The Science of the Shulba: A Study in Early Hindu Geometry c15.124

c15.124

Now tracing the early history of the rational rectangle (15, 36, 39), we find it first in the *Taittirīya Saṃhitā* (*c.* 3000 B.C.)² in connexion with the *Mahāvedī*. It has then reappeared in the *Kāṭhaka Saṃhitā*,³ *Maitrāyaṇī Saṃhitā*,⁴ *Kapiṣṭhala Saṃhitā*,⁵ and *Śatapatha Brāhmaṇa*.⁶ It should perhaps be noted that in these works only the sides (15, 36) have been expressly mentioned, but not the diagonal 39. This non-mention of the diagonal has led some modern writers to suspect if the property 15² + 36² = 39² was at all known in the time of the *Saṃhitā* and *Brāhmaṇa*.¹ But such a suspicion seems to be quite unwarranted. For even in later times Baudhāyana has not mentioned the diagonal of the rational rectangles enumerated by him. From a thorough discussion of the point, Bürk concludes: "After all these no doubt can exist regarding the fact that the rational right-angled triangle with perpendicular sides 15 and 36 was really known in the time of the *Taittirīya Saṃhitā* and the *Śatapatha Brāhmaṇa* and was employed in the construction of the *Saumikī-vedi* as in the *Āpastamba Śulba Sūtra* v. 1 and 2."² Such was also the opinion of Cantor.³ Their conclusion will be further corroborated by what has been shown just above about the special sanctity acquired by this rational rectangle amongst the followers of the Āpastamba school. Our interpretation of the word *jñeyābhiḥ* occurring at the end of Āpastamba's enunciation of the theorem of the square of the diagonal of a rectangle (pp. 132 f.) also will lead one strongly to the same conclusion.