# Baudhayana

### Quick Info

Born
India
Died
India

Summary
Baudhayana was the author of one of the earliest Sulbasutras: documents containing some of the earliest Indian mathematics.

### Biography

To write a biography of Baudhayana is essentially impossible since nothing is known of him except that he was the author of one of the earliest Sulbasutras. We do not know his dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year.

He was neither a mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and it would appear an almost certainty that Baudhayana himself would be a Vedic priest.

The mathematics given in the Sulbasutras is there to enable the accurate construction of altars needed for sacrifices. It is clear from the writing that Baudhayana, as well as being a priest, must have been a skilled craftsman. He must have been himself skilled in the practical use of the mathematics he described as a craftsman who himself constructed sacrificial altars of the highest quality.

The Sulbasutras are discussed in detail in the article Indian Sulbasutras. Below we give one or two details of Baudhayana's Sulbasutra, which contained three chapters, which is the oldest which we possess and, it would be fair to say, one of the two most important.

The Sulbasutra of Baudhayana contains geometric solutions (but not algebraic ones) of a linear equation in a single unknown. Quadratic equations of the forms $ax^{2} = c$ and $ax^{2} + bx = c$ appear.

Several values of π occur in Baudhayana's Sulbasutra since when giving different constructions Baudhayana uses different approximations for constructing circular shapes. Constructions are given which are equivalent to taking π equal to $\large\frac{676}{225}\normalsize$ (where $\large\frac{676}{225}\normalsize$ = 3.004), $\large\frac{900}{289}\normalsize$ (where $\large\frac{900}{289}\normalsize$ = 3.114) and to $\large\frac{1156}{361}\normalsize$ (where $\large\frac{1156}{361}\normalsize$ = 3.202). None of these is particularly accurate but, in the context of constructing altars they would not lead to noticeable errors.

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
$√2 = 1 + \large\frac{1}{3}\normalsize + \large\frac{1}{(3\times 4)}\normalsize - \large\frac{1}{(3\times 4\times 34)}\normalsize = \large\frac{577}{408}\normalsize$
which is, to nine places, 1.414215686. This gives √2 correct to five decimal places. This is surprising since, as we mentioned above, great mathematical accuracy did not seem necessary for the building work described. If the approximation was given as
$√2 = 1 + \large\frac{1}{3}\normalsize + \large\frac{1}{(3\times 4)}\normalsize$
then the error is of the order of 0.002 which is still more accurate than any of the values of π. Why then did Baudhayana feel that he had to go for a better approximation?

### References (show)

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