Geometry forms the foundation of the disciplines of architecture, town planning, design and mathematical thought and for the very same reason has found copious elaboration among ancient civilisations. However, the Indian genius of the art of construction of Vedic ritual altars (‘Yajna Vedis’) marked the zenith of the development of geometry in the ancient world, going as far back as the 9th century BCE (latest). The development of these geometric ideas was closely intertwined with the applicational perspective and spiritual significance and hence is contained in the Vedic corpus of the Kalpasutra literature as an integral part of Vedanga (the limbs of Vedic study). What starts as an exercise to design and construct varied shapes and sizes of Vedic altars, gradually appears to encapsulate ideas far beyond mundane figures and delves into concepts like the transformation of areas, novel constructions of squares, rectangles and circles, irrational numbers, approximations using finite series and also the famed Pythagorean Theorem (yes, the theorem was elucidated in India, way before Pythagoras appeared on the earth).
Classification and Preliminaries
The elucidation of geometry in the Vedic corpus is contained in the Sulba Sutras. The Sulba Sutras are geometric manuals concerned with the construction of Vedis (Fire Altars) for the performance of Yajnas (oblations to deities) and ritual sacrifices, integral to the Vedic Hindu religion. They are components of Kalpa Anga/Sutras, one of the six Vedangas(limbs of Vedas) and particularly embedded in the Srauta Sutras or literature concerned with ceremonies and rituals enjoined in the Vedas. Each Srauta Sutra likely had its own Sulba Sutra although only nine have been identified, all belonging to the Yajurveda school. These include the Baudhayana, Katyayana, Manava, and Apstambha. Of special interest are the Baudhayana, Apstambha and Katyayana Sulba Sutras with the former two belonging to Krishna Yajurveda and the latter to Shukla Yajurveda.
Objective of the Sulba Sutras
Vedic geometry of the Sulba Sutras was developed to enable the construction of such designs and dimensions as mentioned in the Brahmanas and Vedas for conducting certain Vedic Rituals. The Baudhayana Sulba Sutra begins with:
अथेमे अग्निचया ||
Now, the construction of altars associated with the different sacrificial fires is presented here.
What many don’t know today is that an Ashwamedha Yajna and a Vajpayee Yajna require different altars. In a similar fashion, altars needed for various objectives in ‘Kamyesti Yajnas’ (Rituals for fulfilling certain wishes) varied both in dimensions and shapes. A falcon-shaped altar (Syena-Citi) was envisioned for granting heaven to the doer while a tortoise-shaped altar (Kurma-Citi) was for conquering Brahmaloka(usually in Ashwamedha). Hence these exquisite shapes needed advanced geometry to be constructed. Instructions on doubling areas, comparing areas of various figures (eg. construction of an altar of square shape but the same area as a given circle), trapeziums of given diagonal and height and so on, given in Brahmanas need the geometry of Sulba Sutras to be translated into reality. The word śulba stems from the root śulb which means ‘to measure’ and the term is used interchangeably to denote the art of measurement and cord to measure although the latter is more often called ‘rajju’. Sulba Sutras were however manuals of instructions and hence don’t contain any axiomatic statements or proofs. This absence does not indicate that proofs were not known or methods were devised by trial and error but rather that the Sutras needed to be as concise as possible to be accurately transmitted and that such proofs have no utility in actual applications of geometric truths.
Basic Geometrical Formulations
Some elementary postulates that find mention in the Sulba Sutras are:
- A given straight line can be divided into any number of equal parts.
- A circle can be divided into any number of parts by drawing diameters.
- Each diagonal of a rectangle divides it into two equal parts.
- The diagonals of a rectangle bisect one another and divide the rectangle into four ‘parts two and two vertically opposite of which are equal in all respects.
- The diagonals of a rhombus bisect each other at right angles.
- A triangle can be divided into many equal and similar parts by dividing the sides into an equal number of parts and then joining the points of division two and two.
- A quadrilateral fonned by the lines joining the middle points of the sides of a rectangle is a rhombus whose area is half that of the rectangle.
Combining Areas of Squares1
To combine the areas of two squares of unequal areas, the prescription given by various Sulba Sutras is as follows:
नानाचतुरश्रे समस्यन्कनीयसः करण्या वर्षीयसो वृध्रमुल्लिखेत् । वृध्रस्याक्ष्णयारज्जुः समस्तयोः पार्श्वमानी भवति ||
(Desirous of combining different squares, may you mark the rectangular portion of the larger square with a side (karaṇyā) of the smaller one (kanīyasaḥ). The diagonal (akṣṇayārajjuḥ) of this rectangle (vṛddhra) is the side (pārśvamānī) of the sum of the two squares)
ABCD and IHGC are the two unequal squares desired to be combined. The method prescribes marking the points F and E on AB and DC such that AF = DE = CG, and then drawing the diagonal AE of the resulting rectangle AFED. It then clearly states that AE is the required side of the desired square whose area is equal to the sum of the two given squares. Writing the formulation, one can understand the rationale for the above method.
ABCD + IHGC = △ADE + △AFE + △EJH + △EGH + FBIJ = △KIH + △AFE + △EJH + △ABK + FBIJ
= AEHK. ( denotes area of square and △ denotes area of triangle)
This sutra becomes extremely important as it serves as a simple geometrical proof for the Pythagoras Theorem which we will subsequently discuss in the next part (Yes, the theorem was first elaborated in 800BCE by the authors of the Sulba Sutras and not Pythagoras).
Construction of a square whose area is the difference of two given squares
चतुरश्राच्चतुरश्रं निर्जिहीर्षन्यावन्निर्जिहीर्षेत्तस्य करण्या वर्षीयसो वृध्रमुल्लिखेत् । वृध्रस्य पार्श्वमानीमक्ष्णयेतरत्पार्श्वमुपसंहरेत् । सा यत्र निपतेत्तदपच्छिन्द्यात् । छिन्नया निरस्तम् ||
(Wishing to deduct a square from a square, one should cut off a segment by the side of the square to be removed. One of the lateral sides of the segment is drawn diagonally across to touch the other lateral side. The portion of the side beyond this point should be cut off.)
ABCD is a large square, from which a square of side x is desired to be removed. To this end, mark F and E on AB and DC such that AF = DE = x. Join EF. Now, a rope whose length is equal to the side AD = FE of the larger square is fixed at E and stretched and rotated to describe the arc of the circle denoted by the dashed curve starting at F in the figure. Thus, GD corresponds to the side of the square whose area is the difference between the two given squares.
This is as, in the right-angled triangle GDE, GD2=GE2-DE2=AD2-x2
We will continue the discussion in the next part on Pythagoras Theorem, approximation of the Square Root of 2 and other interesting discoveries by the Vedic genius.
[1] A Brief Introduction to the Mathematics of the Śulbasūtras By Aditya Kolachana, K. Mahesh, and K. Ramasubramanian, AMTI Bulletin.
