Bibliography

(A selection of references which I have found useful. Not all are referenced in the text.)

Sources with translations and/or annotations. (No reference is given for the R. gveda; there are a large number of transla- tions and interpretations in a large number of languages. Precise references to chapter, poem and verse for questions related to number names can be found in Bhagyashree Bavare and P. P. Divakaran, “Genesis and Early Evolution of Decimal Enumeration: Evidence from Number Names in R. gveda”, Ind. J. Hist. Sci. vol. 48.4 (2013) p. 535).

[AB-C] The Aryabhat¯ .¯ıya translated into English with commentary by Wal- ter Eugene Clark, University of Chicago Press, Chicago (1930).

[AB-S] Aryabhat¯ .¯ıya of Aryabhat¯ .a, critically edited with English transla- tion, notes, etc. by Kripa Shankar Shukla in collaboration with K. V. Sarma, Indian National Science Academy, New Delhi (1976). [BM-H] The Bakhshali Manuscript, with translation and commentary by Takao Hayashi, Egbert Forsten, Groningen (1995).

[GK-S] Gan. itakaumud¯ı of N¯ar¯ayan. aPan. d. ita, translated with notes by Parmanand Singh, published periodically in instalments in Gan. ita Bh¯arat¯ı, (1998 - 2002). [Ne-W] The Mathematical Papers of Isaac Newton (vols. I - VIII), edited with notes and translations from the Latin, where necessary, by D. T. Whiteside, The University Press, Cambridge (1967 onwards). [PS-K] Pa˜ncasiddh¯antik¯a of Var¯ahamihira with translation and notes by T. S. Kuppanna Sastry, P.P.S.T. Foundation, Madras (1993). [SiSi-P] L’Œuvre math´ematique et astronomique de Bhaskaracarya: Le Siddhantasiromani I-II, with translation and commentary in French by Fran¸cois Patte, Librairie Droz, Paris (2004).

© Hindustan Book Agency 2018 423 P. P. Divakaran, The Mathematics of India, Sources and Studies in the History of Mathematics and Physical Sciences, https://doi.org/10.1007/978-981-13-1774-3 424 Bibliography

[SuSi-B] The Sˆurya Siddhˆanta, translation with notes by Ebenezer Burgess, Motilal Banarsidass, Delhi (1989, reprint). [SuSu-SB] The Sulbas¯´ utras of Baudh¯ayana, Apastamba,¯ K¯aty¯ayana and M¯anava, with translations and commentary by S. N. Sen and A. K. Bag, Indian National Science Academy, New Delhi (1983).

[TaiSam-K] The Veda of the Black Yajus School entitled Taittir¯ıyasam. hit¯a, translated by A. B. Keith, Harvard University Press, Cambridge (1914).

[TaSa-RS] Tantrasa˙ngraha of N¯ılakan. tha Somay¯aj¯ı, with translation and com- mentary by K. Ramasubramanian and M. S. Sriram, Hindustan Book Agency, New Delhi (2011).

[YB-S] Gan. ita-Yukti-Bh¯as.¯a of Jyes.t.hadeva, vol.I (Mathematics), Malay- alam text with English translation by K. V. Sarma (and explana- tory notes by K. Ramasubramanian, M. D. Srinivas and M. S. Sriram), Hindustan Book Agency, New Delhi (2008).

[YB-TA] Yuktibh¯as.¯a (Part I - General Mathematics), Malayalam text with vy¯akhy¯a in Malayalam by Ramavarma (Maru) Tampuran and A. R. Akhileshvara Ayyar, Mangalodayam Ltd., Trichur (1948).

To these has to be added Colebrooke’s translation with notes of L¯ıl¯avat¯ı and B¯ıjagan. ita and chapters XII and XVIII of Br¯ahmasphut.asiddh¯anta: [Co] Henry Thomas Colebrooke, Algebra with Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bh´ascara, Sharada Publishing House, Delhi (2005, reprint).

Sources in the original (without translation or detailed commentary).

[ABB] Aryabhat¯ .¯ıya with the bh¯as.ya of Nilakantha (on the Gan. ita, K¯alakriy¯a and Gola chapters); Parts I and II, K. Sambasiva Sastri (ed.), Gov- ernment of Travancore, Trivandrum (1930, 1931); Part III, Suranad Kunjan Pillai (ed.), University of Kerala, Trivandrum (1957).

[JM] Jyotirm¯ım¯am. s¯a of Nilakantha, K. V. Sarma (ed.), Vishveshvaranand Institute, Hoshiarpur (1977).

[SD] Siddh¯antadarpan. a of Nilakantha with his own analytic commentary, K. V. Sarma (ed.), Vishveshvaranand Institute, Hoshiarpur (1976).

[TaSa-S] Yuktid¯ıpik¯a of Shankara is part of Tantrasam. graha of Nilakantha (with Yuktid¯ıpik¯a and Laghuvivr.ti), K. V. Sarma (ed.), Vishvesh- varanand Institute, Hoshiarpur (1977). Bibliography 425

History of mathematics and related subjects

Books [DS] Bibhutibhushan Datta and Avadhesh Narayan Singh, History of Hindu Mathematics vols. I and II, Bharatiya Kala Prakashan, Delhi (2004, reprint). [If] Georges Ifrah, Histoire universelle des chiffres, Robert Laffont, Paris (1994). [Jo] George Gheverghese Joseph The Crest of the Peacock (third edi- tion), Princeton University Press, Princeton and Oxford (2011). [Pl] Kim Plofker, Mathematics in India, Princeton University Press, Princeton and Oxford (2009). [SA] T. A. Sarasvati Amma, Geometry in Ancient and Medieval India, Motilal Banarsidass, Delhi (1999). [Sa-HKSHA] K. V. Sarma, History of the Kerala School of Hindu Astronomy, Vishveshvaranand Institute, Hoshiarpur (1972). [St-RM] Frits Staal, Ritual and Mantras: Rules Without Meaning, Motilal Banarsidass, Delhi (1996, Indian edition). [St-Veda] Frits Staal, Discovering the Vedas. Origins, Mantras, Rituals, In- sights, Penguin Books, New Delhi (2008). [Th-EI] Romila Thapar, Early India from the Origins to AD 1300,Pen- guin Books India, New Delhi (2003). [Th-CP] Romila Thapar, Cultural Pasts. Essays in Early Indian History, Oxford University Press, New Delhi (2000).

Collections [Agni] Frits Staal (ed.) Agni. The Vedic Ritual of the Fire Altar vol. II, Motilal Banarsidass, Delhi (2001, Indian edition). [CHIM] G´erard G. Emch, R. Sridharan and M. D. Srinivas (eds.) Con- tributions to the History of Indian Mathematics, Hindustan Book Agency, New Delhi (2005). [MEMCII] Victor J. Katz (ed.) The Mathematics of Egypt, Mesopotamia, China, India and Islam. A sourcebook, Princeton University Press, Princeton and Oxford (2007). [SHIM] C. S. Seshadri (ed.) Studies in the History of Indian Mathematics, Hindustan Book Agency, New Delhi (2010). [Sunya] A. K. Bag and S. R. Sarma (eds.) The Concept of S¯´unya, Indira Gandhi National Centre for the Arts, New Delhi (2003). 426 Bibliography

Finally, there is the invaluable web-based archive of J. J. O’Connor and E. F. Robertson, unusually good on Indian mathematics. The master URL is http://www-history.mcs.st-andrews.ac.uk/index.html with links leading to various sub-indices. Index

Abel resummation formula, 326, Anuyogadv¯ara S¯utra, 123 339 Apastamba, 14, 44, 47–50, 58–60, absolutely convergent, 328 63, 67, 69, 70, 78, 131, Achaemenid, 34 191, 237, 394, 395 Achyuta, 258, 267, 271–273, 275, Apollonius, 397 276, 281, 420, 421 Aramaic, 34 addition formula, 285, 292, 297, Aranyakas, 30 302 Archimedes, 178, 198, 221, 311, addition theorem, 194, 197, 295, 357, 382, 397, 398, 420 297, 299–301, 311, 353, arctangent series, 243, 262, 295, 405 296, 311, 313, 327–329 Adhikari, R., 87 arddhajy¯a, 303 ¯adyasam. khy¯a, 100 ardhajy¯a, 40, 154, 194, 400 Agni, 116 area and volume of the sphere, Ahichhatra, 278, 281 202, 205, 206, 296, 319, Ajanta, 163 329, 332, 357 ¯ak¯a´sa, 127 area theorem, 241, 246, 252, 253, Akbar, 220 298, 414 Alattiyur, 258, 265, 266, 273, 276, area-scaling transformations, 55 282, 420 Aristotle, 135, 403 Alberuni, 218, 219, 384, 387 arithmetic series, 170, 171, 180, Alexander, 32, 34, 144, 145, 149, 184 383 Arjuna, 124 Alexandria, 149–151, 153, 154, 172, Artha´s¯astra, 146, 147, 154 178, 197, 209, 245, 417 Aryabhata, 1, 3, 7, 8, 14, 17, 18, Alexandrian, 15, 151, 152, 156, 40, 54, 96, 100, 110, 123, 245, 246 125, 134, 143, 146–148, algebraic identities, 54, 60, 69, 228, 150, 151, 153, 155–162, 236, 240, 244, 247, 294, 164–167, 170, 171, 361, 402 175–189, 191–213, Almagest, 153 216–218, 220, 222, 225, Amitabha, 163, 166, 384 226, 230, 240, 242, 245, Amoghavarsha, 215, 216 246, 254, 258, 259, an. u, 146 261–263, 265, 267–269, anum¯ana, 270 271, 275, 286–290, an. upr¯ayam, 319, 347, 356 292–297, 299, 303, 311, © Hindustan Book Agency 2018 427 P. P. Divakaran, The Mathematics of India, Sources and Studies in the History of Mathematics and Physical Sciences, https://doi.org/10.1007/978-981-13-1774-3 428 Index

316, 319, 322, 331–333, astrology, 151, 153 335, 336, 340, 345, 347, asymptotic behaviour, 323, 341, 350, 353, 355, 357, 373, 361, 362 382, 384–387, 389, 393, asymptotic limit, 321, 335, 336 394, 396, 401–405, 410, asymptotically dominant, 341 412, 413, 417, 419 Atharvaveda,30 Aryabhata II, 188, 189, 208, 384 atomic numbers, 101, 104–109, Aryabhata school, 259 112, 116, 118, 120, 127, Aryabhata’s rule, 195–199, 202, 129, 130, 364, 409 204, 211, 301, 339, 403 Avanti, 214, 218, 383 Aryabhatan, 206, 208, 214–217, avyakta r¯a´si, 392 236, 245, 246, 259, 265, axioms, 97, 98, 102, 406 266, 276, 291, 293, 295, Ayyar, Akhilesvara, 186, 259, 260 300, 346, 356, 384, 391, 395, 417 Br.haspati, 389 Aryabhat¯ .¯ıya, 6, 7, 9, 11, 14, 41, 42, Babylonia, 90, 416 53, 54, 62, 110, 121, 143, Babylonian, 74, 79, 91, 94, 121, 148, 154, 156, 157, 160, 131, 132, 197, 391, 397, 161, 164, 166, 167, 398, 417, 419 170–172, 175–177, 180, Bakhshali manuscript, 10, 41, 64, 185, 190, 191, 193, 194, 123, 126, 128, 129, 133, 198, 203, 205, 206, 134, 167, 170, 171, 182, 208–210, 212, 213, 216, 184, 219, 305, 320, 365, 220, 240, 259, 263, 391, 392, 395 267–269, 276, 288, 292, Bakhshali square root, 172, 292, 394, 395, 403 307, 371, 411, 416 ¯ Aryabhat.¯ıyabh¯as.ya, 7, 14, 15, 157, bakul¯adhis..thita . . . vih¯ara, 285 161, 175, 178, 181, 185, Balagangadharan, K., 260 198, 216, 261, 263, base, 93, 104, 105, 111 267–271, 273, 285, 293, based numbers, 97, 104, 309 294, 297, 302, 309, 310, basic π series, 313, 328, 329, 359, 322, 377, 378, 390, 400, 372–374, 376 401 Baudhayana, 14, 44, 46, 47, 49, Ary¯¯ as..ta´sata,41 55–57, 60, 61, 63–70, 73, asam. khyeya, 391 84, 86, 88, 89, 130, 131, Asanga, 162, 164 152, 191, 382, 393–395 ¯asanna, 202, 203, 226, 269, 270 Bavare, Bhagyashree, 113, 116, Ashmaka, 161, 162, 164 118, 217 Ashoka, 29, 33, 34, 145–147, 215, Bharata, 139, 140, 257, 342, 382 277 Bhartrhari, 38, 100, 101, 105, 167, a´smakajanapadaj¯ata, 161 170, 268, 287, 389, 393 a´smak¯ıya, 161, 164, 286 Bhaskara I, 2, 14, 154, 156, 161, As..t¯adhy¯ay¯ı, 15, 31, 38, 40, 42, 117, 164, 168, 169, 184, 185, 118, 135, 143, 212, 268, 187–189, 202, 203, 205, 389 206, 208, 211, 213, Index 429

216–218, 266, 276–278, 216, 218, 220–254, 258, 282–284, 295, 319, 332, 259, 262, 288, 290, 292, 384, 414 295, 298, 299, 301, 309, Bhaskara II, 3, 7, 9, 11, 14, 51, 52, 319, 327, 332, 352, 353, 96, 102, 121, 123, 134, 359, 362, 363, 384, 385, 149, 164, 169, 170, 179, 387, 389, 392, 394, 396, 184, 186, 191, 197, 202, 408, 413, 414, 417–419 206–208, 213, 214, Brahmagupta property, 250 220–222, 229, 232–236, Brahmagupta quadrilateral, 238, 240, 243, 245, 248, 238–241, 243–254, 327, 254, 258, 259, 269, 275, 396 283–288, 290, 292, 293, Brahmagupta’s identity, 223, 224 296, 301, 303, 319, 331, brahman, 123, 165 343, 346, 355, 359, 363, Brahmanas, 5, 30, 36, 88, 113, 116, 369, 384, 385, 387, 390, 382 392, 394, 400–404, 414, Br¯ahmasphut.asiddh¯anta, 7, 14, 420 133, 158, 206, 221, 225, Bhattotpala, 159 227–229, 231, 236, 237, bh¯avan¯a, 223–225, 227–234, 244, 240, 241, 246, 332, 395, 301, 359, 414 400 Bhoja, 220 Brahmi, 5, 34, 73, 95, 128, 129, bhuj¯a-kot.i-kar.n. a-ny¯aya, 46, 405 145, 168, 388 bh¯utasam. khy¯a, 110, 127, 153, 168, Brahmi numbers, 5, 10, 95, 128, 373 147, 168 B¯ıjagan. ita, 9, 14, 16, 18, 42, 51, Brahmi script, 173, 388 52, 96, 134, 186, 214, 222, Brahui, 28 226, 229, 289, 293, 359, Brhaspati, 389, 393 363, 392 Br.hatsam. hit¯a, 151, 209 B¯ıjagan. it¯avatam. sa, 288 Buddha, 31, 35, 124, 134, 162, 163, bindu, 127, 133, 168 384, 404, 405 binomial coefficients, 139, 181, Buddhagupta, 164 289, 341, 342, 385 Buddhism, 31, 32, 144, 145, 162, binomial expansion, 172 163, 166, 206, 209, 215, binomial series, 311, 314, 354 220, 277, 285, 386, 404, bivalence, 405–408 405 Bos, H. J. M., 212 Burgess, Ebenezer, 41, 121, 193 Brahm¯a, 148, 158, 165 brahma, 165, 166 cakrav¯ala, 214, 227–230, 232, 234, Brahmagupta, 1–3, 7–9, 11, 13, 14, 235, 276, 292, 402, 414 17, 41, 48, 53, 54, 96, 121, Calicut, 257, 258, 267 125, 133, 134, 148, 149, Cantor, Moritz, 394 151, 153, 156, 158, 159, c¯ap¯ıkaran. am, 262, 328 166, 168, 169, 184, 185, Caraka, 419 187, 189, 197, 199, 202, Cartesian geometry, 351, 398 205, 206, 208, 213, 214, Cauchy, 135 430 Index chandah. -´s¯astra,36 236, 237, 240–254, 289, Chandah. s¯utra, 128, 135, 136, 140, 292, 295, 298, 300, 327, 149 332, 352, 376, 385, 396, chandas, 135 404, 414 chandasi, 36, 117 Chandragupta Maurya, 145, 146 D’Alembert, 135, 313, 393 Chera, 216, 217, 279 Daivaj˜na, 283–285, 290, 302, 385, China, 178, 417 420 Chinese remainder theorem, 187 Damodara, 264, 266, 272, 275, 284, Chitrabhanu, 182, 229, 274, 282 285 chord table, 192, 193, 197, 398, 417 Damodaran, 271 Coed`es, Georges, 129 Dani, S.G., 67, 69 cognitive psychology, 35 Datta and Singh, 5, 12, 95, 129, Colebrooke, H.T., 11, 15, 16, 41, 186, 222, 223, 228, 229 121, 193, 220, 221, 223, Datta, B., 7, 60, 69 242, 247, 259, 394, 422 Dauben, Joseph W., 187 combinatorics, 14, 30, 137, 181, 191, 226, 290, 338, 341, Davis, Samuel, 11, 193, 259 342, 352, 385 Deccan, 147, 215, 283, 286 composition, 39, 40, 225 Dehaene, S., 111 compound numbers, 80, 81, 101, derivative, 211, 314–316, 327, 334, 105, 106, 117, 118, 129 335, 350, 351, 354, 415 conditional convergence, 325 Descartes, 95, 97 consonant square, 20, 31, 35, 109, descent, 138, 139, 184, 235, 377, 388 408, 410, 412 continued fraction, 150, 186, 187, dhana, 133, 390 225, 226, 235, 294, 295, Dhar, 220 368–372, 376, 377, 407, Dholavira, 26 408 diagonal theorem, 54–57, 60, 86, continuum limit, 326 154, 177, 190, 191, 230, convergence, 262, 310, 315, 320, 237, 240–243, 245, 246, 325, 329, 359, 411, 415 248, 295, 298, 327, 385, convergent series, 310, 374 396–398, 414, 417, 419 coordinate system, 46 diagonal triples, 47, 48, 57, 86, 90, cube root, 101, 176, 177 236, 237, 240, 327, 396 curvature, 62, 207, 211, 296, 303, difference equation, 143, 262, 334, 329, 344, 345, 350, 357, 338, 410, 411 398 difference formulae, 333 cyclic algorithm, 234 difference/differential equation, cyclic figures, 210, 236, 288, 298 335, 336 cyclic geometry, 14, 55, 208, 299, differential equation, 296, 335, 336, 396, 414 355, 414, 415 cyclic method, 214 differentials, 262, 319, 335, 351 cyclic quadrilaterals, 13, 41, 197, differentiation, 262, 313, 329, 351 199, 206, 214, 221, 229, Digganik¯aya, 162 Index 431

Diophantine, 9, 103, 150, 171, 180, Eudoxus, 398 184, 185, 187, 214, 221, Euler, 193, 225, 229, 235, 349, 419 228, 229, 236, 363 excluded middle, 4, 377, 405, 408 discrete calculus, 54, 331, 352 discrete differentiation, 202 Fairservis, Walter A., 80 discrete fundamental theorem, 212, Fermat, 184, 225, 234, 235 225, 338 Fibonacci, 95, 187 discrete harmonic equation, 335, Filliozat, Pierre-Sylvain, 242 336, 345, 351 finite calculus, 207 discrete integral equation, 338, 339 finite differences, 316, 317 discrete integration, 156, 202, 207, finite differential, 207 316, 321, 335, 345 finite integration, 179, 202, 331, discrete integration by parts, 315, 402 326, 339, 352 finite recursion, 410 distributions, 353 first differences, 331, 345 Divakaran, Odile, 163 first guess, 64, 68, 69, 171, 210, Divakaran, P. P., 70, 113, 116, 118, 247, 305, 307, 323, 326, 217, 263, 309, 341 337, 339, 361, 362, 365, 371 division algorithm, 4, 98, 99, 103, fluents, 348, 349 104, 151, 235, 410 fluxion, 259, 262, 315, 348 division by infinity, 325, 335, 350, Fourier, 193 351, 390, 414 full chords, 236 division principle, 99 functional differences, 225, 331 Doniger, Wendy, 114 functional equation, 310, 361, Dravidian, 28 371–374, 415 Drgganita, 265, 403 . . fundamental solution, 222 Drkkarana, 272, 273 . . fundamental theorem of calculus, Dutta, Amartya, 186, 225 212, 262, 313, 315, 326, dvikaran¯ı, 59, 107 . 336, 346, 349, 350, 352 Dvivedi, Sudhakara, 301 Gandhara, 29, 34, 144, 146, 147, Elamkulam Kunjan Pillai, 278, 162–164, 166, 173, 383 280, 281 Ganesha, 177, 179 Elements, 394 Gangal, Kavita, 87 Empran, 281, 286 Gan. itakaumud¯ı, 14, 248, 254, 266, Emprantiri, 219, 281, 284 342, 401 epicycles, 151 Gan. itap¯ada, 157, 160, 161, 164, Euclid, 44, 48, 53, 54, 88, 89, 91, 165, 170, 171, 175, 177, 132, 151, 193, 210, 240, 180, 181, 183, 184, 357, 382, 394–401, 403, 189–192, 195, 199, 412, 416, 417, 420 202–205, 208–210, 214, Euclidean algorithm, 103, 114, 240, 242, 258, 269–271, 151, 184, 187, 189, 209, 285, 289, 292, 294, 322, 295, 410 331, 332, 336, 347, 401, Euclidean division, 186, 187, 189 403 432 Index

Gan. itas¯arasam. graha, 14, 215 Hunas, 147, 162, 163, 173, 283 g¯ayatr¯ı, 108 geometric algebra, 54, 171 I-Tsing, 36 geometric series, 270, 310, 311, Ilanjippalli, 284–286 314, 315, 337, 340, 367, Indo-European, 30, 33, 95, 394, 372, 374, 411 418, 419 G¯ıtik¯ap¯ada, 157–160, 164, 165, induction, 98, 99, 102, 182, 326, 176, 183, 190, 201, 204 342 Gol¯adhy¯aya, 214, 236, 301 induction axiom, 235, 412 Golap¯ada, 157–160, 164–166, 194, inductive procedure, 327 214 inductive proof, 99, 292, 322, 324, Gonda, Jan, 115 342, 390, 412 Govinda, 208, 216, 217, 277, 282, Indus script, 28 285, 384 Indus Valley, 45, 57, 73–81, 85–88, greatest common divisor, 186, 187, 90, 91, 93, 94, 112, 189 144–146, 162, 180, 191, Greek geometry, 74, 88–90, 95, 209, 214, 383, 385, 395, 132, 178, 192, 197, 351, 417 394, 397 infinite geometric series, 70, 310 Gregory, 262, 313, 349 infinite sequences, 310, 311, 351, Gregory-Leibniz series, 314, 349 366, 394 Gupta, R. C., 123, 301 infinite series, 3, 183, 261, 270, Guptas, 147, 162, 163, 166, 167, 296, 305, 307–310, 315, 215 320, 321, 329, 332, 355, 373, 376, 390, 408, 415 Halayudha, 139, 181 infinitesimal, 319, 329, 334, 346, half-angle formula, 155, 156, 195, 394 201, 202, 291, 295, 301, infinitesimal calculus, 135, 201, 303, 338 211, 262, 263, 308, 385, half-chord, 40, 154, 156, 190, 192, 391, 414, 418 195, 197, 200, 203, 207, infinitesimal limit, 338, 351, 352, 209, 236, 240, 295, 299, 402 303, 313, 318, 319, 327, infinitesimal trigonometry, 295 333, 341, 343, 344, 346, infinitesimalisation, 3, 296, 335, 347, 396 390, 413 Harappa, 26, 28, 76, 80 infinitesimally small, 325, 356, 357 Haridatta, 110, 160, 217, 265 infinity, 98, 126, 134, 223, 230, harmonic equation, 336, 346 309, 319, 323, 325, 351, Hayashi, Takao, 123, 167, 169, 171 389–391, 415 Heath, T. L., 221, 357 instantaneous velocity, 355, 356 Heron, 172, 245 integral, 315, 321, 322, 324, 326, Heron’s formula, 69, 242, 245, 411 327, 345, 350 Hipparchus, 197, 198, 417 integral calculus, 357 hr.daya-rajju, 246 integral equation, 336–339, 350, Huen-Tsang, 161, 164, 278 411, 415 Index 433 integral geometry, 236, 237, 242, 341, 345, 347, 356, 327 361–368, 370, 374, 378, integral right triangle, 238 387, 390, 400, 402, 404, integral solution, 227–233, 412 411–415, 420, 421 integral triples, 397 Jyotirm¯ım¯am. s¯a, 268 integration, 262, 309, 313–315, Jyotpatti, 301, 302 318, 319, 321, 327, 329, 335, 339, 342, 346, 347, Ka, 165 351, 353, 354, 412 Kadamba, 278 integration by parts, 326, 350 K¯alakriy¯ap¯ada, 157, 158, 160, 161, interpolation formula, 275, 291, 166, 186, 214 292, 296, 303, 304, 306, kali days, 150, 161, 267, 272 308, 332, 352, 353 kalpa, 176 interpolation series, 305, 306, 308, Kalpas¯utra, 88, 387 311, 352, 353, 411 Kanheri, 163 interpolator, 186 Kannauj, 161 irrational, 132, 178, 226, 269, 315, Karan. apaddhati, 276, 348, 349 328, 361, 377, 408 Karla, 163 irrationality, 226, 377, 378, kat.apay¯adi, 19, 109, 110, 136, 202, 406–408 217, 267, 272, 277, 294, irrationality conjecture, 270, 328 373 irrationality of π, 311, 320, 376 Katyayana, 44, 45, 58, 60, 63, 69, isosceles trapezia, 78 191, 237, 395 isosceles triangle, 56, 237 Kautilya, 146 iterated refining, 70, 171, 183, 270, Kaye, G. R., 167, 219, 395 305, 306, 337, 361, 368, Kelallur, 264, 266, 271 411 Kenoyer, J. Mark, 74, 76, 77, 80 Kern, Hendrik, 41, 193, 220, 259, Jainism, 31, 32, 145, 404 394 j¯atya triangles, 238–240, 243 Kesavan Veluthat, 279 Jayadeva, 222, 226, 232, 235, 248, kha, 127 276, 384, 412, 419 khah¯ara, 134 j¯ıveparaspara-ny¯ayam, 18, 292, Khan. d. akh¯adyaka, 14, 206, 241, 332 297, 302, 405 Kharoshti, 29, 34, 144, 173 Jones, William, 11, 30, 193, 394, Kichenassamy, Satyanad, 246–248, 418 252 jy¯a, 40, 154, 303, 400 Kollam, 217, 219, 259, 383 jy¯ardha, 40, 190, 194, 195, 204 Kollapuri, 217 Jyeshthadeva, 8, 15, 37, 101, 123, Kriy¯akramakar¯ı, 274, 275, 289, 164, 166, 254, 259, 341, 373 262–264, 266, 267, ks.an. a, 356 271–275, 284, 288, 289, ks.etragan. ita,45 293, 296, 297, 299, 300, Kudallur, 286–288, 387, 389 303–306, 322, 326, 328, Kusumapura, 146, 147, 161, 165, 329, 333, 334, 338, 339, 213, 384 434 Index kut..taka, 150, 171, 184, 186–189, 273–276, 279, 281, 282, 206, 209, 215, 216, 221, 284–293, 295–303, 305, 226, 229, 234, 235, 292, 306, 311, 313, 332–334, 370, 377, 410, 412 339, 341, 347–356, 359, 363, 364, 367, 368, 370, Laghubh¯askar¯ıya, 216, 222, 276 372, 373, 375, 377, 382, Laghubh¯askar¯ıyavivaran. a, 14, 216, 385, 387, 389–391, 279 413–415, 418, 420 Laghuvivr.ti, 274 Madhavan Empran, 287 Lagrange, 225, 229, 235, 419 Madhavan Emprantiri, 282 Lalitavist¯ara, 124, 163, 388, 389 Magadha, 124, 144–146, 161, 163, Lambert, 377 164 Lanka, 152, 156, 158, 159, 185 Mah¯abh¯arata, 6, 33, 113, 123 Laplace, 96 Mah¯abh¯asya, 15, 31, 42, 212, 268, latitude, 46 . 271 lattice, 82 least common multiple, 186, 187 Mah¯asiddh¯anta, 188 Legendre, 377 Mahavira, 14, 215, 240, 277, 283, Leibniz, 212, 262, 313–315, 289, 290, 384 348–350, 353, 394, 418 Mahayana, 123, 124, 144, 146, Leibniz rule, 350 161–164, 166, 384, 389, Leonardo of Pisa, 95 405 L¯ıl¯avat¯ı, 3, 14, 16, 96, 102, 134, Mahmud of Ghazni, 217, 219, 232, 214, 236, 240, 263, 274, 384 275, 283–285, 289, 294, Mahodayapuram, 216, 219, 259, 385, 392 279, 402 linear approximation, 64, 210 Malafouris, L., 111 linear Diophantine equation, 186, Malava, 147, 153, 220, 383, 384 188, 233, 235, 294, 410 Manava, 44, 45, 47, 49, 54, 60, 61, linear Diophantine problem, 222, 63, 67, 237, 395 232 Mar¯ıci, 301, 302 linear equations, 170, 171, 180, mathematical induction, 3, 103, 183, 226 182, 235, 261, 315, 321, linearisation, 262, 307, 329 322, 412, 415 local linearisation, 353, 418 Matilal, B. K., 405 lok¯ayata, 166, 268, 288, 387 m¯atra, 35, 38 lopa, 127, 128 mature Harappan phase, 77–79, Lothal, 26, 27 81, 82, 178 Madhava, 2, 3, 7, 9, 13, 15, 42, 96, Maurya, 145, 147 135, 140, 161, 175, 176, mean value theorem, 356 181–183, 190, 195, 197, Mehrgarh, 25, 27 198, 201, 202, 205, 208, Melpattur Narayana Bhattatiri, 211, 217, 219, 220, 226, 275 236, 243, 254, 255, 258, Meluhha, 26 260–267, 270, 271, Menander, 145 Index 435

Mesopotamia, 26, 75, 79, 86, 90, 348–351, 353–355, 363, 91, 93–95, 126, 130, 144, 366, 370, 394, 414, 418 151, 208, 353, 417 Nila school, 3, 7, 50, 96, 110, 122, metagrammatical, 39 125, 159, 160, 178, 179, metalanguage, 42 181, 193, 195, 201, 204, metalinguistic marker, 41, 135 212, 216, 226, 229, 237, metamathematical, 39, 269, 377, 246, 259–263, 265, 276, 390, 400, 404, 406, 412 277, 288, 289, 291, 293, metaphysics of calculus, 313, 315, 302, 304, 306, 307, 311, 325 322, 336, 341, 348, 349, metarules, 39 359, 370, 384, 390, 411, metatext, 42 413–415, 418, 420 method of exhaustion, 311, 357 Nilakantha, 6–8, 15, 49–51, 58, 70, minimal solution, 221, 223, 235 101, 132, 157, 159–161, Mitanni kingdom, 86 164, 170, 175, 178, 181, Mohenjo Daro, 26, 28, 76, 80, 83 198–201, 203, 204, 212, Monier-Williams, Monier, 119 216, 226, 258, 259, 261, Mount Meru, 152, 166 263–276, 282, 285–288, Muhammad Ghori, 219, 283, 384 292–294, 297, 300, 302, Mumford, David, 132, 133, 353, 303, 306, 309–311, 315, 416 320, 322, 328, 337, 339, Munishvara, 301, 302 340, 354, 356, 364, 374, Murti, T. R. V., 405 377, 378, 387, 390, 396, Muziris, 216, 421 400–404, 407, 408, 410, 411, 413, 415, 419–421 Nalanda, 36, 161, 164, 219, 384 Nilakantha’s conjecture, 273, 361 Namputiris, 385 nominal composition, 19, 40, 42, Narasimha, Roddam, 400, 402 70, 107, 117, 119, 127, Narayana, 14, 140, 181, 214, 216, 409, 410 226, 229, 233–235, 237, nominal representation, 106, 113 241–246, 248–252, 254, nominalism, 334, 389–391, 415 264, 266, 274, 277, 282, not, 406–408 284, 287–290, 292, 295, number cognition, 111 298, 301, 332, 342, 343, number theory, 55, 184, 235, 236, 352, 363, 385, 396, 401, 410, 413 414 number-matching, 114, 122 Narayanan, M. G. S., 279 Ny¯aya, 134, 405, 406 Nasik, 163 N¯at.ya´s¯astra, 139 Needham, Joseph, 129 Old Babylonia, 79, 81, 86, 90, 93, negation, 405, 408 100, 103, 112, 126 negative numbers, 98, 132, 133, Old Tamil, 28, 30, 80, 81 170, 377 operations, 97–99, 101, 103, 170, Newton, 95, 96, 212, 259–262, 308, 294 310, 313, 315, 335, oral tradition, 9, 110, 118, 401 436 Index padap¯at.ha, 31, 32, 37, 38, 42, 117, 360, 372, 374–377, 411, 121, 125, 127, 137, 145, 415 376, 383 Pingala, 14, 19, 31, 36, 41, 42, 111, Pait¯amaha, 148 128, 134–140, 173, 181, Pallavas, 279 235, 249, 268, 289, 342, Pa˜ncasiddh¯antik¯a, 133, 146, 148, 352, 382, 385, 388, 391, 149, 155–157, 160, 178, 393 193, 194, 213, 384 place-value, 96, 100, 101, 103, 107, Pandit, M. D., 127 108, 110, 114, 121, 126, Panini, 9, 14, 15, 30, 32, 35, 36, 128, 130, 131, 137, 151, 38–41, 97, 106, 117, 309 120–122, 127, 128, place-value principle, 103, 104, 134–137, 144, 162, 169, 132, 150, 291, 292, 387 173, 212, 352, 388–390, place-value representation, 364 393, 394 Plato, 403 Paninian, 117, 118, 120, 138, 157, Playfair, John, 11, 193, 194, 196, 393 197, 219, 297, 394, 422 Parakroda, 272, 274 Plofker, Kim, 12, 130, 205, 306 parallelism, 54, 89, 396 polynomial representation, 107, Parameshvara, 159, 160, 220, 258, 118, 121, 127, 169, 389 259, 264–267, 271, 272, polynomials, 3, 13, 95, 96, 99, 104, 275–277, 282–285, 287, 105, 134, 176, 229, 230, 297, 298, 385, 403, 420 234, 236, 293, 324, 329, 341, 359, 362–366, 390, Para˙n˙n¯od, 271, 273 . . 393, 409, 415 paribh¯as¯a, 169, 210 . positional notation, 94, 95, 100 Pascal triangle, 138, 139 power series, 3, 95, 183, 204, 259, Pataliputra, 147, 161 261, 289, 293, 305, 306, Patanjali, 15, 31, 32, 42, 173, 212, 308, 309, 314, 315, 320, 268, 288, 389, 393, 406, 327, 331, 332, 335–337, 409 341, 342, 345, 348–350, P¯at.¯ıgan. ita, 208 353, 355, 363, 366, 376, Patte, F., 16, 220 411, 415, 416 Pauli´sa, 148 pr¯ac¯ı,45 Peano, 97, 98, 102, 103, 398, 399, Prakrit, 34, 145, 162 406, 412, 415 pr¯ati´s¯akhya, 37, 38, 70, 125, 137, Peano axioms, 98 376, 409 Pell’s equation, 225, 226, 229, 235 pre-Brahmagupta quadrilateral, permutations, 240, 249, 414 239, 240, 243, 244, 246, perpendicularity, 54, 78, 89 247, 249, 250, 300, 327, π series, 190, 243, 262, 270, 273, 385, 396 276, 289, 293–296, 309, predecessor, 98 311, 315, 316, 325, 327, primitive triples, 48, 49 332–335, 337, 339, 341, principle of adjacent chords, 18 343, 345, 347, 349, 359, Proclus, 416 Index 437

Prometheus, 117 rational triple, 48, 49 proof by contradiction, 4, 405, 406 Ravi, 264, 285 prosody, 19, 31, 35, 36, 135, 279, rectification, 262, 313, 316, 325, 289, 290, 341, 342 328, 329, 348 Prthudaka, 208, 242, 245 recursion, 40, 138, 139, 235, 390, Ptolemy, 153, 178, 192–195, 197, 408, 410, 412 198, 383, 398, 417 recursive algorithm, 196 Ptolemy’s theorem, 242, 243, 245, recursive approximations, 64 246, 251, 300 recursive constructions, 309, 409, pulverisation, 184 410 p¯urn. a, 127, 391 recursive corrections, 211 Purushapura, 144, 164 recursive definition, 342 Putumana Somayaji, 276, 349 recursive descent, 138 Pythagoras, 53, 210, 241, 394, 397, recursive iteration, 151 399 recursive principle, 4 Pythagoras’ theorem, 399 recursive procedure, 103, 137 Pythagorean (diagonal) triples, 78, recursive reasoning, 307, 309, 322, 84, 209, 230 329 Pythagorean paradigm, 78, 91 recursive reduction, 410 recursive refining, 68, 70, 210, 262, quadratic Diophantine equation, 292, 304, 309, 320, 337, 17, 206, 214, 221, 222, 340, 411 224–226, 228, 237, 246, recursive solution, 196, 350, 410 248, 288, 363, 390, 414, recursive structure, 70, 201, 342, 417, 419 370, 394, 409 quadratic equation, 168, 170, 180, reductio ad absurdum, 234, 357, 182, 184, 226, 307 361, 377, 404, 405, 410 quadratic forms, 55, 235 regular hexagon, 155, 177, 179, quadrature, 262, 314, 335, 347–349 311, 328 Raja, K. Kunjunni, 264, 265, 271, regular polygon, 62, 77, 78, 177, 276 205, 236, 295 Rajagopal, C. T., 260, 349, 350, reification, 416 370, 418 remainder identity, 361, 362 Rakhigarhi, 26 Renou, Louis, 87, 116 Ramanujan, Srinivasa, 349, 378, R. gveda, 5, 14, 30–32, 35, 40, 42, 422 64, 73, 87, 88, 93–95, 102, Ramasubramanian, K., 16, 260, 106, 108, 111, 113–122, 355 126, 127, 130, 131, 137, R¯am¯ayan. a, 6, 33, 113 148, 165, 381, 382, 388, Rangachari, M. S., 370 404, 405, 409, 410 Rashtrakuta, 215, 216, 384 Riemann integral, 414 rational functions, 3, 13, 42, 95, r.n. a, 133, 146, 390 293, 314, 329, 359, 415 r.n. asam. khy¯a,18 rational right triangles, 48, 55 Robson, Eleanor, 90 rational solutions, 222, 227–233 Romaka, 148 438 Index rule of three, 18, 54, 64, 66, 134, Saurasiddh¯anta, 193 180, 183, 198, 210, 211, Sayana, 117, 118, 121, 283, 290 291, 294, 303, 315, 317, scaling, 47, 48, 54–56, 211, 237, 331, 339, 343, 350, 357, 238, 300, 318 398 second derivative, 211, 337, 345 rules without meaning, 393 second difference equation, 292, 331, 335 ´sabda,35 second differences, 210, 211, 303, Sadratnam¯al¯a, 349 335, 343–345, 350 sam¯asa, 40, 107, 118, 119, 127 Seidenberg, A., 74, 89–91 S¯amaveda,30 sexagesimal, 398 sam. kalita-sam. kalitam, 181, 321, Shah, Jayant, 136, 342 341 Shakalya, 32 sam. kalitam, 321, 339, 351, 411 Shakas, 147 sam. kalitam representation, 340 Shakraditya, 164 sam. khy¯a, 116, 117 Shankara, 181, 202, 217, 264, 267, s¯am. khya, 166 271, 273–275, 281, 284, sam. sk¯aram, 171, 262, 292, 304, 289, 293, 305, 334, 337, 305, 307–310, 315, 320, 352, 360, 364, 366, 368, 323, 326, 327, 337, 373, 375, 390, 421 360–362, 366–372, 374, Shankara Varman, 201, 276, 349, 411, 415, 416 373 samsk¯aram series, 335 Shankaranarayana, 14, 208, 216, sandhi, 37, 38, 70, 107, 119, 120, 217, 276, 277, 279, 282, 389 285, 384 Sangamagrama, 270, 275, 285, 286, Shetti, M., 217 288, 387 Shukapuram, 258, 282 Sa˙´ nkaran¯ar¯ayan.¯ıya, 216, 217 Shukla, K.S., 195, 202 Sanskritisation, 215, 217, 219, 257, Siddhanta, 32, 133, 146–153, 156, 265, 276, 279, 285, 286 157, 168, 187, 206, 209, Sarasvati, 29 213, 383, 384, 417 Sarasvati Amma, T.A., 12, 34, 74, Siddh¯antadarpan. a, 50, 170, 267, 250, 258, 260, 266, 290, 268 322, 349, 400, 418 Siddh¯antadarpan. avy¯akhy¯a, 269 Sarma, K. V., 16, 37, 195, 260, Siddh¯anta´siroman. i, 14, 170, 207, 264, 265, 267, 271, 272, 214, 236, 301, 302, 332, 276, 277, 294, 319, 320, 355 352, 400 Siddhartha, 35, 124, 389 Sarma, S. A. S., 123, 125 similar triangles, 47, 54, 61, 250, Sarma, S. R., 140 300, 317, 328 Sarnath, 163 sine and cosine series, 181, 262, S¯´arngadeva, ˙ 140 289, 293, 295, 296, 306, Sarukkai, Sundar, 405 329, 349, 353, 372 Sastry, Kuppanna, 148, 149, 156 sine difference formula, 297, 300, Satapatha´ Br¯ahman. a, 130, 131 303, 356, 403, 411 Index 439 sine differences, 195, 196, 199, 202, 179–182, 191, 204, 209, 292, 331, 335, 356 211, 219, 228, 229, 236, sine series, 260, 292, 296, 303, 315, 237, 240, 244, 246, 247, 326, 332, 333, 338, 345, 253, 269, 270, 305, 318, 350, 352, 354, 385, 388, 350, 382, 383, 385–387, 390, 411, 415 393–397, 401, 410, 411, sine table, 110, 179, 180, 190, 417 193–195, 198, 200, 202, sum of sums, 139, 170, 180–182, 203, 207, 209, 210, 216, 269, 289, 340, 342 236, 266, 285, 295–297, Sundararaja, 58 301, 307, 316, 332, 336, Sundareswaran, N. K., 261 339, 373, 411, 414, 417 Sundar¯ı, 222 Sinha, S., 74, 82 Sung-Yun, 162 Skandagupta, 163, 164 ´s¯unya, 127, 128, 388 smr.ti,36 ´s¯unyabindu, 128, 168 soma,61 ´s¯unyapr¯ayam, 334, 335, 356, 391 sphere formulae, 348, 349, 356, Suryadeva, 276–278, 282, 283 404, 414 S¯uryasiddh¯anta, 11, 121, 148, 193, square root, 60, 63, 64, 69, 101, 196, 202, 204, 259 107, 131, 138, 169, 171, syllabary, 19, 35 172, 176–179, 196, 223, symmetric trapezium, 61, 77, 248, 226, 242, 247, 295, 305, 298, 299 307, 308, 311, 320, 335, 377, 407, 411 Taittir¯ıya Upanis.ad, 123 ´ Srautas¯utra, 36, 44 Taittir¯ıyasam. hit¯a, 5, 14, 30, 36, 88, Sridhara, 169, 208 110, 113–115, 118, 119, Sridharan, R., 140 121–125, 127, 130, 383, Sridharan, Raja, 140 389, 391, 406 Srinivas, M. D., 140, 260, 355, 400, Takakusu, 36 402 Tampuran, Rama Verma (Maru), Sripati, 208, 222, 230, 232 186, 260 Sriram, M. S., 16, 260 Tampuran-Ayyar, 260, 271, 278, Staal, Frits, 116, 127, 129, 393, 409 294, 299, 300, 350, Stokes’ theorem, 353 368–370, 373 Su´sruta, 419 Tantrasam. graha, 7, 15, 160, 164, succession, 98–100, 102, 292, 322, 259, 260, 263, 264, 267, 363, 415 271, 273, 274, 294, 297, Sukh¯avat¯ı, 166 300, 302, 304, 347–349, Sulbas¯´ utra, 1, 3, 5, 6, 11, 14, 36, 354, 364, 368, 413, 421 43–52, 54, 55, 57–64, Tathagatagupta, 164 67–71, 73, 74, 78, 81, tatk¯ala, 356 84–86, 88–91, 93, 102, Taxila, 32, 144, 146, 153, 162, 164, 107, 113, 124, 130–133, 167, 220, 245, 383 144, 149, 151, 152, 154, Taylor series, 306, 308, 311, 352, 157, 167, 171, 172, 353 440 Index tetralemma, 405, 408 Ujjain, 146, 147, 149, 150, 152, tetravalence, 405 153, 213, 220, 383, 384 Thales’ theorem, 240, 241 Umasvati, 54 Thapar, Romila, 144 unit cell, 82 theorem of the diagonal, 46, 48–50, universal set, 406, 407 52–57, 59, 60, 70, 84–86, Upanishads, 5, 31, 32, 101, 113, 89, 90, 154, 177, 180, 190, 165, 382 191, 230, 236, 237, upapatti, 10, 269, 294, 346, 375, 240–243, 245, 246, 248, 402–404 253, 268, 269, 291, 294, 295, 327, 385, 396–398, V¯ac, 36 403, 405, 414, 417, 419 V¯agbhat.a, 419 theorem of the two diagonals, 241 V¯akyapad¯ıya, 100, 170 theorem of three diagonals, 298 Valabhi, 213–215, 217, 218, 278, Thibaut, G., 394 281, 384 third diagonal, 237, 248, 249, 251, van Brummelen, Glen, 197, 198 252, 254, 289, 295, 298, van der Waerden, B. L., 417–419 396 Varahamihira, 133, 146, 148, 149, Tirunavaya, 257, 273, 280, 282, 286 151–154, 156, 159, 164, 194, 201, 202, 206, 209, trair¯a´sikam, 18, 54, 64, 65, 183 213, 277, 289, 384 transposed quadrilateral, 248, 249, vargaprakrti, 221, 225, 226, 252, 298 . 228–230, 233, 235 transposition invariance, 244, 247, variable, 13, 95, 96, 99, 133, 134, 251, 252, 298 308, 311, 314–317, 319, transpositions, 170, 239, 240, 321, 327–329, 351, 363 243–245, 247–250, 254, V¯asan¯a, 170, 207, 355 298, 414 Vasco da Gama, 267, 421 tricaturbhuja, 41, 242, 245, 246, V¯asistha Siddh¯anta, 133, 148 400 .. Vasubandhu, 164, 173, 389 trigonometric functions, 3, 140, Vatasseri, 265, 266 154, 179, 191–193, 196, Ved¯a˙nga, 31, 44, 46, 287 201, 202, 210, 211, 213, Ved¯a˙nga Jyotisa, 143, 149 236, 254, 261, 263, 295, . Venkatachalam, K., 80 303, 305, 311, 332, 342, Venv¯aroha, 285 355 . Vijayanagar, 283, 284, 290, 385 trigonometric series, 260, 261, 274 visama quadrilaterals, 238, 242, trikaran¯ı, 59, 107 . . 400 Trikkandiyur, 258, 266, 271, 273, viskambh¯ardha, 246 275 . vy¯akarana,40 Triprangode, 258, 273 . Tulu coast, 281, 287, 290 Warder,A.K.,220 Tulu country, 284–286 Weil, Andr´e, 225 Whish, Charles, 11, 220, 259, 272, Udayadivakara, 222, 232, 235, 276, 276, 348, 349, 355, 394, 277, 282 422 Index 441

Whiteside, D. T., 260, 308, 242, 245, 247–250, 252, 349–351, 354, 355, 370, 253, 259, 260, 262, 263, 372 266, 269–275, 278, 284, 285, 289, 291–300, Yajurveda, 5, 30, 36, 44, 73, 87, 302–304, 306, 307, 113, 410 309–311, 314–329, yantravalaya, 216 332–343, 345–347, Yavanapura, 149, 152 349–352, 356, 357, y¯avat t¯avat, 134, 169, 415 360–371, 373–378, 386, yuga, 110, 150, 157, 158, 161, 176, 390, 391, 393, 398, 400, 184, 185, 206, 209 401, 403, 409, 411–414 yukti, 10, 269, 274, 289, 293, 294, Yuktid¯ıpik¯a, 272–274, 297, 349, 297, 329, 341, 366, 368, 364, 368, 390 370, 377, 401, 403, 404, 407 Zamorin, 257, 258, 266, 276, 280, Yuktibh¯as.¯a, 8, 11, 13, 15–17, 20, 282, 284, 420, 421 37, 42, 51, 54, 70, 97, Zeno, 135 100–102, 126, 134, 160, zero, 96, 98, 126–130, 133, 134, 164, 166, 169, 171, 181, 168, 169, 214, 364 182, 186, 205, 210–212, Zvelebil, Kamil, 80, 81