Katapayadi System 1

The Kat.apay¯adi system* 1.1 The Kat.apay¯adi (कटपयािद) system is an ancient method of alphabetical notation where each consonant of the Sanskrit alphabet is given a numerical value. The system is described in an anonymous line thus: k¯adi nava, .ta¯ di nava, p¯adi pa˜nca, yady as..tau, कािद नव, टािद नव, पािद प, य ् अौ, “the nine [consonants] starting with ka, the nine starting with .ta, the five starting with pa and the eight from ya [successively denote the numbers 1 to 9].” But the line does not say how the zero is to be represented. The ´ nkara Varman (शर वमन ्) in 1819 AD, 1 Sadratnam¯al¯a (समाला), composed by Sa˙ gives a more comprehensive definition: “na, n˜ a and the vowels are zero. The letters (of the consonant groups) commencing with ka, .ta, pa and ya are digits. In conjunct letters the last consonant is to be taken as the digit. A consonant not attached to a vowel is to be ignored.” This may be graphically shown in the following table:

१ २ ३ क ख ग ट ठ ड प फ ब य र ल

४ घ ढ भ व

५ ङ ण म श

1

2

3

4

ka

kha

ga

.ta

.tha

pa ya

६ ७ ८ ९ ० च छ ज झ ञ त थ द ध झ ष

स

ह

5

6

7

8

9

0

gha

n˙ a

ca

cha

ja

jha

n˜ a

d.a

d.ha

n.a

ta

tha

da

dha

na

pha

ba

bha

ma

ra

la

va

s´a

s.a

sa

ha See footnote: 2

Though neither of the definitions expressly states, the numerals represented in this system are read from the right to the left. This is a neat and elegant method of expressing *

1

2

Sreeramula Rajeswara Sarma, Kat.apay¯adi Notation on a Sanskrit Astrolabe, in: Indian Journal of History of Science, 34, No. 4, 1999, pp. 273 – 287. The Sadratnam¯al¯a, “Garland precious Gems”, is a well-knit astronomical manual (karan.a) in Sanskrit ´ . kara Varman (1774 – 1839), an astronomer-prince of Malabar in North Kerala. composed by San Though the work has been written only in 1819, at a time when western mathematics and astronomy had been introduced in India and the author himself seems to have some knowledge thereof, the work has been set out purely in the traditional style prevalent in Kerala. The work is divided into six chapters and consists of 212 verses. It has also the advantage of having been commented, in Malayalam in detail, by the author himself who supplies the rationale of several matters and also works out ´ nkaravarman शरवममहाराजिवरिचता समाला śaṅkaravarmamahārāexamples. (Sadratnam¯al¯a of Sa˙ javiracitā sadratnamālā. Critically Edited by K. V. Sarma. Indian National Science Academy, New Delhi, 2001 = Indian Journal of History of Science, Vol. 36, Nos. 3 – 4, 2001, p. 1.) K. V. Sarma, l. c., pp. 5 – 6: Numeral Notation in Sadratnam¯al¯a; p. 6: Ka-t.a-pa-y¯adi Table of Integers: ya

ra

la

va

s´a

s.a

sa

ha

[

.la

all vowels

]

2

The Kat.apayadi ¯ system long numbers, more so because the chronograms, 3 apart from the numerical value they represent, are otherwise also meaningful. An oft quoted example is of N¯ar¯ayana Bhat.t.a (नारायण भ), a great Sanskrit poet of Kerala, closing his devotional poem N¯ar¯ayan.¯ıyam (नारायणीयम ्) with the expression a¯ yur¯arogyasaukhyam. On the one hand, it is a prayer for longevity (¯ayur, आयरु ्), health (¯arogya, आरोय) and happiness (saukhyam, सौम ्); on the other it is a chronogram expressing the date of composition, viz., 17,12,211 [?] civil days from the beginning of the Kali era. 4 1.2 What is the anitquity of this system and the geographical extent of its use? Perhaps the earliest occurrence of this notation is in the Candra-V¯akyas [Moon Sentences, च: m. la Lune [“qui brille”], वा: n. parole, discours, langage] of Vararuci ( वरिच) who ¯ is said to have lived in the fourth century AD. 5 In his commentary on the Aryabhat . ¯ıya ¯ ( आयभटीय), S¯uryadeva Yajvan persuasively argues that Aryabhat .a ( आयभट) must have known the Kat.apay¯adi system, thereby implying that the system was already prevalent in the fifth century AD. However, the first positive and datable occurrence is its use by Haridatta (हिरद) in his Grahac¯aranibandhana (हचारिनबन), composed in 683 AD. 6

3

4

5

6

Cf. Richard Salomon, Indian epigraphy: a guide to the study of inscriptions in Sanskrit, Prakrit, and the other Indo-Aryan Llnaguages, New York: OUP, 1998, p. 173 note 32: In some later southern Indian inscriptions, chronograms were composed according to the kat.apay¯adi system; see BIP 86-7, SIE 222 and 234, and BPLM 123). Such kat.apay¯adi chronograms follow the usual right-to-left principle; vowels have no value, and only the last consonant in a conjunct is counted. Thus, e.g., ´ ´aliv¯aha´sake ... d¯asavandya-mite in the Honnehal.l.i inscription (EI 34, 205-6) represents Saka S¯ 1478 (ya =1, va = 4, sa = 7, and da = 8. For other examples, see EI 3, 1894 – 95, 38, II. 40 – 41; EI 4, 1896 – 97, 203 – 4, etc. i. e. from the midnight between 17th and 18th of February, 3102 BC (Julian), the day following the midnight being a Friday [JD 588466 (at noon)] (S. Balachandra Rao, Indian Astronomy. An Introduction, Hyderabad 2000, p. 52). Candrav¯aky¯ani (Moon Sentences of Vararuci). Ed. as Appendix II in V¯akyakaran.a. Ed. T. S. Kuppanna Sastri and K. V. Sarma, Madras: Kuppuswami Sastri Research Institute, 1962, pp. 125 – 134. Vararuci I is the father figure in the astronomical tradition of Kerala. He is supposed to have lived in the first half of the 4th century, this date having been arrived at on the basis of the dates of birth and death of his eldest son Melattol Agnihotri, which are given, according to tradition, in the Kali chronograms yaj˜nasth¯anam suraks.yam (12,70,701) and purudh¯ıh. sam¯as´ryah. (12,57,921) which fall, respectively, in A.D. 343 and 378. The manuscript tradition of the land ascribes to Vararuci the authorship of the 248 Candra-v¯akyas (‘moon-sentences’), popularly called Vararuci-v¯akyas, beginning with g¯ır nah. s´reyah. (गीर ् नः ेयः = “Our Song is richest”) and also the promulgation of the kat.apay¯aadi notation of depicting numbers which has been used in the composition of the said V¯akyas. P. D. Sharma, Hindu Astronomy, Delhi 2004, p. 141. Cf. M. S. Sriran in: Helaine Selin (Ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, 22008, p. 1160: Vararuci (date unconfirmed), who is credited with the authorship of 248 Candrav¯akyas (sentences for computation of the moon’s longitude) by the manuscript tradition, is considered to be the father figure in the astronomical tradition of Kerala. Cf. K. Chandra Hari, Date of Haridatta, Promulgator of the Parahita System of Astronomy in Kerala, in: Indian Journal of History of Science, 37, No. 3, 2002, pp. 223 – 236; K. V. Sarma in: Helaine Selin (Ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, 11997, p. 394 s. v.; P. D. Sharma, Hindu Astronomy, Delhi 2004, p. 212f.

The Kat.apayadi ¯ system ´ nkara Varman (शर वमन )् employs the facile It* deserves to be mentioned here that Sa˙ Kat.apay¯adi system of numerical notation in depicting numbers in the Sadratnam¯al¯a. He also gives a deft definition for this system which is very popularly used in Kerala. The definition runs: na-˜na¯ v aca´s ca s´u¯ ny¯ani sam . khy¯ah. ka-t.a-pa-y¯a-dayah. / mi´sre t¯up¯antyahal sam khy¯ a na ca cintyo hal asvarah. //3.3// . कटपयािदसङ्ािनयमः

नञावच शूािन संाः कटपयादयः । िमे तूपाहल ् संा न च िचो हलरः ॥ ३॥ kaṭapayādisaṅkhyāniyamaḥ nañāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ miśre tūpāntyahal saṃkhyā na ca cintyo halasvaraḥ “na, n˜ a and ac-s (vowels) represent zero. The (nine) integers are represented by the consonant-group beginning with ka (i. e., ka, kha, ga, gha, n˙ a, ca cha, ja, and jha), the group beginning with .ta (i. e., .ta, .tha, d.a, d.ha, n., ta, tha, da and dha), the group beginning with ya (i. e. ya, ra, la, va, s´a, s.a, sa and ha). In a conjunct consonant, the last of the consonants alone will count. And, a consonant standing without a vowel is to be ignored.” “N and n˜ and the vowels are ciphers ; the numbers are k etc., .t etc., p, etc., and y, etc. : in a conjunct consonant it is the last consonant which is the number ; and no attention is to be paid to a consonant which has no vowel.” 7

The facility of the system lies in the fact that several letters are available to represent the same integer, hereby providing the possibility of the same number being formed by the adjacent placement of different letters. Thus a number can be formed by differently spelt words to suit their being fitted into any metre, during verse formation, to express any specific number and, at the same time, make such expressions have literal meanings, thus making them easily remembered and, through them, the number represented by them as well. Thus the moon-sentence rudras tu namyah. (स ् त ु नः = “But Rudra is to be saluted”) is easily remembered since the expression carries a meaning. And, the words da´sa¯ nana (दशानन) [8-5-0-0] and d¯ama (दाम) [8-5], both representing the number 58, can be fitted in verses of different metres.

*

7

´ nkaravarman शरवममहाराजिवरिचता समाला śaṅkaravarmamahārājaviracitā Sadratnam¯al¯a of Sa˙ sadratnamālā. Critically Edited by K. V. Sarma. Indian National Science Academy, New Delhi, 2001 = Indian Journal of History of Science, Vol. 36, Nos. 3 – 4, 2001, p. 5 – 7. J. F. Fleet, The Katapayadi System of Expressing Numbers, in: Journal of the Royal Asiatic Society of Great Britain and Ireland, July, 1911, pp. 788 – 794, here p. 788.f.

3

4

The Kat.apayadi ¯ system ˙ To the definition† thus given we must add that in accordance with a certain rule a˙nk¯an¯am v¯amato gatih. (अानां वामतो गितः), 8 which applies rigorously to this system and to that of numerical words, the numbers must he stated with the lowest figure, the unit, on the left, but are to he applied in the opposite direction, with the unit on the right. It must also be noted that ... there is sometimes a confusion between l and .l. The results are as shown in the table on page 791 below. Rules. 1. Initial vowels, a to au, are ciphers. 2. In a conjunct consonant, only the last member of the combination has value. 3. A consonant without a vowel – that is, a final consonant at the end of a formula or a sentence ; e.g., the t of a¯ s¯ıt and the m of puram – has no value. This applies to also Visarga and Anusv¯ara. 4. The numbers are to be applied in the opposite direction to that in which they are stated.

† J. F. Fleet, The Katapayadi System of Expressing Numbers, in: Journal of the Royal Asiatic Society of Great Britain and Ireland, (July, 1911), pp. 788 – 794, here p. 789 and p. 791. 8 Richard Salomon, Indian Epigraphy, Oxford 1998, p. 173: “numerals run leftward”; M. D. Pandit, Mathematics as Known to the Vedic Samhitas, Delhi 1993, p. 153: “the understanding of the numerals in the reverse way.”

The Kat.apayadi ¯ system A third system* for verbal representation of numbers was the so-called kat.apay¯adi notation, where the thirty-three Sanskrit consonants are mapped not to consecutive numbers but to decimal digits, as shown in table 4.3. Here, the consonants k, .t, p, and y all stand for the digit 1 – hence the name kat.apay¯adi, “beginning with k, .t, p, and y.” As in the concrete number system, numbers are usually read starting with the least significant digit. Vowels have no numerical significance in the kat.apay¯adi convention, and neither do consonants that appear in conjunction with a following consonant or at the end of a word. Only a consonant that is immediately followed by a vowel has a numerical value. The flexibility of the system means that authors can encode digit sequences in actual Sanskrit words. For example, the word bhavati (भवित), “becomes,” is decoded as the sequence 4-4-6, implying the number 644. It is unclear when and bow the kat.apay¯adi system originated. 9 We do know that it forms the basis of the so-called v¯akya or “sentence” genre of astronomical texts that were popular in Kerala in southern India. In these texts, planetary positions at regular intervals were encoded in kat.apay¯adi sentences. The first such work known is traditionally considered to be the Candra-vaky¯ani or “Moon-sentences” of Vararuci, who is traditionally although somewhat doubtfully assigned to the fourth century CE. Sometime in the early first millennium is probably a reasonable estimate for the date of origin of kat.apay¯adi notation. A variant of the kat.apay¯adi system appears in the Mah¯a-siddh¯anta (महािसा) of 10 ¯ the second Aryabhat .a (आयभट ितीय, a¯ ryabhat.a dvit¯ıya) around the eleventh century , where it is described as follows: The digits starting from unity are the sounds beginning with k, t, p, and y, in the order of the sounds. Both n˜ and n are zero. Mah¯a-siddh¯anta 1.2.

* 9

10

Kim Plofker, Mathematics in India, Princeton and Oxford, 2009, p. 75 – 76.. ¯ ¯ A commentator on the Aryabhat . ¯ıya, S¯uryadeva Yajvan, claimed that Aryabhat .a’s own system was derived from kat.apay¯adi, but S¯uryadeva lived in the thirteenth century and may have been just ¯ guessing about the information available to Aryabhat .a. A claim that the seventh-century [the first] Bh¯askara (भार थम, bh¯askara prathama) employed kat.apay¯adi notation appears to have been based on an interpolated later verse. M. M. Pt. Sudhakar Dvivedi, Mahasiddhanta (A Treatise on Astronomy) of Aryabhata, 1995. ¯ There are eighteen chapters in the ‘Mah¯asiddh¯anta’ of Aryabhat .a. In the first chapter which treats of ‘Madhyam¯adhik¯ara’ [madhyam¯a: the average or ‘mean’ position of planets; adhik¯ara: chapter], representation of numbers by alphabet is first explained, and then the revolutions and other properties of the planets, and the number of solar and other days in a Kalpa [kalpa: one thousand times a Mah¯ayuga (43,20,000 = 432 × 104 years), i. e. 432 × 107 years], have been shown. In the eleventh Sloka [ोक, s´loka: a metered and often rhymed poetic verse or phrase, especially a verse in anus..tubh meter, i. e. consisting of four quarter-verses of eight syllables each] of this chapter on the revolutions of planets, the numbers of revolutions of the Great Bear and Ayanagraha [ayanagraha: the planet’s longitude as corrected for ecliptic-deviation], in a Kalpa, are given to be 1599998 and 578159 respectively.

5

6

The Kat.apayadi ¯ system Kat.apay¯adi system* The third system of expressing numbers is used in south India, especialy in Kerala. The name of this system, kat.apay¯adi (‘that which begins with ka, .ta, pa, and ya’), is easily understood from Table IV. In this system one syllable represents one number and vowels play no part except in the initial position. In case more than two consonants are clustered only the last consonant has a numerical value. In other words, a consonant which is not followed by a vowel has no numerical value. Neither the visarga nor the anusv¯ara has numerical value. The numerals expressed in this system are read in the reverse order, namely, the first (i. e leftmost) syllable stands for the number in the lowest decimal place and the last (i. e. rightmost) syllable for that in the highest decimal place. In this system it is quite easy to express numbers by a word or sentence which is meaningful in Sanskrit. For example, s´ar¯ıra (शरीर, ‘body’) = 225, bh¯askara (भार, ‘sun’) = 214, and n¯ılar¯upa (नीलप, ‘blue color’) = 1230. The earliest text that uses this system is Haridatta’s Grahac¯aranibandhana which is dated A.D. 683. 11 This is the basic text of the Parahita (परिहत, ‘useful to laymen’) system of astronomy prevalent in south India 12 . This system was followed by the v¯akya system of astronomy in Kerala. The special feature of the latter is to give astronomical tables in ‘sentences’ (v¯akyas). The earliest existing v¯akya is the Candrav¯akhyas 13 (‘Sentences for the Moon’), which consists of 248 v¯akyas, each giving the daily lunar position in signs, degrees, and minutes. 14 The first three v¯akyas are: गीर ् नः ेयः g¯ır nah. s´reyah. धेवनः ी dhenavah. s´r¯ı ् स त ु नः rudras tu namyah.

‘Our song is richest.’ 0s 12°03' ‘Cows are fortune.’ 0s 24°09' ‘But Rudra is to be saluted.’ 1s 06°22'

्, bhavet sukham, which means ‘There be happiness’, ु The last v¯akya is भवेत ् सखम s besides 0 27°44' as number.

*

11 12 13

14

Michio Yano, Oral and Written Transmission of the Exact Sciences in Sanskrit, in: Journal of Indian Philosophy 34, 2006, pp. 143 – 160, here pp. 150 – 152. Grahac¯aranibandhana or Parahitacagan.ita of Haridatta, ed. by K. V. Sarma, Madras 1954. D. Pingree, Jyotih.s´a¯ stra, 1981, p. 47 Kunhan Raja, Candrav¯akya of Vararuci, reprint from Haricarita, Adya Library Studies No 63, Adyar Library, 1948. This is what O. Neugebauer reported in his The Exact Sciences in Antiquity, 2nd ed. page 166. The period of nine months which is roughly equal to 248 days was known in Babylonian astronomy, too.

The Kat.apayadi ¯ system The* Kat.apay¯adi system was used by Indian astronomers in order to encode complicated numbers (mathematical and astronomical constants) into words and verses that were easy to remember. The system was developed in Kerala in South-India, probably in the ninth century CE. The Kat.apay¯adi system works as follows. To each digit between 0 and 9, a small group of consonants of the Sanskrit alphabet is assigned (see the list below). This Kat.apay¯adi system can be compared to, but is more complicated than the modern telephone keypad system 1 = ABC, 2 = DEF, and so on. To encode a digit between 0 and 9, we choose any Sanskrit consonant from the group belonging to this digit. For the digit 1, for example, we can choose among the four possibilities k, t, p, and y (this is why the system is called Kat.apay¯adi). Of course, each consonant can encode at most one digit. We list the possible encodings in detail (abbreviations such as d.h represent a single Sanskrit consonant): 1 = k, .t, p, y; 2 = kh, .th, ph, r; 3 = g, d., b, l; 4 = gh, d.h, bh, v; 5 = n˙ , n, m, s´; 6 = c, t, s.; 7 = ch, th, s; 8 = j, d, h; 9 = jh, dh; 0 = n˜ , n. To make words and verses, the following rules are added: • If two successive consonants (such as ‘vr’) are placed in a word, only the last consonant r counts, and the v is ignored. • A double consonant such as jj counts only once. • Vowels and consonants that do not occur in the list do not have a numerical meaning, so they can be inserted anywhere without changing the numerical value of the word or verse. Thus the system was very flexible, and it was not difficult to compose words or verses for complicated (series of) numbers. The digits were encoded in reverse order. A fifteenth century example from Kerala in South-India is the verse: vidv¯am .s tunnabalah. kav¯ıs´anicayah. sarv¯artha´s¯ılasthiro nirviddh¯an.ganarendraru. 15 The verse means: *

15

Jan Hogendijk, Vedic Mathematics and the Calculations of Guru T¯ırthaj¯ı, in: Pi in the Sky, no. 8 (December 2004), pp. 24-27. Gold, D. and D. Pingree: A hitherto unknown Sanskrit work concerning M¯adhava’s derivation of the power series for sine and cosine, in: Historia Scientiarum, 42 (1991), 49-65, here p. 53; cf. David Pingree, The Logic of Non-Western Science: Mathematical Discoveries in Medieval India, in: Daedalus, Vol. 132, No. 4, On Science (Fall, 2003), pp. 45 – 53, here p. 50.

7

8

The Kat.apayadi ¯ system The wise ruler whose army has been struck down gathers together the best of advisers and remains firm in his conduct in all matters; then he shatters the (rival) king whose army has not been destroyed. 16 This verse consists of five Sanskrit words, and the letters that carry numerical values [consonants immediately preceding vowels] are vv = 44, tnbl = 6033, kv´sncy = 145061, sv(th)´sl(th)r = 7475372, nv(dh)gnrrr = 04930222. Because the digits were encoded in reverse order, the five words encode the sexagesimal numbers 44/60 3, 33/60 2 + 06/60 3, 16/60 + 05/60 2 + 41/60 3, 273/60 + 57/60 2 + 47/60 3, 2220/60 + 39/60 2 + 40/60 3, which astronomers wanted to memorize because they occur in an expression used for computation of the Indian sine, equivalent to a modern Taylor series. These numbers are written in terms of the first (three powers of 1/60, and they are rounded values of (in ( )10 )8 ( )6 ( )4 ( )2 modern terms) 90 π , 90 π , 90 π , 90 π , and 90 π respectively. 11! 2

9!

2

7!

2

5!

2

3!

2

´ ´ ı [1] Chandra Hari, K., 1999: A critical study of Vedic mathematics of Sankarac¯ arya Sr¯ ¯ Bh¯arat¯ı Kr.s.n.a T¯ırthaj¯ı Mahar¯aj. Indian Journal of History of Science 34, 1-17. [2] Gold, D. and D. Pingree, 1991: A hitherto unknown Sanskrit work concerning M¯adhava’s derivation of the power series for sine and cosine. Historia Scientiarum, 42, 49-65. [3] Gupta, It. C., 1994: Six types of Vedic Mathematics. Gan.ita Bh¯arat¯ı 16, 5-15. ´ ı Bh¯arat¯ı Kr.s.n.a T¯ırthaj¯ı Mah¯ar¯aja, 1992: Vedic Mathematics. [4] Jagadguru Sw¯am¯ı Sr¯ Delhi: Motilal Banarsidas, revised edition. ´ [5] Sen, S. N. and A. K. Bag, 1983: The Sulbas¯ utras. New Delhi: Indian National Science Academy. Cf. Kim Plofker, Mathematics in India, Princeton and Oxford, 2009, p. 237: In the words of the first Sanskrit sentence, the significant kat.apay¯adi consonants (those immediately preceding vowels) correspond to the following numerical equivalents: v 4

v 4

t 6

n 0

b 3

l 3

k 1

v 4

s´ 5

n 0

c 6

y 1

s 7

th 7

s´ 5

l 3

th 7

r 2

n 0

v 4

dh 9

g 3

n 0

r 2

r 2

r 2

v 4

This represents the following set of five coefficients for the Sine series: [0; 0,] 44.

16

[0;] 33, 06.

16; 05, 41.

Gold, D. and D. Pingree l. c. pp. 57-58.

273; 57, 47.

2220; 39, 40.

The Kat.apayadi ¯ system Another* extraordinary verse written by M¯adhava (माधव, 1350 – 1425) 17 employs the kat.apay¯adi system in which the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 are represented by the consonants that are immediately followed by a vowel; this allows the mathematician to create a verse with both a transparent meaning due to the words and an unrelated numerical meaning due to the consonants in those words. Madhava’s verse is: vidv¯am . s tunnabalah. kav¯ıs´anicayah. sarv¯artha´s¯ılasthiro nirviddh¯an.ganarendraru The verbal meaning is: “The ruler whose army has been struck down gathers together the best of advisors and remains firm in his conduct in all matters; then he shatters the (rival) king whose army has not been destroyed.” The numerical meaning is five sexagesimal numbers: 0; 0, 44 0 ; 33, 6 16; 5, 41 273; 57, 47 2220; 39, 40. These five numbers equal, with R = 3437; 44, 48 (where R is the radius): 11 a11 = 5400 10 R 11! 9 a9 = 5400 8 R 9! 7 a7 = 5400 6 R 7! 5 a5 = 5400 R4 5! 3 a3 = 5400 2 R 3!

These numbers are to be employed in the formula (with ϑ1 = ϑ ): 5400

[ 3 1

[ 2 1

2 1

[

2[ 1 9

2 1

]] ]]

sin ϑ = ϑ – ϑ a3 – ϑ a5 – ϑ a7 – ϑ a – ϑ a11

and this formula is a simple transformation of the first six terms in the infinite power series for sine found independently by Newton in 1660: 5 7 9 3 ϑ11 sin ϑ = ϑ – ϑ2 + ϑ4 – ϑ6 + ϑ8 – 10 R 3! R 5! R 7! R 9! R 11!

Not surprisingly, M¯adhava also discovered the infinite power series for the cosine and the tangent that we usually attribute to Gregory. *

17

David Pingree, The Logic of Non-Western Science: Mathematical Discoveries in Medieval India, in: Daedalus, Vol. 132, No. 4, On Science (Fall, 2003), pp. 45 – 53, here p. 50. hi.wikipedia; D. Pingree, Census of the exact sciences in Sanskrit, Series A, Vol. 4, 1981, p. 414: An Empr¯an Br¯ahman.a residing in the Ila˜nn˜ ipal.l.i house in Sa˜ngamagr¯ama (= Irinj¯alakkud.a) near Cochin in Kerala, M¯adhava (fl. ca. 1380/1420) was the teacher of Parame´svara (ca. 1380/1460). He was a brilliant mathematician – evidently the first to investigate various series for the solution of the value of π and of various trigonometrical functions.

9

10

The Kat.apayadi ¯ system Little† is known about M¯adhava’s personal history and education. He was born probably in the second half of the fourteenth century, and worked for some decades before and after 1400 in an illam at Iri˜njalakkud.a [sic!] (ഇരി ാല ട, iri˜nn˜ a¯ lakkut.a) near modern Kochi (Malayalam: െകാ ി, Kocci, pronounced [koˈtʃːi], before 1996 known as Cochin; Hindi कोचीन, kocīna). The only writings of M¯adhava currently known to survive are some astronomical treatises, some of which are dated in the first few years of the fifteenth century. But he is now most renowned for his discoveries in trigonometric power series, preserved only in a few isolated verses. These verses, along with other parts of M¯adhava’s work, were studied and expounded upon in an illam not far from M¯adhava’s home by M¯adhava’s own pupil Parame´svara and Parame´svara’s son D¯amodara. There D¯amodara taught N¯ılakan.t.ha and Jyes.t.hadeva, students from other nearby illams, both of whom in their turn gave instruction to another scholar named ´ nkara V¯ariyar, who worked near the middle of the sixteenth century. Sa˙ The power series that M¯adhava’s followers so carefully elucidated were equivalent to what we know as Maclaurin series expansions for the sine, cosine, and arctangent. In particular, M¯adhava found what is essentially Leibniz’ infinite series for the ratio of the circumference of a circle to its diameter, and also derived a numerical value equivalent to π = 3.14159265359.

Nothing‡ is known about M¯adhava’s own background except that he was a Br¯ahman.a of the caste known as Empr¯antiri, somewhat inferior in status to the Namp¯utiris, and that his home or illam Ila˜nn˜ ipal.l.i (Sanskrit Bakulavih¯ara) was in Sa˙ngamagr¯ama [sic!] (modern Irinjalakuda [sic!], lat. 10°20' N, near Kochi/Cochin). His most famous mathematical achievements are the M¯adhava-Leibniz series for π/4 and the M¯adhavaNewton power series for the Sine and Cosine, but this work now survives only in a few verses recorded by later members of his school. There are, however, several ´ existing astronomical texts by M¯adhava in which Saka-era dates corresponding to 1403 and 1418 CE are mentioned, so it is generally presumed that he was born sometime around the middle of the fourteenth century. His successors frequently referred to him in Sanskrit as “Gola-vid,” “one who knows the sphere” (गोल m. balle; sph`ere; globe; िवद ् in fine compositi adj. m. n. f. qui sait, qui connaˆit; sachant, connaisseur). The only known direct pupil of M¯adhava was a Namp¯utiri named Parame´svara, whose illam Vat.a´ss´eri (Sanskrit Vat.a´sren.i, lat. 10°51' N) lay on the Bharathapuzha (or Nil.a¯ ) River. Kim Plofker, Mathematics in India, Princeton and Oxford, 2009, p. 238: (

sin ϑ = ϑ –

(

(

ϑ3 – ϑ5 – ϑ7 – ... R2 3! R4 5! R6 7!

) ))

† Kim Plofker, Mathematics in India, in: Victor J. Katz, Annette Imhausen (Eds.), The mathematics of Egypt, Mesopotamia, China, India, and Islam: a sourcebook, Princeton 2007, p. 481f. ‡ Kim Plofker, Mathematics in India, Princeton and Oxford, 2009, p. 218f.