7. arithmetic & number theoretic recreations a. Fibonacci numbers

Sefer ha-Echad, p. 25, has a reference to areas of squares which Müller thinks may refer to magic squares.

Sefer ha-Schem, pforte VI, p. 49, discusses the order 3 square. A note says to see Fig. 6, which appears on p. 80 and is the 492 form. Müller's notes, p. 64, observe that the magic square of order 3 is essentially unique and makes some mystic comments about this.

Abraham ibn Ezra. Ta'hbula. c1150. ??NYS. Some source says this has magic squares, but Lévi's comments in 7.B indicate that this book is only concerned with the Josephus problem. Steinschneider's description of Tachbula, pp. 123 124 of the above cited article, makes no mention of a magic square.

Anon. Arabic MS, Fatih 3439. c1150. ??NYS. Described in Sesiano I. Construction of squares of almost all orders. Describes: a method of ibn al Haytham (c1000) for odd orders; a method of al Isfarâ’inî (c1100) for evenly even orders; a method of ibn al Haytham for oddly even squares which only works for order  2 (mod 8). Suter, p. 93, mentions ibn al Haitam -- see above, c1000.

Tshai Yuan Ting. Lo Shu diagram, c1160. ??NYS -- Biggs cites this as being in Needham, but the only references to Tshai in Needham refer to indeterminate analysis (p. 40) and geology (p. 599). Paul Carus [Reflections on magic squares, IN: W. S. Andrews, op. cit. in 4.B.1.a, pp. 113-128, esp. p. 123] says that Ts'ai Yüan-Ting (1135-1198) gives the Lo-Shu diagram 'but similar arithmetical diagrams are traceable as reconstructions of primitive documents among scholars that lived' during 1101-1125. Datta & Singh cite this and say this is the earliest Chinese interpretation of the Lo-Shu as a magic square. This ignores Tai the Elder, I Wei Chhien Tso Tu, and Xu Yiu, though the first two are a bit vague.

(Ahmed (the h should have an underdot) ibn ‘Alî ibn Jûsuf) el Bûnî, (Abû'l ‘Abbâs, el Qoresî) = Abu l‘Abbas al Buni (??= Muhyi'l Dîn Abû’l-‘Abbâs al Bûnî -- can't relocate my source of this form.) Kitâb et chawâşs [NOTE: ş denotes an s with an underdot.] (= Kitab al Khawass or Sharkh ismellah el a‘zam??) (The Book of Magic Properties). c1200. Suter, p. 136, mentions magic squares. ??NYS -- described in: Carra de Vaux; Une solution arabe du problème des carrés magiques; Revue Hist. Sci. 1 (1948) 206 212. Construction of squares of all orders by bordering. Hermelink refers to two other books of al Buni, ??NYS.

al Buni. Sams al ma‘ârif = Shams al ma‘ârif al kubrâ = Šams al-ma‘ārif. c1200. ??NYS. Ahrens-1 describes this briefly and incorrectly. He expands and corrects this work in Ahrens-2, which mainly deals with 3 x 3 and 4 x 4, the various sources and the accumulated errors in most of the squares. He notes that a 4 x 4 can be based on the pattern of two orthogonal Latin squares of order 4, and Al-Buni's work indicates knowledge of such a pattern, exemplified by the square (discussed by Hayashi under Varāhamihira, c550)

8, 11, 14, 1; 13, 2, 7, 12; 3, 16, 9, 6; 10, 5, 4, 15 considered (mod 4). Al Buni gives several 4 x 4's, including that of Ikhwān al Şafā’ (the Ş should be an S with a dot under it), c983, which does not have the above pattern. He also has Latin squares of order 4 using letters from a name of God. He goes on to show 7 Latin squares of order 7, using the same 7 letters each time -- though four are corrupted. (Throughout, the Latin squares also have 'Latin' diagonals.) These are arranged so each has a different letter in the first place. It is conjectured that these are associated with the days of the week or the planets. In Ahrens-1, Ahrens reported that he had recently been told that Al-Buni had an association of magic squares of orders 3 through 9 with the planets, but he had not been able to investigate this. In Ahrens-2, he is clear that al-Buni has no such association -- indeed, there is no square of order 9 anywhere in the standard edition of the works of al-Buni. But Folkerts says such an association was made by the Arabs, perhaps referring to the Nadrûnî, below. See 14C & 15C entries below.

Cammann 4, p. 184, says this text is "deliberately esoteric ... to confuse people" and the larger squares are so garbled as to be incomprehensible. On pp. 200 201, he says this has the knight's move method for odd orders. Later it was noted that any number could be in the centre and 1 was popular, giving the 'unit centred' square of symbolic importance. These squares are also pandiagonal. Al Buni gives many variant 4 x 4 squares with the top row spelling some magical word -- e.g. one of the 99 names of God. He mentions a "method of the Indians", possibly the lozenge method described in Narayana, 1356.

BM Persian MS Add. 7713. 1211? Described in Cammann 4, pp. 196ff. On p. 201, Cammann says p. 23 gives unit centred squares of orders 5 & 9, pp. 112 114 gives a rule for singly even order and p. 164 has an order 20 square. This also has odd order lozenge squares -- see Narayana, 1356. It also has some examples of a form of the system of broken reversions.

Persian MS. 1212. Garrett Collection, No. 1057, Princeton Univ. See Cammann 1 & Cammann 4, p. 196. ??NYS

Cammann 4, pp. 196ff, says the above two MSS show new developments and describes them. Diagonal rules for odd orders first appear here and give an associated square with centre (n² + 1)/2 which acquired mystic significance as a symbol of Allah.

(Jahjâ (the h should have a dot under it) ibn Muhammed ibn ‘Abdân ibn ‘Abdelwâhid, Abû Zakarîjâ Neġm ed dîn,) known as Ibn el Lubûdî (= Najm al Din (or Abu Zakariya) al Lubudi. c1250. Essay on magic squares dedicated to al Mansur. ??NYS. Mentioned in Suter, p. 146.

Yang Hui. Hsü Ku Chai Ch'i Suan Fa (= Xugu Zhaiqi Suanfa) (Continuation of Ancient Mathematical Methods for Elucidating the Strange [Properties of Numbers]) (Needham, vol. 5:IV, p. 464, gives: Choice Mathematical Remains collected to preserve the Achievements of Old). 1275. IN:  Lam Lay Yong; A Critical Study of the Yang Hui Suan Fa; Singapore Univ. Press, 1977. Book III, chap. 1, Magic Squares, pp. 145 151 and commentary, pp. 293 322. This is the only source for older higher order squares in China. (See Cammann 1 and Cammann 3 for details of constructions.) Bordered squares of order 5 and 7. Magic squares of orders 3 through 10, the last being only semimagic. Methods are given for orders 3 and 4 only. Gives some magic circles and other forms. (Lam's commentary, p. 313, corrects the first figure on p. 150. Lam also discusses the constructions.)

Li & Du, pp. 166 167, say that 6 x 6 and 7 x 7 'central' (= bordered) squares arrived from central Asia at about this time. The 6 x 6 example on their p. 172 and the 7 x 7 example on their p. 167 are bordered.

Cammann 4 says one of the order 8 squares is based on a Hindu construction.

Jaina square. Inscription at Khajuraho, India. 12 13C. Ahrens-1 (p. 218) says it first appears as: Fr. Schilling; Communication to the Math. Gesellschaft in Göttingen [Mitteilung zur Math. Ges. in Göttingen], 31 May 1904; reported in: Jahresber. Deutschen Math. Verein. 13 (1904) 383 384. (Schilling is reporting a communication from F. Kielhorn.)

7, 12, 1, 14; 2, 13, 8, 11; 16, 3, 10, 5; 9, 6, 15, 4.

It is pandiagonal and associated. Cammann 4, p. 273, says this is the same as the square reported by Rivett-Carnac (qv above) and there claimed to be 11C? Cammann feels it may derive from an Islamic source. See also: Smith History II 594 and Singh, op. cit. at Nâgârjuna above. Datta & Singh give this and date it as 11C.

Five cast iron plates with 6 x 6 magic squares, late 13C(??), were found at Xian in 1956. The numerals are similar to East Arabic numerals so these reflect the Arabic influence on the Mongol dynasty. Li & Du, p. 172, reproduces one. This is on display in the new Provincial History Museum in Xian. Jerry Slocum has given me a facsimile, in reduced size, of the same one. Can any one supply more information about the others??

Datta & Singh give another Jaina square, from 'not later than the fourteenth century', 'probably a very old one'. This is the following, but with all entries multiplied by five to give a magic constant of 170, which 'is closely connected with an ancient Jaina mythology'.

5, 16, 3, 10; 4, 9, 6, 15; 14, 7, 12, 1; 11, 2, 13, 8.

Folkerts discusses an anonymous and untitled astrological-magic treatise which appears to derive from the court of Alfonso the Wise in Madrid, where a similar work of al-Magriti (al-Mağrīţī [NOTE: ţ denotes t with an underdot.]) (10C) had been translated in the 13C and developed under the name Picatrix. Picatrix refers to astrological amulets but gives no instance of a magic square on one. Though there is no known direct connection, Folkerts considers this treatise to be in the tradition of the Picatrix and names it 'Picatrix-Tradition'. He finds seven MSS of it, from early 14C onward. This specifically associated magic squares and planets according to Folkerts' System I.

Folkerts discusses these associations and calls them Systems I and II. For n = 3, 4, ..., 9, they are as follows.

I: Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon.

II: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn.

System I is almost universal, only a 1446 Arabic MS and Cardano use system II.

Not all writers use the same squares, but there are generally only two examples for each order. Folkerts gives a table of these and which authors used which squares. Basically, there are two sets of squares, one used by Picatrix-Tradition and Pacioli, the other by Agrippa and Cardano (in reverse).

Folkerts says this association was done by the Arabs, but Nadrûnî (cf below) is the earliest Arabic source I have.

In Codex Vat. Reg. lat. 1283 is a 13C(?) fragment with a 5 x 5 magic square associated to Mars. Paolo dell'Abbaco's Trattato di Tutta l'Arte dell'Abacho, 1339, op. cit. in 7.E, has order 6 and 9 squares with their associations (sun and moon) given. A 15C Frankfurt MS (UB, Ms. lat. oct. 231), has some examples and Paracelsus (1572) copied squares from different sources.

Folkerts then discusses various constructions due to al-Buni, Moschopoulos, al Haitham, etc. The 15C Frankfurt MS is the first attempt at a theory, followed by Ries and Stifel.

Μαvoυηλ Μoσχoπoυλoυ (Manuel Moschopoulos -- variously spelled in Greek and variously transliterated). c1315. MS 2428, Bibliothèque Nationale, Paris.