7. arithmetic & number theoretic recreations a. Fibonacci numbers

Chap. 42, section 39, ff. H.v.r   H.vi.r (p. 55). Gives association of planets with magic squares in System II. He is almost unique in using this System, though his squares are the same as Agrippa's. See comments under al Buni and Folkerts.

Chap. 66, section 72, ff. FF.v.r - FF.v.v (p. 157). Shows how to construct a 5 x 5 magic square from the natural 5 x 5 array.

Michael Stifel. Arithmetica Integra. Nuremberg, 1544. ??NYS -- discussed in Cammann 4, p. 194. Pp. 25 26a shows some some bordered squares. Consequently he is sometimes credited with inventing the concept, but see Ikhwān al Şafā’ [NOTE: Ş denotes S with an underdot.] (c983), al Buni (c1200), Yang Hui (1275), ‘Abdelwahhâb (c1340), Sûfî Kemal al Tustarî (1448) and ibn Yūnis, above.

M. Mersenne. Novarum observationum physico mathematicarum. Paris, 1647. Vol. 3, chap. 24, p. 211. ??NYS. States Frenicle's result. (MUS II #29.)

Isomura Kittoku. Ketsugi-shō. 1660, revised in 1684. ??NYS -- described in Smith & Mikami, pp. 65-77. He gives magic squares of orders up to order 10. The order 9 square contains the order 3 square, in the 618 form, in the top middle section. He gives magic circles with n rings of 2n about a central value of 1, for n = 2 - 6. The values are symmetrically arranged, so corresponding pairs add to 2n² + 3 and each ring adds up to n (2n² + 3), while each diameter adds to one more than this. In the 1684 edition, he gives some magic wheels, but these are simply a way of depicting magic squares, though it is not clear where the diagonals are.

Muramatsu Kudayū Mosei. Mantoku Jinkō ri. 1665. ??NYS -- described in Smith & Mikami, pp. 79-80. Gives a magic square of order 19. Gives a magic circle of Isomura's type for n = 8. Smith & Mikami, p. 79, gives Muramatsu's diagram with a transcription on p. 80. The central 1 is omitted and the corresponding pairs no longer add to 131, but the pairs adding to 131 lie on the same radius.

Bernard Frénicle de Bessy. Des Quarrez ou Tables Magiques, including: Table generale des quarrez de quatre. Mem. de l'Acad. Roy. des Sc. 5 (1666 1699) (1729) 209 354. (Frénicle died in 1675. Ollerenshaw & Bondi cite a 1731 edition from The Hague??) (= Divers Ouvrages de Mathématique et de Physique par Messieurs de l'Académie des Sciences; ed. P. de la Hire; Paris, 1693, pp. 423 507, ??NYS. (Rara, 632). = Recueil de divers Ouvrages de Mathematique de Mr. Frenicle; Arkstèe & Merkus, Amsterdam & Leipzig, 1756, pp. 207-374, ??NX.)

Shows there are 880 magic squares of order 4 and lists them all. Cammann 4, p. 202, asserts that they can all be derived from one square!!

The list of squares has been reprinted in the following.

M. Gerardin. Sphinx Oedipe -- supplement 4 (Sep Oct 1909) 129 154. ??NX

K. H. de Haas. Frenicle's 880 basic Magic Squares of 4 x 4 cells, normalized, indexed, and inventoried (and recounted as 1232). D. van Sijn & Zonen, Rotterdam, 1935, 23pp.

Seki Kōwa. Hōjin Yensan. MS revised in 1683. Known also as his Seven Books. ??NYS -- described in Smith & Mikami, pp. 116-122. Describes how to border squares of all sizes. Gives an easy method for writing down a magic circle of Isomura's type.

Thomas Hyde. Mandragorias seu Historia Shahiludii, .... (= Vol. 1 of De Ludis Orientalibus, see 4.B.5 for vol. 2.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. Prolegomena curiosa. The initial material and the Prolegomena are unpaged but the folios of the Prolegomena are marked (a), (a 1), .... The material is on (d 4).v - (e 1).v, which are pages 32-34 if one starts counting from the beginning of the Prolegomena.

Seems to believe magic squares come from Egypt and gives association of orders of squares with planets as in Pacioli and Agrippa, but he only gives one example of a magic square -- an 8 x 8 which is associated.

W. Leybourn. Pleasure with Profit. 1694. Prob. 10, pp. 4-5. Gives the 294 form and then says that each line can be rearranged four ways, e.g. 294, 492, 924, 942. He writes these out for all eight lines, but I can't see any pattern in the way he chooses his four of the six possible permutations.

Ozanam. 1694. 1696: Prob. 9: Des quarrez magiques, 36-41. Prob. 9: Of magical squares, 1708: 33-36. Prob. 12: Des quarrez magiques, 1725: 82-102. Chap. 12: Des quarrés magiques, 1778: 217-244. Chap. 12: Of magic squares: 1803: 211-240; 1814: 183-207 & 366-367; 1840: 94-105. Extended discussion, but contains little new -- except some comments on Franklin's squares -- see Ozanam-Hutton (1803). Associates squares with planets, as done by Pacioli.

Wells. 1698. No. 119, pp. 209-210. Studies the 3 x 3 square carefully, showing that the centre cell must be 5 and the sum of each pair of adjacent side cells is double the value in the opposite corner -- e.g. 9 + 7 is twice 8. I don't recall ever seeing this result before.

Philippe de la Hire. Sur les quarrés magiques. Mémoires de l'Académie Royale des Sciences (1705 (1706)) 377 378. Gives a method for singly even squares, but it uses so many transpositions that it is hard to see if it works in general. ??NYS -- described in Cammann 4, p. 286.

Muhammed ibn Muhammmed. A Treatise on the Magical Use of the Letters of the Alphabet. Arabic MS of 1732, described and partly reproduced in: Claudia Zaslavsky; Africa Counts; Prindle, Weber & Schmidt, Boston, 1973; chap. 12, pp. 137 151. Several of his magic squares are deliberately defective, presumably because of the Islamic belief that only God can create something perfect. I do not recall any other mention of this feature.

Minguet. 1733. Pp. 169-172 (1755: 122-123; 1864: 158-160; not noticed in 1822, but probably about p. 180.) Magic squares of order three with various sums, made by laying out cards.

Benjamin Franklin. 1736-1737. Discovery of some large magic squares and circles. He described these in letters to Peter Collinson whose originals do not survive. I. Bernard Cohen [Benjamin Franklin Scientist and Statesman; DSB Editions, Scribner's, 1975, pp. 18-19] dates them as above, citing Franklin's Autobiography, but my copy is an abridged edition without this material -- ??. He also reproduces them. They were first published in the following.

James Ferguson. Tables and Tracts, Relative to Several Arts and Sciences. A. Millar & T. Cadell, London, 1767. Pp. 309-317. ??NYS. Ferguson may be indicating that he is the first person to whom Franklin showed them.

B. Franklin. Experiments and Observations on Electricity. 4th ed., London, 1769. Two letters to Peter Collinson, pp. 350-355(??). ??NYS, but reprinted in: Albert Henry Smyth; The Writings of Benjamin Franklin; Vol. II, Macmillan, 1907, pp. 456 461 and Plates VII (opp. p. 458) and VIII (opp. p. 460).

The squares are of order 8 and 16, but are only semi-magic (see Ozanam-Hutton (1803) and Patel (1991)), and the circle has 8 rings and 8 radii. Franklin said he could make these squares as fast as he could write down the numbers!

Dilworth. Schoolmaster's Assistant. 1743. Part IV: Questions: A short Collection of pleasant and diverting Questions, p. 168. Problem 4. Asks for a 3 x 3 magic square.

Caietanus Gilardonus. 9 x 9 square on a marble plaque on the Villa Albani, near Rome, dated 1766. The square and the accompanying inscription are given in: E. V. R.; Arranged squares; Knowledge 1 (27 Jan 1882) 273, item 231. These are also given by Catalan; Mathesis 1, p. 151 (??NYS) and Lucas; L'Arithmétique Amusante; 1895; pp. 224-225. Lucas says they were discovered in 1881, and that the villa is now owned by Prince Torlonia and is outside the Porta Salaria.

Catel. Kunst-Cabinet. 1790.