top of page

Śulbasūtras

 

Introduction

The Sulbasutra, or ancient Indian geometry, principally consists of instructions for building fire altars, which are intrinsically linked to Vedic ceremonies in India. In common usage, the words Sulba and Rajju in Sanskrit mean the same thing: a cord or rope. The etymological significance of the name "Shulba" is "measuring" or "the act of measuring," as it is derived from the root "shulb," which means "to measure." (Datta 1932). The Shulbasutras are organized in a way that begins with the most basic geometric and arithmetic constructions, such as praci (East-West lines) and progresses to the specifics of how to make citis, and most crucially, citis with complex structures. All extant sulbasutras belong to Yajurveda.

 

Why Sulbasutra Geometry has immense importance in mathematics?

​

The wholeness and consistency of their geometrical results and constructions

Yajnavalkya claims in Shatapatha Brahmana 10.4.3.9

एषा हैव सा विद्या यदग्निः एतदु हैव तत्कर्म यदग्निः||Shatapatha Brahmana 10.4.3.9||

or "the fire altar represents both knowledge and ritual". (Kak, 2011). Shulbasutra mentions the importance of making accurate measurements for the construction of citis.  According to Baudhayana Shulbasutra, even a slight variation in the measurement can lead to an adverse effect on the purpose of sacrifice. There are exactly the right geometrical constructions to the precise degree of accuracy necessary for the shulbakaras to build the citis. Two points particularly stand out in Sulbasutra's mathematics: first, they offered the construction process an accuracy level suitable for the creation of citis. Additionally, they also offer approximate values of root 2, root 3, etc.

​

Beauty of fire-altars 

Each fire altar is made up of layers of bricks that have been artistically and neatly assembled. Not only simpler structures but complex one, such falcons with wings, a chariot wheel with spokes, or a tortoise with extended heads and legs, were made. The falcon is a majestic bird that can soar to heaven, the wheel is a representation of the "wheel of life," and the tortoise is a symbol of stability and perseverance in these elegant designs.

 

Sulba Sutras have a much deeper significance

One of the many significant phrases utilised in Sulbasutra Geometry that gives the Sutras new depth is the word "citi." Commonly translated as some sort of altar or platform for rituals (Agni) in the Sulba Sutras, the word is also related to the word for "awareness" (cit). It's also common to hear the term "Vedi," which refers to the land on which the citi is built. However, because Veda can also refer to "pure knowledge" or "full knowledge," the term "Vedi" is used to describe a highly enlightened individual. Third, the purusa is the height of a man with his arms outstretched or 120 angulas (another Vedic measurement system) in length (Baudhayana Sulba Sutra I, 1-21). However, purusa is defined as "the uninvolved witnessing quality of intelligence, the unified . . . self-referral state of intelligence at the basis of all creativity" (Price, J.F., 2000). Similarly, the term 'Atman' is used for the body of citi. It is the main part of citi excluding wings and tails which literally means the individual self or the eternal soul.  Therefore, the Sulba Sutras' extended function as a description of consciousness is not hard to infer.

There are ten Sulbasutras available, Baudhayana Sulbasutra(500 B.C.), Apastambha Sulbasutra(400B.C.), Hiranyakeshin Sulbasutra (375 B.C),  Manava Sulabasutra(450 B.C.),  Maitrayana Sulbasutra, Varaha Sulabsutra,  Vadhula Sulabsutra, Mashaka Sulabsutra, Vaikhanasha Sulabsutra(350 B.C.), and Katyayana Sulabasutra(200B.C). 

Linear Measurements
Screenshot_20230113_165556.png

Sulbasutras were meant to address the laws pertaining to mathematics and geometry in relation to the measurement of ritual enclosures and the building of various types of altars.   Fire altars were of two types namely, the perpetual (Nitya) and Optional (Kamya). And it is interesting to note that optional fires are constructed with different shapes but the covered area remains fixed for each citi 7.5 square purusha.

For construction of altars, Shulbakaras also prepare bricks of different sizes and shapes to cover a fixed area of 7.5 Purusha.  It was an arduous task, to get the desired shape with just 200 bricks more or less in the first construction. In order to do this,  they replace the original bricks by half, quarter, or one-eighth to fulfill the desired number of 200 bricks in each layer.  Adhya (or " the half") is a brick with the same shape as the half of a rectangle or square cut along its diagonal; padya (or " the quarter") is a brick with the same shape as a quarter of a square; and the formation of a ubhayi brick provides a source for the discovery of a rational scalene triangle. “ Eighth part of panchami  should be combined as there will be a brick having three corners” ( Datta,1932). The square bricks Chaturthi (one-fourth), Panchami(one-fifth), sashti(one-sixth), and their subdivided bricks were manufactured and each was named separately and details of dimensions both in rational and irrational numbers are given.  

Some bricks and their dimensions are listed below:

Caturthi (one-fourth, square brick, 30*30 angula)

  1. caturthi(square quarter) 30*30 angula

  2. ardha(triangular half) = 30*30*30 √2 angula

  3. trasra padya(triangular quarter)= 30* 15 √2* 15√2 angula

  4. caturasra padya(four-sided quarter)= 22(1/2) *  15* 15/2 *15 √2 angula

  5. hamshamukhi (pentagonal half brick) = 15 √2 *7(1/2) 15* 7(1/2)* 15 √2 angula

Panchami (one- fourth square brick, 1/5  purusha*1/5 purusha; 24 angula * 24 angula)

  1. panchami (one fifth square brick) = 24* 24 angula

  2. ardyardha-panchami(rectangular brick, side longer by one half) = 24 * 36 angula

  3. panchami-sapada(rectangular  brick, side longer by one quarter) = 24 * 30angula

  4. panchami -ardha(triangular half bricks) =  24*24*24√2 angula

  5. pnchami – padya(triangular  quarter bricks)=  24* 12√2*12√2 angula

  6. adhyardhardha(triangular half brick of adhyardha) =  36* 24* 12√13 angula

  7.  dirghapadya(triangular quarter bricks of adhyardha with large  base) = 36* 6√13*6√13 angula

  8. shulapadya(triangular quarter brick of adhyardha with shorter base) = 24*6√13*6√13 angula

  9. ubhayi (triangular brick when half brick of the above two are attached 30*12√2*6√13 angula

  10. panchami-ashtami (one eight triangular brick of Panchami) = 12*12* 12√2 angula[ Bag, 1990]

For the manufacturing of bricks of different shapes, Shulbakaras started with the construction of simple geometrical figures such as squares, circles, triangles, rectangles, rhombi, trapeziums, etc. And each brick of elemental structure is used for the construction of complex structures, viz, Kamya citis. The construction in Shulbasutra starts with the laying out of the east-west line which is usually termed as the line of symmetry in every geometrical construction in shulbasutras. We find from Sulbasutra, the (nitya) altar- Grhapatya must be of the form of a square, or circle the altar for Ahavaniya should be always square, and that of the Dakshina- semicircular. The area of each, however, must be the same and equal to one square vyama (1 vyama = 96 angula). Hence, the construction of these altars can be understood easily by understanding the construction methods of simple geometrical figures.

There are several forms of citis constructed for various purposes.

Perpetual (Nitya) Fire Altar
Optional (Kamya) Fire Altar

Importance in mathematics-

 Geometrical facts outlined in the construction methods-

  1. Baudhayana Shulbasutra insisted on the symmetrical construction of citi. For this purpose, he gave an example of the symmetrical arrangement of bones in the body of the animal.

  2. Theorems of the squares of the diagonal (different combinations of areas are based on this).

  3. Numerous rational right triangles and a few irrational ones with close approximations of their hypotenuses. (given in Maitrayana shulbasutra).

  4. A circle is the locus of points at a constant distance from a given point.( Baudhayana method of construction of squares).

  5. Drawing perpendicular with the help of intersecting circles (Baudhayana, Apastambha,  Katyayana )

  6.  The perpendicular bisector of a line is the locus of points equidistant from the two extremities of the line.

  7. In an isosceles triangle, the base is perpendicular to the line that runs from the triangle's vertex to its midpoint. 

  8. The tangent to a circle is perpendicular to the radius at the point of contact. (Apastambha Shulabsutra)

  9. Any number of discrete segments can be extracted from a finite straight line.

  10. The diagonal of the rectangle or a square bisect it.

  11.  The diagonals of the rectangle bisect one another and they divide the rectangle into 4 equal parts.

  12. The diagonals of the rhombus bisect each other at right angles (Katyayana shulbasutra)

  13.  The area of an isosceles triangle is equal to half of the area of the rectangle with sides equal to the base and altitude of the triangle.

  14.  The figure formed by joining the middle points of the adjacent sides of a square is itself a square and its area is half the area of the original square.[ Amma, 2007]

  15. A triangle can be divided into a number of equal and similar parts by dividing the sides into an equal number of parts and then joining the points of division two and two.

  16. An isosceles triangle is divided into two equal halves by the line joining the vertex with the middle point of the opposite side.

  17. A triangle formed by joining the extremities of any side of a square to the middle point of the opposite side is equal to half the square.

  18. A quadrilateral formed by the lines joining the middle points of the sides of a rectangle is a rhombus whose area is half that of the rectangle.

  19. A parallelogram and a rectangle which are on the same base and within the same parallels are equal to one another.

  20. The maximum square that can be described within a circle is the one that has its corner on the circumference of the circle. [Dutta, 1932]

 

Here, Geogebra applets and VBA (Visual basic for application) tools are used to represent some geometrical operations used by Shulbakaras to construct fire altars.

​

References

Amma, T.S., 1999. Geometry in ancient and medieval India. Motilal Banarsidass Publ..

Bag, A.K., 1990. Ritual Geometry in India and its Parallelism in other Cultural areas. Indian journal of history of science, 25(1-4), pp.4-19.

Datta, B.B., (1932). Ancient Hindu Geometry (The Science of the Sulba). Calcutta University. Reprint: Cosmo Publications, Delhi, 1993.

Kularia, D.P., Hooda, A. (2018). Baudhayana Sulbasutram. Abhishek Prakashan, Delhi.

Kularia, D.P., (2009). Katyayana  Sulbasutram. Devesh Publications, New Delhi. Reprint: 2018.

Kularia, D.P., Nandal, R. (2018). Apastambha Sulbasutram. Abhishek Prakashan, Delhi.

Kulkarni, R. P. 1983. Geometry According to Sulba Sutra. Vaidika Samsodhana Mandala. Pune.

Kulkarni, R. P. 1987. Layout and construction of citis. Bhandarkar oriental research institute. Poona.

Price, J.F., 2000. Applied Geometry of the Sulba Sutras. MAA NOTES, pp.46-58.

Sen SN, Bag A.K. 1983. The Sulbasutras, Indian National Science Academy, New Delhi.

Thibaut, G., 1875. The Śulvasútras. CB Lewis, Baptist mission Press.

Yogi, M.M., 1994. Vedic Knowledge for Everyone: maharishi vedic university-introduction. Maharishi Vedic University Press.

bottom of page