# Baudhayana

By J J O'Connor and E F Robertson

Baudhayana was a mathematician who lived around 800 BCE in what is now modern day India. He is the author of earliest Sulbasutra-s which contains calculation of value of pi, Pythagoras theorem, calculating square root of 2 and circling the square. He is credited with calculating pi and what is now called the "Pythagoras theorem" before Pythagoras had developed it.

Little else is known about him except that he was the author of one of the earliest Sulbasutra-s. He was a man of very considerable learning and probably wrote the Sulbasutra to provide rules for religious rites.

## Contents

## Use of mathematics in construction of Altars

The mathematics given in the Sulbasutra-s enables the accurate construction of altars needed for sacrifices. It is clear from the writing that Baudhayana, as well as being a Vedic priest, must have been a skilled craftsman. He was skilled in the practical use of the mathematics and construction of sacrificial altars of the highest quality.

## Sulbasutra

Baudhayana's Sulbasutra is the oldest surviving Sulbasutra. In one chapter, it contains geometric solutions of a linear equation with a single unknown variable. Quadratic equations of the forms ax<super>2</super> = c and ax<super>2</super> + bx = c are also described.

### Value of Pi

Several values of π (pi) also occur in Baudhayana's Sulbasutra. Specifically, Baudhayana uses different approximations for π when constructing circular shapes. Constructions are given which are equivalent to taking π equal to 676/225 (where 676/225 = 3.004), 900/289 (where 900/289 = 3.114) and to 1156/361 (where 1156/361 = 3.202).

### Value of square root of 2

An interesting, and quite accurate, approximate value for √2 is given in Chapter 1 verse 61 of Baudhayana's Sulbasutra. The Sanskrit text gives in words what we would write in symbols as

√2 = 1 + 1/3 + 1/(3×4) - 1/(3×4×34)= 577/408

which is, to nine places, 1.414215686 and is correct to five decimal places. If the approximation was given as

√2 = 1 + 1/3 + 1/(3×4)

then the error is of the order of 0.002 which is still more accurate than any of the values of π. Thus, it is unclear as to why Baudhayana felt the need for a better approximation for √2 vs π and implies that better approximations of π could have been known at the time but are not provided in this document.

## References

- G G Joseph, The crest of the peacock (London, 1991).
- R C Gupta, Baudhayana's value of √2, Math. Education 6 (1972), B77-B79.
- S C Kak, Three old Indian values of π, Indian J. Hist. Sci. 32 (4) (1997), 307-314.
- G Kumari, Some significant results of algebra of pre-Aryabhata era, Math. Ed. (Siwan) 14 (1) (1980), B5-B13.