The ''Shulva Sutras'' or ''Śulbasūtras'' (
Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
: शुल्बसूत्र; ': "string, cord, rope") are
sutra
''Sutra'' ( sa, सूत्र, translit=sūtra, translit-std=IAST, translation=string, thread)Monier Williams, ''Sanskrit English Dictionary'', Oxford University Press, Entry fo''sutra'' page 1241 in Indian literary traditions refers to an aph ...
texts belonging to the
Śrauta
Śrauta is a Sanskrit word that means "belonging to śruti", that is, anything based on the Vedas of Hinduism. It is an adjective and prefix for texts, ceremonies or person associated with śruti. The term, for example, refers to Brahmins who spec ...
ritual and containing geometry related to
fire-altar construction.
Purpose and origins
The Shulba Sutras are part of the larger corpus of texts called the
Shrauta Sutras
Kalpa ( sa, कल्प) means "proper, fit" and is one of the six disciplines of the Vedānga, or ancillary science connected with the Vedas – the scriptures of Hinduism. This field of study is focused on the procedures and ceremonies associ ...
, considered to be appendices to the
Vedas
upright=1.2, The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the '' Atharvaveda''.
The Vedas (, , ) are a large body of religious texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the ...
. They are the only sources of knowledge of
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
from the
Vedic period
The Vedic period, or the Vedic age (), is the period in the late Bronze Age and early Iron Age of the history of India when the Vedic literature, including the Vedas (ca. 1300–900 BCE), was composed in the northern Indian subcontinent, betw ...
. Unique fire-altar shapes were associated with unique gifts from the Gods. For instance, "he who desires heaven is to construct a fire-altar in the form of a falcon"; "a fire-altar in the form of a tortoise is to be constructed by one desiring to win the world of Brahman" and "those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus".
[, p. 387, "Certain shapes and sizes of fire-altars were associated with particular gifts that the sacrificer desired from the gods: 'he who desires heaven is to construct a fire-altar in the form of a falcon'; 'a fire-altar in the form of a tortoise is to be constructed by one desiring to win the world of Brahman'; 'those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus' en and Bag 1983, 86, 98, 111"]
The four major Shulba Sutras, which are mathematically the most significant, are those attributed to
Baudhayana
The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
,
Manava
Manava (c. 750 BC – 690 BC) is an author of the Hindu geometric text of ''Sulba Sutras.''
The Manava Sulbasutra is not the oldest (the one by Baudhayana is older), nor is it one of the most important, there being at least three Sulbasu ...
,
Apastamba
''Āpastamba Dharmasūtra'' (Sanskrit: आपस्तम्ब धर्मसूत्र) is a Sanskrit text and one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. It is one o ...
and
Katyayana.
[, p. 387] Their language is late
Vedic Sanskrit
Vedic Sanskrit was an ancient language of the Indo-Aryan subgroup of the Indo-European language family. It is attested in the Vedas and related literature compiled over the period of the mid- 2nd to mid-1st millennium BCE. It was orally preser ...
, pointing to a composition roughly during the 1st millennium
BCE
Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
.
[ The oldest is the sutra attributed to Baudhayana, possibly compiled around 800 BCE to 500 BCE.][ Pingree says that the Apastamba is likely the next oldest; he places the Katyayana and the Manava third and fourth chronologically, on the basis of apparent borrowings.][, p. 4] According to Plofker, the Katyayana was composed after "the great grammatical codification of Sanskrit by Pāṇini
, era = ;;6th–5th century BCE
, region = Indian philosophy
, main_interests = Grammar, linguistics
, notable_works = ' (Sanskrit#Classical Sanskrit, Classical Sanskrit)
, influenced=
, notable_ideas=Descript ...
in probably the mid-fourth century BCE", but she places the Manava in the same period as the Baudhayana.[, p.18]
With regard to the composition of Vedic texts, Plofker writes,The Vedic veneration of Sanskrit as a sacred speech, whose divinely revealed texts were meant to be recited, heard, and memorized rather than transmitted in writing, helped shape Sanskrit literature in general. ... Thus texts were composed in formats that could be easily memorized: either condensed prose aphorisms (''sūtras,'' a word later applied to mean a rule or algorithm in general) or verse, particularly in the Classical period. Naturally, ease of memorization sometimes interfered with ease of comprehension. As a result, most treatises were supplemented by one or more prose commentaries ..."
There are multiple commentaries for each of the Shulba Sutras, but these were written long after the original works. The commentary of Sundararāja on the Apastamba, for example, comes from the late 15th century CE and the commentary of Dvārakãnātha on the Baudhayana appears to borrow from Sundararāja. According to Staal, certain aspects of the tradition described in the Shulba Sutras would have been "transmitted orally", and he points to places in southern India where the fire-altar ritual is still practiced and an oral tradition preserved. The fire-altar tradition largely died out in India, however, and Plofker warns that those pockets where the practice remains may reflect a later Vedic revival rather than an unbroken tradition.[ Archaeological evidence of the altar constructions described in the Shulba Sutras is sparse. A large falcon-shaped fire altar (''śyenaciti''), dating to the second century BCE, was found in the excavations by G. R. Sharma at ]Kausambi
Kosambi (Pali) or Kaushambi (Sanskrit) was an important city in ancient India. It was the capital of the Vatsa kingdom, one of the sixteen mahajanapadas. It was located on the Yamuna River about southwest of its confluence with the Ganges at ...
, but this altar does not conform to the dimensions prescribed by the Shulba Sutras.[
]
The content of the Shulba Sutras is likely older than the works themselves. The ''Satapatha Brahmana
The Shatapatha Brahmana ( sa, शतपथब्राह्मणम् , Śatapatha Brāhmaṇam, meaning 'Brāhmaṇa of one hundred paths', abbreviated to 'SB') is a commentary on the Śukla (white) Yajurveda. It is attributed to the Vedic ...
'' and the ''Taittiriya Samhita
The ''Taittirīya Shakha'' (Sanskrit, loosely meaning 'Branch or School of the sage Tittiri'), is a ''shakha'' (i.e. 'branch', 'school', or rescension) of the Krishna (black) Yajurveda. Most prevalent in South India, it consists of the ''Taitti ...
'', whose contents date to the late second millennium or early first millennium BCE, describe altars whose dimensions appear to be based on the right triangle with legs of 15 ''pada'' and 36 ''pada'', one of the triangles listed in the Baudhayana Shulba Sutra.
Several Mathematicians and Historians mention that the earliest of the texts were written beginning in 800 BCE by Vedic Hindus based on compilations of an oral tradition dating back to 2000 BCE. It is possible, as proposed by Gupta, that the geometry was developed to meet the needs of ritual. Some scholars go farther: Staal hypothesizes a common ritual origin for Indian and Greek geometry, citing similar interest and approach to doubling and other geometric transformation problems. Seidenberg, followed by van der Waerden, sees a ritual origin for mathematics more broadly, postulating that the major advances, such as discovery of the Pythagorean theorem, occurred in only one place, and diffused from there to the rest of the world. Van der Waerden mentions that author of Sulbha sutras existed before 600 BCE and could not have been influenced by Greek geometry. While Boyer mentions Old Babylonian
Old Babylonian may refer to:
*the period of the First Babylonian dynasty (20th to 16th centuries BC)
*the historical stage of the Akkadian language
Akkadian (, Akkadian: )John Huehnergard & Christopher Woods, "Akkadian and Eblaite", ''The Camb ...
mathematics (c. 2000 BCE–1600 BCE) as a possible origin, however also states that Shulba sutras contain a formula not found in Babylon sources. KS Krishnan mentions that Shulba sutras predates Mesopotamian Pythagoras triples. Seidenberg argues that either "Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source". Seidenberg suggests that this source might be Sumer
Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of c ...
ian and may predate 1700 BC. In contrast, Pingree cautions that "it would be a mistake to see in he altar builders'
He or HE may refer to:
Language
* He (pronoun), an English pronoun
* He (kana), the romanization of the Japanese kana へ
* He (letter), the fifth letter of many Semitic alphabets
* He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...
works the unique origin of geometry; others in India and elsewhere, whether in response to practical or theoretical problems, may well have advanced as far without their solutions having been committed to memory or eventually transcribed in manuscripts." Plofker also raises the possibility that "existing geometric knowledge asconsciously incorporated into ritual practice".
List of Shulba Sutras
# Apastamba
''Āpastamba Dharmasūtra'' (Sanskrit: आपस्तम्ब धर्मसूत्र) is a Sanskrit text and one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. It is one o ...
# Baudhayana
The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
# Manava
Manava (c. 750 BC – 690 BC) is an author of the Hindu geometric text of ''Sulba Sutras.''
The Manava Sulbasutra is not the oldest (the one by Baudhayana is older), nor is it one of the most important, there being at least three Sulbasu ...
# Katyayana
# Maitrayaniya (somewhat similar to Manava text)
# Varaha
Varaha ( sa, वराह, , "boar") is an avatar of the Hindu god Vishnu, in the form of a boar. Varaha is generally listed as third in the Dashavatara, the ten principal avatars of Vishnu.
Varaha is most commonly associated with the lege ...
(in manuscript)
# Vadhula (in manuscript)
# Hiranyakeshin (similar to Apastamba Shulba Sutras)
Mathematics
Pythagorean theorem and Pythagorean triples
The sutras contain statements of the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, both in the case of an isosceles
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
and in the general case, as well as lists of Pythagorean triples
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
.
In Baudhayana, for example, the rules are given as follows:
1.9. The diagonal of a square produces double the area f the square
..BR> 1.12. The areas f the squaresproduced separately by the lengths of the breadth of a rectangle together equal the area f the squareproduced by the diagonal.
1.13. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.
Similarly, Apastamba's rules for constructing right angles in fire-altars use the following Pythagorean triples:
*
*
*
*
In addition, the sutras describe procedures for constructing a square with area equal either to the sum or to the difference of two given squares. Both constructions proceed by letting the largest of the squares be the square on the diagonal of a rectangle, and letting the two smaller squares be the squares on the sides of that rectangle. The assertion that each procedure produces a square of the desired area is equivalent to the statement of the Pythagorean theorem. Another construction produces a square with area equal to that of a given rectangle. The procedure is to cut a rectangular piece from the end of the rectangle and to paste it to the side so as to form a gnomon
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields.
History
A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the ol ...
of area equal to the original rectangle. Since a gnomon is the difference of two squares, the problem can be completed using one of the previous constructions.
Geometry
The ''Baudhayana Shulba sutra'' gives the construction of geometric shapes such as squares and rectangles.[, pp. 388-391] It also gives, sometimes approximate, geometric area-preserving transformations from one geometric shape to another. These include transforming a square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
into a rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
, an isosceles
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
trapezium, an isosceles triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
, a rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
, and a circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, and transforming a circle into a square.[
In these texts approximations, such as the transformation of a circle into a square, appear side by side with more accurate statements. As an example, the statement of circling the square is given in Baudhayana as:
]2.9. If it is desired to transform a square into a circle, cord of lengthhalf the diagonal f the squareis stretched from the centre to the east part of it lying outside the eastern side of the square with one-third f the part lying outside
F, or f, is the sixth Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Let ...
added to the remainder f the half diagonal the equiredcircle is drawn.[, p. 391]
and the statement of squaring the circle is given as:
2.10. To transform a circle into a square, the diameter is divided into eight parts; one uch
Uch ( pa, ;
ur, ), frequently referred to as Uch Sharīf ( pa, ;
ur, ; ''"Noble Uch"''), is a historic city in the southern part of Pakistan's Punjab province. Uch may have been founded as Alexandria on the Indus, a town founded by Alexan ...
part after being divided into twenty-nine parts is reduced by twenty-eight of them and further by the sixth f the part left
F, or f, is the sixth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''.
His ...
less the eighth f the sixth part
2.11. Alternatively, divide he diameter
He or HE may refer to:
Language
* He (pronoun), an English pronoun
* He (kana), the romanization of the Japanese kana へ
* He (letter), the fifth letter of many Semitic alphabets
* He (Cyrillic), a letter of the Cyrillic script called ''He'' ...
into fifteen parts and reduce it by two of them; this gives the approximate side of the square esired
The constructions in 2.9 and 2.10 give a value of π as 3.088, while the construction in 2.11 gives π as 3.004.
Square roots
Altar construction also led to an estimation of the square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
as found in three of the sutras. In the Baudhayana sutra it appears as:
2.12. The measure is to be increased by its third and this hird
The hird (also named "Håndgangne Menn" in Norwegian), in Scandinavian history, was originally an informal retinue of personal armed companions, hirdmen or housecarls, but came to mean not only the nucleus ('Guards') of the royal army, but also d ...
again by its own fourth less the thirty-fourth part f that fourth this is he value of
He or HE may refer to:
Language
* He (pronoun), an English pronoun
* He (kana), the romanization of the Japanese kana へ
* He (letter), the fifth letter of many Semitic alphabets
* He (Cyrillic), a letter of the Cyrillic script called ''He'' ...
the diagonal of a square hose side is the measure
A hose is a flexible hollow tube designed to carry fluids from one location to another. Hoses are also sometimes called ''pipes'' (the word ''pipe'' usually refers to a rigid tube, whereas a hose is usually a flexible one), or more generally ' ...
which leads to the value of the square root of two as being:
:[, p. 200]
Indeed, an early method for calculating square roots can be found in some Sutras, the method involves the recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
formula: for large values of x, which bases itself on the non-recursive identity for values of ''r'' extremely small relative to ''a''.
It has also been suggested, for example by Bürk that this approximation of √2 implies knowledge that √2 is irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
. In his translation of Euclid's ''Elements'', Heath outlines a number of milestones necessary for irrationality to be considered to have been discovered, and points out the lack of evidence that Indian mathematics had achieved those milestones in the era of the Shulba Sutras.[, p. 364: "As einrichVogt says, three stages had to be passed through before the irrationality of the diagonal of a square was discovered in any real sense. (1) All values found by direct measurement of calculations based thereon have to be recognized as being inaccurate. Next (2) must supervene the conviction that it is ''impossible'' to arrive at an accurate arithmetical expression of the value. And lastly (3) the impossibility must be proved. Now there is no real evidence that the Indians, at the date in question, had even reached the first stage, still less the second or third."]
See also
*Kalpa (Vedanga)
Kalpa ( sa, कल्प) means "proper, fit" and is one of the six disciplines of the Vedānga, or ancillary science connected with the Vedas – the scriptures of Hinduism. This field of study is focused on the procedures and ceremonies associ ...
Citations and footnotes
References
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Translations
* "The Śulvasútra of Baudháyana, with the commentary by Dvárakánáthayajvan", by George Thibaut, was published in a series of issues of ''The Pandit. A Monthly Journal, of the Benares College, devoted to Sanskrit Literature''. Note that the commentary is left untranslated.
** (1875) 9''
(108): 292–298
** (1875–1876) 10''
(109): 17–22
(110): 44–50
(111): 72–74
(114): 139–146
(115): 166–170
(116): 186–194
(117): 209–218
** (new series) (1876–1877) 1''
(5): 316–322
(9): 556–578
(10): 626–642
(11): 692–706
(12): 761–770
* "Kátyáyana's Śulbapariśishta with the Commentary by Ráma, Son of Súryadása", by George Thibaut, was published in a series of issues of ''The Pandit. A Monthly Journal, of the Benares College, devoted to Sanskrit Literature''. Note that the commentary is left untranslated.
** (new series) (1882) 4''
(1–4): 94–103
(5–8): 328–339
(9–10): 382–389
(9–10): 487–491
* Transcription and analysis in .
*
{{Indian mathematics
Indian mathematics
Pi
Sutras (Hinduism)