Magic squares are like a magic mirror which reflects a ray of the symmetry of the divine norm immanent in all things, in the immeasurable immensity of the cosmos not less than in the mysterious depths of the human mind.

Paul Carus

 One of the outstanding things which Alice discovered when she entered through the Looking-glass was a curious country divided by hedges and streams into endless squares. It was like "a great huge game of chess that's being played – all over the world – if this is the world at all, you know". In her delight she wished to be one of the players – "I wouldn't mind being a Pawn, if only I might join – though, of course, I should like to be a Queen best." She was duly informed by the Red Queen that when she got to the Eighth Square she would indeed be a Queen, an accomplishment made less than probable by the fact that however quickly the "players ran, they still remained in the same place. But there they were all laid out before her and the moves, if there were to be any, would have to be made in squares.

 Squares, once they are established, remain as cells delineated by perfectly perpendicular lines which are spaced to create precisely commensurable magnitudes. The jingle square is a perfect measure of progress in a duo-dimensional world, as Alice readily understood. When you make a move – forward, backward or diagonally – there is no better unit with which one may calibrate direction and the pattern of movement. Corresponding to the number 4, the square signifies all equal, four-part divisions of number, form and movement. Its four corners indicate the thrust of the four directional points which extend outwards, losing themselves in infinity, but which intersect at the middle point of the square. In this way, the square becomes a cross that divides height and breadth, forming the magical and scientific quaternary which, "when it is inscribed within the perfect square, is the basis of the occultist".

 Plato taught that numbers are models of ideas and of relations established between them. Ideas are thus pattern constituted of numbers, which are themselves intermediaries between the sensible realms. In other words, they introduce quantity into the sensible world. The idea of fourness or square or squaring is related to many archetypal sets of fours to be found in the manifest world. The question may arise as to whether this idea of fourness originated in the elementary stages of arithmetical reasoning "when concrete associations of number were more real than abstract". Fourness, or squareness, being associated with the earth, may well reflect a mental pattern which directly expresses the unfolding manifest universe. The Pythagoreans showed that 1 + 2 + 3 + 4 = 10, the number of the sacred decad whose bottom line is not only a necessary factor producing the number 10 but an expression of matter. The progression from l to 4 signifies the involution of Divine Mind in matter, and it seems that the idea of fourness or squareness is concurrent with this completed involvement. The arcane square, however, is not to be thought of as merely an earthly symbol for, as Theon of Smyrna suggested in the first century A.D., its ultimate nature is to be traced to the great mystery of the divine Tetraktis from which proceed the sets of quaternary divisions in the phenomenal world. It is doubtful whether Alice had these considerations in mind as she gazed across the endless chess-board in Looking-glass land, but their breadth and depth very likely contributed to the enlarged curiosity her author displayed in his fascination with number, pattern and form.

 Theon listed ten sets of four things related to the square: the numbers 1, 2, 3 and 4; the magnitudes of point, line, surface and solid; the elements of fire, air, water and earth; the corresponding, figures of the pyramid, octahedron, icosahedron and cube; the living stages of seed, growth in length, in breadth and in thickness; the faculties of reason, knowledge, opinion and sensation; the seasons of spring, summer, autumn and winter; the ages of infancy, youth, manhood and old age; the social increments of man, village, city and nation; and, last but not the least, the lunar phases within four weeks. This list could easily be expanded and the square could be studied in relation to many quadratic patterns such as those suggested by the cross and the swastika. But the square itself is the symbol of organization and construction, of firmness and stability. In this the square contrasts, with the dynamism of odd-number forms like the triangle. According to ancient Egyptian hieroglyphics, the square signified achievement and a square-shaped spiral represents constructive, materialized energy. The idea of construction is reflected in the Latin root quadra which, prefixed by ex (out + square), meant to square out something' or 'to make a thing square'. It is from this exquadra that the Old French and Middle English took its esquarre which ultimately became 'square'.

 Because it is the symbol of organized matter and because its umber is even, the square has traditionally been associated with that which is feminine. In Graeco-Roman times it was considered the emblem of Venus-Aphrodite and linked with female reproductive power. The qualities of stability and definition manifested by the square doubtless imply a close relationship with matter. The quaternary is both material and passive. It is the base of the four-sided pyramid and is informed by the fiery triangles above 14. The fact that there are four of these is an important subsequent consideration but the square base itself corresponds to the earth. Similarly, the Greek fret pattern which commonly decorated the squared eaves of temple structures, the base of vases and the hems of gowns, symbolized the contact of spirit with the earth. In Christian architecture, the more ethereal symbol of the circle was often connected with squared columns by an octagonal element which was believed to indicate a half-way point in the squaring of the circle. However the combinations were expressed, the same basic idea of spiritual involution in substance was indicated. The static nature of the square with its immutable perfection reflected an intelligent, orderly expression of measure and anticipated growth. The ancient Hindus constructed their temples on a square base, believing it to be a perfect measure for man, a reflection of Purusha through the pairs of opposites and the four castes.

 No wonder, then, that chess and other games are usually played on square boards, resembling a quadrille or square dance which permits compact movements of precise measure. The beauty of the 'allemande left, do-si-do and promenade' lies in the patterns prescribed by a square. Movement which is so perfectly enclosed provides a sense of great satisfaction precisely because it is so exacting. Within such a foursquare framework, equality and balance are demonstrated and convey a sense of limit and crisp stability which readily lends itself to the notion of ordered measurement. This is naturally associated with widely popular notions of fairness and honesty. A square meal is balanced; but if one squares one's shoulders or looks another square in the eye, one conveys the sort of balance which we translate in moral terms as honesty. In Anthony and Cleopatra Shakespeare expressed this sense of meaning when he wrote "I have not kept my square; but that to come shall be done by the rule." Theodore Roosevelt used this idea in a famous speech made on behalf of the common man. He declared that "A man who is good enough to shed his blood for his country Is good enough to be given a square deal afterwards. More than that no man is entitled to, and less than that no man shall have." This was seen as a plea for protection against oppression by great financial and commercial powers and was an affirmation of interdependence over independence. The so-called 'square deal' was meant to ensure an equitable distribution of national wealth, and its logic was based upon the notion of balance, fairness and honesty.

 A Chinese legend tells of a certain King Yu who ruled the ancient kingdom of Cathay over four thousand years ago. One day, while walking about his palace, he saw at his feet a mystical turtle. It had a pattern on its shell which came to be known as Lo-Shu and which appears to this day on charms and amulets. The pattern was a square, which contained nine subdivisions of square cells, each of which enclosed dots of numbers ranging from 1 to 9. The king found that when he added the columns of dots they always yielded the same sum. His astonishment increased considerably when he found that this was also true of the horizontal and diagonal rows. Whichever way he added up, he arrived at a sum of 15. The first record of a fourth order square is at Khajuraho, on a gate leading to a Jaina temple and this, like the Lo-Shu, is considered to be a magic square. These remarkable square sets of integers were introduced to the West only in the fifteenth century and individuals like Cornelius Agrippa were fascinated with them, constructing 3, 4, 5, 6, 7, 8 and 9 order squares.

 Though of no immediate practical purpose, magic squares seem to teach an important lesson, as palpable instances of the symmetry of mathematics. They throw light upon the often obscured order that pervades the universe, an prefer even traceable in human culture and thought. They are a visible proof of the intrinsic harmony of the law of number, which harmony gives great joy to anyone who would attempt to follow the precepts of Pythagoras and locate the key that may unlock the mysteries of the universe. The law of number contains an immanent order which, upon examination, reveals itself to be intrinsically necessary. This explains the marvellous consistency of the laws of Nature. The magic square be said to serve as an interpreter of the cosmic order, its quadratic form suggesting the archetypal mode in which number magnifies in the manifest world.

 Yes, by our Tetraktis which giveth the soul the fount and source of ever-flowing nature!

 The number 729 which, according to Plutarch, belongs to the sun, was of great importance in the Pythagorean system. This number can be derived from a progression of the Tetraktis which is 1:2::4:8 and 1:3::9:27. If one constructs a magic square with 27 by 27 cells upon a plan of a checkerboard and arranges the numbers 1 to 729 first in numerical order, then by shifting the nine largest squares (9 x 9) into positions indicated in the familiar 3 x 3 square, and repeats the process with the subdivision of the 9 x 9 squares and so on down, one arrives at the following combination: 365 white squares (days) and 364 dark squares (nights) with the number 365 occupying the central cell of the system. The columns, horizontals and diagonals of the central 3x3 square will yield 1095, or three years, the time period during which Pythagorean initiates were vowed to silence. A persistent assertion of the importance of such a period was made during the Renaissance when 1095 was known as 'the number of Silence'. The columns, horizontals and diagonals of the 9 x 9 square yield the sum of 3285, or nine years, while that of 27 x 27 produce 9855 or 27 years.

 Such a combination of numbers resembles what Plato described to be the characteristics of the Tetraktis, that which is concerned with days and nights and months and years. The square was taken by the Greeks to be, at one level, the symbol of the Four Sacred Powers of the Tetraktis. These powers must be related to the four levels of ideation expressed as point, line, surface and solid, all of which are symbolized in the Pythagorean decad. The sequences of even and odd numbers described on either side of the Tetraktis in Greek mathematics indicate sets of these four powers, but their relationship with one another is a mystery not easily unveiled. Certain 'authorities' have claimed that some people are constitutionally incapable of thinking mathematically. This is odd, for it would seem that anyone who thinks logically is also thinking mathematically. From this standpoint, the learning of special symbolism is a mere device to facilitate logical thinking. If the Tetraktis is archetypal in relation to the very formula by which ideation manifests, then it would seem that orderly thought processes patterned along the lines of these Four Sacred Powers would be innate to the human mind.

 Socrates, in Plato's Meno, successfully demonstrates this supposition by leading a slave boy through a carefully prepared series of questions related to a diagram of squares within a square. In responding to the questions, the boy comes to recognize that double the square of any straight line is not the square on double the line, but the square on the diagonal of the original square. It is obvious here that Plato is primarily interested not in the Pythagorean theorem or squares as such, but in a line of reasoning, a logical procedure. The slave boy is able to demonstrate that the mind, having contemplated the patterns of rational and irrational numbers in the world of ideas, 'remembers' them when confronted with their actual geometric illustration. This anamnesis, then, involves principles of formulation or limits, without which forms could not be distinguished or even exist. Without this quality of measurableness, the universe would have no articulate structure.

 During the European Renaissance, Giorgio in his Harmonia Mundi reasserted the Pythagorean concept that our knowledge of the Divine and of natural things comes from number. He believed that all number comes from 4, which 'contains' the decad and, hence, all numbers. He felt that the very modes of human understanding were embraced by the four senses of allegory: literal, allegorical, tropological and analogical. This connects with the idea that logical thought reflects a hidden universal order and that there can be a consciously intelligent mirroring of this order in the rational faculty of human individuals. Plato brilliantly demonstrated this when he turned the mathematical logic of earlier geometry into a philosophical method of enquiry.

 The idea of universal order begins with Deity or the Demiurge. In the Greek scheme the Demiurge was symbolized by the fourth letter of the alphabet, Δ (delta) or D. This corresponds to Deity symbolized by four syllables. In his Revelations, the Gnostic Marcos described a Supreme Tetrad which "came down into me (him) from the region which cannot be seen or named, in a female form, because the world would have been unable to bear her appearance under a male figure". This Supreme Tetrad, whose female form is Secret Wisdom veiled, revealed to him the generation of the universe. From the Great Book of the Mysteries comes a similar idea embodied in the notion that "Seven Lords created Seven men; three Lords were holy and good, four less heavenly and full of passion ... the chhayas (phantoms) of the Father were they". This is very much like the Mexican cosmology which identifies four 'good men' who are the real ancestors of the human race and whose creation was wrought by the Creative Powers. Perhaps there is an echo here of the Four Sacred Powers of the Tetraktis expressed through the square. But the Tetraktis is the Formless Square, giving forth only the idea of universal order. Ancient cosmogonies teach that the 'Son' of the immaculate Celestial Virgin is born again on earth and becomes Humanity (past, present and future). Thus, the triad becomes Tetraktis (the Perfect Square) and a six-faced cube on earth. The Ancient of Days gets transformed into Tetragrammaton through the Logos, and the cube reflects the generational power of the square sparked by the six-sided principle of cosmic creation.

 Pythagoras associated the square with the Soul, not in the sense of earthly matter but in the fact that the phenomenal world received its culminating expression in man. Thus, man is the mystic square; in his metaphysical aspect he is the Tetraktis, and a cube on the creative plane. This is what is meant by "the culmination of Deity on Earth whose body is the cross of flesh". The Logos Spirit of Universal Ideation is the apex of the Pythagorean triangle which, when complete, becomes the Tetraktis, the Triangle in the Square which is "the dual symbol of the four-lettered Tetragrammaton in the manifested Kosmos, and of its radical triple RAY in the unmanifested, or its noumenon". From this one can see that the genesis of Kosmos resulted from the triangle and the square. Three figures arose from the triangle: the pyramid, octahedron and icosahedron (fire, air and water), while the cube (earth) arose from the square. The triangle and the square together are 7 which, when multiplied by 4, yields the lunar cycle of 28 that so obviously has to do with growth rates on the earth. It is interesting that 28 is the sum of two odd cubes: 3 x 3 x 3 = 27 added to 1 x 1 x 1 = 1. One needs to bear in mind here that odd numbers are of limit, the formal principle of the universe.

 Squaring a number gives it extension. It gives area and surface. One begins to perceive something about the interaction between triangles and squares upon recalling the Pythagorean theorem which explains that the area of the square over the hypotenuse of a right triangle is the sum of the areas of the squares erected its sides. Pythagoras pointed out that "the plane around a point completely filled by six equilateral triangles or four squares or three regular hexagons". The idea of being completely filled is basic in considering universal order, and one may take it that growth involves some sort of movement of energy along the lines of the aforementioned bodies. The importance of the square in this progression lies, as in the case of the quadrille, in the enforced symmetry, the commensurability and the framework that lays the basis of Law in a law-governed universe. The equilateral and equiangular aspects of the square are the necessary characteristics from which all other subsequent regularities and irregularities can be measured.

  Thinking of the square and squaring, one thinks of things doubled. In thinking of double, the mind is drawn to contemplate the number 2 and the nature of duality. This consideration was embraced by the ancient Greeks who discussed at length the nature of the dyad, as they called it. "The dyad", they said, "is the principle of numerical multiplicity and geometrical extension. It produces all the even numbers by multiplication, and all the odd numbers by the function of limit, which stops, equalizes, and stabilizes the propensity of the dyad to multiply." Pythagoras, in placing even and odd sequences on either side of the Tetraktis, revealed his awareness of this necessary interaction even in the construction of the World Soul. First there is the 1 and then 2 (double of 1), then 4 (double of 2), then 8 (double of 4). Then there is the 1 and 3 (triple of 1), then 9 (triple of 3), then 27 (triple of 9). In this process one sees that integers are derived from more fundamental principles (like that of doubling) and that multiplication (and squaring) is logically independent of addition. With doubling we can move from the point to the line and square. But there is a distinction to be made between what one might call twoness and twiceness. This may be seen as the difference between Plato's auto-dyad and the dyad. Twoness is the essence of the number 2. It is changeless as a principle and independent of becoming. It is Being. Twiceness involves the ability of any number to proceed from itself to another number and to be integrated into measure and formulae. This is Becoming.

 The first and most important study is of numbers in themselves; not of corporeal numbers, but of the whole genesis of the odd and even, and the extent of their influence on the nature of things.

The Republic

 From Being to Becoming the idea of static perfection and immutable equality shifts to include the odd and irrational. To increase a square by integers, Pythagoras successively added odd, masculine numbers. 2n + 1 makes the next higher square, (n + 1)2. The odd numbers thus added were called gnomons (a word originally used to describe an upright stick which cast a shadow on the hemispherical surface of a sundial to mark the hours. The term literally means 'marker' or 'pointer' and connotes perpendicularity). That which is odd is out of the ordinary and clearly indicates potential change. In the universal process of becoming these numbers act as increments of change and as binders. Between two square numbers there is one mean proportional number; between two cubes there are two mean proportional numbers. This third (odd) number acts as the 'lock' or binder between what might be called the building blocks of the universe. This role cannot be played by even numbers which, when divided, are empty in the centre and therefore weak. Thus, in the building of the square, the odd number is always master. It moves beyond the dyad to the three, the four and eventually the cube.

 It is important, however, to recall that the Demiurge or Tetraktis was described by Pythagoras as 'the August Four' and 888 was a symbol of the generation of reality by the number 4. That this doubling process involves even and odd numbers is not surprising in the light of the interrelating triangle and square. But this does not diminish the principle of doubling itself, and anyone familiar with the process of meiosis can testify to the correlative expression of both characteristics. They interrelate in tandem as expressions of the One. The Sacred Decad generates the number 55, which is the sum of 1, 2, 4 and 8 added together with 1, 3, 9 and 27. The Greek sign for 55 was EN (the name of the One) and in this we see that the One, the Decad, 55, and the duple and triple ratios are all related. Plato discussed this in the Timaeus and showed that five squares in succession (1, 4, 9, 16 and 25) equal 55 or the One.

 The Sacred Four of the Tetraktis is formed by the Triad within the circle, "the Square within the Circle being the most potent of all the magical figures". The idea of squaring the circle, then, is ultimately a great mystery concealed in the androgynous nature of the Logos itself. The line that drops down from the apex of the Triad to form the first cross simultaneously creates the Perfect Square. The problem of squaring the circle has fascinated men for ages. When in prison, Anaxagoras tried to solve it and Hippocrates attempted its solution by squaring certain lines on the circle, Iamblichus said that Apollonius squared the circle by means of a certain curve ('the Nicomedes curve') and referred to a tract in which he discussed a cylindrical helix capable of squaring the circle. Others, of course, have focussed upon the area of the surface leading to the discovery of p, the ratio of the diameter to the circumference of the circle. In occult teaching this ratio describes the 'Ring-Pass-Not' beyond which there is neither height, breadth or thickness, none of the attributes identified with the square. It is said that "Sat is born from Asat, and ASAT is begotten by Sat: the perpetual motion in a circle, truly; yet a circle that can be squared only at the supreme Initiation, at the threshold of Paranirvana."

 In identifying the square as a symbol of equity and honesty, men have intuitively perceived that it is a reflection of Truth. But the square in perpetual motion becomes a circle which attains a unity over and above the obstacles of the number 4 and the equiangular nature of the static order of the material world. The accomplishment of this is possible only in man, the mystic square or the square informed by the Triad. When his quaternary becomes filled with the light of the Triad, it becomes dynamic in a transcending fashion. It revolves, as it were, on its axis so as to resemble the diamond shape of the soul and the pull of the overbrooding Triad awakens a conscious awareness of the vertical Tetraktis line. At some point in the balance of things between the above and below, the individual becomes aware of himself as being at once the upward and downward pointing triangle of all creation. As Mahatma K. H. said, "The two interlaced triangles are the Buddhangums [doctrines] of Creation. They contain the 'squaring of the circle,' the 'philosophical stone,' the great problem of Life and Death, and the Mystery of Evil."

 Thus, the square lays the basis of manifest life and the discovery of the Perfect Square within leads ultimately to the mystery of the squaring of the circle. In attempting to apply the qualities of symmetry, order, proportionality; equity and stability to one's relationships in the world, it is necessary to embrace the gnomon and continually to increase the magnitude of the square. If the 'square deal' spoke to the hearts of men, it was because it affirmed interdependence and a continual extension of the principles of equity. If all work in a way that demonstrates the integrity of the square, and if this is not confused with a static imitation of perfection, the fire of spirit can quicken and inform the material vestures so that they are alchemized and reassemble in more universal expressions of the archetypes. As Plato pointed out, the theory of innate knowledge independent of our experience holds "as much about two equal lines as about absolute beauty, the absolute just and good and all things whatever". It is in that realm, where the soul recollects, that the Tetraktis, the Perfect Square, the ultimate magic square, may be realized, with the benediction of the Heavenly Man.