TowardS the end of the Vedic period, and more or less simultaneously with the production of the principal Upanishads, concise, technical, and usually aphoristic texts were composed about various subjects relating to the proper and timely performance of the Vedic sacrificial rituals. These were eventually labelled as Vedangas.
The four major Shulba Sutras, which are mathematically the most significant, are those attributed to Baudhayana, Manava, Apastamba and Katyayana. Their language is the late Vedic Sanskrit, pointing to a composition roughly during the 1st millennium BCE. The oldest is the sutra attributed to Baudhayana, possibly compiled around 800 BCE to 500 BCE. Pingree says that the Apastamba is likely the next oldest; he places the Katyayana and the Manava third and fourth chronologically, on the basis of apparent borrowings. According to Plofker, the Katyayana was composed after "the great grammatical codification of Sanskrit by Pini in probably the mid-fourth century BCE", but she places the Manava in the same period as the Baudhayana
List of Shulbha Sutras
- Maitrayaniya (somewhat similar to Manava text)
- Varaha (in manuscript)
- Vadhula (in manuscript)
- Hiranyakeshin (similar to Apastamba Shulba Sutras)
Pythagorean theorem and Pythagorean Triples
- The sutras contain statements of the Pythagorean theorem, both in the case of an isosceles right triangle and in the general case, as well as lists of Pythagorean triples. In Baudhayana, for example, the rules are given as follows:
- The diagonal of a square produces double the area [of the square].
- The areas [of the squares] produced separately by the lengths of the breadth of a rectangle together equal the area [of the square] produced by the diagonal.
- This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.
- Similarly, Apastamba's rules for constructing right angles in fire-altars use the following Pythagorean triples
In addition, the sutras describe procedures for constructing a square with area equal either to the sum or to the difference of two given squares. Both constructions proceed by letting the largest of the squares be the square on the diagonal of a rectangle, and letting the two smaller squares be the squares on the sides of that rectangle. The assertion that each procedure produces a square of the desired area is equivalent to the statement of the Pythagorean theorem. Another construction produces a square with area equal to that of a given rectangle. The procedure is to cut a rectangular piece from the end of the rectangle and to paste it to the side so as to form a gnomon of area equal to the original rectangle. Since a gnomon is the difference of two squares, the problem can be completed using one of the previous constructions.
Geometric Figures and their Inter-conversions
Many geometrical figures (probably 1st time) were seen in Sulbasutras for carrying out their rituals in the form of Altars. These included circle, square, rectangle, trapezium, isosceles trapezium, isosceles triangle, rhombus, etc. Also their conversion, keeping the areas same, like square to circle, rectangle to square, square to circle, circle to square, etc. were seen.
And hence in these conversions, value of pi was also calculated.
Example: Conversion of circle to square keeping area constant.
Constructing a square of side 13/15 times the diameter of the given circle
This corresponds to taking = 4 (13/15)2 = 676/225 = 3.00444.
More correct value of pi was calculated as 3.125 by Manava.
Calculation of Irrational numbers was also seen in Sulbasutras.
Eg: Squareroot of 2 (irrational number).
It was stated as Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth.
2 = 1 + 1/3 + 1/(3 4) 1/(3 4 34) = 577/408 = 1.4142
There is yet again to discover more from vedic system.