MAGIC SQUARES

[yanom at JPNKSUVX.EARN]

Because Takao Hayashi has some trouble in sending e-mail to Indology,
he asked me to transfer his reply.  I got this today through
a Japnese commercial e-mail network.   Michio YANO
----------------------------
Dear lovers of Indian magic squares,
 
  Sorry, but for me e-mails seem to take longer time than regular ones.
  I am sure that my knowledge about the history of Indian magic squares
is quite insufficient, and on the other hand I am not very sure about
what kind of information is needed by each researcher, but I hope the
following bibliographical information is amusing to those who are
interested in them.
  As Wujastyk guessed, Pingree told me before about the existence of
magic squares written on leaves of various Indian manuscripts, but
I did not ask him about their details because at that time I was
interested only in older ones.  Wujastyk's list will be valuable
for writing a whole history of Indian magic squares, and now I am
certainly interested in it.  I am also interested in Allen's collection
of "a large number of sources in print and MS in Sanskrit, Prakrit,
and vernaculars."
  I do not know exactly what is the magic hexagon talked about before
in this circle, but there is a hexagonal formation of the first 48
natural numbers with the constant sum 98 in the last chapter (called
bhadraga\nita) of N\ar\aya\na's Ga\nitakaumud\i (A.D. 1356), which
mathematically treats magic squares as well as magic figures of various
shapes in a quite general manner.
 
===== Secondary sources for Indian magic squares =====
 
***G. Abe, 1985, N\ar\ayana pandita no h\ojin.  Sugakushi Kenkyu 104,
32-34. [Comments on N\ar\aya\na's magic squares.]
***G. Abe, 1991, Var\ahamihira no 4 h\ojin to Aru-B\un\i no 4 h\ojin ni
tsuite. Sugakuhsi Kenkyu 131, 3-11.  [On the relationship between
Var\ahamihira's magic square of order four and al-B\un\i's.]
***A.A.K. Ayyangar, 1954, Magic Squares.  Mathematics Student 20, 1952, 166.
 
[= Indian Magic Squares, Scripta Mathematica 20, 202.  Contains a
magic square of order four "which is quoted in verse by many elderly
persons in the village parts of India.]
***S. Cammann, 1968/69, Islamic and Indian Magic Squares.  History
of Religions 8, 181-209, and 271-299. [The latter half is a study of
Indian magic squares based mainly on the last chapter of N\ar\aya\na's
Ga\nitakaumud\i.  Cammann was an expert of Chinese magic squares, and
wrote many articles on them.]
***A. Cunningham, 1871, Four Reports Made during the Years 1862-63-64-65.
 Archaeological Survey of India 2.  [Contains (on p. 434) a report on
a magic square of order four incised on the front wall of a Jaina temple
called Jinan\atha in Khajuraho.  The figures used in it can be palaeogra-
phically ascribed to the 12th or the 13th century.]
***W. Goonetilleke, 1882, The Americal Puzzle.  The Indian Antiquary 11,
83-84. [Construction method, stated in the Kak\sapu\ta, of magic squares
of order four with any optional sum.]
***G.A. Grierson, 1881, An American Puzzle.  The Indian Antiquary 10, 89-90.
 
[Construction method, stated in Raghunandana's Sm\rtitattva (ca. A.D.
1500), of magic squares of order four.]
***T. Hayashi, 1986, H\ojinzan.  Epist\em\e (Tokyo: Asahi Press) II, 3,
pp. i-xxxiv.  [A Japanese translation with mathematical comments of the
last chapter of N\ar\aya\na's Ga\nitakaumud\i, with an Introduction which
includes a Japanese translation of a small section on magic squares of
Thakkura Pher\u's Ga\nitas\ara (ca. A.D. 1315).  S.R. Sarma of Aligarh
University is preparing an English translation of the Ga\nitas\ara.
T. Kusuba of Osaka University of Economics is making an edition and
an English translation of the last two chapters of the Ga\nitakaumud\i
for his dissertation to be submitted to Brown University.
  Both Sarma and Kusuba happen to be working with Pingree at Brown up
to March.]
***T. Hayashi, 1987, Var\ahamihira's Pandiagonal Magic Square of the
Order Four.  Historia Mathematica 14, 159-166.  [A study of a modified
magic square of order four (with the constant sum 18) utilized by Var\a-
hamihira (in his B\rhatsa\mhit\a, ca. A.D. 550) for making various
kinds of perfumes.  Its existence was already known to A.N. Singh 1936.]
***T. Hayashi, 1988, Magic Squares in Indian Medical Works.  The Science
and Engineering Review of Doshisha University 28, 231-243.  [Comments in
Japanese on the magic squares reported by Rosu 1987.]
***T. Hayashi, 1988, A Preliminary Study in the History of Magic
Squares before the Seventeenth Century.  Bulletin of the National Museum
of Ethnology 13, 615-719.  [A survey, based mainly on secondary sources,
of magic squares known to the ancient and medieval worlds.  Classifies
the construction methods which are explicitly stated in classical texts.
The main text in Japanese with an English introduction.]
***A. Hirayama, 1987, N\ar\ayana no h\ojinzan.  Sugakushi Kenkyu 112,
1-12.  [Comments on some of N\ar\aya\na's magic squares.]
***H.R. Kapadia, 1934, A Note on Jaina Hymns and Magic Squares.  Indian
Historical Quarterly 10, 148-153.  [Reports three hymns, M\anadevas\uri's
Sattarisayathutta, etc. which are associated with magic squares.]
***F. Kielhorn, 1892, Inscriptions from Khajuraho.  Epigraphia Indica 1,
135-136. [See Cunningham above.]
***A. Rosu, 1987, Etudes ayurvediques III: Les carres magiques dans la
medicine indienne.  Studies on Indian Medical History (ed. by Meulenbeld
& Wujastyk), Groningen Oriental Studies 2, 103-112.  [A study of magic
squares of order three employed, since around A.D. 900, in \ayurveda
for making delivery easy.]
***A. Rosu, 1989, Les carres magiques indiens et l'histoire des idees
en Asie.  ZDMG 139, 120-158.  [Most informative article on Indian magic
squares with special reference to their significances in the medical and
tantric traditions, and to their relationships with Islamic and Chinese
magic squares.]
***R. Shortreede, 1842, On an Ancient Indian Magic Square Cut in a Temple
at Gwalior.  Journal of the Asiatic Society of Bengal, N.S. 11 (1),
292-293.  [A magic square of order four.  The date of the temple is
ascribed to A.D. 1483.]
***A.N. Singh, 1936, The History of Magic Squares in India.  Proceed-
ings of the International Congress of Mathematicians, pp. 275-276.
[Mentions, without details, magic squares of the Kak\sapu\ta, of Var\a-
hamihira, of Khajuraho, of Jaina hymns, of N\ar\aya\na, of Dharm\ananda,
and of Sundaras\uri.]
***P. Singh, 1982, Total Number of Perfect Magic Squares: N\ar\aya\na's
Rule.  Mathematics Education 16, A, 32-37.  [Gives a conjecture for
N\ar\aya\na's method of calculating the total number, 384, of the
pandiagonal magic squares of order four.]
***P. Singh, 1986, N\a\raya\na's Treatment of Magic Squares.  Indian
Journal of History of Science 21, 123-130.  [Mathematical comments on
N\ar\aya\na's magic squares.]
***T. Vijayaraghavan, 1941, On Jaina Magic Squares.  Mathematics Student
9, 97-102.  [On mathematical peculiarities of the so-called Jaina magic
square.  See Cunningham above.]
===============================================
Takao Hayashi,
Doshisha University,
Kyoto 602 Japan.