A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the

degree symbol The degree symbol or degree sign, , is a typographical symbol that is used, among other things, to represent degrees of arc (e.g. in geographic coordinate systems), hours (in the medical field), degrees of temperature or alcohol proof. The sym ...

), is a measurement of a plane

angle In Euclidean 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. Angles formed by two rays lie in the plane that contains the rays. Angles ...

in which one full rotation is 360 degrees. It is not an

SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...

—the SI unit of angular measure is the

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

—but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians.

History

Equilateral chord with length equal to radius

The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient

astronomers An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either obse ...

noticed that the sun, which follows through the

ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...

path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient

calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A date is the designation of a single and specific day within such a system. A calendar is also a phy ...

s, such as the

Persian calendar The Iranian calendars or Iranian chronology ( fa, گاه‌شماری ایرانی, ) are a succession of calendars invented or used for over two millennia in Iran, also known as Persia. One of the longest chronological records in human history, ...

and the

Babylonian calendar The Babylonian calendar was a lunisolar calendar with years consisting of 12 lunar months, each beginning when a new crescent moon was first sighted low on the western horizon at sunset, plus an intercalary month inserted as needed by decree. Th ...

, used 360 days for a year. The use of a calendar with 360 days may be related to the use of

sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

numbers. Another theory is that the Babylonians subdivided the circle using the angle of an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

as the basic unit, and further subdivided the latter into 60 parts following their

numeric system. The earliest trigonometry, used by the

Babylonian astronomers Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia. Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellations m ...

and their

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

successors, was based on

chords Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...

of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard

divisions, was a degree.

Aristarchus of Samos Aristarchus of Samos (; grc-gre, Ἀρίσταρχος ὁ Σάμιος, ''Aristarkhos ho Samios''; ) was an ancient Greek astronomer and mathematician who presented the first known heliocentric model that placed the Sun at the center of the ...

and

Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos''; BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the e ...

seem to have been among the first

Greek scientists Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

to exploit Babylonian astronomical knowledge and techniques systematically.

Timocharis Timocharis of Alexandria ( grc-gre, Τιμόχαρις or Τιμοχάρης, ''gen.'' Τιμοχάρους; c. 320–260 BC) was a Greek astronomer and philosopher. Likely born in Alexandria, he was a contemporary of Euclid. Work What little is kn ...

, Aristarchus,

Aristillus Aristyllus ( el, Ἀρίστυλλος; fl. c. 261 BC) was a Greek astronomer, presumably of the school of Timocharis (c. 300 BC). He was among the earliest meridian-astronomy observers. Six of his stellar declinations are preserved at ...

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60

arc minute A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...

Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandr ...

used a simpler

system dividing a circle into 60 parts. Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s,The divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360. making it one of only 7 numbers such that no number less than twice as much has more divisors . Furthermore, it is divisible by every number from 1 to 10 except 7.Contrast this with the relatively unwieldy 2520, which is the

least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by ...

for every number from 1 to 10. This property has many useful applications, such as dividing the world into 24

time zone A time zone is an area which observes a uniform standard time for legal, commercial and social purposes. Time zones tend to follow the boundaries between countries and their subdivisions instead of strictly following longitude, because it ...

s, each of which is nominally 15° of

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

, to correlate with the established

24-hour The modern 24-hour clock, popularly referred to in the United States as military time, is the convention of timekeeping in which the day runs from midnight to midnight and is divided into 24 hours. This is indicated by the hours (and minutes) pass ...

day A day is the time period of a full rotation of the Earth with respect to the Sun. On average, this is 24 hours, 1440 minutes, or 86,400 seconds. In everyday life, the word "day" often refers to a solar day, which is the length between two ...

convention. Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere, and that it was rounded to 360 for some of the mathematical reasons cited above.

Subdivisions

For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in

astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

or for

geographic coordinates The geographic coordinate system (GCS) is a spherical or ellipsoidal coordinate system for measuring and communicating positions directly on the Earth as latitude and longitude. It is the simplest, oldest and most widely used of the various ...

(

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 ...

and

), degree measurements may be written using decimal degrees (''DD notation''); for example, 40.1875°. Alternatively, the traditional

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...

subdivisions can be used: one degree is divided into 60 ''minutes (of arc)'', and one minute into 60 ''seconds (of arc)''. Use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the ''

arcminute A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...

'' and ''

arcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...

'', are represented by a single prime (′) and

double prime The prime symbol , double prime symbol , triple prime symbol , and quadruple prime symbol are used to designate units and for other purposes in mathematics, science, linguistics and music. Although the characters differ little in appearance fr ...

(″) respectively. For example, . Additional precision can be provided using decimal fractions of an arcsecond. Maritime charts are marked in degrees and decimal minutes to facilitate measurement; 1 minute of latitude is 1

nautical mile A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute ( of a degree) of latitude. Tod ...

. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25). The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by

al-Kashi Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer an ...

and other ancient astronomers, but is rarely used today. These subdivisions were denoted by writing the

Roman numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...

for the number of sixtieths in superscript: 1^I for a "

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

" (minute of arc), 1^II for a

second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ea ...

, 1^III for a

third Third or 3rd may refer to: Numbers * 3rd, the ordinal form of the cardinal number 3 * , a fraction of one third * 1⁄60 of a ''second'', or 1⁄3600 of a ''minute'' Places * 3rd Street (disambiguation) * Third Avenue (disambiguation) * Hi ...

, 1^IV for a fourth, etc. Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system.

SI prefixes A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...

can also be applied as in, e.g., millidegree, microdegree, etc.

Alternative units

In most

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

work beyond practical geometry, angles are typically measured in

s rather than degrees. This is for a variety of reasons; for example, the

trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...

s have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2'' '' radians, so 180° is equal to radians, or equivalently, the degree is a

mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

: 1° = . The turn (corresponding to a cycle or revolution) is used in

technology Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, scien ...

and

science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...

. One turn is equal to 360°. With the invention of the

metric system The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Intern ...

, based on powers of ten, there was an attempt to replace degrees by decimal "degrees" in France and nearby countries,These new and decimal "degrees" must not be confused with decimal degrees. where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = gon). This was called or '' grad''. Due to confusion with the existing term ''grad(e)'' in some northern European countries (meaning a standard degree, of a turn), the new unit was called in

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...

(whereas the "old" degree was referred to as ), likewise in Danish, Swedish and

Norwegian Norwegian, Norwayan, or Norsk may refer to: *Something of, from, or related to Norway, a country in northwestern Europe * Norwegians, both a nation and an ethnic group native to Norway * Demographics of Norway *The Norwegian language, including ...

(also ''gradian''), and in Icelandic. To end the confusion, the name ''gon'' was later adopted for the new unit. Although this idea of metrification was abandoned by Napoleon, grades continued to be used in several fields and many

scientific calculator A scientific calculator is an electronic calculator, either desktop or handheld, designed to perform mathematical operations. They have completely replaced slide rules and are used in both educational and professional settings. In some are ...

s support them. Decigrades () were used with French artillery sights in World War I. An angular mil, which is most used in military applications, has at least three specific variants, ranging from to . It is approximately equal to one

milliradian A milliradian ( SI-symbol mrad, sometimes also abbreviated mil) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjusting t ...

( ). A mil measuring of a revolution originated in the

imperial Russian army The Imperial Russian Army (russian: Ру́сская импера́торская а́рмия, tr. ) was the armed land force of the Russian Empire, active from around 1721 to the Russian Revolution of 1917. In the early 1850s, the Russian Ar ...

, where an equilateral chord was divided into tenths to give a circle of 600 units. This may be seen on a lining plane (an early device for aiming

indirect fire Indirect fire is aiming and firing a projectile without relying on a direct line of sight between the gun and its target, as in the case of direct fire. Aiming is performed by calculating azimuth and inclination, and may include correcting aim ...

artillery) dating from about 1900 in the St. Petersburg Museum of Artillery.

Notes

References

External links