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# Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. [1]

147 relations: Acoustics, Al-Battani, Almagest, Ancient Greek, Angle, Applied mathematics, Archimedes, Architecture, Aryabhata, Astronomer, Astronomy, Astronomy in the medieval Islamic world, Āryabhaṭa's sine table, Babylonia, Bartholomaeus Pitiscus, Basilios Bessarion, Biology, Brook Taylor, Byzantine Empire, Calculus, Cartography, Chemistry, Chinese mathematics, Chord (geometry), Circumscribed circle, Civil engineering, Clark University, Colin Maclaurin, Complex number, Computer graphics, Cryptography, Crystallography, CT scan, Curvature, De Moivre's formula, De revolutionibus orbium coelestium, Degree (angle), E (mathematical constant), Economics, Electrical engineering, Electronics, Elliptic geometry, Euclid, Euler's formula, Floating-point unit, Fourier transform, Gemma Frisius, Generalized trigonometry, Geodesy, Geography, ... Expand index (97 more) »

## Acoustics

Acoustics is the branch of physics that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound.

## Al-Battani

Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī (Arabic: محمد بن جابر بن سنان البتاني) (Latinized as Albategnius, Albategni or Albatenius) (c. 858 – 929) was an Arab astronomer, astrologer, and mathematician.

## Almagest

The Almagest is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy. One of the most influential scientific texts of all time, its geocentric model was accepted for more than 1200 years from its origin in Hellenistic Alexandria, in the medieval Byzantine and Islamic worlds, and in Western Europe through the Middle Ages and early Renaissance until Copernicus.

## Ancient Greek

The Ancient Greek language includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD.

## Angle

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

## Applied mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.

## Archimedes

Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

## Architecture

Architecture is both the process and the product of planning, designing, and constructing buildings or any other structures.

## Aryabhata

Aryabhata (IAST) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.

## Astronomer

An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside the scope of Earth.

## Astronomy

Astronomy (from ἀστρονομία) is a natural science that studies celestial objects and phenomena.

## Astronomy in the medieval Islamic world

Islamic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age (9th–13th centuries), and mostly written in the Arabic language.

## Āryabhaṭa's sine table

Āryabhaṭa's sine table is a set of twenty-four numbers given in the astronomical treatise Āryabhaṭīya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa (476–550 CE), for the computation of the half-chords of certain set of arcs of a circle.

## Babylonia

Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia (present-day Iraq).

## Bartholomaeus Pitiscus

Bartholomaeus Pitiscus (also Barthélemy, Bartholomeo, August 24, 1561 &ndash; July 2, 1613) was a 16th-century German trigonometrist, astronomer and theologian who first coined the word trigonometry.

## Basilios Bessarion

Basilios (or Basilius) Bessarion (Greek: Βασίλειος Βησσαρίων; 2 January 1403 – 18 November 1472), a Roman Catholic Cardinal Bishop and the titular Latin Patriarch of Constantinople, was one of the illustrious Greek scholars who contributed to the great revival of letters in the 15th century.

## Biology

Biology is the natural science that studies life and living organisms, including their physical structure, chemical composition, function, development and evolution.

## Brook Taylor

Brook Taylor (18 August 1685 &ndash; 29 December 1731) was an English mathematician who is best known for Taylor's theorem and the Taylor series.

## Byzantine Empire

The Byzantine Empire, also referred to as the Eastern Roman Empire and Byzantium, was the continuation of the Roman Empire in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinople (modern-day Istanbul, which had been founded as Byzantium).

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Cartography

Cartography (from Greek χάρτης chartēs, "papyrus, sheet of paper, map"; and γράφειν graphein, "write") is the study and practice of making maps.

## Chemistry

Chemistry is the scientific discipline involved with compounds composed of atoms, i.e. elements, and molecules, i.e. combinations of atoms: their composition, structure, properties, behavior and the changes they undergo during a reaction with other compounds.

## Chinese mathematics

Mathematics in China emerged independently by the 11th century BC.

## Chord (geometry)

A chord of a circle is a straight line segment whose endpoints both lie on the circle.

## Circumscribed circle

In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon.

## Civil engineering

Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works such as roads, bridges, canals, dams, airports, sewerage systems, pipelines, and railways.

## Clark University

Clark University is an American private research university located in Worcester, Massachusetts.

## Colin Maclaurin

Colin Maclaurin (Cailean MacLabhruinn; 1 February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Computer graphics

Computer graphics are pictures and films created using computers.

## Cryptography

Cryptography or cryptology (from κρυπτός|translit.

## Crystallography

Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure).

## CT scan

A CT scan, also known as computed tomography scan, makes use of computer-processed combinations of many X-ray measurements taken from different angles to produce cross-sectional (tomographic) images (virtual "slices") of specific areas of a scanned object, allowing the user to see inside the object without cutting.

## Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

## De Moivre's formula

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) and integer it holds that where is the imaginary unit.

## De revolutionibus orbium coelestium

De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) is the seminal work on the heliocentric theory of the Renaissance astronomer Nicolaus Copernicus (1473–1543).

## Degree (angle)

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees.

## E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

## Economics

Economics is the social science that studies the production, distribution, and consumption of goods and services.

## Electrical engineering

Electrical engineering is a professional engineering discipline that generally deals with the study and application of electricity, electronics, and electromagnetism.

## Electronics

Electronics is the discipline dealing with the development and application of devices and systems involving the flow of electrons in a vacuum, in gaseous media, and in semiconductors.

## Elliptic geometry

Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

## Floating-point unit

A floating-point unit (FPU, colloquially a math coprocessor) is a part of a computer system specially designed to carry out operations on floating point numbers.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## Gemma Frisius

Gemma Frisius (born Jemme Reinerszoon; December 9, 1508 &ndash; May 25, 1555), was a Dutch physician, mathematician, cartographer, philosopher, and instrument maker.

## Generalized trigonometry

Ordinary trigonometry studies triangles in the Euclidean plane R2.

## Geodesy

Geodesy, also known as geodetics, is the earth science of accurately measuring and understanding three of Earth's fundamental properties: its geometric shape, orientation in space, and gravitational field.

## Geography

Geography (from Greek γεωγραφία, geographia, literally "earth description") is a field of science devoted to the study of the lands, the features, the inhabitants, and the phenomena of Earth.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## George of Trebizond

George of Trebizond (Γεώργιος Τραπεζούντιος; 1395–1486) was a Greek philosopher, scholar and humanist.

## Germany

Germany (Deutschland), officially the Federal Republic of Germany (Bundesrepublik Deutschland), is a sovereign state in central-western Europe.

The gradian is a unit of measurement of an angle, equivalent to \frac of a turn, \frac of a degree, or \frac of a radian.

## Greek mathematics

Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.

## Greek scholars in the Renaissance

The migration waves of Byzantine scholars and émigrés in the period following the Crusader sacking of Constantinople and the end of the Byzantine Empire in 1453, is considered by many scholars key to the revival of Greek and Roman studies that led to the development of the Renaissance humanism and science.

## Hellenistic period

The Hellenistic period covers the period of Mediterranean history between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire as signified by the Battle of Actium in 31 BC and the subsequent conquest of Ptolemaic Egypt the following year.

## Heron's formula

In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulae for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.

## Hipparchus

Hipparchus of Nicaea (Ἵππαρχος, Hipparkhos) was a Greek astronomer, geographer, and mathematician.

## Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

## Hypotenuse

In geometry, a hypotenuse (rarely: hypothenuse) is the longest side of a right-angled triangle, the side opposite of the right angle.

## Identity (mathematics)

In mathematics an identity is an equality relation A.

## Image compression

Image compression is a type of data compression applied to digital images, to reduce their cost for storage or transmission.

## Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

## Indian mathematics

Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century.

## Indus Valley Civilisation

The Indus Valley Civilisation (IVC), or Harappan Civilisation, was a Bronze Age civilisation (5500–1300 BCE; mature period 2600–1900 BCE) mainly in the northwestern regions of South Asia, extending from what today is northeast Afghanistan to Pakistan and northwest India.

## Inscribed angle

In geometry, an inscribed angle is the angle formed in the interior of a circle when two secant lines (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle.

## Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

## Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

## Islamic Golden Age

The Islamic Golden Age is the era in the history of Islam, traditionally dated from the 8th century to the 14th century, during which much of the historically Islamic world was ruled by various caliphates, and science, economic development and cultural works flourished.

## James Gregory (mathematician)

James Gregory FRS (November 1638 – October 1675) was a Scottish mathematician and astronomer.

## Latin translations of the 12th century

Latin translations of the 12th century were spurred by a major search by European scholars for new learning unavailable in western Europe at the time; their search led them to areas of southern Europe, particularly in central Spain and Sicily, which recently had come under Christian rule following their reconquest in the late 11th century.

## Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles.

## Law of sines

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles.

## Law of tangents

In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.

## Lénárt sphere

A Lénárt sphere is a teaching and educational research model for spherical geometry.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Light

Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum.

## List of triangle topics

This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrices, or concretely in physical space.

## List of trigonometric identities

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined.

## Lothal

Lothal is one of the southernmost cities of the ancient Indus valley civilization, located in the Bhāl region of the modern state of Gujarāt and first inhabited 3700 BCE.

## Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

## Mathematical table

Mathematical tables are lists of numbers showing the results of calculation with varying arguments.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mathematics in medieval Islam

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta).

## Mechanical engineering

Mechanical engineering is the discipline that applies engineering, physics, engineering mathematics, and materials science principles to design, analyze, manufacture, and maintain mechanical systems.

## Medical imaging

Medical imaging is the technique and process of creating visual representations of the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology).

## Meteorology

Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting.

## Mnemonic

A mnemonic (the first "m" is silent) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory.

## Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x&minus;1, is a number which when multiplied by x yields the multiplicative identity, 1.

## Music theory

Music theory is the study of the practices and possibilities of music.

## Nasir al-Din al-Tusi

Muhammad ibn Muhammad ibn al-Hasan al-Tūsī (محمد بن محمد بن حسن طوسی&lrm; 18 February 1201 – 26 June 1274), better known as Nasir al-Din Tusi (نصیر الدین طوسی; or simply Tusi in the West), was a Persian polymath, architect, philosopher, physician, scientist, and theologian.

Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.

## Nicaea

Nicaea or Nicea (Νίκαια, Níkaia; İznik) was an ancient city in northwestern Anatolia, and is primarily known as the site of the First and Second Councils of Nicaea (the first and seventh Ecumenical councils in the early history of the Christian Church), the Nicene Creed (which comes from the First Council), and as the capital city of the Empire of Nicaea following the Fourth Crusade in 1204, until the recapture of Constantinople by the Byzantines in 1261.

## Nicolaus Copernicus

Nicolaus Copernicus (Mikołaj Kopernik; Nikolaus Kopernikus; Niklas Koppernigk; 19 February 1473 – 24 May 1543) was a Renaissance-era mathematician and astronomer who formulated a model of the universe that placed the Sun rather than the Earth at the center of the universe, likely independently of Aristarchus of Samos, who had formulated such a model some eighteen centuries earlier.

## Nubia

Nubia is a region along the Nile river encompassing the area between Aswan in southern Egypt and Khartoum in central Sudan.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Oceanography

Oceanography (compound of the Greek words ὠκεανός meaning "ocean" and γράφω meaning "write"), also known as oceanology, is the study of the physical and biological aspects of the ocean.

## Optics

Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it.

## Outline of physical science

Physical science is a branch of natural science that studies non-living systems, in contrast to life science.

## Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

## Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

## Pharmacy

Pharmacy is the science and technique of preparing and dispensing drugs.

## Phonetics

Phonetics (pronounced) is the branch of linguistics that studies the sounds of human speech, or—in the case of sign languages—the equivalent aspects of sign.

## Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.

## Polygon

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

## Precalculus

In mathematics education, precalculus is a course that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus.

## Programming language

A programming language is a formal language that specifies a set of instructions that can be used to produce various kinds of output.

## Ptolemy

Claudius Ptolemy (Κλαύδιος Πτολεμαῖος, Klaúdios Ptolemaîos; Claudius Ptolemaeus) was a Greco-Roman mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology.

## Ptolemy's table of chords

The table of chords, created by the astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy.

## Pure mathematics

Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.

## Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.

## Ratio

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.

## Rational trigonometry

Rational trigonometry is a proposed reformulation of metrical planar and solid geometries (which includes trigonometry) by Canadian mathematician Norman J. Wildberger, currently an associate professor of mathematics at the University of New South Wales.

## Regiomontanus

Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus, was a mathematician and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg.

## Right angle

In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn.

## Right triangle

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).

A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning.

## Scientific calculator

A scientific calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science, engineering, and mathematics.

## Seismology

Seismology (from Ancient Greek σεισμός (seismós) meaning "earthquake" and -λογία (-logía) meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies.

## Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

## Shape

A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture or material composition.

## Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other.

## Sine

In mathematics, the sine is a trigonometric function of an angle.

## Skinny triangle

A skinny triangle in trigonometry is a triangle whose height is much greater than its base.

## Slide rule

The slide rule, also known colloquially in the United States as a slipstick, is a mechanical analog computer.

## Small-angle approximation

The small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero.

## Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

## Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere.

## Spherical trigonometry

Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere.

## Sumer

SumerThe name is from Akkadian Šumeru; Sumerian en-ĝir15, approximately "land of the civilized kings" or "native land".

## Surveying

Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them.

## Surya Siddhanta

The Surya Siddhanta is the name of a Sanskrit treatise in Indian astronomy from 6th Century BCE.

## Synthesizer

A synthesizer (often abbreviated as synth, also spelled synthesiser) is an electronic musical instrument that generates electric signals that are converted to sound through instrument amplifiers and loudspeakers or headphones.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## Triangle

A triangle is a polygon with three edges and three vertices.

## Triangulation

In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to it from known points.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Trigonometric series

A trigonometric series is a series of the form: It is called a Fourier series if the terms A_ and B_ have the form: where f is an integrable function.

## Trigonometric tables

In mathematics, tables of trigonometric functions are useful in a number of areas.

## Ultrasound

Ultrasound is sound waves with frequencies higher than the upper audible limit of human hearing.

## Unit circle

In mathematics, a unit circle is a circle with a radius of one.

## Uses of trigonometry

Amongst the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory.

## Video game development

Video game development is the process of creating a video game.

## Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.

## Western Europe

Western Europe is the region comprising the western part of Europe.

## References

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