The Science of the Shulba: A Study in Early Hindu Geometry c24: Index

The Science of the Śulba: A Study in Early Hindu Geometry c24

Language: English

Index

c24.1

*Abhyantara-viśeṣa*, 197n. *Agni*, 27, 30 *et seq.*, 39-40, 60n-61n, 120, 152; Enlargement of, 158 *et seq.* (See *Kāmya Agni*, and *Ekaśata-vidha Agni*). *Agni-cayana*, 28, 29. *Agni-cit*, 28, 28n. *Āgnīdhrīya*, 34, 42. *Āhavanīya*, 21, 27, 203 *et seq.*; Altar of, 21, 31. *Alaja*, 23. Algebra, Early history, 165; Geometrical, 165 *et seq.*; Problems of, in ancient India, 2. Al-Khowārizmī, 111n. Allman, 111. Altar, 2, 20 *et seq.*; Science of construction, 164; date of the science, 29; geometry traced in, 30. (See Falcon-shaped Altar, Fire Altar, and Vedic Altar). *Aṁsa*, 212. Āpastamba, Authority mentioned, 26-27; Bürk’s edition, 20; Commentaries of, 10, 13, 19; *Śrauta-sūtra* of, 1, 13, 25n, 29; *Śulba-sūtra* of 3-4, 13, 25; -*Sūtra-paribhāṣā*, 13. *Āpastambīya Śulba-bhāṣya*, 10. *Āraṇyaka*, 25. *Ardha-daśama*, 213. *Ardha-navama*, 213. *Ardhyā*, 44. Area, combination, 71 *et seq.*; To draw a square equal to, *n* times a given square, 71-74; *n*th part of a given square, 74-76; sum and difference of two given squares, 76-80; two given pentagons. 80-82; two given triangles, 80; Definition of, 95; Problems of, in ancient India, 2; Transformation of, 83 *et seq.*; square, from isosceles triangle, 92-93, from rectangle, 83-85, from rhombus, 94, into rectangle, 85-90, square and rectangle, into isosceles trapezium, 90-92, into rhombus, 93-94, into triangle, 92. (See also Measures.) Āryabhaṭa I, 16, 18. *Āryabhaṭīya*, 16. *Aṣṭadaśa-karaṇī*, 188. *Aṣṭa-vidha*, 170. *Aśvamedha-vedi*, 34, 152 *et seq.*, 162 *et seq.* *Aśvamedhika Agni*, 172-73. *Aśvamedhikī-vedi*, 107, 188. *Atharva-veda*, schools of, 1n.

c24.2

Babylonian mathematician, 213. Bakhshālī Manuscript, 208. Baudhāyana, commentaries, 10; His sources, 25, 26; *Śrauta-sūtra* of, 1; *Śulba-sūtra* of, 2-3, 25 *et seq.* *Bāhya-viśeṣa*, 197n. *Bhāga*, 212. *Brāhmaṇa*, 25 *et seq.*, 29 *et seq.*; Primary functions, 29; Source of geometry in, 25-27. (See also *Maitrāyaṇīya Brāhmaṇa*, *Śatapatha Brāhmaṇa*, and *Taittirīya Brāhmaṇa*.) *Bhāradvāja Gṛhya Sūtra*, 13. Bhāskara (Second), 12; *Bījagaṇita* of, 165. Brahmagupta, 17. Bürk, 86 *et seq.*; 107 *et seq.*

c24.3

Caland, 86n. Cantor, 111. *Caturasra-śyenacit*, 109-110, 112, 120, 152. *Caturbhāgona*, 213. *Catur-guṇa*, 212. *Caturtha-saviśeṣārdha*. 213. *Caturtha-saviśeṣa-saptama*, 213. *Catuṣkaraṇī*, 188. Chinese mathematicians, 218.

c24.4

Circle; Area of, 95; Division, equal, 42-43; Doubling, 21; Magnitudes, 201; Squaring and vice versa, 18-19, 21, 27, 40, 143-51. (See also, Postulates, Quadrature, Segment, and Square.) Commentators, 10-11, 13, 16-19. Compass, Possibility of, 34 Congruence theorems, 44. Constructions, 52-70; Parallelogram, 67-70; Perpendicular, 42-55; Rectangle, 62-63; Square, 55-62; Trapezium, isosceles, 63-66. (See Altar.) Cylinder, Volume of a, 101.

c24.5

Dakṣiṇa, 21, 120, 203. Dakṣiṇāgni, 27, 31, 203 *et seq.* Darśapaurṇamāsa-sūtra, 13. Dārśapaurṇamāsikī-vedi, 30, 34. Darśa Sacrifice, 21. Demonstration, 50-51. Dhiṣṇiya, 42, 150. Diagonal, bisects a rectangle and mutually, 43, 45. (See also Postulates, Pythagorean Theorem, Rectangle and Rhombus.) Dimension. (See Spatial Dimensions.) Double-producer, 71. Droṇa, 23. Droṇa-cit, 150, 214. Dvādaśa, 212. Dvārakanātha Yajvā, 10, 18-19, 87 *et seq.* Dvi-guṇa, 212. Dvi-karaṇī, 188. Dvirajvāvihāraṇa, 64. Dvi-saptama, 213.

c24.6

Eggeling, 158n. Egyptian mathematician, 213. Ekādaśinī-vedi, 162 *et seq.* Ekarajvāvihāraṇa, 64. Ekaśata-vidha, 169 *et seq.* Ekaśata-vidha, Agni, 160. Ekavidha Agni, 171. Ekaviṃśa-vidha, 172 *et seq.* Equation, Solution of. (See Quadratic equation, Simple Equation.)

c24.7

Factorisation, 218-20. Figures, construction of similar, 24; Division of, 24; Rectilinear, 222. (See also Similar Figures.) Fire altar, 152 *et seq.*; Enlargement of, 158 *et seq.*, 174 *et seq.*, 203 *et seq.*; Falcon-shaped, 22-23, 67, 183 *et seq.*, 215-216; bending the wings, 37-38; construction of, 154 *et seq.*, 158n; enlargement of, 166 *et seq.*; form of, 134; magnitude, 34; measurement, 34-37. (See Altars, Āhavanīya, Dakṣiṇa, Gārhapatya.) Fraction, 212-17; Division of, 214; Fraction of a, 213; General fraction, 213; Operations with, 213-17; Squaring of a, 216; Terminology of, 212-13; Unit fraction, 212 *et seq.*

c24.8

Garbe, 25. Gārhapatya, 21, 27, 203 *et seq.*; Altar of, 21, 31, 150. Gārhapatya-citi, 31. Gārhapatya-vedi, 180-81. Geometry, Hindu, before 300 B.C., 6-7; Before 3000 B. C., 27 *et seq.*; Esoteric, 2n; Growth of, 20; Influence upon Greek Geometry, 9; Manuals of, 1n; Name of 1, 7, 88n; Origin, 2-6, 20, 25 *et seq.*, 30; Practical, 2n; Problem of, in Śulba-sūtra, 2; Propositions of, in the Śulba, 21, 22, 24; (q. v.). Gopāla, 10. Gṛhya-sūtra, 1n.

c24.9

Haṃsamukhī, 82 Hankel, 50-51, 104, 106, 107. Health, 111. Hemādri, 13. Hindu Geometers, 50. Hindu Geometry. (See Geometry,) Hiraṇyakeśī Śulba, 2, 29. Horse Sacrifice, 173. Hotrīya, 34.

c24.10

Indeterminate Equations, 180-86. Indeterminate problems, 178 et seq. Irrationality, of √2. (See Surds.) Irrationals, first discovery by the Hindus, 115 et seq. Iṣṭi-yajña, 21.

c24.11

Jainas, Canonical works of the, 201-2. Jānu, 101. Junge, 106.

c24.12

Kalpa-sūtra, 1, 1n. Kāmya Agni, 22, 24, 38, 120, 121. Kapardisvāmī, 10, 13, 19. Kapiṣṭhala Saṃhitā, 28. Kaṅka, 28. Karaṇī, 188. (See Surds.) Karavindusvāmī, 10, 14-17. Karma-dīpikā, 11. Kalā, 212. Kāṭhaka Saṃhitā, 28. Kātyāyana, 1, 4-5, 10-11, 26. Koṭihoma-vedi, 152. Kṛṣṇa-yajur-veda, 2. Kuṇḍakṛti, 11. Kūrma, 23.

c24.13

Lakṣahoma-vedi, 152. Leonardo Febonacci of Pisa, 180. Līlāvatī, 12. Line, 222; Angular units of, 69. (See also Straight Line.)

c24.14

Magnitudes, 2-3. (See also Circle, Spatial Relations and Magnitudes.) Mahādeva, King of Devagiri, 13. Mahā-vedi, 22, 64, 96-97, 98, 153. (See also Saumiki-vedi) Mahāvīra, 180. Mahīdhara, 10; Works of, 11.

c24.15

Maitrāyaṇa, Brāhmaṇa, 25; Śrauta-sūtra of, 2; Sulba-sūtra of, 6; Sulba, 12; Saṃhitā, 98, 28n. Mānava, Śrauta-sūtra of, 2; Śulba-sūtra of, 5-6; commentary of Sivadāsa, 12. Mantramahodadhi, Age of, 11. Mārjālīya, 34, 43. Maśaka, 2. Mathematics, Jaina school of, 201n; Problems of Hindu, 2. Measures, Areas and Volumes of, 95 et seq.; kinds of, 8. Mensuration, 95.

c24.16

Nata, 67. Nemicandra, 201. Nirūdhapaśubandha, 21; -vedi, 137. Nitya, 20. Nitya Agni, 21. Nyañchana, 58.

c24.17

Oldenberg, 27.

c24.18

Pādya, 14. Paitṛki-vedi, 22, 75, 113. Pākayājñikī-vedi, 152. Pakṣeṣṭaka, 67-68. Pañcadaśa-bhāga, 22. Pañcaguṇa, 212. Pañcakarṇa : vide Pentalateral. Pañcama (Pañcama-bhāga), 212. Pañcamī, 44n. Pañcāgni, 62. Parallelogram, Construction of, 67-70; Measurement of area of, 96. Paricāyya, 23. Paricāyya-cit, 150. Pārśvamāni, 221. Paśu-yajña, 21. Pātīnī, 62. Paurṇamāsa sacrifice, 21. Pentagon, 80-82. Pentalateral, 80-82. Perpendicular, Construction of, 52-55. Pitr-yajña, 98.

c24.20

Postulates, 41 *et seq.* ; Circle, divisibility, 42; Circle, maximum square described within a, 49; Diagonals, mutual bisection; 44; Diagonals of a rhombus, perpendicular bisection, 45; Rectangle, bisection by diagonal, 43; Rectangle, relation with a parallelogram, p. 48; Rectangle, relation with the rhombus inscribed, 48; Square, relation with the square inscribed, 47; Square, relation with the triangle inscribed, 47; Straight line, equal divisibility, 41; Triangle, bisection of an isosceles, 46; Triangle, equal and similar division. 42. (See also *Śulba.*)

c24.21

*Prācī*, 221.

c24.22

*Prāgvaṃśa*, 22.

c24.23

*Pramāṇa*, 196.

c24.24

*Praüga*, 23.

c24.25

*Praüga-citi*, 121.

c24.26

Prism, Volume of a, 101.

c24.27

Proclus, 178.

c24.28

Progression, Arithmetical, 217-18.

c24.29

Problems, Geometrical, 22-23; Solutions of, 52 *et seq.* (See Constructions, and Indeterminate Problems.)

c24.30

*Pṛthvī*, 221.

c24.31

*Puruṣa*, 102.

c24.32

*Puruṣābhāsa*, 157.

c24.33

Pyramid, Volume of the Frustum of, 101-3.

c24.34

Pythagorean Theorem, 24, 71, 104, *et seq.*; Arithmetical character *vs.* Geometrical character, 107-8; Bhāskara's proof, 118; Chinese origin, 118; Converse, 105-6; Early History, 119-22; Early knowledge and employment, 32-33; Geometrical proof of the Hindus, 115-19; Hindu *vs.* Greek origin, 106 *et seq.*, 118.

c24.35

Q. E. F., Corresponding term in Hindu constructions, 50.

c24.36

Quadratic Equation, 165 *et seq.*: Solutions of complete, 166 *et seq.*; Discrepancies, 168 *et seq.*; Solutions of pure, 165; ax²+bx=c, 165; x²=1+m/972, 165; x²=1+2m/15, 166; 7x²+½x=7½+m, 166 *et seq.*

c24.37

Quadrature, of a circle, 40.

c24.38

Quadrilateral, 47, 48. (See also Square.)

c24.39

*Rāma*, 10-12.

c24.40

*Rāmacandra*, 13.

c24.41

*Rathacakra*, 23.

c24.42

*Rathacakra-citi*, 120, 150.

c24.43

Rational Rectangle, 109, 123 *et seq.*, 178 *et seq.*; Derivation, Āpastamba's formula, 136; by the Hindus, 136; Early History of, 138-39; Euclid's formula, 135-36; Hindu discovery, 124-27; Hindu knowledge of general rules, 127-39; Number of, known to the Hindus, 127; Plato's formula, 134-36; Pythagoras's discovery, 127; Pythagoras's formula, 133-36. (See also Rational Rectangle.)

c24.44

Rational Right-angled Triangle, 178-80.

c24.45

Rectangle, construction, given sides, 22, 62-63; Equal to a square, 23; Diagonical bisection, 43; Measurement of, 95, 96 *et seq.*; Square on the diagonal equal to sum of the squares on the sides, 24; Transformation into isosceles trapezium, 90-92; rhombus, 93-94; square, 83-85, 206; triangle, 92. (See also Construction, Postulates, Rational Rectangle, Rhombus and Square.)

c24.46

Rectilinear figure, measurement of, 95; construction, having given area, 98.

c24.47

*Ṛg-Veda*, *Saṃhitā*, 27 *et seq.*; Schools of, 1n.

c24.48

Rhombus, Construction, equal to square, 23; Diagonals bisect mutually, 45; Formation by joining the middle points of the sides of a rectangle, 48. (See also Postulates, Rectangle and Square.)

c24.49

Right-angle. (See Straight Line.) Right-angled Triangle, 105; Construction, Kapardisvāmi's, 13, Karavindasvāmi's, 14-16; Rāma's, 11. Rodét, 208. Root, Hindu name of, 188. Rule of Three, Bhāskara's (Second), 12; Līlāvatī's, 12. Ruler, use in construction, 33.

c24.50

Sacrifice, Darśa, 21; Iṣṭi, 2; Nirūḍhapaśubandha, 21; Obligatory, 20; Optional, 20, 23; Paśu, 21; Paurṇamāsa, 21; Soma yajña, 21; Vedic, 20 et seq. Sadas, 34. Ṣaḍguruśiṣya, 18. Samara-sāra, 11. Samara-sāra-saṃgraha, 11. Śmaśāna, 23. Sāma-Veda, Schools of, 1n. Saṃhitā: see Kapiṣṭhala Saṃhitā, Kāṭhaka Saṃhitā, Maitrāyaṇīya Saṃhitā, Ṛg-Veda Saṃhitā, Taittirīya Saṃhitā, Vājasaneya Saṃhitā. Samūhya 23. Samūhya-cit, 150. Śankarabhaṭṭa, 12. Śāṅkhyāyana Gṛhya-paddhati, 11. Saptama-karaṇī 188. Sapta-ridha, 170. Śaradā-tilaka Tantra, 11. Sarvābhyāsa, 157. Śatapatha Brāhmaṇa, 27n, 28 et seq., 28 n. Saumikī-vedī, 22, 64-67. (See Mahā-vedī.) Sautrāmaṇī-vedī, 22, 75, 98, 107, 152 et seq., 186. Saviśeṣa, 188, 196-202. Sāyaṇa, 12. Scalene triangle, Source of discovery by the Hindus, 45. Schopenhauer, 107. Segment of a circle, Hindu method of finding area, 17. Series, Early reference, 217; Progressive, 217-18. Similar Figures, 40, 152 et seq.; Construction, given a side, 162. (See also Trapezium.)

c24.51

Simple Equation, solution of, 177. Śivadāsa, Commentator of Mānava Sulba, 12. Śmaśāna-cit, 101 et seq. Ṣoḍaśī, 197n. Soma-yajña, 21. Spatial dimensions, 215. Spatial relations and magnitude, 2 et seq.; 25 et seq. Square, 27, 40; Area, on its diagonal is double its area, 22, 109 et seq.; History of the theorem, 40, 105 et seq., 114-15; Area, on the diagonal of a rectangle equal to sum of squares on sides, 24, 27. (See Pythagorean Theorem.) Area, double of a triangle on the same base and with equal altitude, 47. To construct area equal to 108 sq. padas, 98-99; equal to a simple multiple or submultiple of another, 23; equal to the sum or difference of two equal, 23; To construct, a circle equal in area of a, 21, 140-43; and vice versa, 21, 140-43; To construct, having given the side, 21, 33, 55-62; To construct a triangle or rhombus equal to a, 23. Enlargement of a, 176; Mensuration of a, 176 et seq.; Relation with the quadrilateral formed by joining the middle points, 97; Size, maximum, within a circle, 49; To transform, into an isosceles trapezium, 90-92; from an isosceles triangle, 92-93; into a rectangle, 23, 85; from a rectangle, 23; into a rhombus, 93, from a rhombus, 94, into a triangle, 92; (See also Areas, combination of, Circle, Construction, Postulates, Pythagorean Theorem, Rectangle and Rhombus.) Śrauta-sūtra, 1-2. Śruti, source of Geometry in, 26-27. Straight line, to draw at right angles to another, 52-55; Equal division of a, 40-41. (See Postulates.) Śulapāṇi, 13. Śulba, Definition of, 7-8; Early origin of matter in, 39; Postulates of, 41 et seq.; -proof vs.

c24.52

Euclidian proof, 50; Technical terms of, 221-26. (See *Śulba-sūtra.*) *Śulba-dīpikā*, 10. *Śulba-mīmāṁsā*, 10 *Śulba-Pradīpa*, 10. *Śulba Pradīpikā* 10. *Śulba-sūtra*, of Āpastamba, 3-4; of Baudhāyana, 2-3; Classification of, 6-7; Commentators of, *q.v.*; Definition of, 1-7; Number of, 1, 1n; π evaluated according to, 18; Problems of Geometry in, 2, 2n; Study of, 12; Time of, 6-7. *Śulba-sūtra-vivaraṇa*, 10. *Śulba-sūtra-vṛtti*, 10. *Śulba-vārttika*, 11. *Śulba-vyākhyā*, 10. Sundararāja, 10, 87 *et seq.*; Age of, 17-18. *Suparṇa-citi*, 34n. *Suparṇa Garutman* (or *Suparṇa*), 34. Surds, 185 *et seq.*; Evaluation of some, 205; Formula, approximate, 206-11; Hindu knowledge of, 185; Hindu knowledge of the irrationality of √2, 195-203; Rationalisation of simple, 185; *Śulba* name of, 188; Value of √2, 188-89; Hindu origin of, 189-94; Value of √3, 194-95. *Śyena*, 34. *Śyena-cit*, 22-23. (See Altar, falcon-shaped; *Caturasra-Śyena-cit*, *Vakrapakṣa-Śyena-cit*). Symmetry, Line of, 221.

c24.53

*Taittirīya Brāhmaṇa*, 25. *Taittirīya Saṁhitā*, 25-26, 28-31. Terminology, 212-13. Terms, Technical, 221-26. Theorem, Geometrical, 21-22, 24; Pythagorean, 24, 104 *et seq.* Thibaut, 86 *et seq.*, 108 *et seq.* *Tiryaṅmānī*, 221. Transformation, Geometrical, 23. Trapezium, Area of, to find the, 22; Isosceles, to construct an, face, base and altitude given, 22, 31-32, 63-67; similar and equal to a third of another, 153; similar and double its area, 154; similar and n times its area, 154; similar and equal to multiple or sub-multiple of another, 22; similar to a given one and having a given altitude, 163; Isosceles, having an area of 324 sq. padas, 99-101; Measurement of, 95-96; Isosceles, to transform a square or rectangle into an, 90-92. (See also Construction, Rectangle and Square.) *Trayodaśa*, 212. Triangle, area of a, is half the square on the same base and equal altitude, 47; To construct a, equal to a square, 92; Division of a, into equal and similar parts, 46; Isosceles, to bisect an, 46-47; Measurement of, 95, 96. (See also Postulates, Rectangle, Right-angled Triangle, Scalene triangle and Square.) *Tri-aṣṭama*, 213. *Triguṇa*, 212. *Trī-karaṇī*, 109, 188. *Triśatikā*, Age of, 11. *Tṛtīya-karaṇī*, 109, 188.

c24.54

*Ubhadataḥ praüga*, 23. *Ubhayı̄*, 45. *Uparava*, 34. *Uttara vedi*, 33, 34.

c24.55

Vādhula, 2. *Vājasaneya Saṁhitā*, 28. *Vakrapakṣa-Śyena-cit*, 38, 112 *et seq.* *Vakrapakṣa-Vyastapuccha-Śyena*, 23. *Vakrapakṣa-Vyastapucchā-Śyena-cit*, 197n Varāha, 2, 6. Vedāṅga, Meaning of, 1. *Vedārtha-dīpikā*, 13. Vedi, Configuration of, 221; Enlargement of, 158 *et seq.*; Line of symmetry, 221; Relative position, 30; Time of existence, 39. Vedic Altars, 20 *et seq.*