Sexagesimal

Sexagesimal,also known asbase 60,^[1]is anumeral systemwithsixtyas itsbase.It originated with the ancientSumeriansin the 3rd millennium BC, was passed down to the ancientBabylonians,and is still used—in a modified form—for measuringtime,angles,andgeographic coordinates.

The number 60, asuperior highly composite number,has twelvefactors,namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 areprime numbers.With so many factors, manyfractionsinvolving sexagesimal numbers are simplified. For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute. 60 is the smallest number that is divisible by every number from 1 to 6; that is, it is thelowest common multipleof 1, 2, 3, 4, 5, and 6.

In this article, all sexagesimal digits are represented as decimal numbers, except where otherwise noted. For example, the largest sexagesimal digit is "59".

Origin[edit]

According toOtto Neugebauer,the origins of sexagesimal are not as simple, consistent, or singular in time as they are often portrayed. Throughout their many centuries of use, which continues today for specialized topics such as time, angles, and astronomical coordinate systems, sexagesimal notations have always contained a strong undercurrent of decimal notation, such as in how sexagesimal digits are written. Their use has also always included (and continues to include) inconsistencies in where and how various bases are to represent numbers even within a single text.^[2]

The most powerful driver for rigorous, fully self-consistent use of sexagesimal has always been its mathematical advantages for writing and calculating fractions. In ancient texts this shows up in the fact that sexagesimal is used most uniformly and consistently in mathematical tables of data.^[2]Another practical factor that helped expand the use of sexagesimal in the past even if less consistently than in mathematical tables, was its decided advantages to merchants and buyers for making everyday financial transactions easier when they involved bargaining for and dividing up larger quantities of goods. In the late 3rd millennium BC, Sumerian/Akkadian units of weight included thekakkaru(talent,approximately 30 kg) divided into 60manû(mina), which was further subdivided into 60šiqlu(shekel); the descendants of these units persisted for millennia, though the Greeks later coerced this relationship into the more base-10 compatible ratio of ashekelbeing one 50th of amina.

Apart from mathematical tables, the inconsistencies in how numbers were represented within most texts extended all the way down to the most basiccuneiformsymbols used to represent numeric quantities.^[2]For example, the cuneiform symbol for 1 was an ellipse made by applying the rounded end of the stylus at an angle to the clay, while the sexagesimal symbol for 60 was a larger oval or "big 1". But within the same texts in which these symbols were used, the number 10 was represented as a circle made by applying the round end of the style perpendicular to the clay, and a larger circle or "big 10" was used to represent 100. Such multi-base numeric quantity symbols could be mixed with each other and with abbreviations, even within a single number. The details and even the magnitudes implied (sincezero was not used consistently) were idiomatic to the particular time periods, cultures, and quantities or concepts being represented. While such context-dependent representations of numeric quantities are easy to critique in retrospect, in modern times we still have dozens of regularly used examples of topic-dependent base mixing, including the recent innovation of adding decimal fractions to sexagesimal astronomical coordinates.^[2]

Usage[edit]

Babylonian mathematics[edit]

The sexagesimal system as used in ancientMesopotamiawas not a pure base-60 system, in the sense that it did not use 60 distinct symbols for itsdigits.Instead, thecuneiformdigits usedtenas a sub-base in the fashion of asign-value notation:a sexagesimal digit was composed of a group of narrow, wedge-shaped marks representing units up to nine (,,,,...,) and a group of wide, wedge-shaped marks representing up to five tens (,,,,). The value of the digit was the sum of the values of its component parts:

Numbers larger than 59 were indicated by multiple symbol blocks of this form inplace value notation.Because there was no symbol forzeroit is not always immediately obvious how a number should be interpreted, and its true value must sometimes have been determined by its context. For example, the symbols for 1 and 60 are identical.^[3]^[4]Later Babylonian texts used a placeholder () to represent zero, but only in the medial positions, and not on the right-hand side of the number, as in numbers like13200.^[4]

Other historical usages[edit]

In theChinese 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 ofPlato'sRepublicinvolves an allegory of marriage centered on the number 60⁴=12960000and 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.^[5]

Ptolemy'sAlmagest,a treatise onmathematical astronomywritten in the second century AD, uses base 60 to express the fractional parts of numbers. In particular, histable of chords,which was essentially the only extensivetrigonometric tablefor more than a millennium, has fractional parts of a degree in base 60, and was practically equivalent to a modern-day table of values of thesinefunction.

Medieval astronomers also used sexagesimal numbers to note time.Al-Birunifirst subdivided the hour sexagesimally intominutes,seconds,thirdsandfourthsin 1000 while discussing Jewish months.^[6]Around 1235John of Sacroboscocontinued this tradition, although Nothaft thought Sacrobosco was the first to do so.^[7]The Parisian version of theAlfonsine tables(ca. 1320) used the day as the basic unit of time, recording multiples and fractions of a day in base-60 notation.^[8]

The sexagesimal number system continued to be frequently used by European astronomers for performing calculations as late as 1671.^[9]For instance,Jost BürgiinFundamentum Astronomiae(presented toEmperor Rudolf IIin 1592), his colleague Ursus inFundamentum Astronomicum,and possibly alsoHenry Briggs,used multiplication tables based on the sexagesimal system in the late 16th century, to calculate sines.^[10]

In the late 18th and early 19th centuries,Tamilastronomers were found to make astronomical calculations, reckoning with shells using a mixture of decimal and sexagesimal notations developed byHellenisticastronomers.^[11]

Base-60 number systems have also been used in some other cultures that are unrelated to the Sumerians, for example by theEkari peopleofWestern New Guinea.^[12]^[13]

Modern usage[edit]

Modern uses for the sexagesimal system include measuringangles,geographic coordinates,electronic navigation, andtime.^[14]

Onehourof time is divided into 60minutes,and one minute is divided into 60 seconds. Thus, a measurement of time such as 3:23:17(3 hours, 23 minutes, and 17 seconds)can be interpreted as a whole sexagesimal number (no sexagesimal point), meaning3 × 60²+ 23 × 60¹+ 17 × 60⁰seconds.However, each of the three sexagesimal digits in this number (3, 23, and 17) is written using thedecimalsystem.

Similarly, the practical unit of angular measure is thedegree,of which there are360(six sixties) in a circle. There are 60minutes of arcin a degree, and 60arcsecondsin a minute.

YAML[edit]

In version 1.1^[15]of theYAMLdata storage format, sexagesimals are supported for plain scalars, and formally specified both for integers^[16]and floating point numbers.^[17]This has led to confusion, as e.g. someMAC addresseswould be recognised as sexagesimals and loaded as integers, where others were not and loaded as strings. In YAML 1.2 support for sexagesimals was dropped.^[18]

Notations[edit]

InHellenistic Greekastronomical texts, such as the writings ofPtolemy,sexagesimal numbers were written usingGreek alphabetic numerals,with each sexagesimal digit being treated as a distinct number. Hellenistic astronomers adopted a new symbol for zero,—°,which morphed over the centuries into other forms, including the Greek letter omicron, ο, normally meaning 70, but permissible in a sexagesimal system where the maximum value in any position is 59.^[19]^[20]The Greeks limited their use of sexagesimal numbers to the fractional part of a number.^[21]

In medieval Latin texts, sexagesimal numbers were written usingArabic numerals;the different levels of fractions were denotedminuta(i.e., fraction),minuta secunda,minuta tertia,etc. By the 17th century it became common to denote the integer part of sexagesimal numbers by a superscripted zero, and the various fractional parts by one or more accent marks.John Wallis,in hisMathesis universalis,generalized this notation to include higher multiples of 60; giving as an example the number49‵‵‵‵36‵‵‵25‵‵15‵1°15′2″36‴49⁗;where the numbers to the left are multiplied by higher powers of 60, the numbers to the right are divided by powers of 60, and the number marked with the superscripted zero is multiplied by 1.^[22]This notation leads to the modern signs for degrees, minutes, and seconds. The same minute and second nomenclature is also used for units of time, and the modern notation for time with hours, minutes, and seconds written in decimal and separated from each other by colons may be interpreted as a form of sexagesimal notation.

In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots:primeorprimus,secondeorsecundus,tierce,quatre,quinte,etc. To this day we call the second-order partof an hourorof a degreea "second". Until at least the 18th century,⁠1/60⁠of a second was called a "tierce" or "third".^[23]^[24]

In the 1930s,Otto Neugebauerintroduced a modern notational system for Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integer and fractional portions of the number and using a comma (,) to separate the positions within each portion.^[25]For example, the meansynodic monthused by both Babylonian and Hellenistic astronomers and still used in theHebrew calendaris 29;31,50,8,20 days. This notation is used in this article.

Fractions and irrational numbers[edit]

Fractions[edit]

In the sexagesimal system, anyfractionin which thedenominatoris aregular number(having only 2, 3, and 5 in itsprime factorization) may be expressed exactly.^[26]Shown here are all fractions of this type in which the denominator is less than or equal to 60:

1⁄2= 0;30

1⁄3= 0;20

1⁄4= 0;15

1⁄5= 0;12

1⁄6= 0;10

1⁄8= 0;7,30

1⁄9= 0;6,40

1⁄10= 0;6

1⁄12= 0;5

1⁄15= 0;4

1⁄16= 0;3,45

1⁄18= 0;3,20

1⁄20= 0;3

1⁄24= 0;2,30

1⁄25= 0;2,24

1⁄27= 0;2,13,20

1⁄30= 0;2

1⁄32= 0;1,52,30

1⁄36= 0;1,40

1⁄40= 0;1,30

1⁄45= 0;1,20

1⁄48= 0;1,15

1⁄50= 0;1,12

1⁄54= 0;1,6,40

1⁄60= 0;1

However numbers that are not regular form more complicatedrepeating fractions.For example:

1⁄7= 0;8,34,17(the bar indicates the sequence of sexagesimal digits 8,34,17 repeats infinitely many times)

1⁄11= 0;5,27,16,21,49

1⁄13= 0;4,36,55,23

1⁄14= 0;4,17,8,34

1⁄17= 0;3,31,45,52,56,28,14,7

1⁄19= 0;3,9,28,25,15,47,22,6,18,56,50,31,34,44,12,37,53,41

1⁄59= 0;1

1⁄61= 0;0,59

The fact that the two numbers that are adjacent to sixty, 59 and 61, are both prime numbers implies that fractions that repeat with a period of one or two sexagesimal digits can only have regular number multiples of 59 or 61 as their denominators, and that other non-regular numbers have fractions that repeat with a longer period.

Irrational numbers[edit]

Babylonian tabletYBC 7289showing the sexagesimal number1;24,51,10approximating√2

The representations ofirrational numbersin any positional number system (including decimal and sexagesimal) neither terminate norrepeat.

Thesquare root of 2,the length of thediagonalof aunit square,was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as

1;24,51,10=1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}={\frac {30547}{21600}}\approx 1.41421296\ldots

^[27]

Because√2≈1.41421356... is anirrational number,it cannot be expressed exactly in sexagesimal (or indeed any integer-base system), but its sexagesimal expansion does begin 1;24,51,10,7,46,6,4,44... (OEIS:A070197)

The value of $π$ as used by theGreekmathematician and scientistPtolemywas 3;8,30 =3 +⁠8/60⁠+⁠30/60²⁠=⁠377/120⁠≈3.141666....^[28]Jamshīd al-Kāshī,a 15th-centuryPersianmathematician, calculated 2 $π$ as a sexagesimal expression to its correct value when rounded to nine subdigits (thus to⁠1/60⁹⁠); his value for 2 $π$ was 6;16,59,28,1,34,51,46,14,50.^[29]^[30]Like√2above, 2 $π$ is an irrational number and cannot be expressed exactly in sexagesimal. Its sexagesimal expansion begins 6;16,59,28,1,34,51,46,14,49,55,12,35... (OEIS:A091649)

References[edit]

^Pronounced/sɛksəˈdʒɛsɪməl/and/sɛkˈsædʒɪnəri/;see"sexagesimal",Oxford English Dictionary(Online ed.),Oxford University Press(subscription orparticipating institution membershiprequired)
^^a ^b ^c ^dNeugebauer, O. (1969), "The Exact Sciences In Antiquity",Acta Historica Scientiarum Naturalium et Medicinalium,9,Dover: 17–19,ISBN 0-486-22332-9,PMID 14884919
^Bello, Ignacio; Britton, Jack R.; Kaul, Anton (2009),Topics in Contemporary Mathematics(9th ed.), Cengage Learning, p. 182,ISBN 9780538737791.
^^a ^bLamb, Evelyn (August 31, 2014),"Look, Ma, No Zero!",Scientific American,Roots of Unity
^Barton, George A. (1908), "On the Babylonian origin of Plato's nuptial number",Journal of the American Oriental Society,29:210–219,doi:10.2307/592627,JSTOR 592627.McClain, Ernest G.;Plato (1974), "Musical" Marriages "in Plato's" Republic "",Journal of Music Theory,18(2): 242–272,doi:10.2307/843638,JSTOR 843638
^Al-Biruni(1879) [1000],The Chronology of Ancient Nations,translated by Sachau, C. Edward, pp. 147–149
^Nothaft, C. Philipp E. (2018),Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe,Oxford: Oxford University Press, p. 126,ISBN 9780198799559,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.
^Nothaft, C. Philipp E. (2018),Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe,Oxford: Oxford University Press, p. 196,ISBN 9780198799559,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.
^Newton, Isaac(1671),The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines.,London:Henry Woodfall(published 1736), p. 146,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 ofSixty,and certain numeral Coefficients not exceeding fifty-nine.
^Folkerts, Menso; Launert, Dieter; Thom, Andreas (2016), "Jost Bürgi's method for calculating sines",Historia Mathematica,43(2): 133–147,arXiv:1510.03180,doi:10.1016/j.hm.2016.03.001,MR 3489006,S2CID 119326088
^Neugebauer, Otto(1952), "Tamil Astronomy: A Study in the History of Astronomy in India",Osiris,10:252–276,doi:10.1086/368555,S2CID 143591575;reprinted inNeugebauer, Otto(1983),Astronomy and History: Selected Essays,New York:Springer-Verlag,ISBN 0-387-90844-7
^Bowers, Nancy (1977),"Kapauku numeration: Reckoning, racism, scholarship, and Melanesian counting systems"(PDF),Journal of the Polynesian Society,86(1): 105–116, archived fromthe original(PDF)on 2009-03-05
^Lean, Glendon Angove (1992),Counting Systems of Papua New Guinea and Oceania,Ph.D. thesis,Papua New Guinea University of Technology,archived fromthe originalon 2007-09-05.See especiallychapter 4 Archived2007-09-28 at theWayback Machine.
^Powell, Marvin A. (2008). "Sexagesimal System".Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures.pp. 1998–1999.doi:10.1007/978-1-4020-4425-0_9055.ISBN 978-1-4020-4559-2.
^"YAML Ain't Markup Language (YAML™) Version 1.1".
^"Integer Language-Independent Type for YAML Version 1.1".
^"Floating-Point Language-Independent Type for YAML™ Version 1.1".
^Oren Ben-Kiki; Clark Evans; Brian Ingerson (2009-10-01),"YAML Ain't Markup Language (YAML™) Version 1.2 (3rd Edition, Patched at 2009-10-01) §10.3.2 Tag Resolution",The Official YAML Web Site,retrieved2019-01-30
^Neugebauer, Otto(1969) [1957], "The Exact Sciences in Antiquity",Acta Historica Scientiarum Naturalium et Medicinalium,9(2 ed.),Dover Publications:13–14, plate 2,ISBN 978-0-486-22332-2,PMID 14884919
^Mercier, Raymond,"Consideration of the Greek symbol 'zero'"(PDF),Home of Kairos
^Aaboe, Asger(1964),Episodes from the Early History of Mathematics,New Mathematical Library, vol. 13, New York: Random House, pp. 103–104
^Cajori, Florian(2007) [1928],A History of Mathematical Notations,vol. 1, New York: Cosimo, Inc., p. 216,ISBN 9781602066854
^Wade, Nicholas (1998),A natural history of vision,MIT Press, p. 193,ISBN 978-0-262-73129-4
^Lewis, Robert E. (1952),Middle English Dictionary,University of Michigan Press, p. 231,ISBN 978-0-472-01212-1
^Neugebauer, Otto;Sachs, Abraham Joseph;Götze, Albrecht (1945),Mathematical Cuneiform Texts,American Oriental Series, vol. 29, New Haven: American Oriental Society and the American Schools of Oriental Research, p. 2
^Neugebauer, Otto E.(1955),Astronomical Cuneiform Texts,London: Lund Humphries
^Fowler, David;Robson, Eleanor(1998), "Square root approximations in old Babylonian mathematics: YBC 7289 in context",Historia Mathematica,25(4): 366–378,doi:10.1006/hmat.1998.2209,MR 1662496
^Toomer, G. J.,ed. (1984),Ptolemy's Almagest,New York: Springer Verlag, p. 302,ISBN 0-387-91220-7
^Youschkevitch, Adolf P., "Al-Kashi", in Rosenfeld, Boris A. (ed.),Dictionary of Scientific Biography,p. 256.
^Aaboe (1964),p. 125

External links[edit]

"Facts on the Calculation of Degrees and Minutes"is an Arabic language book bySibṭ al-Māridīnī, Badr al-Dīn Muḥammad ibn Muḥammad(b. 1423). This work offers a very detailed treatment of sexagesimal mathematics and includes what appears to be the first mention of the periodicity of sexagesimal fractions.

[1] Pronounced/sɛksəˈdʒɛsɪməl/and/sɛkˈsædʒɪnəri/;see"sexagesimal",Oxford English Dictionary(Online ed.),Oxford University Press(subscription orparticipating institution membershiprequired)

[Neugebauer-2] Neugebauer, O. (1969), "The Exact Sciences In Antiquity",Acta Historica Scientiarum Naturalium et Medicinalium,9,Dover: 17–19,ISBN 0-486-22332-9,PMID 14884919

[3] Bello, Ignacio; Britton, Jack R.; Kaul, Anton (2009),Topics in Contemporary Mathematics(9th ed.), Cengage Learning, p. 182,ISBN 9780538737791.

[lmnz-4] Lamb, Evelyn (August 31, 2014),"Look, Ma, No Zero!",Scientific American,Roots of Unity

[5] Barton, George A. (1908), "On the Babylonian origin of Plato's nuptial number",Journal of the American Oriental Society,29:210–219,doi:10.2307/592627,JSTOR 592627.McClain, Ernest G.;Plato (1974), "Musical" Marriages "in Plato's" Republic "",Journal of Music Theory,18(2): 242–272,doi:10.2307/843638,JSTOR 843638

[al-Biruni-6] Al-Biruni(1879) [1000],The Chronology of Ancient Nations,translated by Sachau, C. Edward, pp. 147–149

[7] Nothaft, C. Philipp E. (2018),Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe,Oxford: Oxford University Press, p. 126,ISBN 9780198799559,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.

[8] Nothaft, C. Philipp E. (2018),Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe,Oxford: Oxford University Press, p. 196,ISBN 9780198799559,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.

[9] Newton, Isaac(1671),The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines.,London:Henry Woodfall(published 1736), p. 146,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 ofSixty,and certain numeral Coefficients not exceeding fifty-nine.

[folkerts-10] Folkerts, Menso; Launert, Dieter; Thom, Andreas (2016), "Jost Bürgi's method for calculating sines",Historia Mathematica,43(2): 133–147,arXiv:1510.03180,doi:10.1016/j.hm.2016.03.001,MR 3489006,S2CID 119326088

[11] Neugebauer, Otto(1952), "Tamil Astronomy: A Study in the History of Astronomy in India",Osiris,10:252–276,doi:10.1086/368555,S2CID 143591575;reprinted inNeugebauer, Otto(1983),Astronomy and History: Selected Essays,New York:Springer-Verlag,ISBN 0-387-90844-7

[12] Bowers, Nancy (1977),"Kapauku numeration: Reckoning, racism, scholarship, and Melanesian counting systems"(PDF),Journal of the Polynesian Society,86(1): 105–116, archived fromthe original(PDF)on 2009-03-05

[13] Lean, Glendon Angove (1992),Counting Systems of Papua New Guinea and Oceania,Ph.D. thesis,Papua New Guinea University of Technology,archived fromthe originalon 2007-09-05.See especiallychapter 4 Archived2007-09-28 at theWayback Machine.

[14] Powell, Marvin A. (2008). "Sexagesimal System".Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures.pp. 1998–1999.doi:10.1007/978-1-4020-4425-0_9055.ISBN 978-1-4020-4559-2.

[15] "YAML Ain't Markup Language (YAML™) Version 1.1".

[16] "Integer Language-Independent Type for YAML Version 1.1".

[17] "Floating-Point Language-Independent Type for YAML™ Version 1.1".

[18] Oren Ben-Kiki; Clark Evans; Brian Ingerson (2009-10-01),"YAML Ain't Markup Language (YAML™) Version 1.2 (3rd Edition, Patched at 2009-10-01) §10.3.2 Tag Resolution",The Official YAML Web Site,retrieved2019-01-30

[19] Neugebauer, Otto(1969) [1957], "The Exact Sciences in Antiquity",Acta Historica Scientiarum Naturalium et Medicinalium,9(2 ed.),Dover Publications:13–14, plate 2,ISBN 978-0-486-22332-2,PMID 14884919

[Mercier-20] Mercier, Raymond,"Consideration of the Greek symbol 'zero'"(PDF),Home of Kairos

[21] Aaboe, Asger(1964),Episodes from the Early History of Mathematics,New Mathematical Library, vol. 13, New York: Random House, pp. 103–104

[22] Cajori, Florian(2007) [1928],A History of Mathematical Notations,vol. 1, New York: Cosimo, Inc., p. 216,ISBN 9781602066854

[23] Wade, Nicholas (1998),A natural history of vision,MIT Press, p. 193,ISBN 978-0-262-73129-4

[24] Lewis, Robert E. (1952),Middle English Dictionary,University of Michigan Press, p. 231,ISBN 978-0-472-01212-1

[25] Neugebauer, Otto;Sachs, Abraham Joseph;Götze, Albrecht (1945),Mathematical Cuneiform Texts,American Oriental Series, vol. 29, New Haven: American Oriental Society and the American Schools of Oriental Research, p. 2

[26] Neugebauer, Otto E.(1955),Astronomical Cuneiform Texts,London: Lund Humphries

[27] Fowler, David;Robson, Eleanor(1998), "Square root approximations in old Babylonian mathematics: YBC 7289 in context",Historia Mathematica,25(4): 366–378,doi:10.1006/hmat.1998.2209,MR 1662496

[28] Toomer, G. J.,ed. (1984),Ptolemy's Almagest,New York: Springer Verlag, p. 302,ISBN 0-387-91220-7

[29] Youschkevitch, Adolf P., "Al-Kashi", in Rosenfeld, Boris A. (ed.),Dictionary of Scientific Biography,p. 256.

[30] Aaboe (1964),p. 125

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