Context of Development
In Indian civilization, we saw that Geometry originated in a very remote age in connection with the construction of the fire altars for the Vedic rituals. In the course of time, geometry grew beyond its original altar-specific purposes or the bounds of practical utility and began to be cultivated as a science for its own sake.
This happened in the Vedic Era when different schools of geometry were founded. More notable ones amongst them were the schools of Baudhāyana, Āpastamba and Kātyāyana. Though the geometrical propositions treated in all of them were almost the same, and there were many things common in the methods of their solution, still there were other things to distinguish one school from another. Even in the solution of elementary propositions such as the construction of a square, rectangle or an isosceles trapezium, different schools had preferential liking for differential methods. The difference appears most marked in the solution of the problems of the division of figures.
The word śulba (or sulva) means a cord or rope and sūtra describes the style of the writing, a compressed aphoristic presentation in which all inessential words (including, often, the verbs) are dropped. The geometry described in these “Cord Sutras” is the geometry that can be done with (inextensible) cords. The two basic elements out of which the geometry is built up are therefore the circle (produced by fixing one end and taking the other end of a stretched cord round) and the straight line (produced by stretching the cord end to end). In other words, they deal with ‘compass and ruler’ geometry.So far seven different Śulbasūtra texts have been identified by scholars.
Of them, the Baudhāyana-śulbasūtra is considered to be the most ancient one (prior to 800 BCE). This assessment is based upon the style, completeness, and certain archaic usages that are not that frequently found in later texts. Baudhāyana-śulbasūtra also presents a very systematic and detailed treatment of several topics that are skipped in later texts. It is made up of three chapters constituting about 520 sūtras (113 + 83 + 323). It is quite remarkable to just know the contents of the Baudhāyana-śulbasūtra.
Table: Topics covered in the Baudhāyana-śulbasūtra
In the last article, we saw that the large altars of which the fundamental one was of the shape of a falcon, had to be built with 200 bricks. While this was mainly enabled by Śulba-sūtras, it is important to understand some selective mathematical breakthroughs achieved much ahead of the rest of the word by the Vedic civilization of India. In the Vedic Era, the interesting geometrical achievements made are the following ﹣
Construction of rectilinear (square, trapezia, etc.) and curvilinear (circles, vedis, etc.) geometrical objects
Enunciation of geometric principles and practical application of them
Transformation of one geometrical object into another by applying these principles
Obtaining the value of surds by means of geometrical construction
Estimating the value of surds in the form of a sequence of rational numbers
Different methods proposed by scholars to arrive at these expression for the value of surds ﹣ in particular √2 and Pi
In the forthcoming articles we shall delve deep into the two key mathematical aspects of this era ﹣theorem of Śulba and how √2 and Pi were born.