Gnomonics (from the
ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
word γνώμων, , meaning 'interpreter, discerner') is the study of the design, construction and use of
sundials
A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat p ...
.
The foundations of gnomonics were known to the ancient Greek
Anaximander
Anaximander (; grc-gre, Ἀναξίμανδρος ''Anaximandros''; ) was a pre-Socratic Greek philosopher who lived in Miletus,"Anaximander" in ''Chambers's Encyclopædia''. London: George Newnes, 1961, Vol. 1, p. 403. a city of Ionia (in moder ...
(ca. 550 BCE), which augmented the science of shadows brought back from Egypt by
Thales of Miletus
Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him ...
.
Gnomonics was used by Greek and Roman architects from 25 BCE for the design of buildings.
Modern gnomonics has its root in the nascent European astronomy of the 16th Century. The first works, in Latin, were published by
Sebastian Münster
Sebastian Münster (20 January 1488 – 26 May 1552) was a German cartographer and cosmographer. He also was a Christian Hebraist scholar who taught as a professor at the University of Basel. His well-known work, the highly accurate world map, '' ...
in 1531 and
Oronce Fine in 1532, rapidly followed by books in French. At the end of the 17th century, gnomonics developed notably in the application of
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
. Several methods, both graphical and analytical, were published in books which allowed the creation of
sundials
A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat p ...
of greater or lesser precision to be placed on buildings and in gardens.
In his ''Histoire de la Gnomonique ancienne et moderne'',
Jean-Étienne Montucla
Jean-Étienne Montucla (5 September 1725 – 18 December 1799) was a French mathematician and historian.
Montucla was born at Lyon, France.
In 1754 he published an anonymous treatise on quadrature, ''Histoire des recherches sur la quadrature d ...
sums up gnomonics in these words:
Analytical gnomonics
Coordinate system transforms - Change of bases
The
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
of the Sun in the
horizontal coordinate system
The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane to define two angles: altitude and azimuth.
Therefore, the horizontal coordinate system is sometimes called as th ...
can be determined by successive
changes of bases.
Expression as transformation matrices
A
transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...
from a system B to a system B' allows for calculating the coordinates of a point or
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
in system B' when its coordinates are known is system B.
For example, to change the system by rotating by an angle α around the Z axis, the coordinates in the new system can be calculated from those in the old system as:
Similarly, for rotation of an angle α around the X axis:
And for rotation by the angle α around the Y axis:
Model of the apparent movement of the Sun
The Cartesian coordinates of the Sun in the horizontal system of coordinates can be calculated using
change of basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are considere ...
matrices:
where:
:
Latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
of the place of observation
: Local mean
sidereal time
Sidereal time (as a unit also sidereal day or sidereal rotation period) (sidereal ) is a timekeeping system that astronomers use to locate celestial objects. Using sidereal time, it is possible to easily point a telescope to the proper coord ...
:
Axial tilt
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orbi ...
:
Ecliptic longitude
The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System bodi ...
of the Sun
Projection of the shadow of a vertical gnomon
Let
be the Cartesian coordinates, in the local coordinate system, of the end of a vertical gnomon of length
.
The coordinates of the extremity of the shadow in the horizontal plane can be obtained with an
affine transform
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
parallel to the line by
and
.
Inclined and declined sundial
The Cartesian coordinates of the Sun in the system of coordinates bound to an inclined sundial of given declination are:
*
where:
:
declination
In astronomy, declination (abbreviated dec; symbol ''δ'') is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declination's angle is measured north or south of the ...
of the plane of the sundial
: inclination of the sundial, that is, the angle of the normal with respect to the zenith
Other uses
Gnomonic projection
A gnomonic map projection is a map projection which displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achie ...
is a
map projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...
where the vanishing point is in the centre of a spheroid.
References
Sources
*
*
*
*
*
*
External links
* at the Société Astronomique de France website
* , Minutes of meetings of the ''Commission des Cadrans solaires du Québec (CCSQ)'' from 1995 to 2014, available as PDF files
* In May 2018, a French-language {{cite web, url=http://www.cadrans-solaires.info, title=online learning course on the theory and construction of sundials was launched by a member of the Sundials Commission of the
Société astronomique de France
The Société astronomique de France (SAF; ), the French astronomical society, is a non-profit association in the public interest organized under French law (Association loi de 1901). Founded by astronomer Camille Flammarion in 1887, its purpose ...
.
Sundials