Modes of creation of a technical vocabulary

the case of Sanskrit mathematics

Pierre-Sylvain Filliozat

1a. Mathematical knowledge contained in the Sanskrit language. Names of numbers

1b. Manipulation of language. Acceptance and extensive use of synonymy

1c. Rhetorical manipulations. Use of metonymy

2. Pāṇini’s concept of ordinal and fraction

3. Relation with practice: case of karaṇī in the śrauta ritual

4. Freedom of invention: case of karaṇī in bījagaṇita

5. Role of writing, case of kapāṭasaṃdhi and tricaturbhuja

6. Conclusion

eka 1 1

daśa 10 10

śata 100 102

sahasr

a

1000 103

ayuta 10 000 104

lakşa 100 000 105

niyuta 1 000

000

106

koţi 10 000

000

107

arbuda 100 000 000 108

vŗnda 1 000 000 000 109

kharva 10 000 000 000 101

0

nikharva 100 000 000 000 101

1

śańkha 1 000 000 000 000 101

2

padma 10 000 000 000 000 101

3

sagara 100 000 000 000

000

101

4

antya 1000 000 000 000

000

101

5

madhya 10 000 000 000 000

000

101

6

parārdh

a

100 000 000 000

000 000

101

7

guṇo maurvyāmapradhāne rūpādau sūda indriye |tyāge śauryādisaṃdhyādisattvādyāvṛttirajjuṣu |śuklādāvapi vaṭyāṃ ca iti medinīBowstring, secondary, property such as colour etc., cook, sense organ, generosity, quality such as valour etc., way of dealing with an enemy such as negotiation etc., substance of primordial matter such as sattva etc., repetition, cordguṇa āmantraṇe, hana hiṃsāgatyoḥvadhādau viyat khasya khaṃ khena ghāte khahāro bhavet khena bhaktaś ca rāśiḥ ||Bhāskara, Bīja 14

In multiplication (/murder) etc. of zero (/void) the result is zero (/sky), in multiplication (/blow) by zero (/void)zero (/void). A number (/heap) divided by zero (/void) will be number having a zero (/void) divisor.

saṃvatsaré-saṃvatsare ha vā 'syāgnihotráṃ mahátokthéna sáṃpadyate“Every year, indeed, his libation in fire amounts to the grand litany.”(Śatapathabrāhmaṇa 11.3.3.20)grand litany: recitation 3 times of 80 stanzas of three lines 2times a day = 720 = 2*360 in a year

tāni saṃvatsaré dáśa ca sahásrāņy astaú ca śatāni sám apadyataIn the year they [the muhūrtas] amounted to ten thousand eight hundred.(Śatapathabrāhmaṇa 10.4.2.20)

dakṣiṇā gāyatrīsaṃpannā brāhmaṇasya / jagatyā rājñaḥ /bṛhatī-saṃpannāḥ paśu-kāmasya /“The fee amounts to the gāyatrī for the

Brahmin, to the jagatī for the nobleman, to the bṛhatī for the cattle-seeker.”

(Kātyāyanaśrautasūtra 22.11.21-25)The gāyatrī is a stanza of 3 lines of 8 syllables.

Its name refers here to the number 24. The jagatī is a stanza of 4 lines of 12 syllables = 48. The bṛhatī is a stanza of 2 lines of 8 syllables, 1 of 12 and 1 of 8 = 36.

dīrghasyākṣṇayārajjuḥ pārśvamānī tiryaṅmānī ca yatpṛthagbhūte kurutastadubhayaṃ karoti |(Āpastamba 1.4)

“The diagonal cord of a rectangle produces both the squares that the flank cord and the transverse cord produce separately.”

paitṛkyāṃ dvipuruṣaṃ caturaśraṃ kṛtvā karaṇīmadhyeṣu śaṅkavaḥ sa samādhiḥ |(Kātyāyana 2.6)“Regarding the paitṛkī, after making a square with a side of two puruṣas, pins are fixed in the middle of the producing cords.”

The square of a sum of square roots is :

Technical names are given for: a + b called mahatī “large” and

called laghu “light” The sum is the addition of these two elements,

operated like the addition of two integers. Ka, abbreviation of karaṇī, is used in writing. For the

addition of ka8 and ka2: mahatī =8 + 2 = 10, laghu = , sum

= ka18. This operation is possible only if the product of a and

b is a square. The choice of the words mahatī and laghu is based on a mathematical fact: for all numbers a and b :

abbaba 22

ab2

baab 2

8282

vinyasyādho guṇyaṃ kapāṭasaṃdhikrameṇa guṇarāśeḥ |

guṇayed vilomagatyānulomamārgeṇa vā kramaśaḥ ||utsāryotsārya tataḥ kapāṭasaṃdhir bhaved idaṃ

karaṇam |

“After placing the multiplicand below, in the manner of adjusting verandah panels one should multiply by the multiplier, sliding step by step in reverse movement or in direct course. Therefore this operation will be the adjustment of verandah panels.”

tribhujasya vadho bhujayordviguṇitalamboddhṛto hṛdayarajjuḥ |sā dviguṇā tricaturbhujakoṇaspṛgvṛttaviṣkambhaḥ ||(Brahmasphuṭasiddhānta 12.27)

“The product of two sides divided by twice the altitude is the circum-radius of the trilateral; twice that is the diameter of the circle touching the vertices of the triquadrilateral.”

sthūlaphalaṃ tricaturbhuja-bāhupratibāhuyogadalaghātaḥ |bhujayogārdhacatuṣṭaya-

bhujonaghātāt padaṃ sūkṣmam ||“The product of half the sides and counter sides of a tricaturbhuja is the gross area; the square root of the product of four sets of half the sum of the sides lessened by the sides is its exact [area].”

s being the half perimeter of a tricaturbhuja of sides a, b, c, d, the area is:

dscsbsas