MS3. MAGIC SQUARES

MAGIC SQUARES AND ASTROLOGY

The famous Cornelius Agrippa(1486-1535) constructed Magic Squares of the orders 3, 4, 5, 6, 7, 8 & 9 which were associated by him, with the 7 astrological "Planets", namely Saturn, Jupiter, Mars, Sun, Venus, Mercury and the Moon. A magic square of order 4 appears in one of Albercht Durer's best engraving "Melancholia 1_ (1514). His square is like the one from India, slightly altered (by rotation) and is shown below:
16 03 02 13
05 10 11 08
09 06 07 12
04 15 14 01 (please note 0's have been placed by me, they are not there in the original magic square.)_This square is also known as the Magic Square of Jupiter and is supposed to be intimately connected with all the attributes which, traditionally, astrologers have associated with Jupiter namely - Happiness, Prosperity, Good Fortune, A Long Life and so on. This square appears in the background in the Durer's engraving of the spirit of Melancholia. Albercht Durer, a 16th Century Artist, who was considered to be the greatest of his time, held the belief, still held by some present day occulists, that numerological techniques could be used to induce mood changes, the lifting, for example, of acute depression, the psychological disorder, which our ancestors referred to as melancholia. The Magic Square of Saturn, reproduced below, is said to have the reverse effect.
4 9 2
3 5 7
8 1 6
The Magic Square of Saturn- the harbinger of Love is reproduced below:
22 47 16 41 10 35 04
05 23 48 17 42 11 29
30 06 24 49 18 36 12
13 31 07 25 43 19 37
38 14 32 01 26 44 20
21 39 08 33 02 27 45
46 15 40 09 34 03 28

MAGIC SQUARES AND POPE URBAN V111:

Pope Urban V111 was a dedicated student of astrology of his time and had come to the conclusion that the influence of planetary positions of Mars and Saturn in his horoscope, combined with an impending eclipse of the Sun, would have to be countered in some way or the other. He consulted a Dominician Friar named Campanella, one of the most learned man of his time, who advised that the beneficial planetary effects of Jupiter and Venus shold be used to neutralize the harmful effects of Mars and Saturn. The two squares were hung on his wall. The supposed influences of the squares were supplemented by the Pope listening to music, which he believed to be of a Jupiterian or Venusian type, scenting his apartment with roses and drinking distilled liqueurs flavoured with aromatic plants, which the astrologers of his time believed to be under the domain of Jupiter and Venus. Perhaps all these worked, certainly the Sun's eclipse took place, without the death of Pope Urban V111, who lived on for another 16 years.

ARE THEY REALLY MAGICAL?
WELL do these squares have really some magical properties? I believe they do have, for example, the total number of 5x5 squares is over 60 millions, using numbers from 1 to 25 and to give a total of 65, but, if you are asked to construct one, I am sure, you will either not succeed or take a long time! And what if you are asked to produce a 16x16 square, using numbers from 1 to 256, yielding a total of 2056? (Let me tell you the number of solutions will be in trillions.) Well don't despair, by the time you digest this paper, you will, probably, only take that much time as is required to draw the square and write down the numbers in the 256 cells of the given square - so aren't they really magical? Considering the stupendous results achieved by the engineering and the other applied sciences with the assistance of mathematics, it must be accepted that the forms of thought are wonderful indeed! As such, it is not at all surprising, that the primitive thinkers of mankind, when the importance of the laws of formal thought, in some way or another, first dawned in their minds, attributed to them supernatural powers. Somewhere herein lies the origin of the word MAGIC SQUARES! However Prof. De Morgan has been reported as having said: "Though the question of Magic Squares be in itself of no use, yet it belongs to a class of problems which call into action a beneficial species.... Without laying down any rules for their construction we shall content ourselves with destroying their magic quality and showing that the non-existence of such squares would be much more surprising than their existence". Truth, however, is not such a simple thing and, perhaps lies in between. When we discuss methods of formation of these squares we will find that the squares do need different solutions and the laws are indeed difficult to formulate. We have already seen how the numbers of the 3x3 square and the 4x4 square have some interesting properties. More as we go on to the other squares!