*Dedicated to Sri Chandrasekharendra Sarasvati*

*who died in his hundredth year while this paper was being written.*

by

David W. Henderson^{1}

Department of Mathematics, Cornell University

Ithaca, NY, 14853-7901, USA (e-mail: dwh2@cornell.edu)

Several Sanskrit texts collectively called the *Sulbasutra *were written by the Vedic Hindus starting before 600 B.C. and are thought^{2} to be compilations of oral wisdom which may go back to 2000 B.C. These texts have prescriptions for building fire altars, or *Agni*. However, contained in the *Sulbasutra* are sections which constitute a geometry textbook detailing the geometry necessary for designing and constructing the altars. As far as I have been able to determine these are the oldest geometry (or even mathematics) textbooks in existence. It is apparently the oldest applied geometry text.

The *Sulbasutra** ^{3}* contain the following prescription for finding the length of the diagonal of a square:

Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. The increased length is a small amount in excess (*savi**´**e**¸**a*)^{4}.

Thus the above passage from the *Sulbasutram* gives the approximation:

and the *Sulbasutram*'s value expressed in decimals is

There have been several speculations^{5} as to how this value was obtained, but no one as far as I can determine has noticed that there is a step-by-step method (based on geometric techniques in the *Sulbasutram*) that will not only obtain the approximation:

but can also be continued indefinitely to obtain as accurate an approximation as one wishes.

This method will in one more step obtain:

The interested reader can check that this approximation is accurate to eleven decimal places.

In the *Sulbasutram* the *agni* are described as being constructed of bricks of various sizes. Mentioned often are square bricks of side 1 *pradesa* (span of a hand, about 9 inches) on a side. Each *pradesa* was equal to 12 *angula* (finger width, about 3/4 inch) and one *angula* was equal to 34 sesame seeds laid together with their broadest faces touching^{6}. Thus the diagonal of a *pradesa* brick had length:

1 *pradesa* + 4 *angula* + 1 *angula* - 1 sesame thickness.

*Dissecting Rectangles and A ^{2 }+ B^{2 }= C^{2}*

(3,4,5), (5,12,13), (7,24,25), (8,15,17), (9,12,15), (12,35,37), (15,36,39)

which the *Sulbasutram *used in its various methods for constructing right angles.

*Construction of the Savi ´e¸a for the Square Root of Two*

We can get directly to by considering the following dissection:

2[1154(1154/2)-1] = (1154)^{2 }- 2 = 1,331,714

and thus that the next approximation (*savi**´**e**¸**a*) is

The difference between 2·1 and the square of this *savi**´**e**¸**a* is

This is equivalent to being approximated by 1.44.

you can make slight modifications in the above method to find:

*Comparing with the Divide-and-Average (D&A) Method*

1 + (1/3) + (1/4)(1/3) 1.414215686 ½[(17/12)+ 2(12 (577/408) 1.414213562 (665857/470832) 1.414213562 = (886731088897/627013566048)

**D&A - calculator****D&A - Fractions****Baudhayana's Method**
*a*_{1 }= 1.41666666717/12
*a*_{2 }= ½(*a*_{1 }+ (2*/a*_{1})) =*/*17)] =*k*_{2 }= 2[(3·4)+4+1] = 34*c*_{2 }= - (1/34)(1/4)(1/3)
*a*_{3 }= ½(*a*_{2 }+ (2*/a*_{2})) =½[(577/408)+2(408/577)] =
*k*_{3 }= (34)^{2}-2 = 1154*c*_{3 }= -(1/1154)(1/34)(1/4)(1/3)
*a*_{4 }= ½(*a*_{3 }+ (2*/a*_{3})) = ½[(665857/470832)+2(470832/665857)]
*k*_{4 }= (1154)^{2}-2 = 1331714*c*_{4 }= -(1/1331714) *c*_{3}

Notice that in Baudhayana's fourth representation of the *savi**´**e**¸**a* for the square root of 2:

Baudhayana's method can not come even close to the D&A method in terms of ease of use with a computer and its applicability to finding the square root of any number. However, the *Sulbasutra* contains many powerful techniques, which, in specific situations have a power and efficiency that is missing in more general techniques. Numerical computations with the decimal system in either fixed point or floating point form has many well-known problems.^{7} Perhaps we will be able to learn something from the (apparently) first applied geometry text in the world and devise computational procedures that combine geometry and numerical techniques.

^{2}
See for example, A. Seidenberg, The Ritual Origin of Geometry, *Archive for the History of the Exact Sciences*,* *1(1961), pp. 488-527.

^{3}
*Baudhayana Sulbasutram*, i. 61-2. *Apastamba Sulbasutram*, i. 6. *Katyayana Sulbasutram*, II. 13.

^{4}
This last sentence is translated by some authors as "The increased length is called *savi**´**e**¸**a*". I follow the translation of "*savi**´**e**¸**a*" given by B. Datta on pp. 196-202 in *The Science of the Sulba*, University of Calcutta, 1932; see also G. Joseph (*The Crest of the Peacock, *I.B. Taurus, London, 1991) who translates the word as "a special quantity in excess".

^{5}
See Datta *Op.cit.* for a discussion of several of these, some of which are also discussed in G. Joseph, *Op. cit.*

^{6}
Baudhayana Sulbasutram, i. 3-7.

^{7}
See, for example, P.R. Turner's "Will the 'Real' Real Arithmetic Please Stand Up?" in *Notices of AMS*, Vol. 34, April 1991, pp. 298-304.