The Sulbasutra- Baudhayana, Apastamba, Katyayana and Manava with Text, English Translation and Commentary (An Old and Rare Book)

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In the last century, and at the beginning of the present, George Thibaut and A. Bark brought to the notice of the scholarly world the mathematical significance of two zina texts, namely the Sulba-strar by Baudhayana and Apastamba. These two works are manuals for the construction of various types of altars for fire-sacrifice, an ancient cult fully established in the time of the Samhits. Their mathematical importance lies in a precise statement of the squared relationship of the two sides and the diagonal of a rectangle, the realization of the irrationality of numbers like 2, efforts to determine their approximate values, formulation of rules for combining rectilinear figures, transforming squares into rectangles, triangles, trapeziums and circles and vice versa, the use of simple fractions and approximate values of. All these arose from the mensuration of laying altars of a fixed square area but of different shapes and filling them with a fixed number of bricks necessarily of different geometrical shapes. The Greek tradition of the development of geometry from the more ancient Egyptian mensuration, the rope- stretcher's art, appears to have its counterpart in the scriptural tradition of the halbtras. That similar tradition obtained in other culture areas has become increasingly clear with advancing knowledge of the cuneiform texts and ancient Chinese mathematical sources. I am happy that the authors of the present monograph have presented the full critical texts and translations of the four alba-stras by Baudhayana, Apastamba, Karyayana and Manava and modern commentaries on them. A carefully prepared introduction has dealt with: the genesis of these texts as part of the ritual literature (Kalpasatras) constituting one of the six Veddigar; the vexed question of the antiquity of these sitrar; the philosophical significance of performing fire sacrifices on various types of altars; and other reated questions. Regarding the dating of Baudhayana, Manava, Apastamba and others a balanced view has been maintained between the opinions of European sanskritists like Buhler, Keith, Macdonell, Renou and Filliozat and those of Indian scholars like Kane and Ramgopal. Though the date of Baudhayana, the earliest of sitra writers, still varies over a range of two to three centuries, there is no doubt that the tradition of altar construction and the related sacrifices goes back to the period of the Saphitts and the Brahmapas antedating the satras, as the numerous passages in the Taittiriya, Maitrayaai, and Kepijthala Samhitas and the Satapatha and other Brahmanas amply indicate.

The present monograph on the sulbasitrar by Baudhayana, Apastamba, Katyayana and Mánava was planned by the History of Science Unit established at Calcutta by the History of Science Board of the Indian National Science Academy (then called the National Institute of Sciences of India) and continued by the National Commission for the Compilation of History of Sciences in India. The Unit under the supervision of one of us (S.N.S.), with which the other author (A.Κ.Β.) was then associated as a Research Scholar, functioned at first at the Asiatic Society and subsequently at the Calcutta office of the Academy in the same premises of the Society. This happy location permitted the free and frequent use of the Society's rich library and its manuscript holdings. The authors also had the opportunity of consulting the MS No. Th. 184 of the Manasa-fulba kept at the National Library, Calcutta and a microfilm transcription of the MS. No. 536 of the same text available at the Bombay Branch of the Asiatic Society. We express our sincere thanks to the Librarians of the Asiatic Society and the National Library, Calcutta and the Bombay Branch of the Asiatic Society for permitting us to make use of the aforesaid manuscripts in their respective holdings. We also place on record that late Nagendranath Vedantatirtha, Curator of the Asiatic Society rendered us his ungrudging help in elucidating some of the difficult passages of the sutras, particularly of the Manava- fulba. We further express our grateful thanks to the Academy for providing a Senior Research Fellowship to one of us (A.K.B.) and contingent grants to meet the expenses of the project.

The Vedangas, 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 (siksa), ritual (kalpa), grammar (vyakarana), etymology (nirukta), metrics (chanda) and astronomy (jyotisa). 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 sutra 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 Vedangas and was also adopted by the writers of the Arthasastra, the Kamasastra, the Natyasastra and so on.

The Kalpasatras, concerned principally with the rituals constituting the chief contents of the Brahmanas, are supposed to be the first Vedangas to have received systematic treatment. The Kalpasatras are again available in four different classes, e.g. the frauta, the grhya, the dharma and the fulba. The Srautasūtras deal with frauta- sacrifices abundantly discussed in the Brahmagas 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 Sulbasutras which are sometimes classified as a separate branch of the Srautasūtras, but which are often found attached to the Srautastras.

The Sulbasutras 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 fulba (also spelt as fulva) means a 'cord', a 'rope' or a 'string', and its root sulb signifies 'measuring' or 'act of measurement. 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 sulbavids.