HOME
The Info List - Sundial



--- Advertisement ---


A SUNDIAL is a device that tells the time of day by the apparent position of the Sun in the sky . In the narrowest sense of the word, it consists of a flat plate (the _dial_) and a _gnomon _, which casts a shadow onto the dial. As the Sun appears to move across the sky, the shadow aligns with different _hour-lines_, which are marked on the dial to indicate the time of day. The _style_ is the time-telling edge of the gnomon, though a single point or _nodus_ may be used. The gnomon casts a broad shadow; the shadow of the style shows the time. The gnomon may be a rod, wire, or elaborately decorated metal casting. The style must be parallel to the axis of the Earth\'s rotation for the sundial to be accurate throughout the year. The style's angle from horizontal is equal to the sundial's geographical latitude .

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 Sun * 3 History * 4 Terminology * 5 Sundials in the Southern Hemisphere

* 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 Equation of time correction

* 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 Diptych (tablet) sundial * 11.2 Multiface dials * 11.3 Prismatic dials

* 12 Unusual sundials

* 12.1 Benoy dials * 12.2 Bifilar sundial * 12.3 Digital sundial * 12.4 Globe dial * 12.5 Noon marks * 12.6 Sundial cannon

* 13 Meridian lines * 14 Sundial mottoes * 15 Using a sundial as a compass * 16 See also * 17 Notes

* 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 North . Portable dials are self-aligning: for example, it may have two dials that operate on different principles, such as a horizontal and analemmatic dial, mounted together on one plate. In these designs, their times agree only when the plate is aligned properly.

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 Greenwich , England but within the same time-zone, shows an _earlier_ time than the official time. It will show "noon" after the official noon has passed, since the sun passes overhead later. This correction is often made by rotating the hour-lines by an angle equal to the difference in longitudes.

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 Aberdeen, Scotland or Sitka, Alaska .

The principles of sundials are understood most easily from the Sun 's apparent motion. The Earth rotates on its axis, and revolves in an elliptical orbit around the Sun. An excellent approximation assumes that the Sun revolves around a stationary Earth on the celestial sphere , which rotates every 24 hours about its celestial axis. The celestial axis is the line connecting the celestial poles . Since the celestial axis is aligned with the axis about which the Earth rotates, the angle of the axis with the local horizontal is the local geographical latitude .

Unlike the fixed stars , the Sun changes its position on the celestial sphere, being at a positive declination in spring and summer, and at a negative declination in autumn and winter, and having exactly zero declination (i.e., being on the celestial equator ) at the equinoxes . The Sun's celestial longitude also varies, changing by one complete revolution per year. The path of the Sun on the celestial sphere is called the ecliptic . The ecliptic passes through the twelve constellations of the zodiac in the course of a year. Sundial in Singapore Botanic Gardens . The fact that Singapore is located almost at the equator is reflected in its design.

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 North or South Poles) a circle .

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 History of sundials . World's oldest sundial, from Egypt's Valley of the Kings (c. 1500 BC)

The earliest sundials known from the archaeological record are shadow clocks (1500 BC) from ancient Egyptian astronomy and Babylonian astronomy . Presumably, humans were telling time from shadow-lengths at an even earlier date, but this is hard to verify. In roughly 700 BC, the Old Testament describes a sundial — the "dial of Ahaz" mentioned in Isaiah 38:8 and 2 Kings 20:11. The Roman writer Vitruvius lists dials and shadow clocks known at that time. A canonical sundial is one that indicates the canonical hours of liturgical acts. Such sundials were used from the 7th to the 14th centuries by the members of religious communities. Italian astronomer Giovanni Padovani published a treatise on the sundial in 1570, in which he included instructions for the manufacture and laying out of mural (vertical) and horizontal sundials. Giuseppe Biancani 's _Constructio instrumenti ad horologia solaria_ (ca. 1620) discusses how to make a perfect sundial. They have been commonly used since the 16th century.

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 Perth , Australia . Magnify to see that the hour marks run counterclockwise. Note graph of Equation of Time, needed to correct sundial readings.

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 Northern Hemisphere becomes a vertical direct north sundial in the Southern Hemisphere . To position a horizontal sundial correctly, one has to find true North or South . The same process can be used to do both. The gnomon, set to the correct latitude, has to point to the true South in the Southern hemisphere as in the Northern Hemisphere it has to point to the true North. The hour numbers also run in opposite directions, so on a horizontal dial they run counterclockwise rather than clockwise.

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 Perth , Australia , which is at latitude 32 degrees South, would function properly if it were mounted on a south-facing vertical wall at latitude 58 (i.e. 90–32) degrees North, which is slightly further North than Perth, Scotland . The surface of the wall in Scotland would be parallel with the horizontal ground in Australia (ignoring the difference of longitude), so the sundial would work identically on both surfaces. Correspondingly, the hour marks, which run counterclockwise on a horizontal sundial in the southern hemisphere, also do so on a vertical sundial in the northern hemisphere. (See the first two illustrations at the top of this article.) On horizontal northern-hemisphere sundials, and on vertical southern-hemisphere ones, the hour marks run clockwise.

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 North and its _height_ (its angle with the horizontal) must equal the local latitude. To adjust the style height, the sundial can often be tilted slightly "up" or "down" while maintaining the style's north-south alignment.

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 Alaska , China , and Spain . For more details and examples, see Skewing of time zones .

EQUATION OF TIME CORRECTION

_ The Equation of Time - above the axis the equation of time is positive, and a sundial will appear fast_ relative to a clock showing local mean time. The opposites are true below the axis. Main article: Equation of time The Whitehurst it does not depend on the local latitude of the sundial. It does, however, change over long periods of time, centuries or more, because of slow variations in the Earth's orbital and rotational motions. Therefore, tables and graphs of the equation of time that were made centuries ago are now significantly incorrect. The reading of an old sundial should be corrected by applying the present-day equation of time, not one from the period when the dial was made.

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 Universal Time . Sunquest sundial, designed by Richard L. Schmoyer, at the Mount Cuba Observatory in Greenville, Delaware

The Sunquest sundial , designed by Richard L. Schmoyer in the 1950s, uses an analemmic inspired gnomon to cast a shaft of light onto an equatorial time-scale crescent. Sunquest is adjustable for latitude and longitude, automatically correcting for the equation of time, rendering it "as accurate as most pocket watches".

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 Time was not used. After the invention of good clocks, sundials were still considered to be correct, and clocks usually incorrect. The Equation of Time was used in the opposite direction from today, to apply a correction to the time shown by a clock to make it agree with sundial time. Some elaborate "equation clocks ", such as one made by Joseph Williamson in 1720, incorporated mechanisms to do this correction automatically. (Williamson's clock may have been the first-ever device to use a differential gear.) Only after about 1800 was uncorrected clock time considered to be "right", and sundial time usually "wrong", so the Equation of Time became used as it is today.

SUNDIALS WITH FIXED AXIAL GNOMON

The 1959 Carefree sundial in Carefree, Arizona has a 62-foot (19 m) gnomon, possibly the largest sundial in the United States.

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 North and South, and making an angle with the horizontal equal to the geographical latitude. This axis is aligned with the celestial poles , which is closely, but not perfectly, aligned with the (present) pole star Polaris . For illustration, the celestial axis points vertically at the true North Pole , where it points horizontally on the equator . At Jaipur , a famous location for sundials, gnomons are raised 26°55" above horizontal, reflecting the local latitude.

On any given day, the Sun appears to rotate uniformly about this axis, at about 15° per hour, making a full circuit (360°) in 24 hours. A linear gnomon aligned with this axis will cast a sheet of shadow (a half-plane) that, falling opposite to the Sun, likewise rotates about the celestial axis at 15° per hour. The shadow is seen by falling on a receiving surface that is usually flat, but which may be spherical, cylindrical, conical or of other shapes. If the shadow falls on a surface that is symmetrical about the celestial axis (as in an armillary sphere, or an equatorial dial), the surface-shadow likewise moves uniformly; the hour-lines on the sundial are equally spaced. However, if the receiving surface is not symmetrical (as in most horizontal sundials), the surface shadow generally moves non-uniformly and the hour-lines are not equally spaced; one exception is the Lambert dial described below.

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: Schema for horizontal dials and Equation of time

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 Roman numerals ). If the hour-lines are not all marked in a single day, the clock must be adjusted every day or two to take account of the variation of the equation of time.

EQUATORIAL SUNDIALS

An equatorial sundial in the Forbidden City , Beijing. 39°54′57″N 116°23′25″E / 39.9157°N 116.3904°E / 39.9157; 116.3904 ( Forbidden City equatorial sundial) The gnomon points true North and its angle with horizontal equals the local latitude . Closer inspection of the full-size image reveals the "spider-web" of date rings and hour-lines.

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 London dial and Whitehurst "> Horizontal sundial in Minnesota . June 17, 2007 at 12:21. 44°51′39.3″N, 93°36′58.4″W

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 North ) on the plane, and t_ is the number of hours before or after noon. For example, the angle H H {displaystyle H_{H}} of the 3pm hour-line would equal the arctangent of sin L, since tan 45° = 1. When L equals 90° (at the North Pole ), the horizontal sundial becomes an equatorial sundial; the style points straight up (vertically), and the horizontal plane is aligned with the equatorial plane; the hour-line formula becomes H H {displaystyle H_{H}} = 15° × t, as for an equatorial dial. A horizontal sundial at the Earth's equator , where L equals 0°, would require a (raised) horizontal style and would be an example of a polar sundial (see below). Detail of horizontal sundial outside Kew Palace in London, United Kingdom

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 North and its angle with the horizontal equals the sundial's geographical latitude L. A sundial designed for one latitude can be adjusted for use at another latitude by tilting its base upwards or downwards by an angle equal to the difference in latitude. For example, a sundial designed for a latitude of 40° can be used at a latitude of 45°, if the sundial plane is tilted upwards by 5°, thus aligning the style with the Earth's rotational axis. Many ornamental sundials are designed to be used at 45 degrees north. Some mass-produced garden sundials fail to correctly calculate the _hourlines_ and so can never be corrected. A local standard time zone is nominally 15 degrees wide, but may be modified to follow geographic or political boundaries. A sundial can be rotated around its style (which must remain pointed at the celestial pole) to adjust to the local time zone. In most cases, a rotation in the range of 7.5 degrees east to 23 degrees west suffices. This will introduce error in sundials that do not have equal hour angles. To correct for daylight saving time , a face needs two sets of numerals or a correction table. An informal standard is to have numerals in hot colors for summer, and in cool colors for winter. Since the hour angles are not evenly spaced, the equation of time corrections cannot be made via rotating the dial plate about the gnomon axis. These types of dials usually have an equation of time correction tabulation engraved on their pedestals or close by. Horizontal dials are commonly seen in gardens, churchyards and in public areas.

VERTICAL SUNDIALS

Two vertical dials at Houghton Hall Norfolk UK 52°49′39″N 0°39′27″E / 52.827469°N 0.657616°E / 52.827469; 0.657616 ( Houghton Hall vertical sundials) . The left and right dials face South and East, respectively. Both styles are parallel, their angle to the horizontal equaling the latitude. The East-facing dial is a polar dial with parallel hour-lines, the dial-face being parallel to the style.

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 West are called _vertical direct dials_. It is widely believed, and stated in respectable publications, that a vertical dial cannot receive more than twelve hours of sunlight a day, no matter how many hours of daylight there are. However, there is an exception. Vertical sundials in the tropics which face the nearer pole (e.g. north facing in the zone between the Equator and the Tropic of Cancer) can actually receive sunlight for more than 12 hours from sunrise to sunset for a short period around the time of the summer solstice. For example, at latitude 20 degrees North, on June 21, the sun shines on a north-facing vertical wall for 13 hours, 21 minutes. Vertical sundials which do _not_ face directly South (in the northern hemisphere) may receive significantly less than twelve hours of sunlight per day, depending on the direction they do face, and on the time of year. For example, a vertical dial that faces due East can tell time only in the morning hours; in the afternoon, the sun does not shine on its face. Vertical dials that face due East or West are _polar dials_, which will be described below. Vertical dials that face North are uncommon, because they tell time only during the spring and summer, and do not show the midday hours except in tropical latitudes (and even there, only around midsummer). For non-direct vertical dials — those that face in non-cardinal directions — the mathematics of arranging the style and the hour-lines becomes more complicated; it may be easier to mark the hour lines by observation, but the placement of the style, at least, must be calculated first; such dials are said to be _declining dials_. "Double" sundials in Nové Město nad Metují , Czech Republic; the observer is facing almost due north.

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 North and its angle with the horizontal will equal the sundial's geographical latitude; on a direct south dial, its angle with the vertical face of the dial will equal the colatitude , or 90° minus the latitude.

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 South (far left) shows all the hours from 6am to 6pm, and has converging hour-lines symmetrical about the noon hour-line. By contrast, a West-facing dial (far right) is polar, with parallel hour lines, and shows only hours after noon. At the intermediate orientations of South-Southwest, Southwest, and West-Southwest , the hour lines are asymmetrical about noon, with the morning hour-lines ever more widely spaced. Two sundials, a large and a small one, at Fatih Mosque , Istanbul dating back to the late 16th century. It is on the southwest facade with an azimuth angle of 52° N.

A _declining dial_ is any non-horizontal, planar dial that does not face in a cardinal direction, such as (true) North , South , East or West . As usual, the gnomon's style is aligned with the Earth's rotational axis, but the hour-lines are not symmetrical about the noon hour-line. For a vertical dial, the angle H VD {displaystyle H_{text{VD}}} between the noon hour-line and another hour-line is given by the formula below. Note that H VD {displaystyle H_{text{VD}}} is defined positive in the clockwise sense w.r.t. the upper vertical hour angle; and that its conversion to the equivalent solar hour requires careful consideration of which quadrant of the sundial that it belongs in. tan H VD = cos L cos D cot ( 15 t ) s o sin L sin D {displaystyle tan H_{text{VD}}={frac {cos L}{cos Dcot(15^{circ }times t)-s_{o}sin Lsin D}}}

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 South ( D = 0 {displaystyle D=0^{circ }} ), this formula reduces to the formula given above for vertical south-facing dials, i.e. tan H V = cos L tan ( 15 t ) {displaystyle tan H_{text{V}}=cos Ltan(15^{circ }times t)}

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 South or North ( D = 0 {displaystyle D=0^{circ }} or D = 180 {displaystyle D=180^{circ }} , respectively), the angle B = 0 {displaystyle B=0^{circ }} and the substyle is aligned with the noon hour-line.

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 North or true South) and is neither horizontal nor vertical nor equatorial. For example, such a sundial might be found on a roof that was not oriented in a cardinal direction.

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