7. arithmetic & number theoretic recreations a. Fibonacci numbers

Greek version with discussion in: Siegmund Günther; Vermischte Untersuchungen zur Geschichte der mathematischen Wissenschaften; Teubner, Leipzig, 1876; reprinted by Sändig, Wiesbaden, 1968. Chap. IV: Historische Studien über die magischen Quadrate, pp. 188-276. Section 5, pp. 195-203 is the Greek.

Greek and French in: Paul Tannery; Le traité de Manual Moschopoulos sur les carrés magiques; Annuaire de l'Assoc. pour l'Encouragement des Études Grecques en France 20 (1886) 88 118. (= Mémoires (??*) Scientifiques, Paris, 1916 1946, vol. 4, pp. 27? 61, ??NYS.) English translation by: J. C. McCoy; Manuel Moschopolous's treatise on magic squares; SM 8 (1941) 15 26.

Gives diagonal rules for odd order and two rules for evenly even order. These rules are sometimes attributed to Moschopoulos, but see the MSS at 1211? & 1212. Various examples up through order 9.

Paolo dell'Abbaco. Trattato di Tutta l'Arta dell'Abacho. 1339. Op. cit. in 7.E. B 2433, ff. 20v - 21r, gives order 6 and order 9 magic squares. The latter may be associated with the moon, but my copy is not quite legible. Dario Uri (email of 31 Oct 2001) said he had found this MS (B 2433), which has 6x6 and 8x8 magic squares, but the latter must be a misreading.

‘Abdelwahhâb ibn Ibrâhîm, ‘Izz eddîn el Haramî el (the H should have a dot under it) Zenġânî = ‘Abd al Wahhâb ibn Ibrâhîm al Zinjânî. Arabic MS, Feyzullah Ef. 1362. c1340. ??NYS -- Described in Sesiano II. Suter, p. 144, doesn't mention magic squares. Construction of bordered squares of all orders.

Muhammad [the h should have an underdot) ibn Yūnis. Compendium on construction of bordered magic squares in MS Hüsrev Pasa 257 in the Süleymaniye Library, Istanbul, ff. 32v-37v. Translated and discussed in: Jacques Sesiano; An Arabic treatise on the construction of bordered magic squares; Historia Scientiarum 42 (1991) 13-31. The actual MS was compiled in the 12th century of the Hegira, i.e. c18C), but the treatise is undated. Sesiano compares the methods with other medieval Arabic material, e.g. al-Buni, al-Karagi, al-Buzjani, al Zanjani, so he seems to think it dates from a similar period.

Narayana Pandita (= Nārāyaņa Paņdita [NOTE: ņ denotes n with an overdot and the d should have an underdot.]). Gaņita[NOTE: ņ denotes n with an underdot.] Kaumudī (1356). Edited by P. Dvivedi, Indian Press, Benares, 1942. Part II: Introduction -- magic squares, pp. xv xvi (in English); Chap. 14: Bhadra gaņita [NOTE: ņ denotes an n with an underdot.], esp. pp. 384 392 (in Sanskrit). Shows orders 6, 10, 14. Shows the 8 forms of order 3. Obviously an extensive section -- is there an English translation of this material??. (Editor refers to earlier sources: Bhairava and Ŝiva Tāndava [the d should have a dot under it] Tantras, ??NYS. Cammann 4 cites other MS sources. Singh, op. cit. under Nâgârjuna, c1C, above, says this is the first mathematical treatment. He says it classifies into odd, evenly even and oddly even; gives the superposition method of de la Hire; gives knight's move method for 4n and filling parallel to diagonal for odd, attributing both to previous authors.

Cammann 4, pp. 274 290 discusses this in more detail. He gives another diagonal rule, sometimes beginning and ending at the middle of a side. He then gives a quite different rule based on use of x + y with x = 0, n, 2n, ..., (n 1)n, y = 1, 2, ..., n with both sets of values cycling in the row, then reversing the xs. E.g., for n = 5, his first row of y values is: 4 5 1 2 3 and the second is: 5 1 2 3 4. His first two rows of x values are: 15  20  0  5 10 and 20 0 5 10 15. Reversing the xs and adding gives rows: 14 10 1 22 18 and 20 11 7 3 24. This process gives a central lozenge (or diamond) pattern of the odds and has an extended knight's move pattern. He extends this to doubly even squares. He also gives the 'method of broken reversions' for singly even squares in three forms -- cf. C. Planck; The Theory of Reversions, IN: W. S. Andrews, op. cit. in 4.B.1.a, pp. 295 320.

Datta & Singh give a lengthy (51pp) description of Narayana's work, including about 19 other magic figures. Many of Narayana's methods are novel.

Ahrens-1 gives references to further Arabic mentions of magic squares, usually as amulets, notably to ibn Khaldun (c1370). He also gives many 14C and later examples of 3 x 3 and 4 x 4 squares, with rearrangement and/or constants added, used for magical purposes.

Nadrûnî. Qabs al Anwâr. pre-1384. ??NYS -- described in Ahrens-1, but not mentioned in Ahrens-2. Ahrens only knows of this from a modern article in Arabic. This gives the association of planets by System I. See Folkerts, above.

Arabic MS, 1446, ??NYS. Discussed in Ahrens-1 and Ahrens-2, citing: W. Ahlwardt; Verzeichniss der arab. Handschr. der Königl. Bibliothek zu Berlin; Berlin, 1891; Vol. III, pp. 505-506 (No. 4115). This gives the System II association of planets with magic squares, later given by Cardan in 1539, with the unique addition of a 10 x 10 square for the zodiac coming after Saturn.

Dharmananda. 15C Jaina scholar. Datta & Singh present his 8 x 8 square and say his method works for the evenly even case in general, extending Narayana.

Sundarasūri. c15C Jaina scholar. Datta & Singh say he gives some novel methods, extending Narayana.

Jagiellonian MS 753. 15C Latin MS in Cracow. Described in Cammann 4, pp. 291 297. Earliest European set of magic squares of orders 3 through 9 associated with the planets in System I -- but see Folkerts above, c13C. The order 4 square is Dürer's. These squares later appear in Paracelsus.

Sûfî Kemal al Tustarî. Ghayat al Murâd. 1448. MS at Columbia. ??NYS -- cited by Cammann 4, p. 192. On p. 196 Cammann says this represents a Persian Sufi tradition which was lost in sectarian warfare and the Mongol invasion. On p. 201 he says this has a unit centred square of order 7. On pp. 205 206 are squares of orders 20, 29, 30. He describes two bordering methods beyond al Buni's.

Hindu square in a temple at Gwalior Fort, 1483. Cited by Cammann 4, p. 275, where the original source is cited -- ??NYS.

Pacioli. De Viribus. c1500. Ff. 118r - 118v, 121r - 122v (some folios are wrongly inserted in the middle). C.A. [i.e. Capitolo] LXXII. D(e). Numeri in quadrato disposti secondo astronomi ch' p(er) ogni verso fa'no tanto cioe per lati et per Diametro figure de pianeti et amolti giuochi acomodabili et pero gli metto (Of numbers arranged in a square by astronomers, which total the same in all ways, along sides and along diagonals, as symbols of the planets and suitable for many puzzles and how to put them ??). Gives magic squares of orders 3 through 9 associated with planets in System I, usually attributed to Agrippa (1533), but see Folkerts, above. Ff. 121v and 122r have spaces for diagrams, but they are lacking. He gives the first two lines of the order 4 square as 16,  3, 2, 13; 5, 10, 11, 8; so it must be the same square as shown by Dürer, below.

Albrecht Dürer. Melencolia. 1514. Two impressions are in the British Museum. 4 x 4 square with 15, 14 in the bottom centre cells. Surprisingly, this is the same as the 4 x 4 appearing in Ikhwān al Şafā’ [NOTE: Ş denotes S with an underdot.] (c983), with the two central columns interchanged and the whole square reflected around a horizontal midline, i.e.

16, 3, 2, 13; 5, 10, 11, 8; 9, 6, 7, 12; 4, 15, 14, 1. This is the same as that described by Pacioli. There is some belief that the association with Jupiter relates to the theme of the picture. This is the first printed 4 x 4 magic square.

Riese. Rechnung. 1522. 1544 ed. -- pp. 106 107; 1574 ed. -- pp. 71v 72v. Gives 3 x 3 square in 672 form and how to construct other 3 x 3 forms. Also gives a 4 x 4 square, like Dürer's but with inner columns interchanged.

Riese. Rechenung nach der lenge .... Op. cit. under Riese, Die Coss. 1525. ??NYR. Cammann 4, p. 294, says pp. 103r 105v gives a diagonal rule for odd orders. A quick look shows the material starts on p. 102v.

Cornelius Agrippa von Nettesheim. De Occulta Philosophia. Cologne, 1531, ??NYS. Included in his Opera, vol. 1, and available in many translations. 2nd Book, Chap. 22. Gives association of planets with magic squares in System I -- as previously done by Pacioli, c1500, but with different squares. See Folkerts, above, and Cammann 4, p. 293-294. The squares do not appear in the 1510 draft version of this book. Bill Kalush has kindly sent Chap. 22 from a 1913 English version, but it doesn't have any squares -- perhaps it was from the wrong Book??. He gives each square twice, with Arabic and Hebrew numerals. His 3 x 3 is the 492 version. His 4 x 4 is the same as that of Ikhwān al Şafā’ [NOTE: Ş denotes S with an underdot.], c983.

Cardan. Practica Arithmetice. 1539.