The Kaṭapayādi system of numerical notation and its spread outside Kerala
[Le système de notation numérique Kaṭapayādi et sa diffusion en dehors du Kérala]
Revue d'histoire des mathématiques, Tome 18 (2012) no. 1, pp. 37-66.

Alors que l’étude de la transmission des idées scientifiques entre l’Inde et le reste du monde a sa propre importance, il est également nécessaire d’examiner la transmission de celles-ci à l’intérieur du pays, entre les régions, de la langue sanskrite vers les langues régionales et vice-versa. Cet article se propose de cartographier la diffusion du système Kaṭapayādi de notation numérique, largement répandu au Kérala, aux others parties de l’Inde, et montre que ce très important outil de notation mathématique, alors que bien connu en Inde du Nord, y a été rarement employé. D’autre part, cet article refute l’affirmation de Bibhutibhusan Datta et d’Avadhesh Narayan Singh sur l’existence de quatre variantes distinctes du système Kaṭapayādi et montre qu’en réalité il n’y en avait que deux, l’une d’entre elles étant limitée à un seul texte, d’origine inconnue.

While the study of the transmission of scientific ideas from and to India has its own importance, it is also necessary to examine the transmission of ideas within India, from one region to another, from Sanskrit to regional languages and vice versa. This paper attempts to map the spread of the Kaṭapayādi system of numerical notation, widely popular in Kerala, to other parts of India, and shows that this very useful tool of mathematical notation, though well known in northern India, was rarely employed there. The paper further refutes the contention of Bibhutibhusan Datta and Avadhesh Narayan Singh that there existed four distinct variants of the Kaṭapayādi system and shows that there were only two genuine variants of the system, one of these being limited to just one single text of unknown provenance.

Mots clés : Akṣarasaṃkhyā, Bhālūmal, Bhāskara II, Bhūtasaṃkhyā, Harṣa, inscriptions, Kaṭapayādi system, magic squares, Rāmacandra Vājapeyin, Sanskrit astrolabes, Sanskrit celestial globes
@article{RHM_2012__18_1_37_0,
     author = {Rajeswara Sarma, Sreeramula},
     title = {The {\protect\emph{Kaṭapay\={a}di}} system of numerical notation and its spread outside {Kerala}},
     journal = {Revue d'histoire des math\'ematiques},
     pages = {37--66},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {18},
     number = {1},
     year = {2012},
     mrnumber = {2987370},
     zbl = {1264.01008},
     language = {en},
     url = {http://www.numdam.org/item/RHM_2012__18_1_37_0/}
}
TY  - JOUR
AU  - Rajeswara Sarma, Sreeramula
TI  - The Kaṭapayādi system of numerical notation and its spread outside Kerala
JO  - Revue d'histoire des mathématiques
PY  - 2012
SP  - 37
EP  - 66
VL  - 18
IS  - 1
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/RHM_2012__18_1_37_0/
LA  - en
ID  - RHM_2012__18_1_37_0
ER  - 
%0 Journal Article
%A Rajeswara Sarma, Sreeramula
%T The Kaṭapayādi system of numerical notation and its spread outside Kerala
%J Revue d'histoire des mathématiques
%D 2012
%P 37-66
%V 18
%N 1
%I Société mathématique de France
%U http://www.numdam.org/item/RHM_2012__18_1_37_0/
%G en
%F RHM_2012__18_1_37_0
Rajeswara Sarma, Sreeramula. The Kaṭapayādi system of numerical notation and its spread outside Kerala. Revue d'histoire des mathématiques, Tome 18 (2012) no. 1, pp. 37-66. http://www.numdam.org/item/RHM_2012__18_1_37_0/

[Al-Bīrūnī 1964] Al-Bīrūnī – Alberuni’s India, ed. with translation and notes by Edward C. Sachau, Delhi: S. Chand & co., 1964.

[Āryabhaṭa I 1976] Āryabhaṭa I – Āryabhaṭīya of Āryabhaṭa, critically edited with Introduction, English translation, notes, comments and indexes by Kripa Shankar Shukla and K. V. Sarma, New Delhi: Indian National Science Academy, 1976.

[Āryabhaṭa II 1910] Āryabhaṭa II – Mahásiddhánta, A Treatise on Astronomy by Āryabhaṭ, ed. with his own commentary by Sudhakara Dvivedi, Benares: Braj Bhushan Das & co., 1910.

[Bhāskara I 1949] Bhāskara IThe Laghubhāskarīya with the Commentary Vivaraṇa of Śaṅkaranārāyaṇa, Trivandrum Sanskrit Series, vol. 162, Trivandrum, 1949. | Zbl

[Bhāskara II 1981] Bhāskara Ii Śiṣyadhīvṛddhidam of Lallācārya, with the Commentary Vivaraṇa by Śrīmad Bhāskarācārya, Varanasi: Sampurnanand Sanskrit University, 1981.

[Chatterjee 1975] Chatterjee (Bina)Al-Bīrūnī and Brahmagupta, Indian Journal of History of Science, 10 (1975), p. 161–165. | MR | Zbl

[Datta 1930–31] Datta (Bibhutibhusan)Aṅkānām vāmato gatiḥ (in Bengali), Sāhitya Pariṣat Patrikā, 2 (BS 1337 = 1930–31), 1930–31.

[Datta & Singh 1962] Datta (Bibhutibhusan) & Singh (Avadhesh Narayan)) – History of Hindu Mathematics: A Source Book, Bombay: Asia Publishing House, 1962; first published in 1935 and 1938. | MR | Zbl

[Devasthali 1943] Devasthali (G. V.)Harṣa, the author of the Aṅka-yantra-cintāmaṇi & his Relatives, in B. C. Law Volume, vol.1, Calcutta, 1943, p. 496–503.

[Devasthali 1944] Devasthali (G. V.)A Descriptive Catalogue of the Sanskrit and Prakrit Mss. in the Library of the University of Bombay, Bombay, 1944.

[Fleet 1911] Fleet (J. F.)The Katapayadi System of Expressing Numbers, Journal of the Royal Asiatic Society, 1911, p. 788–794.

[Fleet 1912] Fleet (J. F.)The Katapayadi Notation of the Second Aryasiddhanta, Journal of the Royal Asiatic Society, 1912, p. 459–462.

[Goonetilleke 1882] Goonetilleke (Wm.)The American Puzzle, Indian Antiquary, 1882, p. 83–84.

[Grierson 1881] Grierson (George A.)An American Puzzle, Indian Antiquary, 1881, p. 89–90.

[Harṣa] Harṣa – Aṅkayantracintamani; MS, U, 25.16. of the Bhandarkar Collection of MSS presented by Prof. H. D. Velankar to the Bombay University Library.

[Hultzsch 1894–1895] Hultzsch (E.)Satyamangalam Plates of Devarāya II Saka Samvat 1346, Epigraphia Indica, 3 (1894–1895), p. 35–41.

[Jacquet 1835] Jacquet (E.)Mode d’expression symbolique des nombres, employé par les Indiens, les Tibétains et les Javanais, Nouveau Journal Asiatique, 14 (1835), p. 5–42, 97–130.

[Joseph 1991] Joseph (George Gheverghese)The Crest of the Peacock: Non-European Roots of Mathematics, London: Penguin Books, 1991. | MR | Zbl

[Kak 2000] Kak (Subhash)Indian Binary Numbers and the Kaṭapayādi Notation, Annals of the Bhandarkar Oriental Research Institute, 81 (2000), p. 269–272.

[Kochhar 1991] Kochhar (R. K.)French Astronomers in India during 17th-19th centuries, Journal of the British Astronomical Association, 101.20 (1991), p. 95–100.

[Lewis et al. 1971 Lewis, Ménage, Pellat & Schacht] Lewis (B.), Ménage (V.-L.), Pellat (C.) & Schacht (J.), eds. – Encyclopædia of Islam (The), vol.3, Leiden: E. J. Brill, 1971.

[Lüders 1900–1901] Lüders (H.)Two Pillar Inscriptions of the Time of Krishnaraya of Vijayanagara, Epigraphia Indica, 6 (1900–1901), p. 108–133.

[Madhavan 1991] Madhavan (C.)Origin of Kaṭapayādi System of Numerals, Technical Report, Pūrṇatrayī, Govt. Sanskrit College, Tripunithura, 1991; p. 35–42.

[Minkowski 2008] Minkowski (Christopher)Meanings Numerous and Numerical: Nīlakaṇṭha and Magic Squares in the Ṛgveda, in Kulikov (L.) & Rusanov (M.), eds., Indologica: T. Ya. Elizarenkova Memorial Volume, Moscow, 2008, p. 315–328.

[Misra 1973] Misra (Nilamani), ed. – An Alphabetical Catalogue of Sanskrit Manuscripts in the Collection of the Orissa State Museum, Bhubaneswar, Part 1, Bhubaneswar, 1973.

[Nārāyaṇa] Nārāyaṇa (Melpatūr Nārāyaṇa Bhaṭṭatiri) – Nārāyaṇīyam; digitized by the Department of Sahitya, Rashtriya Sanskrit Sanskthan (Deemed University), Guruvayoor Campus, Puranattukara, Thrissur, Kerala, accessible at http://www.sanskrit.nic.in/DigitalBook/N/Narayaneeyam.pdf.

[Nemicandra 1974a] Nemicandra (Siddhānta-Cakravartin)Gommaṭasāra, Jīvakāṇḍa, with the Hindi gloss by Ratan Chand Jain Mukhtār, Rajasthan, 1974.

[Nemicandra 1974b] Nemicandra (Siddhānta-Cakravartin)Trilokasāra, with the commentary by Mādhava-candra Traividya-deva and Hindi gloss by Āryikā Viśuddhamati Mātājī, Rajasthan, 1974.

[Ojha 1971] Ojha (Guarishankar Hirachand)Bhāratīya Prācīna Lipimālā: The Palaeography of India, Delhi: Munshiram Manoharlal, 1971; original edition: Delhi, 1894; revised and enlarged second edition 1918.

[Pingree 1981] Pingree (David)Jyotiḥśāstra: Astral and Mathematical Literature, Wiesbaden: Otto Harrassowitz, 1981.

[Pingree 1992] Pingree (David)On the Date of the Mahāsiddhānta of the second Āryabhaṭa, Gaṇita Bhāratī, Bulletin of the Indian Society for History of Mathematics, 14 (1992), p. 55–56. | MR | Zbl

[Pingree 1994] Pingree (David)Census of the Exact Sciences in Sanskrit, vol.5, Philadelphia: American Philosophical Society, 1994.

[Plofker 2009] Plofker (Kim)Mathematics in India, Princeton: Princeton Univ. Press, 2009. | DOI | MR | Zbl

[Raja 1963] Raja (K. Kunjunni)Astronomy and Mathematics in Kerala (An Account of the Literature), Brahmavidyā: The Adyar Library Bulletin, 27 (1963), p. 118–167.

[Rāmacandra 1762] Rāmacandra – Samarasara with the commentary by Bharata, 1762; manuscript “Jyotiṣa 39,” Aligarh Muslim University, Aligarh, copied on Saṃvat 1819 Jyeṣṭha-śuddha-pūrṇimā = Sunday 6 June 1762.

[Rāmacandra 1982–1983] Rāmacandra – Śrīrāmacandra-somayāji-praṇītaḥ Samarasāraḥ Bharata-Saṃskr̥ta-ṭīkopetaḥ, Jammu, 1982–1983.

[Rāmacandra 1986] Rāmacandra – Samarasāram of Somayāji Śrī Rāmachandra, with Sanskrit commentary and Hindi translation by Rāmajanma Mishra, Kashi Sanskrit Series, vol. 234, Varanasi, 1986.

[Randel 2003] Randel (Ron Michael), ed. – The Harvard Dictionary of Music, Cambridge: Belknap Press, 4th edition, 2003.

[Roşu 1986] Roşu (Arion)Mantra et yantra dans la médecine et l’alchemie indiennes, Journal asiatique, 274 (1986), p. 203–268.

[Roşu 1987] Roşu (Arion)Les carrés magiques dans la médecine indienne, in Meulenbeld (G. Jan) & Wujastyk (Dominik), eds., Studies on Indian Medical History, Groningen: John Benjamins Pub Co, 1987, p. 103–112.

[Roşu 1989] Roşu (Arion)Les carrés magiques indiens et l’histoire des idées en Asie, Zeitschrift der Deutschen Morgenländischen Gesellschaft, 139 (1989), p. 120–158.

[Śaṅkaranārāyaṇa 1949] Śaṅkaranārāyaṇa – The Laghubhāskarīya with the Commentary Vivaraṇa of Śaṅkaranārāyaṇa, Trivandrum Sanskrit Series, no.1620, Trivandrum, 1949.

[Śaṅkaravarman 2001] Śaṅkaravarman – Sadratnamālā of Śaṅkaravarman, Indian Journal of History of Science, 36 (2001). | MR | Zbl

[Sarma 1972] Sarma (K. V.)A History of the Kerala School of Hindu Astronomy (In Perspective), Hoshiarpur: Vishveshvaranand Institute, 1972.

[Sarma 2003] Sarma (K. V.)Word and Alphabetical Numerical Systems in India, in Bag (A. K.) & Sarma (S. R.), eds., The Concept of Śūnya, New Delhi: Indira Gandhi National Centre for the Arts, Indian National Science Academy and Aryan Books International, 2003, p. 37–71.

[Sarma 1966] Sarma (Sreeramula Rajeswara)The Pūrvaganḥita of Āryabhaṭa’s (II) Mahāsiddhānta, edited and translated, Thesis, Philipps-Universität Marburg, 1966.

[Sarma 1992] Sarma (Sreeramula Rajeswara)Astronomical Instruments in Mughal Miniatures, Studien zur Indologie und Iranistik, 16–17 (1992), p. 235–276; reprinted in [Sarma 2008, p. 76–121].

[Sarma 1994] Sarma (Sreeramula Rajeswara)Indian Astronomical and Time-Measuring Instruments: A Catalogue in Preparation, Indian Journal of History of Science, 29 (1994), p. 507–528; reprinted in [Sarma 2008, p. 1–46]. | MR

[Sarma 1999] Sarma (Sreeramula Rajeswara)Kaṭapayādi Notation on a Sanskrit Astrolabe, Indian Journal of History of Science, 34 (1999), p. 273–287; reprinted in [Sarma 2008, p. 257–272]. | MR

[Sarma 2001] Sarma (Sreeramula Rajeswara)Some Indo-Persian Astronomical Instruments of the early Nineteenth Century, Khuda Bakhsh Library Journal, 123 (2001), p. 1–16.

[Sarma 2004] Sarma (Sreeramula Rajeswara)Magic Square for 2004, Indian Journal of History of Science, 39 (2004), p. 143–144.

[Sarma 2006] Sarma (Sreeramula Rajeswara)On the Life and Works of Rāmacandra Vājapeyin, in Tripathi (Radha Vallabh), ed., Śrutimahatī: Glory of Sanskrit Tradition (Professor R. K. Sharma Felicitation Volume), New Delhi: Pratibha Prakashan, 2006, p. 645–661.

[Sarma 2008] Sarma (Sreeramula Rajeswara)The Archaic and the Exotic: Studies in the History of Indian Astronomical Instruments, New Delhi: Manohar, 2008.

[Sarma 2009a] Sarma (Sreeramula Rajeswara)Early Transmission of Indian Mathematics, in Gheverghese (Joseph George), ed., Kerala Mathematics: History and its Possible Transmission to Europe, Delhi: B. R. Publishing Corporation, 2009, p. 205–231. | Zbl

[Sarma 2009b] Sarma (Sreeramula Rajeswara)On the Rationale of the Maxim aṅkānāṃ vāmato gatiḥ, Gaṇita Bhāratī, Bulletin of the Indian Society for History of Mathematics, 31 (2009), p. 65–86. | MR

[Shukla 1976] Shukla (Kripa Shankar)Āryabhaṭīya of Āryabhaṭa, with the Commentary of Bhāskara I and Someśvara, New Delhi: Indian National Science Academy, 1976.

[Sircar 1965] Sircar (D. C.)Indian Epigraphy, Delhi: Motilal Banarsidass, 1965.

[Subbarayappa & Sarma 1985] Subbarayappa (B. V.) & Sarma (K. V.)Indian Astronomy: A Source-Book (Based primarily on Sanskrit Texts), Bombay: Nehru Centre, 1985.

[Sundararāja 1962] Sundararāja – Vākyakaraṇa with Sundararāja’s Laghuprakāśikā, Madras: KSR Institute, 1962.

[Sūryadeva 1976] Sūryadeva (Yajvan)Āryabhaṭīya of Āryabhaṭa, with the Commentary of Sūryadeva Yajvan, New Delhi: Indian National Science Academy, 1976.

[Tīrthajī 1991] Tīrthajī (Jagadguru Swāmī Śrī Bhāratī Kṛṣṇa)Vedic Mathematics, or Sixteen Simple Mathematical Formulae from the Vedas, Delhi: Motilal Banarsidass, 1991. | MR | Zbl

[Varmâ 1873] Varmâ (Râma)Inscriptions in the Pagodas of Tirukurangadi, in Tinnevelli, and Śucindram, in South Travancore, Indian Antiquary, 2 (1873), p. 360–362.

[Warren 1825] Warren (John)Kala Sankalita: A collection of memoirs on the various modes according to which the nations of the southern parts of India divide time: to which are added three general tables, wherein may be found ... the beginning, character and roots of the Tamul, Tellinga and Mahommedan Civil Years, concurring, viz. the two former with the European Years of the XVIIth, XVIIIth and XIXth Centuries, and the latter with those from A.D. 622 (A. H. 1) to 1900, Madras: College Press, 1825.

[Weber 1863] Weber (A.)Bibliographische Anzeigen (Râmâyaṇa, Bombay 1859. 1860. und einige andere neue Bombayer Drucke), Zeitschrift der Deutschen Morgenländischen Gesellschaft, 17 (1863), p. 771–785.

[Whish 1827] Whish (C. M.)On the Alphabetical Notation of the Hindus, Transactions of the Literary Society of Madras, 1 (1827), p. 54–62.

[Wood 1985] Wood (Ananda E.)Knowledge before Printing and After: the Indian Tradition in Changing Kerala, Delhi: Oxford Univ. Press, 1985.

[Yano 2006] Yano (Michio)Oral and written Transmission of the Exact Sciences in Sanskrit, Journal of Indian Philosophy, 34 (2006), p. 143–160.