The Science of the Shulba: A Study in Early Hindu Geometry c9.61

c9.61

It will be noticed that every one of the above propositions is demonstrable. Branding of them as postulates raises the important question of the character of the early Hindu geometry as regards the matter of demonstration. Of course the propositions of the *Sulba* are not proved after the manner of Euclid by purely deductive reasoning. On the other hand it is not wholly empirical without any semblance of demonstration. In fact we find a kind of proof in case of the propositions of the subtraction of one square from or its addition with another square and the mensuration of an isosceles trapezium. After the enunciation as a general proposition of the theorem of the square of the diagonal, Baudhāyana observes that the truth of it will be "realised" in case of certain rational rectangles enumerated. This is really an attempt for a kind of demonstration. What is much more noteworthy in this connexion is the fact that after a description of the geometrical construction for a proposition the Śulbakāras are often found to have remarked *sa samādhiḥ* or "This is the construction."¹ The significance of such an observation is obvious. It emphasizes that the construction which was required to be made, has been thus actually made, and indeed corresponds to the expression *Quod Erat Faciendum* (or "What it was required to do") occurring at the end of a proposition of Euclid's *Elements*. Further it discloses a rational and demonstrative attitude of the mind of the early Hindu geometer. With reference to a similar remark occurring in the works of the celebrated Hindu mathematician of the twelfth century of the Christian era, Bhāskara II, Hankel observed: "The small word 'see' along with the figure together with the necessary auxiliary lines supplies the Brāhmaṇas with the 'proof of the Greeks' concluding with solemn words 'what was to be proved.' All that a practised mind could recognise by means of assiduous consideration of a figure was admitted as certain."¹