Editing Magic square (section)

===Middle East, North Africa, Muslim Iberia===
[[File:16th century arabic magic square.jpg|thumb|A 6×6 magic square from ''Book of Wonders'' (from 16th century manuscript).]]
Although the early history of magic squares in Persia and Arabia is not known, it has been suggested that they were known in pre-Islamic times.<ref>J. P. Hogendijk, A. I. Sabra, ''The Enterprise of Science in Islam: New Perspectives'', Published by MIT Press, 2003, {{isbn|0-262-19482-1}}, p. xv.</ref>
It is clear, however, that the study of magic squares was common in [[Islamic Golden Age|medieval Islam]], and it was thought to have begun after the introduction of [[chess]] into the region.<ref>[[Helaine Selin]], [[Ubiratan D'Ambrosio]], ''Mathematics Across Cultures: The History of Non-Western Mathematics'', Published by Springer, 2001, {{isbn|1-4020-0260-2}}, p. 160.</ref><ref name="Sesiano2003">{{cite journal| last = Sesiano | first= Jacques| title=Construction of magic squares using the knight's move in Islamic mathematics | journal=Archive for History of Exact Sciences | volume = 58 | issue = 1 | pages= 1–20| date=November 2003 | url=http://doc.rero.ch/record/316928/files/407_2003_Article_71.pdf | doi= 10.1007/s00407-003-0071-4| s2cid= 123219466}}</ref><ref name="Sesiano1997">{{cite book| last = Sesiano | first= Jacques| chapter=Magic squares in Islamic mathematics | title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures | pages=1259–1260 | date=1997 }}</ref> The first dateable appearance of a magic square of order 3 occurs in [[Jabir ibn Hayyan|Jābir ibn Hayyān]]'s (fl. c. 721 – c. 815) ''Kitab al-mawazin al-Saghir'' (The Small Book of Balances) where the magic square and its related numerology is associated with alchemy.<ref name="Swetz2008"/> While it is known that treatises on magic squares were written in the 9th century, the earliest extant treaties date from the 10th-century: one by [[Buzjani|Abu'l-Wafa al-Buzjani]] ({{circa|998}}) and another by Ali b. Ahmad al-Antaki ({{circa|987}}).<ref name="Sesiano2003"/><ref name="Sesiano2007">{{cite book| last = Sesiano | first= Jacques| title=Magic squares in the tenth century: Two Arabic treatises by Antaki and Buzjani| publisher=Springer | date=2007 }}</ref><ref>Sesiano, J., ''Abūal-Wafā\rasp's treatise on magic squares'' (French), Z. Gesch. Arab.-Islam. Wiss. 12 (1998), 121–244.</ref> These early treatises were purely mathematical, and the Arabic designation for magic squares used is ''wafq al-a'dad'', which translates as ''harmonious disposition of the numbers''.<ref name="Sesiano1997"/> By the end of 10th century, the two treatises by Buzjani and Antaki makes it clear that the Middle Eastern mathematicians had understood how to construct bordered squares of any order as well as simple magic squares of small orders (''n'' ≤ 6) which were used to make composite magic squares.<ref name="Sesiano2003"/><ref name="Sesiano2007"/> A specimen of magic squares of orders 3 to 9 devised by Middle Eastern mathematicians appear in an encyclopedia from [[Baghdad]] {{circa|983}}, the [[Rasa'il Ikhwan al-Safa]] (the ''[[Encyclopedia of the Brethren of Purity]]'').<ref name="Cammann1969a">{{cite journal| last = Cammann | first= Schuyler| title=Islamic and Indian Magic Squares, Part I | journal=History of Religions| volume = 8 | issue = 3 | pages= 181–209| date=February 1969 | doi= 10.1086/462584| s2cid= 162095408}}</ref> The squares of order 3 to 7 from Rasa'il are given below:<ref name="Cammann1969a"/>

{{col-begin|width=auto;margin:0.5em auto}}
{{col-break|valign=bottom}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
|-
| 2 || 7 || 6
|-
| 9 || 5 || 1
|-
| 4 || 3 || 8
|}

{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:8em;height:8em;table-layout:fixed;"
|-
| style="background-color: silver;"|4 || 14 || 15 || style="background-color: silver;"|1
|-
| 9 || style="background-color: silver;"|7 || style="background-color: silver;"|6 || 12
|-
| 5 || style="background-color: silver;"|11 || style="background-color: silver;"|10 || 8
|-
| style="background-color: silver;"|16 || 2 || 3 || style="background-color: silver;"|13
|}

{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:10em;height:10em;table-layout:fixed;"
|-
| style="background-color: silver;"|21 || 3 || 4 || 12 || style="background-color: silver;"|25
|-
| 15 || style="background-color: silver;"|17 || 6 || style="background-color: silver;"|19 || 8
|-
| 10 || 24 || style="background-color: silver;"|13 || 2 || 16
|-
| 18 || style="background-color: silver;"|7 || 20 || style="background-color: silver;"|9 || 11
|-
| style="background-color: silver;"|1 || 14 || 22 || 23 || style="background-color: silver;"|5
|}

{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:12em;height:12em;table-layout:fixed;"
|-
| style="background-color: silver;"|11 || style="background-color: silver;"|22 || 32 || 5 || style="background-color: silver;"|23 || style="background-color: silver;"|18
|-
| style="background-color: silver;"|25 || style="background-color: silver;"|16 || 7 || 30 || style="background-color: silver;"|13 || style="background-color: silver;"|20
|- 
| 27 || 6 || 35 || 36 || 4 || 3
|-
| 10 || 31 || 1 || 2 || 33 || 34
|-
| style="background-color: silver;"|14 || style="background-color: silver;"|19 || 8 || 29 || style="background-color: silver;"|26 || style="background-color: silver;"|15
|-
| style="background-color: silver;"|24 || style="background-color: silver;"|17 || 28 || 9 || style="background-color: silver;"|12 || style="background-color: silver;"|21
|}

{{col-break|valign=bottom|gap=1em}}
{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:14em;height:14em;table-layout:fixed;"
|-
| style="background-color: silver;"|47 || style="background-color: silver;"|11 || style="background-color: silver;"|8 || style="background-color: silver;"|9 || style="background-color: silver;"|6 || style="background-color: silver;"|45 || style="background-color: silver;"|49
|-
| style="background-color: silver;"|4 || 37 || 20 || 17 || 16 || 35 || style="background-color: silver;"|46
|- 
| style="background-color: silver;"|2 || 18 || style="background-color: silver;"|26 || style="background-color: silver;"|21 || style="background-color: silver;"|28 || 32 || style="background-color: silver;"|48
|-
| style="background-color: silver;"|43 || 19 || style="background-color: silver;"|27 || style="background-color: silver;"|25 || style="background-color: silver;"|23 || 31 || style="background-color: silver;"|7
|-
| style="background-color: silver;"|38 || 36 || style="background-color: silver;"|22 || style="background-color: silver;"|29 || style="background-color: silver;"|24 || 14 || style="background-color: silver;"|12
|-
| style="background-color: silver;"|40 || 15 || 30 || 33 || 34 || 13 || style="background-color: silver;"|10
|-
| style="background-color: silver;"|1 || style="background-color: silver;"|39 || style="background-color: silver;"|42 || style="background-color: silver;"|41 || style="background-color: silver;"|44 || style="background-color: silver;"|5 || style="background-color: silver;"|3
|}
{{col-end}}

The 11th century saw the finding of several ways to construct simple magic squares for odd and evenly even orders; the more difficult case of oddly even case (''n = 4k + 2'') was solved by [[Ibn al-Haytham]] with ''k'' even (c. 1040), and completely by the beginning of 12th century, if not already in the latter half of the 11th century.<ref name="Sesiano2003"/> Around the same time, pandiagonal squares were being constructed. Treaties on magic squares were numerous in the 11th and 12th century. These later developments tended to be improvements on or simplifications of existing methods. From the 13th century, magic squares were increasingly put to occult purposes.<ref name="Sesiano2003"/> However, much of these later texts written for occult purposes merely depict certain magic squares and mention their attributes, without describing their principle of construction, with only some authors keeping the general theory alive.<ref name="Sesiano2003"/> One such occultist was the Algerian [[Ahmad al-Buni]] (c. 1225), who gave general methods on constructing bordered magic squares; some others were the 17th century Egyptian Shabramallisi and the 18th century Nigerian al-Kishnawi.<ref name="Sesiano1994">{{cite journal| last = Sesiano | first= Jacques| title=Quelques methodes arabes de construction des carres magiques impairs (some Arabic construction methods of odd magical squares)| journal=Bulletin de la Société Vaudoise des Sciences Naturelles | language=fr | volume = 83 | issue = 1 | pages= 51–76 | date=2004 }}</ref>

The magic square of order three was described as a child-bearing charm<ref>Peter, J. Barta, The Seal-Ring of Proportion and the magic rings (2016), pp.&nbsp;6–9.</ref><ref name="Needham1987">{{cite book| last = Needham | first= Joseph | title=Theoretical Influences of China on Arabic Alchemy| publisher=UC Biblioteca Geral 1 | date=1987 | url=https://books.google.com/books?id=4b2zC7fL558C}}</ref> since its first literary appearances in the alchemical works of [[Jabir ibn Hayyan|Jābir ibn Hayyān]] (fl. c. 721 – c. 815)<ref name="Needham1987"/><ref>Jābir ibn Hayyān, Book of the Scales. French translation in: Marcelin Berthelot (1827–1907), Histoire de sciences. La chimie au moyen âge, Tom. III: L'alchimie arabe. Paris, 1893. [rprt.. Osnabruck: O. Zeller, 1967], pp.&nbsp;139–162, in particular: pp.&nbsp;150–151</ref> and [[Al-Ghazali|al-Ghazālī]] (1058–1111)<ref>al-Ghazālī, Deliverance From Error (al-munqidh min al-ḍalāl ) ch. 145. Arabic: al-Munkidh min al-dalal. ed. J. Saliba – K. Ayyad. Damascus: Maktab al-Nashr al-'Arabi, 1934, p. 79. English tr.: Richard Joseph McCarthy, Freedom and Fulfillment: An annotated translation of al-Ghazali's al-Munkidh min al-Dalal and other relevant works of al-Ghazali. Boston, Twayer, 1980. He refers a book titled 'The Marvels of Special Properties' as his source. This square was named in the Orient as the ''Seal of Ghazali'' after him.</ref> and it was preserved in the tradition of the planetary tables. The earliest occurrence of the association of seven magic squares to the virtues of the seven heavenly bodies appear in Andalusian scholar [[Abū Ishāq Ibrāhīm al-Zarqālī|Ibn Zarkali]]'s (known as Azarquiel in Europe) (1029–1087) ''Kitāb tadbīrāt al-kawākib'' (''Book on the Influences of the Planets'').<ref name="Comes2016"/> A century later, the Algerian scholar Ahmad al-Buni attributed mystical properties to magic squares in his highly influential book ''Shams al-Ma'arif'' (''The Book of the Sun of Gnosis and the Subtleties of Elevated Things''), which also describes their construction. This tradition about a series of magic squares from order three to nine, which are associated with the seven planets, survives in Greek, Arabic, and Latin versions.<ref>The Latin version is Liber de septem figuris septem planetarum figurarum Geberi regis Indorum. This treatise is the identified source of Dürer and Heinrich Cornelius Agrippa von Nettesheim. Cf. Peter, J. Barta, The Seal-Ring of Proportion and the magic rings (2016), pp.&nbsp;8–9, n. 10</ref> There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.<ref name="Sesiano2004">{{cite book| last = Sesiano | first= Jacques| title=Les carrés magiques dans les pays islamiques| publisher=PPUR presses polytechniques | language=fr | date=2004 }}</ref><ref name="Schimmel1993">{{cite book| last = Schimmel | first= Annemarie| title=The mystery of numbers| publisher=Oxford University Press | location = New York | date=1993 }}</ref>