Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya

280 Pages·2006·3.059 MB·English

by  Agathe Keller

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J. Gray, Milton Keynes
Agathe Keller
Expounding the
Mathematical Seed
Volume 2: The Supplements
A Translation of Bhaskara I on the Mathematical Chapter
of the Aryabhatiya
Birkhäuser Verlag
Basel · Boston · Berlin
Author
Agathe Keller
Rehseis CNRS
Centre Javelot
2 place Jussieu
75251 Paris Cedex 05
e-mail: [email protected]
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ISBN 3-7643-7292-3 Birkhäuser Verlag, Basel – Boston – Berlin
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Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
How to read this book? . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Abbreviations and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Supplements 1
A BAB.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
A.1 Arithmetical squaring and its geometrical interpretation . . 2
A.2 Squares and cubes of greater numbers . . . . . . . . . . . . 7
A.3 Squaring and cubing with fractions . . . . . . . . . . . . . . 12
B BAB.2.4-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
B.1 Extracting square-roots . . . . . . . . . . . . . . . . . . . . 15
B.2 Extracting cube-roots . . . . . . . . . . . . . . . . . . . . . 18
C BAB.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
C.1 Area of a triangle. . . . . . . . . . . . . . . . . . . . . . . . 22
C.2 Volume of a pyramid . . . . . . . . . . . . . . . . . . . . . . 27
D BAB.2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
D.1 Area of a circle . . . . . . . . . . . . . . . . . . . . . . . . . 31
D.2 Volume of a sphere . . . . . . . . . . . . . . . . . . . . . . . 32
D.3 Procedure followed in examples . . . . . . . . . . . . . . . . 34
E BAB.2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
E.1 General rule. . . . . . . . . . . . . . . . . . . . . . . . . . . 34
E.2 Description of the field . . . . . . . . . . . . . . . . . . . . . 35
E.3 Bha¯skara’s interpretation . . . . . . . . . . . . . . . . . . . 36
E.4 Procedure followed in examples . . . . . . . . . . . . . . . . 39
vi Contents
F BAB.2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
F.1 Ab.2.9.ab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
F.2 Ab.2.9.cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
G BAB.2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
G.1 A¯ryabhat.a’s verse . . . . . . . . . . . . . . . . . . . . . . . 47
G.2 The “ten karan.¯ıs” theory . . . . . . . . . . . . . . . . . . . 48
G.3 Steps used to refute the “ten karan.¯ıs” theory . . . . . . . . 52
H BAB.2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
H.1 Bha¯skara’s understanding of Ab.2.11. . . . . . . . . . . . . 54
H.2 The steps of the diagrammatic procedure . . . . . . . . . . 58
H.3 A chord of the same length as the arc it subtends . . . . . . 69
I BAB.2.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
I.1 A specific interpretation of the rule . . . . . . . . . . . . . . 70
I.2 Understanding the procedure . . . . . . . . . . . . . . . . . 73
I.3 Rversed sine . . . . . . . . . . . . . . . . . . . . . . . . . . 75
J BAB.2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
J.1 What Bha¯skara says of compasses . . . . . . . . . . . . . . 75
J.2 Parame´svara’s descriptions of a pair of compasses. . . . . . 76
K BAB.2.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
K.1 A¯ryabhat.a’s verse . . . . . . . . . . . . . . . . . . . . . . . 78
K.2 Understanding Bha¯skara’s astronomical extension. . . . . . 80
K.3 Different types of gnomons . . . . . . . . . . . . . . . . . . 84
L BAB.2.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
L.1 Understanding the rule . . . . . . . . . . . . . . . . . . . . 89
L.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
M BAB.2.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
M.1 A¯ryabhat.a’s rule . . . . . . . . . . . . . . . . . . . . . . . . 92
M.2 Astronomical misinterpretations . . . . . . . . . . . . . . . 95
M.3 U¯jjayin¯ı, Lan˙ka¯ and Sthane´svara . . . . . . . . . . . . . . . 98
N BAB.2.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
N.1 The “Pythagoras Theorem” . . . . . . . . . . . . . . . . . . 100
N.2 Two arrows and their half-chord . . . . . . . . . . . . . . . 101
O BAB.2.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Contents vii
P BAB.2.19-22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
P.1 Ab.2.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
P.2 Ab.2.20: The number of terms . . . . . . . . . . . . . . . . 111
P.3 Ab.2.21: Progressive sums of natural numbers . . . . . . . . 111
P.4 Ab.2.22: Sum of squares and cubes . . . . . . . . . . . . . . 113
Q BAB.2.23-24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Q.1 BAB.2.23:Knowingtheproductfromthesumofthesquares
and the square of the sum . . . . . . . . . . . . . . . . . . . 114
Q.2 BAB.2.24: Finding two quantities knowing their difference
and product . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
R BAB.2.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
R.1 The rule given by A¯ryabhat.a . . . . . . . . . . . . . . . . . 116
R.2 Procedure followed by Bha¯skara in examples . . . . . . . . 117
R.3 Verification with a Rule of Five . . . . . . . . . . . . . . . . 117
S BAB.2.26-27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
S.1 Rule of Three . . . . . . . . . . . . . . . . . . . . . . . . . . 118
S.2 Rule of Five and the following . . . . . . . . . . . . . . . . 123
S.3 The Reversed Rule of Three . . . . . . . . . . . . . . . . . . 125
T BAB.2.28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
T.1 Notations and references . . . . . . . . . . . . . . . . . . . . 128
T.2 Computing the time with the Rsine of the sun’s altitude . . 128
T.3 Which procedure is reversed? . . . . . . . . . . . . . . . . . 129
U BAB.2.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
V BAB.2.30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
V.1 General resolution of first order equations . . . . . . . . . . 133
V.2 Debts and wealth . . . . . . . . . . . . . . . . . . . . . . . . 135
W BAB.2.31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
W.1 Understanding the verse . . . . . . . . . . . . . . . . . . . . 137
W.2 Bha¯skara’s distinctions and explanations . . . . . . . . . . . 137
W.3 Finding the longitude of the meeting point . . . . . . . . . 139
X BAB.2.32-33: The pulverizer. . . . . . . . . . . . . . . . . . . . . . 142
X.1 Two different problems . . . . . . . . . . . . . . . . . . . . 142
X.2 Procedure for the pulverizer “with remainder” . . . . . . . 143
X.3 Procedure of the pulverizer without remainder . . . . . . . 155
X.4 Astronomical applications . . . . . . . . . . . . . . . . . . . 160
viii Contents
Appendix: Some elements of Indian astronomy 186
1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
3 Movement of planets . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4 Time cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5 Orbits and non-integral residues of revolutions . . . . . . . . . . . 194
Glossary 197
1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
2 Peculiar and metaphoric expressions . . . . . . . . . . . . . . . . . 221
3 Measure units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
3.1 Units of length . . . . . . . . . . . . . . . . . . . . . . . . . 222
3.2 Measures of weight . . . . . . . . . . . . . . . . . . . . . . . 223
3.3 Coins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
3.4 Time units . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
3.5 Subdivisions of a circle . . . . . . . . . . . . . . . . . . . . . 224
4 Names of planets, constellations, zodiac signs . . . . . . . . . . . . 224
5 Days of the week . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6 Gods and mythological figures. . . . . . . . . . . . . . . . . . . . . 226
7 Cardinal directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Bibliography 227
A Primary sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Index 235
List of Figures
1 Bh¯askara’s diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Bh¯askara’s diagram in Shukla’s edition . . . . . . . . . . . . . . . . 3
3 Bh¯askara’s diagram in a manuscript . . . . . . . . . . . . . . . . . 3
4 Counting sub-surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 4
5 Counting strokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6 Ganes.a’s ‘proof’ of the ‘Pythagoras Theorem’ . . . . . . . . . . . . 6
7 Equilateral and isoceles triangles . . . . . . . . . . . . . . . . . . . 23
8 Any triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
9 An equilateral pyramid with a triangular base . . . . . . . . . . . . 27
10 A S´r.n˙gataka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
11 Rule of Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
12 An isoceles trapezium . . . . . . . . . . . . . . . . . . . . . . . . . 34
13 Fields inside a trapezium . . . . . . . . . . . . . . . . . . . . . . . 38
14 A rectangle and an equilateral triangle . . . . . . . . . . . . . . . . 41
15 One rectangle and any triangle . . . . . . . . . . . . . . . . . . . . 42
16 Two rectangles and any triangle . . . . . . . . . . . . . . . . . . . 43
17 Drum field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
18 Tusk field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
19 Chord in a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
20 Fields inside a circle . . . . . . . . . . . . . . . . . . . . . . . . . . 47
21 Circle and bow fields . . . . . . . . . . . . . . . . . . . . . . . . . . 50
22 Circle and arrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
23 Chedkyaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
24 Trilaterals and quadrilaterals . . . . . . . . . . . . . . . . . . . . . 55