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The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava: With Text, English Translation and Commentary c5: Introduction

The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava: With Text, English Translation and Commentary c5

Language
English

Contents

Introduction

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The *Vedāṅgas*, that important group of literature often referred to as the appendages of the *Vedas*, constitute an important source in the history of science in ancient India. This is evident from such subjects as phonetics (*śikṣā*), ritual (*kalpa*), grammar (*vyākaraṇa*), etymology (*nirukta*), metrics (*chanda*) and astronomy (*jyotiṣa*). These branches of study arose within the Vedic schools themselves as a necessary condition for mastering the *Vedas*. This class of literature was written in the *sūtra* or aphoristic style, a form of expression characterized by great precision, brevity and economy of words, the like of which is not met with in the entire literature of the world. The style has been developed to sum up only the pith of the learning in short sentences using nouns often compounded at great length and avoiding the use of verbs as far as possible. The style became a dominant feature of the various branches of the *Vedāṅgas* and was also adopted by the writers of the *Arthaśāstra*, the *Kāmaśāstra*, the *Nāṭyaśāstra* and so on.

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The *Kalpasūtras*, concerned principally with the rituals constituting the chief contents of the *Brāhmaṇas*, are supposed to be the first *Vedāṅga* to have received systematic treatment.a The *Kalpasūtras* are again available in four different classes, e.g. the *śrauta*, the *gṛhya*, the *dharma* and the *śulba*. The *Śrautasūtras* deal with *śrauta*-sacrifices abundantly discussed in the *Brāhmaṇas* and are naturally concerned with direction for the laying of the sacrificial fires for the fire-sacrifice (*agnihotra*), the new and the full-moon, the seasonal, the *soma* and other sacrifices. These are, as Winternitz has pointed out, our most important source for the understanding of the Indian sacrifice-cult. Through their preoccupation with the laying and construction of the various sacrificial altars and fires, these also constitute an important source of Vedic, and possibly the earliest, mathematics in India. Of special importance as far as concerns mathematics, geometry in particular, are the *Śulbasūtras* which are sometimes classified as a separate branch of the *Śrautasūtras*, but which are often found attached to the *Śrautasūtras*.

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The *Śulbasūtras* are of special importance because these deal specifically with rules for the measurements and constructions of the various sacrificial fires and altars and consequently involve geometrical propositions and problems relating to rectilinear figures, their combinations and transformations, squaring the circle and circling the square as well as arithmetical and algebraic solutions of problems arising out of such measurements and constructions. The word *śulba* (also spelt as *śulva*) means a 'cord', a 'rope' or a 'string', and its root *śulb* signifies 'measuring' or 'act of measurement'.b It is interesting to note that, among the Egyptians, geometry of surveying was considered to be the science of the 'rope-stretchers' (*harpedonap'tae*) who thus appear to be the Egyptian counterpart of the Indian *śulbavids*.

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a Winternitz, I, pt. 1, 237; also see Sen, chapter on 'A Survey of Source Materials', *A Concise History of Science in India*, p. 23-24

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b Datta (2), 8.

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Quite appropriately, therefore, the *Śulbasūtras* represent the *Brāhmaṇa* geometry or mensuration, the *śulba-vijñāna*, as mentioned in the *Mānava* and other *śulbas* and in their commentaries. It is also evident that the *śulbavid*, the expert geometer was held in high esteem in the learned priestly circles.

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Since *Śulbasūtras* form part of the *Śrautasūtras*, inspite of their separate classification under the *Kalpasūtras*, one would expect a *śulba* section attached to each *Śrautasūtra*. And there are *Śrautasūtras* belonging to all the four *Saṃhitās*. But what we possess today are a small number of *Śulbasūtras* attached to the *śrautas* belonging only to the various schools of the *Yajurveda*. Of them the Black Yajurvedins of the *Taittirīya* school were the most active and prolific producers of the *sūtra* text, and it is no wonder that the most comprehensive, as also the largest number of, *Śulbasūtras* were produced by the scholars of this school, such as Baudhāyana, Āpastamba, Vādhula and Hiraṇyakeśin. Of the *Maitrāyaṇī* school, Mānava and Vārāha are known to have written works on the *śulba*. In this subject *Kāṭhaka-Kapiṣṭhala* school is represented by Laugākṣi. Of the White Yajurvedins, Kātyāyana, another prolific writer of the *sūtras* is credited with a small but scientifically executed *śulba* work. Maśaka, the *sūtrakāra* of the *Sāmaveda* school probably also compiled a *śulba* text attached to the *Śrautasūtras* of that school. The initiative of the Yajurvedins in producing works of this kind is not surprising when we bear in mind that they were the principal custodians of the knowledge of sacrificial formulas and specialized in sacrificial performances.

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Of these various *śulba* works, those due to Baudhāyana, Āpastamba, Kātyāyana and Mānava are best known, and others are known through references. We tried to locate manuscripts of some of these latter ones, but without success and had to be content with the *Śulbasūtras* by the four aforesaid scholars. The need for a monograph dealing with the different *Śulbasūtras*, in one volume so as to present a comprehensive view of the subject, inspite of the excellent studies of some of these texts by distinguished scholars, has long been felt, and has been further reinforced by the fact that the works of Thibaut and Bürk published towards the end of the 19th and the beginning of the present century are now very difficult to obtain. That the works of Thibaut and Bürk constituted the main inspiration of our humble efforts presented in these pages need hardly be overestimated.

Place and Time of the Authors of the Śulbasūtras

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There is a good deal of uncertainty and differences of opinion about the place and time of the *śulbakāras*. The *Śrautasūtras* and the *Śulbasūtras* are silent about these questions as are other Vedic and post-Vedic texts. Georg Bühler who considered the question of date and time of Baudhāyana and Āpastamba was inclined to believe that both of them hailed from the Andhra country.a He argued that the followers of Baudhāyana and Āpastamba had lived in south India since early times, that Baudhāyana manuscripts had been found in the south, and that the *Mahārṇava*, an early work mentioned Andhra country as the native place of Āpastamba. Bühler also pointed out that the name of Āpastamba had been found on several land-grants of the south and that both Baudhāyana and Āpastamba referred in their *Dharma-sūtras* to the manners and customs of the people of their times inhabiting the northern parts of India,—a reference rather unusual for authors hailing from the north. Recently Ramgopal has shown that, in their *Śrautasūtras* and *Dharmasūtras*, Baudhāyana gave ample evidence of his close familiarity with *Āryāvarta*, the doab between the Ganges and the Yamuna and its surroundings and Āpastamba with Kurupāñcāla country and its vicinity.a About the places of origin of Kātyāyana, Mānava and other *śulbakāras*, nothing definite can be said.

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a Bühler, *SBE*, 14, pp. xliii; 2, pp. xxx.

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If their native places are open to question so are their dates. Bühler placed Āpastamba around the fifth century B.C. but did not himself consider this date as conclusive or anything more than tentative. He was only definite that both Baudhāyana and Āpastamba had lived before the third century B.C. Between these two *sūtrakāras*, all scholars agree, Baudhāyana is anterior to Āpastamba.

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Instead of working downwards from the high antiquity of Baudhāyana and Āpastamba whatever that antiquity might be, other scholars have preferred to consider the dates of later writers like Pāṇini, Kātyāyana, and Patañjali and work upwards to arrive at the dates of Baudhāyana and Āpastamba. Here again, Pāṇini's date itself is debatable and no better datable than other ancient texts. Nevertheless, scholars are generally agreed that Pāṇini most probably lived in the fourth century B.C. Keith argued that Pāṇini's date depended essentially on the date to be assigned to the *Mahābhāṣya* of Patañjali. Renou and Filliozat and Keith placed Patañjali's date around *c.* 150 B.C. on the ground that the *Mahābhāṣya referred to a sacrifice by Puṣyamitra who reigned around c.* 185 or 178 B.C.b In his *Veda of the Black Yajurveda*, Keith has observed that by 140 B.C. Pāṇini's work attained a commanding position as is evident by the clear proof of the elaborate way in which it was commented upon in Dākṣāyaṇi's *Saṃgraha*, by Kātyāyana and others and concluded that Pāṇini's date could hardly by any chance be later than 300 B.C., nor there could be any reason to deny that he might have lived about 350 B.C.

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This brings us to the date of Kātyāyana who flourished after Pāṇini and before Patañjali. Admitting that there is no direct proof, Keith suggested 250 B.C. as the most probable date for Kātyāyana. Eggeling, on the other hand, placed Kātyāyana in the fourth century B.C. after considering a still earlier date for Pāṇini.c Macdonell, in his *Bṛhaddevatā*d favoured a date *c.* 350 B.C. for Kātyāyana, more or less in agreement with Eggeling.

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Bühler, Keith and other scholars all agreed that the irregular forms persistently used by Baudhāyana and Āpastamba in composing the whole texts of the *Śrauta-*, the *Gṛhya-* and the *Dharma-sūtras* could have hardly been possible after Pāṇini's grammar reached its accepted position. Bühler, therefore, considered Āpastamba anterior to Pāṇini by about 150-200 years, which would place the former in the fifth-sixth century B.C. In his *Āpastamba Śrautasūtra*, Garbe, in general agreement with Bühler, assigned Āpastamba to the fifth century B.C. Keith considered such a high date for Āpastamba improbable from consideration of language and would not take him beyond say 300 or 350 B.C. By Keith's own agreement with Bühler that Āpastamba could be anterior to Pāṇini and his statement that there is absolutely no conclusive evidence of the date of these early *sūtrakāras*, it is somewhat baffling to understand how Āpastamba could be placed around *c.* 350 B.C.

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a Ramgopal, 93-100.

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b Renou and Filliozat, 86-91; Keith (2), 426-28; Keith (1), Preface clxviii to clxxii.

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c *SBE*, 12, xxxix.

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d I, p, xxii

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Regarding Baudhāyana and Mānava, both have been considered definitely older than Āpastamba, as their archaic style of writing would indicate. Mānava whose works were used by Āpastamba wrote more or less in the *Brāhmaṇa* style, and Richard Garbe considered his *sūtra* as the oldest, but that, as Keith points out, was before the text of Baudhāyana was known. Baudhāyana's archaic *Brāhmaṇa* style, is clearly discernible in his *uttarati* as also in all his *sūtra* writings. Considering all this, Keith's conservative estimate for Baudhāyana's date was fifth century B.C. while Caland was prepared to place him in the sixth century B.C. in his *Über das Rituelle Sūtra des Baudhāyana*.a Mānava doubtless should be placed somewhere between Baudhāyana and Āpastamba. P. V. Kane in his *History of Dharmasūtras* dated the *Śrautasūtras* of Baudhāyana, Āpastamba and Kātyāyana between 800 B.C. and 400 B.C. Ramgopal who considered the sequence between Baudhāyana and Āpastamba, Āpastamba and Pāṇini, Pāṇini and Kātyāyana and Kātyāyana and Patañjali, more or less agreed with Kane and concluded that the principal *sūtras* were composed between *c.* 800 B.C. and 500 B.C.

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The foregoing discussion will make it abundantly clear that we are still far from narrowing the date range differing by centuries and putting the early *sūtra* works on a firmer chronological basis. What is generally accepted is this relative chronological position in order of anteriority, e.g. Baudhāyana, Mānava, Āpastamba, Pāṇini, Kātyāyana. Hiraṇyakeśī, Vārāha, Vādhula and so on. This is also somewhat borne out by the tradition of the *Taittirīya* school which would place the early *sūtrakāras* in the following order of age: Baudhāyana, Bharadvāja, Āpastamba, Satyāṣāḍhā, Hiraṇyakeśin and Vaikhānasa. Whatever date one might wish to assign to the *Kalpasūtras*, the *Śrautasūtras* and the *Śulbasūtras*, we must agree with Thibaut that these *sūtras* only give a systematized account of sacrificial rites which had been practised during long preceding ages.b The rules for the sides of the various altars, detailed arrangement of the sacrificial ground, the positioning of the fires, altars, tents and sheds, the shapes of the fire-altars in the form of the falcon and other birds are all mentioned and discussed in various ways and at various places in the *Brāhmaṇas* as we shall see in what follows. The manner in which the measurements and transformations had to be carried out, in other words, the geometry and mensuration involved in their construction, it is true, are not discussed in the *Brāhmaṇas* and cannot be expected either in this class of literature. But there can hardly be any doubt that what Baudhāyana, Āpastamba, Mānava, Kātyāyana and others tried to codify in a systematic manner in their *Śulbasūtra* manuals must have for a long time formed the common property of all *adhvaryus* and priests specialized in the performance of sacrificial rites.

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a pp. 7 ff.

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b Thibaut (1) pt. 1, 270

References to Sacrificial Altars and Fires in Saṃhitās and Brāhmaṇas

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The Vedic cult knew no temple. The ceremonies were performed either in the sacrificer's house or on a nearby plot of ground. This ground must be flat and covered with grass. For the performance of sacrifices, certain *vedis* and *agnis* had to be constructed. A *vedi* is a specified raised area on which the sacrifice is to be performed and on which persons performing the ceremony, namely the sacrificer, the *hotā*, the *adhvaryu*, the *ṛtvik* and others are to take their seats. Some of the main *vedis* include the *mahāvedi*, the *darśapūrṇamāsa vedi*, the *śautrāmaṇi vedi*, the *paitṛki vedi*, the *uttara vedi*, and the *aśvamedha vedi*. An *agni* is a raised altar, generally made of bricks, for keeping the fire. The fire-altars were of two types, the *nitya* (or perpetual) and the *kāmya* (or optional). The three perpetual fire-altars were the *gārhapatya*, the *āhavanīya* and the *dakṣiṇāgni* and meant for daily sacrifices. The *kāmya agnis* intended for wish fulfilment, included the *śyenacit*, the *vakrapakṣa-vyasta-puccha-śyena*, the *kaṅkacit*, the *alajacit*, the *praugacit*, the *ubhayata praugacit*, the *droṇacit*, the *rathacakracit*, the *śmaśānacit*, the *kūrmacit*, and so on.

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The mention of the *gārhapatya* fire occurs at several places in the *Ṛg-veda*.a In another place, there is a reference to three places (*trisadhasthe*) of the *agni*, implying the *gārhapatya*, the *āhavanīya* and the *dakṣiṇāgni*.b A reference to the form of *śyena* is found in the *Ṛg-veda* where *agni* is frequently called a bird (*vayas*).c The *Ṛg-veda* also contains references to altars and their constructions of which a few examples are as follows: ‘Let the priests decorate the altar (*vedi*), let them kindle the fire to the east',d ‘May the measure-lengths (*yūpa*) of the sacrificial posts be to our felicity; may the sacred grass (*oṣadhi*) be (stream) for our happiness; may the altar (*vedi*) be (raised for) our happiness’.e

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In the *Taittirīya Saṃhitā*,f it is so ordained that the *gārhapatya citi* is to be constructed with 21 bricks arranged in an identical manner. Similar passages are found in the *Maitrāyaṇī Saṃhitā*, *Kāṭhaka Saṃhitā* and the *Kapiṣṭhala Saṃhitā*.g In the *Taittirīya Saṃhitā*, it is said,h "He who constructs (the *gārhapatya citi*) for the first time should construct in five layers . . . . . He who constructs for the second time should construct in three layers . . . . . ”. The *saumiki vedi* (also called *mahāvedi*), as described in the *Taittirīya Saṃhitā*, is in the form of an isosceles trapezium having its face 24 *prakramas* (or *padas*) long, base 30 *prakramas* and altitude 36 *prakramas*.a The measures are also given in other *saṃhitās*.b Although elaborate descriptions of rites and ceremonies in connection with the construction of the various altars such as the *dārśapaurṇamāsa vedi*, the *uttara vedi*, the *āgnīdhriya* the *hotrīya*, the *mārjālīya*, the *sadas* (tent), the *uparavas* etc. are found in the *Taittirīya* and other *saṃhitās*, their measurements and constructional details are rarely given.

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a *gārhapatyena santya ṛtunā* (*RV.* 1.15.12); *asthurino gārhapatyāni santu* (*RV.* 6.15.19); *iha priyam prajayā to saṃṛdhyatāmasmiṅgṛhe gārhapatyāya jāgṛhi ; enā patyā tanvaṃ saṃ sṛjasvadhā jīvṛi vidathamā vadāthaḥ* (*RV.* 10.85.27)

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b *jajñasya ketum prathamam purohitamagniṃ naratriṣadhasthe samīdhīre* (*RV.* 5.11.2).

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c *RV.* 1. 164.52; 10.14.5. Compare also with 1.58.5; 1.41.7; 2.2.4; 6.3.7; 10.8.3 etc.

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d *araṃ kṛṇvantu vedim samagnimindhatām puraḥ* (*RV* 1.170. 4).

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e *śaṃ na soma bhavatu brahma śaṃ naḥ śaṃ no grāvāṇaḥ śamu santu yajñāḥ śaṃ naḥ svarūṇām mitayo bhavantu śaṃ naḥ prasvaḥ śamvasu vediḥ* (*RV* 7. 35. 7).

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f *pratiṣṭhyā vā ekaviṃśaḥ pratiṣṭhā gārhapatyā ekaviṃśasvaiva pratiṣṭhyām gārhapatyamanu prati tiṣṭhati* (*Taitt. S.* 5. 2. 3. 5).

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g *Mait. S.* 3. 2. 3; *Kāṭh. S.* 20. 1; *KPS.* 32. 3.

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h *pañcacitikam cinvīta prathamam cinvānaḥ . . . . . tricitikam cinvīta dvitīyam cinvāna . . . . . *(*Taitt. S.* 5. 2. 3. 6).

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The standard form of an optional fire-altar was that of a certain bird. This bird was called *śyena* in the *Taittirīya Saṃhitā*c and *suparṇa garutman* or well winged eagle in the *Vājasaneya Saṃhitā*.d The spatial magnitudes of the falcon-shaped fire-altar were also given in almost all the earlier works from the *Taittirīya Saṃhitā* onwards. The measurements were made with units like *aratni*, *vyāma*, *puruṣa* etc. The area on which these fire-altars were drawn covered 7½ sq. *puruṣas*.e A complete list of the various *kāmya agni* together with a statement of the objects for the attainment of which each of them has to be constructed, as found in the *Taittirīya Saṃhitā*,f is given below :

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| *Agni* | Desired objects |
| *Chandaścit* (in the form of bird ) | Desiring cattle |
| *Śyenacit* ( -do- ) | „ heaven |
| *Kaṅkacit* ( -do- ) | „ „ |
| *Alajacit* ( -do- ) | Desiring support from the heaven |
| *Prauga* (in the form of an isosceles triangle) | Annihilation of rivals |
| *Ubhayata prauga* (in the form of a rhombus) | „ |
| *Rathacakracit* (in the form of a chariot wheel) | „ |
| *Droṇacit* (in the form of a trough) | Gaining food |
| *Paricāyyacit* (in the form of a circle) | Desiring a region |
| *Śmaśānacit* (in the form of a pyre) (isosceles trapezium) | Attaining the place where the forefathers have gone (*pitṛloka*). |

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a *triṃśat padāni paścāt tiraścī bhavati ṣaṭtriṃśat prācī, caturviṃśatiḥ purastāt, tiraścī daśadaśa saṃpadyate (Taitt. S.* 6. 2. 4. 5).

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b *Mait. S.* 3. 8. 4 ; *Kāṭh. S.* 25. 3 ; *KPS.* 3. 8. 6.

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c *Taitt. S.* 5. 4. 11. 1.

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d *Vāj. S.* 12. 4.

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e *Taitt. S.* 5. 2. 5. 1 & *Mait. S.* 3. 2. 4.

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f *Taitt. S.* 5. 4. 11. 1-3. *chandaścitaṃ cinvīta paśukāmaḥ paśavo vai chandāṃsi paśumāneva bhavati, śyenacitaṃ cinvīta suvargakāmaḥ śyenovai vayaśāṃ pratiṣṭhā śyena eva bhūtvā suvargaṃ patati, kaṅkacitaṃ cinvīta yaḥ kāmayeta śīrṣaṇvānamum-ṣmilloke syāmiti śīrṣaṇvānevāmumṣmilloke bhavatyalajacitaṃ cinvīta, catuḥsītaṃ pratiṣṭhākāmascatasro diśo dikṣveva prati tiṣṭhati, praugacitaṃ cinvīta bhrātṛvyavān praiva bhrātṛvyān nudata ubhayataḥ-praugaṃ cinvīta yaḥ kāmayeta prajātān bhrātṛvyān....rathacakracitaṃ cinvīta grāmakāmo śmaśānacitaṃ cinvīta yaḥ kāmayeta pitṛloka......(Taitt. S.* 5. 4. 11. 1-3).

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As regards the height of the *agni* and the number of bricks to be used in its construction, *Taittirīya Saṃhitā*a observes:

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“He should pile (the fire) of a thousand (bricks) when first piling (it); this world is commensurate with a thousand; verily he conquers this world. He should pile (it) of two thousand, when piling a second time, the atmosphere is commensurate with two thousand; verily he conquers the atmosphere. He should pile (it) of three thousand; verily he conquers the yonder world. Knee-deep should he pile (it) when piling for the first time, verily with the *gāyatrī* he mounts this world; naval-deep should he pile (it) when piling for the second time, verily with the *triṣṭubh*, he mounts the atmosphere; neck-deep should he pile (it) when piling for the third time, verily with the *jagatī*, he mounts the yonder world”.

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An expert in this science was called *agnicit* (constructor of the *agni*). The term appears in the *Taittirīya Saṃhitā*b and the *Maitrāyaṇī Saṃhitā*.c

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In the *Yajurveda*, we find an elaborate and tedious rite of the *agnicayana* or the construction of the fire-altar, associated with highly speculative philosophy.d The same mystic significance is found in different schools of this *Saṃhitā*, e.g. the *Taittirīya*, the *Maitrāyaṇīya*, the *Kāṭhaka-Kapiṣṭhala* and the *Vājasaneya*. This shows that the *agnicayana* rite and its philosophy had already taken definite shape in the time of the *Yajurveda*. The existence of different masters of this science with independent views is also referred to in the *Taittirīya Saṃhitā*.e

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The relative positions of the three *nitya* fires, the *gārhapatya*, the *āhavanīya* and the *dakṣiṇāgni* are also described in the *Śatapatha Brāhmaṇa*.f The *gārhapatya* fire is represented like a man lying on his back with head ‘towards the east’. The first clear description of the *gārhapatya* as a circle of one square *vyāma* and of the *āhavanīya* as a square of the same size appears in the *Śatapatha Brāhmaṇa*.g The *gārhapatya* fire is to be constructed with 21 bricks.h

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In the *Śatapatha Brāhmaṇa*, the same measure of the *mahāvedi* as given in the *Taittirīya Saṃhitā* has been adopted.i The *kāmya agnis* such as the *suparṇa garutman*, the *droṇacit*, the *rathacakracit*, the *kaṅkacit*, the *praugacit*, the *ubhāyata praugacit* etc. have been described here.j These *kāmya citis* all measure 7½ sq. *puruṣas*. Regarding the areas of the fire-altars, the *Śatapatha Brāhmaṇa* observes, “According to one (school), *ekavidha agni* should be constructed first, then by an increment of one (*square puruṣa*) successively upto a construction of an unlimited size. But indeed the *agni* (or *Prajāpati*) was to be constructed first as *saptavidha* (7½ sq. *puruṣas* and then by the increment of one square *puruṣa*) in succession is to be made upto *ekaśatavidha* (101½ sq. *puruṣas*)”a Compare this with Baudhāyana’s and Āpastamba’s statements in their *śulbasūtras*b that one fold means 1½ sq. *puruṣas*, two-fold means 2½ sq. *puruṣas*, seven-fold means 7½ sq. *puruṣas* and so on. This has also been explained by other *śulbakāras*.

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a *sahasraṃ cinvīta prathamaṃ cinvānaḥ, sahasrasaṃmito vā ayaṃ loka imameva lokamabhi jayati dvisāhasraṃ cinvīta dvitīyaṃ cinvāno, dvisāhasraṃ vā antarikṣamantarikṣamevābhi jayati, trisāhasraṃ cinvīta tṛtīyaṃ cinvānastrisāhasro vā asau lokomumeva lokamabhi jayati jānudaghnaṃ cinvīta prathamaṃ cinvāno gāyatriyaivemaṃ lokamabhyārohati nābhidaghnaṃ cinvīta dvitīyaṃ cinvānastriṣṭuvaivāntarikṣamabhyārohati grīvādaghnaṃ cinvīta tṛtīyaṃ cinvāno jagatyairāmuṃ lokamabhyārohati* (Taitt. S. 5. 6. 8. 2-3).

c5.s2.27

b *Taitt. S.* 5. 2. 5. 5-6; *TS* 5. 7. 6. 1.

c5.s2.28

c *Mait. S.* 3. 4. 8.

c5.s2.29

d Keith ( 1 ), cxxv.

c5.s2.30

e *Taitt. S.* 5. 2. 8. 1-2; 5. 3. 8. 1; 5. 5. 2. 1.

c5.s2.31

f *Śat. Br.* 1. 7. 3. 23-25.

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g *Śat. Br.* 7. 1. 1. 37.

c5.s2.33

h *Śat. Br.* 7. 1. 1. 34.

c5.s2.34

i *Śat. Br.* 7. 1. 1. 1.

c5.s2.35

j *SBR.* 3. 5. 1. 1-6.

General Characteristics of Śulbasūtras, Their Geometry, Arithmetic and Algebra

c5.s3.1

In our section on ‘Commentaries’, we have discussed in detail the *sūtras* of the four texts by Baudhāyana, Āpastamba, Kātyāyana and Mānava. In presenting the texts, their translations and commentaries, we have not followed the chronological order in which these were written as would appear from our foregoing discussions, but in order of their importance and completeness.

c5.s3.2

Baudhāyana’s *sūtras* are not only the earliest but represent the most systematic, logical and detailed treatment of the subject inspite of their archaic and highly condensed *sūtra* style. It opens with the various units of linear measurements and then develops the geometry of rectilinear figures, triangles and circles, their transformations from one type to the other, methods of arriving at areas by combination or difference of given areas, the irrational number like √ 2 and the value of π albeit indirectly. Then the measurements of the three perpetual fires, the *gārhapatya*, the *āhavanīya* and the *dakṣiṇāgni*, the sacrificial altars such as the *mahāvedi*, the *uttara vedi* the *sautrāmaṇī vedi*, the *paitṛki vedi*, the *dārśapaurṇamāsika vedi*, the *paśubandha vedi*, various sacrificial fires such as the *dhiṣṇyas*, the *āgnīdhriya*, and the *mārjālīya*, the *sadas* tent, the *havirdhāna* shed for the placement of the soma carts and various pits like the *utkara*, the *uparava* and the *cātvāla* are given along with their relative distances from one another. Although the plan of the sacrificial ground and placement therein of the various fires, altars, tents, sheds and pits is not described in separate *sūtras*, the manner in which their measurements, relative distances and directions for their placements are given makes it quite easy for one to visualize the picture of the sacrificial ground and at once appreciate how closely and meticulously the *śulbavid* was following the time-honoured Vedic sacrificial practices. About the fire-altars for the various wish fulfilments (*kāmyaciti*), Baudhāyana gives the measurements, constructions and methods of laying bricks of different geometrical shapes in alternate layers following the strict injunction that the edges of bricks in contiguous layers must not meet. Such fire-altars include those in the form of a falcon, both rectilinear and with curved wings and extended tail, of a kite and *alaja* bird, of an isosceles triangle and a rhombus, of a chariot-wheel without and with spokes, of a square and circular trough, of a pyre, and finally one in the form of a tortoise. The facile use of geometry presented in the opening chapters is abundantly clear in the constructional procedures of these fire-altars.

c5.s3.3

a *taddhaike, ekavidhaṁ prathamam vidadhātyathaikottaramā parimitavidhānna tathā kuryāt saptavidho vā ‘agre prajāpatirasyjvata] . . . . . saptavidhameva prathamam vidadhītāthaikottaramaikaśatavidhādekaśatavidhaṁ tu nātividadhīta . . . . . /(*Śat. Br.* 10. 2. 3. 17-18).*

c5.s3.4

b *Bśl.* 5.1, 5.8. *Āśl.* 8.3.

c5.s3.5

Āpastamba follows more or less the same procedure and provides the same rules and techniques; but in our view, coming after Baudhāyana and with his text before him, he has not shown any improvement upon Baudhāyana. About the fire-altars, he has discussed different types of *śyenacit*, rectilinear as well as those with curved wings and extended tail and given different arrangements with different types of bricks, but has not gone into the details of other types which he mentions presumably because he has no alternative method of arrangement to suggest. As far as the use of terms and expressions are concerned, there is a remarkable resemblance between Baudhāyana and Āpastamba.

c5.s3.6

Kātyāyana’s treatment is succinct and systematic. He emphasizes the geometry behind the construction of altars and fires and gives a clear exposition of it. He deals with a few altars and *agnis* but refrains from considering the *kāmyacitis* as the latter are discussed separately in a chapter of his *Śrautasūtras*.

c5.s3.7

The *Mānava-śulbasūtra*, a part of the *Śrautasūtra* by the same author, although following the common tradition of the *śulbakāras*, gives methods and details often very difficult to comprehend. In many cases the details are either lacking or incomplete and can be understood only by reference to Baudhāyana, Āpastamba and Kātyāyana. To us the very arrangement and the treatment of the subject have appeared far from systematic. In our judgement, the work does not measure up to the standard attained by Baudhāyana, Āpastamba and Kātyāyana.

c5.s3.8

*Geometry*

c5.s3.9

Several methods of constructing a square on a given straight line have been given. Rectangles are rectilinear figures of which the two sides are different. For the construction of such rectilinear figures the squared relationship between the diagonal and the two sides has been given at various places in the *śulbasūtras*, which we summarize below:

c5.s3.10

a) n²+(¾n)² = (5/4n)² (*Bśl.* 1.5; *Mśl.* 1.11—1.12)
(i) n = 4, 4²+3² = 5² (*Bśl.* 1.13)
(ii) n = 12, 12²+9² = 15² (*Kśl.* 2.5)
(iii) n = 20, 20²+15² = 25² (*Āśl.* 5.3)
(iv) n = 16, 16²+12² = 20² (*Āśl.* 5.3)
b) n²+(5/12n)² = (13/12n)² (*Bśl.* 1.8; *Āśl.* 1.2; *Kśl.* 1.4)
(i) n = 1, 1²+(5/12)² = (13/12)² (*Bśl.* 1.8 *Āśl.* 1.2; *Kśl* 1.4-1.5)
(ii) n = 36, 36²+15² = 39² (*Bśl.* 1.13; *Āśl.* 5.4)
(iii) n = 188, 188²+(78⅓)² = (203⅔)² (*Āśl.* 6.5)
(iv) n = 6, 6²+(2½)² = (6½)² (*Āśl.* 6.6; *Mśl.* 2.4)

c5.s3.11

(v) n=5, 5²+(2 1/12)²=(5 5/12)² (*Āśl.* 6.7)
(vi) n=10, 10²+(4⅙)²=(10⅚)² (*Āśl.* 6.8)
(vii) n=27, 27²+(11¼)²=(29¼)² (*Āśl.* 7.3) (viii) n=18, 18²+(7½)²=(19½)² (*Āśl.* 7.1—7.2)
(ix) n=12, 12+5²=13² (*Bśl.* 1.13; *Āśl.* 5.4)
(x) n=96, 96²+40²=104² (*Mśl.* 1.4-1.6;)
c) 7²+24²=25² (*Bśl.* 1.13)
d) 8²+15²=17² (*Bśl.* 1.13; *Āśl.* 5.5)
e) 12²+35²=37² (*Bśl.* 1.13; *Āśl.* 5.5)
f) 1²+3²=(√10)² (*Kśl.* 2.4; *Mśl.* 3.5)
g) 2²+6²=(√40)² (*Kśl.* 2.5)
h) 1²+(√2)²+(√3)² (*Kśl.* 2.10)
i) na²=[((n+1)a)/2]²-[((n-1)a)/2]², where a= rational integer (*Kśl.* 6.7)
j) 6²+(4½)²=(7½)² (*Mśl.* 2.5)
k) 1²+(√10)²=11 (*Mśl.* 12.5)

c5.s3.12

*Theorem of Square on the Diagonal of a Square or a Rectangle*

c5.s3.13

The above-mentioned squared relationships are followed or preceded by the general statement that the diagonal of a rectangle produces by itself both (the areas) produced separately by its two sides. This is the so-called theorem known after Pythagoras. A question has often been asked whether such a definition resulted from empirical guess work or was based on a proof of some kind. Such questions are of course to be expected from scholars firmly entrenched in the Euclidean tradition in geometry. As is well known, mathematics, including geometry, in ancient India did not follow the Euclidean tradition, and usually gave the rules, leaving their proofs to be explained by the teachers to the pupils of their respective schools. Several scholars who discussed this issue of proof have shown that such proof is implicit in the very operations with rectilinear figures; we have discussed the matter in detail in our commentaries in connection with the rule and further elaboration here is unnecessary. George Sarton, while considering the question of Hindu influence on Pythagorean derivation of the theorem of squares, referred to Gaston Milhaud's claim that Pythagorean geometry may have been partly inspired by Hindu models. This argument was based on the high antiquity of Baudhāyana and Āpastamba. But Sarton, presumably on the basis of Keith's views on the dates of the *śulbakāras*, pleaded inability to accept such high antiquity and observed: 'It is highly probable that the *śulbasūtras* date from a period posterior to 500 B.C. and pre-Christian. They are probably post-Pythagorean'.a

c5.s3.14

This brings us to the important question of the origin of the theorems and also whether Pythagoras himself was the discoverer of it.

c5.s3.15

The tradition attributing the theorem to Pythagoras is due to Cicero (*c.* 50 B.C.) Diogenes Laertius (second century A.D.), Athenaeus (*c.* A.D. 300), Heron (third century A.D.), and Proclus (*c.*A.D. 460), and therefore started about five centuries after the death of Pythagoras. Junge pointed out that the Greek literature of the first five centuries after Pythagoras contained no mention of the discovery of this or any other important geometrical theorem by the great philosopher and furthermore emphasized uncertainties in the statements of Plutarch and Proclus. Although various attempts have been made to justify the tradition and trace the proof to Pythagoras, no record of proof has come down to us earlier than that given by Euclid (Theorem 47, BK 1). As to the relation 4²+3²=5² from which the theorem of rational triangle is derivable, very ancient Egyptian knowledge is attested by the Kahun papyrus of the twelfth dynasty (*c.* 2000 B.C.), but its association with rational triangles does not seem indicated in this or other Egyptian papyrii.a As to the antiquity of Pythagorean theorem in China, it is stated, though not proved, in the arithmetical classic *Chou Pei Suan Ching* (third or fourth century B.C.); the numerical relationship 4, 3 and 5 between the sides and the diagonal of a rational rectangle is also given in this text. The old Babylonians of the second millennium B.C. left records on their cuneiform tablets of similar squared relationships indicating practical use of the theorem of squares. No general statement in the form of a theorem is of course found. Neugebauer is of the view that Pythagoras derived his number theorem of the universe as well as the theorem known after his name from such Babylonian cuneiform tablets.b

c5.s3.16

a Sarton, I, 74-75.

c5.s3.17

As we have stated, methods have been given for transforming rectilinear figures from squares to rectangles, of transforming squares into circles, of developing isosceles triangle and rhombus from squares and so on. Various geometrical shapes like parallelograms, five-sided rectilinear figures are mentioned in various ways in connection with the construction of bricks with which to cover the sacrificial altars.

c5.s3.18

*Irrational numbers, π, Fractions, Surds*

c5.s3.19

What is of great significance is the treatment of irrational numbers like √2 and statement of their accurate values. The manner in which such accurate values were possibly obtained by the *śulbakāras* has been fully discussed in our notes. Neugebauer has shown that these values are identical with those found in certain Babylonian cuneiform texts, given in sexagesimal system. He tried to imply that the Indian value after all represented the Babylonian one expressed only in decimal system or more accurately in fraction. As we have shown, there is certainly no proof of such an assertion and the Indian value is certainly derivable from the methods contained in the *śulbasūtras* themselves.

c5.s3.20

In connection with the pits for stacking the sacrificial poles in, Baudhāyana has given the ratio of the circumference to the diameter as 3. We have shown from the rules for transforming squares into circles, as given by Baudhāyana, that the *śulbakāras* in all probability knew of more accurate value of π.

c5.s3.21

a Heath, (2), I, 352. Sen, Chapter on 'Mathematics', *A Concise History of Science in India*, 148-149,

c5.s3.22

b Neugebauer, 28-42 \* The numbering within parenthesis refers to Thibaut's edition as printed in the *Pandit*.

c5.s3.23

The *śulbakāras* were familiar with the use of fractions and manipulated with them in various ways, specially in connection with the construction of bricks for the fire-altars. The terms used by them are significant as well as interesting, of which few examples are given :

c5.s3.24

| *caturbhāgona* | =1-¼=¾ | (*Bśl.* 1.5) |
| *ardhāṣṭama* | =7½ | (*Bśl.* 5.1, 5.6) |
| *ardhadaśama* | =9½ | (*Bśl.* 5.1) |
| *ardhanavama* | =8½ | (*Bśl.* 5.1) |
| *caturtha-saviśeṣārdha* | =½ (¼√2) | (*Āśl.* 19.4) |
| *caturtha-saviśeṣa-saptama* | =1/7(¼√2) | (*Āśl.* 19.7) |

c5.s3.25

In our notes we have explained that elementary knowledge of operating with surds was also possessed by these geometers. It will certainly be not proper to say that the *śulbakāras* dealt with algebra as is known from later Indian mathematicians. But it cannot be denied that germs of algebraic equation are embedded in many of their rules and operations. Such is the case with quadratic equation as also with indeterminate equation of the first degree. As we have amply stressed, these texts were compiled primarily as manuals for the construction of sacrificial fires and altars. Geometry, mensuration, arithmetic, and germs of algebra came out only incidentally. Nevertheless, the gleams we obtain of their knowledge of these subjects can hardly fail to excite our admiration when we remember the time of their compilation.

Source Materials and Plan of Work

c5.s4.1

In our edition of the *Baudhāyana-śulba*, we have used Thibaut's edition as printed in the *Paṇḍit* and Caland's edition of the *śulba* attached to Baudhāyana's *Śrautasūtra*. Bürk's plan in the break-up of the *sūtras* and their numbering has been followed. This plan appeared to us systematic and logical inasmuch as the breaking up of the *sūtras* and their numbering were generally guided by considerations of self-contained statements. It may also be noted that Caland punctuated the *sūtras* from similar considerations although no numbering was used. We have, however, retained Thibaut's numbering within parenthesis in order that scholars already accustomed to Thibaut's edition may not experience any difficulty. Our edition of the *Āpastamba-śulba* is based on Bürk's *Das Āpastamba-śulbasūtra* and the Mysore edition of the same text. Bürk's arrangement and numbering have been mostly retained; only a few *sūtras* have been regrouped from considerations of self-consistency. In such regroupings also Bürk's number has been given within parenthesis. For the *Kātyāyana-śulba*, we have used the editions by Madana Pāṭhaka, the Kāśī Sanskrit series and the MS. No. G. 6145 of the Asiatic Society, Calcutta. In our edition of the *Mānava-śulba*, Van Gelder's edition of the *śulba* attached to the *Śrautasūtra* and the MS. No. Th. 184 of the National Library, Calcutta have been used and Gelder's arrangement and grouping retained.

c5.s4.2

As to commentaries, Dvārakānātha's *Śulbamīmāṃsā* on Baudhāyana, the commentaries of Kapardisvāmī, Karavindasvāmī and Sundararāja on Āpastamba, *Karkabhāṣya* and Mahīdhara's *Śulbasūtravṛtti* on Kātyāyana, and MS. No. 536 of the Bombay Branch of the Asiatic Society, a commentary on the *Mānava-śulba*, have been used. Other manuscripts used by previous editors like Thibaut, Bürk and Van Gelder have been referred to in the foot-notes.

c5.s4.3

We have given the texts, translations and our own commentaries in separate parts, always referring to the number used in our edition wherever necessary. In writing the commentaries again, the *sūtras*, singly as well as in groups, have been treated as found convenient for purposes of elucidation. In the case of the *Āpastamba-śulba*, a number of chapters dealing with the same topic, e.g. the construction of *śyenacit*, have been dealt with together for the same reason.

Linked texts (13)