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 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
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
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
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 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.
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
Such a combination of numbers resembles what Plato described to be the characteristics of the
Socrates, in Plato's
During the European Renaissance, Giorgio in his
The idea of universal order begins with Deity or the
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 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
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
It is important, however, to recall that the
The Sacred Four of the
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
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 |