The magic square has a rich history, which most likely journeyed from China to India, then to the Arab countries and after that to Europe.
The earliest appearance dates back to China around 2200 B.C. A Chinese legend claimed that while the Chinese Emperor Yu was walking along the Yellow River, he became aware of a tortoise with a unique diagram on its shell. The Emperor decided to call the unusual numerical pattern lo shu.
Magic Squares can be traced in Chinese literature as far back as 2800 B.C.
The legend of "Lo Shu" or "scroll of the river Lo" tells the story of a huge flood that destroyed crops and land. The people offered a sacrifice to the river god for one of the flooded rivers, the Lo river, to calm his anger.
Every time the river flooded, there emerged a turtle that would walk around the sacrifice. It was not until a child noticed a unique pattern on the turtles shell (Figure 2) that told the people how many sacrifices to make for the river god to accept their sacrifice.
There were circular dots of numbers that were arranged in a 3-by-3 grid pattern such that the sum of the numbers in each column, row, and diagonal equaled the same sum: fifteen.
Fifteen became the number of sacrifices needed in order to make the river god happy. This number is equal to the number of days in each of the 24 cycles of the Chinese solar year.
The oldest magic square of order four was found inscribed in Khajuraho, India dating to the eleventh or twelfth century. This magic square is also known as the diabolic or panmagic square, where, in addition to the rows, columns, and diagonals the broken diagonals also have the same sum.
History
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.
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 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.
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 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 on the systematic methods for constructing larger magic squares 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 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 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 (circa 1300), Chen 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 the first magic square of order ten, and lastly Bao Qishou's Binaishanfang ji (circa 1880), who gave various three dimensional magic configurations.
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 Islamic, the Indian, or the 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.
This is possibly because of the Chinese scholars' enthrallment 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.
Middle East: Persia, Arabia, North Africa, Muslim Iberia
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 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 datable appearance of magic square of order 3 occur in the alchemical works of Jābir ibn Hayyān (fl. c. 721– c. 815).
While it is known that treatises on magic squares were written in the 9th century, the earliest extant treaties we have date from the 10th-century: one by Abu'l-Wafa al-Buzjani (circa 998) and another by Ali b. Ahmad al-Antaki (circa 987).
These early treatise were purely mathematical, and the Arabic designation for magic squares is wafq al-a'dad which translates as harmonious disposition of the numbers.
By the end of 10th century, the Islamic 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.
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 (circa 1040), and completely by 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 on wards, 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 the general theory alive.
One such occultist was the Egyptian Ahmad al-Buni(circa 1225), who gave general methods on constructing bordered magic squares; another one was 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 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 Egyptian 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.
There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.
India
The 3×3 magic square has been a part of rituals in India since ancient times, and still is today. For instance, the Kubera-Kolam, a magic square of order three, is commonly painted on floors in India.
It is essentially the same as the Lo Shu Square, but with 19 added to each number, giving a magic constant of 72 (below, square on the left).
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; however passage on planets could not have been written earlier than 400 CE.
The first datable instance of 3×3 magic square in India occur in a medical text Siddhayog (ca. 900 CE) by Vrnda, which was prescribed to women in labor in order to have easy delivery.
The earliest unequivocal occurrence of magic square is found in a work called Kaksaputa, composed by the alchemist Nagarjuna around 1st 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. Incidentally, the special Nagarjuniya square cannot be constructed from the method he expounds.
The Nagarjuniya square is a pan-diagonal magic square, where the broken diagonals (e.g. 16+22+34+28, 18+24+32+26, etc) sum to 100. It is also an instance of a most perfect magic square, where every 2×2 sub-square, four corners of any 3×3 sub-square, four corners of the 4×4 square, the four corners of any 2×4 or 4×2 sub-rectangle, and the four corners of oblong diagonals (18+24+32+26 and 10+16+34+40) all sum to 100.
Furthermore, the corners of eight trapezoids (16+18+32+34, 44+22+28+6, etc) all sum to 100. 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, we obtain the adjacent square.
The oldest datable 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 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 square of Varahamihira as given above has sum of 18. Here the numbers 1 to 8 appear twice in the square. It is a pan-diagonal magic square. It is also an instance of most perfect magic square. Four different magic squares can be obtained by adding 8 to one of the two sets of 1 to 8 sequence.
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.
His book also contains a method for constructing a magic square of order four when a constant sum is given. It also contains the Nagarjuniya square.
Around 12th-century, a 4×4 magic square was inscribed on the wall of Parshvanath temple in Khajuraho, India. Several Jain hyms teach how to make magic squares, although they are undatable.
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 (ca. 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.[28] 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.[28] 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.[28]
Below is Pheru's square of order six.
The next comprehensive work on magic figures was taken up by Narayana Pandit, who in the fourteenth chapter of his Ganita Kaumudi (1356) gives general methods for the constructions of all sorts of magic squares with the principles governing such constructions. It consists of 55 verses for rules and 17 verses for examples.
Narayana gives the method to make a magic squares of order four using knight's move; enumerates the number of pan-diagonal magic squares of order four, 384, including every variation made by rotation and inversion; 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 odd squares when the sum is given.
While Narayana recounts some older methods of construction, his folding method seems to be his own invention, which was later re-discovered by De la Hire. 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.
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.
Latin Europe
 |
|
belongs to a treatise on magic squares
and shows the Sigillum Iovis associated with Jupiter
|
Unlike in Persia and Arabia, we have better documentation of how the magic squares were transmitted to Europe. Around 1315, influenced by Islamic sources, the Greek Byzantine scholar Manuel Moschopoulos wrote a mathematical treatise on the subject of magic squares, leaving out the mysticism of his Muslim 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.
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.
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 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 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.
Europe after 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 a 4×4 square in his famous engraving Melencolia I. Paracelsus' contemporary Heinrich Cornelius Agrippa von Nettesheim published his famous book De occulta philosophia in 1531, where he devoted a chapter to the planetary squares.
The same set of squares given by Agrippa reappear in 1539 in Practica Arithmetice by Girolamo Cardano. 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 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.
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), and then later 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).
In Divers ouvrages de mathematique et de physique published posthumously in 1693, Bernard Frenicle de Bessy demonstrated that there were exactly 880 distinct magic squares of order four.
De la Hire also introduced concentric bordered square 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.
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.
In the 19th century, Bernard Violle gave the most 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 square.
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 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.
This iron plate, inscribed with Arabic numbers in a six by six grid was excavated from beneath the cornerstone of the palace of Prince Anxi in the eastern suburbs of Xi’an, China (1275 A.D.)
Such plates were buried in the corner of the foundation to ward off evil spirits.
In India the 3×3 magic square has been a part of rituals in India since Vedic times, and still is today. Magic squares were used in the conventional mathematical context, alchemical and medicinal recipes as well as a magical means. In Vrnda’s medical work Siddhayoga, 900 A.D. he prescribes a magic square the order of three to be employed by a woman in labor to ease childbirth.2 There is also a well known 10th century 4×4 magic square on display in the Parshvanath Jain temple in Khajuraho.
In the Islamic Arabic speaking nations, the mathematical properties of magic squares were already developed by the 9th and 10th century A.D. The magic square (known in Arabic as waqf) appeared in Islamic literature at around 9th century A.D. It was attributed to the writings of Jabir ibn Hayyan, in the Jabirean corpus, and used as a charm to ease childbirth.
Thabit Ibn Qurra was a famous Harranian Sabian Arab mathematician, astronomer, physician, and philosopher. He translated many Greek texts in Baghdad, under the Abbasid caliphate and soon wrote original works on the magic square in latter half of the 9th century A.D.
The science of magic squares reached its pinnacle in the 11th and 12th centuries. From the 13th century, magical and divinatory applications began to replace the mathematical study of the magic square.
The Arab mathematician and sufi mystic Ahmad ibn ‘Ali al-Buni, attributed magical properties to the square with references to the use of magic squares in astrological calculations in his 13th century treaties, The Shams al-Ma’arif. The Shams al-Ma’arif is a manual on Arabic magic and for achieving esoteric mysticism through magic squares, numerology, astrology, alchemy and amulets. Al-Buni combined the magic square with astrology and assigned them with the planets. Al Buni’s work is still very much a point of reference for taweez makers in the Indian subcontinent, North Africa, the Yoruba healers of Nigeria and also the Arab countries today.
 |
| Pages from Al Buni’s occult manuscript, Shams al-Ma’arif |
In West Africa there was also substantial interest in magic squares, which were interwoven throughout West African culture. The squares held particular religious importance and were adorned on clothing, masks, and religious artifacts. In the early 18th century, Muhammad ibn Muhammad, a well-known astronomer, mathematician, mystic, and astrologer in Muslim West Africa, took an interest in magic squares. In one of his manuscripts, he gave examples of, and explained how to construct, odd order magic squares.
Abraham ben Meir bin Ezra (c. 1090-1167), a Jewish philosopher and astrologer, was born in Toledo during the Golden Age of Muslim Spain. He translated many Arabic works into Hebrew and had a deep interest in magic squares and numerology in general. He traveled widely throughout Italy and beyond, and may have been the one of the people responsible for the introduction of magic squares into Europe.
Magic squares were introduced into Europe in 1300 AD by Manuel Moschopoulos, Greek Byzantine scholar. He wrote a mathematical treatise on the subject of the magic squares, building on the work of Al-Buni who preceded him. In contrast, his work was purely mathematical to that of the Arabic manuscripts.
The Italian mathematician and Franciscan friar, Luca Pacioli wrote De viribus quantitatis (On The Powers Of Numbers) between 1496 and 1508, which contains a large collection of examples of magic squares. With Pacioli there is a move towards the western mystical practice concerning magic squares.
Heinrich Cornelius Agrippa (1486 – 1535) was an influential writer of renaissance esoterica. In his work “de occulta philosophia, Book II ” he constructed magic squares from orders 3 to 9. A Magic Square is given for each planet, and sigils are drawn using the square to represent the Angel (Intelligence), Demon (Spirit), and the Seal of each Planet.
The most famous European work involving magic squares in art is Albrecht Durer’s engraving ‘Melancolia’, from 1514 which contains a heavily coded 4 x4 magic square influenced by alchemical ideas and symbolism.
.jpg)
