[Pramāṇa](https://indike.org/pramana/) › [Studies](https://indike.org/pramana/ins) › [Science, technology and education](https://indike.org/pramana/ins.science) › [The Science of the Shulba: A Study in Early Hindu Geometry](https://indike.org/pramana/ins.science.datta-1932) › c13 # The Science of the Shulba: A Study in Early Hindu Geometry c13: Chapter VIII. Areas and Volumes The Science of the Śulba: A Study in Early Hindu Geometry c13 - **Language:** English ## Contents - [c13.s1 Figures with a given area](https://indike.org/pramana/ins.science.datta-1932.c13.s1) — 21 parts - [c13.s2 Volume of a Prism or Cylinder](https://indike.org/pramana/ins.science.datta-1932.c13.s2) — 2 parts - [c13.s3 Approximate Volume of the Frustum of a Pyramid](https://indike.org/pramana/ins.science.datta-1932.c13.s3) — 15 parts ### Chapter VIII. Areas and Volumes ### [c13.2](https://indike.org/pramana/ins.science.datta-1932.c13.2) The unit of area is defined by Āpastamba thus: ### [c13.3](https://indike.org/pramana/ins.science.datta-1932.c13.3) "By means of a measure is produced a measure."¹ That is, the unit of surface measure or area is the area of a square on a side of unit length. ### [c13.4](https://indike.org/pramana/ins.science.datta-1932.c13.4) ¹ \*ĀpŚl\*, iii. 4. ### [c13.5](https://indike.org/pramana/ins.science.datta-1932.c13.5) In the \*Śulba\* we find express rules for the mensuration of squares and isosceles trapeziums only. It is certain that mensuration of certain other elementary figures, such as triangles and rectangles, was also known in that time. The area of the circle was only roughly approximate. Other kinds of rectilinear figures, particularly the Fire-altars of various shapes enumerated before, were used to be mensurated by breaking them up into elementary triangles, squares and rectangles. ### [c13.6](https://indike.org/pramana/ins.science.datta-1932.c13.6) \*The number of square units in the area of a square is obtained by multiplying the number of linear units in a side by itself.\* ### [c13.7](https://indike.org/pramana/ins.science.datta-1932.c13.7) Āpastamba and Kātyāyana state: ### [c13.8](https://indike.org/pramana/ins.science.datta-1932.c13.8) "As many units of a measure as are in a cord, so many rows (or series) of squares (of that measure) there will be in a square on that cord as a side."² ### [c13.9](https://indike.org/pramana/ins.science.datta-1932.c13.9) ² \*ĀpŚl\*, iii. 7; \*KŚl\*, iii. 9. ### [c13.10](https://indike.org/pramana/ins.science.datta-1932.c13.10) FIG. 54 ### [c13.11](https://indike.org/pramana/ins.science.datta-1932.c13.11) This result was of course generalised, so that the area of any square was used to be calculated by multiplying its side by itself. ### [c13.12](https://indike.org/pramana/ins.science.datta-1932.c13.12) \*The number of square units in the area of a rectangle is obtained by multiplying together the numbers of linear units in the length and breadth of the rectangle.\* ### [c13.13](https://indike.org/pramana/ins.science.datta-1932.c13.13) Fig. 55 ### [c13.14](https://indike.org/pramana/ins.science.datta-1932.c13.14) Let ABCD be a rectangle whose length AB contains p units and whose breadth AD has q units. On dividing AB and AD and drawing parallel lines, it is found that the rectangle ABCD is divided into pq unit squares. Therefore its area is pq units of area. ### [c13.15](https://indike.org/pramana/ins.science.datta-1932.c13.15) It should be noted that we do not find in the \*Śulba\* any explicit rule for the mensuration of a rectangle. But there is no doubt that it was used to be done in the above way. Similarly from the method of constructing the \*ubhayī\* bricks (Fig. 7) which are of the shape of a scalene triangle, and of the parallelograms (Figs. 32-5; \*cf.\* also Fig. 14), it is clear that the following formulae also were known: ### [c13.16](https://indike.org/pramana/ins.science.datta-1932.c13.16) Area of a triangle = ½ (base) × (altitude), Area of a parallelogram = (base) × (altitude). ### [c13.17](https://indike.org/pramana/ins.science.datta-1932.c13.17) \*The area of a trapezium.\* ### [c13.18](https://indike.org/pramana/ins.science.datta-1932.c13.18) How to find the area of a trapezium has been demonstrated by Āpastamba in the course of determination of the area of the \*Mahāvedi\* which is of the shape of an isosceles trapezium whose altitude, face and base are respectively 36, 24 and 30 padas (or prakramas). He says: ### [c13.19](https://indike.org/pramana/ins.science.datta-1932.c13.19) "The \*Mahāvedi\* measures (in area) one-thousand less twenty-eight (square) padas. Draw a straight line from the south-eastern corner (of the \*vedi\*) to a point 12 padas towards the south-western corner. Place the portion thus cut off on the other (\*i.e.\*, northern) side of the \*vedi\* after inverting it. It (the \*Mahāvedi\*) will then become a rectangle. After that construction the area will be apparent."¹ ### [c13.20](https://indike.org/pramana/ins.science.datta-1932.c13.20) ¹ \*ĀpŚl\*, v. 7. ### [c13.21](https://indike.org/pramana/ins.science.datta-1932.c13.21) Fig. 56 ### [c13.22](https://indike.org/pramana/ins.science.datta-1932.c13.22) Let \*ABCD\* be the isosceles trapezium. Cut off WH = EB. Join \*BH\*. Place the triangle \*BHC\* after inversion in the position \*DGA\*. Then, the rule says, as is also obvious, ### [c13.23](https://indike.org/pramana/ins.science.datta-1932.c13.23) The area of the trapezium \*ABCD\* \begin{align\*} &= the area of the rectangle GBHD, \\ &= DH × BH, \\ &= 1/2\ (AB + DC) × BH, \\ &= 1/2\ (face + base) × altitude. \end{align\*} ### [c13.24](https://indike.org/pramana/ins.science.datta-1932.c13.24) This result also follows at once from the method indicated in the \*Śatapatha Brāhmaṇa\* and by Baudhāyana for the transformation of a square or a rectangle into an isosceles trapezium.² ### [c13.25](https://indike.org/pramana/ins.science.datta-1932.c13.25) ² \*Vide supra\*, p. 90f. ### Figures with a given area ### [c13.s1.1](https://indike.org/pramana/ins.science.datta-1932.c13.s1.1) For the construction of an altar of proper size and shape, as prescribed by the Holy Scriptures, it is sometimes necessary to describe a rectilinear figure having a given area. For instance, it is prescribed that the \*vedi\* of the \*Pitṛyajña\* must be a square of an area equal to one-ninth of that of the \*Mahāvedi\*.¹ Now the area of the \*Mahāvedi\* is given to be 972 square padas. So the problem becomes to construct a square having an area of 108 square padas. Again it is said that the \*Sautrāmaṇikī-vedi\* is of the form of an isosceles trapezium having an area equal to the third part of the \*Mahāvedi\*, that is, to 324 square padas. According to Āpastamba and Kātyāyana, this isosceles trapezium must be similar to the shape of the \*Mahāvedi\*. This case will be treated separately under "Similar figures." But Baudhāyana seems to have left the shape of the trapezium unrestricted. Hence there arises the problem: to find an isosceles trapezium having an area of 324 square padas. According to the tradition of some schools, it is necessary to construct square-shaped altars of area varying from 1½ to 6½ square puruṣas.² ### [c13.s1.2](https://indike.org/pramana/ins.science.datta-1932.c13.s1.2) ¹ \*BŚl\*, i. 81. ### [c13.s1.3](https://indike.org/pramana/ins.science.datta-1932.c13.s1.3) \*To construct a square having an area of 108 square padas.\* ### [c13.s1.4](https://indike.org/pramana/ins.science.datta-1932.c13.s1.4) Baudhāyana gives the following method: ### [c13.s1.5](https://indike.org/pramana/ins.science.datta-1932.c13.s1.5) "One-third producer (\*i.e.\*, the side of a square whose area is one-third the area) of the square made with the third part of the \*Mahāvedi\* is that (the side of the \*Paitṛkī-vedi\*). Its area is the ninth part of the area (of the \*Mahāvedi\*)."³ ### [c13.s1.6](https://indike.org/pramana/ins.science.datta-1932.c13.s1.6) ³ \*BŚl\*, i. 82; compare \*Paṇḍit\*, X (O. S.). p. 46f ### [c13.s1.7](https://indike.org/pramana/ins.science.datta-1932.c13.s1.7) 108 = 324/3 = 18²/3. So the required square will be one-third of a square on a side 18 padas long. The method for this latter construction has been taught before. Let AB be a straight line 18 padas long. Divide it into three equal parts. Let AC be one such part. Describe the square ACDE. Join AD. Describe a circle with ### [c13.s1.8](https://indike.org/pramana/ins.science.datta-1932.c13.s1.8) ² \*BŚl\*, ii. 14; \*ĀpŚl\*, viii. 3. ### [c13.s1.9](https://indike.org/pramana/ins.science.datta-1932.c13.s1.9) Fig. 57 ### [c13.s1.10](https://indike.org/pramana/ins.science.datta-1932.c13.s1.10) centre A and radius AD cutting AB at F. Join EF. Then EF is a side of the square having an area of 108 square padas. For ### [c13.s1.11](https://indike.org/pramana/ins.science.datta-1932.c13.s1.11) EF² & = EA² + AF², / & = EA² + AD², / & = 3AC², / & = 1/3 AB². / \therefore EF² & = 108 square padas. ### [c13.s1.12](https://indike.org/pramana/ins.science.datta-1932.c13.s1.12) This construction is the same as to find a square three times a square of 36 sqaure padas. For 108 = 3 × 6². If 108 would have been a square number or expressible as the sum of the two square numbers, the solution would have of course been very easy. ### [c13.s1.13](https://indike.org/pramana/ins.science.datta-1932.c13.s1.13) \*To construct an isosceles trapezium having an area of 324 square padas.\* ### [c13.s1.14](https://indike.org/pramana/ins.science.datta-1932.c13.s1.14) Baudhāyana says: ### [c13.s1.15](https://indike.org/pramana/ins.science.datta-1932.c13.s1.15) "If a square be formed with the third part of the \*Mahāvedi\*, its sides will be each 18 padas long. Then by making it longer on one side and shorter on the other, the sides should be determined optionally as necessary."¹ ### [c13.s1.16](https://indike.org/pramana/ins.science.datta-1932.c13.s1.16) ¹ \*BŚl\*, i. 86-7. ### [c13.s1.17](https://indike.org/pramana/ins.science.datta-1932.c13.s1.17) The rule does not explain clearly how the sides of the square formed with the third part of the \*Mahāvedi\*, are to be varied, so that the modified figure may assume the shape of an isosceles trapezium, but still retaining the same area. It seems, however, that the altitude is left unchanged, only the face and base being varied. ### [c13.s1.19](https://indike.org/pramana/ins.science.datta-1932.c13.s1.19) Fig. 58 ### [c13.s1.20](https://indike.org/pramana/ins.science.datta-1932.c13.s1.20) Let \*ABCD\* be the square whose side \*AB\* is 18 padas long. Suppose \*EFGH\* to be the modified form. Also suppose EF=18x and HG=18y. Since the area must remain the same, we must have $18((18x+18y)/2)=324,$ or $x+y=2.$ ### [c13.s1.21](https://indike.org/pramana/ins.science.datta-1932.c13.s1.21) Thus we can easily obtain any number of isosceles trapeziums having the same altitude and area. ### [c13.s1.22](https://indike.org/pramana/ins.science.datta-1932.c13.s1.22) We have explained before Baudhāyana's method for the transformation of a square (or a rectangle) into an isosceles trapezium which shall have a given face. That is different from the one supposed here. But this method has one advantage over that; it does not disturb the east-west line and so its adoption, we think, to have been more probable. If the altitude also vary, we get an indeterminate equation of the form $z(x+y)=2$ where 18z is the altitude of the resulting isosceles trapezium. ### Volume of a Prism or Cylinder ### [c13.s2.1](https://indike.org/pramana/ins.science.datta-1932.c13.s2.1) We have so far treated of the shape and size of the base of the various kinds of Fire-altars and have observed that the size in all cases has been prescribed to be the same, namely, 7½ square puruṣas. Every Fire-altar, save and except the one of the shape of the cemetery (\*Śmaśāna-cit\*), has a perfectly horizontal surface whose height is prescribed to be one \*jānu\* (=32 aṅgulis) for the first construction, two \*jānus\* for the second construction, and so on. Again the number of layers of bricks at successive constructions increases as multiples of five; and the shape and size of any layer of a particular Fire-altar at any construction is the same as those of its base. Hence all the Fire-altars are right prisms or circular cylinders. And the early Hindu geometers evidently knew the formula ### [c13.s2.2](https://indike.org/pramana/ins.science.datta-1932.c13.s2.2) $Volume of a prism or cylinder=(base)×(height).$ ### Approximate Volume of the Frustum of a Pyramid ### [c13.s3.1](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c13.s3.3) ¹ The quotation is from \*TS\*, v. 4. 11.3. ### [c13.s3.4](https://indike.org/pramana/ins.science.datta-1932.c13.s3.4) ² \*BŚr\*, xvii. 30. ### [c13.s3.5](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c13.s3.6) ¹ \*BŚl\*, iii. 253-4 ; compare also Thibaut, \*Śulvasūtra\*, p. 40. ### [c13.s3.7](https://indike.org/pramana/ins.science.datta-1932.c13.s3.7) The height of this Fire-altar has been specified by Baudhāyana thus : ### [c13.s3.8](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c13.s3.9) ² \*BŚr\*, xvii. 30. ### [c13.s3.10](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c13.s3.12) ³ \*BŚl\*, iii. 266-8.. ### [c13.s3.13](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.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](https://indike.org/pramana/ins.science.datta-1932.c13.s3.15) Volume of the frustum = ((a+a')/2)((b+b')/2)h. ## Linked texts (5) - [Āpastambaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.apastamba.3.4) — Āpastamba, Sanskrit - [Baudhāyanaśrautasūtra](https://indike.org/pramana/in.vedanga.kalpa.srauta.baudhayana.17.30) — Baudhāyana, Sanskrit - [Taittirīyasaṃhitā](https://indike.org/pramana/in.veda.samhita.krsnayajur*taittiriya.5.4.11.3) — Sanskrit - [Kātyāyanaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.katyayana.3.9) — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit - [Baudhāyanaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.baudhayana) — Baudhāyana, Sanskrit --- Source: [https://indike.org/pramana/ins.science.datta-1932.c13](https://indike.org/pramana/ins.science.datta-1932.c13) · Pramāṇa