The Sulbasutras-1 In this edition we shall look at Śulba-sūtras which form the key evidence for significant development of mathematics in the Vedic Era.

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Overview of Śulbasūtras

The class of texts called Śulbasūtras form a part of Kalpa, which is one of the Vedāṅgas. The term Vedāṅga is used to refer to six branches of knowledge namely śikṣā, vyākaraṇaṃ, kalpaḥ, niruktaṃ, jyotiṣaṃ and chandaḥ. In ancient times, all these branches used to be studied by every Vedic priest either after completing studies of the Veda, or simultaneously along with it.

The Vedic priests had developed a class of manuals that would assist them in the construction of altars (called Vedis) used for performing sacrifices. The word śulba stems from the root śulb which means ‘to measure’. Since all the measurements were done using ropes or chords in the very early times — traces of which can be found in practice even today — it seems the word in due course was synonymously employed to refer to the chords themselves.

 

Some of the geometrical constructions such as the śyenaciti prescribed by the Śulbakāras (the authors of the Śulbasūtras) are quite complex. In fact, there are a number of constraints that need to be fulfilled in the construction of śyenaciti such as, the number of bricks in each layer should be constant (200), the area of all the bricks put together must be equal to a specified number, and so on.  

The citīs or vedis cannot be simply constructed without having a mastery over certain techniques that include the procedures for determining the east-west direction at a given location, for drawing straight lines that are at right angles to each other, for constructing a square whose side is surd times an integer, for finding the area of certain geometrical objects, and more such mathematical principles. We shall see some of these impressive principles and methods.

Finding the cardinal directions

Having chosen the location at which the sacrificial altar is to be constructed, the first thing that needs to be done is the determination of the east-west direction at that location. In fact, determining the exact east-west line at a given location, is a prerequisite for all constructions, be it a residence, a temple, a sacrificial altar or a fire-place. The procedure for determining it is described by an ingenious way in the Śulbasūtras:

समे शङ्कुं निखाय शङ्कुसम्मितया रज्ज्वा मण्डलं परिलिख्य यत्र लेखयोः शङ्क्वग्रच्छाया निपपति तत्र शङ्कू निहन्ति, सा प्राची।

(Kt. Su. I 2)

Fixing the gnomon (śaṅku) on level ground and drawing a circle with a cord measured by the gnomon, he fixes pins at points on the line (of the circumference) where the shadow of the tip of the gnomon falls. That is the east direction (prācı̄).

Asking the question as to why perform this experiment with śaṅku 5 in order to determine the direction, and not simply look at the sunrise and sunset and be with it, the commentator Mahı̄dhara observes that “Since the rising points are many, varying from day to day, the [cardinal] east point cannot be known [from the sunrise point]. Therefore it has been prescribed that the east be determined by fixing a śaṅku.” 

Fundamentally, owing to the tilt of the earth and the latitudinal position, one may observe that the rising point of the Sun has a northward and a southward movement across a year, with the days of equinox as the inflection points. This makes the determination of the cardinal directions very difficult. This is a concept we shall delve further into while looking at ayanāmśa in Jyotiṣa

We shall learn in detail about the breakthroughs achieved in geometrical theorems in Śulba-sūtras in our next article.