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 in which one

full rotation A turn is a unit of plane angle measurement equal to radians, 360 degrees or 400 gradians. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, etc. The closely related terms ''cycle'' and ''rev ...

is 360 degrees. It is not an SI unit—the SI unit of angular measure is the radian—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 noticed that the sun, which follows through the ecliptic 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 physi ...

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, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers. Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit, and further subdivided the latter into 60 parts following their sexagesimal numeric system. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree.

Aristarchus of Samos Aristarchus of Samos (; grc-gre, Ἀρίσταρχος ὁ Σάμιος, ''Aristarkhos ho Samios''; ) was an ancient Greek astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or ...

and Hipparchus 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 scientists ...

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

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 Alexandria ...

used a simpler sexagesimal 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 divisors,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 for every number from 1 to 10. This property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, 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) pas ...

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 so ...

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 or for geographic coordinates ( latitude and longitude), degree measurements may be written using decimal degrees (''DD notation''); for example, 40.1875°. Alternatively, the traditional sexagesimal unit 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'' 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 na ...

'', 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. Today ...

. 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 and other ancient astronomers, but is rarely used today. These subdivisions were denoted by writing the Roman numeral for the number of sixtieths in superscript: 1^I for a " prime" (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 ...

, 1^III for a third, 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 work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions 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 Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * 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

(corresponding to a cycle or revolution) is used in technology and science. One turn is equal to 360°. With the invention of the metric system, 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 (whereas the "old" degree was referred to as ), likewise in Danish,

Swedish Swedish or ' may refer to: Anything from or related to Sweden, a country in Northern Europe. Or, specifically: * Swedish language, a North Germanic language spoken primarily in Sweden and Finland ** Swedish alphabet, the official alphabet used by ...

and Norwegian (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 calculators 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 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 artillery) dating from about 1900 in the St. Petersburg Museum of Artillery.

Notes

References

External links