Baudhayana
Encyclopedia
Baudhāyana, was an India
n mathematician
, who
was most likely also a priest. He is noted as the author of the earliest Sulba Sūtra
—appendices to the Vedas
giving rules for the construction of altars—called the , which contained several important mathematical results. He is older than the other famous mathematician Āpastambha
. He belongs to the Yajurveda
school.
He is accredited with calculating the value of pi
to some degree of precision, and with discovering what is now known as the Pythagorean theorem
.
sūtra
s related to performing Vedic
sacrifice
s has followers in some Smārta
brāhmaṇa
s (Iyers) and some Iyengar
s of Tamil Nadu
, Kongu 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 Kṛiṣhṇa
had done tarpaṇa on the day before Amāvasyā
; they call themselves Baudhāyana Amāvasya
.
This appears 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
:
). His sūtra i.58 gives this construction:
Explanation:
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:
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.
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, who
was most likely also a priest. He is noted as the author of the earliest Sulba Sūtra
Sulba Sutras
The Shulba Sutras or Śulbasūtras are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.- Purpose and origins :...
—appendices to the Vedas
Vedas
The Vedas are a large body of texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature and the oldest scriptures of Hinduism....
giving rules for the construction of altars—called the , which contained several important mathematical results. He is older than the other famous mathematician Āpastambha
Apastamba
The Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
. He belongs to the Yajurveda
Yajurveda
The Yajurveda, a tatpurusha compound of "sacrificial formula', + ) is the third of the four canonical texts of Hinduism, the Vedas. By some, it is estimated to have been composed between 1400 and 1000 BC, the Yajurveda 'Samhita', or 'compilation', contains the liturgy needed to perform the...
school.
He is accredited with calculating the value of pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
to some degree of precision, and with discovering what is now known as the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
.
The Shrautasūtra
His shrautaSrauta
' traditions are conservative ritualistic traditions of the historical Vedic religion in Hinduism, based on the body of Śruti literature...
sūtra
Sutra
Sūtra is an aphorism or a collection of such aphorisms in the form of a manual. Literally it means a thread or line that holds things together and is derived from the verbal root siv-, meaning to sew , as does the medical term...
s related to performing Vedic
Vedas
The Vedas are a large body of texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature and the oldest scriptures of Hinduism....
sacrifice
Sacrifice
Sacrifice is the offering of food, objects or the lives of animals or people to God or the gods as an act of propitiation or worship.While sacrifice often implies ritual killing, the term offering can be used for bloodless sacrifices of cereal food or artifacts...
s has followers in some Smārta
Smartism
Smarta Sampradaya is a liberal or nonsectarian denomination of the Vedic Hindu religion which accept all the major Hindu deities as forms of the one Brahman, in contrast to Vaishnavism, Shaivism, and Shaktism, the other three major Hindu sects, which revere Vishnu, Shiva, and Shakti,...
brāhmaṇa
Brahmana
The Brāhmaṇas are part of the Hindu śruti literature. They are commentaries on the four Vedas, detailing the proper performance of rituals....
s (Iyers) and some Iyengar
Iyengar
Iyengar or Ayyangar is a caste given to Hindu Brahmins of Tamil origin who follow the Visishtadvaita philosophy propounded by Sri Ramanujacharya. They are found mostly in Tamil Nadu as they are generally native to the Tamil Nadu state of the Republic of India...
s of Tamil Nadu
Tamil Nadu
Tamil Nadu is one of the 28 states of India. Its capital and largest city is Chennai. Tamil Nadu lies in the southernmost part of the Indian Peninsula and is bordered by the union territory of Pondicherry, and the states of Kerala, Karnataka, and Andhra Pradesh...
, Kongu of Tamil nadu, Yajurvedis
Yajurveda
The Yajurveda, a tatpurusha compound of "sacrificial formula', + ) is the third of the four canonical texts of Hinduism, the Vedas. By some, it is estimated to have been composed between 1400 and 1000 BC, the Yajurveda 'Samhita', or 'compilation', contains the liturgy needed to perform the...
or Namboothiris of Kerala
Kerala
or Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
, Gurukkal brahmins
Gurukkal brahmins
Gurukkal is Tamil plural form of the Sanksrit word "Guru" - Spiritual Teacher. These are the brahmins who acted as gurus to medieval king like Rajaraja I and brought many kings in to the fold of saivism.-Migration of Brahmins :...
, among others. The followers of this sūtra follow a different method and do 24 Tila-tarpaṇa, as Lord Kṛiṣhṇa
Krishna
Krishna is a central figure of Hinduism and is traditionally attributed the authorship of the Bhagavad Gita. He is the supreme Being and considered in some monotheistic traditions as an Avatar of Vishnu...
had done tarpaṇa on the day before Amāvasyā
Amavasya
Amavasya is the Indian name for a New moon. The word Amavasya is common to many Indian languages especially Sanskrit, Hindi, Assamese, Kannada, Bengali, Konkani, Marathi, Oriya, Telugu, and Gujarati...
; they call themselves Baudhāyana Amāvasya
Amavasya
Amavasya is the Indian name for a New moon. The word Amavasya is common to many Indian languages especially Sanskrit, Hindi, Assamese, Kannada, Bengali, Konkani, Marathi, Oriya, Telugu, and Gujarati...
.
The Dharmasūtra
The Vivaraṇa of Govindasvami is an important commentary on the .Pythagorean theorem
The most notable of the rules (the Sulbasūtra-s do not contain any proofs of 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
DiagonalA diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
produces an areaAreaArea is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
which the vertical and horizontal sides make together.
This appears to be referring to a rectangle, although some interpretations consider this to refer to a square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
. In either case, it states that the square of the hypotenuse
Hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the...
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
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
.
Baudhāyana also provides a non-axiomatic demonstration using a rope measure of the reduced form of the Pythagorean theorem for an isosceles right triangle
Right triangle
A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...
:
- 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 circleSquaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
). 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 thirty-fourth. That is its diagonal approximately.
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
Apastamba
The Dharmasutra of Āpastamba forms a part of the larger Kalpasūtra of Āpastamba. It contains thirty praśnas, which literally means ‘questions’ or books. The subjects of this Dharmasūtra are well organized and preserved in good condition...
(c. 600 BC) and Kātyāyana
Katyayana
Kātyāyana was a Sanskrit grammarian, mathematician and Vedic priest who lived in ancient India.-Works:He is known for two works:...
(c. 200 BC), authors of other sulba sūtras, extend some of Baudhāyana's ideas. Āpastamba provides a more general proof of the Pythagorean theorem.
See also
- Indian mathematicsIndian mathematicsIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
- Indian mathematicians
- Sulba SutrasSulba SutrasThe Shulba Sutras or Śulbasūtras are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.- Purpose and origins :...