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# Sundial

[suhn-dahy-uhl, -dahyl]

A sundial is a device that measures time by the position of the Sun. In common designs such as the horizontal sundial, the sun casts a shadow from its style (a thin rod or a sharp, straight edge) onto a flat surface marked with lines indicating the hours of the day. As the sun moves across the sky, the shadow-edge progressively aligns with different hour-lines on the plate. Such designs rely on the style being aligned with the axis of the Earth's rotation. Hence, if such a sundial is to tell the correct time, the style must point towards true North (not the north or south magnetic pole) and the style's angle with horizontal must equal the sundial's geographical latitude. However, many sundials do not fit this description, and operate on different principles.

## Introduction

Sundials can be categorized in several ways. First, some sundials use a spot of light, or a line of light, to indicate the time, where others use the edge or tip of a shadow. In the former case, the spot of light may be formed by allowing the sun's rays through a small hole or reflecting them from a small circular mirror; a line of light may be formed by allowing the rays through a thin slit or focusing them through a cylindrical lens. In the other case, the shadow-casting object — the sundial's gnomon — may be a thin rod, or any object with a sharp tip or a straight edge. Second, 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. Third, sundials may use many types of surfaces to receive the spot or line of light, the shadow-tip or shadow-edge. Planes are the most common surface, but partial spheres, cylinders, cones and even more complicated shapes have been used for greater accuracy or intriguing aesthetics. Fourth, sundials differ in their portability and their need of 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. In contrast, other dials are self-aligning; for example, two dials that operate on different principles, such as a horizontal and analemmatic dial, may be mounted together on one plate, such that their times agree only when the plate is aligned properly. Sundials indicate the local solar time, unless otherwise corrected. To obtain the standard clock time, three types of corrections need to be made. First, the solar time needs to be corrected for the longitude of the sundial relative to the longitude at which the official time zone is defined. 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, since the sundial is further in the west. This correction is often made by rotating the hour-lines by an angle equaling the difference in longitudes. Second, the practice of daylight saving time shifts the official time away from solar time by an hour or, in rare cases, by another amount. This correction is usually made by numbering the hour-lines with two sets of numbers. Third, the orbit of the Earth is not perfectly circular and its rotational axis not perfectly perpendicular to its orbit, which together produce small variations in the sundial time throughout the year. This correction — which may be as great as 15 minutes — is described by the equation of time. A more sophisticated sundial design is required to incorporate this correction automatically; alternatively, a small plaque can be affixed to the sundial giving the offsets at various times of the year.

## Apparent motion of the Sun

The principles of sundials can be understood most easily from an ancient model of the Sun's motion. Science has established that the Earth rotates on its axis, and revolves in an elliptic orbit about the Sun; however, meticulous astronomical observations and physics experiments were required to establish this. For navigational and sundial purposes, it is an excellent approximation to assume that the Sun revolves around a stationary Earth on the celestial sphere, which rotates every 23 hours and 56 minutes about its celestial axis, the line connecting the celestial poles. Since the celestial axis is aligned with the axis about which the Earth rotates, its angle with the local horizontal equals the local geographical latitude. Unlike the fixed stars, the Sun changes its position on the celestial sphere, being at positive declination in summer, at negative declination in winter, and having exactly zero declination (i.e., being on the celestial equator) at the equinoxes. The path of the Sun on the celestial sphere is known as the ecliptic, which passes through the twelve constellations of the zodiac in the course of a year.

This model of the Sun's motion helps to understand the principles of 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 perhaps the most commonly seen design and, in such cases, the same set of 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 circularly symmetric about the gnomon (as in the armillary sphere). In other cases, the hour-lines are not spaced evenly, even though the shadow is rotating 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 which reflect from a small mirror, trace out a cone that is 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, as seen in shepherd's dials, sundial rings, and vertical gnomons such as obelisks. Alternatively, sundials may change the angle and/or position of the gnomon relative to the hour lines, as in the analemmatic dial or the Lambert dial.

## Terminology

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 the 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 with the axis of the celestial sphere, and therefore 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 perpendicularly with 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 somber 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

One half of the Earth's globe is either north or south of the Equator. A sundial at a particular latitude in one hemisphere must be reversed for use at the reciprocal latitude in the other hemisphere. A vertical direct south sundial in the Northern Hemisphere becomes a vertical direct north sundial in the Southern Hemisphere ; an opposite. 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. Also the hour numbers on a horizontal dial run anti-clockwise rather than clockwise. Sundials are not as common in the Southern hemisphere as in the North. One proposition for this is that when Europeans arrived the mechanical clock was accurate enough for their purposes of time keeping and there was no need to erect sundials.

Most of the sundials described below use shadow to indicate time, whether it be the shadow-edge of the style, or the shadow-point of the nodus. However, light may be used in equivalent ways. Nodus-based sundials may use a small hole or mirror to isolate a single ray of light; the former are sometimes called aperture dials. The oldest example is perhaps the antiborean sundial (antiboreum), a spherical nodus-based sundial that faces true North; a ray of sunlight enters from the South through a small hole located at the sphere's pole and falls on the hour and date lines inscribed within the sphere, which resemble lines of longitude and latitude, respectively, on a globe.

Light may also be used to replace the shadow-edge of a gnomon. Whereas the style usually casts a sheet of shadow, an equivalent sheet of light can be created by allowing the sun's rays through a thin slit, reflecting them from a long, slim mirror (usually half-cylindrical), or focusing them through a cylindrical lens. For illustration, the Benoy Dial uses a cylindrical lens to create a sheet of light, which falls as a line on the dial surface. Benoy dials can be seen throughout Great Britain, such as

## Sundials with fixed axial gnomon

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, the gnomons of such sundials are raised 26°55" above horizontal, since that is the local latitude.

Some types of sundials are designed with a fixed gnomon that is not aligned with the celestial poles, such as a vertical obelisk. Such sundials are covered below under the section, "Nodus-based sundials".

### Equatorial sundials

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). The uniformity of their spacing makes this type of sundial easy to construct. Both sides of the equatorial dial must be marked, since the shadow will be cast from below in winter and from above in summer. 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

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 theta = sin lambda tan(15^{circ} times t)

where λ is the sundial's geographical latitude, θ 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 θ of the 3pm hour-line would equal the arctangent of sin(λ), since tan(45°) = 1. When λ 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 place; the hour-line formula becomes θ = 15° × t, as for an equatorial dial. However, a horizontal sundial is impractical on the Earth's equator, where λ equals 0°, the style would lie flat in the plane and cast no shadow.

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 λ. A sundial designed for one latitude can be used in another latitude, provided that the sundial is tilted 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.

### Vertical sundials

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 theta = cos lambda tan(15^{circ} times t)

where λ is the sundial's geographical latitude, θ 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 θ of the 3pm hour-line would equal the arctangent of cos(λ), since tan(45°) = 1. Interestingly, the shadow moves counter-clockwise on a South-facing vertical dial, whereas it runs clockwise on horizontal and equatorial dials.

Dials that face due South, North, East or West are called vertical direct dials. If the face of a vertical dial does not face due South, the hours of sunlight that the dial receives may be limited. For example, a vertical dial that faces due East will 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 rarely used, since they tell time only before 6am or after 6pm, by local solar time. For non-direct vertical dials — those that face in non-cardinal directions — the mathematics of arranging the hour-lines becomes more complicated, and is often done by observation; such dials are said to be declining dials.

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 does not face in a cardinal direction such as due South, the hour lines must be corrected. Since the gnomon's style is aligned with the Earth's rotation axis, it points true North and its angle with the horizontal equals the sundial's geographical latitude; consequently, its angle with the vertical face of the dial equals the colatitude, or 90°-latitude.

### Polar dials

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^{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 infinity; this occurs when the sun's rays become parallel to the plane.

### Vertical declining dials

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 θ between the noon hour-line and another hour-line is given by the formula


tan theta = frac{cos lambda}{sin eta sin lambda + cos eta cot(15^{circ} times t)}

where λ is the sundial's geographical latitude, t is the time before or after noon, and η is the angle of declination from true South. When such a dial faces South (η=0°), this formula reduces to the formula given above, tan θ = cos λ tan(15° × 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 β between the substyle and the noon hour-line is given by the formula


tan beta = sin eta cot lambda ,

If a vertical sundial faces true South or North (η=0° or 180°, respectively), the correction β=0° and the substyle is aligned with the noon hour-line.

### Reclining dials

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 theta = sin(lambda + chi) tan(15^{circ} times t)

where χ is the desired angle of reclining, λ is the sundial's geographical latitude, θ 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 θ of the 3pm hour-line would equal the arctangent of sin(λ+χ), since tan(45°) = 1. When χ equals 90° (in other words, a South-facing vertical dial), we obtain the vertical formula above, since sin(λ+90°) = cos(λ).

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.

### Reclining-declining dials

Some sundials both decline and recline, in that their shadow-receiving plane is not oriented with a cardinal direction (such as true North) 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 complicated. The angle θ between the noon hour-line and another hour-line has two components θ = θ1 + θ2, described by the formulae


tan theta_{1} = tan eta tan chi ,


tan theta_{2} = frac{cos chi cos eta sin lambda + sin chi cos lambda - cos chi sin eta cot(15^{circ} times t)}{sin eta sin lambda + cos eta cot(15^{circ} times t)}

where λ is the sundial's geographical latitude, t is the time before or after noon, and χ and η are the angles of inclination and declination, respectively.

As in the simpler declining dial, the gnomon-substyle is not aligned with the noon hour-line. The general formula for the angle β between the substyle and the noon-line is given by


tan beta = sin chi sin eta frac{tan lambda cos chi + sin chi cos eta}{cos chi - tan lambda cos eta sin chi}

### Spherical sundials

The surface receiving the shadow need not be a plane, but can can have any shape, provided that the sundial maker is willing to mark the hour-lines. If the style is aligned with the Earth's rotational axis, a spherical shape is convenient since the hour-lines are equally spaced, as they are on the equatorial dial above; the sundial is equiangular. This is the principle behind the armillary sphere and the equatorial bow sundial. However, some equiangular sundials — such as the Lambert dial described below — are based on other principles.

In the equatorial bow sundial, the gnomon is a bar, slot or stretched wire parallel to the celestial axis. The face is a semicircle (corresponding to the equator of the sphere, with markings on the inner surface. This pattern, built a couple of meters wide out of temperature-invariant steel invar, was used to keep the trains running on time in France before World War I.

Among the most precise sundials ever made are two equatorial bows constructed of marble found in Yantra mandir. This collection of sundials and other astronomical instruments was built by Maharaja Jai Singh II at his then-new capital of Jaipur, India between 1727 and 1733. The larger equatorial bow is called the Samrat Yantra (The Supreme Instrument); standing at 27 meters, its shadow moves visibly at 1 mm per second, or roughly a hand's breadth (6 cm) every minute.

### Cylindrical, conical, and other non-planar sundials

Other non-planar surfaces may be used to receive the shadow of the gnomon. For example, the gnomon may be aligned with the celestial poles and located also along the symmetry axis of a cone or a cylinder. Due to the symmetry, the hour lines on such surfaces will be equally spaced, as on an equatorial dial or an armillary sphere. The conical dial is very old, and was the basis for one type of chalice sundial; the style was a vertical pin within a conical goblet, within which were inscribed the hour lines.

As an elegant alternative, the gnomon may be located on the circumference of a cylinder or sphere, rather than at its center of symmetry. In that case, the hour lines are again spaced equally, but at double the usual angle, due to the geometrical inscribed angle theorem. This is the basis of some modern sundials, but it was also used in ancient times; in one type, the edges of a half-cylindrical gnomon served as the styles.

Just as the armillary sphere is largely open for easy viewing of the dial, such non-planar surfaces need not be complete. For example, a cylindrical dial could be rendered as a helical ribbon-like surface, with a thin gnomon located either along its center or at its periphery.

The most common reason for a sundial to differ 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 sundials must have been designed for that latitude and their style must be parallel to the Earth's rotational axis; the style must be aligned with true North and its angle with the horizontal must equal the local geographical latitude. To align the style, the sundial can sometimes be tilted slightly on its north south axis.

### 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 time zone can cover 60° of longitude, so any point within that zone will experience time difference with the reference longitude, equivalent to 4 minutes of time per degree. For illustration, sunsets and sunrises occur at a later "official" time in the far western edge of a time-zone, compared to those observed at the far eastern edge. As an example, if a sundial is located at 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 the equatorial, spherical or Lambert dials, this correction can be made by rotating the dial surface by an angle equalling 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.

### Equation of time correction

Although the Sun appears to rotate nearly uniformly about the Earth, it is not perfectly uniform, due to the ellipticity of the Earth's orbit (the fact that the Earth's orbit about the Sun is not perfectly circular) and the tilt (obliquity) of the Earth's rotational axis relative to the plane of its orbit. Therefore, sundials time varies from standard clock time. On four days of the year, the correction is effectively zero, but on others, it can be as much as a quarter-hour early or late. The amount of correction is described by the equation of time. This correction is universal; it does not depend on the local latitude of the sundial.

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 that corrects apparent solar time to mean solar time or another standard time. Heliochronometers usually indicate the minutes to within 1 minute of Universal Time. See this discussion of the limits of Sundial Accuracy

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 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.

## Movable-gnomon sundials

In addition to the sundials have a gnomon that is designed to be moved over the course of the year. In other words, the position of the gnomon relative to the center of the hour lines can vary. The advantage of such dials is that the gnomon need not be aligned with the celestial poles and may even be perfectly vertical (the analemmatic dial). A second advantage is that such dials, when combined with a fixed-gnomon sundial, allow the user to determine true North with no other aid; the two sundials are correctly aligned if and only if the time on the two sundials agrees. This is a useful property for portable sundials.

### Universal equinoctial ring dial

A universal equinoctial ring dial (sometimes called a ring dial for brevity, although the term is ambiguous) is a portable version of an armillary sundial, or was inspired by the mariner's astrolabe. It was likely invented by William Oughtred around 1600 and became common throughout Europe.

In its simplest form, the style is a thin slit that allows the sun's rays to fall on the hour-lines of a equatorial ring. As usual, the style is aligned with the Earth's axis; to do this, the user may orient the dial towards true North and suspend the ring dial vertically from the appropriate point on the meridian ring. Such dials may be made self-aligning with the addition of a more complicated central bar, instead of a simple slit-style. This bar could pivot about its end points and held a perforated slider that was positioned to the month and day according to a scale scribed on the bar. The time was determined by rotating the bar towards the sun so that the light shining through the hole fell on the equatorial ring. This forced the user to rotate the instrument, which had the effect of aligning the instrument's vertical ring with the meridian.

When not in use, the equatorial and meridian rings can be folded together into a small disk.

In 1610, Edward Wright created the sea ring, which mounted a universal ring dial over a magnetic compass. This permitted mariners to determine the time and magnetic variation in a single step.

### Analemmatic sundials

An analemmatic sundial uses a vertical gnomon and its hour lines are the vertical projection of the hour lines of a circular equatorial sundial onto a flat plane. Therefore, the analemmatic sundial is an ellipse, where the short axis is aligned North-South and the long axis is aligned East-West. The noon hour line points true North, where as the hour lines for 6am and 6pm point due West and East, respectively; the ratio of the short to long axes equals the sine sin(Φ) of the local geographical latitude, denoted Φ. All the hour lines converge to a single center; the angle θ of a given hour line with the noon hour is given by the formula


tan theta = frac{tan (15^{circ} times t)}{sin phi }

where t is the time (in hours) before or after noon. However, the vertical gnomon does not always stand at the center of the hour lines; rather, to show the correct time, the gnomon must be moved northwards from the center by the distance


Y = W cos phi tan delta ,

where W is the width of the ellipse and δ is the Sun's declination at that time of year. The declination measures how far the sun is above the celestial equator; at the equinoxes, δ=0 whereas it equals roughly ±23.5° at the summer and winter solstices.

Accurate dials of this type fit nicely in a public square, using a ball at the tip of a flagpole as the nodus, with the face painted on or inlaid in the pavement. A less accurate version of the sundial is to lay out the hour marks on concrete, and then let the user stand in a square marked with the month. In middle latitudes, the ellipse with the hour-marks should be about six meters wide, so the shadow of the head of the beholder will fall near it most of the time. The month squares are arranged to correct the sundial for the time of year. The user's head then forms the gnomon of the dial. If the sundial is molded into the concrete, it resists vandalism and is engaging and reasonably accurate.

### Lambert dials

The Lambert dial is another movable-gnomon sundial. In contrast to the elliptical analemmatic dial, the Lambert dial is circular with evenly spaced hour lines, making it an equiangular sundial, similar to the equatorial, spherical, cylindrical and conical dials described above. The gnomon of a Lambert dial is neither vertical or aligned with the Earth's rotational axis; rather, it is tilted northwards by an angle α = 45° - (Φ/2), where Φ is the geographical latitude. Thus, a Lambert dial located at latitude 40° would have a gnomon tilted away from vertical by 25° in a northerly direction. To read the correct time, the gnomon must also be moved northwards by a distance


Y = R tan alpha tan delta ,

where R is the radius of the Lambert dial and δ again indicates the Sun's declination for that time of year.

## Altitude-based sundials

Altitude dials measure the height of the sun in the sky, rather than its rotation about the celestial axis. They are not oriented towards true North, but rather towards the sun and generally held vertically. The sun's elevation is indicted by the position of a nodus, either the shadow-tip of a gnomon, or a spot of light. The time is read from where the nodus falls on a set of hour-curves that vary with the time of year. Since the sun's altitude is the same at times equally spaced about noon (e.g., 9am and 3pm), the user had to know whether it were morning or afternoon. Many of these dials are portable and simple to use, although they are not well-suited for travelers, since their hour-curves are specific for a given latitude.

The length of a human shadow (or of any vertical object) can be used to measure the sun's elevation and, thence, the time. The Venerable Bede gave a table for estimating the time from the length of one's shadow in feet, on the assumption that a monk's height is six times the length of his foot. Such shadow lengths will vary with the geographical latitude and with the time of year. For example, the shadow length at noon is short in summer months, and long in winter months.

Chaucer evokes this method a few times in his Canterbury Tales, as in his Parson's Tale

It was four o'clock according to my guess,
Since eleven feet, a little more or less,
my shadow at the time did fall,
Considering that I myself am six feet tall.

An equivalent type of sundial using a vertical rod of fixed length is known as a backstaff dial.

### Shepherd dials –Timesticks

A shepherd's dial — also known as a shepherds' column dial, pillar dial, cylinder dial or chilindre — is a portable cylindrical sundial with a gnomon that juts out perpendicularly. When held vertically and pointed at the Sun, it measures the altitude of the Sun, from which the hour can be calculated if the day is known. The hour curves are inscribed on the cylinder for reading the time. Shepherd's dials are sometimes hollow, so that the gnomon can be stored within when not in use.

Shepherd's dials appear in several works of literature. For example, in the Chaucer's Canterbury Tales, the monk says,

"Goth now your wey," quod he, "al stille and softe,
And lat us dyne as sone as that ye may;
for by my chilindre it is pryme of day."

Similarly, the shepherd's dial is evoked in Shakespeare's Henry VI, Part 3,

O God! methinks it were a happy life
To be no better than a homely swain;
To sit upon a hill, as I do now,
To carve out dials, quaintly, point by point,
Thereby to see the minutes, how they run--
How many makes the hour full complete,
How many hours brings about the day,
How many days will finish up the year,
How many years a mortal man may live.

The cylindrical shepherd's dial can be unrolled into a flat plate. In one simple version, the front and back of the plate each have three columns, corresponding to pairs of months with roughly the same solar declination (June-July, May-August, April-September, March-October, February-November, and January-December). The top of each column has a hole for inserting the shadow-casting gnomon, a peg. Only two times are marked on the column below, one for noon and the other for mid-morning/mid-afternoon.

Timesticks, clock spear, or shepherds' time stick, are based on the same principles as dials. The time stick is carved with eight vertical time scales for a different period of the year, each bearing a time scale calculated according to the relative amount of daylight during the different months of the year. Any reading depends not only on the time of day but also on the latitude and time of year. A peg gnomon is inserted at the top in the appropriate hole or face for the season of the year, and turned to the Sun so that the shadow falls directly down the scale. Its end displays the time.

### Ring dials

In a ring dial (also known as an Aquitaine or a perforated ring dial), the ring is hung vertically and oriented sideways towards the sun. A beam of light passes through a small hole in the ring and falls on hour-curves that are inscribed on the inside of the ring.

### Card dials (Capuchin dials)

Card dials are another form of altitude dial. A card is aligned edge-on with the sun and tilted so that a ray of light passes through an aperture onto a specified spot, thus determining the sun's altitude. A weighted string hangs vertically downwards from a hole in the card, and carries a bead or knot. The position of the bead on the hour-lines of the card gives the time. In more sophisticated versions such as the Capuchin dial, there is only one set of hour-lines, i.e., the hour lines do not vary with the seasons. Instead, the position of the hole from which the weighted string hangs is varied according to the season.

## Nodus-based sundials

Another type of sundial follows the motion of a single point of light or shadow, which may be called the nodus. For example, the sundial may follow the sharp tip of a gnomon's shadow, e.g., the shadow-tip of a vertical obelisk (e.g., the Solarium Augusti) or the tip of the horizontal marker in a shepherd's dial. Alternatively, sunlight may be allowed to pass through a small hole or reflected from a small (e.g., coin-sized) circular mirror, forming a small spot of light whose position may be followed. In such cases, the rays of light trace out a cone over the course of a day; when the rays fall on a surface, the path followed is the intersection of the cone with that surface. Most commonly, the receiving surface is a geometrical plane, so that the path of the shadow-tip or light-spot traces out a conic section such as a hyperbola or an ellipse. The collection of hyperbolae was called a pelekonon (axe) by the Greeks, because it resembles a double-bladed ax, narrow in the center (near the noonline) and flaring out at the ends (early morning and late evening hours).

### Reflection sundials

Isaac Newton developed a convenient and inexpensive sundial, in which a small mirror is placed on the sill of a south-facing window. The mirror acts like a nodus, casting a single spot of light on the ceiling. Depending on the geographical latitude and time of year, the light-spot follows a conic section, such as the hyperbolae of the pelikonon. Using the ceiling as a sundial surface has several advantages; it exploits unused space and may be very accurate, if large.

## Multiple dials

Sundials are sometimes combined into multiple dials. If two or more dials that operate on different principles — say, such as an analemmatic dial and a horizontal or vertical dial — are combined, the resulting multiple dial becomes self-aligning. In other words, the direction of true North need not be determined; the dials are oriented correctly when they read the same time. This is a significant advantage in portable dials. However, the most common forms combine dials based on the same principle, and thus are not self-aligning.

### Diptych (tablet) sundial

The diptych consisted of two small flat faces, joined by a hinge. Diptychs usually folded into little flat boxes suitable for a pocket. The gnomon was a string between the two faces. When the string was tight, the two faces formed both a vertical and horizontal sundial. These were made of white ivory, inlaid with black lacquer markings. The gnomons were black braided silk, linen or hemp string. With a knot or bead on the string as a nodus, and the correct markings, a diptych (really any sundial large enough) can keep a calendar well-enough to plant crops. A common error describes the diptych dial as self-aligning. This is not correct for diptych dials consisting of a horizontal and vertical dial using a string gnomon between faces, no matter the orientation of the dial faces. Since the string gnomon is continuous, the shadows must meet at the hinge; hence, any orientation of the dial will show the same time on both dials.

A common multiple dial is to place sundials on every face of a Platonic solid, usually a cube. Extremely ornate sundials can be composed in this way, by applying a sundial to every surface of a solid object. In some cases, the sundials are formed as hollows in a solid object, e.g., a cylindrical hollow aligned with the Earth's rotational axis (in which the edges play the role of styles) or a spherical hollow in the ancient tradition of the hemisphaerium or the antiboreum. (See the History section below.) In some cases, these multiface dials are small enough to sit on a desk, whereas in others, they are large stone monuments.

Such multiface dials have the advantage of receiving light (and, thus, telling time) at every hour of the day. They can also be designed to give the time in different time-zones simultaneously. However, they are generally not self-aligning, since their various dials generally use the same principle to tell time, that of a gnomon-style aligned with the Earth's axis of rotation. Self-aligning dials require that at least two independent principles are used to tell time, e.g., a horizontal dial (in which the style is aligned with the Earth's axis) and an analemmatic dial (in which the style is not). In many cases, the multiface dials are erected never to be moved and, thus, need be aligned only once.

### Prismatic dials

Prismatic dials are a special case of polar dials, in which the sharp edges of a prism of a concave polygon serve as the styles and the sides of the prism receive the shadow. Examples include a three-dimensional cross or star of David on gravestones.

## Unusual sundials

### Bifilar sundial

Discovered by the German mathematician Hugo Michnik, the bifilar sundial has two non-intersecting threads parallel to the dial. Usually the second thread is orthogonal to the first.

### Digital sundial

A digital sundial uses light and shadow to 'write' the time in numerals rather than marking time with position. One such design uses two parallel masks to screen sunlight into patterns appropriate for the time of day.

### Analog calculating sundial

A horizontal sundial with a face cut on a cardioid keeps clock time, while still resembling a conventional garden sundial. The cardioid shape connects the intersections between the solar-time marks of a conventional sundial, and the equal-angles of a true clock-time face. The place where The shadow crosses the cardioid's edge, and the clock time can be read from the underlying clock-time dial. The sundial is adjusted for daylight saving time by rotating the underlying equal-angle clock-time face. The sun-time face does not move.

### Globe dial

The globe dial is a sphere aligned with the Earth's rotational axis, and equipped with a spherical vane. Similar to sundials with a fixed axial style, a globe dial determines the time from the Sun's azimuthal angle in its apparent rotation about the earth. This angle can be determined by rotating the vane to give the smallest shadow. This style of sundial was popularized by Thomas Jefferson at his home in Monticello.

### Noon marks

The simplest sundials do not give the hours, but rather note the exact moment of 12:00 noon. In centuries past, such dials were used to correct mechanical clocks, which were sometimes so inaccurate as to lose or gain significant time in a single day.

In U.S. colonial-era houses, a noon-mark can often be found carved into a floor or windowsill. Such marks indicate local noon, and they provide a simple and accurate time reference for households that do not possess accurate clocks. In modern times, some Asian countries, post offices have set their clocks from a precision noon-mark. These in turn provided the times for the rest of the society. The typical noon-mark sundial was a lens set above an analemmatic plate. The plate has an engraved figure-eight shape., which corresponds to plotting the equation of time (described above) versus the solar declination. When the edge of the sun's image touches the part of the shape for the current month, this indicates that it is 12:00 noon.

### Noon cannon

A noon cannon, sometimes called a 'meridian cannon', is a specialized sundial that is designed to create an 'audible noonmark', by automatically igniting at noon a quantity of gunpowder. These were novelties rather than precision sundials, sometimes installed in parks in Europe mainly in the late 18th or early 19th century. They typically consist of a horizontal sundial, which has in addition to a gnomon a suitably mounted lens, set up to focus the rays of the sun at exactly noon on the firing pan of a miniature cannon loaded with some gunpowder (but no bullet). To function properly the position and angle of the lens must be adjusted seasonally.

### Bernhardt's sundial

Martin Bernhardt created a special gnomon for an equatorial sundial that allows to read of the time at one scale without knowing the date and taking into account the equation of time. The precicision of such a sundial is less then a minute.

## History

The earliest sundials known from the archaeological record are obelisks (3500 BC) and shadow clocks (1500 BC) from ancient Egypt and Babylon. 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 II Kings 20:11 (possibly the earliest account of a Sun-Dial that is anywhere to be found in history) — which was likely of Egyptian or Babylonian design. Sundials are believed to have existed in China since ancient times, but very little is known of their history.

The ancient Greeks developed many of the principles and forms of the sundial. Sundials are believed to have been introduced into Greece by Anaximander of Miletus, c. 560 BC. According to Herodotus, the Greeks sundials were initially derived from the Babylonian counterparts. The Greeks were well-positioned to develop the science of sundials, having founded the science of geometry, and in particular discovering the conic sections that are traced by a sundial nodus. The mathematician and astronomer Theodosius of Bithynia (ca. 160 BC-ca. 100 BC) is said to have invented a universal sundial that could be used anywhere on Earth.

The Romans adopted the Greek sundials, so much so that Plautus complained in one of his plays about his day being "chopped into pieces" by the ubiquitous sundials. Writing in ca. 25 BC, the Roman author Vitruvius listed all the known types of dials in Book IX of his De Architectura, together with their Greek inventors. All of these are believed to be nodus-type sundials, differing mainly in the surface that receives the shadow of the nodus.

The Romans built a very large sundial in 10 BC, the Solarium Augusti, which is a classic nodus-based obelisk casting a shadow on a planar pelekinon.

The Greek dials were inherited and developed further by the Islamic Caliphate cultures and the post-Renaissance Europeans. Since the Greek dials were nodus-based with straight hour-lines, they indicated unequal hours — also called temporary hours — that varied with the seasons, since every day was divided into twelve equal segments; thus, hours were shorter in winter and longer in summer. The idea of using hours of equal time length throughout the year was the innovation of Abu'l-Hasan Ibn al-Shatir in 1371, based on earlier developments in trigonometry by Muhammad ibn Jābir al-Harrānī al-Battānī (Albategni). Ibn al-Shatir was aware that "using a gnomon that is parallel to the Earth's axis will produce sundials whose hour lines indicate equal hours on any day of the year." His sundial is the oldest polar-axis sundial still in existence. The concept later appeared in Western sundials from at least 1446.

The onset of the Renaissance saw an explosion of new designs. 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, with accompanying illustrations.

The oldest sundial in England is incorporated into the Bewcastle Cross ca. 800 AD. The dial is divided into four tides, covering the parts of the working day in areas influenced by the Vikings, a maritime culture which noted the passage of time in the progression of the two high and two low tides each day.

The custom of measuring time by one's shadow has persisted since ancient times. In Aristophanes' play, Assembly of Women, Praxagora asks her husband to return when his shadow reaches . The Venerable Bede also gave instructions to his follows, how to interpret their shadow lengths to know what time it is

## Sundial mottoes

Sundials are associated with the passage of time, and it has become common to inscribe a motto into a sundial, often one that prompts the viewer to reflect on the transience of the world and the inevitability of death, e.g., "Do not kill time, for it will surely kill thee." A more cheerful popular motto is "I count only the sunny hours." Another is "I am a sundial, and I make a botch of what is done far better by a watch." Various collections of sundial mottoes have been published over the past few centuries.

## Bibliography

• Earle AM Sundials and Roses of Yesterday. Rutland, VT: Charles E. Tuttle. ISBN 0-8048-0968-2, Reprint of the 1902 book published by Macmillan (New York).
• A.P.Herbert, Sundials Old and New,Methuen & Co. Ltd, 1967.
• Mayall RN, Mayall MW Sundials: Their Construction and Use. 3rd ed., Cambridge, MA: Sky Publishing.
• Hugo Michnik, Theorie einer Bifilar-Sonnenuhr, Astronomishe Nachrichten, 217(5190), p.81-90, 1923
• Rohr RRJ Sundials: History, Theory, and Practice. translated by G. Godin, New York: Dover. Slightly amended reprint of the 1970 translation published by University of Toronto Press (Toronto). The original was published in 1965 under the title Les Cadrans solaires by Gauthier-Villars (Montrouge, France).
• Frederick W. Sawyer, Bifilar gnomonics, JBAA (Journal of the British Astronomical association), 88(4):334–351, 1978
• Gerard L'E. Turner, Antique Scientific Instruments, Blandford Press Ltd. 1980 ISBN 0-7137-1068-3
• Make A Sundial, (The Education Group British Sundial Society) Editors Jane Walker and David Brown, British Sundial Society 1991 ISBN 0-9518404-0
• Waugh AE Sundials: Their Theory and Construction. New York: Dover Publications.
• "Illustrating Shadows", Simon Wheaton-Smith, ISBN 0-9765286-8-1, LCN: 2005900674
• also see "Illustrating More Shadows", Simon Wheaton-Smith, both books are over 300 pages long.