Editing Sexagesimal (section)

===Other historical usages===
[[File:sexagenary_cycle_years.svg|thumb|Combinations of the five elements and twelve animals of the [[Chinese zodiac]] form the {{nowrap|60-year}} [[sexagenary cycle]].]]
In the [[Chinese calendar]], a system is commonly used in which days or years are named by positions in a sequence of ten stems and in another sequence of 12 branches. The same stem and branch repeat every 60 steps through this cycle.

Book VIII of [[Plato]]'s ''[[The Republic (Plato)|Republic]]'' involves an allegory of marriage centered on the number 60<sup>4</sup> = {{val|12,960,000}} and its divisors. This number has the particularly simple sexagesimal representation 1,0,0,0,0. Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.<ref>{{citation
 | last = Barton | first = George A.
 | title = On the Babylonian origin of Plato's nuptial number
 | year = 1908
 | journal = Journal of the American Oriental Society
 | volume = 29
 | pages = 210–219
 | doi = 10.2307/592627
 | jstor = 592627}}. {{citation
 | last = McClain |first = Ernest G.
 | author-link = Ernest G. McClain
 | year = 1974
 | title = Musical "Marriages" in Plato's "Republic"
 | journal = Journal of Music Theory
 | volume = 18
 | issue = 2
 | pages = 242–272
 | author2 = Plato
 | jstor = 843638}}</ref>

[[Ptolemy]]'s ''[[Almagest]]'', a treatise on [[mathematical astronomy]] written in the second century AD, uses base 60 to express the fractional parts of numbers. In particular, his [[table of chords]], which was essentially the only extensive [[trigonometric table]] for more than a millennium, has fractional parts of a degree in base&nbsp;60, and was practically equivalent to a modern-day table of values of the [[Sine and cosine|sine]] function.

Medieval astronomers also used sexagesimal numbers to note time. [[Al-Biruni]] first subdivided the hour sexagesimally into [[minute]]s, [[second]]s, [[third (angle)|third]]s and [[fourth (angle)|fourth]]s in 1000 while discussing Jewish months.<ref name="al-Biruni">{{ citation | author=Al-Biruni | year=1879 | orig-year=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 | author-link=Al-Biruni}}</ref> Around 1235 [[John of Sacrobosco]] continued this tradition, although Nothaft thought Sacrobosco was the first to do so.<ref>{{ Citation | last = Nothaft | first = C. Philipp E. | date = 2018 | title = Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe | publisher = Oxford University Press | place = Oxford | page = 126 | isbn = 9780198799559 | quote = Sacrobosco switched to sexagesimal fractions, but rendered them more congenial to computistical use by applying them not to the day but to the hour, thereby inaugurating the use of hours, minutes, and seconds that still prevails in the twenty-first century.}}</ref> The Parisian version of the [[Alfonsine tables]] (ca. 1320) used the day as the basic unit of time, recording multiples and fractions of a day in base-60 notation.<ref>{{ Citation | last = Nothaft | first = C. Philipp E. | date = 2018 | title = Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe | publisher = Oxford University Press | place = Oxford | page = 196 | isbn = 9780198799559 | quote = One noteworthy feature of the Alfonsine Tables in their Latin-Parisian incarnation is the strict 'sexagesimalization' of all tabulated parameters, as … motions and time intervals were consistently dissolved into base-60 multiples and fractions of days or degrees.}}</ref>

The sexagesimal number system continued to be frequently used by European astronomers for performing calculations as late as 1671.<ref>{{citation|last= Newton |first= Isaac |author-link= Isaac Newton |title= The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines. |year= 1671 |publication-date= 1736 |publisher= [[Henry Sampson Woodfall|Henry Woodfall]] |location= [[London]] |page= 146 |url= https://books.google.com/books?id=WyQOAAAAQAAJ&pg=PA146 |quote= The most remarkable of these is the Sexagenary or Sexagesimal Scale of Arithmetick, of frequent use among Astronomers, which expresses all possible Numbers, Integers or Fractions, Rational or Surd, by the Powers of ''Sixty'', and certain numeral Coefficients not exceeding fifty-nine.}}</ref> For instance, [[Jost Bürgi]] in ''[[Fundamentum Astronomiae]]'' (presented to [[Emperor Rudolf II]] in 1592), his colleague Ursus in ''Fundamentum Astronomicum'', and possibly also [[Henry Briggs (mathematician)|Henry Briggs]], used multiplication tables based on the sexagesimal system in the late 16th century, to calculate sines.<ref name="folkerts">{{citation
 | last1 = Folkerts | first1 = Menso
 | last2 = Launert | first2 = Dieter
 | last3 = Thom | first3 = Andreas
 | arxiv = 1510.03180
 | doi = 10.1016/j.hm.2016.03.001
 | issue = 2
 | journal = Historia Mathematica
 | mr = 3489006
 | pages = 133–147
 | title = Jost Bürgi's method for calculating sines
 | volume = 43
 | year = 2016| s2cid = 119326088
 }}</ref>

In the late 18th and early 19th centuries, [[Tamil people|Tamil]] astronomers were found to make astronomical calculations, reckoning with shells using a mixture of decimal and sexagesimal notations developed by [[Hellenistic#Indian reference|Hellenistic]] astronomers.<ref>{{citation
| title=Tamil Astronomy: A Study in the History of Astronomy in India
| first=Otto
| last=Neugebauer
| author-link= Otto Neugebauer 
| journal=Osiris
| volume=10
| pages=252–276
| year=1952
| doi=10.1086/368555
| s2cid=143591575
}}; reprinted in {{Citation
| first=Otto
| last=Neugebauer
| author-link= Otto Neugebauer 
| title = Astronomy and History: Selected Essays
| place = New York
| publisher = [[Springer-Verlag]]
| year = 1983
| bibcode=1983ahse.book.....N
| isbn = 0-387-90844-7}}</ref>

Base-60 number systems have also been used in some other cultures that are unrelated to the Sumerians, for example by the [[Ekari people]] of [[Western New Guinea]].<ref>{{citation
 |title       = Kapauku numeration: Reckoning, racism, scholarship, and Melanesian counting systems
 |first       = Nancy
 |last        = Bowers
 |journal     = Journal of the Polynesian Society
 |volume      = 86
 |issue       = 1
 |pages       = 105–116
 |year        = 1977
 |url         = http://www.ethnomath.org/resources/bowers1977.pdf
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20090305190121/http://www.ethnomath.org/resources/bowers1977.pdf
 |archive-date = 2009-03-05
}}</ref><ref>{{citation
 |first       = Glendon Angove
 |last        = Lean
 |title       = Counting Systems of Papua New Guinea and Oceania
 |publisher   = Ph.D. thesis, [[Papua New Guinea University of Technology]]
 |year        = 1992
 |url         = http://www.uog.ac.pg/glec/thesis/thesis.htm
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20070905170848/http://www.uog.ac.pg/glec/thesis/thesis.htm
 |archive-date = 2007-09-05
}}. See especially [http://www.uog.ac.pg/glec/thesis/ch4web/ch4.htm chapter 4] {{webarchive|url=https://web.archive.org/web/20070928061154/http://www.uog.ac.pg/glec/thesis/ch4web/ch4.htm |date=2007-09-28 }}.</ref>