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Jyotihshastra: Astral and Mathematical Literature c3: Chapter I. Śulbasūtras

Jyotiḥśāstra: Astral and Mathematical Literature c3

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English

Chapter I. Śulbasūtras

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In the performance of Vedic *śrauta* rituals an essential prerequisite is the piling up of the fire altar (*agnicayana*). These altars (*citis*) take the form of various objects; the forms mentioned in Taittirīyasaṃhitā 5, 4, 11 (after the *chandaściti*¹ or "meter altar") and the sacrificers who should erect them are:

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¹ An imaginary altar constructed in the sacrificer's mind, but with the recitation of the appropriate mantras.

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1. *śyenaciti* or "hawk altar"² by one desiring heaven (*suvarga*);
2. *kaṅkaciti* or "heron altar" by one desiring a head in the other world;
3. *alajaciti* or "*alaja*-bird altar" with four furrows by one desiring support;
4. *praügaciti* or "triangle altar" by one desiring to repel his foes;
5. *ubhayataḥ praügaciti* or "triangle on both sides altar" by one desiring to repel both present and future foes;
6. *rathacakraciti* or "chariot-wheel altar" by one wishing to defeat his foes;
7. *droṇaciti* or "trough altar" by one desiring food;
8. *samūhyaciti*³ or "things to be gathered together altar" by one desiring cattle;
9. *paricāyyaciti*⁴ or "circle altar" by one desiring a village;
10. *śmaśānaciti*⁴ or "cemetery altar" by one desiring the world of the fathers (*pitṛloka*).

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² Called *suparṇaciti* in ŚB. 6, 7, 2, 8. This is the principle altar shape of which the others were regarded as derivatives.

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³ Called the *samuhyapurīṣaciti* in ŚB.

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⁴ Omitted by ŚB.

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A few other altar-shapes are described in other Brāhmaṇas, where also are prescribed the rituals to be performed at these altars. The Śrautasūtras belonging to the Yajurveda often include as appendices treatises that give rules concerning the geometry involved in the construction of these altars. These treatises are known as the Śulbasūtras.⁵

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⁵ See B. DATTA, The Science of the Śulba, Calcutta 1932; A. MICHAELS, Beweisverfahren in der vedischen Sakralgeometrie, Wiesbaden 1978; and T. A. SARASVATI AMMA, Geometry in Ancient and Medieval India, Delhi—Varanasi—Patna 1979, pp. 14—60. Still informative is G. THIBAUT, "On the Śulvasūtras," JASB, NS 44 (1875), 227—275.

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The Śrautasūtras containing Śulbasūtras are those of Baudhāyana,⁶ in which the Śulbasūtra is praśna 30;⁷ of Āpastamba,⁸ in which the Śulbasūtra is also praśna 30;⁹ of Vādhūla,¹⁰ whose Śulbasūtra is said to survive in a manuscript at Madras;¹¹ of Mānava,¹² in which the Śulbasūtra is adhyāya 10¹³ (a recension of this is entitled the Maitrāyaṇīyaśulbasūtra¹⁴); of Varāha,¹⁵ whose Śulbasūtra survives in a manuscript at Mysore;¹⁶ and of Kātyāyana,¹⁷ in which the Śulbasūtra is pariśiṣṭa 7.¹⁸ The last of these belongs to the Śuklayajurveda (the Vājasaneyisaṃhitā), the first five to the Kṛṣṇayajurveda (Baudhāyana, Āpastamba, and Vādhūla to the Taittirīyasaṃhitā, and Mānava and Varāha to the Maitrāyaṇīyasaṃhitā).

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⁶ J. GONDA, The Ritual Sūtras, Wiesbaden 1977, pp. 514–518.

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⁷ Edited with the commentary, Śulbadīpikā, of Dvārakānātha Yajvan by G. F. THIBAUT, “The Sulvasutra of Baudhāyana with the Commentary of Dvārakānāthayajvan,” The Pandit 9–10 and NS 1 (1874/75–1876/77); by W. CALAND, BI 163, vol. 3, Calcutta 1913, pp. 389ff.; and by S. PRAKASH and R. S. SHARMAN, New Delhi 1968.

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⁸ GONDA, pp. 520–521.

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⁹ CESS A1, 50a. Edited with a German translation and a commentary by A. BÜRK, “Das Āpastamba-Śulba-Sūtra,” ZDMG 55 (1901), 543–591, and 56 (1902), 327–391; and, with the commentaries of Kapardisvāmin, Karavinda, and Sundararāja, by D. SRINIVASACHAR and S. NARASIMHACAR, MSS 73, Mysore 1931, and by S. PRAKASH and R. S. SHARMA, New Delhi 1968.

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¹⁰ GONDA, pp. 522–524. The Āpastamba is closely related to a section (praśna 25) of the Satyāṣāḍhaśrautasūtra; see MICHAELS, pp. 173–180.

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¹¹ N. K. MAJUMDAR, “On the Different Śulba Sūtras,” PAIOC 2 (1923), pp. 561 to 564.

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¹² GONDA, pp. 525–526.

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¹³ Edited and translated by J. M. VAN GELDER, The Mānava Śrautasūtra, 2 vols., New Delhi 1961–1963. See also N. K. MAZUMDAR, “Mānava Śulba Sūtram,” JDL/U Calcutta 8 (1922), 327–342.

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¹⁴ DATTA, p. 6.

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¹⁵ GONDA, p. 527.

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¹⁶ DATTA, pp. 6 and 230.

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¹⁷ GONDA, p. 528–529.

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¹⁸ An incomplete edition with the Śulbasūtravṛtti of Rāma was published by G. F. THIBAUT, “Katyayana Śulbapariśiṣṭa, with the commentary of Rāma, son of Sūryadāsa,” The Pandit, NS 4 (1882); edited with the commentaries of Karka and of Mahidhara by G. S. NENE and A. S. DOGRE, KSS 120, Benares 1936; and by S. D. Khadilkar, Poona 1974.

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Precise dating of any of these texts is impossible. The earliest, that of Baudhāyana, was perhaps written before 500 B.C., and the remainder presumably antedate the Christian era. It was, indeed, during this period also, probably in the second century B.C., that the most striking śyenaciti of which remains survive was built in Kauśāmbī.¹⁹ The Āpastamba appears to be the second oldest of the major Śulbasūtras, and the Kātyāyana, which consists of a sūtra section (to a large extent repeating sūtras of the Āpastamba verbatim), followed by a verse section, is among the latest; the Mānava has apparently copied some verses from the Kātyāyana.

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¹⁹ G. R. SHARMA, The Excavations at Kauśāmbī (1957–59), Allahabad 1960, pp. 87–126; on the interpretation of this monument see D. SCHLINGLOFF, “Menschenopfer in Kauśāmbī?,” IIJ 11 (1969), 175–189.

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Each of the basic altars must be constructed with five layers of bricks, and there must be a fixed number of bricks in each layer; moreover, the bricks in the second and fourth layers must not be directly above or below those in the first, third, and fifth layers. And the surface covered by the altar, regardless of its shape, must cover an area of seven and one half square puruṣas, or, for certain purposes, that area increased by specified numbers of square puruṣas, or it must be multiplied by a given factor. Finally, the altar must be correctly oriented with respect to the cardinal directions. The task faced by the authors of the Śulbasūtras was to prescribe rules for laying out these altars with only a rope (rajju or śulba) of determined length and posts or gnomons (śaṅku).²⁰ The geometrical problems that were solved by these altar-builders are indeed impressive, but it would be a mistake to see in their works the unique origin of geometry;²¹ others in India and elsewhere, whether in response to practical or theoretical problems, may well have advanced as far without their solutions having been committed to memory or eventually transcribed in manuscripts.

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²⁰ A. K. BAG, “The knowledge of Geometrical Figures, Instruments, and Units in the Śulbasūtras,” EW 21 (1971), 111–119.

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²¹ A. SEIDENBERG, “The Origin of Mathematics,” AHES 18 (1978), 301–342.

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The solutions utilized by the sūtrakāras involve the knowledge of a number of specific right-angled triangles (e.g., 3, 4, 5; 5, 12, 13; 7, 24, 25; 8, 15, 17; 12, 35, 37; and 15, 36, 39) as well as the general rule that the square on the diagonal of a rectangle (square or oblong) is equal to the sum of the squares on two sides; of the approximation

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and the radius, r, of a circle whose area is approximately equal to a square of side x:

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r = x/2 + (x/2 · √2 - x/2)/3 .

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They also give particular solutions to certain indeterminate equations, though without any hint at the method by which they arrived at them.²²

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²² For specific aspects of the geometry of the Śulbasūtras one should consult, in addition to the books and articles previously cited, those listed in the article on Āpastamba in CESS A1, 50a, as well as R. C. GUPTA, “Baudhāyana’s Value of √2,” ME 6 (1972), B 77–79, and R. P. KULKARNI, “The Value of π Known to Śulbasūtrakāras,” IJHS 13 (1978), 32–41.

c3.34

The Baudhāyanaśulbasūtra in the edition by Prakash and Sharman contains ten adhyāyas divided into 21 khaṇḍas (4, 3, 2, and 6 khaṇḍas in the first four adhyāyas respectively, 1 khaṇḍa in each of the remaining six adhyāyas); these correspond to three adhyāyas in Thibaut’s edition:

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| Thibaut | Prakash and Sharman |
| *adhyāya* 1 | *adhyāya* 1 = *khaṇḍas* 1–4. |
| *adhyāya* 2 | *adhyāya* 2 = *khaṇḍas* 5–7. |
| *adhyāya* 3 | *adhyāyas* 3–10 = *khaṇḍas* 8–21. |

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The total number of *sūtras* is 519. The work begins with definitions of the measures used in the constructions and the geometrical propositions employed in designing them; there follows a description of the spatial relations in different altars; and finally there come the rules for constructing the *gārhapatyaciti* and *chandaściti* (*khaṇḍa* 7), the *śyenacit* (*khaṇḍas* 8–9), the *vakrapakṣa vyastapuccha* (*khaṇḍas* 10–11), the *kaṅkacit* (*khaṇḍa* 12), the *ajalacit* (*khaṇḍa* 13), the *praügacit* (*khaṇḍa* 14), the *ubhayataḥ praüga* (*khaṇḍa* 15), the *rathakracit* (*khaṇḍa* 16), the *droṇacit* (*khaṇḍas* 17–18), the *śmaśānacit* (*khaṇḍa* 19), and the *kūrmacit* (*khaṇḍas* 20–21). The Baudhāyana was commented on by Dvārakānātha Yajvan,²³ whose *Śulbadīpikā* was composed after Āryabhaṭa, and by Veṅkateśvara Dīkṣita, who wrote his *Śulbamīmāṃsā* in the Vijayanagara kingdom in about 1600.

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²³ CESS A 3, 123a.

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The Āpastamba, which is divided into six *paṭalas* and twenty-one *adhyāyas* (3 *adhyāyas* in *paṭalas* 1, 3, and 5, 4 in *paṭalas* 2, 4, and 6), presents essentially the same material as does the Baudhāyana, though with fewer variations on the basic forms of the altars, in 223 *sūtras*. It was commented on by Kapardisvāmin,²⁴ who apparently lived before 1150 (Kapardibhāṣya); by Karavindasvāmin,²⁵ who has followed Kapardisvāmin (*Śulbapradīpikā*) and quotes from the Āryabhaṭīya; by Gopāla²⁶ (*Rahasyaprakāśa*); and by Sundararāja, the famous Kerala astronomer of the end of the fifteenth century (Sundararājīya).

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²⁴ CESS A 2, 19b.

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²⁵ CESS A 2, 24a.

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²⁶ CESS A 2, 130b.

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The first section of the Kātyāyana, which consists of ninety *sūtras*, is divided into six *kaṇḍikās*. It teaches, in language often identical with or similar to the Āpastamba’s, the geometrical propositions (including the first detailed instructions for determining the cardinal directions from the shadows cast by a gnomon), the measures employed in constructing the altars, and their spatial relationships. The second section, which consists of more than 40 verses, repeats some of the material given in the *sūtra* section, and adds to it descriptions of the rope, the gnomon, and the expert builder of altars. There are commentaries on the Kātyāyana by Karka,²⁷ who wrote before 1260/1270 (Karkabhāṣya); by Rāma Vājapeyin, who composed three of them²⁸ (the *Śulbavārttika*, a metrical gloss on the Kātyāyana and on the Karkabhāṣya, in 1434; the Śulbavārttikaṭīkā, a commentary on the preceding; and the Śulbasūtravṛtti, a prose commentary based on the Śulbavārttika) at Naimiṣa in Uttarapradeśa; and by Mahīdhara²⁹ at Vārāṇasī in 1590 (Śulbasūtravivaraṇa).

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²⁷ CESS A 2, 24a.

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²⁸ S. L. KATRE, “Three Works by Rāma Vājapeyin Pertaining to Kātyāyana’s Śulbasūtra,” PAIOC 13 (1946), 72–78.

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²⁹ CESS A 4.

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Finally, the Mānava, which is divided into seven khaṇḍas, describes the rope and the gnomon, the measures, the spatial relations of altars, the sacrificial fees, and the method of constructing the suparṇaciti. The closely related Maitrāyaṇīya, which is divided into four khaṇḍas, arranges this material in a different fashion. The Mānava was commented on by Śivadāsa at Vārāṇasī, the Maitrāyaṇīya by his younger brother, Śaṅkarabhaṭṭa; the two brothers both quote from Rāma Vājapeyin, and therefore lived after 1450.

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