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71-519: Magic squares that include repeated entries do not fall under this definition and are referred to as "trivial". Some well-known examples, including the Sagrada Família magic square and the Parker square , are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square ). The mathematical study of

142-445: A singular matrix , but associative magic squares of odd order can be singular or nonsingular. Cheng Dawei Cheng Dawei (程大位, 1533–1606), also known as Da Wei Cheng or Ch'eng Ta-wei , was a Chinese mathematician and writer who was known mainly as the author of Suanfa Tongzong (算法統宗) ( General Source of Computational Methods ). He has been described as "the most illustrious Chinese arithmetician ." Almost all that

213-582: A 1765 letter of Benjamin Franklin – is also associative, with each pair of opposite numbers summing to 17. The numbers of possible associative n  ×  n magic squares for n = 3,4,5,..., counting two squares as the same whenever they differ only by a rotation or reflection, are: The number zero for n = 6 is an example of a more general phenomenon: associative magic squares do not exist for values of n that are singly even (equal to 2 modulo 4). Every associative magic square of even order forms

284-480: 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. The same set of squares given by Agrippa reappear in 1539 in Practica Arithmetice by Girolamo Cardano , where he explains the construction of

355-537: A magic square typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. Magic squares are generally classified according to their order n as: odd if n

426-452: A method of constructing 4×4 magic square using a primary skeleton square, given an odd or even magic sum. The Nagarjuniya square is given below, and has the sum total of 100. The Nagarjuniya square is a 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

497-408: A number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations. 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

568-644: 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). In 1897, Emroy McClintock published On

639-614: A similar collection written around the same time by the Byzantine scholar Manuel Moschopoulos . 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. Japan and China have similar mathematical traditions and have repeatedly influenced each other in

710-423: 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

781-488: 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 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

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852-414: Is borne out by the contents of his work General Source of Computational Methods which is essentially a compilation of problems from earlier works. The General Source of Computational Methods was first published in 1592. It is essentially a general arithmetic for the abacus . Though there is nothing particularly original about this book, it was republished several times and became widely popular. Beyond

923-467: Is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an n  ×  n square, filled with the numbers from 1 to n , this common sum must equal n  + 1. These squares are also called associated magic squares , regular magic squares , regmagic squares , or symmetric magic squares . For instance, the Lo Shu Square –

994-406: 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. The construction of 4th-order magic square is detailed in a work titled Kaksaputa , composed by the alchemist 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

1065-564: Is believed to be the first seen in European art. The square associated with Jupiter appears as a talisman used to drive away melancholy. It is very similar to Yang Hui 's square, which was created in China about 250 years before Dürer's time. As with every order 4 normal magic square, the magic sum is 34. But in the Durer square this sum is also found in each of the quadrants, in the center four squares, and in

1136-551: 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. 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

1207-491: Is known about his life is contained in a passage written in the Preface of the book by one of his descendants when the book was being reprinted: Cheng Dawei was not a professional mathematician. From the description cited above, one could deduce that he must have traveled widely. Also must have been well off since he purchased books without asking the price. Again we can see that he was an avid collector of books on mathematics. This

1278-456: Is known that treatises on magic squares were written in the 9th century, the earliest extant treaties date from the 10th-century: one by Abu'l-Wafa al-Buzjani ( c.  998 ) and another by Ali b. Ahmad al-Antaki ( c.  987 ). 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

1349-520: Is odd, evenly even (also referred to as "doubly even") if n is a multiple of 4, oddly even (also known as "singly even") if n is any other even number. This classification is based on different techniques required to construct odd, evenly even, and oddly even squares. Beside this, depending on further properties, magic squares are also classified as associative magic squares , pandiagonal magic squares , most-perfect magic squares , and so on. More challengingly, attempts have also been made to classify all

1420-464: 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 . 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. It is clear, however, that

1491-615: The Chautisa Yantra ( Chautisa , 34; Yantra , lit. "device"), since its magic sum is 34. It is one of the three 4×4 pandiagonal magic squares and is also an instance of the most-perfect magic square . The study of this square led to the appreciation of pandiagonal squares by European mathematicians in the late 19th century. Pandiagonal squares were referred to as Nasik squares or Jain squares in older English literature. The order four normal magic square Albrecht Dürer immortalized in his 1514 engraving Melencolia I , referred to above,

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1562-563: The Rasa'il Ikhwan al-Safa (the Encyclopedia of the Brethren of Purity ). The squares of order 3 to 7 from Rasa'il are given below: 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 evenly-odd case ( n = 4k + 2 ) was solved by Ibn al-Haytham with k even (c. 1040), and completely by

1633-600: The Sagrada Família church in Barcelona , conceptualized by Antoni Gaudí and designed by sculptor Josep Subirachs , features a trivial order 4 magic square: The magic constant of the square is 33, the age of Jesus at the time of the Passion . Structurally, it is very similar to the Melancholia magic square , but it has had the numbers in four of the cells reduced by 1. Associative magic square An associative magic square

1704-763: 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. 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). One of Seki Takakazu 's Seven Books ( Hojin Yensan ) (1683)

1775-569: 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. 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. The contents of Yang Hui's treatise were collected from older works, both native and foreign; and he only explains

1846-615: The Influences of the Planets ) written by Ibn Zarkali of Toledo, Al-Andalus, as planetary squares by 11th century. 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. Ibn Zarkali's work was translated as Libro de Astromagia in the 1280s, due to Alfonso X of Castille. In the Alfonsine text, magic squares of different orders are assigned to

1917-866: The Luo Shu principle is clearly evident. 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. After Yang Hui, magic squares frequently occur in Chinese mathematics such as in Ding Yidong's Dayan suoyin ( c.  1300 ), Cheng Dawei 's Suanfa tongzong (1593), Fang Zhongtong's Shuduyan (1661) which contains magic circles, cubes and spheres, Zhang Chao's Xinzhai zazu ( c.  1650 ), who published China's first magic square of order ten, and lastly Bao Qishou's Binaishanfang ji ( c.  1880 ), who gave various three dimensional magic configurations. However, despite being

1988-666: 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. There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs. Unlike in Persia and Arabia, better documentation exists of how

2059-459: The Zhou dynasty. 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. The 3×3 magic square was referred to as the "Nine Halls" by earlier Chinese mathematicians. The identification of

2130-478: The abacus with its tables which must be learnt by heart, the history of Chinese mathematics, mathematical recreations and mathematical curiosities of all types." The book provided a poem to solve a remainder problem. In the Huangshan City in southern Anhui province in, there is a museum for abacuses named after Cheng Dawei. There are more than 1,000 abacuses and 3,000 copies of related materials displayed in

2201-401: The authors of the venerable classic, Cheng Dawei was not afraid of superfluity or verbosity. His book is an encyclopaedic hotch-potch of ideas which contains everything from A to Z relating to the Chinese mystique of numbers (magic squares, ... generation of the eight trigrams, musical tubes), how computation should be taught and studied, the meaning of technical arithmetical terms, computation on

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2272-606: The beginning of 12th century, if not already in the latter half of the 11th century. 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. 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

2343-438: The construction of third and fourth-order magic squares, while merely passing on the finished diagrams of larger squares. 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. The above magic squares of orders 3 to 9 are taken from Yang Hui's treatise, in which

2414-478: The continuous numbering method to construct odd ordered squares published by Agrippa. However, due to the religious upheavals of that time, these work were unknown to the rest of Europe. In 1624 France, 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

2485-554: The corner squares (of the 4×4 as well as the four contained 3×3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle ), the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14), the sum of the middle two entries of the two outer columns and rows (5+9+8+12 and 3+2+15+14), and in four kite or cross shaped quartets (3+5+11+15, 2+10+8+14, 3+9+7+15, and 2+6+12+14). The two numbers in

2556-525: The end of 15th century. The planetary squares had disseminated into northern Europe by the end of 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

2627-516: The even ordered squares were constructed using borders. He also showed that interchanging rows and columns of a magic square produced new magic squares. 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

2698-492: 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

2769-599: 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 c.  983 , the Encyclopedia of the Brethren of Purity ( Rasa'il Ikhwan al-Safa ). By the end of 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. In India, all

2840-475: 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

2911-524: The general theory alive. 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. The magic square of order three was described as a child-bearing charm since its first literary appearances in the alchemical works of Jābir ibn Hayyān (fl. c. 721 – c. 815) and al-Ghazālī (1058–1111) and it

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2982-410: The history of magic squares. 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 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

3053-431: 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. Below are some of the magic squares constructed by Narayana: 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

3124-611: The late 17th century, when Philippe de la Hire rediscovered his treatise in the Royal Library of Paris. 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

3195-610: The letters Fermat boasts of being able to construct 1,004,144,995,344 magic squares of order 8 by his method. An early account on the construction of bordered squares was given by Antoine Arnauld in his Nouveaux éléments de géométrie (1667). 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

3266-522: The limited circle of mathematicians, it must have reached a vast popular audience. Its popularity must have continued to modern times as can be seen from a remark by a contemporary historian of Chinese mathematics: "Nowadays, various editions of the book can still be found in China and old people still recite the versified formulas and talk about the difficult problems in it." The book contains 595 problems divided into 12 chapters. According to Jean Claude Martzloff, historian of Chinese mathematics, "... unlike

3337-513: The magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection. There is a well-known 12th-century 4×4 normal magic square inscribed on the wall of the Parshvanath temple in Khajuraho , India. This is known as

3408-520: The magic squares of a given order as transformations of a smaller set of squares. Except for n ≤ 5, the enumeration of higher order magic squares is still an open challenge. The enumeration of most-perfect magic squares of any order was only accomplished in the late 20th century. Magic squares have a long history, dating back to at least 190 BCE in China. At various times they have acquired occult or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized

3479-505: The magic squares was initiated by Nushizumi Yamaji. 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 ( c.  900 CE ) by Vrnda, which

3550-520: 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

3621-480: The magic squares: Greeks, Babylonians, Egyptians, and Pre-Columbian Americans. 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

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3692-419: The middle of the bottom row give the date of the engraving: 1514. It has been speculated that the numbers 4,1 bordering the publication date correspond to Durer's initials D,A. But if that had been his intention, he could have inverted the order of columns 1 and 4 to achieve "A1514D" without compromising the square's properties. Dürer's magic square can also be extended to a magic cube. The Passion façade of

3763-462: 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. Legends dating from as early as 650 BCE tell the story of the Lo Shu (洛書) or "scroll of the river Lo". According to the legend, there was at one time in ancient China a huge flood. While the great king Yu was trying to channel

3834-446: 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 Parshvanath temple in Khajuraho , India. Several Jain hymns teach how to make magic squares, although they are undateable. 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

3905-562: 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 ( c.  966 CE ) The square of Varahamihira as given above has sum of 18. Here

3976-508: The numbers . 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. A specimen of magic squares of orders 3 to 9 devised by Middle Eastern mathematicians appear in an encyclopedia from Baghdad c.  983 ,

4047-413: The numbers 1 to 8 appear twice in the square. It is a 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

4118-497: The numbers into cells according to the pattern of a standard square of order four. For odd squares, Pheru gives the method using horse move or knight's move . Although algorithmically different, it gives the same square as the De la Loubere's method. 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

4189-529: The odd ordered squares using "diamond method", which was later reproduced by Bachet. 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

4260-703: 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

4331-411: 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

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4402-709: 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. 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 . It is interesting to observe that Paolo Dagomari, like Pacioli after him, refers to

4473-510: 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

4544-446: The study of magic squares was common in medieval Islam , and it was thought to have begun after the introduction of chess into the region. The first dateable appearance of a magic square of order 3 occurs in 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. While it

4615-460: The subject was treated as a part of recreational mathematics. 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

4686-421: The unique 3 × 3 magic square – is associative, because each pair of opposite points form a line of the square together with the center point, so the sum of the two opposite points equals the sum of a line minus the value of the center point regardless of which two opposite points are chosen. The 4 × 4 magic square from Albrecht Dürer 's 1514 engraving Melencolia I – also found in

4757-400: The water out to sea, a turtle emerged from it with a curious pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15. According to the legend, thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods. The Lo Shu Square , as

4828-660: 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, and Naonobu Ajima . 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

4899-414: Was prescribed to women in labor in order to have easy delivery. The oldest dateable fourth order magic square in the world is found in an encyclopaedic work written by 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

4970-641: 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 Ibn Zarkali 's (known as Azarquiel in Europe) (1029–1087) Kitāb tadbīrāt al-kawākib ( Book on the Influences of the Planets ). 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

5041-416: 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. By this time the earlier mysticism attached to the magic squares had completely vanished, and

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