the shadow of
the bead falling on one of these curve lines would mark a division of
time for that day, and thus we should have a sun-dial which would divide
each period of daylight into twelve equal parts. These equal parts were
called _temporary hours_; and, since the duration of daylight varies
from day to day, the temporary hours of one day would differ from those
of another; but this inequality would probably be disregarded at that
time, and especially in countries where the variation between the
longest summer day and the shortest winter day is much less than in our
climates.
The dial of Berossus remained in use for centuries. The Arabians, as
appears from the work of Albategnius, still followed the same
construction about the year A.D. 900. Four of these dials have in modern
times been found in Italy. One, discovered at Tivoli in 1746, is
supposed to have belonged to Cicero, who, in one of his letters, says
that he had sent a dial of this kind to his villa near Tusculum. The
second and third were found in 1751--one at Castel-Nuovo and the other
at Rignano; and a fourth was found in 1762 at Pompeii. G. H. Martini in
his _Abhandlungen von den Sonnenuhren der Alten_ (Leipzig, 1777), says
that this dial was made for the latitude of Memphis; it may therefore
be the work of Egyptians, perhaps constructed in the school of
Alexandria.
Herodotus recorded that the Greeks derived from the Babylonians the use
of the gnomon, but the great progress made by the Greeks in geometry
enabled them in later times to construct dials of great complexity, some
of which remain to us, and are proof not only of extensive knowledge but
also of great ingenuity.
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.