A magic square is a series of numbers arranged in an equal number of cells constituting a square figure, the enumeration of all of whose columns, vertically, horizontally and diagonally, wilt give the same sum. The Oriental philosophers, and especially the Jewish Talmudists, have indulged in many fanciful speculations in reference to these magic squares, many of which were considered as talismans. The accompanying figure of nine squares containing the nine digits so arranged as to make fifteen when counted in every way, was of peculiar import.
There was no talisman more sacred than this among the Orientalists, when arranged as in Figure 1-6. Thus designed, they called it by the name of the planet Saturn, ZaHaL, because the sum of the 9 digits in the square was equal to 45 (1+2+3+4+5+6+7+8 +g) which is the numerical value of the letters in the word ZaHaL, in the Arabic alphabet. The talmudists also esteemed it as a sacred talisman because 15 is the numerical value of the letters of the word JaH, which is one of the forms of the Tetragrammaton.
The Hermetic Philosophers called these magic squares Tables of the Planets, and attributed to them many occult virtues. The Table of Saturn consisted of 9 squares, and has Just been given. The Table of Jupiter consisted of 16 squares of numbers, whose total value is 136, and the sum of them added, horizontally, perpendicularly, and diagonally, in rows, is always 34; as in Figure 3. So the Table of Mars consists of 25 squares, of the Sun of 36, of Venus of 49, of Mercury of 64, and of the Moon of 81. These magic squares and their values have been used in the symbolism of numbers in some of the advanced Degrees of Freemasonry.
This subject should not be dismissed as a purely imaginative study. The matter has for many years engaged the attention of mathematicians of the highest quality. The Magic Square has been worn as an emblem or talisman insuring good luck to the possessor and evidently it formed an essential part in the early symbolism connected with the Craft. That singular picture by Albrecht Durer of the sixteenth century, Malancolia, shows a Magic Square with many other symbols easily recognized by members of the Masonic institution. The history of the Magie Square goes back hundreds of years and there has been undoubtedly through this period a superstitious, as vrell as a scientific, esteem for this device. They have not been worked out to their present perfection in any other than by systematic methods. The earliest known writer on the subject was a Greek, Emanuel Moscopulus, who flourished in the fourth or fifth century. Since that time there have been many laborers upon this work.
One of the very interesting of these Magic Squares is referred to above by Doctor Mackey. This occurs in a book by Agrippa (De Occulta Philosophie, logo) and is quoted on page 279 of George Falkener's Gaines Ancient arid Moderns By first arrangement the numerals from 1 to 16 in four rows as in Figure 4 it will be seen that by leaving the numerals unchanged at each corner of the large square, namely 1, 4, 16, and 13, and also at the inner square of 6, 7, 10, and 11, and substituting the other pairs of numerals, reversing them at the time, we have in Figure 5, this remarkable Magic Square reversed, which Brother Mackey has called the Table of Jupiter. The combinations of this figure are surprising, amounting to fifty-six arrangements, each totaling thirty-four. The four horizontals, as 1+15+14+4=34, 12+6+7+9=34, etc; and the four perpendicular columns, as 1+12+8+13 = 34, and 15+6+10+3=34, etc.; the diagonals,1+6+11+16= 34, and4+7+10+13=34; the diamonds, 1+7+16+ 10=34, and 4+11+13+6=34; the squares, 1+4+ 16+13=34, and 6+7+11+10=34; the oblongs, 15+ 14+2+3 =34, 12+9+5+8=34, and the romboids, 1+15+16+2=34, and 4+9+13+8=34, etc.
The method of working out a Magic Square with an uneven number of cells was suggested by De la Loubere. The several steps may be considered as follows: In assigning consecutive numbers, proceed in an oblique direction up and to the right as 4, 5, 6, as in Figure 6. When this would carry a number out of the Magic Square, write that number in the cell at the opposite end of the column or row, as shown by the numbers in the margin of Figure 6. When the application of the first of these rules in the present paragraph would place a number in a cell already occupied, write the new number in the cell beneath the one last filled. For instance, the cell above and to the right of 3 being occupied, 4 is written under 3. Treat the marginal square at the upper nght-hand corner marked x as an occupied cell and apply the rule given in the last sentence. Begin by putting 1 in the top cell of the middle column. A comparison of Figure 6 will show that it is a reflection of Figure 1 given by Doctor Mackey.
One of the most successful of all students of the subject unquestionably was Brother Benjamin Franklin. Two of his efforts, an 8xS and a 16x16, are today unsurpassed as purely remarkably successful attempts at the making of Magic Squares. A communication to an English friend by Brother Franklin appears in the work entitled Letters and Papers on Philosophical Subjects by Benjamin Franklin, printed in 1769. This letter is in part as follows:
According to your request I now send you the arithmetical curiosity of which this is the history. Being one day in the country at the house of our common friend, the late learned Mr. Logan, he showed me a folio French book filled with magic squares, wrote, if I forget not by one Mr. Frenicle, in which he said the author had discovered great ingenuity and dexterity in the management of numbers; and though several other foreigners had distinguished themselves in the same way, he did not recollect that any one Englishman had done anything of the kind remarkable.
I said it was perhaps a mark of the good sense of our mathematicians that they would not spend their time in things that were merely domiciles novae, incapable of any useful application. He answered that many of the arithmetical or mathematical questions publicly proposed in England were equally trifling and useless. Perhaps the considering and answering such questions, I replied, may not be altogether useless if it produces by practice an habitual readiness and exactness in mathematical disquisitions, which readiness may, on many occasions be of real use. In the same way, says he, may the making of these squares be of use. I then confessed to him that in my younger days, having once some leisure (which I still think I might have employed more usefully) I had amused myself in making this kind of magic squares, and, at length had acquired such a knack at it, that I could fill the cells of any magic square of reasonable size with a series of numbers as fast as I could write them, disposed in such a manner that the sums of every row, horizontal, perpendicular, or diagonal, should be equal; but not being satisfied with these, which I looked on as common and easy things, I had imposed on myself more difficult tasks, and succeeded in making other magic squares with a varietal of properties, and much more curious. He then showed me several in the same book of an uncommon and more curious kind, but as I thought none of them equal to some I remembered to have made, he desired me to let him see them; and accordingly the next time I visited him, I carried him a square of 8 which I found among my old papers, and which I will now give you with an account of its properties Figure 7-9. The properties are:
1. That every straight rov., horizontal or vertical, of 8 numbers added together, make 260, and half of each row, half of 260.
2. That the bent row of 8 numbers ascending and descending diagonally, viz., from 16 ascending to 10 and from 23 descending to 17 and every one of its parallel bent rows of 8 numbers make 260, etc., etc. And lastly the four corner numbers with the four middle numbers make 260. So this magical square seems perfect in its kind, but these are not all its properties, there are five other curious ones which at some time I will explain to you.
This Magic Square by Franklin is given here as Figure 7.
Brother Paul Carus has investigated the means by which Brother Franklin may have worked out his system of Magic Squares but it is really somewhat a question even now with all the later studies that have been given to the subject whether any one has perfected an ability capable of preparing a means of producing these designs with the facility that Brother Franklin mentions. Those who wish to examine the subject further will find it discussed in the Encyclopedia Britannica, in Magic Squares and Cubes, by W. S. Andrews, containing chapters by Brother Paul Carus and others, and in a Scrap Book of Elementary Mathematics by William F. White, as well as in Mathematical Recreations by Professor W. W. R. Ball.
This subject is somewhat allied as a mathematical curiosity with two other figures which come down to us through the Middle Ages, the Magic Pentagon or the Five Pointed Star, as a symbol of the School of Pythagoras, as in Figure 8, and the Magic Hexagram, Figure 9, commonly called the Shield of David and frequently used on synagogues, as Brother Carus points out. these two designs, Figures 8 and 9, have a peculiarity that is not perhaps noticed at the first glance- They can be drawn by one stroke of the pencil, beginning at any point. If they be compared in this respect with any square having two diagonals the difference can soon be tested as the square is not capable of being drawn as a complete figure, including the two diagonals, with one stroke. In order to better illustrate the operation of drawing Figures 8 and 9, numerals have been attached to illustrate the movement of the pencil in tracing them out. Of course, they can be begun at any place in any one of the lines composing the figures.
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