In the previous article, we have looked at the lesser known fact that a clear enunciation of the so-called ‘Pythagorean’ theorem, which must rightly be named the Śulba Theorem (called bhujā-koṭi-karṇa-nyāya in the later literature) has been described in Baudhāyana-śulbasūtra. In this edition we shall look at further breakthroughs and contributions found in the ancient texts of Śulba-sūtras.
The Śulba Theorem in Mānava-śulbasūtra
The presentation of the theorem in Mānava-śulbasūtra differs from Bodhāyana Śulbasūtra both in form and in style. Here it is given in the form of a verse as follows:
आयामं आयामगुणं विस्तारं विस्तरेण तु।
समस्य वर्गमूलं यत् तत् कर्णं तद्विदो विदुः॥ |
Phrase in Mānava-śulbasūtra | Meaning |
आयामं आयामगुणं | the length multiplied by itself |
विस्तारं विस्तरेण तु | and indeed the breadth by itself |
समस्य वर्गमूलं | the square root of the sum |
तत् कर्णं | that is the hypotenuse |
तद्विदो विदुः | those versed in the discipline say so |
Using modern notation the result may be expressed as:
āyāmā2 + vistārā2 = karṇa
In no uncertain terms, it is clear that the statements in both Mānava-śulbasūtra and Bodhāyana Śulbasūtra predate the period of Pythagoras. Without well-reasoned understanding this theorem continues to be credited to Pythagoras ( c. 570 – c. 495 BC) and even touted as a “Greek miracle” by certain short-sighted claims.
A list of theorems in Śulbasūtras
The following theorems are either expressly stated or the results are implied in the methods of construction of the altars of different shapes and sizes:
- The diagonals of a rectangle bisect each other. They divide the rectangle into four parts, two and two (vertically opposite) – all of which are equal in all respects.
- The diagonals of a rhombus bisect each other at right angles.
- An isosceles triangle is divided into two equal halves by the line joining the vertex to the middle point of the base.
- The area of a square formed by joining the middle points of the sides of a square is half that of the original one.
- 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.
- A parallelogram and rectangle on the same base and within the same parallels have the same area.
- The square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides.
- If the sum of the squares on two sides of a triangle be equal to the square on the third side, then the triangle is right-angled.
Construction knowledge repository
As explained in earlier articles, the Śulba-sūtras, which form a part of the Vedic literature, deal with the construction of fire altars for yajña purposes. Their construction requires a thorough knowledge of the properties of the square, the rectangle, the rhombus, the trapezium, the triangle and the circle. The knowledge prevalent of those times are astounding and are hardly known to Indians and students of science around the world. The geometrical knowledge repository in these texts have been well researched by serious scholars and involved in the constructions are the following:
- To divide a line into any number of equal parts.
- To divide a circle into any number of equal areas by drawing diameters.
- To divide a triangle into a number of equal and similar areas.
- To draw a straight line at right angles to a given line.
- To draw a straight line at right angles to a given straight line from a given point on it.
- To construct a square on a given side.
- To construct a rectangle of given sides.
- To construct an isosceles trapezium of given altitude, face and base.
- To construct a parallelogram having given sides at a given inclination.
- To construct a square equal to the sum of two different squares.
- To construct a square equivalent to two given triangles.
- To construct a square equivalent to two given pentagons.
- To construct a square equal to a given rectangle.
- To construct a rectangle having a given side and equivalent to a given square.
- To construct an isosceles trapezium having a given face and equivalent to a given square or rectangle.
- To construct a triangle equivalent to a given square.
- To construct a square equivalent to a given isosceles triangle.
- To construct a rhombus equivalent to a given square or rectangle.
- To construct a square equivalent to a given rhombus.
It is indeed enthralling to know in depth about the rich scientific heritage of the Indian civilization. In the forthcoming articles we shall delve more into greater understanding of Jyotiṣa and Gaṇitā in India across different ages.