The Science of the Shulba: A Study in Early Hindu Geometry c4: Contents

The Science of the Śulba: A Study in Early Hindu Geometry c4

Language: English

| | | | PAGE |
| Orthodox texts on the *Śulba* | ... | ... | 1 |
| Their relations to the *Veda* | ... | ... | 2 |
| *Baudhāyana Śulba* | ... | ... | 2 |
| *Āpastamba Śulba* ... | ... | ... | 3 |
| *Kātyāyana Śulba* ... | ... | ... | 4 |
| *Mānava Śulba* ... | ... | ... | 5 |
| *Maitrāyaṇīya Śulba* | ... | ... | 6 |
| *Vārāha Śulba* ... | ... | ... | 6 |
| Hindu names for geometry | ... | ... | 7 |
| Their significance ... | ... | ... | 8 |
| Influence ... | ... | ... | 9 |

c4.4

CHAPTER II

c4.5

COMMENTATORS

c4.6

| Available commentaries on the *Śulba* | ... | ... | 10 |
| Uncertainty about their dates | ... | ... | 10 |
| Mahīdhara ... | ... | ... | 11 |
| Rāma Vājapeyī ... | ... | ... | 11 |
| Śivadāsa and Śaṅkara | ... | ... | 12 |
| Kapardisvāmī ... | ... | ... | 13 |
| Karavindasvāmī ... | ... | ... | 14 |
| Sundararāja ... | ... | ... | 17 |
| Dvārakanātha Yajvā | ... | ... | 18 |
| Earlier commentaries lost | ... | ... | 19 |

c4.7

CHAPTER III

c4.8

GROWTH AND DEVELOPMENT OF THE ŚULBA

c4.9

| | | | PAGE |
| Vedic Sacrifices | ... | ... | 20 |
| Propositions of the *Śulba*: a brief *résumé* | ... | ... | 21 |
| Their earlier origin | ... | ... | 25 |
| References in the *Ṛg-veda* | ... | ... | 27 |
| *Taittirīya Saṃhitā* | ... | ... | 30 |
| *Śatapatha Brāhmaṇa* | ... | ... | 33 |
| Dimensions of the *Caturasra-śyena-citi* | ... | ... | 34 |
| Measurement by rod | ... | ... | 36 |
| Measurement by cord | ... | ... | 37 |
| Number of bricks to be used in the construction | ... | ... | 39 |
| Number of layers of the *Vedi* | ... | ... | 40 |

c4.10

CHAPTER IV

c4.11

POSTULATES

c4.12

| Postulates of the *Śulba* defined | ... | ... | 41 |
| Division of finite straight lines | ... | ... | 41 |
| Division of circles ... | ... | ... | 42 |
| Division of rectangles | ... | ... | 43 |
| Simple cases of congruence theorems | ... | ... | 44 |
| Principle of juxtaposition of two right triangles | ... | ... | 45 |
| Division of triangles | ... | ... | 46 |
| Maximum square within a circle | ... | ... | 49 |
| Demonstration in Hindu geometry | ... | ... | 50 |

c4.13

CHAPTER V

c4.14

CONSTRUCTIONS

c4.15

| Drawing of perpendiculars | ... | ... | 52 |
| Construction of squares : | | | |
| Method I | ... | ... | 55 |

c4.16

| Method II | ... | ... | ... | 56 |
| Method III | ... | ... | ... | 57 |
| Method IV | ... | ... | ... | 58 |
| Method V | ... | ... | ... | 59 |
| Oldest method of constructing a square | ... | | ... | 60 |
| Construction of rectangles | ... | ... | ... | 62 |
| Construction of isosceles trapeziums | ... | ... | ... | 63 |
| Construction of the *Mahāvedi* : | | | | |
| Method I | ... | ... | ... | 64 |
| Method II | ... | ... | ... | 65 |
| Method III | ... | ... | ... | 65 |
| Method IV | ... | ... | ... | 66 |
| Construction of parallelograms : | ... | | ... | 67 |
| First Method | ... | ... | ... | 68 |
| Alternative Method | ... | ... | ... | 69 |
| Two cases of it | ... | ... | ... | 70 |

c4.17

CHAPTER VI

c4.18

COMBINATION OF AREAS

c4.19

| Multiple of a square | ... | ... | ... | 72 |
| Kātyāyana's method | ... | ... | ... | 73 |
| Submultiple of a square | ... | ... | ... | 74 |
| Two methods of it | ... | ... | ... | 75 |
| Addition of unequal squares | ... | ... | ... | 76 |
| Subtraction of squares | ... | ... | ... | 78 |
| Combination of triangles | ... | ... | ... | 80 |
| Combination of pentagons | ... | ... | ... | 80 |

c4.20

CHAPTER VII

c4.21

TRANSFORMATION OF AREAS

c4.22

| Transformation of a rectangle into a square | | ... | 83 |
| Transformation of a square into a rectangle | | ... | 85 |

c4.23

Transformation into an isosceles trapezium ... 90 Transformation into a triangle ... 92 Transformation of a triangle into a square ... 92 Transformation into a rhombus ... 93

c4.24

CHAPTER VIII

c4.25

AREAS AND VOLUMES

c4.26

Unit of area defined ... 95 Area of a square ... ... 95 Area of a rectangle ... ... 96 Triangles and parallelograms... ... 96 Trapeziums ... ... 96 Figures with a given area ... ... 98 Construction of a square having a given area ... 98 Construction of an isosceles trapezium having a given area ... ... 99 Volume of a prism or a cylinder ... 101 Approximate formula for the frustum of a pyramid ... 101

c4.27

CHAPTER IX

c4.28

THE THEOREM OF THE SQUARE OF THE DIAGONAL

c4.29

Theorem of the Square of the Diagonal ... 104 Converse Theorem ... ... 105 Did ancient Hindus know a proof ... ... 106 Hypotheses as regards particular proofs ... 108 General proofs ... ... ... 115 Early history of the theorem ... ... 119

c4.30

CHAPTER X

c4.31

RATIONAL RECTANGLES

c4.32

| | | | PAGE |
| Rational rectangles found in the *Śulba* | ... | ... | 123 |
| Method of their discovery | ... | ... | 124 |
| Calculation of the gnomon | ... | ... | 127 |
| Interpretation of two Sanskrit words | ... | ... | 128 |
| General formulæ for finding rational rectangles : | | | |
| First formula | ... | ... | 133 |
| Second formula | ... | ... | 134 |
| Third formula | ... | ... | 135 |
| Further generalisation | ... | ... | 136 |
| Rectangle (15, 36, 39) | ... | ... | 136 |

c4.33

CHAPTER XI

c4.34

SQUARING THE CIRCLE

c4.35

| Circling the square | ... | ... | 140 |
| Squaring the circle | ... | ... | 144 |
| Alternative Method | ... | ... | 146 |
| Dvārakanātha's corrections | ... | ... | 148 |
| Values of π | ... | ... | 148 |
| Early history | ... | ... | 149 |

c4.36

CHAPTER XII

c4.37

SIMILAR FIGURES

c4.38

| Similar Figures | ... | ... | 152 |
| Similar isosceles trapeziums | ... | ... | 153 |
| Similar Fire-altars: | | | |
| Methods of Baudhāyana and Āpastamba | ... | ... | 154 |
| Methods of Kātyāyana | ... | ... | 156 |

c4.39

| | | | PAGE |
| Early methods of enlargement | ... | ... | 158 |
| Two methods of enlargement of the Fire-altar | ... | 160 |
| Principle of similarity | ... | ... | 161 |
| Application of areas | ... | ... | 162 |

c4.40

CHAPTER XIII

c4.41

GEOMETRICAL ALGEBRA

c4.42

| Geometrical solution of pure quadratic equation | ... | 165 |
| Solution of complete quadratic equation | ... | 166 |
| Solution of simple equation | ... | ... | 177 |

c4.43

CHAPTER XIV

c4.44

INDETERMINATE PROBLEMS

c4.45

| Rational right-angled triangles | ... | ... | 178 |
| Kātyāyana's solution | ... | ... | 178 |
| Rational right triangles having a given leg | ... | 179 |
| Simultaneous Indeterminate Equations | ... | 180 |
| Baudhāyana's solutions | ... | ... | 182 |
| Āpastamba's solutions | ... | ... | 185 |
| Other Indeterminate Problems | ... | ... | 186 |