Degree (angle)#History

File:Equilateral chord with length equal to radius.svg (red). One sixtieth of this arc is a degree. Six such chords complete the circle.{{cite book |author=Euclid |title=Euclid's Elements of Geometry |title-link=Euclid's Elements |date=2008 |publisher=Princeton University Press |isbn=978-0-6151-7984-1 |edition=2 |language=en |translator-last1=Heiberg |translator-first1=Johan Ludvig |trans-title=Euclidis Elementa, editit et Latine interpretatus est I. L. Heiberg, in aedibus B. G. Teubneri 1883–1885 |chapter=Book 4 |author-link=Euclid |translator-last2=Fitzpatrick |translator-first2=Richard |translator-link1=Johan Ludvig Heiberg (poet)}} [https://books.google.com/books?id=7HDWIOoBZUAC]]]

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 calendars, such as the Persian calendar 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.{{cite book |author-first=James Hopwood |author-last=Jeans |author-link=James Hopwood Jeans |title=The Growth of Physical Science |publisher=Cambridge University Press (CUP) |date=1947 |url=https://archive.org/details/in.ernet.dli.2015.210060 |page=[https://archive.org/details/in.ernet.dli.2015.210060/page/n25 7]}}{{cite book |author-first=Francis Dominic |author-last=Murnaghan |author-link=Francis Dominic Murnaghan (mathematician) |title=Analytic Geometry |date=1946 |page=2}} 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 and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically.{{cite journal |url=http://www.dioi.org/cot.htm#dqsr |author-first=Dennis |author-last=Rawlins |title=On Aristarchus |journal=DIO - the International Journal of Scientific History }}{{cite book |author-first=Gerald James |author-last=Toomer |author-link=Gerald James Toomer |title=Hipparchus and Babylonian astronomy}} Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes.{{cn|date=May 2024}} Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.{{Citation needed|reason=Reliable source needed for this last sentence|date=January 2021}}

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 {{OEIS|id=A072938}}.{{cite web |url=http://www.brefeld.homepage.t-online.de/teilbarkeit.html |language=de |title=Teilbarkeit hochzusammengesetzter Zahlen |trans-title=Divisibility highly composite numbers |author-first=Werner |author-last=Brefeld}} 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 day 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.

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.{{Cite book |title=Introduction to Geographic Information Systems |last=Chang |first=Kang-Tsung |publisher=McGraw-Hill Education |isbn=978-1-259-92964-9 |edition=9th |location=New York |publication-date=2019 |pages=24 |lccn=2017049567}} These subdivisions, also called the arcminute and arcsecond, are represented by a single prime (′) and double prime (″) respectively. For example, {{nowrap|40.1875° {{=}} 40° 11′ 15″}}. 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. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25).{{cite book |author-last=Hopkinson |author-first=Sara |title=RYA day skipper handbook - sail |date=2012 |isbn=9781-9051-04949 |publisher=The Royal Yachting Association |location=Hamble |page=76}}

The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by al-Kashi{{citation needed|date=April 2020}} 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, 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.{{cite book |author-first=Graham H. |author-last=Flegg |title=Numbers Through the Ages |pages=156–157 |publisher=Macmillan International Higher Education |date=1989 |isbn=1-34920177-4}}

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

{{See also|Angle#Measuring angles|l1=Measuring angles}}

File:Degree-Radian Conversion.svg

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 (360°) is equal to 2{{pi}} radians, so 180° is equal to {{pi}} radians, or equivalently, the degree is a mathematical constant: 1° = {{frac|{{pi}}|180}}.

One turn (corresponding to a cycle or revolution) 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° = {{frac|10|9}} gon). This was called {{lang|fr|grade (nouveau)}} or grad. Due to confusion with the existing term grad(e) in some northern European countries (meaning a standard degree, {{sfrac|1|360}} of a turn), the new unit was called {{lang|de|Neugrad}} in German (whereas the "old" degree was referred to as {{lang|de|Altgrad}}), likewise {{lang|da|nygrad}} in Danish, Swedish and Norwegian (also gradian), and {{lang|is|nýgráða}} 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 ({{frac|1|4,000}}) 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 {{frac|1|6,400}} to {{frac|1|6,000}}. It is approximately equal to one milliradian ({{circa}} {{frac|1|6,283}}). A mil measuring {{frac|1|6,000}} of a revolution originated in the imperial Russian army, 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.

{{Table of angles}}

• Compass
• Degree of curvature
• Degrees per second
• Geographic coordinate system
• Gradian
• Meridian arc
• Square degree
• Square minute
• Square second
• Steradian

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{{cite book |author=Al-Biruni |author-link=Al-Biruni |date=1879 |orig-date=1000 |title=The Chronology of Ancient Nations |url=https://books.google.com/books?id=pFIEAAAAIAAJ&pg=PA147 |pages=147–149 |translator-last=Sachau |translator-first=C. Edward}}

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{{Commons category|Degree (angle)}}

• {{cite web |url=http://www.mathopenref.com/degrees.html |title=Degrees as an angle measure}}, with interactive animation
• {{cite web |title=° Degree of Angle |author-last1=Gray |author-first1=Meghan |author-last2=Merrifield |author-first2=Michael |author-last3=Moriarty |author-first3=Philip |url=http://www.sixtysymbols.com/videos/degree.htm |work=Sixty Symbols |publisher=Brady Haran for the University of Nottingham |date=2009}}

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