Sundial
Overview

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 onto a surface marked with lines indicating the hours of the day. The style is the time-telling edge of the gnomon
Gnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....

, often a thin rod or a sharp, straight edge. As the sun moves across the sky, the shadow-edge aligns with different hour-lines.
Discussions
Encyclopedia
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 onto a surface marked with lines indicating the hours of the day. The style is the time-telling edge of the gnomon
Gnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....

, often a thin rod or a sharp, straight edge. As the sun moves across the sky, the shadow-edge aligns with different hour-lines. All sundials must be aligned with the axis of the Earth's rotation to tell the correct time. In most designs, the style must point towards true celestial north (not the north magnetic pole or south magnetic pole). That is, the style's horizontal angle must equal the sundial's geographical latitude.

It is common for inexpensive decorative sundials to have incorrect hour angles, and these cannot be adjusted to tell correct time.

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## Introduction

There are different types of sundials: Some sundials use a line of light to indicate the time. Others use the edge of a shadow. 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
Cylindrical lens
A cylindrical lens is a lens which focuses light which passes through on to a line instead of on to a point, as a spherical lens would. The curved face or faces of a cylindrical lens are sections of a cylinder, and focus the image passing through it onto a line parallel to the intersection of the...

.

Gnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....

— may be a thin rod, or any 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.

Sundials also may use many types of surfaces to receive the light or shadow. Planes
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

are the most common surface, but partial sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

s, cylinders
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

, cones
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, the precise vertical direction (e.g., by a level or plumb-bob), and the direction to true North
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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
Solar time
Solar time is a reckoning of the passage of time based on the Sun's position in the sky. The fundamental unit of solar time is the day. Two types of solar time are apparent solar time and mean solar time .-Introduction:...

, unless corrected for some other time. To obtain the official clock time, three types of corrections need to be made.

First, 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
Equation of time
The equation of time is the difference between apparent solar time and mean solar time. At any given instant, this difference will be the same for every observer...

. 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.

Second, the solar time must be corrected for the longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

of the sundial relative to the longitude of the official time zone. For example, a sundial located west of Greenwich
Greenwich
Greenwich is a district of south London, England, located in the London Borough of Greenwich.Greenwich is best known for its maritime history and for giving its name to the Greenwich Meridian and Greenwich Mean Time...

, England but within the same time-zone, shows a later 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.

Last, to adjust for daylight saving time
Daylight saving time
Daylight saving time —also summer time in several countries including in British English and European official terminology —is the practice of temporarily advancing clocks during the summertime so that afternoons have more daylight and mornings have less...

, 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

The principles of sundials can be understood most easily from the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

's apparent motion. Scientists have proven that the Earth rotates on its axis, and revolves in an elliptic orbit about the Sun; however, meticulous observations and experiments were needed. For the purposes of a sundial, an excellent approximation assumes that the Sun revolves around a stationary Earth on the celestial sphere
Celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...

, which rotates every 23 hours and 56 minutes about its celestial axis. The celestial axis is the line connecting the celestial pole
Celestial pole
The north and south celestial poles are the two imaginary points in the sky where the Earth's axis of rotation, indefinitely extended, intersects the imaginary rotating sphere of stars called the celestial sphere...

s. 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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

.

Unlike the fixed stars, the Sun changes its position on the celestial sphere, being at a positive declination
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...

in summer, at a negative declination in winter, and having exactly zero declination (i.e., being on the celestial equator
Celestial equator
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...

) at the equinox
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...

es. The path of the Sun on the celestial sphere is called the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

. The ecliptic passes through the twelve constellations of the zodiac
Zodiac
In astronomy, the zodiac is a circle of twelve 30° divisions of celestial longitude which are centred upon the ecliptic: the apparent path of the Sun across the celestial sphere over the course of the year...

in the course of a year.
This model of the Sun's motion helps to understand sundials. If the shadow-casting gnomon is aligned with the celestial pole
Celestial pole
The north and south celestial poles are the two imaginary points in the sky where the Earth's axis of rotation, indefinitely extended, intersects the imaginary rotating sphere of stars called the celestial sphere...

s, 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
Armillary sphere
An armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...

).

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
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

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
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

, such as a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

, ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

or (at the North or South Poles) a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

.

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 and/or position of the gnomon relative to the hour lines, as in the analemmatic dial or the Lambert dial.

## History

The earliest sundials known from the archaeological record are the obelisks (3500 BC) and shadow clocks (1500 BC) from ancient Egyptian astronomy
Egyptian astronomy
Egyptian astronomy begins in prehistoric times, in the Predynastic Period. In the 5th millennium BCE, the stone circles at Nabta Playa may have made use of astronomical alignments...

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
Old Testament
The Old Testament, of which Christians hold different views, is a Christian term for the religious writings of ancient Israel held sacred and inspired by Christians which overlaps with the 24-book canon of the Masoretic Text of Judaism...

describes a sundial — the "dial of Ahaz" mentioned in and . The Roman writer Vitruvius lists dials and shadow clocks known at that time. Italian astronomer Giovanni Padovani
Giovanni Padovani was an Italian mathematician and astronomer. He lived in Verona and was a student of Pietro Pitati...

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
Giuseppe Biancani
Giuseppe Biancani was an Italian Jesuit astronomer, mathematician, and selenographer, after whom the crater Blancanus on the Moon is named...

Constructio instrumenti ad horologia solaria (ca. 1620) discusses how to make a perfect sundial,

## 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 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 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
Motto
A motto is a phrase meant to formally summarize the general motivation or intention of a social group or organization. A motto may be in any language, but Latin is the most used. The local language is usual in the mottoes of governments...

. 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

A sundial at a particular latitude in one hemisphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

must be reversed for use at the opposite latitude in the other hemisphere.
A vertical direct south sundial in the Northern Hemisphere
Northern Hemisphere
The Northern Hemisphere is the half of a planet that is north of its equator—the word hemisphere literally means “half sphere”. It is also that half of the celestial sphere north of the celestial equator...

becomes a vertical direct north sundial in the Southern Hemisphere
Southern Hemisphere
The Southern Hemisphere is the part of Earth that lies south of the equator. The word hemisphere literally means 'half ball' or "half sphere"...

; an opposite. To position a horizontal sundial correctly, one has to find true North
North
North is a noun, adjective, or adverb indicating direction or geography.North is one of the four cardinal directions or compass points. It is the opposite of south and is perpendicular to east and west.By convention, the top side of a map is north....

or South
South
South is a noun, adjective, or adverb indicating direction or geography.South is one of the four cardinal directions or compass points. It is the opposite of north and is perpendicular to east and west.By convention, the bottom side of a map is 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 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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid north...

and South, and making an angle with the horizontal equal to the geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. This axis is aligned with the celestial pole
Celestial pole
The north and south celestial poles are the two imaginary points in the sky where the Earth's axis of rotation, indefinitely extended, intersects the imaginary rotating sphere of stars called the celestial sphere...

s, which is closely, but not perfectly, aligned with the (present) pole star
Pole star
The term "Pole Star" usually refers to Polaris, which is the current northern pole star, also known as the North Star.In general, however, a pole star is a visible star, especially a prominent one, that is approximately aligned with the Earth's axis of rotation; that is, a star whose apparent...

Polaris
Polaris
Polaris |Alpha]] Ursae Minoris, commonly North Star or Pole Star, also Lodestar) is the brightest star in the constellation Ursa Minor. It is very close to the north celestial pole, making it the current northern pole star....

. For illustration, the celestial axis points vertically at the true North Pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...

, where it points horizontally on the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

. At Jaipur
Jaipur
Jaipur , also popularly known as the Pink City, is the capital and largest city of the Indian state of Rajasthan. Founded on 18 November 1727 by Maharaja Sawai Jai Singh II, the ruler of Amber, the city today has a population of more than 3.1 million....

, a famous location for sundials, gnomons are raised 26°55" above horizontal, reflecting 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 equinox
Equinox
An equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...

es 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
Declination
In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...

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

where λ is the sundial's geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

(and the angle the style makes with horizontal), θ is the angle between a given hour-line and the noon hour-line (which always points towards true North
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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
Inverse trigonometric function
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions with suitably restricted domains .The notations sin−1, cos−1, etc...

of sin
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

(λ), since tan(45°) = 1. When λ equals 90° (at the North Pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...

), 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
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

, 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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid north...

and its angle with the horizontal equals the sundial's geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

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

where λ is the sundial's geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, 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
Inverse trigonometric function
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions with suitably restricted domains .The notations sin−1, cos−1, etc...

of cos
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

(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, cupola
Cupola
In architecture, a cupola is a small, most-often dome-like, structure on top of a building. Often used to provide a lookout or to admit light and air, it usually crowns a larger roof or dome....

s 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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid north...

and its angle with the horizontal equals the sundial's geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

; consequently, its angle with the vertical face of the dial equals the colatitude
Colatitude
In spherical coordinates, colatitude is the complementary angle of the latitude, i.e. the difference between 90° and the latitude.-Astronomical use:The colatitude is useful in astronomy because it refers to the zenith distance of the celestial poles...

, or 90°-latitude.

### Pocket Sundials

This portable folding German sundial has a string gnomon (pointer), adjustable for accuracy at any latitude. As shadows fall across the sundial, the smaller dials show Italian and Babylonian hours. The dial also indicates the length of the day and the position of the sun in the zodiac.

### 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

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 +∞
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" 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
North
North is a noun, adjective, or adverb indicating direction or geography.North is one of the four cardinal directions or compass points. It is the opposite of south and is perpendicular to east and west.By convention, the top side of a map is north....

, South
South
South is a noun, adjective, or adverb indicating direction or geography.South is one of the four cardinal directions or compass points. It is the opposite of north and is perpendicular to east and west.By convention, the bottom side of a map is south....

, East
East
East is a noun, adjective, or adverb indicating direction or geography.East is one of the four cardinal directions or compass points. It is the opposite of west and is perpendicular to north and south.By convention, the right side of a map is east....

or West
West
West is a noun, adjective, or adverb indicating direction or geography.West is one of the four cardinal directions or compass points. It is the opposite of east and is perpendicular to north and south.By convention, the left side of a map is 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

where λ is the sundial's geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, t is the time before or after noon, and η is the angle of declination from true South
South
South is a noun, adjective, or adverb indicating direction or geography.South is one of the four cardinal directions or compass points. It is the opposite of north and is perpendicular to east and west.By convention, the bottom side of a map is 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

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.

The height of the gnomon, γ (that is the angle the style makes to the plate) is

### 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

where χ is the desired angle of reclining, λ is the sundial's geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, 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
Inverse trigonometric function
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions with suitably restricted domains .The notations sin−1, cos−1, etc...

of sin
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

(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
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

(λ+90°) = cos
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

(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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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

where λ is the sundial's geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, 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

### Spherical sundials

The surface receiving the shadow need not be a plane, but 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
Invar
Invar, also known generically as FeNi36 , is a nickel steel alloy notable for its uniquely low coefficient of thermal expansion . The name, Invar, comes from the word invariable, referring to its lack of expansion or contraction with temperature changes.It was invented in 1896 by Swiss scientist...

, 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
Marble
Marble is a metamorphic rock composed of recrystallized carbonate minerals, most commonly calcite or dolomite.Geologists use the term "marble" to refer to metamorphosed limestone; however stonemasons use the term more broadly to encompass unmetamorphosed limestone.Marble is commonly used for...

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
Jaipur
Jaipur , also popularly known as the Pink City, is the capital and largest city of the Indian state of Rajasthan. Founded on 18 November 1727 by Maharaja Sawai Jai Singh II, the ruler of Amber, the city today has a population of more than 3.1 million....

, India between 1727 and 1733. The larger equatorial bow is called the Samrat Yantra (The Supreme Instrument); standing at 27 meters
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

, 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
Inscribed angle
In geometry, an inscribed angle is formed when two secant lines of a circle intersect on the circle....

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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid north...

and its angle with the horizontal must equal the local geographical latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. 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
Daylight saving time
Daylight saving time —also summer time in several countries including in British English and European official terminology —is the practice of temporarily advancing clocks during the summertime so that afternoons have more daylight and mornings have less...

, 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
Time zone
A time zone is a region on Earth that has a uniform standard time for legal, commercial, and social purposes. In order for the same clock time to always correspond to the same portion of the day as the Earth rotates , different places on the Earth need to have different clock times...

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
Local mean time
Local mean time is a form of solar time that corrects the variations of local apparent time, forming a uniform time scale at a specific longitude...

. 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
Equation of time
The equation of time is the difference between apparent solar time and mean solar time. At any given instant, this difference will be the same for every observer...

. 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
Solar time
Solar time is a reckoning of the passage of time based on the Sun's position in the sky. The fundamental unit of solar time is the day. Two types of solar time are apparent solar time and mean solar time .-Introduction:...

to mean solar time or another standard time
Standard time
Standard time is the result of synchronizing clocks in different geographical locations within a time zone to the same time rather than using the local meridian as in local mean time or solar time. Historically, this helped in the process of weather forecasting and train travel. The concept...

. Heliochronometers usually indicate the minutes to within 1 minute of Universal Time
Universal Time
Universal Time is a time scale based on the rotation of the Earth. It is a modern continuation of Greenwich Mean Time , i.e., the mean solar time on the Prime Meridian at Greenwich, and GMT is sometimes used loosely as a synonym for UTC...

.

An analemma may be added to many types of sundials to correct apparent solar time to mean solar time or another standard time
Standard time
Standard time is the result of synchronizing clocks in different geographical locations within a time zone to the same time rather than using the local meridian as in local mean time or solar time. Historically, this helped in the process of weather forecasting and train travel. The concept...

. These usually have hour lines shaped like "figure eights" (analemma
Analemma
In astronomy, an analemma is a curve representing the angular offset of a celestial body from its mean position on the celestial sphere as viewed from another celestial body relative to the viewing body's celestial equator...

s) according to the equation of time
Equation of time
The equation of time is the difference between apparent solar time and mean solar time. At any given instant, this difference will be the same for every observer...

. 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.

## 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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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
Mariner's astrolabe
The mariner's astrolabe, also called sea astrolabe, was an inclinometer used to determine the latitude of a ship at sea by measuring the sun's noon altitude or the meridian altitude of a star of known declination. Not an astrolabe proper, the mariner's astrolabe was rather a graduated circle with...

. It was likely invented by William Oughtred
William Oughtred
William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...

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 an 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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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. These bars are sometimes an addition to a set of Gemma's rings. 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
Edward Wright (mathematician)
Edward Wright was an English mathematician and cartographer noted for his book Certaine Errors in Navigation , which for the first time explained the mathematical basis of the Mercator projection, and set out a reference table giving the linear scale multiplication factor as a function of...

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

Analemmatic sundials are a common feature at science museum
Science museum
A science museum or a science centre is a museum devoted primarily to science. Older science museums tended to concentrate on static displays of objects related to natural history, paleontology, geology, industry and industrial machinery, etc. Modern trends in museology have broadened the range of...

s , planetarium
Planetarium
A planetarium is a theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky, or for training in celestial navigation...

s and public parks. They exploit the fact that the sun travels in a predictable pattern over the course of a year called the analemma
Analemma
In astronomy, an analemma is a curve representing the angular offset of a celestial body from its mean position on the celestial sphere as viewed from another celestial body relative to the viewing body's celestial equator...

and trace the projection of an object's shadow to measure time, not only the hours, as in normal sundials, but also weeks and months.

### 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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. 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

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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

.

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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

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
An equivalent type of sundial using a vertical rod of fixed length is known as a backstaff dial
Backstaff
The backstaff or back-quadrant is a navigational instrument that was used to measure the altitude of a celestial body, in particular the sun or moon...

.

### 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 knife-like gnomon that juts out perpendicularly. It is normally dangled from a rope or string so the cylinder is vertical. The gnomon can be twisted to be above a month or day indication on the face of the cylinder. This corrects the sundial for the equation of time. The entire sundial is then twisted on its string so that the gnomon aims toward the sun, while the cylinder remains vertical. The tip of the shadow indicates the time on the cylinder. The hour curves inscribed on the cylinder permit one to read the time. Shepherd's dials are sometimes hollow, so that the gnomon can fold 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,
Similarly, the shepherd's dial is evoked in Shakespeare's Henry VI, Part 3
Henry VI, part 3
Henry VI, Part 3 or The Third Part of Henry the Sixt is a history play by William Shakespeare believed to have been written in 1591, and set during the lifetime of King Henry VI of England...

,
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. Often 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. To adjust for the equation of time, the hole is usually on a loose ring within the ring so that the hole can be adjusted to reflect the current month.

### 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
Obelisk
An obelisk is a tall, four-sided, narrow tapering monument which ends in a pyramid-like shape at the top, and is said to resemble a petrified ray of the sun-disk. A pair of obelisks usually stood in front of a pylon...

(e.g., the Solarium Augusti
Solarium Augusti
The Solarium Augusti was a sundial in ancient Rome, the largest of the ancient era. It was erected by Emperor Augustus, with a 30-metre Egyptian red granite obelisk that he had imported from Heliopolis...

) 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
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

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
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

such as a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

or an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

. 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).

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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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.

### Reflection sundials

Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

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
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

and time of year, the light-spot follows a conic section, such as the hyperbolae of the pelikonon. If the mirror is parallel to the Earth's equator, and the ceiling is horizontal, then the resulting angles are those of a conventional horizontal sundial. Using the ceiling as a sundial surface exploits unused space, and the dial may be large enough to be very accurate.

## 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
True north
True north is the direction along the earth's surface towards the geographic North Pole.True geodetic north usually differs from magnetic north , and from grid 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
Diptych
A diptych di "two" + ptychē "fold") is any object with two flat plates attached at a hinge. Devices of this form were quite popular in the ancient world, wax tablets being coated with wax on inner faces, for recording notes and for measuring time and direction.In Late Antiquity, ivory diptychs with...

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
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

, usually a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

. 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
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

of a concave polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

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.

### Benoy dials

The Benoy Dial was invented by Walter Gordon Benoy of Collingham in Nottinghamshire. 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
Cylindrical lens
A cylindrical lens is a lens which focuses light which passes through on to a line instead of on to a point, as a spherical lens would. The curved face or faces of a cylindrical lens are sections of a cylinder, and focus the image passing through it onto a line parallel to the intersection of the...

. 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
• Carnfunnock Country Park Antrim
Antrim, County Antrim
Antrim is a town in County Antrim in the northeast of Northern Ireland, on the banks of the Six Mile Water, half a mile north-east of Lough Neagh. It had a population of 20,001 people in the 2001 Census. The town is the administrative centre of Antrim Borough Council...

Northern Ireland
Northern Ireland
Northern Ireland is one of the four countries of the United Kingdom. Situated in the north-east of the island of Ireland, it shares a border with the Republic of Ireland to the south and west...

• Upton Hall British Horological Institute
British Horological Institute
The British Horological Institute is the representative body of the horological industry in the United Kingdom.-History:...

Newark-on-Trent
Newark-on-Trent
Newark-on-Trent is a market town in Nottinghamshire in the East Midlands region of England. It stands on the River Trent, the A1 , and the East Coast Main Line railway. The origins of the town are possibly Roman as it lies on an important Roman road, the Fosse Way...

Nottinghamshire
Nottinghamshire
Nottinghamshire is a county in the East Midlands of England, bordering South Yorkshire to the north-west, Lincolnshire to the east, Leicestershire to the south, and Derbyshire to the west...

UK
• Within the collections of St Edmundsbury Heritage Service Bury St Edmunds UK
• Longleat Warminster
Warminster
Warminster is a town in western Wiltshire, England, by-passed by the A36, and near Frome and Westbury. It has a population of about 17,000. The River Were runs through the town and can be seen running through the middle of the town park. The Minster Church of St Denys sits on the River Were...

Wiltshire
Wiltshire
Wiltshire is a ceremonial county in South West England. It is landlocked and borders the counties of Dorset, Somerset, Hampshire, Gloucestershire, Oxfordshire and Berkshire. It contains the unitary authority of Swindon and covers...

UK
• Jodrell Bank
Jodrell Bank
The Jodrell Bank Observatory is a British observatory that hosts a number of radio telescopes, and is part of the Jodrell Bank Centre for Astrophysics at the University of Manchester...

Science Centre and Arboretum
Arboretum
An arboretum in a narrow sense is a collection of trees only. Related collections include a fruticetum , and a viticetum, a collection of vines. More commonly, today, an arboretum is a botanical garden containing living collections of woody plants intended at least partly for scientific study...

• Birmingham Botanical Gardens
Birmingham Botanical Gardens
The Birmingham Botanical Gardens are botanical gardens situated in Edgbaston, Birmingham, England. The gardens are close to the centre of Birmingham and open every day of the year, from sunrise to sunset. They are located at ....

Edgbaston
Edgbaston
Edgbaston is an area in the city of Birmingham in England. It is also a formal district, managed by its own district committee. The constituency includes the smaller Edgbaston ward and the wards of Bartley Green, Harborne and Quinton....

UK
• Science Museum
Science museum
A science museum or a science centre is a museum devoted primarily to science. Older science museums tended to concentrate on static displays of objects related to natural history, paleontology, geology, industry and industrial machinery, etc. Modern trends in museology have broadened the range of...

UK - (inventory number 1975-318)

### 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
Digital sundial
A digital sundial is a clock that indicates the current time with numerals formed by the sunlight striking it. Like a classical sundial, the device is purely passive and contains no moving parts. It uses no electricity nor other artificial sources of energy....

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
Cardioid
A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp...

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.

### 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 analemma
Analemma
In astronomy, an analemma is a curve representing the angular offset of a celestial body from its mean position on the celestial sphere as viewed from another celestial body relative to the viewing body's celestial equator...

tic plate. The plate has an engraved figure-eight shape., which corresponds to plotting the equation of time
Equation of time
The equation of time is the difference between apparent solar time and mean solar time. At any given instant, this difference will be the same for every observer...

(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 a quantity of gunpowder at noon. 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
Gnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....

a suitably mounted lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

, set up to focus the rays of the sun at exactly noon on the firing pan of a miniature cannon
Cannon
A cannon is any piece of artillery that uses gunpowder or other usually explosive-based propellents to launch a projectile. Cannon vary in caliber, range, mobility, rate of fire, angle of fire, and firepower; different forms of cannon combine and balance these attributes in varying degrees,...

Gunpowder
Gunpowder, also known since in the late 19th century as black powder, was the first chemical explosive and the only one known until the mid 1800s. It is a mixture of sulfur, charcoal, and potassium nitrate - with the sulfur and charcoal acting as fuels, while the saltpeter works as an oxidizer...

(but no bullet
Bullet
A bullet is a projectile propelled by a firearm, sling, or air gun. Bullets do not normally contain explosives, but damage the intended target by impact and penetration...

). To function properly the position and angle of the lens must be adjusted seasonally.

## Meridian lines

A horizontal line aligned on a meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

with a gnomon facing the noon-sun is termed a meridian line and does not indicate the time, but instead the day of the year. Historically they were used to accurately determine the length of the solar year
Tropical year
A tropical year , for general purposes, is the length of time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice...

. Examples are the Bianchini
Francesco Bianchini
Francesco Bianchini was an Italian philosopher and scientist. He worked for the curia of three popes, including being camiere d`honore of Clement XI, and secretary of the commission for the reform of the calendar, working on the method to calculate the astronomically correct date for Easter in a...

meridian line in Santa Maria degli Angeli e dei Martiri
Santa Maria degli Angeli e dei Martiri
The Basilica of St. Mary of the Angels and the Martyrs is a titular basilica church in Rome, built inside the frigidarium of the Baths of Diocletian. The Cardinal priest of the is William Henry Keeler.- The basilica :...

in Rome
Rome
Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...

, and the Cassini
Giovanni Domenico Cassini

line in San Petronio Basilica
San Petronio Basilica
The Basilica of San Petronio is the main church of Bologna, Emilia Romagna, northern Italy. It dominates the Piazza Maggiore. It is the fifth largest church in the world, stretching for 132 meters in length and 60 meters in width, while the vault reaches 45 meters inside and 51 meters in the facade...

at Bologna
Bologna
Bologna is the capital city of Emilia-Romagna, in the Po Valley of Northern Italy. The city lies between the Po River and the Apennine Mountains, more specifically, between the Reno River and the Savena River. Bologna is a lively and cosmopolitan Italian college city, with spectacular history,...

.

## Sundial mottoes

The association of sundials with time has inspired their designers over the centuries to display mottoes as part of the design. Often these cast the device in the role of memento mori
Memento mori
Memento mori is a Latin phrase translated as "Remember your mortality", "Remember you must die" or "Remember you will die". It names a genre of artistic work which varies widely, but which all share the same purpose: to remind people of their own mortality...

, inviting the observer to reflect on the transience of the world and the inevitability of death. "Do not kill time, for it will surely kill thee." Other mottoes are more whimsical: "I count only the sunny hours," and "I am a sundial and I make a botch / of what is done far better by a watch." Collections of sundial mottoes have often been published through the centuries.

• Foucault pendulum
Foucault pendulum
The Foucault pendulum , or Foucault's pendulum, named after the French physicist Léon Foucault, is a simple device conceived as an experiment to demonstrate the rotation of the Earth. While it had long been known that the Earth rotated, the introduction of the Foucault pendulum in 1851 was the...

• Francesco Bianchini
Francesco Bianchini
Francesco Bianchini was an Italian philosopher and scientist. He worked for the curia of three popes, including being camiere d`honore of Clement XI, and secretary of the commission for the reform of the calendar, working on the method to calculate the astronomically correct date for Easter in a...

• Horology
Horology
Horology is the art or science of measuring time. Clocks, watches, clockwork, sundials, clepsydras, timers, time recorders and marine chronometers are all examples of instruments used to measure time.People interested in horology are called horologists...

• Moondial
Moondial
Moondial is also the name of a television series.Moondials are time pieces similar to a sundial. The most basic moondial, which is identical to a sundial, is only accurate on the night of the full moon...

• Nocturnal
Nocturnal (instrument)
A nocturnal is an instrument used to determine the time based on the position of a certain star in the night sky. Sometimes called a "horologium nocturnum" or nocturlabe , it is closely related to the sun dial. A nocturnal is typically a navigational instrument...

— device for determining time by the stars at night.
• Scottish sundial
Scottish sundial
Scottish sundials of the renaissance period are not just more numerous than in any other country, they are also stylistically unique. This is particularly notable when the size and wealth of Scotland at the time are taken into account. They are free standing stone sculptures of the 17th and 18th...

— the ancient renaissance sundials of Scotland.
• Tide (time)
Tide (time)
Tide is an obsolete or archaic term for time, period or season.It survives in compounds such as Yuletide, eventide, shrovetide, Eastertide, noontide, etc....

— divisions of the day on early sundials.
• Wilanów Palace Sundial, created by Johannes Hevelius
Johannes Hevelius
Johannes Hevelius Some sources refer to Hevelius as Polish:Some sources refer to Hevelius as German:*Encyplopedia Britannica * of the Royal Society was a councilor and mayor of Danzig , Pomeranian Voivodeship, in the Polish-Lithuanian Commonwealth...