Magic Squares in the Tenth Century: Two Arabic Treatises by Anṭākī and Būzjānī

398 Pages·2017·4.67 MB·English

by  Jacques Sesiano (auth.)

Sources and Studies in the History of Mathematics
and Physical Sciences
Jacques Sesiano
Magic Squares
in the Tenth
Century
Two Arabic Treatises by Ant. ākī
and Būzjānī
Sources and Studies in the History of Mathematics
and Physical Sciences
Series editor
Jed Z. Buchwald
Associate editor
J.L. Berggren
J. Lützen
J. Renn
Advisory Board
C. Fraser
T. Sauer
A. Shapiro
Sources and Studies in the History of Mathematics and Physical Sciences
was inaugurated as two series in 1975 with the publication in Studies of Otto
Neugebauer’s seminal three-volume History of Ancient Mathematical Astronomy,
which remains the central history of the subject. This publication was followed the
next year in Sources by Gerald Toomer’s transcription, translation (from the Arabic),
and commentary of Diocles on Burning Mirrors. The two series were eventually
amalgamated under a single editorial board led originally by Martin Klein (d. 2009)
and Gerald Toomer, respectively two of the foremost historians of modern and
ancient physical science. The goal of the joint series, as of its two predecessors, is
to publish probing histories and thorough editions of technical developments in
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from pre-classical antiquity through the last century, ranging from Babylonian
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will interest scholars in the history of mathematics and physics, mathematicians,
physicists, engineers, and anyone who seeks to understand the historical
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More information about this series at http://www.springer.com/series/4142
Jacques Sesiano
Magic Squares in the Tenth
Century
Two Arabic Treatises by Anṭākī and Būzjānī
Jacques Sesiano
Département de mathématiques
Station 8
Ecole polytechnique fédérale
CH-1015 Lausanne
[email protected]
ISSN 2196-8810 ISSN 2196-8829 (electronic)
Sources and Studies in the History of Mathematics and Physical Sciences
ISBN 978-3-319-52113-8 ISBN 978-3-319-52114-5 (eBook)
DOI 10.1007/978-3-319-52114-5
Library of Congress Control Number: 2017935465
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Table of contents
Part I: Tenth-century construction methods
Introduction .................................................3
§1. General notions on magic squares.........................3
§2. The two texts from the tenth century .....................9
A. Text A ............................................9
1. Author and manuscript.............................9
2. The Introduction to Arithmetic by Nicomachos ........11
3. Contents of text A................................13
B. Text B ...........................................16
Chapter I. Ordinary magic squares .............................19
§1. Construction of odd-order squares .......................20
A. Particular case of order 3............................20
1. Uniqueness of the square of order 3..................20
2. Construction of the square of order 3 ‘by displacement’ .20
3. A construction ‘without displacement’ ...............21
B. Square of order 5...................................21
C. Later developments.................................23
§2. Construction of even-order squares.......................26
A. Square of order 4 ..................................26
1. Constructions by displacement......................26
2. A construction without displacement ................27
B. Square of order 6...................................30
C. Square of order 8...................................31
D. Allusion to a generalization ..........................31
1. Evenly even orders ...............................32
2. Evenly odd orders ................................34
E. Later developments.................................36
1. Equalization rules ................................36
2. Case of evenly even orders .........................37
3. Case of evenly odd orders..........................40
v
vi Table of contents
Chapter II. Bordered magic squares ............................43
§1. Construction of odd-order bordered squares ...............43
A. Empirical discovery of a method......................43
1. Square of order 5 .................................43
2. Higher odd-order squares ..........................46
B. Description of the general method ....................48
C. Mathematical basis for this method ...................49
§2. Construction of even-order bordered squares...............51
A. Construction of bordered squares of evenly even orders...51
1. Description of the method in A and B ...............51
2. Description of the general method...................53
3. Mathematical basis for this method .................54
(α) General theory ................................54
(β) The tenth-century method ......................58
B. Construction of bordered squares of evenly odd orders ...59
1. Particular case of order 6 ..........................59
2. Description of a general method in A ................61
3. Other description of the general method .............61
4. Mathematical basis for this method .................62
Chapter III. Separation by parity ..............................67
§1. Structure of the rhomb according to B....................67
A. The rows of the rhomb..............................67
B. Largest square within the rhomb .....................68
§2. The partially empirical construction of B .................69
§3. Filling the rhomb according to A ........................72
§4. Placing the even numbers around the rhomb...............74
A. Situation after filling the rhomb ......................74
1. Determining the number of remaining empty cells .....74
2. Determining the sum required ......................75
B. Rules for placing the even numbers....................78
C. Case of the order n = 4t+1 (with t ≥ 2)...............85
1. First border .....................................85
2. Other borders....................................86
3. Recapitulation ...................................90
4. Completing the construction .......................92
D. Case of the order n = 4t+3 (with t ≥ 1) ..............95
1. First border .....................................95
Table of contents vii
2. Other borders....................................95
3. Recapitulation ...................................99
4. Completing the construction ......................101
E. Particular case of order 5 ...........................102
Chapter IV. Composite magic squares .........................104
§1. Subsquares having same size and unequal sums ...........104
§2. Subsquares having same size and equal sums .............107
§3. Parts having different sizes ............................108
A. Method of the cross ...............................108
B. Method of the central square........................113
Part II: Translation
Editorial procedure .........................................117
Text A ...................................................119
Chapter I of Book III ....................................120
Chapter II. Science of the magic square .....................143
§1. Science of odd-order (squares)........................143
§2. Science of even-order (squares) .......................164
Composite magic squares ...............................170
Chapter III. Determining the hidden number.................180
Text B....................................................207
(Introduction) ..........................................208
(Construction of ordinary magic squares)....................210
(Construction of bordered magic squares) ...................222
(Particular cases of odd-order squares) ......................240
(Particular cases of even-order squares) .....................250
Opuscule on the magic square by al-Kharaq¯ı ....................253
Part III: Arabic texts
Editorial procedure .........................................257
Text A ...................................................259
Chapter I of Book III ....................................262
Chapter II. Science of the magic square .....................282
§1. Science of odd-order (squares)........................282
§2. Science of even-order (squares) .......................299
Composite magic squares ...............................303
viii Table of contents
Chapter III. Determining the hidden number.................316
Text B....................................................335
(Introduction) ..........................................335
(Construction of ordinary magic squares)....................337
(Construction of bordered magic squares) ...................349
(Particular cases of odd-order squares) ......................368
(Particular cases of even-order squares) .....................378
Opuscule on the magic square by al-Kharaq¯ı ....................383
Bibliography...............................................385
Index .....................................................389
Part I
Tenth-century construction methods