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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, , 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 odd 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 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.
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. 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. 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 and another by Ali b.
Ahmad al-Antaki circa 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. 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. Unlike in Persia and Arabia, we have better documentation of how the magic squares were transmitted to Europe.
Around , 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.
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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. 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 surface again in Florence, Italy in the 14th century.
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. 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 , although in highly garbled form. Paracelsus' contemporary Heinrich Cornelius Agrippa von Nettesheim published his famous three volume book De occulta philosophia in , where he devoted Chapter 22 of Book II to the planetary squares shown below.
The same set of squares given by Agrippa reappear in in Practica Arithmetice by Girolamo Cardano.
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The tradition of planetary squares was continued into the 17th century by Athanasius Kircher in Oedipi Aegyptici In Germany, mathematical treaties concerning magic squares were written in 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 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. Concentric bordered squares were also studied by De la Hire in , while Sauveur introduced magic cubes and lettered squares, which was taken up later by Euler in , who is often credited for devising them. In d'Ons-le-Bray rediscovered the method of constructing doubly even and singly even squares using bordering technique; while in Benjamin Franklin published a semi-magic square which had the properties of eponymous Franklin square.
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 , On the General Properties of Nasik Squares , On the General Properties of Nasik Cubes , On the construction of Nasik Squares of any order He showed that it is impossible to have normal singly-even pandiagonal magic square.
Frederick A. 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 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 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.
This is known as the Chautisa Yantra since its magic sum is 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 square associated with Jupiter appears as a talisman used to drive away melancholy. The two numbers in the middle of the bottom row give the date of the engraving: The numbers 1 and 4 at either side of the date correspond respectively to the letters "A" and "D," which are the initials of the artist.
While having the same pattern of summation, this is not a normal magic square because two numbers 10 and 14 are duplicated and two 12 and 16 are absent, failing the rule that an n x n magic square must contain each of the positive integers 1 through n 2. Such squares are not generally mathematically interesting and may be described, in a slightly deprecative sense, as trivial.
Lee Sallows has pointed out that, due to Subirachs's ignorance of magic square theory, the renowned sculptor made a needless blunder, and supports this assertion by giving several examples of non-trivial 4 x 4 magic squares showing the desired magic constant of Parker 's square  is an attempt to create the 3x3 magic square of squares an prized unsolved problem since Euler .