A SUNDIAL is a device that tells the time of day by the apparent
position of the
In a broader sense, a sundial is any device that uses the Sun's altitude or azimuth (or both) to show the time. In addition to their time-telling function, sundials are valued as decorative objects, literary metaphors, and objects of mathematical study. It is common for inexpensive, mass-produced decorative sundials to have incorrectly aligned gnomons and hour-lines, which cannot be adjusted to tell correct time. CONTENTS * 1 Introduction
* 2 Apparent motion of the
* 6 Adjustments to calculate clock time from a sundial reading * 6.1 Summer (daylight saving) time correction
* 6.2 Time-zone (longitude) correction
* 6.3
* 7 Sundials with fixed axial gnomon * 7.1 Empirical hour-line marking * 7.2 Equatorial sundials * 7.3 Horizontal sundials * 7.4 Vertical sundials * 7.5 Polar dials * 7.6 Vertical declining dials * 7.7 Reclining dials * 7.8 Declining-reclining dials/ Declining-inclining dials * 7.8.1 Empirical method * 7.9 Spherical sundials * 7.10 Cylindrical, conical, and other non-planar sundials * 8 Movable-gnomon sundials * 8.1 Universal equinoctial ring dial * 8.2 Analemmatic sundials * 8.3 Foster-Lambert dials * 9 Altitude-based sundials * 9.1 Human shadows * 9.2 Shepherd dials – Timesticks * 9.3 Ring dials * 9.4 Card dials (Capuchin dials) * 10 Nodus-based sundials * 10.1 Reflection sundials * 11 Multiple dials * 11.1
* 12 Unusual sundials * 12.1 Benoy dials
* 12.2
* 13 Meridian lines
* 14
* 18 References * 18.1 Citations * 18.2 Bibliography * 19 External links * 19.1 National organisations * 19.2 Historical * 19.3 Other INTRODUCTION There are several different types of sundials. Some sundials use a shadow or the edge of a shadow while others use a line or spot of light to indicate the time. The shadow-casting object, known as a _gnomon_, may be a long thin rod or other object with a sharp tip or a straight edge. Sundials employ many types of gnomon. The gnomon may be fixed or moved according to the season. It may be oriented vertically, horizontally, aligned with the Earth's axis, or oriented in an altogether different direction determined by mathematics. Given that sundials use light to indicate time, a line of light may be formed by allowing the sun's rays through a thin slit or focusing them through a cylindrical lens . A spot of light may be formed by allowing the sun's rays to pass through a small hole or by reflecting them from a small circular mirror. Sundials also may use many types of surfaces to receive the light or shadow. Planes are the most common surface, but partial spheres , cylinders , cones and other shapes have been used for greater accuracy or beauty. Sundials differ in their portability and their need for orientation.
The installation of many dials requires knowing the local latitude ,
the precise vertical direction (e.g., by a level or plumb-bob), and
the direction to true
Sundials indicate the local solar time , unless corrected for some other time. To obtain the official clock time, three types of corrections need to be made. Firstly, the orbit of the Earth is not perfectly circular and its rotational axis not perfectly perpendicular to its orbit. The sundial's indicated solar time thus varies from clock time by small amounts that change throughout the year. This correction — which may be as great as 15 minutes — is described by the equation of time . A sophisticated sundial, with a curved style or hour lines, may incorporate this correction. Often instead, simpler sundials are used, with a small plaque that gives the offsets at various times of the year. Secondly, the solar time must be corrected for the longitude of the
sundial relative to the longitude of the official time zone. For
example, a sundial located _west_ of
Lastly, to adjust for daylight saving time , the sundial must shift the time away from solar time by some amount, usually an hour. This correction may be made in the adjustment plaque, or by numbering the hour-lines with two sets of numbers. APPARENT MOTION OF THE SUN Top view of an equatorial sundial. The hour lines are spaced
equally about the circle, and the shadow of the gnomon (a thin
cylindrical rod) rotates uniformly. The height of the gnomon is 5/12
the outer radius of the dial. This animation depicts the motion of the
shadow from 3 a.m. to 9 p.m. (not accounting for Daylight Saving Time)
on or around Solstice, when the sun is at its highest declination
(roughly 23.5°). Sunrise and sunset occur at 3am and 9pm,
respectively, on that day at geographical latitudes near 57.05°,
roughly the latitude of
The principles of sundials are understood most easily from the
Unlike the fixed stars , the
This model of the Sun's motion helps to understand sundials. If the shadow-casting gnomon is aligned with the celestial poles , its shadow will revolve at a constant rate, and this rotation will not change with the seasons. This is the most common design. In such cases, the same hour lines may be used throughout the year. The hour-lines will be spaced uniformly if the surface receiving the shadow is either perpendicular (as in the equatorial sundial) or circular about the gnomon (as in the armillary sphere ). In other cases, the hour-lines are not spaced evenly, even though the
shadow rotates uniformly. If the gnomon is _not_ aligned with the
celestial poles, even its shadow will not rotate uniformly, and the
hour lines must be corrected accordingly. The rays of light that graze
the tip of a gnomon, or which pass through a small hole, or reflect
from a small mirror, trace out a cone aligned with the celestial
poles. The corresponding light-spot or shadow-tip, if it falls onto a
flat surface, will trace out a conic section , such as a hyperbola ,
ellipse or (at the
This conic section is the intersection of the cone of light rays with the flat surface. This cone and its conic section change with the seasons, as the Sun's declination changes; hence, sundials that follow the motion of such light-spots or shadow-tips often have different hour-lines for different times of the year. This is seen in shepherd's dials, sundial rings, and vertical gnomons such as obelisks. Alternatively, sundials may change the angle or position (or both) of the gnomon relative to the hour lines, as in the analemmatic dial or the Lambert dial. HISTORY For more details on this topic, see
The earliest sundials known from the archaeological record are shadow
clocks (1500 BC) from ancient
TERMINOLOGY A London type horizontal dial . The western edge of the gnomon is used as the style before noon, the eastern edge after that time. The changeover causes a discontinuity, the noon gap, in the time scale. In general, sundials indicate the time by casting a shadow or throwing light onto a surface known as a _dial face_ or _dial plate_. Although usually a flat plane, the dial face may also be the inner or outer surface of a sphere, cylinder, cone, helix, and various other shapes. The time is indicated where a shadow or light falls on the dial face, which is usually inscribed with hour lines. Although usually straight, these hour lines may also be curved, depending on the design of the sundial (see below). In some designs, it is possible to determine the date of the year, or it may be required to know the date to find the correct time. In such cases, there may be multiple sets of hour lines for different months, or there may be mechanisms for setting/calculating the month. In addition to the hour lines, the dial face may offer other data—such as the horizon, the equator and the tropics—which are referred to collectively as the dial furniture. The entire object that casts a shadow or light onto the dial face is known as the sundial's _gnomon_. However, it is usually only an edge of the gnomon (or another linear feature) that casts the shadow used to determine the time; this linear feature is known as the sundial's _style_. The style is usually aligned parallel to the axis of the celestial sphere, and therefore is aligned with the local geographical meridian. In some sundial designs, only a point-like feature, such as the tip of the style, is used to determine the time and date; this point-like feature is known as the sundial's _nodus_. Some sundials use both a style and a nodus to determine the time and date. The gnomon is usually fixed relative to the dial face, but not always; in some designs such as the analemmatic sundial, the style is moved according to the month. If the style is fixed, the line on the dial plate perpendicularly beneath the style is called the _substyle_, meaning "below the style". The angle the style makes with the plane of the dial plate is called the substyle height, an unusual use of the word _height_ to mean an _angle_. On many wall dials, the substyle is not the same as the noon line (see below). The angle on the dial plate between the noon line and the substyle is called the _substyle distance_, an unusual use of the word _distance_ to mean an _angle_. By tradition, many sundials have a motto . The motto is usually in the form of an epigram: sometimes sombre reflections on the passing of time and the brevity of life, but equally often humorous witticisms of the dial maker. A dial is said to be _equiangular_ if its hour-lines are straight and spaced equally. Most equiangular sundials have a fixed gnomon style aligned with the Earth's rotational axis, as well as a shadow-receiving surface that is symmetrical about that axis; examples include the equatorial dial, the equatorial bow, the armillary sphere, the cylindrical dial and the conical dial. However, other designs are equiangular, such as the Lambert dial, a version of the analemmatic dial with a moveable style. SUNDIALS IN THE SOUTHERN HEMISPHERE Southern-hemisphere sundial in
A sundial at a particular latitude in one hemisphere must be reversed
for use at the opposite latitude in the other hemisphere. A vertical
direct south sundial in the
Sundials which are designed to be used with their plates horizontal
in one hemisphere can be used with their plates vertical at the
complementary latitude in the other hemisphere. For example, the
illustrated sundial in
ADJUSTMENTS TO CALCULATE CLOCK TIME FROM A SUNDIAL READING The most common reason for a sundial to differ greatly from clock
time is that the sundial has not been oriented correctly or its hour
lines have not been drawn correctly. For example, most commercial
sundials are designed as _horizontal sundials_ as described above. To
be accurate, such a sundial must have been designed for the local
geographical latitude and its style must be parallel to the Earth's
rotational axis; the style must be aligned with true
SUMMER (DAYLIGHT SAVING) TIME CORRECTION Some areas of the world practice daylight saving time , which shifts the official time, usually by one hour. This shift must be added to the sundial's time to make it agree with the official time. TIME-ZONE (LONGITUDE) CORRECTION A standard time zone covers roughly 15° of longitude, so any point within that zone which is not on the reference longitude (generally a multiple of 15°) will experience a difference from standard time equal to 4 minutes of time per degree. For illustration, sunsets and sunrises are at a much later "official" time at the western edge of a time-zone, compared to sunrise and sunset times at the eastern edge. If a sundial is located at, say, a longitude 5° west of the reference longitude, its time will read 20 minutes slow, since the sun appears to revolve around the Earth at 15° per hour. This is a constant correction throughout the year. For equiangular dials such as equatorial, spherical or Lambert dials, this correction can be made by rotating the dial surface by an angle equaling the difference in longitude, without changing the gnomon position or orientation. However, this method does not work for other dials, such as a horizontal dial; the correction must be applied by the viewer. At its most extreme, time zones can cause official noon, including
daylight savings, to occur up to three hours early (the sun is
actually directly overhead at official clock time of 3 pm). This
occurs in the far west of
EQUATION OF TIME CORRECTION _ The Equation of
In some sundials, the equation of time correction is provided as a
plaque affixed to the sundial. In more sophisticated sundials,
however, the equation can be incorporated automatically. For example,
some equatorial bow sundials are supplied with a small wheel that sets
the time of year; this wheel in turn rotates the equatorial bow,
offsetting its time measurement. In other cases, the hour lines may be
curved, or the equatorial bow may be shaped like a vase, which
exploits the changing altitude of the sun over the year to effect the
proper offset in time. A _heliochronometer_ is a precision sundial
first devised in about 1763 by Philipp Hahn and improved by Abbé
Guyoux in about 1827. It corrects apparent solar time to mean solar
time or another standard time . Heliochronometers usually indicate the
minutes to within 1 minute of
The
An analemma may be added to many types of sundials to correct apparent solar time to mean solar time or another standard time . These usually have hour lines shaped like "figure eights" (analemmas ) according to the equation of time . This compensates for the slight eccentricity in the Earth's orbit and the tilt of the Earth's axis that causes up to a 15-minute variation from mean solar time. This is a type of dial furniture seen on more complicated horizontal and vertical dials. Prior to the invention of accurate clocks, in the mid-17th Century,
sundials were the only timepieces in common use, and were considered
to tell the "right" time. The Equation of
SUNDIALS WITH FIXED AXIAL GNOMON The 1959
The most commonly observed sundials are those in which the
shadow-casting style is fixed in position and aligned with the Earth's
rotational axis, being oriented with true
On any given day, the
Some types of sundials are designed with a fixed gnomon that is not aligned with the celestial poles like a vertical obelisk. Such sundials are covered below under the section, "Nodus-based sundials". EMPIRICAL HOUR-LINE MARKING See also:
The formulas shown in the paragraphs below allow the positions of the
hour-lines to be calculated for various types of sundial. In some
cases, the calculations are simple; in others they are extremely
complicated. There is an alternative, simple method of finding the
positions of the hour-lines which can be used for many types of
sundial, and saves a lot of work in cases where the calculations are
complex. This is an empirical procedure in which the position of the
shadow of the gnomon of a real sundial is marked at hourly intervals.
The equation of time must be taken into account to ensure that the
positions of the hour-lines are independent of the time of year when
they are marked. An easy way to do this is to set a clock or watch so
it shows "sundial time" which is standard time , plus the equation
of time on the day in question. The hour-lines on the sundial are
marked to show the positions of the shadow of the style when this
clock shows whole numbers of hours, and are labelled with these
numbers of hours. For example, when the clock reads 5:00, the shadow
of the style is marked, and labelled "5" (or "V" in
EQUATORIAL SUNDIALS An equatorial sundial in the
The distinguishing characteristic of the _equatorial dial_ (also called the _equinoctial dial_) is the planar surface that receives the shadow, which is exactly perpendicular to the gnomon's style. This plane is called equatorial, because it is parallel to the equator of the Earth and of the celestial sphere. If the gnomon is fixed and aligned with the Earth's rotational axis, the sun's apparent rotation about the Earth casts a uniformly rotating sheet of shadow from the gnomon; this produces a uniformly rotating line of shadow on the equatorial plane. Since the sun rotates 360° in 24 hours, the hour-lines on an equatorial dial are all spaced 15° apart (360/24). H E = 15 t ( h o u r s ) {displaystyle H_{E}=15^{circ }times t(hours)} The uniformity of their spacing makes this type of sundial easy to construct. If the dial plate material is opaque, both sides of the equatorial dial must be marked, since the shadow will be cast from below in winter and from above in summer. With translucent dial plates (e.g. glass) the hour angles need only be marked on the sun-facing side, although the hour numberings (if used) need be made on both sides of the dial, owing to the differing hour schema on the sun-facing and sun-backing sides. Another major advantage of this dial is that equation of time (EoT) and daylight saving time (DST) corrections can be made by simply rotating the dial plate by the appropriate angle each day. This is because the hour angles are equally spaced around the dial. For this reason, an equatorial dial is often a useful choice when the dial is for public display and it is desirable to have it show the true local time to reasonable accuracy. The EoT correction is made via the relation : C o r r e c t i o n = E o T ( m i n u t e s ) + 60 D S T ( h o u r s ) 4 {displaystyle Correction^{circ }={frac {EoT(minutes)+60times Delta DST(hours)}{4}}} Near the equinoxes in spring and autumn, the sun moves on a circle that is nearly the same as the equatorial plane; hence, no clear shadow is produced on the equatorial dial at those times of year, a drawback of the design. A _nodus_ is sometimes added to equatorial sundials, which allows the sundial to tell the time of year. On any given day, the shadow of the nodus moves on a circle on the equatorial plane, and the radius of the circle measures the declination of the sun. The ends of the gnomon bar may be used as the nodus, or some feature along its length. An ancient variant of the equatorial sundial has only a nodus (no style) and the concentric circular hour-lines are arranged to resemble a spider-web. HORIZONTAL SUNDIALS For a more detailed description of such a dial, see
In the _horizontal sundial_ (also called a _garden sundial_), the plane that receives the shadow is aligned horizontally, rather than being perpendicular to the style as in the equatorial dial. Hence, the line of shadow does not rotate uniformly on the dial face; rather, the hour lines are spaced according to the rule tan H H = sin L tan ( 15 t ) {displaystyle tan H_{H}=sin Ltan(15^{circ }times t)} Or in other terms: H H = tan 1 {displaystyle H_{H}=tan ^{-1}} where L is the sundial's geographical latitude (and the angle the
style makes with horizontal), H H {displaystyle H_{H}} _ is
the angle between a given hour-line and the noon hour-line (which
always points towards true
The chief advantages of the horizontal sundial are that it is easy to
read, and the sun lights the face throughout the year. All the
hour-lines intersect at the point where the gnomon's style crosses the
horizontal plane. Since the style is aligned with the Earth's
rotational axis, the style points true
VERTICAL SUNDIALS Two vertical dials at
In the common _vertical dial_, the shadow-receiving plane is aligned vertically; as usual, the gnomon's style is aligned with the Earth's axis of rotation. As in the horizontal dial, the line of shadow does not move uniformly on the face; the sundial is not _equiangular_. If the face of the vertical dial points directly south, the angle of the hour-lines is instead described by the formula tan H V = cos L tan ( 15 t ) {displaystyle tan H_{V}=cos Ltan(15^{circ }times t)} where L is the sundial's geographical latitude , H V {displaystyle H_{V}} _ is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and t_ is the number of hours before or after noon. For example, the angle H V {displaystyle H_{V}} _ of the 3pm hour-line would equal the arctangent of cos L, since tan 45° = 1. Interestingly, the shadow moves counter-clockwise_ on a South-facing vertical dial, whereas it runs clockwise on horizontal and equatorial north-facing dials. Dials with faces perpendicular to the ground and which face directly
South, North, East, or
Vertical dials are commonly mounted on the walls of buildings, such
as town-halls, cupolas and church-towers, where they are easy to see
from far away. In some cases, vertical dials are placed on all four
sides of a rectangular tower, providing the time throughout the day.
The face may be painted on the wall, or displayed in inlaid stone; the
gnomon is often a single metal bar, or a tripod of metal bars for
rigidity. If the wall of the building faces _toward_ the South, but
does not face due South, the gnomon will not lie along the noon line,
and the hour lines must be corrected. Since the gnomon's style must be
parallel to the Earth's axis, it always "points" true
POLAR DIALS Polar sundial at Melbourne Planetarium In _polar dials_, the shadow-receiving plane is aligned _parallel_ to the gnomon-style. Thus, the shadow slides sideways over the surface, moving perpendicularly to itself as the sun rotates about the style. As with the gnomon, the hour-lines are all aligned with the Earth's rotational axis. When the sun's rays are nearly parallel to the plane, the shadow moves very quickly and the hour lines are spaced far apart. The direct East- and West-facing dials are examples of a polar dial. However, the face of a polar dial need not be vertical; it need only be parallel to the gnomon. Thus, a plane inclined at the angle of latitude (relative to horizontal) under the similarly inclined gnomon will be a polar dial. The perpendicular spacing _X_ of the hour-lines in the plane is described by the formula X = H tan ( 15 t ) {displaystyle X=Htan(15^{circ }times t)} where _H_ is the height of the style above the plane, and _t_ is the time (in hours) before or after the center-time for the polar dial. The center time is the time when the style's shadow falls directly down on the plane; for an East-facing dial, the center time will be 6am, for a West-facing dial, this will be 6pm, and for the inclined dial described above, it will be noon. When _t_ approaches ±6 hours away from the center time, the spacing _X_ diverges to +∞ ; this occurs when the sun's rays become parallel to the plane. VERTICAL DECLINING DIALS Effect of declining on a sundial's hour-lines. A vertical dial,
at a latitude of 51° N, designed to face due
A _declining dial_ is any non-horizontal, planar dial that does not
face in a cardinal direction, such as (true)
where L {displaystyle L} _ is the sundial's geographical
latitude ; t_ is the time before or after noon; D {displaystyle
D} is the angle of declination from true south , defined as positive
when east of south; and s o {displaystyle s_{o}} is a
switch integer for the dial orientation. A partly south-facing dial
has an s o {displaystyle s_{o}} value of + 1; those partly
north-facing, a value of -1. When such a dial faces
When a sundial is not aligned with a cardinal direction, the substyle of its gnomon is not aligned with the noon hour-line. The angle B {displaystyle B} between the substyle and the noon hour-line is given by the formula tan B = sin D cot L {displaystyle tan B=sin Dcot L} If a vertical sundial faces true
The height of the gnomon, that is the angle the style makes to the plate, G {displaystyle G} , is given by : sin G = cos D cos L {displaystyle sin G=cos Dcos L} RECLINING DIALS Vertical reclining dial in the Southern Hemisphere, facing due north, with hyperbolic declination lines and hour lines. Ordinary vertical sundial at this latitude (between tropics) could not produce a declination line for the summer solstice. The sundials described above have gnomons that are aligned with the Earth's rotational axis and cast their shadow onto a plane. If the plane is neither vertical nor horizontal nor equatorial, the sundial is said to be _reclining_ or _inclining_. Such a sundial might be located on a South-facing roof, for example. The hour-lines for such a sundial can be calculated by slightly correcting the horizontal formula above tan H R V = cos ( L + R ) tan ( 15 t ) {displaystyle tan H_{RV}=cos(L+R)tan(15^{circ }times t)} where R {displaystyle R} _ is the desired angle of reclining relative to the local vertical, L is the sundial's geographical latitude, H R V {displaystyle H_{RV}} is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and t_ is the number of hours before or after noon. For example, the angle H R V {displaystyle H_{RV}} of the 3pm hour-line would equal the arctangent of cos (L + R), since tan 45° = 1. When R equals 0° (in other words, a South-facing vertical dial), we obtain the vertical dial formula above. Some authors use a more specific nomenclature to describe the orientation of the shadow-receiving plane. If the plane's face points downwards towards the ground, it is said to be _proclining_ or _inclining_, whereas a dial is said to be _reclining_ when the dial face is pointing away from the ground. Many authors also often refer to reclined, proclined and inclined sundials in general as inclined sundials. It is also common in the latter case to measure the angle of inclination relative to the horizontal plane on the sun side of the dial. In such texts, since I = 90° + R, the hour angle formula will often be seen written as : tan H R V = sin ( L + I ) tan ( 15 t ) {displaystyle tan H_{RV}=sin(L+I)tan(15^{circ }times t)} The angle between the gnomon style and the dial plate, B, in this type of sundial is : B = 90 ( L + R ) {displaystyle B=90^{circ }-(L+R)} Or : B = 180 ( L + I ) {displaystyle B=180^{circ }-(L+I)} DECLINING-RECLINING DIALS/ DECLINING-INCLINING DIALS Some sundials both decline and recline, in that their
shadow-receiving plane is not oriented with a cardinal direction (such
as true
The formulae describing the spacing of the hour-lines on such dials are rather more complicated than those for simpler dials. There are various solution approaches, including some using the methods of rotation matrices, and some making a 3D model of the reclined-declined plane and its vertical declined counterpart plane, extracting the geometrical relationships between the hour angle components on both these planes and then reducing the trigonometric algebra. One system of formulas for Reclining-Declining sundials: (as stated by Fennewick) The angle H RD {displaystyle H_{text{RD}}} between the noon hour-line and another hour-line is given by the formula below. Note that H RD {displaystyle H_{text{RD}}} advances counterclockwise with respect to the zero hour angle for those dials that are partly south-facing and clockwise for those that are north-facing. tan H RD = cos R cos L sin R sin L cos D s o sin R sin D cot ( 15 t ) cos D cot ( 15 t ) s o sin D sin L {displaystyle tan H_{text{RD}}={frac {cos Rcos L-sin Rsin Lcos D-s_{o}sin Rsin Dcot(15^{circ }times t)}{cos Dcot(15^{circ }times t)-s_{o}sin Dsin L}}} within the parameter ranges : D L ) {displaystyle -90^{circ } H RD = sin I cos L + cos I sin L cos D + s o cos I sin D cot ( 15 t ) cos D cot ( 15 t ) s o sin D sin L |