Baudhāyana (fl. c. 800 BCE)

Baudhāyana was the author of the Baudhayana sūtras, which cover dharma, daily ritual, mathematics, etc. He belongs to the Yajurveda school, and is older than the other sūtra author Āpastamba.

He was the author of the earliest Sulba Sūtra—appendices to the Vedas giving rules for the construction of altars—called the Baudhāyana Śulbasûtra. These are notable from the point of view of mathematics, for containing several important mathematical results, including giving a value of pi to some degree of precision, and stating a version of what is now known as the Pythagorean theorem.

Sequences associated with primitive Pythagorean triples have been named Baudhayana sequences. These sequences have been used in cryptography as random sequences and for the generation of keys.

The Sūtras of Baudhāyana

The Sûtras of Baudhāyana are associated with the Taittiriya Śākhā (branch) of Krishna (black) Yajurveda. The sutras of Baudhāyana have six sections,

1. the Śrautasûtra, probably in 19 Praśnas (questions),
2. the Karmāntasûtra in 20 Adhyāyas (chapters),
3. the Dvaidhasûtra in 4 Praśnas,
4. the Grihyasutra in 4 Praśnas,
5. the Dharmasûtra in 4 Praśnas and
6. the Śulbasûtra in 3 Adhyāyas.

The Shrautasūtra

Main article: Baudhayana Shrauta Sutra

His shrauta sūtras related to performing Vedic sacrifices has followers in some Smārta brāhmaṇas (Iyers) and some Iyengars of Tamil Nadu, Yajurvedis or Namboothiris of Kerala, Gurukkal Brahmins, among others. The followers of this sūtra follow a different method and do 24 Tila-tarpaṇa, as Lord Krishna had done tarpaṇa on the day before amāvāsyā; they call themselves Baudhāyana Amavasya.

The Dharmasūtra

The Dharmasūtra of Baudhāyana like that of Apastamba also forms a part of the larger Kalpasutra. Likewise, it is composed of praśnas which literally means ‘questions’ or books. The structure of this Dharmasūtra is not very clear because it came down in an incomplete manner. Moreover, the text has undergone alterations in the form of additions and explanations over a period of time. The praśnas consist of the Srautasutra and other ritual treatises, the Sulvasutra which deals with vedic geometry, and the Grhyasutra which deals with domestic rituals.

Authorship and Dates

Āpastamba and Baudhāyana come from the Taittiriya branch vedic school dedicated to the study of the Black Yajurveda. Robert Lingat states that Baudhāyana was the first to compose the Kalpasūtra collection of the Taittiriya school followed by Āpastamba. Kane assigns this Dharmasūtra an approximate date between 500 to 200 BC.

Commentaries

There are no commentaries on this Dharmasūtra with the exception of Govindasvāmin's Vivaraṇa. The date of the commentary is uncertain but according to Olivelle it is not very ancient. Also the commentary is inferior in comparison to that of Haradatta on Āpastamba and Gautama.

Organization and Contents

This Dharmasūtra is divided into four books. Olivelle states that Book One and the first sixteen chapters of Book Two are the ‘Proto-Baudhayana’ even though this section has undergone alteration. Scholars like Bühler and Kane agree that the last two books of the Dharmasūtra are later additions. Chapter 17 and 18 in Book Two lays emphasis on various types of ascetics and acetic practices.

The first book is primarily devoted to the student and deals in topics related to studentship. It also refers to social classes, the role of the king, marriage, and suspension of Vedic recitation. Book two refers to penances, inheritance, women, householder, orders of life, ancestral offerings. Book three refers to holy householders, forest hermit and penances. Book four primarily refers to the yogic practices and penances along with offenses regarding marriage.

Pythagorean Theorem

It is also referred to as Baudhayana theorem. The most notable of the rules (the Sulbasūtra-s do not contain any proofs for the rules which they describe, since they are sūtra-s, formulae, concise) in the Baudhāyana Sulba Sūtra says:

dīrghasyākṣaṇayā rajjuḥ pārśvamānī, tiryaḍam mānī, cha yatpṛthagbhūte kurutastadubhayāṅ karoti.

A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.

The lines are to be referring to a rectangle, although some interpretations consider this to refer to a square. In either case, it states that the square of the hypotenuse equals the sum of the squares of the sides. If restricted to right-angled isosceles triangles, however, it would constitute a less general claim, but the text seems to be quite open to unequal sides. If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem. Baudhāyana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles right triangle:

The cord which is stretched across a square produces an area double the size of the original square.

Circling the Square

Another problem tackled by Baudhāyana is that of finding a circle whose area is the same as that of a square (the reverse of squaring the circle). His sūtra i.58 gives this construction:

Draw half its diagonal about the centre towards the East-West line; then describe a circle together with a third part of that which lies outside the square.

Explanation:

• Draw the half-diagonal of the square, which is larger than the half-side by .
• Then draw a circle with radius , or , which equals .
• Now , so the area .

Square Root of 2

Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6) gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:

samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet
tac caturthenātmacatustriṃśonena saviśeṣaḥ

The diagonal [lit. "doubler"] of a square. The measure is to be increased by a third and by a fourth decreased by the 34th. That is its diagonal approximately.

That is,

which is correct to five decimals.

Other theorems include: diagonals of rectangle bisect each other, diagonals of rhombus bisect at right angles, area of a square formed by joining the middle points of a square is half of original, the midpoints of a rectangle joined forms a rhombus whose area is half the rectangle, etc.

Note the emphasis on rectangles and squares; this arises from the need to specify yajña bhūmikās—i.e. the altar on which a rituals were conducted, including fire offerings (yajña). Āpastamba (c. 600 BC) and Kātyāyana (c. 200 BC), authors of other sulba sūtras, extend some of Baudhāyana's ideas. Āpastamba provides a more general proof of the Pythagorean theorem.