Two thousand years of numerical magic squares

Two thousand years of numerical magic squares Vladimir

Karpenko

A magic square is any square-shaped array of numbers or letters exhibiting certain properties: in numerical squares it is the constant sum of numbers in each row, column, or diagonal. This is the constant of a square, while the number of cells in a row gives its order. Any set of numbers can be used if the condition of a constant summation is met. In natural magic squares only natural numbers from 1 to n* are used. The same set of natural numbers written sequentially into a square grid leads to a natural square which is not magic. Numerical magic squares attracted attention in the distant past. They have been used in charms, talismans, and in other objects destined to magical use. More rarely, these squares appeared in philosophical or alchemical speculations as expressions of certain ideas, or even as an instruction to experiments. Now studied as problems in number theory, numerical magic squares once had deeper significance, far from purely mathematical questions. A Turtle from the River Lo ‘The Master said, The phoenix does not come; the river gives forth no chart. It is all over with me!’ The 8th saying from the chapter IX of Lun Yii (Conversation and Discourses of Confucius[l]) has always attracted attention. No wonder; Confucius usually expressed his thoughts less mysteriously. Legend has it that Chinese emperor Ta Yu [Yii the Great, one of three legendary emperors who allegedly ruled 2205-2197 BC] was presented with two charts or diagrams[2]. Ho Thu. drawn in a green colour, was brought by a dragon-horse that appeared out of the Yellow River; better known Lo Shu was said to be delineated in red on the carapace of a turtle swimming in the river Lo. Both peculiar gifts were numerical constructions (figure 1 (a, b)). A mysterious sign Confucius had been waiting for was apparently just Lo Shu. the smallest numerical magic square. There is no doubt that the Lo Shu square is the oldest magic square known. The first indisputable reference listing its numbers in regular order is from the first century AD, but indirect references may be traced back to the writings of Tsou Yen, alchemist and philosopher active in the fourth century BC[3]. Ancient Chinese natural philosophy was based on the five elements (wu hsing), and V. Karpenko,

Ph.D.

Is Associate Professor of Physical Chemistry, Charles University, Prague. His research is concerned with the physrcal chemistry of human blood proteins but for the past 20 years he has worked also in the history of chemistry and alchemy. Endesvour, New Series, Volume 18, No. 4. 1994. Copyright ‘8 1994 Elsevier Science Ltd. Printed in Great Britain. All riahts reserved. 0160-9x7/94 $7.00 + 0.00. Pergamon

Figure squares

1

(a) The Ho Thu diagram; can be derived, but they

are

(b) The Lo rotations

Shu and

square; reflections

seemingly of the

eight third-order only possible

type. on two complementary principles yin and yang. The elements - earth, water, metal, fire, and wood - were supposed to be the basic constituents of matter. As generalized properties they were expected to undergo mutual transformations: earth to metal, since the Earth apparently gave birth to metals; metal to water, because, on proper treatment, metal can turn to liquid state; etc. Yet the elements were mutually antagonistic as well: fire is extinguished by water; water stopped by an earthen dam; etc. Yin and yang are well known in a graphic presentation (figure 2); yin represents the female principle - darkness, passivity - while yang as the male principle is connected with light and energy. The first ‘magic’ property of the Lo Shu is that even and odd numbers alternate around the centre. In China even numbers were considered yin, odd numbers yang. Each of these pairs stood for one of the elements; central 5 thus acquired crucial position (figure 3). This number symbolized the element earth and, simultaneously, the

most prominent of the five geographical directions, the centre, added in ancient China to east, west, south, and north. China was supposed to be the centre of the world, the Empire of Centre. Since the world was thought to be square, the Lo Shu summarized the basis of Chinese natural philosophy as well as geographical conceptions. How did the Chinese arrive at this square? It can be easily constructed by the trial-anderror approach, but in China transposition method starting from a natural square (figure 4(a)) was introduced. For a long time the Lo Shu was not a true magic square (figure 4 (b)) and was simply revered as a symbol of the Earth. Instead of direct transformation by interchanging the numbers 2 and 8 this figure was first rotated 180” (figure 4(c)). This extra rotation was connected with I Ching [4] (Book of Changes), a divinatory book containing 64 hexagrams, symbols consisting always of six lines. Solid ones represented yang, broken symbolized yin. The origin of I Ching was traditionally attributed to the legendary Emperor Fu Hsi

147

I 18

6 Water

Figure 3 The arrangement of the five elements in the Lo Shu square.

Figure 2 Yin (black) and yang (white). No dividing line intersecting the centre can be drawn which would separate white or dark field only. Traditionally, these principles are written in the order yin-yang so that the passive, female partner precedes the active male one. It is supposed to be a remnant of matriarchal society.

2

6

7

5 8

3 4

4

9

8

4

9

2

3

5

7

3

5

I

2

1

6

8

1

6

9

(a) Figure square

4 (a)-(c) was used

(b) Three steps in the construction in its nonmagic variant (b).

lived from 2953 to 2838 BC. Hexagrams have been derived from eight trigrams arranged traditionally in a circle. The early trigram circle could symbolize the passing of Time. as well as directions of Space, or it served as a symbol for Sky and Heaven. This symbol could be quite naturally completed by its counterpart Earth with the Lo Shu square as its symbol. The seemingly complicated way in which this square was constructed reflects transformations of trigrams; they changed their places in the circle several times [5]. The extraordinary importance of 5 was stressed by the position of this number in the centre of the whole Universe: it was in the centre of Earth represented by the Lo Shu placed in the centre of the heavenly sphere [6] symbolized by the eight trigrams (figure 5). who reputedly

148

of the Lo Shu square;

(c) originally

this

In its classical form the trigram circle fufilled a function of the indicator of the Eight Directions, with south pointing to the top. Eight trigrams were used in this way to mark directions on the Chinese geomancer’s compasses that are still in use in Hong Kong and on Taiwan. Even the modern instruments have on their dials, above each trigram, an arrangement of small dots representing the corresponding number of the Lo Shu. Forgotten past Because of its importance. the Lo Shu square was kept secret for centuries. Its use as a divination board continued down into the Tang Dynasty (AD 6 18-907) and until the end of this Dynasty this square, although mentioned in writings, was never publicly shown. It was openly published shortly after

Figure 5 The final shape of the eight trigrams with numbers of the Lo Shu square.

the fall of the Tang dynasty. By that time. however, the original meanings of this square were largely forgotten [3]. In 1275 the scholar-antiquarian Yang Hui published a small collection of magic squares, but he was obviously quite ignorant of their original meaning, and apparently unaware of the basic principles underlying their construction. These squares, nevertheless, are the only ones of the orders higher than three known from China before the thirteenth century. Four years after this book appeared, the Mongols overran China. and Kublai Khan. the Mongol Emperor of China, ordered in 1281 AD that all the Taoist books in his realm should be burned. Yang Hui’s book thus became important as the only source on magic squares. The squares of the third to ninth orders delineated in Yang Hui’s treaties are quite original, as can be shown by the example of the ninth-order square. the most thorough ever evolved by the Old Chinese mathematicians [3]. This square was derived in a way quite different from later methods and exhibits an undoubted relationship to the Lo Shu. In the first step, a natural square of nine was constructed: numbers from 1 to 81 were written in nine vertical columns. Then, from each horizontal line, a little magic square was constructed with the smallest digit placed in the centre of the bottom row. These squares were put together so that in the final ninth-order square digits I to 9

I

Figure 6 separated

29

)

I 74

1

13

36

81

18

21

39

51

23

41

59

25

43

61

66

3

48

68

5

50

70

7

52

35

80

17

28

73

10

33

78

15

26

44

62

19

37

55

24

42

60

71

8

53

64

1

46

69

6

51

11

square was given by Yang Hui. Nine of its subsquares the numbers of the Lo Shu square are in bold italics.

occupied the same position, relative to each other, as in the Lo Shu square (figure 6). In the resulting ninth-order square, digits of each number can be added until a single digit remains. For example, in the subsquare at the top left, 31 is added as 3 + 1 = 4; 76 needs two summations 7 + 6 = 13, and 1 + 3 = 4, etc. This process always leads to the smallest number of the given subsquare, here 4; for the subsquare next to the right it is 9, etc. In the Lo Shu the number 9 represented the highest powers of yang; now there were even nine degrees of this good luck for determining somebody’s fortune in divination. Similarly 1 was considered baleful. There are even more remarkable properties hidden in this square. If 9 is repeatedly subtracted from the constant sum of this square, 369, the result is again 9; if digits of the total sum of this square, 3321, are added, 9 is obtained as well. Analogous links to the Lo Shu are apparent in all other squares published by Yang Hui. These squares were known in China in the distant past, probably when the Lo Shu was in full flourish. Dreams of gold The further spread of magic squares in the ancient world was linked with the development of papermaking; attempts to construct squares usually needed much of the writing material. Invented in China, paper was introduced into Central Asia by captured Chinese specialists, and found its way later into the Arabic world and thence to Europe. Here favourable conditions for magic squares existed, because the Arabs used positional numbers borrowed from Indians.

of the then known metals was subordinated to one of the planets, that included also the Sun and the Moon. In the seventh century AD these pairs were as follows [7]: Sun gold; Moon - silver; Mercury - mercury; Venus - copper; Mars - iron; Jupiter tin; and Saturn - lead. In harrfinian planetary worship a certain number of steps always lead to the throne of an idol made of the corresponding metal (Table 1). This number could have been the order of a magic square related to the given planet [8], but there is no direct proof of this hypothesis. Jfibir ibn Hayyan was an outstanding figure in Arab alchemy, but it is not certain whether he, as an individual, lived at all; if he did, it was then between CAD 72 1 and 8 15. In his work strong mystical tendencies are felt, obviously because of his inclination to sufism [9]. He is perhaps analogous to Hippocrates, identified with medical works by many hands. Arabic alchemy originated from earlier Hellenistic science based on Greek natural philosophy, particularly on Aristotle’s theory. Four elements - earth, water, fire and air - were considered basic constituents of matter. Jribir, however, refined these theories into a rather complicated system in which numbers played an important role. Jabirian numerology and its application to proposed laboratory operations can be demonstrated on a passage from his treatise K&b al-muwcizin (Book of the Balances). A numerical value for basic properties (derived from Aristotelian theory; e.g. coldness, humidity, etc.) was assigned to each letter of the Arabic alphabet. When a word for a substance was analysed letter by letter, the numerical values of the letters determined the composition of the substance. Silver (fidda), for example, should have contained the following proportion of properties [lo] :

I

16

The ninth-order by thick lines;

1

I

31

are

It is not too easy to trace the origin of magic squares in the Middle East. Indirect evidence points to harr&iun culture, another source of information is the work of the Arabic alchemist Jabir ibn HayyBn, though some writing attributed to him may be the work of others. In the Syrian town Harrti, lying in the western bend of the Upper Euphrates, the last outpost of Sumerian and Babylonian civilizations persisted. Perhaps here, resulting from the interest of Mesopotamian cultures in astronomy and astrology, links between metals and planets originated: each Result

of analysis

Heat

14 danaqs

of letters Total

Complement 5; danaqs

1; dirhams (=

3+ dirhams

Humidity

14 danaqs -

34 dirhams

52 dirhams

53 dirhams

(=

5 x 1:)

Dryness

-

94 dirhams

94 dirhams

(=

8 x I$)

19% dirhams

(=

17 x 1;)

Coldness

Sum TABLE

1 THE

LINK

The number of steps to the throne/ the order of the square

BETWEEN

NUMBERS

AND Harran Picatrix Cardano

3 x 14)

PLANETS Agrippa Paracelsus Esch Mezareph

3

Moon

Saturn

4

Mercury

Jupiter

5

Venus

Mars

6

Sun

Sun

7

Mars

Venus

8

Jupiter

Mercury

9

Saturn

Moon

149

This analysis shows unambiguously that the wish was father to the thought - the complements were chosen intentionally so that as a basic unit 1 l/6 dirham was obtained. Then for all four named properties the resulting quantities stood in proportion 1:3:5:8. Choice of these numbers was not accidental, as is obvious from a similar analysis made with lead and gold. It should be noted that when describing metals as such Jabir always attributed two ‘inner’ and two ‘outer’ qualities to each of them. Further analysis of his Book of the Balances revealed the following interrelations between properties of lead and gold [ 1 I]. Lead: outer qualities 3 8 inner qualities: 1 5

parts of coldness parts of dryness part

of heat

parts of humidity

together 17 parts Gold:

outer

quabties

3 parts

Figure 8 Persian method for the construction of odd-order squares. The numbers can be written in downward as well as upward directions. In later modifications 1 was placed at the centre of the top or bottom row.

of heat

8 parts of humidity inner qualities: 1 part of coldness 5 Darts of drvness together 17 parts If the qualities are summed up in this way, no significant differences between the two metals are found. According to Jabir’s analysis both should have contained 17 parts of certain qualities. the proportion of which decided the nature of the given metal. Then, by proper treatment (e.g. alchemical) these qualities could have been changed. with resulting transmutation of lead into gold. This analysis was performed with the word usrub for lead, but this substance has yet another Arabic name. rasas, which would lead to entirely different results. In every numerical analysis of words the numbers I, 3,5. and 8 appeared. Their relation to the third-order

square

is obvious

[8]:

they are obtained by the gnomonical analysis of this square (figure 7). In Jabirian theories these numbers were also linked to Aristotelian elements: l-fire, 3-earth. S-water,

R-air.

Jabir’s

work

represents

one

approach to magic squares, namely their use in alchemical speculations. Slightly later, about AD 989, an Arabic encyclopedia produced by an Islamic brotherhood called the Ikhu*cin as-S@ (Brethren of Purity) appeared. In this work, for the first time in the Middle East, magic

Figure 9 Knight’s move method for the odd-order squares; as in the previous case (figure 8) moves can be directed downward or upward. squares of the orders three published [ 121. The Brethren

new system of constructing the odd-order squares, a dynamic one that can be described in terms of moves of chess-men. The Arabic system was later widely adopted in the Middle East and was fully developed by the thirteenth century as evidenced by Persian manuscripts. While the Chinese started from natural squares. transposing individual numbers, in the Persian method the 1 is placed immediately below the centre cell and the following numbers are written either diagonally, or in positions corresponding to the move of a knight in chess [13] (figures 8, 9). Muslims have considered magic squares to be symbolic

Figure 7 Gnomonical analysis of the third-order square. In this method an L shape or gnomon (shown by thick lines) is removed from the original square.

150

to nine were introduced a

models

of the Universe

and

squares constructed in the above way become symbols of Life as an endless flux: when all numbers are written in this square, the last move would lead back to 1, to the start. A rare square with 1, the symbol for Allah the Creator, in the centre acquired extraordinary significance (figure 10). The movement starts at the centre and finishes there,

which

corresponds

to the Sufi

idea

that Allah is both Source and Destination 1141.

Figure 10 constructed method.

Magic square with 1 in centre by the knight’s moves

Experimenting with fourth-order squares, Persian scholars knew hundreds of them. Except for the third-order square, this kind became the most popular in the Islamic world. Lower order squares were drawn on various talismans and charms; were engraved on the blades of swords to increase their cutting power; or engraved on metal bowls to purify their content, etc. Typical was the use of the third-order squares in talismans for easier delivery of babies. The question of priority in magic squares between Arabs and Indians cannot be answered, since no evidence can prove the claim of either side. An attempt was made to show that a fourth-order magic square was used by Varahamihira, an Indian authority on astronomy, astrology, and divination [ 151. In his voluminous work, Brhatsamhita, dated about 550 AD he described the composition of a certain kind of perfume. Numbers given in this recipe can be arranged into a fourth-order square identical with the most popular one in the Islamic world. European epilogue Magic squares reached Latin Europe rather late [ 161, in this respect manuscripts of Picatrix from the thirteenth century are particularly important. This crucial work on astronomy and astrology was translated from an Arabic original written allegedly in the tenth century. Picatrix had obviously brought the first examples of magic squares up to the ninth order, but gave no instruction how to construct them. Squares of individual orders were linked to planets in the same way as in Harran (Table 1) and advice was added from which material. and at what time, amulets bearing these squares should be made. Deeper interest in magic squares can be observed from the fourteenth century onwards. At that time a Byzantine scholar Manuel Moschopoulos described in a letter to his friend the principles of construction of magic squares [17]. He employed the Persian continuous and knight’s move method for the odd-order squares; the Indian method for the fourth and eight order squares. These methods, rediscovered by B.

Figure 11 The page from Cardano’s square constructed by the diagonal of the same order (figure 6).

treatise on magic squares. The ninth-order method can be compared with the Chinese

square

TABLE 2 FOR EACH ‘SEAL’ BOTH PARACELSUS AND AGRIPPA RECOMMENDED A METAL ON WHICH CORRESPONDING SQUARES SHOULD BE ENGRAVED (MERCURY SHOULD BE ALLOYED WITH LEAD). AGRIPPA’S CONCEPT, HOWEVER, WAS BROADER: DEPENDING ON THE CONSTELLATION OF PLANET, FAVOURABLE OR UNFAVOURABLE, THE METAL COULD BE CHOSEN, WITH DIFFERENT RESULTING EFFECTS FROM SUCH TALISMAN Metal The order the square

3 4 5 6 7 8 9

of

of the talismanic

Paracelsus

medal Agrippa Constellation

Pb

of the planet

Favourable

Unfavourable

Pb

not stated

Sn

Aglcoral

not stated

Fe

Fe

cu

Au

Au

not stated

CU

Ag

cu

AglSnlbrass

not stated

&I

Pb

Hg (with Ag

Pb)

de MCziriac and S. de Laloubtre in the seventeenth century, are sometimes presented under their names [ 181. Yet even the complete instructions given by Moschopoulos did not increase interest in magic squares. Only two centuries later did further works on this topic appear, the Occulta philosophia by Agrippa von Nettesheim dated 1531, and eight years later, the mathematical treatise Practica arithmetice et Mensurandi singularis by Girolamo Cardano (figure 11). In both books there are identical magic squares of the orders three to nine, but related to the planets in opposite sequence (Table 1). Then, 1567, the third work appeared Archidoxa Magica, attributed to Paracelsus. The Seventh book of this treatise describes seals of the planets, actually talismans made of the corresponding metal and bearing engraved magic squares. The sequence of the planets and squares is the same as in Agrippa’s book (Table 1). Both magic treatises attracted extraordinary attention; Agrippa’s work was among the most printed books of the sixteenth century. Cardano’s book did not become so popular, nor were German mathematical works written by Michael Stifter in 1544, and by Adam Riese around 1550. The works of Agrippa and Paracelsus started the period of magical applications of numerical squares. Agrippa presented magic squares both in Arabic numerals and in Hebrew notation with letters of the Jewish alphabet used as numbers (figure 12). This approach was obviously a reflection of cabalistic influence, the roots of which can be traced back to the Toledan scholar Abraham ibn Ezra, who, in the twelfth century, published the magic square of three with a mystical interpretation [16]. These thoughts were later developed by his successors in religio-mathematical speculations of Kabbalah [ 191. In Agrippa’s and Paracelsus’ books, the squares were constructed by Islamic methods, as is obvious from diagonal construction of the odd-order squares. Yet no direct link has been traced between Agrippa, Paracelsus and Islamic sources. Especially in the seventeenth century, talismans - often medals with magic squares (figure 13) - became quite common [20, 2 11. Two basic kinds of medals can be distinguished - with Arabic numerals and with Hebrew letters, respectively. In the latter type no links to the Jewish community can be established [22]. Only limited information on the metal used in these medals is available; it seems, however, that in the majority of examples, it corresponds with Agrippa’s system (Table 2). The symbolism of these medals is quite complicated: often the names of the symbols of planets appear as given by Agrippa, figures like Saturn with scythe, or a lion representing the Sun accompany a square. In Europe, one particular square attracted particular attention: the fourth-order square of Jupiter (figure 14) on Albrecht Diirer’s engraving Melencolia I. The two central

151

turbing its magic properties [18]. The use of magic squares in Europe was quite different from that in the Chinese or Islamic worlds. In those regions magic squares have played a role in philosophical speculations on the verge of science. In the Arabic world they even crossed this border and the numbers derived from them have assumed a quantitative role in alchemy. They led to attempted quantification of qualities like hotness or dryness that are difficult to define and impossible to prepare in a pure state. In spite of their attractiveness magic squares have never appeared in European alchemy. The only exception seems to be a Jewish work Esch Mezareph (The Purifying Fire) written not later than the sixteenth century [24]. In this treatise, which is in part alchemical, magic squares are connected with both planets and metals. Perhaps Esch Mezareph continued a line of Jewish cabalistic tradition beginning with Abraham ibn Ezra, if not earlier. Here the Sephiroth, spheres of light representing the ten basic forces of the cosmos [19], are related to planets and metals; the number of individual sephiroth are in the same sequence as the order of squares and metals given by Agrippa (Table 1). Magic squares are a part of the history of mathematics, and a marginal episode in the history of natural philosophy and science. They appeared in all major cultural regions and have influenced temporarily some field of science; in China, natural philosophy, in the Arabic world even experimental alchemy. Only in Latin Europe did they remain almost exclusive an aspect of mysticism. The question is, why? It was in Europe that alchemy has reached its last flourish, but surprisingly without the attractive support of numerical magic squares.

Figure 12 Page from Agrippa’s treatise; in general, even-order squares are difficult to construct. The exceptions are orders that are multiples of four 1181. numbers in the bottom row form 15 14, the date of this engraving. Diirer’s square prompted numerous speculations about its origin and meaning. The explanations vary; mostly this square is supposed to be an original way to date the engraving, yet there are more complicated interpretations as well. According to one of them [23] this square should be related to Arabic numeral mysticism: the constant of this square, 34, corresponds with the numerical value of rrgtil, one of Allah’s names. In the meaning ‘time-limit’ it should be understood as the symbol of the transient nature of human life,

152

like the hour-glass and the balance shown in this engraving. Another explanation finds a link between this square and Diirer’s sojourn in Italy 150% 1507, particularly his stay in Bologna. Between 1496 and 1508 Lucca Pacioli wrote.there the treatise De viribus quantiruris about magic squares. His fourth-order square is identical with Diirer’s. Since this type of square is easy to construct it sounds quite plausible that Diirer could have adopted any of the related squares and reshaped it to get the date 15 14. In this type of square any two rows or columns can be interchanged without dis-

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16

3

2

13

5

10

II

8

9

6

I

12

4

15

14

1

Figure 14 The fourth-order square from the engraving Melencolia I. In all, 880 different squares of this order can be derived.

Figure 13 Page from Historische Miinzbelustigungen by J.D. Kijhler (NOrnberg 1736) with talismanic medal bearing the sixth-order square. This medal is now in the collection of Germanisches Nationalmuseum, Niirnberg (Med. 5954). [ 151 Hayashi, 1987. (16) Folkerts, 1981.

T. Hisforia M.

Sudhoffs

Marhemafica. Archiv.

14, 159, 65,

313.

I171 Ahrens, W. ‘Mathematische Unterhaltungen und Spiele’ B.C. Teubner Verl.. 1918. 1181 Benson. W.H. and Jacoby, 0. ‘New Recreations with Magic Squares’ Dover

Publ.. 1976. 1191 Halevi. Z’ev ben Shimon ‘Kabbalah’, Thames and Hudson, 1991. 1201 Ahrens, W. Das Weltall, 15. 81, 1915. 1211 Ahrens, W. Cosmos, 12, 115. 1915. 122) Ahrens, W. ‘Hebrlische Amulette mit magischen Zahlenquadraten’ L.Lamm Verl.. 1916. 1231 Fischer. L. Zeir. d. Deucschen Morpnl~ndischen Ges., 103, 308, 1953. [24] Scholem, G. Monatsch. fir Ceschichte und Wissensrhaf

des

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69,

Neue

Folge 33. 13, 95. 1925.

153