Editing Magic square (section)

===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"/>

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{| 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
|}
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{| 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
|}
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{| 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
|}
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{| 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
|}
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{| 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
|}
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{| 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
|}
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{| 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
|}
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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.