Pramāṇa · Search · About · API

The Science of the Shulba: A Study in Early Hindu Geometry c20.s1: Approximate Value of √2

The Science of the Śulba: A Study in Early Hindu Geometry c20.s1

Language
English

Approximate Value of √2

c20.s1.1

Baudhāyana and Āpastamba say:

c20.s1.2

"Increase the measure (of which the *dvi-karaṇī* is to be found) by its third part, and again by the fourth part (of this third part) less by the thirty-fourth part of itself (*i.e.*, of this fourth part). (The value thus obtained is called) the *saviśeṣa*."⁴

c20.s1.3

⁴ “प्रमाणं तृतीयेन वर्धयेत्तच्च चतुर्थेनात्मचतुस्त्रिंशोनेन। सविशेषः।”—*BŚl*, i. 61-2; *ĀpŚl*, i. 6.

c20.s1.4

Kātyāyana defines the rule in nearly identical words.⁵ Thus if d be the *dvi-karaṇī* of a, that is, if d be the side of a square whose area is double that of the square on a, then, according to the rule, $d = a + a/3 + a/3.4 - a/3.4.34.$

c20.s1.5

⁵ “करणीं तृतीयेन वर्धयेत्तच्च स्वचतुर्थेनात्मचतुस्त्रिंशोनेन सविशेष इति विशेषः।”—*KŚl*, ii. 13.

c20.s1.6

Now, it has been stated before, the diagonal of a square is its *dvi-karaṇī*. So this rule gives the relation between the diagonal and side of a square. Indeed the above rule is particularly meant to define that relation. Thus we get $√ 2 = 1 + ⅓ + 1/3.4 - 1/3.4.34.$

c20.s1.7

In terms of decimal fractions, this works out √ 2 = 1·4142156. According to modern calculation √ 2 = 1·414213. Thus it is clear that the ancient Hindus attained, a very remarkable degree of accuracy in calculating an approximate value of √ 2.

Linked texts (3)