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The Science of the Shulba: A Study in Early Hindu Geometry c13.s3: Approximate Volume of the Frustum of a Pyramid

The Science of the Śulba: A Study in Early Hindu Geometry c13.s3

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Approximate Volume of the Frustum of a Pyramid

c13.s3.1

The cemetery-shaped Fire-altar is in fact a frustum of a pyramid. Its base is an isosceles trapezium whose dimensions are stated by Baudhāyana to be as follows:

c13.s3.2

"He should construct the *Śmaśāna-cit* ('the Fire-altar of the shape of the cemetery') who desires: 'May I gain prosperity in the *Pitṛloka* (the world of the fathers):'¹ it has been (taught). Six puruṣas are the length of the east-to-west line, three the length of the eastern side and two the length of the western side. This is the body (of that Fire-altar)."²

c13.s3.3

¹ The quotation is from *TS*, v. 4. 11.3.

c13.s3.4

² *BŚr*, xvii. 30.

c13.s3.5

It should be remarked that the unit puruṣa in this passage is not the ordinary puruṣa of 120 aṅgulis, but a reduced unit whose length is equal to a side of a square equivalent to one half of the ordinary square puruṣa.¹ So the area of the trapezium is equal to 15 reduced square puruṣas or 7½ ordinary square puruṣas.

c13.s3.6

¹ *BŚl*, iii. 253-4 ; compare also Thibaut, *Śulvasūtra*, p. 40.

c13.s3.7

The height of this Fire-altar has been specified by Baudhāyana thus :

c13.s3.8

“Its (*Śmaśāna-cit*) measure is, when neck-deep on the east, navel-deep on the west ; when navel-deep on the east, knee-deep on the west ; when knee-deep on the east, ankle-deep on the west ; and when ankle-deep on the east, it is on a level with the ground on the west.”²

c13.s3.9

² *BŚr*, xvii. 30.

c13.s3.10

Notwithstanding this difference in height on the two sides of the Fire-altar, its cubic content has to be kept intact. To effect that in practice the following device is generally adopted :

c13.s3.11

“ Increase the (usual) vertical measure of the Fire-altar by its one-fifth. Then divide the total height into three parts, and make bricks with (the height equal to) the fourth, ninth or fourteenth part of two of these parts. Construct four, nine or fourteen layers with them. Divide the remaining part (having constructed it with one layer of bricks of the height of one-third the total height) by the diagonal (plane) inclined down towards the west and remove the half (*i.e.*, the upper portion).”³

c13.s3.12

³ *BŚl*, iii. 266-8..

c13.s3.13

It has been prescribed that the nth construction of the Fire-altar shall have a height of n jānus and comprise of 5n layers of bricks. Increasing the height by its one-fifth, we get 6n/5 jānus. Two-thirds of them gives 4n/5 jānus. Up to this height, the altar is constructed in (5n-1)

c13.s3.14

layers, so that the height of each brick is equal to (5n-1)th part of 4n/5. One-third of the increased altitude is 2n/5 jānus. Next layer of the Fire-altar is constructed with bricks of height 2n/5 jānus. Then the upper portion of this layer is cut off by the diagonal plane as directed. Hence the altitude of the altar is now 6n/5 jānus on the east, 4n/5 jānus on the west ; so that its average altitude is (6n/5+4n/5)/2 or n jānus. This device will be easily recognised to be based on the following approximate formula for calculating the volume of the frustum of a pyramid: If (a, b) be the length and breadth of the rectangular base of the solid, (a', b') the corresponding sides of the face parallel to it and h the height, then

c13.s3.15

Volume of the frustum = ((a+a')/2)((b+b')/2)h.

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