In the Sanskrit of the Sacred Indian Texts

Tonietti, Tito M.

doi:10.1007/978-3-0348-0672-5_4

Tito M. Tonietti⁵

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 46))

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Abstract

The deepest layers of Greek sciences emerge from the texts of philosophers. From these, century after century, the books of specialists like Euclid derived. In the more ancient classical China, we have seen that books were written or commentated by imperial functionaries, with the aim of practical results for the administration, such as the calendar; in them we find the more interesting results for mathematical sciences. On the contrary, for India, traces of scientific procedures and reasonings are to be sought in the original texts of the Hindu religious traditions, that is to say, in the Veda [Wisdom] and their commentaries.

Having abandoned attachment to the fruits of action,

ever content, depending on nothing, though engaged

in karma [action], verily he does not do anything.

Bhagavad Gita , 4,20

Do your allotted work, but renounce its fruits

-be detached and work- have no desire for reward and work.

Gandhi , “The Message of the Gita”

Passage, O soul, to India!

Eclaircise the myths Asiatic – the primitive fables.

Not you alone, proud truths of the world!

Not you alone, ye facts of modern science!

But myths and fables of eld – Asia’s, Africa’s fables,

The far-darting beams of the spirit! the unloos’d dreams!

The deep diving bibles and legends,

The daring plots of the poets – the elder religions;

Walt Whitman

Chapter elaborated together with Giacomo Benedetti , who is responsible for various direct translations from Sanskrit into Italian.

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Notes

1.
Thibaut 1875, p. 227; reprint Thibaut 1984, pp. 3 and 33. The rite even contemplated an officiant, called a brahman, whose task was only that of indicating the errors committed and the way to correct them; Chandogya-Upaniṣad, IV, XVI–XVII; ed. 1995, pp. 271–274. Malamoud 2005, p. 111.
2.
Seidenberg 1981, p. 271.
3.
Śatapatha Brahmana I, 2, 5, 15–16 and 7; Seidenberg 1960/62, p. 520.
4.
Taittirya Samhita V, 4, 11; Seidenberg 1960/62, p. 507.
5.
Śatapatha Brahmana VI, 1, 1, 1–3; X, 2, 3, 18; Seidenberg 1960/62, p. 492.
6.
Śatapatha Brahmana XII, 7, 3, 4; Seidenberg 1960/62, p. 495.
7.
Unit of length corresponding to a man with his arms raised.
8.
Śatapatha Brahmana X, 2, 1, 1–8; X, 2, 2, 7–8; Seidenberg 1960/62, pp. 507–508.
9.
Quoted in van der Waerden 1983, p. 13.
10.
Śatapatha Brahmana X, 2, 3, 11; Seidenberg 1960/62, p. 508.
11.
Seidenberg 1960/62, p. 505. In his introduction to Thibaut 1984, pp. i–xxii, Debiprasad Chattopadhyaya tried to backdate the geometrical art of vedic altars to the brick constructions of the culture that developed in the valley of the Indus during the second millennium B.C., but he was unconvincing in various points. Here, we cannot wonder how much implicit geometry still emanated from the ruins of Mohenjo-daro, but rather how much precision brickmakers needed to have, as different subjects from the brahmana, and whether they could write in Sanskrit.
12.
For the Śulvasutra , we use the edition (with relative verse numbers) of S.N. Sen and A.K. Bag , who also offer us an English translation: Śulvasutra s 1983. But various critical passages cited in our chapter have been translated directly from the Sanskrit ex novo by Giacomo Benedetti . B.[audhayana] 1.9; A.[pastamba] 1.5; K.[atyayana] 2.8; Ś.[ulvasutras 1983], pp. 78, 101, 121. Cf. Thibaut 1984 (1874–1877), p. 73. Cf. Seidenberg 1960/62, p. 524.
13.
Above, Sect. 2.3.
14.
B. 1.10; A. 2.2; K. 2.10. Ś. pp. 78, 102, 122. Thibaut 1984, p. 74. Cf. Seidenberg 1960/62, p. 524.
15.
B. 1.12; A. 1.4; K. 2.7. Ś. pp. 78, 101, 121. Thibaut 1984, p. 74. Cf. Seidenberg 1960/62, p. 524.
16.
A. 3.1; B. 2.4. Ś. pp. 103, 79. Cf. Seidenberg 1960/62, pp. 517–518; Seidenberg 1977/78, pp. 334–335.
17.
Seidenberg 1960/62, pp. 517–518. Seidenberg 1977/78, pp. 334–335. Śulvasutra s 1983, pp. 157–158.
18.
Euclid 1956, p. 340.
19.
B. 2.5; A. 2.7; K. 3.2. Ś. pp. 79, 102–103, 122. Thibaut 1984, p. 77. Cf. Seidenberg 1960/62, p. 524; Seidenberg 1977/78, p. 318.
20.
A. 3.9; Ś. p. 103. Cf. Seidenberg 1975, pp. 290–291.
21.
Euclid 1956, pp. 379–380.
22.
A. 3.10; Ś. p. 103. Cf. Seidenberg 1975, pp. 290–291.
23.
B. 2.10; Ś. p. 80. Thibaut 1984, p. 78.
24.
B. 2.11; A. 3.3; K. 3.12; Ś. pp. 80, 103, 123. Thibaut 1984, pp. 33–36.
25.
K. 2.9; B. 2.12; A. 1.6; Ś. pp. 121, 80, 101. Sen and Bag interpret saviśeṣa as deriving from the word saśeṣa, which means “with the rest, incomplete”. But if the sentence is divided differently, as sa viśeṣa, the text would simply say, “that is the diagonal”, and the approximation would disappear. Cf. Thibaut 1984, pp. 18 and 79.
26.
Seidenberg 1960/62, p. 515; Euclid 1956, pp. 363–364. Śulvasutra s 1983, pp. 168–169.
27.
Fraction of the puruṣa.
28.
A. 5.7; Ś. p. 105. Cf. Seidenberg 1960/62, p. 519.
29.
Euclid 1956, p. 361.
30.
B. 5.6; Ś. p. 83. Thibaut 1984, pp. 62 and 87–88. Cf. Seidenberg 1960/62, p. 525.
31.
K. 5.4 e 5.5; Ś. p. 124. Cf. Hayashi 2001, p. 729.
32.
Seidenberg 1960/62; 1975; 1977/78.
33.
Staal 1999.
34.
Thibaut 1984, pp. 3–4. Cf. Seidenberg 1977/78, p. 306.
35.
B. 1.13; Ś. p. 78. Thibaut 1984, p. 74. Cf. van der Waerden 1983, p. 9.
36.
A. 5.3; Ś. pp. 105, 238.
37.
A. 5.4; Ś. pp. 105, 238.
38.
A. 5.5 e 5.6; Ś. pp. 105, 237–238.
39.
B. 3.9; Ś. p. 81. Thibaut 1984, p. 81.
40.
B. 4.15; Ś. p. 82. Thibaut 1984, p. 86.
41.
A. 1.1; Ś. p. 101.
42.
B. 1.1–2; Ś. p. 77. Thibaut 1984, p. 69.
43.
A. 4.5; Ś. p. 104.
44.
A. 10.8; Ś. pp. 109–110.
45.
Ś. pp. 111–119. “The mystic number eighteen”; Bhagavad Gita 1996, at the beginning, page unnumbered. Malamoud 1994, p. 299.
46.
A. 9.8; Ś. p. 109.
47.
Manava, 9.1; Ś. p. 133.
48.
Manava, 13.17; Ś. p. 138.
49.
B. 1.12–13; Ś. p. 78. The geometrical style of the Śulvasutra was underlined by Thibaut, who corrected the arithmetic readings of subsequent Indian commentators, Thibaut 1984, pp. 60–64.
50.
Quoted in Datta & Singh 1935, I, p. 155. Thibaut 1984, pp. 64–66.
51.
Quoted in Datta & Singh 1935, I, p. 169.
52.
Datta & Singh 1935, I, p. 170. Joseph 2003 , pp. 301 and 318.
53.
See Sect. 6.3.
54.
B. 1.10–11; Ś. p. 78. A. 1.5; Ś. p. 101. K. 2.3–4; Ś. p. 121. Thibaut 1984, pp. 73–74.
55.
Datta & Singh 1935, I, p. 170. But isn’t this a bit too anachronistic, and too Greek an interpretation of the term? In K. 2.4, rajjurdaśakarani literally means “chord that constructs [a square whose area is] ten”. Thibaut 1984, pp. 65–66.
56.
Datta & Singh 1935, II, pp. 1–8.
57.
Datta & Singh 1935, I, p. xi.
58.
Datta & Singh 1935, II, p. 4.
59.
Datta & Singh 1935, I, pp. 63–74.
60.
Datta & Singh 1935, I, p. 84.
61.
Datta & Singh 1935, I, p. 13.
62.
Datta & Singh 1935, I, p. 38.
63.
Datta & Singh 1935, I, p. 75.
64.
Datta & Singh 1935, I, pp. 75–82.
65.
Euclid 1969, pp. XLII-1. Cf. Euclid 1956, pp. 153–194.
66.
Datta & Singh 1935, I, p. 203.
67.
Datta & Singh 1935, I, p. 188.
68.
Datta & Singh 1935, I, p. 11.
69.
Datta & Singh 1935, I, p. 86.
70.
Joseph 2003 , p. 249. But we are not, as Joseph believes, in the European paradise of Georg Cantor (1845–1918); cf. Tonietti 1990.
71.
Datta & Singh 1935, I, p. 243.
72.
Datta & Singh 1935, II, pp. 12ff.
73.
Datta & Singh 1935, II, pp. 17ff.
74.
Bharata 1959, p. 165. Cardona 2001, p. 741.
75.
Datta & Singh 1935; Joseph 2003 .
76.
They calculated how to distribute among the seven hells one living creature, or two, three, … ten, countable or uncountable beings; Hayashi 2001, p. 775. Joseph 2003 , p. 252.
77.
Datta & Singh 1935; Joseph 2003 .
78.
Van der Waerden 1983, pp. 113–154. But I am not convinced by Van der Waerden ’s hypothesis that the Indians had been subject to Hellenistic influence, because there is a lack of documentation about them.
79.
Joseph 2003 , p. 291.
80.
Joseph 2003 , pp. 278–282.
81.
Zeuthen 1904a, pp. 107–111.
82.
Datta & Singh 1935, I, pp. 5–6.
83.
Upaniṣad 1995, p. 311.
84.
Staal 2001, p. 629.
85.
Staal 2001, p. 615.
86.
Torella 2001, pp. 642–643.
87.
Torella 2001, pp. 655ff.
88.
Torella 2001, pp. 661ff.
89.
Torella 2001, pp. 666ff. Cf. Balslev 1986.
90.
Staal 2001, pp. 637–638.
91.
Torella 2001, p. 644.
92.
Torella 2001, p. 653.
93.
Torella 2001, pp. 668–669.
94.
Cardona 2001, p. 745.
95.
Dagens 2001, p. 903.
96.
Malamoud 1994, p. 180.
97.
See Sect. 4.3 above.
98.
Influenced by India, in Europe, the mathematician L.E.J. Brouwer avoided the tertium non datur rule; Tonietti 1983a and 1990.
99.
Torella 2001, pp. 670–689.
100.
Comba 2001, p. 835.
101.
Torella 2001, p. 640.
102.
Torella 2001, p. 660.
103.
Torella 2001, p. 661.
104.
Sacred Books of the East , v. 12, p. 153; Seidenberg 1962b, p. 33.
105.
Sani 1991.
106.
Malamoud 2005, pp. 102, 193, 200, 212.
107.
Malamoud 2005, p. 116.
108.
Malamoud 2005, pp. 89–121.
109.
White 2001, pp. 868–869. Cf. Chap. 3 above. However, the ardhanariśvara [androgyne] created by the couple Śiva Parvati, comes close to this. Malamoud 1994, p. 180.
110.
Upaniṣad , “Brhadaranyaka”, IV, 4, 22; cf. Malamoud 1994, pp. 95–96.
111.
Malamoud 1994, pp. 143–167.
112.
Datta & Singh 1935, I, p. 7.
113.
Pingree 2001, p. 694
114.
Perinu 1981, p. 19, n. 21. “The hymns are broken down into mantras, and these mantras often attract attention only because they contain a certain word, a certain morpheme, a certain syllable, regardless of the general sense of the phrase. … Disjointed and dispersed, the mantras quoted by the Brahmana form a collection of divisible, mobile Elements , much more than a textual continuum that draws its meaning from the linking together of words and phrases: …”; Malamoud 1994, p. 298.
115.
Together with atodya [instrumental music]; Pingree 2001, p. 703. In the myth about the rite of the soma, the stringed instrument was used by the gods only in order to ensure the return of the Word, exchanged with the gandharva [see below] to obtain the sacrificial plant. But it appears to be a fraud; it was an instrument used only because the sacrifice with the offer of the soma is based on the Word of the Veda. Malamoud 1994, pp. 181–182.
116.
Bhagavad Gita , 1996.
117.
Minkowski 2001, p. 709.
118.
Cardona 2001, pp. 740–743.
119.
Hayashi 2001, pp. 776ff. Unless these scholars, afflicted by blinkers and ear-muffs, neglected to quote it.
120.
Hayashi 2001, p. 783.
121.
Hayashi 2001, p. 775. Cf. Leibniz 1666 and Tonietti 1999a. See Part II, Chaps. 9 and 10
122.
Hayashi 2001, p. 727.
123.
Bharata 1959, p. vi.
124.
Pingree 2001, p. 720.
125.
Perinu 1981, pp. 18 and 24.
126.
Perinu 1981, p. 21.
127.
Torella 2001, pp. 656–659.
128.
Bharata 1959, pp. 3 and 213.
129.
Bharata 1959, Chap. XV.
130.
Bharata 1959, pp. 3–5.
131.
Bharata 1959, p. 35.
132.
Bharata 1959, p. 47.
133.
Bharata 1959, pp. 51ff.
134.
Bharata 1959, pp. 51–73.
135.
Bharata 1959, pp. 121 and 125.
136.
Bharata 1959, pp. 147–149.
137.
Bharata 1959, pp. 185 and 189.
138.
Bharata 1959, p. 139.
139.
Bharata 1959, pp. 138–139.
140.
Natya Śastraa 1996, Chaps. 28–30, 33. But, in spite of the opinion of a scholar like Alain Daniélou , even here it is not possible to find any relationship with the predominant Greek musical concepts, as we shall soon see. Bharata 1959, p. viii.
141.
Satyanarayana 2005, p. 184. The adjective also means “fearless”.
142.
Perinu 1981, p. 21.
143.
Natya Śastraa , Chaps. 28, 23 and 27–28; ed. 1996, pp. 388–389.
144.
ṣadja [born from six] is the first note.
145.
madhyama [middle] is the fourth note.
146.
Natya Śastraa , Chaps. 33, 29–35; ed. 1996, p. 486.
147.
Natya Śastraa, Chaps. 19, 38–40; ed. 1996, p. 268.
148.
Natya Śastraa , Chap. 1.
149.
See above, Sect. 2.5.
150.
Bharata 1959, p. v.
151.
See Part II, Sect. 11.1. Euler 1739. Tonietti 2002b.
152.
Cf. Perinu 1981, pp. 11 and 141.
153.
A.N. Sanyal : “… the concept of the octave … cannot be accepted for classical Indian music … Neither the ancient traditions nor modern practice show signs of it …”; quoted by Perinu 1981, p. 40, n. 16. Otherwise they would have written aṣtaka [the eight].
154.
Perinu 1981, p. 24.
155.
Perinu 1981, pp. 25, 29–30 and 44, n. 42.
156.
Perinu 1981, p. 42, n. 30.
157.
Malamoud 2005, pp. 204–205.
158.
Malamoud 1994, pp. 181–182. Gandharva was also a kind of marriage based on love, thus defined: “Voluntary union of a girl and her lover, with desire as the cause and sexual pleasure as the end.” Malamoud 1994, p. 172.
159.
Part II, Sect. 11.2; Tonietti 2002b.
160.
Natya Śastra , Chaps. 33, 4–13; ed. 1996, p. 484.
161.
Above, Sect. 2.1.
162.
Satyanarayana 2005, p. 202; Perinu 1981, p. 9.
163.
Boyer 1990, p. 250.
164.
Joseph 2003 , p. 301.
165.
Subbarayappa 2001, p. 796.
166.
Kline 1972, p. 190. Casari 2001, p. 909.
167.
Casari 2001, p. 915.
168.
Speziale 2001, p. 927.
169.
Casari 2001, p. 919.
170.
Speziale 2001, p. 921.
171.
Speziale 2001.
172.
Speziale 2001, p. 926.
173.
Comba 2001.
174.
Speziale 2001.
175.
Joseph 2003 , p. 410.
176.
Subbarayappa 2001, pp. 795–796.
177.
Casari 2001, pp. 915–917.
178.
Pingree 2001.
179.
Casari 2001, p. 913.
180.
Casari 2001, p. 914.
181.
See above, Sect. 4.5.
182.
Casari 2001, p. 908.
183.
See above, Sect. 4.3.
184.
Casari 2001, pp. 913–914.

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Dipartimento di matematica, University of Pisa, Pisa, Italy
Tito M. Tonietti

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Tonietti, T.M. (2014). In the Sanskrit of the Sacred Indian Texts. In: And Yet It Is Heard. Science Networks. Historical Studies, vol 46. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0672-5_4

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