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# 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

### Contents

### [c4.1](https://indike.org/pramana/ins.science.datta-1932.c4.1)

CHAPTER I

### [c4.2](https://indike.org/pramana/ins.science.datta-1932.c4.2)

ŚULBA

### [c4.3](https://indike.org/pramana/ins.science.datta-1932.c4.3)

| | | | 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](https://indike.org/pramana/ins.science.datta-1932.c4.4)

CHAPTER II

### [c4.5](https://indike.org/pramana/ins.science.datta-1932.c4.5)

COMMENTATORS

### [c4.6](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.7)

CHAPTER III

### [c4.8](https://indike.org/pramana/ins.science.datta-1932.c4.8)

GROWTH AND DEVELOPMENT OF THE ŚULBA

### [c4.9](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.10)

CHAPTER IV

### [c4.11](https://indike.org/pramana/ins.science.datta-1932.c4.11)

POSTULATES

### [c4.12](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.13)

CHAPTER V

### [c4.14](https://indike.org/pramana/ins.science.datta-1932.c4.14)

CONSTRUCTIONS

### [c4.15](https://indike.org/pramana/ins.science.datta-1932.c4.15)

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

### [c4.16](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.17)

CHAPTER VI

### [c4.18](https://indike.org/pramana/ins.science.datta-1932.c4.18)

COMBINATION OF AREAS

### [c4.19](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.20)

CHAPTER VII

### [c4.21](https://indike.org/pramana/ins.science.datta-1932.c4.21)

TRANSFORMATION OF AREAS

### [c4.22](https://indike.org/pramana/ins.science.datta-1932.c4.22)

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

### [c4.23](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.24)

CHAPTER VIII

### [c4.25](https://indike.org/pramana/ins.science.datta-1932.c4.25)

AREAS AND VOLUMES

### [c4.26](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.27)

CHAPTER IX

### [c4.28](https://indike.org/pramana/ins.science.datta-1932.c4.28)

THE THEOREM OF THE SQUARE OF THE DIAGONAL

### [c4.29](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.30)

CHAPTER X

### [c4.31](https://indike.org/pramana/ins.science.datta-1932.c4.31)

RATIONAL RECTANGLES

### [c4.32](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.33)

CHAPTER XI

### [c4.34](https://indike.org/pramana/ins.science.datta-1932.c4.34)

SQUARING THE CIRCLE

### [c4.35](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.36)

CHAPTER XII

### [c4.37](https://indike.org/pramana/ins.science.datta-1932.c4.37)

SIMILAR FIGURES

### [c4.38](https://indike.org/pramana/ins.science.datta-1932.c4.38)

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

### [c4.39](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c4.40)

CHAPTER XIII

### [c4.41](https://indike.org/pramana/ins.science.datta-1932.c4.41)

GEOMETRICAL ALGEBRA

### [c4.42](https://indike.org/pramana/ins.science.datta-1932.c4.42)

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

### [c4.43](https://indike.org/pramana/ins.science.datta-1932.c4.43)

CHAPTER XIV

### [c4.44](https://indike.org/pramana/ins.science.datta-1932.c4.44)

INDETERMINATE PROBLEMS

### [c4.45](https://indike.org/pramana/ins.science.datta-1932.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 |

### [c4.46](https://indike.org/pramana/ins.science.datta-1932.c4.46)

CHAPTER XV

### [c4.47](https://indike.org/pramana/ins.science.datta-1932.c4.47)

ELEMENTARY TREATMENT OF SURDS

### [c4.48](https://indike.org/pramana/ins.science.datta-1932.c4.48)

| Simple operations with surds | ... | ... | 187 |  
| Technical terms | ... | ... | 188 |  
| Approximate value of √2 | ... | ... | 188 |  
| Various hypotheses about its origin | ... | ... | 189 |  
| Approximate value of √3 | ... | ... | 194 |

### [c4.49](https://indike.org/pramana/ins.science.datta-1932.c4.49)

Irrationality of √2 ... ... ... 195 Other approximate values of √2 ... ... 202 Approximations to the value of √5 ... ... 203 Evaluation of other surds ... ... 205 Approximate Formula ... ... ... 206

### [c4.50](https://indike.org/pramana/ins.science.datta-1932.c4.50)

CHAPTER XVI

### [c4.51](https://indike.org/pramana/ins.science.datta-1932.c4.51)

FRACTIONS AND OTHER MINOR MATTERS

### [c4.52](https://indike.org/pramana/ins.science.datta-1932.c4.52)

Terminology of fractions ... ... ... 212 Operations with fractions ... ... ... 213 Progressive Series ... ... ... 217 Factorisation ... ... ... 218

### [c4.53](https://indike.org/pramana/ins.science.datta-1932.c4.53)

APPENDIX.—Some technical terms of the \*Śulba\* ... 221 BIBLIOGRAPHY OF THE ŚULBA ... ... 227

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Source: [https://indike.org/pramana/ins.science.datta-1932.c4](https://indike.org/pramana/ins.science.datta-1932.c4) · Pramāṇa