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The Science of the Shulba: A Study in Early Hindu Geometry c20.s4: Other Approximate Values of √2

The Science of the Śulba: A Study in Early Hindu Geometry c20.s4

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Other Approximate Values of √2

c20.s4.1

From the *Mānava Śulba* we obtain certain other noteworthy approximations to the value of √2. By way of some calculations in that work are employed the relations: {lrl}
(1) & 40² + 40² \!\!&= 56² / (2) & 4² + \ 4² \!\!&= (5 2/3)²

c20.s4.2

From these we easily derive the values, {lrl} (1·¹) & \surd2 \!\!&= 7/5 = 1·4 / (2·1) & \surd2 \!\!&= 17/12 = 1·4166\dots

c20.s4.3

It may be noted that 7/5 is the third convergent of √2 expressed as a continued fraction ¹ and the value 17/12 is its fourth convergent. What is still more noteworthy is the fact that the former value is not derivable from the series for √2 stated before. This latter is, however, the eighth convergent of the continued fraction for √2: $577/408 = 1 + ⅓ + 1/3.4 - 1/3.4.34.$

c20.s4.4

¹ √2 = 1 + 1/(2+) 1/(2+)

c20.s4.5

These facts will lead one strongly to suspect if the rudiments of the theory of continued fractions were known to the early Hindus. At any rate, these are very remarkable cases of coincidence.

c20.s4.6

There are certain other rough values of √2 to which we shall refer shortly.

c20.s4.7

*Approximation to the Value of √5.*

c20.s4.8

There seems to have been a serious attempt, though without much success, to find an approximation to the value of the surd √5. The occasion was to define clearly the relative positions of the three principal and oldest known fire-altars, *viz.*, the *Gārhapatya*, *Āhavanīya* and *Dakṣiṇa*. Baudhāyana's rules to determine their positions are these:

c20.s4.9

"With the third part of the length (*i.e.*, the distance between the *Gārhapatya* and *Āhavanīya*) describe three squares closely following one another (from the west towards the east); the place of the *Gārhapatya* is at the north-western corner of the western square; that of the *Dakṣiṇāgni* is at its south-eastern corner; and the place of the *Āhavanīya* is at the north-eastern corner of the eastern square." ¹

c20.s4.10

¹ *BŚl*, i. 67.

c20.s4.11

"Or else divide the distance between the *Gārhapatya* and *Āhavanīya* into five or six (equal) parts; add (to it) a sixth or seventh part; then divide (a cord as long as) the whole increased length into three parts and make a mark at the end of two parts from the eastern end (of the cord). Having fastened the two ends of the cord (to the two) poles at the extremities of the distance between the *Gārhapatya* and *Āhavanīya*, stretch it toward the south, having taken it by the mark and fix a pole at the point reached. This is the place of the *Dakṣiṇāgni*." ²

c20.s4.12

² *BŚl*, i. 68.

c20.s4.13

" 'Or else increase the measure (between the *Gārhapatya* and *Āhavanīya*) by its fifth part; divide (a cord as long as) the whole into five parts and make a mark at the end of two parts from the western extremity (of the cord). Having fastened the two ties at the ends of the east-west line, stretch the cord towards the south having taken it by the mark and fix a pole at the point reached. This is the place of the *Dakṣiṇāgni*.' "¹

c20.s4.14

¹ *Ibid*, i. 69.

c20.s4.15

The second is also given by Āpastamba.² A rule leading to the same result as the first one above, though defined differently, is stated by Kātyāyana ³ and Manu.⁴ Kātyāyana has specified the relative positions of the three fire-altars also in a new way :

c20.s4.16

² *ĀpŚl*, iv. 4.

c20.s4.17

³ *KŚl*, i. 29.

c20.s4.18

⁴ *MāŚl*, iii.

c20.s4.19

" 'Divide the distance between the *Gārhapatya* and *Āhavanīya* into six or seven parts; add a part; then divide (a cord) equal to the total increased length into three parts, etc.' "⁵

c20.s4.20

⁵ *KŚl*, i. 27.

c20.s4.21

The rest of this rule is the same as the latter portion of the second rule above and hence need not be mentioned.

c20.s4.22

Fig. 77

c20.s4.23

Let b denote the distance between the *Gārhapatya* and *Āhavanīya*, that is, AG. Then from the different specifications given above we obtain the following values for AD and GD: $AD = b/3√ 5, ⅘b, 7/9b, 16/21b, 18/25b$ $GD = b/3√ 2, ⅖b, 7/18b, 8/21b, 12/25b$

c20.s4.24

If it be assumed that the relative positions of the three fire-altars were meant to be the one and the same, in all cases though expressed differently, then we shall have the following approximations to the values of √ 5 and √ 2: $√ 5 = 2⅖, 2⅓, 2 2/7, 2 4/25,$ $= 2·4, 2·333, 2·285, 2·16.$ $√ 2 = 1⅕, 1⅙, 1 1/7, 1 11/25,$ $= 1·2, 1·166, 1·142, 1·44.$

c20.s4.25

Since according to modern calculation √ 2 = 1·414213 and √ 5 = 2·23607, none of the above values can be said to be a fair approximation, perhaps except the values √ 5 = 2 2/7 and √ 2 = 1 11/25 which are correct up to the first place of decimals.

c20.s4.26

*Evaluation of Other Surds.*

c20.s4.27

In the *Mānava Śulba*, we find results leading incidentally to the evaluation of two other surds: $36² + 90² = 97²$ $5² + 61² = (7⅞)².$

c20.s4.28

Whence we easily obtain $√ 29 = 5 7/18 = 5·3888$ $√ 61 = 7⅚ = 7·8333$

c20.s4.29

Now correct up to three places of decimals √29 = 5·385... and √61 = 7·810... Hence the above approximations, specially the first one, may be said to be fair for ordinary purpose.

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