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==History== [[File:Yuan dynasty iron magic square.jpg|thumb|right|220px|Iron plate with an order-6 magic square in [[Eastern Arabic numerals]] from China, dating to the [[Yuan Dynasty]] (1271–1368).]] The third-order magic square was known to Chinese mathematicians as early as 190 BCE, and explicitly given by the first century of the common era. The first dateable instance of the fourth-order magic square occurred in 587 CE in India. Specimens of magic squares of order 3 to 9 appear in an encyclopedia from [[Baghdad]] {{circa|983}}, the ''[[Encyclopedia of the Brethren of Purity]]'' (''Rasa'il Ikhwan al-Safa''). By the end of the 12th century, the general methods for constructing magic squares were well established. Around this time, some of these squares were increasingly used in conjunction with magic letters, as in [[Shams Al-ma'arif]], for occult purposes.<ref>The most famous Arabic book on magic, named "Shams Al-ma'arif ({{langx|ar|كتاب شمس المعارف}}), for [[Ahmed bin Ali Al-boni]], who died about 1225 (622 AH). Reprinted in [[Beirut]] in 1985</ref> In India, all the fourth-order pandiagonal magic squares were enumerated by Narayana in 1356. Magic squares were made known to Europe through translation of Arabic sources as occult objects during the Renaissance, and the general theory had to be re-discovered independent of prior developments in China, India, and Middle East. Also notable are the ancient cultures with a tradition of mathematics and numerology that did not discover the magic squares: Greeks, Babylonians, Egyptians, and Pre-Columbian Americans. ===Chinese=== [[File:Suanfatongzong-790-790.jpg|thumb|right|220px|A page displaying 9×9 magic square from Cheng Dawei's ''Suanfa tongzong'' (1593).]] While ancient references to the pattern of even and odd numbers in the 3×3 magic square appear in the ''[[I Ching]]'', the first unequivocal instance of this magic square appears in the chapter called ''Mingtang'' (Bright Hall) of a 1st-century book ''Da Dai Liji'' (Record of Rites by the Elder Dai), which purported to describe ancient Chinese rites of the Zhou dynasty.<ref name="Yoke">{{cite book |last=Yoke |first=Ho Peng | series=Encyopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures | date=2008 | edition=2 | pages=1252–1259| publisher=Springer |doi=10.1007/978-1-4020-4425-0_9350 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=978-1-4020-4559-2 |chapter=Magic Squares in China }}</ref> <ref name="Andrews122">{{cite book |last=Andrews |first=William Symes |title=Magic Squares and Cubes |publisher=Open Court Publishing Company| date=1917| edition=2nd| page=122| url=https://archive.org/details/MagicSquaresAndCubes_754}}</ref><ref name="Cammann">{{cite journal| last = Cammann | first= Schuyler| title=The Evolution of Magic Squares in China | journal=Journal of the American Oriental Society | volume = 80 | issue = 2 | pages= 116–124| date=April 1960 | url= http://www.chinesehsc.org/downloads/cammann/camman_the_evolution_of_magic_squares_in_china.pdf| doi= 10.2307/595587| jstor= 595587}}</ref><ref name="Swetz2008"/> These numbers also occur in a possibly earlier mathematical text called ''Shushu jiyi'' (Memoir on Some Traditions of Mathematical Art), said to be written in 190 BCE. This is the earliest appearance of a magic square on record; and it was mainly used for divination and astrology.<ref name="Yoke"/> The 3×3 magic square was referred to as the "Nine Halls" by earlier Chinese mathematicians.<ref name="Cammann"/> The identification of the 3×3 magic square to the legendary Luoshu chart was only made in the 12th century, after which it was referred to as the Luoshu square.<ref name="Yoke"/><ref name="Cammann"/> The oldest surviving Chinese treatise that displays magic squares of order larger than 3 is [[Yang Hui]]'s ''Xugu zheqi suanfa'' (Continuation of Ancient Mathematical Methods for Elucidating the Strange) written in 1275.<ref name="Yoke"/><ref name="Cammann"/> The contents of Yang Hui's treatise were collected from older works, both native and foreign; and he only explains the construction of third and fourth-order magic squares, while merely passing on the finished diagrams of larger squares.<ref name="Cammann"/> He gives a magic square of order 3, two squares for each order of 4 to 8, one of order nine, and one semi-magic square of order 10. He also gives six magic circles of varying complexity.<ref name="Connor">{{cite web |url= http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Yang_Hui.html|title= Yang Hui|last1= O'Connor|first1= J.J. | last2 = Robertson| first2 = E.F. |website=MacTutor History of Mathematics Archive |access-date= 15 March 2018}}</ref> {{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;" |- | 4 || 9 || 2 |- | 3 || 5 || 7 |- | 8 || 1 || 6 |} {{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;" |- | 2 || 16 || 13 || 3 |- | 11 || 5 || 8 || 10 |- | 7 || 9 || 12 || 6 |- | 14 || 4 || 1 || 15 |} {{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;" |- | 1 || 23 || 16 || 4 || 21 |- | 15 || style="background-color: silver;"|14 || style="background-color: silver;"|7 || style="background-color: silver;"|18 || 11 |- | 24 || style="background-color: silver;"|17 || style="background-color: silver;"|13 || style="background-color: silver;"|9 || 2 |- | 20 || style="background-color: silver;"|8 || style="background-color: silver;"|19 || style="background-color: silver;"|12 || 6 |- | 5 || 3 || 10 || 22 || 25 |} {{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;"|13 || style="background-color: silver;"|22 || 18 || 27 || style="background-color: silver;"|11 || style="background-color: silver;"|20 |- | style="background-color: silver;"|31 || style="background-color: silver;"|'''4''' || 36 || '''9''' || style="background-color: silver;"|29 || style="background-color: silver;"|'''2''' |- | 12 || 21 || style="background-color: silver;"|14 || style="background-color: silver;"|23 || 16 || 25 |- | 30 || '''3''' || style="background-color: silver;"|'''5''' || style="background-color: silver;"|32 || 34 || '''7''' |- | style="background-color: silver;"|17 || style="background-color: silver;"|26 || 10 || 19 || style="background-color: silver;"|15 || style="background-color: silver;"|24 |- | style="background-color: silver;"|'''8''' || style="background-color: silver;"|35 || 28 || '''1''' || style="background-color: silver;"|'''6''' || style="background-color: silver;"|33 |} {{col-end}} {{col-begin|width=auto;margin:0.5em auto}} {{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;"|46 || style="background-color: silver;"|8 || style="background-color: silver;"|16 || style="background-color: silver;"|20 || style="background-color: silver;"|29 || style="background-color: silver;"|7 || style="background-color: silver;"|49 |- | style="background-color: silver;"|3 || 40 || 35 || 36 || 18 || 41 || style="background-color: silver;"|2 |- | style="background-color: silver;"|44 || 12 || style="background-color: silver;"|33 || style="background-color: silver;"|23 || style="background-color: silver;"|19 || 38 || style="background-color: silver;"|6 |- | style="background-color: silver;"|28 || 26 || style="background-color: silver;"|11 || style="background-color: silver;"|25 || style="background-color: silver;"|39 || 24 || style="background-color: silver;"|22 |- | style="background-color: silver;"|5 || 37 || style="background-color: silver;"|31 || style="background-color: silver;"|27 || style="background-color: silver;"|17 || 13 || style="background-color: silver;"|45 |- | style="background-color: silver;"|48 || 9 || 15 || 14 || 32 || 10 || style="background-color: silver;"|47 |- | style="background-color: silver;"|1 || style="background-color: silver;"|43 || style="background-color: silver;"|34 || style="background-color: silver;"|30 || style="background-color: silver;"|21 || style="background-color: silver;"|42 || style="background-color: silver;"|4 |} {{col-break|valign=bottom|gap=1em}} {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;" |- | 61 || 3 || 2 || 64 || style="background-color: silver;"|57 || style="background-color: silver;"|7 || style="background-color: silver;"|6 || style="background-color: silver;"|60 |- | 12 || 54 || 55 || 9 || style="background-color: silver;"|16 || style="background-color: silver;"|50 || style="background-color: silver;"|51 || style="background-color: silver;"|13 |- | 20 || 46 || style="border-left:double; border-top:double;"|47 || style="border-top:double;"|17 || style="background-color: silver; border-top:double;"|24 || style="background-color: silver; border-top:double; border-right:double;"|42 || style="background-color: silver;"|43 || style="background-color: silver;"|21 |- | 37 || 27 || style="border-left:double;"|26 || 40 || style="background-color: silver;"|33 || style="background-color: silver; border-right:double;"|31 || style="background-color: silver;"|30 || style="background-color: silver;"|36 |- | style="background-color: silver;"|29 || style="background-color: silver;"|35 || style="background-color: silver; border-left:double;"|34 || style="background-color: silver;"|32 || 25 || style="border-right:double;"|39 || 38 || 28 |- | style="background-color: silver;"|44 || style="background-color: silver;"|22 || style="background-color: silver; border-left:double; border-bottom:double;"|23 || style="background-color: silver; border-bottom:double;"|41 || style="border-bottom:double;"|48 || style="border-right:double; border-bottom:double;"|18 || 19 || 45 |- | style="background-color: silver;"|52 || style="background-color: silver;"|14 || style="background-color: silver;"|15 || style="background-color: silver;"|49 || 56 || 10 || 11 || 53 |- | style="background-color: silver;"|5 || style="background-color: silver;"|59 || style="background-color: silver;"|58 || style="background-color: silver;"|8 || 1 || 63 || 62 || 4 |} {{col-break|valign=bottom|gap=1em}} {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:18em;table-layout:fixed;" |- | style="background-color: silver;"|31 || style="background-color: silver;"|76 || style="background-color: silver;"|13 || 36 || 81 || 18 || style="background-color: silver;"|29 || style="background-color: silver;"|74 || style="background-color: silver;"|11 |- | style="background-color: silver;"|22 || style="background-color: silver;"|40 || style="background-color: silver;"|58 || 27 || 45 || 63 || style="background-color: silver;"|20 || style="background-color: silver;"|38 || style="background-color: silver;"|56 |- | style="background-color: silver;"|67 || style="background-color: silver;"|'''4''' || style="background-color: silver;"|49 || 72 || '''9''' || 54 || style="background-color: silver;"|65 || style="background-color: silver;"|'''2''' || style="background-color: silver;"|47 |- | 30 || 75 || 12 || style="background-color: silver;"|32 || style="background-color: silver;"|77 || style="background-color: silver;"|14 || 34 || 79 || 16 |- | 21 || 39 || 57 || style="background-color: silver;"|23 || style="background-color: silver;"|41 || style="background-color: silver;"|59 || 25 || 43 || 61 |- | 66 || '''3''' || 48 || style="background-color: silver;"|68 || style="background-color: silver;"|'''5''' || style="background-color: silver;"|50 || 70 || '''7''' || 52 |- | style="background-color: silver;"|35 || style="background-color: silver;"|80 || style="background-color: silver;"|17 || 28 || 73 || 10 || style="background-color: silver;"|33 || style="background-color: silver;"|78 || style="background-color: silver;"|15 |- | style="background-color: silver;"|26 || style="background-color: silver;"|44 || style="background-color: silver;"|62 || 19 || 37 || 55 || style="background-color: silver;"|24 || style="background-color: silver;"|42 || style="background-color: silver;"|60 |- | style="background-color: silver;"|71 || style="background-color: silver;"|'''8''' || style="background-color: silver;"|53 || 64 || '''1''' || 46 || style="background-color: silver;"|69 || style="background-color: silver;"|'''6''' || style="background-color: silver;"|51 |} {{col-end}} The above magic squares of orders 3 to 9 are taken from Yang Hui's treatise, in which the Luo Shu principle is clearly evident.<ref name="Cammann"/><ref name="Swetz2008"/> The order 5 square is a bordered magic square, with central 3×3 square formed according to Luo Shu principle. The order 9 square is a composite magic square, in which the nine 3×3 sub squares are also magic.<ref name="Cammann"/> After Yang Hui, magic squares frequently occur in Chinese mathematics such as in Ding Yidong's ''Dayan suoyin'' ({{circa|1300}}), [[Cheng Dawei]]'s ''[[Suanfa tongzong]]'' (1593), Fang Zhongtong's ''Shuduyan'' (1661) which contains magic circles, cubes and spheres, Zhang Chao's ''Xinzhai zazu'' ({{circa|1650}}), who published China's first magic square of order ten, and lastly Bao Qishou's ''Binaishanfang ji'' ({{circa|1880}}), who gave various three dimensional magic configurations.<ref name="Yoke"/><ref name="Swetz2008"/> However, despite being the first to discover the magic squares and getting a head start by several centuries, the Chinese development of the magic squares are much inferior compared to the Indian, Middle Eastern, or European developments. The high point of Chinese mathematics that deals with the magic squares seems to be contained in the work of Yang Hui; but even as a collection of older methods, this work is much more primitive, lacking general methods for constructing magic squares of any order, compared to a similar collection written around the same time by the Byzantine scholar [[Manuel Moschopoulos]].<ref name="Cammann"/> This is possibly because of the Chinese scholars' enthralment with the Lo Shu principle, which they tried to adapt to solve higher squares; and after Yang Hui and the fall of [[Yuan dynasty]], their systematic purging of the foreign influences in Chinese mathematics.<ref name="Cammann"/> ===Japan=== Japan and China have similar mathematical traditions and have repeatedly influenced each other in the history of magic squares.<ref>[https://core.ac.uk/download/pdf/161528551.pdf The Influence of Chinese Mathematical Arts on Seki Kowa] by Shigeru Jochi, MA, School of Oriental and African Studies, University of London, 1993</ref> The Japanese interest in magic squares began after the dissemination of Chinese works—Yang Hui's ''Suanfa'' and Cheng Dawei's ''Suanfa tongzong''—in the 17th century, and as a result, almost all the ''[[Japanese mathematics|wasans]]'' devoted their time to its study. In the 1660 edition of ''Ketsugi-sho'', Isomura Kittoku gave both odd and even ordered bordered magic squares as well as magic circles; while the 1684 edition of the same book contained a large section on magic squares, demonstrating that he had a general method for constructing bordered magic squares.<ref name="DESmith69">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n74 69]–75| url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> In ''Jinko-ki'' (1665) by Muramatsu Kudayu Mosei, both magic squares and magic circles are displayed. The largest square Mosei constructs is of 19th order. Various magic squares and magic circles were also published by Nozawa Teicho in ''Dokai-sho'' (1666), Sato Seiko in ''Kongenki'' (1666), and Hosino Sanenobu in ''Ko-ko-gen Sho'' (1673).<ref name="DESmith79">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n84 79]–80 | url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> One of [[Seki Takakazu]]'s ''Seven Books'' (''Hojin Yensan'') (1683) is devoted completely to magic squares and circles. This is the first Japanese book to give a general treatment of magic squares in which the algorithms for constructing odd, singly even and doubly even bordered magic squares are clearly described.<ref name="DESmith116">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n121 116]–122| url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> In 1694 and 1695, Yueki Ando gave different methods to create the magic squares and displayed squares of order 3 to 30. A fourth-order magic cube was constructed by Yoshizane Tanaka (1651–1719) in ''Rakusho-kikan'' (1683). The study of magic squares was continued by Seki's pupils, notably by Katahiro Takebe, whose squares were displayed in the fourth volume of ''Ichigen Kappo'' by Shukei Irie, Yoshisuke Matsunaga in ''Hojin-Shin-jutsu'', Yoshihiro Kurushima in ''Kyushi Iko'' who rediscovered a method to produce the odd squares given by Agrippa,<ref name="DESmith178">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n183 178]| url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> and [[Ajima Naonobu|Naonobu Ajima]].<ref name="Michiwaki">{{cite book |last=Michiwaki |first=Yoshimasa | series=Encyopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures | date=2008 | edition=2 | pages=1252–1259| publisher=Springer |doi=10.1007/978-1-4020-4425-0_9154 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=978-1-4020-4559-2 |chapter=Magic Squares in Japanese Mathematics }}</ref><ref name="Mikami">{{cite book |last=Mikami |first=Yoshio | title=Magic squares in Japanese mathematics |publisher=Imperial Academy of Science| date=1917 | location=Tokyo | language=ja | url=https://books.google.com/books?id=sU1tAAAAMAAJ&q=Isomura++Kittoku}}</ref> Thus by the beginning of the 18th century, the Japanese mathematicians were in possession of methods to construct magic squares of arbitrary order. After this, attempts at enumerating the magic squares was initiated by Nushizumi Yamaji.<ref name="Mikami"/> ===India=== [[File:Hindi Manuscript 317, folio 2b Wellcome L0024035.jpg|thumb|right|220px|The 3×3 magic square in different orientations forming a non-normal 6×6 magic square, from an unidentified 19th century Indian manuscript.]] The 3×3 magic square first appears in India in ''Gargasamhita'' by Garga, who recommends its use to pacify the nine planets (''navagraha''). The oldest version of this text dates from 100 CE, but the passage on planets could not have been written earlier than 400 CE. The first dateable instance of 3×3 magic square in India occur in a medical text ''Siddhayog'' ({{Circa|900 CE}}) by Vrnda, which was prescribed to women in labor in order to have easy delivery.<ref name="Hayashi"/> The oldest dateable fourth order magic square in the world is found in an encyclopaedic work written by [[Varāhamihira|Varahamihira]] around 587 CE called ''Brhat Samhita''. The magic square is constructed for the purpose of making perfumes using 4 substances selected from 16 different substances. Each cell of the square represents a particular ingredient, while the number in the cell represents the proportion of the associated ingredient, such that the mixture of any four combination of ingredients along the columns, rows, diagonals, and so on, gives the total volume of the mixture to be 18. Although the book is mostly about divination, the magic square is given as a matter of combinatorial design, and no magical properties are attributed to it. The special features of this magic square were commented on by Bhattotpala ({{Circa|966 CE}})<ref name="Datta"/><ref name="Hayashi">{{cite book |last=Hayashi |first=Takao | date=2008 | edition=2 | pages=1252–1259| publisher=Springer |doi=10.1007/978-1-4020-4425-0_9778 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=978-1-4020-4559-2 |chapter=Magic Squares in Indian Mathematics }}</ref> {{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:8em;height:8em;table-layout:fixed;" |- | 2 || 3 || 5 || 8 |- | 5 || 8 || 2 || 3 |- | 4 || 1 || 7 || 6 |- | 7 || 6 || 4 || 1 |} {{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;" |- | 10 || 3 || 13 || 8 |- | 5 || 16 || 2 || 11 |- | 4 || 9 || 7 || 14 |- | 15 || 6 || 12 || 1 |} {{col-end}} The square of Varahamihira as given above has sum of 18. Here the numbers 1 to 8 appear twice in the square. It is a [[pandiagonal magic square|pan-diagonal magic square]]. Four different magic squares can be obtained by adding 8 to one of the two sets of 1 to 8 sequence. The sequence is selected such that the number 8 is added exactly twice in each row, each column and each of the main diagonals. One of the possible magic squares shown in the right side. This magic square is remarkable in that it is a 90 degree rotation of a magic square that appears in the 13th century Islamic world as one of the most popular magic squares.<ref name="Hayashi1">{{cite journal |last=Hayashi |first=Takao |title=Varahamihira's Pandiagonal Magic Square of the Order Four | journal=Historia Mathematica | date=1987 | volume=14 | issue=2 | pages=159–166| url=https://core.ac.uk/download/pdf/82500954.pdf |doi=10.1016/0315-0860(87)90019-X |doi-access=free }}</ref> The construction of 4th-order magic square is detailed in a work titled ''Kaksaputa'', composed by the alchemist [[Nagarjuna (metallurgist)|Nagarjuna]] around 10th century CE. All of the squares given by Nagarjuna are 4×4 magic squares, and one of them is called ''Nagarjuniya'' after him. Nagarjuna gave a method of constructing 4×4 magic square using a primary skeleton square, given an odd or even magic sum.<ref name="Datta">{{cite journal |last1=Datta |first1=Bibhutibhusan |last2=Singh |first2=Awadhesh Narayan |title=Magic Squares in India |journal=Indian Journal of History of Science |date=1992 |volume=27 |issue=1 |pages=51–120 |url=http://124.108.19.235:12000/jspui/bitstream/123456789/11663/1/Vol27_1_5_BDatta.pdf |access-date=2018-01-16 |archive-url=https://web.archive.org/web/20180117131236/http://124.108.19.235:12000/jspui/bitstream/123456789/11663/1/Vol27_1_5_BDatta.pdf |archive-date=2018-01-17 |url-status=dead }}</ref> The Nagarjuniya square is given below, and has the sum total of 100. {{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:8em;height:8em;table-layout:fixed;" |- | 30 || 16 || 18 || 36 |- | 10 || 44 || 22 || 24 |- | 32 || 14 || 20 || 34 |- | 28 || 26 || 40 || 6 |} {{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;" |- | '''7''' || 1 || '''4''' || 6 |- | '''2''' || 8 || '''5''' || 3 |- | 5 || '''3''' || 2 || '''8''' |- | 4 || '''6''' || 7 || '''1''' |} {{col-end}} The Nagarjuniya square is a [[pandiagonal magic square|pan-diagonal magic square]]. The Nagarjuniya square is made up of two arithmetic progressions starting from 6 and 16 with eight terms each, with a common difference between successive terms as 4. When these two progressions are reduced to the normal progression of 1 to 8, the adjacent square is obtained. Around 12th-century, a 4×4 magic square was inscribed on the wall of [[Parshvanatha temple, Khajuraho|Parshvanath]] temple in [[Khajuraho]], India. Several Jain hymns teach how to make magic squares, although they are undateable.<ref name="Hayashi"/> As far as is known, the first systematic study of magic squares in India was conducted by [[Thakkar Pheru]], a Jain scholar, in his ''Ganitasara Kaumudi'' (c. 1315). This work contains a small section on magic squares which consists of nine verses. Here he gives a square of order four, and alludes to its rearrangement; classifies magic squares into three (odd, evenly even, and oddly even) according to its order; gives a square of order six; and prescribes one method each for constructing even and odd squares. For the even squares, Pheru divides the square into component squares of order four, and puts the numbers into cells according to the pattern of a standard square of order four. For odd squares, Pheru gives the method using [[Knight (chess)|horse move or knight's move]]. Although algorithmically different, it gives the same square as the De la Loubere's method.<ref name="Hayashi"/> The next comprehensive work on magic squares was taken up by [[Narayana Pandit]], who in the fourteenth chapter of his ''Ganita Kaumudi'' (1356) gives general methods for their construction, along with the principles governing such constructions. It consists of 55 verses for rules and 17 verses for examples. Narayana gives a method to construct all the pan-magic squares of fourth order using knight's move; enumerates the number of pan-diagonal magic squares of order four, 384, including every variation made by rotation and reflection; three general methods for squares having any order and constant sum when a standard square of the same order is known; two methods each for constructing evenly even, oddly even, and of squares when the sum is given. While Narayana describes one older method for each species of square, he claims the method of [[Superposition principle|superposition]] for evenly even and odd squares and a method of interchange for oddly even squares to be his own invention. The superposition method was later re-discovered by [[De la Hire]] in Europe. In the last section, he conceives of other figures, such as circles, rectangles, and hexagons, in which the numbers may be arranged to possess properties similar to those of magic squares.<ref name="Datta"/><ref name="Hayashi"/> Below are some of the magic squares constructed by Narayana:<ref name="Datta"/> {{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;" |- | 8 || 1 || 6 |- | 3 || 5 || 7 |- | 4 || 9 || 2 |} {{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;" |- | 1 || 14 || 4 || 15 |- | 8 || 11 || 5 || 10 |- | 13 || 2 || 16 || 3 |- | 12 || 7 || 9 || 6 |} {{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;" |- | 16 || 14 || 7 || 30 || 23 |- | 24 || 17 || 10 || 8 || 31 |- | 32 || 25 || 18 || 11 || 4 |- | 5 || 28 || 26 || 19 || 12 |- | 13 || 6 || 29 || 22 || 20 |} {{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;" |- | 1 || 35 || 4 || 33 || 32 || 6 |- | 25 || 11 || 9 || 28 || 8 || 30 |- | 24 || 14 || 18 || 16 || 17 || 22 |- | 13 || 23 || 19 || 21 || 20 || 15 |- | 12 || 26 || 27 || 10 || 29 || 7 |- | 36 || 2 || 34 || 3 || 5 || 31 |} {{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;" |- | 35 || 26 || 17 || 1 || 62 || 53 || 44 |- | 46 || 37 || 21 || 12 || 3 || 64 || 55 |- | 57 || 41 || 32 || 23 || 14 || 5 || 66 |- | 61 || 52 || 43 || 34 || 25 || 16 || 7 |- | 2 || 63 || 54 || 45 || 36 || 27 || 11 |- | 13 || 4 || 65 || 56 || 47 || 31 || 22 |- | 24 || 15 || 6 || 67 || 51 || 42 || 33 |} {{col-break|valign=bottom|gap=1em}} {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;" |- | 60 || 53 || 44 || 37 || 4 || 13 || 20 || 29 |- | 3 || 14 || 19 || 30 || 59 || 54 || 43 || 38 |- | 58 || 55 || 42 || 39 || 2 || 15 || 18 || 31 |- | 1 || 16 || 17 || 32 || 57 || 56 || 41 || 40 |- | 61 || 52 || 45 || 36 || 5 || 12 || 21 || 28 |- | 6 || 11 || 22 || 27 || 62 || 51 || 46 || 35 |- | 63 || 50 || 47 || 34 || 7 || 10 || 23 || 26 |- | 8 || 9 || 24 || 25 || 64 || 49 || 48 || 33 |} {{col-end}} The order 8 square is interesting in itself since it is an instance of the most-perfect magic square. Incidentally, Narayana states that the purpose of studying magic squares is to construct ''yantra'', to destroy the ego of bad mathematicians, and for the pleasure of good mathematicians. The subject of magic squares is referred to as ''bhadraganita'' and Narayana states that it was first taught to men by god [[Shiva]].<ref name="Hayashi"/> ===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. 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. 139–162, in particular: pp. 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. 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> ===Latin Europe=== [[File:Sigillum Iovis.jpg|thumb|215px|right|This page from [[Athanasius Kircher]]'s ''Oedipus Aegyptiacus'' (1653) belongs to a treatise on magic squares and shows the ''Sigillum Iovis'' associated with Jupiter]] Unlike in Persia and Arabia, better documentation exists of how the magic squares were transmitted to Europe. Around 1315, influenced by Arab sources, the Greek Byzantine scholar [[Manuel Moschopoulos]] wrote a mathematical treatise on the subject of magic squares, leaving out the mysticism of his Middle Eastern predecessors, where he gave two methods for odd squares and two methods for evenly even squares. Moschopoulos was essentially unknown to the Latin Europe until the late 17th century, when Philippe de la Hire rediscovered his treatise in the Royal Library of Paris.<ref>{{Cite web|url=https://www.maa.org/press/periodicals/convergence/the-magic-squares-of-manuel-moschopoulos-introduction|title=The Magic Squares of Manuel Moschopoulos - Introduction {{pipe}} Mathematical Association of America|website=www.maa.org}}</ref> However, he was not the first European to have written on magic squares; and the magic squares were disseminated to rest of Europe through Spain and Italy as occult objects. The early occult treaties that displayed the squares did not describe how they were constructed. Thus the entire theory had to be rediscovered. Magic squares had first appeared in Europe in ''Kitāb tadbīrāt al-kawākib'' (''Book on the Influences of the Planets'') written by Ibn Zarkali of Toledo, Al-Andalus, as planetary squares by 11th century.<ref name="Comes2016"/> The magic square of three was discussed in numerological manner in early 12th century by Jewish scholar Abraham ibn Ezra of Toledo, which influenced later Kabbalists.<ref name="Cammann1969">{{cite journal| last = Cammann | first= Schuyler| title=Islamic and Indian Magic Squares, part II | journal=History of Religions | volume = 8 | issue = 4 | pages= 271–299| date=May 1969 | jstor= 1062018| doi= 10.1086/462589| s2cid= 224806255}}</ref> Ibn Zarkali's work was translated as ''Libro de Astromagia'' in the 1280s,<ref>presently in the Biblioteca Vaticana (cod. Reg. Lat. 1283a)</ref> due to [[Alfonso X]] of Castille.<ref>See ''Alfonso X el Sabio, Astromagia (Ms. Reg. lat. 1283a)'', a cura di A.D'Agostino, Napoli, Liguori, 1992</ref><ref name="Comes2016">{{cite book| last = Comes | first= Rosa | chapter = The Transmission of Azarquiel's Magic Squares in Latin Europe | pages = 159–198 |editor-last1=Wallis |editor-first1=Faith |editor-last2=Wisnovsky |editor-first2=Robert | title = Medieval Textual Cultures: Agents of Transmission, Translation and Transformation | series = Judaism, Christianity, and Islam – Tension, Transmission, Transformation | volume = 6 | publisher = Walter de Gruyter GmbH & Co KG | isbn = 978-3-11-046730-7 | year= 2016 }}</ref> In the Alfonsine text, magic squares of different orders are assigned to the respective planets, as in the Islamic literature; unfortunately, of all the squares discussed, the Mars magic square of order five is the only square exhibited in the manuscript.<ref>Mars magic square appears in figure 1 of "Saturn and Melancholy: Studies in the History of Natural Philosophy, Religion, and Art" by [[Raymond Klibansky]], [[Erwin Panofsky]] and [[Fritz Saxl]], Basic Books (1964)</ref><ref name="Comes2016"/> Magic squares surface again in Florence, Italy in the 14th century. A 6×6 and a 9×9 square are exhibited in a manuscript of the ''Trattato d'Abbaco'' (Treatise of the Abacus) by [[Paolo Dagomari]].<ref>The squares can be seen on folios 20 and 21 of MS. 2433, at the Biblioteca Universitaria of Bologna. They also appear on folio 69rv of Plimpton 167, a manuscript copy of the ''Trattato dell'Abbaco'' from the 15th century in the Library of Columbia University.</ref><ref>In a 1981 article ("Zur Frühgeschichte der magischen Quadrate in Westeuropa" i.e. "Prehistory of Magic Squares in Western Europe", Sudhoffs Archiv Kiel (1981) vol. 65, pp. 313–338) German scholar Menso Folkerts lists several manuscripts in which the "Trattato d'Abbaco" by Dagomari contains the two magic square. Folkerts quotes a 1923 article by Amedeo Agostini in the Bollettino dell'Unione Matematica Italiana: "A. Agostini in der Handschrift Bologna, Biblioteca Universitaria, Ms. 2433, f. 20v–21r; siehe Bollettino della Unione Matematica Italiana 2 (1923), 77f. Agostini bemerkte nicht, dass die Quadrate zur Abhandlung des Paolo dell'Abbaco gehören und auch in anderen Handschriften dieses Werks vorkommen, z. B. New York, Columbia University, Plimpton 167, f. 69rv; Paris, BN, ital. 946, f. 37v–38r; Florenz, Bibl. Naz., II. IX. 57, f. 86r, und Targioni 9, f. 77r; Florenz, Bibl. Riccard., Ms. 1169, f. 94–95."</ref> It is interesting to observe that Paolo Dagomari, like Pacioli after him, refers to the squares as a useful basis for inventing mathematical questions and games, and does not mention any magical use. Incidentally, though, he also refers to them as being respectively the Sun's and the Moon's squares, and mentions that they enter astrological calculations that are not better specified. As said, the same point of view seems to motivate the fellow Florentine [[Luca Pacioli]], who describes 3×3 to 9×9 squares in his work ''De Viribus Quantitatis'' by the end of 15th century.<ref>This manuscript text (circa 1496–1508) is also at the Biblioteca Universitaria in Bologna. It can be seen in full at the address http://www.uriland.it/matematica/DeViribus/Presentazione.html {{Webarchive|url=https://web.archive.org/web/20120301085501/http://www.uriland.it/matematica/DeViribus/Presentazione.html |date=2012-03-01 }}</ref><ref>Pacioli states: ''A lastronomia summamente hanno mostrato li supremi di quella commo Ptolomeo, al bumasar ali, al fragano, Geber et gli altri tutti La forza et virtu de numeri eserli necessaria'' (Masters of astronomy, such as [[Ptolemy]], [[Albumasar]], [[Alfraganus]], [[Jabir ibn Aflah|Jabir]] and all the others, have shown that the force and the virtue of numbers are necessary to that science) and then goes on to describe the seven planetary squares, with no mention of magical applications.</ref> ===Europe after 15th century=== [[File:Siamese Square.jpg|thumb|215px|right|A page from Simon de la Loubère's ''Du Royaume de Siam'' (1691) showcasing the Indian method of constructing an odd magic square.]] The planetary squares had disseminated into northern Europe by the end of the 15th century. For instance, the Cracow manuscript of ''[[Picatrix]]'' from Poland displays magic squares of orders 3 to 9. The same set of squares as in the Cracow manuscript later appears in the writings of [[Paracelsus]] in ''Archidoxa Magica'' (1567), although in highly garbled form. In 1514 [[Albrecht Dürer]] immortalized a 4×4 square in his famous engraving ''[[Melencolia I]].'' Paracelsus' contemporary [[Heinrich Cornelius Agrippa von Nettesheim]] published his famous three volume book ''De occulta philosophia'' in 1531, where he devoted Chapter 22 of Book II to the planetary squares shown below.<ref name="Cammann1969"/> The same set of squares given by Agrippa reappear in 1539 in ''Practica Arithmetice'' by [[Girolamo Cardano]], where he explains the construction of the odd ordered squares using "diamond method", which was later reproduced by Bachet.<ref name="Muurinen2020"/> The tradition of planetary squares was continued into the 17th century by [[Athanasius Kircher]] in ''Oedipi Aegyptici'' (1653). In Germany, mathematical treaties concerning magic squares were written in 1544 by [[Michael Stifel]] in'' Arithmetica Integra'', who rediscovered the bordered squares, and [[Adam Riese]], who rediscovered the continuous numbering method to construct odd ordered squares published by Agrippa. However, due to the religious upheavals of that time, these works were unknown to the rest of Europe.<ref name="Cammann1969"/> {{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;" |- |+ [[Saturn (astrology)|Saturn]]=15 |- | 4 || 9 || 2 |- | 3 || 5 || 7 |- | 8 || 1 || 6 |} {{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;" |- |+ [[Jupiter (astrology)|Jupiter]]=34 |- | 4 || 14 || 15 || 1 |- | 9 || 7 || 6 || 12 |- | 5 || 11 || 10 || 8 |- | 16 || 2 || 3 || 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;" |- |+ [[Mars (astrology)|Mars]]=65 |- | 11 || 24 || 7 || 20 || 3 |- | 4 || 12 || 25 || 8 || 16 |- | 17 || 5 || 13 || 21 || 9 |- | 10 || 18 || 1 || 14 || 22 |- | 23 || 6 || 19 || 2 || 15 |} {{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;" |- |+ [[Sun (astrology)|Sol]]=111 |- | 6 || 32 || 3 || 34 || 35 || 1 |- | 7 || 11 || 27 || 28 || 8 || 30 |- | 19 || 14 || 16 || 15 || 23 || 24 |- | 18 || 20 || 22 || 21 || 17 || 13 |- | 25 || 29 || 10 || 9 || 26 || 12 |- | 36 || 5 || 33 || 4 || 2 || 31 |} {{col-end}} {{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:14em;height:14em;table-layout:fixed;" |- |+ [[Venus (astrology)|Venus]]=175 |- | 22 || 47 || 16 || 41 || 10 || 35 || 4 |- | 5 || 23 || 48 || 17 || 42 || 11 || 29 |- | 30 || 6 || 24 || 49 || 18 || 36 || 12 |- | 13 || 31 || 7 || 25 || 43 || 19 || 37 |- | 38 || 14 || 32 || 1 || 26 || 44 || 20 |- | 21 || 39 || 8 || 33 || 2 || 27 || 45 |- | 46 || 15 || 40 || 9 || 34 || 3 || 28 |} {{col-break|valign=bottom|gap=1em}} {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:16em;height:16em;table-layout:fixed;" |- |+ [[Mercury (astrology)|Mercury]]=260 |- | 8 || 58 || 59 || 5 || 4 || 62 || 63 || 1 |- | 49 || 15 || 14 || 52 || 53 || 11 || 10 || 56 |- | 41 || 23 || 22 || 44 || 45 || 19 || 18 || 48 |- | 32 || 34 || 35 || 29 || 28 || 38 || 39 || 25 |- | 40 || 26 || 27 || 37 || 36 || 30 || 31 || 33 |- | 17 || 47 || 46 || 20 || 21 || 43 || 42 || 24 |- | 9 || 55 || 54 || 12 || 13 || 51 || 50 || 16 |- | 64 || 2 || 3 || 61 || 60 || 6 || 7 || 57 |} {{col-break|valign=bottom|gap=1em}} {| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:18em;height:18em;table-layout:fixed;" |- |+ [[Moon (astrology)|Luna]]=369 |- | 37 || 78 || 29 || 70 || 21 || 62 || 13 || 54 || 5 |- | 6 || 38 || 79 || 30 || 71 || 22 || 63 || 14 || 46 |- | 47 || 7 || 39 || 80 || 31 || 72 || 23 || 55 || 15 |- | 16 || 48 || 8 || 40 || 81 || 32 || 64 || 24 || 56 |- | 57 || 17 || 49 || 9 || 41 || 73 || 33 || 65 || 25 |- | 26 || 58 || 18 || 50 || 1 || 42 || 74 || 34 || 66 |- | 67 || 27 || 59 || 10 || 51 || 2 || 43 || 75 || 35 |- | 36 || 68 || 19 || 60 || 11 || 52 || 3 || 44 || 76 |- | 77 || 28 || 69 || 20 || 61 || 12 || 53 || 4 || 45 |} {{col-end}} In 1624 France, [[Claude Gaspard Bachet de Méziriac|Claude Gaspard Bachet]] described the "diamond method" for constructing Agrippa's odd ordered squares in his book ''Problèmes Plaisants''. During 1640 [[Bernard Frenicle de Bessy]] and [[Pierre Fermat]] exchanged letters on magic squares and cubes, and in one of the letters Fermat boasts of being able to construct 1,004,144,995,344 magic squares of order 8 by his method.<ref name="Muurinen2020">{{cite thesis | type= MSc | title= Fermat, magic squares and the idea of self-supporting blocks | last= Muurinen |first= Ismo | date= 2020 | publisher=University of Helsinki |url= https://helda.helsinki.fi/bitstream/handle/10138/322537/Muurinen_Ismo_gradu_2020.pdf?sequence=2}}</ref> An early account on the construction of bordered squares was given by [[Antoine Arnauld]] in his ''Nouveaux éléments de géométrie'' (1667).<ref>{{cite book | title= A History of Algorithms: From the Pebble to the Microchip |last=Chabert |first= Jean-Luc |date= 1999 | page= 524 | publisher= Springer | isbn= 978-3540633693}}</ref> In the two treatise ''Des quarrez ou tables magiques'' and ''Table générale des quarrez magiques de quatre de côté'', published posthumously in 1693, twenty years after his death, [[Bernard Frenicle de Bessy]] demonstrated that there were exactly 880 distinct magic squares of order four. Frenicle gave methods to construct magic square of any odd and even order, where the even ordered squares were constructed using borders. He also showed that interchanging rows and columns of a magic square produced new magic squares.<ref name="Muurinen2020" /> In 1691, [[Simon de la Loubère]] described the Indian continuous method of constructing odd ordered magic squares in his book ''Du Royaume de Siam'', which he had learned while returning from a diplomatic mission to Siam, which was faster than Bachet's method. In an attempt to explain its working, de la Loubere used the primary numbers and root numbers, and rediscovered the method of adding two preliminary squares. This method was further investigated by Abbe Poignard in ''Traité des quarrés sublimes'' (1704), by [[Philippe de La Hire]] in ''Mémoires de l'Académie des Sciences'' for the Royal Academy (1705), and by [[Joseph Sauveur]] in ''Construction des quarrés magiques'' (1710). Concentric bordered squares were also studied by De la Hire in 1705, while Sauveur introduced magic cubes and lettered squares, which was taken up later by [[Euler]] in 1776, who is often credited for devising them. In 1750 d'Ons-le-Bray rediscovered the method of constructing doubly even and singly even squares using bordering technique; while in 1767 [[Benjamin Franklin]] published a semi-magic square that had the properties of eponymous Franklin square.<ref>{{cite web |url= http://www-history.mcs.st-andrews.ac.uk/Biographies/Franklin_Benjamin.html|title= Benjamin Franklin|last1= O'Connor|first1= J.J. | last2 = Robertson| first2 = E.F. |website=MacTutor History of Mathematics Archive |access-date= 15 December 2018}}</ref> By this time the earlier mysticism attached to the magic squares had completely vanished, and the subject was treated as a part of recreational mathematics.<ref name="Cammann1969"/><ref name="RouseBall1904">{{cite book| last =Rouse Ball | first= W.W. | title=Mathematical Recreations and Essays | chapter= Magic Squares | edition = 4 | pages= 122–142| publisher = Mac Millan and Co., Limited | location= London}}</ref> In the 19th century, Bernard Violle gave a comprehensive treatment of magic squares in his three volume ''Traité complet des carrés magiques'' (1837–1838), which also described magic cubes, parallelograms, parallelopipeds, and circles. Pandiagonal squares were extensively studied by Andrew Hollingworth Frost, who learned it while in the town of Nasik, India, (thus calling them Nasik squares) in a series of articles: ''On the knight's path'' (1877), ''On the General Properties of Nasik Squares'' (1878), ''On the General Properties of Nasik Cubes'' (1878), ''On the construction of Nasik Squares of any order'' (1896). He showed that it is impossible to have normal singly-even pandiagonal magic squares. Frederick A.P. Barnard constructed inlaid magic squares and other three dimensional magic figures like magic spheres and magic cylinders in ''Theory of magic squares and of magic cubes'' (1888).<ref name="RouseBall1904"/> In 1897, Emroy McClintock published ''On the most perfect form of magic squares'', coining the words ''pandiagonal square'' and ''most perfect square'', which had previously been referred to as perfect, or diabolic, or Nasik.
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