Editing Radian (section)

=== Pre-20th century ===
The idea of measuring angles by the length of the arc was in use by mathematicians quite early. For example, [[al-Kashi]] (c. 1400) used so-called ''diameter parts'' as units, where one diameter part was {{sfrac|1|60}} radian. They also used sexagesimal subunits of the diameter part.<ref>{{cite book|first=Paul|last= Luckey|editor-first=A. |editor-last=Siggel|location=Berlin|publisher= Akademie Verlag| orig-year=Translation of 1424 book|year=1953| title=Der Lehrbrief über den kreisumfang von Gamshid b. Mas'ud al-Kasi|trans-title=Treatise on the Circumference of al-Kashi| number=6|pages= 40}}</ref> Newton in 1672 spoke of "the angular quantity of a body's circular motion", but used it only as a relative measure to develop an astronomical algorithm.<ref name="Roche">{{cite book |last1=Roche |first1=John J. |title=The Mathematics of Measurement: A Critical History |date=21 December 1998 |publisher=Springer Science & Business Media |isbn=978-0-387-91581-4 |page=134 |url=https://books.google.com/books?id=eiQOqS-Q6EkC&pg=PA134 |language=en}}</ref>

The concept of ''the'' radian measure is normally credited to [[Roger Cotes]], who died in 1716. By 1722, his cousin Robert Smith had collected and published Cotes' mathematical writings in a book, ''Harmonia mensurarum''.<ref>{{cite web |url = http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html |title = Biography of Roger Cotes |work = The MacTutor History of Mathematics |date = February 2005 |last1 = O'Connor |first1 = J. J. |first2 = E. F. |last2 = Robertson |access-date = 2006-04-21 |archive-date = 2012-10-19 |archive-url = https://web.archive.org/web/20121019161705/http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Cotes.html |url-status = dead }}</ref> In a chapter of editorial comments, Smith gave what is probably the first published calculation of one radian in degrees, citing a note of Cotes that has not survived. Smith described the radian in everything but name – "Now this number is equal to 180 degrees as the radius of a circle to the [[semicircumference]], this is as 1 to 3.141592653589" –, and recognized its naturalness as a unit of angular measure.<ref>{{cite book |last1=Cotes |first1=Roger |title=Harmonia mensurarum |date=1722 |editor-first=Robert|editor-last=Smith|location=Cambridge, England|chapter=Editoris notæ ad Harmoniam mensurarum |pages=94–95 |chapter-url=https://books.google.com/books?id=J6BGAAAAcAAJ&pg=RA2-PA95 |language=la|quote=In Canone Logarithmico exhibetur Systema quoddam menfurarum numeralium, quæ Logarithmi dicuntur: atque hujus systematis Modulus is est Logarithmus, qui metitur Rationem Modularem in Corol. 6. definitam. Similiter in Canone Trigonometrico finuum & tangentium, exhibetur Systema quoddam menfurarum numeralium, quæ Gradus appellantur: atque hujus systematis Modulus is est Numerus Graduum, qui metitur Angulum Modularem modo definitun, hoc est, qui continetur in arcu Radio æquali. Eft autem hic Numerus ad Gradus 180 ut Circuli Radius ad Semicircuinferentiam, hoc eft ut 1 ad 3.141592653589 &c. Unde Modulus Canonis Trigonometrici prodibit 57.2957795130 &c. Cujus Reciprocus eft 0.0174532925 &c. Hujus moduli subsidio (quem in chartula quadam Auctoris manu descriptum inveni) commodissime computabis mensuras angulares, queinadmodum oftendam in Nota III.|trans-quote=In the Logarithmic Canon there is presented a certain system of numerical measures called Logarithms: and the Modulus of this system is the Logarithm, which measures the Modular Ratio as defined in Corollary 6. Similarly, in the Trigonometrical Canon of sines and tangents, there is presented a certain system of numerical measures called Degrees: and the Modulus of this system is the Number of Degrees which measures the Modular Angle defined in the manner defined, that is, which is contained in an equal Radius arc. Now this Number is equal to 180 Degrees as the Radius of a Circle to the Semicircumference, this is as 1 to 3.141592653589 &c. Hence the Modulus of the Trigonometric Canon will be 57.2957795130 &c. Whose Reciprocal is 0.0174532925 &c. With the help of this modulus (which I found described in a note in the hand of the Author) you will most conveniently calculate the angular measures, as mentioned in Note III.}}</ref><ref>{{cite book |last1=Gowing |first1=Ronald |title=Roger Cotes - Natural Philosopher |date=27 June 2002 |publisher=Cambridge University Press |isbn=978-0-521-52649-4 |url=https://books.google.com/books?id=I2Cy4wjj1soC&pg=PA39}}</ref>

In 1765, [[Leonhard Euler]] implicitly adopted the radian as a unit of angle.<ref name="Roche"/> Specifically, Euler defined angular velocity as "The angular speed in rotational motion is the speed of that point, the distance of which from the axis of gyration is expressed by one."<ref>{{cite book |last1=Euler |first1=Leonhard |translator1-last=Bruce |translator1-first=Ian |title=Theoria Motus Corporum Solidorum seu Rigidorum|trans-title= Theory of the motion of solid or rigid bodies|language=latin|at=Definition 6, paragraph 316|url=http://www.17centurymaths.com/contents/euler/mechvol3/tmvol1ch2tr.pdf#page=3}}</ref> Euler was probably the first to adopt this convention, referred to as the radian convention, which gives the simple formula for angular velocity {{math|1=''ω'' = ''v''/''r''}}. As discussed in ''{{section link|#Dimensional analysis}}'', the radian convention has been widely adopted, while dimensionally consistent formulations require the insertion of a dimensional constant, for example {{math|1=''ω'' = ''v''/(''ηr'')}}.{{sfn|Quincey|2021}}

Prior to the term ''radian'' becoming widespread, the unit was commonly called ''circular measure'' of an angle.<ref>Isaac Todhunter, ''Plane Trigonometry: For the Use of Colleges and Schools'', [https://books.google.com/books?id=bo5FAAAAcAAJ&pg=PA10 p. 10], Cambridge and London: MacMillan, 1864 {{OCLC|500022958}}</ref> The term ''radian'' first appeared in print on 5 June 1873, in examination questions set by [[James Thomson (engineer)|James Thomson]] (brother of [[Lord Kelvin]]) at [[Queen's University Belfast|Queen's College]], [[Belfast]]. He had used the term as early as 1871, while in 1869, [[Thomas Muir (mathematician)|Thomas Muir]], then of the [[University of St Andrews]], vacillated between the terms ''rad'', ''radial'', and ''radian''. In 1874, after a consultation with James Thomson, Muir adopted ''radian''.<ref>{{cite book| author-link=Florian Cajori| first=Florian| last=Cajori| orig-year=1st Pub. 1929 | date=1993 | title=History of Mathematical Notations| volume=2| pages=[https://archive.org/details/historyofmathema00cajo_0/page/147 147–148]| publisher=Dover Publications| isbn=0-486-67766-4| url-access=registration| url=https://archive.org/details/historyofmathema00cajo_0/page/147}}</ref><ref>
*{{cite journal| journal=Nature| year=1910| volume= 83| pages=156|doi=10.1038/083156a0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2110|bibcode = 1910Natur..83..156M | s2cid=3958702| url=https://zenodo.org/record/1429528| doi-access=free}}
*{{cite journal| journal=Nature| year=1910| volume= 83| pages=217|doi=10.1038/083217c0| title=The Term "Radian" in Trigonometry| last1=Thomson| first1=James| issue=2112|bibcode = 1910Natur..83..217T | s2cid=3980250| url=https://zenodo.org/record/1429530| doi-access=free}}
*{{cite journal| journal=Nature| year=1910| volume= 83| pages=459–460|doi=10.1038/083459d0| title=The Term "Radian" in Trigonometry| last1=Muir| first1=Thos.| issue=2120|bibcode = 1910Natur..83..459M | s2cid=3971449| url=https://zenodo.org/record/1429528}}</ref><ref>{{cite web|url=http://jeff560.tripod.com/r.html|date=Nov 23, 2009| access-date=Sep 30, 2011|last=Miller|first=Jeff |title= Earliest Known Uses of Some of the Words of Mathematics}}</ref> The name ''radian'' was not universally adopted for some time after this. ''Longmans' School Trigonometry'' still called the radian ''circular measure'' when published in 1890.<ref>Frederick Sparks, ''Longmans' School Trigonometry'', p. 6, London: Longmans, Green, and Co., 1890 {{oclc|877238863}} (1891 edition)</ref>

In 1893 [[Alexander Macfarlane]] wrote "the true analytical argument for the circular ratios is not the ratio of the arc to the radius, but the ratio of twice the area of a sector to the square on the radius."<ref>A. Macfarlane (1893) "On the definitions of the trigonometric functions", page 9, [https://archive.org/details/principlesalgeb01macfgoog/page/n138/mode/2up link at Internet Archive]</ref> However, the paper was withdrawn from the published proceedings of mathematical congress held in connection with [[World's Columbian Exposition]] in Chicago (acknowledged at page 167), and privately published in his ''Papers on Space Analysis'' (1894). Macfarlane reached this idea or ratios of areas while considering the basis for [[hyperbolic angle]] which is analogously defined.<ref>{{wikibooks inline|Geometry/Unified Angles}}</ref>