V. Karpenko - Numeral Magic Squares

7/26/2019 V. Karpenko - Numeral Magic Squares

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AMBIX,Vol. 40, Part 3, November 1993

BETWEEN MAGIC AND SCIENCE: NUMERICAL MAGIC SQUARES

By VLADIMIR KARPENKO*

THEpeculiar properties of numerical magic squares became involved from quite early times

in philosophical speculations and in practical activities. 1 To philosophers, the squares

seemed to promise the discovery (after appropriate analysis) ofnew relations between things

and their properties. However, this application of the squares was relatively limited when

compared with their use in magic. Here they were particularly important because the occult

properties attributed to numbers-or to any other symbols derived from them -seemed to

be enhanced in squares. In magical practices, a certain property might be ascribed to the

square as such-for example the third-order square might be used as a talismanic symbol to

promote easier delivery in childbirth. For this purpose, it was important that the property

could be formulated generally-it did not have to relate to the numbers of any particular

square. In philosophical speculations, the situation was different. In this case certain

relations between specific numbers and particular kinds of matter (or their properties) were

sought

In this way, numerical magic squares survived for centuries somewhere between magic,

philosophy and 'proto-science,' particularly alchemy. They were never particularly

prominent, remaining rather shadowy entities surrounded by superstition. It is not easy to

trace them through the history of the sciences-particularly of alchemy, which itself was

anything but an accessible body of knowledge. Nevertheless, we can discover some cases

where magic squares affected the alchemical speculations and experiments of adepts in the

remote past. With some simplification, this process can be described in two steps. In the

first, significant links between the squares and philosophical principles were sought. Then

the resulting hypothesis was applied to actual, or imaginary, experiments.

MAGIC SQUARES, PLANETS AND METALS

An important aspect of magic squares was their relationship with astrological and

alchemical speculations. Squares were linked with planets via the metals, a practice which

indicates the ancient origins of the squares themselves. By the seventh century A.D., the

pairing ofplanets with metals was firmly established,2 though the roots of this beliefmust be

sought in ancient Mesopotamian astrology. An interesting relationship between planetary

gods and numbers was noted by Stapleton.3 In the Harranian culture, the planet-metal

pairing was institutionalized in temple worship, where idols of the gods were made of the

corresponding metal (with the exception of mercury; here the metal was inside the statue in a

small vessel). In Europe, a relationship between numbers (the order of the square), planets

and metals was recognized somewhat later, in a magical context. Significantly, Saturn (lead)

always appeared on one of the ends of the series. This obviously relates to the ancient belief

that lead was the 'father of metals,' which can be traced to writings by Greek authors as early

as the first century A.D.4 Two centuries later, Zosimos maintained that' ... it is from lead

that the other bodies are derived.'s

* Department of Physical and l\!facromolecular Chemistry, Charles University, Albertov 2030, 128 40 Prague 2,Czech Republic.


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122 VLADIMIR KARPENKO

Zosimos, however, left open the question ofwhether some, at least, of the numbers which

appeared in his writings might be con~ected with the simplest magic square. In his Visions,6

the seven steps mentioned in the first sentence of Lesson Two undoubtedly symbolize the

seven metals. The fifteen steps leading to the altar described in Lesson One are more difficultto explain. The number of steps might reflect the constant of the simplest magic square; but

in this case, it seems that Zosimos adopted different relations between the squares and

metals than are common in Harran (see Table I) . Zosimos stated that lead is the father of

metals, but in the Harranian system, Saturn was associated with the number nine.7

AgrippaEsch mezareph

SaturnjupiterMarsSunVenusMercuryMoon

MoonMercuryVenusSunMars

jupiterSaturn

HarranCardanus

3456

78

9

Table I. The link between numbers and planets

The nurnber of steps to the

throne/ the order of thesquare

As mentioned above, the third-order magic square was generally used in all cultures as a

talisman for easier delivery, making its identification with lead as the father of metals look

more plausible. At the same time, it suggests that the system of numbers/squares and metals

adopted in Harran remained rather limited in use and did not spread significantly in the

ancient world.

ARABIC ALCHEMY

Arabic alchemy yields a rather different picture. This art, nourished by older cultural and

philosophical traditions-such as astrology and the esoteric ideas of the neo-Pythagor-

eans-reached its peak in the works attributed tojabir Ibn Hayyan. He was able to draw on

work by earlier Arabic scholars on magic squares, a treatise on the topic having already been

written by Tabit Ibn Qurra (836-901 A.D.).8 The role of the third-order square can serve as

an illustration of jabirian numerology9 and its attemp.ted application to laboratory

operations. jabir's approach was developed in Kutub al-Mawiizzn, a collection of treatises on

his theory of balances.lO Significantly, the numerical values of the basic Aristotelian

qualities (coldness, humidity, etc.) were ascribed to each letter of the Arabic alphabet. In

this way, the numerical value of the names of substances could be examined. Silver (fidda),for example, should have contained the following proportions of properties. I I


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BETWEEN MAGIC AND SCIENCE: NUMERICAL MAGIC SQUARES 123

HeatColdness

Humidity

Dryness

Result of Analysis ofLetters

I~ danaqI~ danaq

Sum

Complement

5~ danaq3 i dirham

5~ dirham

9~ dirham

Total

1-5dirham

3~ dirham(=3xI~)5~ dirham(=5xI~)9~ dirham(=8XI~)I9~ dirham(=I7xI~)

This analysis suggests that, in this case, the wish was father to the thought.

Complements were chosen intentionally so that I~ dirham was obtained as the basic unit.

Then for all four named properties the resulting quantities stood in the proportion 1:3:5:8.

The choice of these numbers was not accidental, as can be seen from the results of similar

analyses made with lead and gold.12 The Kutub al-Mawiizin reveals the following

interrelations between the properties of lead and gold:

Lead: outer qualities

Lead: inner qualities

together

Gold: outer qualities

Gold: inner qualities

together

3 parts of coldness8 parts of drynessI part of heat5 parts of humidity

17 parts

3 parts of heat8 parts of humidityI part of coldness

5 parts of dryness17 parts

Summing up the qualities in this way, Jabir could not detect any significant differences

between the two metals. According to his analysis, they each should have contained

seventeen pairs of certain qualities, the relative proportions of which decided the nature of

the given metal. With proper treatment, the proportions of the qualities could be changed,

resulting in the transmutation of lead into gold. Jabir's approach was straightforward: the

relation between the above numbers and the third-order square is quite obvious. As

Stapleton pointed out, the numbers I, 3, 5, and 8 can be obtained by the gnomonical analysis

of this square.13

InJabir's theory, these numbers were linked to the Aristotelian elements: 1-

fire, 3-earth, 5-water, 8-air. The third-century-A.D. Neoplatonist, Theodorus of Asine, 14is

said to have arrived at his numbers for the four elements (water 9, earth 7, fire I I and air 13)

by a somewhat similar procedure: I + 8 = 9, 3+ 4 = 7, 9 + 2 = I I and 7+ 6 = 13.

EUROPEAN ALCHEMY

The use of numerology in Arabic alchemy represented the highest flourish of magic squares

in their role as guides to the miraculous transmutation for which the alchemists hoped.


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124 VLADIMiR KARPENKO

However, although European alchemy developed from translations ofArabic sources, magic

squares did not become part of its equipment. This is perhaps surprising; alchemy was

always accompanied by esoteric beliefs, yet European alchemists seemed unaware of this

suggestive aspect. The first European tract to deal with magic squares and some details oftheir construction was a purely mathematical one, written by the Byzantine mathematician

Manuel Moschopoulos who was active around 1300. His principles for the construction of

the odd-order squares were adopted by later authors. 15

However, according to Ahrens,16 it was a work by al-Buni (d. 1225) which may have

been of significant influence on later European authors. AI-Buni's book displayed numerous

magic squares, many of them of the third and fourth order, derived from the basic form

where only the numbers I through n2 are used. In his approach, however, the multiples of

these basic sets were used, or even systems of quadruplets. 17The intention was to arrive at

certain values of the magic constant of the square, corresponding to the numerical value of a

particular word, typically one of the names of God.AI-Buni used combinations of magic squares and planets and, in his work, all squares

were of the seventh order, differing only in their construction. This is further proof that

systems relating squares to planets existed before their first known use by Europeans. (The

influence of still older cultures, like Harran, was apparently not of great importance.) It was

significant for later developments in Renaissance Europe that, by the mid-fifteenth century,

two systems existed in the Arabic world -one with Saturn, and the other with the Moon as

the first planet and smallest square.

The next appearances of magic squares after the work of Moschopoulos occurred almost

simultaneously in the magic treatise of 153 I, Occulta ph ilosophia, by Agrippa von

Nettesheim

18

and in the mathematical treatise of 1539, Practica arithmetice et Mensurandisingularis, by Girolamo Cardano (see Figure I). In both books there are identical magic

squares of the orders from three to nine, attributed to eighteen planets but arranged in an

opposite sequence (see Table Iabove). Numerical magic squares had become the exclusive

property of the manipulators of magic forces. Three decades later in 1567, a similar book

appeared, Archidoxa Magica, which is attributed to Paracelsus. In the Seventeenth book of

this work, the seals of the planets are always associated with their corresponding magic

squares.19 The correlation of squares with planets and metals is the same as in Aprippa's

treatise and, with one exception,20 the squares are identical in both works.

Both Agrippa's and Paracelsus' works influenced European mysticism deeply. Material

proof of this is provided by surviving talismanic medals, bearing numerical magic squares as

their symbols. Sometimes these medals were wrongly attributed to the activity of alchemists.

Even a brief examination of such medals permits certain general conclusions.21 The medals

can be divided into two groups. The first used Arabic numerals; the second used Hebrew

letters instead of numbers (although, as Ahrens points out, these medals were not ofjewish

origin).22 In all the cases examined by the present author, the order of squares on the medals

corresponds with the planets according to Agrippa's system. The planets are represented

symbolically (figurative motives often appear-like Saturn with his scythe, the Sun as a lion,

etc.), and the names attributed by Agrippa to theinteligentia of planets are frequently added.

Unfortunately information about the metal of the medals is rather scarce, so no general

conclusions can be drawn. It seems that approximately equal attention was paid to the

advice of Agrippa and to that of Para eelsus, except that in the case of medals with the sixth

order squares, gold seems to prevail.


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~ 1 9 1 ~ 1~ ISl7 '

1 6 1 1 ) 6 1

Mcrcmiu.~ Jupitft~

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M~rj.1111471131411101,51 4 JI,,~,,481111+1 1111.19'1 ; < ' 1 6 11af .14 91 l S I~6111jII!I 'II 711~14JlliljJ', J. ! I.~~:. I;1. ' 1 11'1 ffIU J I1111; '1 8 IJ J I 1'1714~1~'1"40' 9 1 f f' ,,~ ,

Fig. I. Magic squares for the heavenly bodies, Girolamo Cardano, Practica aritmetice et mensurandi singlllaris, 1539.

Of particular interest is the fourth-order magic square depicted in Durer's Melencolia I.

(see Figure 2) . Two central numbers in the lowest row of this square, 15and 14, form the

date of origin of this engraving. The artist's intention has always been a matter of

speculation.23 Most explanations interpret the fourth order square as a Jupiter amulet,

which corresponds with Agrippa's system (see Table Iabove). In Fischer's view, however,

the square inMelencolia I must have been derived from Arabic numerical mysticism, where

the number 34, the constant of this square, coincides with the numerical value of one of

Allah's names. Fischer supposed that agal (meaning 'time-limit' in the Qunln) should be

read as symbolizing the transient nature of human life. This would also explain other

symbols in this engraving, such as the hour-glass and the balance.

Although this explanation is strikingly original, we should also note that the constuctionof this square is exeedingly simple, and leads to a symmetrical magic square.24 In this type of

square, any two rows or columns can be interchanged without destroying its magical


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126 VLADIMIR KARPENKO

Fig. 2. Detail from Albrect Durer's engraving Melencolia I (1514). [Mansell collection]

properties. In the first step of construction, the numbers in the lower row are in the sequence

14,15. Following the transposition of columns, this becomes 15,14. Because of this simple

coincidence, an explanation based on the date of the engraving also seems quite reasonable.

One further European source should be mentioned separately here-aJewish workEsch

mezareph)25 (" "S 7~ tfN ), written not later than the sixteenth century.26 It is, at least in some

parts, an alchemical treatise. In it, magic squares are connected explicitly not only with the

planets, but with the metals as well. Significantly influenced by theJewish Cabala, this book

rela tes the Sephiroth (spheres of light representing the ten basic forces of the cosmos) with

planets and metals. The numbers of individual Sephiroth are in the same sequence as the

order of squares of the corresponding metal as given by Agrippa.27

Numerical magic squares represent an interesting episode in the evolution of the science

of matter in general, and of alchemy in particular. Since Hellenistic, Arabic and European

alchemy followed a more or less continuous line of development, the question remains: why

did the triad planet-metal-square, having developed during the earlier two periods, lose its

mathematical part-the magic square-in the final European stage?

NOTES AND REFERENCES

I. A numerical magic square consists ofa set of numbers arranged in a square, such that the sums of the numbersalong each horizontal row, each vertical column and each long diagonal yields the same result. In a more

general discussion nothing else need be specified about the numbers in a square. In the present paper, unless

otherwise stated, only "normal squares" will be considered. These consist of the integers from 1to n2-where n

denotes the order of the square. (Other sets of numbers which produce similar results when arranged in a

different geometrical shape, e.g. a cross, may also be discussed under this heading.)

It is obvious that the simplest instance is the third order square. (It appears to be the oldest one known,

several ancient civilizations having discovered it independently.) It might seem that eight different squares of

this order could be constructed, but in fact they are all products of rotations and reflections of a single version.

With higher orders, the picture changes dramatically-there are 880 different fourth-order squares and

275,305,224 squares of the fifth order. See, W. H. Benson and O.Jacoby, New Recreation with Magic Squares (New

York: Dover, 1976). The smallest squares can be constructed by trial and error, but this approach becomes very

difficult as the order of the square increases. From the first appearance of such squares techniques were sought

which would permit the construction of any square using certain simple rules. Such rules were soon found for all

odd-order squares and for even-order squares where n is a multiple of four. Construction of other even-order

squares is not as easy. For the history of magic squares see S. Gunther, Vermischte Untersuchungen zur Geschichte der


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mathematischen Wissenschaften (Leipzig, 1876), ch. 4; M. Cantor, Geschichte der Mathematik (Berlin, 1907), vol. I,

pp. 438, 516; W. Ahrens, Mathematische Unterhaltungen und Spiele (Leipzig: G. Teubner, 1918), vol. 2.

2. Iv!. P. Crosland, Historical Studies in the Language of Chemistry (London: Heinemann, 1962), pp. 79-81.

3. H. E. Stapleton, "The Antiquity of Alchemy," Ambix, 5 (1953), 1-43

4. E. O. von Lippmann, Entstehung und Ausbreitung der Alchemie (Hildesheim: G. alms, 1978), p. 59.

5. J. M. Stillman, The Story of Alchemy and Early Chemistry (New York: Dover, 1960), p. 166.

6. F. Sherwood Taylor, The Alchemists (St. Alban's: Paladin, 1976), p. 57fT.

7. It is not quite clear to what extent the Hellenistic world was familiar with magic squares. Although it seems

improbable that the simplest third-order square was unknown, the square by Theon of Smyrna (2nd century

A.D.) is not magic [Cantor, op. cit. (I )]; the numbers were written here from the top downward and from the left

to the right.

1

2

3

4

5

6

7

8

9

On the other hand al-Quazwini wrote that Archimedes worked out the science of magic squares. See E.

Wiedemann, "Zu den Ivlagischen Quadraten," Der Islam, 8 (1918), p. 94. The Greek scientist was also said to

have recommended the third-order square as a talisman for easier delivery in childbirth, and that a square of

1000 X 1000, put on flags of an army, world bring victory. The second statement is typical of the legends that

have been fabricated in the history of science; nevertheless, it suggests that knowledge of magic squares may

have been considerable in the Hellenistic world.

8. Ibid.

9. Stapleton, op. cit. (3).

10. P. Kraus,}iibir Ibn Hayyiin (Cairo, 1942), vol. 2, pp. 223-36.

I I. Ibid., p. 229. See also, K. Garbers andJ. Weyer, Quellengeschichtliches Lesebuch zur Chemie und A lchemie der Araber im

j\;f itte lalt er (Hamburg: H. Buske, 1980), pp. 30, 86.

12. Kraus, op. cit., (10), p. 229. It should be noted here that when he was describing metals as such, Jabir always

attributed two 'inner' and two 'outer' qualities to each of them; E. J. Holmyard, Alchemy (Harmondsworth:

Penguin Books, 1957), pp. 74fT.

13. H. E. Stapleton, "The Gnomon," Ambix, 6 (1957), 1-9. The gnomon consists ofa stick and its shadow which

together form an L shape like a carpenter's rule. When an L-shaped combination of cells is removed from anymagic square, it is converted into the next lower one. For instance, iffrom a square with 16 cells the utmost right

column of cells and the bottom cells which together form an L shape or gnomon shape are removed, the result is

a square with 9 cells.

14. Stapleton, op. cit. (3).

15. Cantor, op. cit. (I). This procedure, called the first method of Moschopoulos, is closely related to the Indian

method. In both, the numbers are written diagonally, in the first case starting from the cell situated

immediately under the central field of the square, in the second case starting from the central cell in the first-

row. The method of Moschopoulos was used by both Agrippa and Cardanus. Benson and Jacoby, op. cit. (I)

introduce the method of Moschopoulos as the Method of Bachet de Meziriac; and indeed, it appears again in

Bachet de Meziriac, Problemes plaisants et delectables (Paris, 1624), problem XXI, p. 161. The Indian method is

also referred to as Laloubere's and described in S. de Laloubere, Du royaume de Siam (Amsterdam: Henry &la

Veuve de Theodore Boom, 1700), t. 2, pp. 235-88.

16. W. Ahrens, "Die 'magischen Quadrate' al-Buni's," Der Islam, 12 (1922), pp. 235-88.

17 Ibid., p. 167. Quadruplets [5,6,7,8; 9,10, I 1,12; 78, 79,80,81; and 900,901,902,903] appear in the followingexample:

900 10 80 8

7 81 9 901

12 902 6 78

79 5 903 11

18. According to K. A. Nowotny, the editor of the facsimile edition of the 1533 edition ofDe occu lta phi loso phia , the

text to which Agrippa goes back is by Imam Abu Ishak Ibrahim bin Yahya an-Nakkas az-Zarkani, The Book on

the Assigning of the Right Place to the Heavens and their Influence on the Earth and on Man. [See George Sarton, Introduction

to the History of Science, 3 vols. (Baltimore, Ivld.: Carnegie Institution, 1927-I948), vol. I, pp. 758-9.] This Arabic

text remained untranslated in I957 when Nowotny mentioned it in the appendix to his edition of De Occulta

phi loso phia, where he maintained that this, or a similar text, was the basis for the Latin text, De septem quadraturis

pla neta rum excerpted by Agrippa. It could have been accessible to him in the Hrabanus l\Iaurus Codex 5239

(Philos. 418 and 455from the I Ith or 12th century in the Vienna National Library). According to R. Eisler,"Zur Terminologie und Geschichte der Jiidischen Alchemie," Monatschr. fur Geschichte und Wissenschaft des

}ud entu ms, (1926), 70. Jahrgang, Neue Folge 34, p. 196, the Picatrix is the probable source of Agrippa's magic

squares. According to W.-D. Ivliiller-Jahncke, j\;fagie als Wissenschaft im friihen /6. }ahrhundert, Inaugural-
http://www.ingentaconnect.com/content/external-references?article=0002-6980(1953)5L.1[aid=9048204]http://www.ingentaconnect.com/content/external-references?article=0002-6980(1953)5L.1[aid=9048204]http://www.ingentaconnect.com/content/external-references?article=0002-6980(1957)6L.1[aid=9048203]http://www.ingentaconnect.com/content/external-references?article=0002-6980(1957)6L.1[aid=9048203]http://www.ingentaconnect.com/content/external-references?article=0002-6980(1953)5L.1[aid=9048204]http://www.ingentaconnect.com/content/external-references?article=0002-6980(1957)6L.1[aid=9048203]


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128 VLADIMiR KARPENKO

Dissertation, Universitat Marburg, 1973, p. 34, n.l, the author of the Picatrix was probably Pseudo-Majriti, for

whom he gives no date.

19. Both Paracelsus and Agrippa recommended a metal for each 'seal', in which the corresponding square should

be engraved (mercury had to be alloyed with lead). Agrippa's concept, however, was broader. Depending on

the aspect (favourable or unfavourable) of the planet, the metal for such a talisman could be chosen so as tohave different effects. The recommendations of both authors can be compared:

Order of

the square

Parace1sus Agrippa

Aspect of the Planet

Favourable Unfavourable

3

4

5

6

7

89

Pb

Sn

Fe

Au

Cu

Hg (with Pb)Ag

Pb

Ag/coral

Fe

Au

Ag

Ag/Sn/brassAg

not stated

not stated

Cu

not stated

Cu

not statedPb

20. The fifth-order square is constructed in a less common way, similar to the 'knight's move' described by

Benson and Jacoby, op. cit. (I), p. 17 . In this case, the next number is written in the cell situated two places

upward and three to the right. This method had already been described by Moschopoulos; Ahrens, op. cit.,

(I), p. 31.

2I. V. Karpenko, "Cfselne magicke Ctverce na medailfeh [Numerical magic squares on medals]," Numismaticki

listy (Czech), forthcoming.

22. W. Ahrens, Hebriiishce Amulette mit magischen Zahlenquadraten (Berlin: L. Lamm, 1916).

23. L. Fischer, "Zur Deutung des magischen Quadrates in Diirers Melencolia !,"Zeit schri ft dem Deu tsch en

Mo rgen liin disc hen Ge sellscha ft, 103 (1953), p. 308.

24. This is a semi-pandiagonal, Type IIIin Benson and Jacoby, op. cit. (I).

25. G. Scholem, "Alchemie und Kabbala," Monatschr. fur Geschichte und Wissenschaft des Judentums (1925), 69.

Jahrgang, Neue Folge 33, pp. 13~30; 95-110.

26. Eisler, op. cit. (1 8 ).

27 . Was Esch mezareph, then, the swan song of magic squares in alchemy? This system is a matter for debate,

since three different sources offer different views: compare Scholem, op. cit. (25), table on p. 25; H. Gebe1ein,

Alc hem ie (Diederichs Verl., 1991), p. 83; and K. R. H. Frick, Die Erleuchteten (Graz: Akademische Druck-

u.Verlagsanstalt, 1973). The latter two both depict the cabbalistic tree of Hfe. The picture in Frick's book

corresponds with A. Kircher's Oedipus Aegypticus (1562). If, instead of the planetary symbols corresponding

to metals assigned to individual Sephiroth, the symbols of chemical elements are written, there is

considerable but not complete correspondence among the three sources.

Sephiroth Scholem Gebe1ein Frick

2 Chokma Chokma

3 Chokma, Pb Binah, Pb Binah

4 Binah, Nezach, Sn Chessed, Sn Chessed, Sn

5 Tipheret, Fe Geburah, Fe Pechad, Pb

6 Geburah/Tipheret, Au Tipheret, Au Tipheret, Au

7 Hod, Cu Nezach, Cu Nezach, Cu

8 Jessod, Hg Hod, Hg Hod, Cu

9 Chessed, Ag Jessod, Ag Jessod, Hg

IO Malkuth, Ag Malkuth, Ag

V. Karpenko - Numeral Magic Squares

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