160:
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 (
211:
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
1197:, 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' ."
75:
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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
138:. 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. According to Plofker, the Katyayana was composed after "the great grammatical codification of Sanskrit by
1463:, p. 207, "We find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. However all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the
1775:, p. 364: "As Vogt 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
478:
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
219:
tablet containing a table of triplets, however also states that Shulba sutras contain a formula not found in
Babylon sources. KS Krishnan asserts that Shulba sutras predates Mesopotamian Pythagoras triples. Seidenberg argues that either "Old Babylonia got the theorem of Pythagoras from India or that
210:
The origin of the mathematics in the Shulba Sutras is not known. 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
224:
and may predate 1700 BC. In contrast, Pingree cautions that "it would be a mistake to see 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
149:
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
110:
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
159:
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
176:
1467:
is not unlikely. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides, but this form of the
Pythagorean theorem also may have been derived from Mesopotamia. ... So conjectural are the origin and period of the
1001:
1472:
that we cannot tell whether or not the rules are related to early
Egyptian surveying or to the later Greek problem of altar doubling. They are variously dated within an interval of almost a thousand years stretching from the eighth century B.C. to the second century of our
154:
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
851:, 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:
1779:
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
1657:, p. 392, "The 'circulature' and quadrature techniques in 2.9 and 2.10, the first of which is illustrated in figure 4.4, imply what we would call a value of π of 3.088, The quadrature in 2.11, on the other hand, suggests that π = 3.004 (where
1164:, 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.
2029:
1928:
1079:
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gives the construction of geometric shapes such as squares and rectangles. It also gives, sometimes approximate, geometric area-preserving transformations from one geometric shape to another. These include transforming a
863:
2.10. To transform a circle into a square, the diameter is divided into eight parts; one part after being divided into twenty-nine parts is reduced by twenty-eight of them and further by the sixth less the eighth
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2.9. If it is desired to transform a square into a circle, half the diagonal is stretched from the centre to the east ; with one-third added to the remainder , the circle is drawn.
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199:, 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
1725:
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memory or eventually transcribed in manuscripts." Plofker also raises the possibility that "existing geometric knowledge consciously incorporated into ritual practice".
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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.
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2.12. The measure is to be increased by its third and this again by its own fourth less the thirty-fourth part ; this is the diagonal of a square .
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3064:
2956:
1426:
Joseph, George Gheverghese (1997). "What Is a Square Root? A Study of Geometrical Representation in Different Mathematical Traditions".
2851:
2247:"Kátyáyana's Śulbapariśishta with the Commentary by Ráma, Son of Súryadása", by George Thibaut, was published in a series of issues of
1084:
3026:
2171:"The Śulvasútra of Baudháyana, with the commentary by Dvárakánáthayajvan", by George Thibaut, was published in a series of issues of
266:
996:{\displaystyle {\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}={\frac {577}{408}}=1.4142...}
311:
1.12. The areas produced separately by the lengths of the breadth of a rectangle together equal the area produced by the diagonal.
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2.11. Alternatively, divide into fifteen parts and reduce it by two of them; this gives the approximate side of the square .
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Similarly, Apastamba's rules for constructing right angles in fire-altars use the following Pythagorean triples:
551:
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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.
1953:
The Thirteen Books of Euclid's Elements, Translated from the Text of Heiberg, with Introduction and Commentary
1700:), which is already considered only 'approximate.' In 2.12, the ratio of a square's diagonal to its side (our
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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.
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The Śulba Sūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava with Text, English Translation and Commentary
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567:
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2094:
1840:
Delire, Jean Michele (2009). "Chronological inferences from a comparison between commentaries on different
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It has also been suggested, for example by Bürk that this approximation of √2 implies knowledge that √2 is
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Indeed, an early method for calculating square roots can be found in some Sutras, the method involves the
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in probably the mid-fourth century BCE", but she places the Manava in the same period as the Baudhayana.
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1981:
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The four major Shulba Sutras, which are mathematically the most significant, are those attributed to
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Old Babylonia and India got it from a third source". Seidenberg suggests that this source might be
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Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
1947:
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Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
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654:
301:
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2280:"Das Āpastamba-Śulba-Sūtra, herausgegeben, übersetzt und mit einer Einleitung versehen"
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2048:
1919:
1905:
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1825:"Das Āpastamba-Śulba-Sūtra, herausgegeben, übersetzt und mit einer Einleitung versehen"
1443:
649:
195:
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99:
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The Pandit. A Monthly Journal, of the Benares College, devoted to Sanskrit Literature
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The Pandit. A Monthly Journal, of the Benares College, devoted to Sanskrit Literature
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2006:
1985:
1933:
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172:, but this altar does not conform to the dimensions prescribed by the Shulba Sutras.
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2103:
2080:
Seidenberg, A. (1983). "The Geometry of the Vedic Rituals". In Staal, Frits (ed.).
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1968:, A history of Indian literature, vol. VI, Scientific and technical literature
1951:
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541:
2702:
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1074:{\displaystyle {\sqrt {x}}\approx {\sqrt {x-1}}+{\frac {1}{2\cdot {\sqrt {x-1}}}}}
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The content of the Shulba Sutras is likely older than the works themselves. The
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mathematics (c. 2000 BCE–1600 BCE) as a possible origin, the c. 1800 BCE
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164:), dating to the second century BCE, was found in the, 1957-59, excavations by
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3104:
2995:
2927:
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1961:
1923:
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1978:
The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
139:
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2413:
1858:
The Quest for the Origins of Vedic Culture: The Indo-Aryan Migration Debate
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for large values of x, which bases itself on the non-recursive identity
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as found in three of the sutras. In the Baudhayana sutra it appears as:
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future enemies should construct a fire-altar in the form of a rhombus".
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The Shulba Sutras are part of the larger corpus of texts called the
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Cover page of a treaty of Śulbasūtra by the Indian mathematician
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Shanti Swarup Bhatnagar Prize recipients in Mathematical Science
1138:{\displaystyle {\sqrt {a^{2}+r}}\approx a+{\frac {r}{2\cdot a}}}
175:
2377:
1583:
The Crest of the Peacock: The Non-European Roots of Mathematics
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260:
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304:. In Baudhayana, for example, the rules are given as follows:
284:
207:, one of the triangles listed in the Baudhayana Shulba Sutra.
145:
With regard to the composition of Vedic texts, Plofker writes,
134:, pointing to a composition roughly during the 1st millennium
890:
which leads to the value of the square root of two as being:
221:
95:
74:
55:
1727:
is considered to be 1 + 1/3 + 1/(3·4) - 1/(3·4·34) = 1.4142.
82:
excavated from Purola, Uttarkashi; likely belonging to the
2836:
Infinite series expansions for the trigonometric functions
1976:(2007). "Mathematics in India". In Katz, Victor J (ed.).
1907:
The Science of the Sulba. A study in early Hindu geometry
135:
308:
1.9. The diagonal of a square produces double the area .
2284:
Zeitschrift der Deutschen Morgenländischen Gesellschaft
1829:
Zeitschrift der Deutschen Morgenländischen Gesellschaft
504:
1966:
Jyotiḥśāstra : astral and mathematical literature
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and the statement of squaring the circle is given as:
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Ramanujan Institute for Advanced Study in Mathematics
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Altar construction also led to an estimation of the
2251:. Note that the commentary is left untranslated.
2175:. Note that the commentary is left untranslated.
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2304:. New Delhi: Indian National Science Academy.
2148:Geometry and Algebra in Ancient Civilizations
1638:
1636:
1634:
1403:Geometry and Algebra in Ancient Civilizations
800:
300:and in the general case, as well as lists of
2132:The Journal of the Asiatic Society of Bengal
1956:. Vol. I (2 ed.). New York: Dover.
1207:
1205:
1203:
98:. They are the only sources of knowledge of
1237:
1235:
285:Pythagorean theorem and Pythagorean triples
62:ritual and containing geometry related to
2852:Kerala school of astronomy and mathematics
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2317:
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1880:The History of Mathematics: A Brief Course
1631:
1507:
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3080:Homi Bhabha Centre for Science Education
2082:Agni: The Vedic Ritual of the Fire Altar
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1586:. Princeton University Press. p.
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69:
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2053:Archive for History of Exact Sciences
2051:(1978). "The origin of mathematics".
2019:
1946:
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1792:
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289:The sutras contain statements of the
94:, considered to be appendices to the
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2092:(1999). "Greek and Vedic Geometry".
1846:Mathematics and Medicine in Sanskrit
1822:
1760:
1330:
1190:
1188:
275:(similar to Apastamba Shulba Sutras)
27:Texts belonging to the Śrauta ritual
2605:Subbayya Sivasankaranarayana Pillai
2084:. Berkeley: Asian Humanities Press.
24:
3065:Institute of Mathematical Sciences
1401:Van der Waerden, Barten L (1983).
25:
3132:
3075:Harish-Chandra Research Institute
2585:K. R. Parthasarathy (probabilist)
1305:Excavations at Kausambi (1957-59)
1185:
1160:. In his translation of Euclid's
257:(somewhat similar to Manava text)
2144:van der Waerden, Bartel Leendert
2220:(new series) (1876–1877)
2165:
1844:". In Wujastyk, Dominik (ed.).
1766:
1754:
1730:
1693:{\displaystyle s=2r\cdot 13/15}
1648:
1604:
1561:
1549:
1537:
1525:
1513:
1501:
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1454:
1419:
1405:. Springer Verlag. p. 12.
1394:
1383:
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1348:
1336:
1324:
1312:
873:
3060:Chennai Mathematical Institute
2429:Melpathur Narayana Bhattathiri
2292:Transcription and analysis in
2005:. Princeton University Press.
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13:
1:
2300:Sen, S.N.; Bag, A.K. (1983).
1786:
3050:Indian Statistical Institute
2095:Journal of Indian Philosophy
1720:{\displaystyle {\sqrt {2}})}
1149:extremely small relative to
54:: "string, cord, rope") are
7:
3070:Indian Institute of Science
3055:Bhaskaracharya Pratishthana
2933:A. A. Krishnaswamy Ayyangar
2898:Shankar Balakrishna Dikshit
2825:Hindu–Arabic numeral system
2271:(9–10): 487–491
2267:(9–10): 382–389
1860:. Oxford University Press.
1167:
486:
183:around the 2nd century BCE.
49:
10:
3137:
2565:Subrahmanyan Chandrasekhar
2263:(5–8): 328–339
2023:(1997). "Sulbasutras". In
1982:Princeton University Press
1964:(1981), Gonda, Jan (ed.),
1485:Origin of Vedas, Chapter 5
468:{\displaystyle (12,35,37)}
86:period (150 BCE - 250 CE).
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2844:
2810:
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2575:Veeravalli S. Varadarajan
2542:
2401:
2355:
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2259:(1–4): 94–103
1922:(1997). "Baudhāyana". In
599:Madhava's correction term
429:{\displaystyle (8,15,17)}
390:{\displaystyle (5,12,13)}
293:, both in the case of an
130:. Their language is late
2991:Henry Thomas Colebrooke
2580:S. R. Srinivasa Varadhan
1800:A History of Mathematics
1369:, pp. 106, 109–110
778:Other topics related to
58:texts belonging to the
2464:Mādhava of Saṅgamagrāma
2108:10.1023/A:1004364417713
1180:Citations and footnotes
821:Baudhayana Shulba sutra
351:{\displaystyle (3,4,5)}
1912:University of Calcutta
1721:
1694:
1483:Krishnan, K S (2019).
1390:van der Waerden (1983)
1302:Sharma, G. R. (1960).
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857:
503:mathematical constant
469:
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352:
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87:
2675:Brāhmasphuṭasiddhānta
2278:Bürk, Albert (1902).
1999:Plofker, Kim (2009).
1902:Datta, Bibhutibhushan
1823:Bürk, Albert (1901).
1806:John Wiley & Sons
1722:
1695:
1580:Joseph, G.G. (2000).
1428:Mathematics in School
1140:
1076:
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552:Use in other formulae
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431:
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306:
229:List of Shulba Sutras
178:
147:
77:
2951:T. A. Saraswati Amma
2662:Bakhshali manuscript
2630:Kannan Soundararajan
2254:(new series) (1882)
2216:(117): 209–218
2212:(116): 186–194
2208:(115): 166–170
2204:(114): 139–146
2183:(108): 292–298
2128:"On the Śulvasútras"
2002:Mathematics in India
1948:Heath, Sir Thomas L.
1704:
1661:
1085:
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897:
496:a series of articles
441:
402:
363:
324:
3027:Islamic mathematics
2986:Walter Eugene Clark
2780:Vasishtha Siddhanta
2752:Siddhānta Shiromani
2731:Paitamaha Siddhanta
2610:Tilak Raj Prabhakar
2555:Satyendra Nath Bose
2550:Srinivasa Ramanujan
2504:Nilakantha Somayaji
2241:(12): 761–770
2237:(11): 692–706
2233:(10): 626–642
2150:. Springer-Verlag.
1804:(Second ed.).
1613:, pp. 243–246
1546:, pp. 232–238
760:Squaring the circle
695:Chudnovsky brothers
685:Srinivasa Ramanujan
302:Pythagorean triples
291:Pythagorean theorem
70:Purpose and origins
3111:Indian mathematics
2519:Gangesha Upadhyaya
2340:Indian mathematics
2229:(9): 556–578
2225:(5): 316–322
2200:(111): 72–74
2196:(110): 44–50
2192:(109): 17–22
2187:(1875–1876)
2065:10.1007/BF00348435
1884:Wiley-Interscience
1717:
1690:
1135:
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993:
650:Ludolph van Ceulen
465:
426:
387:
348:
196:Taittiriya Samhita
190:Satapatha Brahmana
185:
100:Indian mathematics
88:
3121:Sutras (Hinduism)
3098:
3097:
2904:Sudhakara Dvivedi
2724:Paulisa Siddhanta
2643:
2642:
2499:Jagannatha Samrat
2494:Achyuta Pisharati
2040:978-0-7923-4066-9
1991:978-0-691-11485-9
1939:978-0-7923-4066-9
1848:. pp. 37–62.
1712:
1508:Seidenberg (1983)
1379:Seidenberg (1978)
1133:
1106:
1069:
1066:
1038:
1022:
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150:prose aphorisms (
108:Vedi (fire-altar)
16:(Redirected from
3128:
2738:Romaka Siddhanta
2620:Akshay Venkatesh
2590:M. S. Narasimhan
2560:P.C. Mahalanobis
2353:
2352:
2333:
2326:
2319:
2310:
2309:
2305:
2291:
2161:
2139:
2119:
2085:
2076:
2044:
2016:
1995:
1969:
1957:
1943:
1915:
1897:
1871:
1849:
1836:
1819:
1803:
1781:
1770:
1764:
1758:
1752:
1746:
1740:
1734:
1728:
1726:
1724:
1723:
1718:
1713:
1708:
1699:
1697:
1696:
1691:
1686:
1652:
1646:
1640:
1629:
1623:
1614:
1608:
1602:
1601:
1577:
1571:
1565:
1559:
1553:
1547:
1541:
1535:
1529:
1523:
1517:
1511:
1505:
1499:
1498:
1487:. Notion Press.
1480:
1474:
1458:
1452:
1451:
1423:
1417:
1416:
1398:
1392:
1387:
1381:
1376:
1370:
1364:
1358:
1352:
1346:
1340:
1334:
1328:
1322:
1316:
1310:
1309:
1299:
1293:
1287:
1281:
1275:
1269:
1263:
1257:
1251:
1245:
1239:
1230:
1224:
1215:
1209:
1198:
1192:
1144:
1142:
1141:
1136:
1134:
1132:
1118:
1107:
1099:
1098:
1089:
1080:
1078:
1077:
1072:
1070:
1068:
1067:
1056:
1044:
1039:
1028:
1023:
1018:
1002:
1000:
999:
994:
986:
978:
973:
971:
951:
946:
944:
930:
925:
917:
906:
901:
880:square root of 2
809:
802:
795:
781:
773:
645:Jamshīd al-Kāshī
542:Area of a circle
528:
527:
524:
521:
518:
507:
491:
490:
474:
472:
471:
466:
435:
433:
432:
427:
396:
394:
393:
388:
357:
355:
354:
349:
52:
21:
3136:
3135:
3131:
3130:
3129:
3127:
3126:
3125:
3101:
3100:
3099:
3094:
3042:
3036:
3000:
2974:
2945:C. T. Rajagopal
2892:Bapudeva Sastri
2884:
2878:
2840:
2820:Brahmi numerals
2812:
2806:
2766:Surya Siddhanta
2639:
2615:Manjul Bhargava
2538:
2397:
2342:
2337:
2168:
2158:
2124:Thibaut, George
2041:
2013:
1992:
1940:
1894:
1868:
1816:
1789:
1784:
1771:
1767:
1759:
1755:
1747:
1743:
1735:
1731:
1707:
1705:
1702:
1701:
1682:
1662:
1659:
1658:
1653:
1649:
1641:
1632:
1624:
1617:
1609:
1605:
1598:
1578:
1574:
1566:
1562:
1554:
1550:
1542:
1538:
1530:
1526:
1518:
1514:
1506:
1502:
1495:
1481:
1477:
1459:
1455:
1424:
1420:
1413:
1399:
1395:
1388:
1384:
1377:
1373:
1365:
1361:
1353:
1349:
1341:
1337:
1329:
1325:
1317:
1313:
1300:
1296:
1288:
1284:
1276:
1272:
1264:
1260:
1252:
1248:
1240:
1233:
1225:
1218:
1210:
1201:
1193:
1186:
1182:
1174:Kalpa (Vedanga)
1170:
1122:
1117:
1094:
1090:
1088:
1086:
1083:
1082:
1055:
1048:
1043:
1027:
1017:
1015:
1012:
1011:
977:
955:
950:
934:
929:
916:
900:
898:
895:
894:
876:
865:
839:, an isosceles
813:
779:
771:
739:Indiana pi bill
722:A History of Pi
700:Yasumasa Kanada
525:
522:
519:
516:
514:
505:
489:
442:
439:
438:
403:
400:
399:
364:
361:
360:
325:
322:
321:
312:
310:
309:
287:
282:
269:(in manuscript)
263:(in manuscript)
231:
72:
28:
23:
22:
15:
12:
11:
5:
3134:
3124:
3123:
3118:
3113:
3096:
3095:
3093:
3092:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3046:
3044:
3038:
3037:
3035:
3034:
3029:
3024:
3019:
3014:
3008:
3006:
3002:
3001:
2999:
2998:
2993:
2988:
2982:
2980:
2976:
2975:
2973:
2972:
2966:
2960:
2954:
2948:
2942:
2936:
2930:
2925:
2919:
2916:P. C. Sengupta
2913:
2907:
2901:
2895:
2888:
2886:
2880:
2879:
2877:
2876:
2854:
2848:
2846:
2842:
2841:
2839:
2838:
2833:
2827:
2822:
2816:
2814:
2808:
2807:
2805:
2804:
2797:
2790:
2783:
2776:
2773:Tantrasamgraha
2769:
2762:
2755:
2748:
2741:
2734:
2727:
2720:
2713:
2706:
2699:
2696:Karanapaddhati
2692:
2685:
2682:Ganita Kaumudi
2678:
2671:
2664:
2659:
2651:
2649:
2645:
2644:
2641:
2640:
2638:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2600:Harish-Chandra
2597:
2595:C. S. Seshadri
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2557:
2552:
2546:
2544:
2540:
2539:
2537:
2536:
2531:
2529:Sankara Variar
2526:
2521:
2516:
2511:
2506:
2501:
2496:
2491:
2486:
2481:
2476:
2471:
2466:
2461:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2405:
2403:
2399:
2398:
2396:
2395:
2390:
2385:
2380:
2375:
2370:
2365:
2359:
2357:
2350:
2348:Mathematicians
2344:
2343:
2336:
2335:
2328:
2321:
2313:
2307:
2306:
2297:
2275:
2274:
2273:
2245:
2244:
2243:
2218:
2185:
2167:
2164:
2163:
2162:
2156:
2140:
2120:
2086:
2077:
2059:(4): 301–342.
2049:Seidenberg, A.
2045:
2039:
2025:Selin, Helaine
2017:
2011:
1996:
1990:
1970:
1962:Pingree, David
1958:
1944:
1938:
1924:Selin, Helaine
1916:
1898:
1892:
1872:
1866:
1850:
1837:
1820:
1814:
1794:Boyer, Carl B.
1788:
1785:
1783:
1782:
1765:
1753:
1741:
1737:Plofker (2007)
1729:
1716:
1711:
1689:
1685:
1681:
1678:
1675:
1672:
1669:
1666:
1655:Plofker (2007)
1647:
1643:Plofker (2007)
1630:
1626:Plofker (2007)
1615:
1611:Thibaut (1875)
1603:
1596:
1572:
1560:
1556:Plofker (2007)
1548:
1544:Thibaut (1875)
1536:
1532:Plofker (2009)
1524:
1520:Pingree (1981)
1512:
1500:
1494:978-1645879800
1493:
1475:
1453:
1418:
1411:
1393:
1382:
1371:
1359:
1347:
1335:
1323:
1319:Plofker (2009)
1311:
1294:
1282:
1270:
1266:Pingree (1981)
1258:
1254:Plofker (2009)
1246:
1242:Plofker (2009)
1231:
1227:Pingree (1981)
1216:
1212:Plofker (2007)
1199:
1195:Plofker (2007)
1183:
1181:
1178:
1177:
1176:
1169:
1166:
1145:for values of
1131:
1128:
1125:
1121:
1116:
1113:
1110:
1105:
1102:
1097:
1093:
1065:
1062:
1059:
1054:
1051:
1047:
1042:
1037:
1034:
1031:
1026:
1021:
1004:
1003:
992:
989:
984:
981:
976:
970:
967:
964:
961:
958:
954:
949:
943:
940:
937:
933:
928:
923:
920:
915:
912:
909:
904:
875:
872:
815:
814:
812:
811:
804:
797:
789:
786:
785:
784:
783:
775:
767:
762:
754:
753:
752:Related topics
749:
748:
747:
746:
741:
733:
732:
728:
727:
726:
725:
718:
710:
709:
705:
704:
703:
702:
697:
692:
687:
682:
680:William Shanks
677:
672:
667:
662:
657:
655:François Viète
652:
647:
642:
637:
632:
627:
622:
614:
613:
609:
608:
607:
606:
601:
596:
594:Approximations
591:
589:Less than 22/7
583:
582:
578:
577:
576:
575:
570:
562:
561:
557:
556:
555:
554:
549:
544:
536:
535:
531:
530:
510:
509:
500:
499:
488:
485:
476:
475:
464:
461:
458:
455:
452:
449:
446:
436:
425:
422:
419:
416:
413:
410:
407:
397:
386:
383:
380:
377:
374:
371:
368:
358:
347:
344:
341:
338:
335:
332:
329:
298:right triangle
286:
283:
281:
278:
277:
276:
270:
264:
258:
252:
247:
242:
237:
230:
227:
213:Old Babylonian
132:Vedic Sanskrit
92:Shrauta Sutras
78:Falcon-shaped
71:
68:
66:construction.
46:: शुल्बसूत्र;
26:
9:
6:
4:
3:
2:
3133:
3122:
3119:
3117:
3114:
3112:
3109:
3108:
3106:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3047:
3045:
3039:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3009:
3007:
3005:Other regions
3003:
2997:
2996:David Pingree
2994:
2992:
2989:
2987:
2984:
2983:
2981:
2977:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2910:M. Rangacarya
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2889:
2887:
2883:Historians of
2881:
2874:
2870:
2866:
2862:
2858:
2857:Jantar Mantar
2855:
2853:
2850:
2849:
2847:
2843:
2837:
2834:
2832:
2828:
2826:
2823:
2821:
2818:
2817:
2815:
2809:
2803:
2802:
2798:
2796:
2795:
2791:
2789:
2788:
2784:
2782:
2781:
2777:
2775:
2774:
2770:
2768:
2767:
2763:
2761:
2760:
2756:
2754:
2753:
2749:
2747:
2746:
2742:
2740:
2739:
2735:
2733:
2732:
2728:
2726:
2725:
2721:
2719:
2718:
2714:
2712:
2711:
2707:
2705:
2704:
2700:
2698:
2697:
2693:
2691:
2690:
2686:
2684:
2683:
2679:
2677:
2676:
2672:
2670:
2669:
2665:
2663:
2660:
2658:
2657:
2653:
2652:
2650:
2646:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2547:
2545:
2541:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2474:Mahendra Sūri
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2406:
2404:
2400:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2366:
2364:
2361:
2360:
2358:
2354:
2351:
2349:
2345:
2341:
2334:
2329:
2327:
2322:
2320:
2315:
2314:
2311:
2303:
2298:
2295:
2289:
2286:(in German).
2285:
2281:
2276:
2272:
2268:
2264:
2260:
2257:
2253:
2252:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2223:
2219:
2217:
2213:
2209:
2205:
2201:
2197:
2193:
2190:
2186:
2184:
2181:
2177:
2176:
2174:
2170:
2169:
2159:
2157:9783642617812
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2125:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2096:
2091:
2087:
2083:
2078:
2074:
2070:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2036:
2032:
2031:
2026:
2022:
2018:
2014:
2012:9780691120676
2008:
2004:
2003:
1997:
1993:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1954:
1949:
1945:
1941:
1935:
1931:
1930:
1925:
1921:
1917:
1913:
1909:
1908:
1903:
1899:
1895:
1893:0-471-44459-6
1889:
1885:
1881:
1877:
1873:
1869:
1867:9780195137774
1863:
1859:
1855:
1854:Bryant, Edwin
1851:
1847:
1843:
1838:
1834:
1831:(in German).
1830:
1826:
1821:
1817:
1815:0-471-54397-7
1811:
1807:
1802:
1801:
1795:
1791:
1790:
1778:
1774:
1769:
1762:
1757:
1750:
1745:
1738:
1733:
1709:
1687:
1683:
1679:
1676:
1673:
1670:
1667:
1664:
1656:
1651:
1644:
1639:
1637:
1635:
1628:, pp. 388-391
1627:
1622:
1620:
1612:
1607:
1599:
1597:0-691-00659-8
1593:
1589:
1585:
1584:
1576:
1569:
1564:
1558:, pp. 388–389
1557:
1552:
1545:
1540:
1533:
1528:
1521:
1516:
1509:
1504:
1496:
1490:
1486:
1479:
1471:
1466:
1462:
1457:
1449:
1445:
1441:
1437:
1433:
1429:
1422:
1414:
1408:
1404:
1397:
1391:
1386:
1380:
1375:
1368:
1363:
1356:
1351:
1344:
1339:
1332:
1327:
1320:
1315:
1307:
1306:
1298:
1291:
1286:
1279:
1278:Delire (2009)
1274:
1267:
1262:
1255:
1250:
1243:
1238:
1236:
1228:
1223:
1221:
1213:
1208:
1206:
1204:
1196:
1191:
1189:
1184:
1175:
1172:
1171:
1165:
1163:
1159:
1154:
1152:
1148:
1129:
1126:
1123:
1119:
1114:
1111:
1108:
1103:
1100:
1095:
1091:
1063:
1060:
1057:
1052:
1049:
1045:
1040:
1035:
1032:
1029:
1024:
1019:
1009:
990:
987:
982:
979:
974:
968:
965:
962:
959:
956:
952:
947:
941:
938:
935:
931:
926:
921:
918:
913:
910:
907:
902:
893:
892:
891:
887:
883:
881:
871:
867:
860:
856:
852:
850:
846:
842:
838:
835:
831:
827:
822:
810:
805:
803:
798:
796:
791:
790:
788:
787:
782:
776:
774:
770:Six nines in
768:
766:
765:Basel problem
763:
761:
758:
757:
756:
755:
751:
750:
745:
742:
740:
737:
736:
735:
734:
730:
729:
724:
723:
719:
717:
714:
713:
712:
711:
707:
706:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
670:William Jones
668:
666:
663:
661:
660:Seki Takakazu
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
617:
616:
615:
611:
610:
605:
602:
600:
597:
595:
592:
590:
587:
586:
585:
584:
580:
579:
574:
573:Transcendence
571:
569:
568:Irrationality
566:
565:
564:
563:
559:
558:
553:
550:
548:
547:Circumference
545:
543:
540:
539:
538:
537:
533:
532:
529:
512:
511:
508:
502:
501:
497:
493:
492:
484:
482:
459:
456:
453:
450:
447:
437:
420:
417:
414:
411:
408:
398:
381:
378:
375:
372:
369:
359:
342:
339:
336:
333:
330:
320:
319:
318:
314:
305:
303:
299:
296:
292:
274:
273:Hiranyakeshin
271:
268:
265:
262:
259:
256:
253:
251:
248:
246:
243:
241:
238:
236:
233:
232:
226:
223:
218:
214:
208:
206:
202:
198:
197:
192:
191:
182:
177:
173:
171:
167:
163:
156:
153:
146:
143:
141:
137:
133:
129:
125:
121:
117:
112:
109:
105:
101:
97:
93:
85:
81:
76:
67:
65:
61:
57:
53:
51:
45:
41:
40:
35:
34:
33:Shulva Sutras
19:
3043:institutions
2963:K. S. Shukla
2935:(1892– 1953)
2801:Yavanajataka
2799:
2792:
2785:
2778:
2771:
2764:
2759:Śulba Sūtras
2758:
2757:
2750:
2745:Sadratnamala
2743:
2736:
2729:
2722:
2715:
2708:
2701:
2694:
2689:Kanakkusaram
2687:
2680:
2673:
2666:
2654:
2524:Varāhamihira
2489:Parameshvara
2444:Govindasvāmi
2414:Āryabhaṭa II
2301:
2287:
2283:
2255:
2248:
2221:
2188:
2179:
2172:
2166:Translations
2147:
2135:
2131:
2099:
2093:
2090:Staal, Frits
2081:
2056:
2052:
2033:. Springer.
2028:
2001:
1977:
1974:Plofker, Kim
1965:
1952:
1932:. Springer.
1927:
1906:
1879:
1876:Cooke, Roger
1857:
1845:
1841:
1832:
1828:
1799:
1776:
1773:Heath (1925)
1768:
1756:
1749:Cooke (2005)
1744:
1732:
1650:
1606:
1582:
1575:
1568:Boyer (1991)
1563:
1551:
1539:
1527:
1515:
1503:
1484:
1478:
1469:
1464:
1461:Boyer (1991)
1456:
1431:
1427:
1421:
1402:
1396:
1385:
1374:
1367:Staal (1999)
1362:
1355:Gupta (1997)
1350:
1343:Heath (1925)
1338:
1326:
1314:
1304:
1297:
1290:Staal (1999)
1285:
1273:
1261:
1249:
1161:
1155:
1150:
1146:
1005:
889:
885:
877:
874:Square roots
869:
862:
858:
854:
820:
818:
720:
665:Takebe Kenko
604:Memorization
513:
477:
316:
307:
288:
255:Maitrayaniya
217:Plimpton 322
209:
204:
200:
194:
188:
186:
166:G. R. Sharma
161:
158:
151:
148:
144:
113:
104:Vedic period
89:
79:
47:
38:
37:
32:
31:
29:
2979:Translators
2971:(1919–2005)
2969:K. V. Sarma
2965:(1918–2007)
2959:(1918–1992)
2953:(1918–2000)
2947:(1903–1978)
2941:(1901–1954)
2939:A. N. Singh
2924:(1888–1958)
2922:B. B. Datta
2918:(1876–1962)
2912:(1861–1916)
2906:(1855–1910)
2900:(1853–1898)
2894:(1821–1900)
2885:mathematics
2829:Symbol for
2813:innovations
2710:Lokavibhaga
2656:Aryabhatiya
2454:Jyeṣṭhadeva
2439:Brahmagupta
2424:Bhāskara II
2409:Āryabhaṭa I
2393:Yajnavalkya
2294:Bürk (1901)
2102:: 105–127.
2021:Sarma, K.V.
1920:Gupta, R.C.
1842:Śulbasūtras
1761:Bürk (1901)
1470:Sulbasutras
1465:Sulvasutras
1331:Bürk (1901)
690:John Wrench
675:John Machin
630:Zu Chongzhi
280:Mathematics
39:Śulbasūtras
3105:Categories
2928:T. Hayashi
2811:Pioneering
2794:Yuktibhāṣā
2717:Pātīgaṇita
2668:Bijaganita
2625:Ravi Vakil
2479:Munishvara
2459:Kamalakara
2434:Brahmadeva
2419:Bhāskara I
2368:Baudhayana
2290:: 327–391.
2138:: 227–275.
1835:: 543–591.
1787:References
1777:impossible
1434:(3): 4–9.
1412:0387121595
1158:irrational
731:In culture
716:Chronology
620:Archimedes
560:Properties
240:Baudhayana
116:Baudhayana
64:fire-altar
18:Sulbasutra
2957:S. N. Sen
2865:New Delhi
2787:Veṇvāroha
2648:Treatises
2449:Halayudha
2402:Classical
2373:Katyayana
2363:Apastamba
2073:118671661
1950:(1925) .
1878:(2005) .
1677:⋅
1440:0305-7259
1127:⋅
1109:≈
1061:−
1053:⋅
1033:−
1025:≈
1010:formula:
1008:recursive
991:1.4142...
966:⋅
960:⋅
948:−
939:⋅
908:≈
837:trapezium
834:isosceles
830:rectangle
635:Aryabhata
295:isosceles
250:Katyayana
235:Apastamba
181:Kātyāyana
162:śyenaciti
128:Katyayana
124:Apastamba
106:. Unique
102:from the
2873:Varanasi
2831:zero (0)
2703:Līlāvatī
2570:C.R. Rao
2534:Virasena
2514:Sridhara
2484:Narayana
2469:Mahāvīra
2146:(1983).
2126:(1875).
2116:16466375
1904:(1932).
1856:(2001).
1796:(1991).
1763:, p. 575
1751:, p. 200
1739:, p. 392
1645:, p. 391
1570:, p. 207
1510:, p. 121
1448:30215281
1357:, p. 154
1345:, p. 362
1333:, p. 554
1292:, p. 111
1214:, p. 387
1168:See also
1162:Elements
847:, and a
841:triangle
526:26433...
494:Part of
487:Geometry
222:Sumerian
193:and the
170:Kausambi
44:Sanskrit
3012:Babylon
2845:Centres
2509:Śrīpati
2388:Pingala
2356:Ancient
2178:(1875)
2027:(ed.).
1926:(ed.).
1780:third."
1534:, p. 17
1321:, p 19.
1280:, p. 50
1256:, p. 11
845:rhombus
828:into a
708:History
640:Madhava
625:Liu Hui
515:3.14159
267:Vadhula
203:and 36
152:sūtras,
84:Kuninda
3041:Modern
3032:Europe
3022:Greece
2869:Ujjain
2861:Jaipur
2543:Modern
2383:Pāṇini
2378:Manava
2154:
2114:
2071:
2037:
2009:
1988:
1936:
1890:
1864:
1812:
1594:
1522:, p. 5
1491:
1446:
1438:
1409:
1268:, p. 6
1244:, p.18
1229:, p. 4
849:circle
826:square
744:Pi Day
612:People
498:on the
481:gnomon
261:Varaha
245:Manava
140:Pāṇini
120:Manava
60:Śrauta
3017:China
2112:S2CID
2069:S2CID
1473:era."
1444:JSTOR
832:, an
581:Value
523:23846
520:89793
517:26535
96:Vedas
56:sutra
50:śulba
3090:TIFR
2152:ISBN
2035:ISBN
2007:ISBN
1986:ISBN
1934:ISBN
1888:ISBN
1862:ISBN
1810:ISBN
1592:ISBN
1489:ISBN
1436:ISSN
1407:ISBN
843:, a
819:The
534:Uses
205:pada
201:pada
155:..."
126:and
80:vedi
30:The
2104:doi
2061:doi
1588:229
983:408
980:577
168:at
136:BCE
36:or
3116:Pi
3107::
2871:,
2867:,
2863:,
2288:56
2282:.
2269:,
2265:,
2261:,
2239:,
2235:,
2231:,
2227:,
2214:,
2210:,
2206:,
2202:,
2198:,
2194:,
2189:10
2136:44
2134:.
2130:.
2110:.
2100:27
2098:.
2067:.
2057:18
2055:.
1984:.
1980:.
1910:.
1886:.
1882:.
1833:55
1827:.
1808:.
1688:15
1680:13
1633:^
1618:^
1590:.
1442:.
1432:26
1430:.
1234:^
1219:^
1202:^
1187:^
1153:.
969:34
460:37
454:35
448:12
421:17
415:15
382:13
376:12
122:,
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2859:(
2332:e
2325:t
2318:v
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2256:4
2222:1
2180:9
2160:.
2118:.
2106::
2075:.
2063::
2043:.
2015:.
1994:.
1942:.
1914:.
1896:.
1870:.
1818:.
1715:)
1710:2
1684:/
1674:r
1671:2
1668:=
1665:s
1600:.
1497:.
1450:.
1415:.
1308:.
1151:a
1147:r
1130:a
1124:2
1120:r
1115:+
1112:a
1104:r
1101:+
1096:2
1092:a
1064:1
1058:x
1050:2
1046:1
1041:+
1036:1
1030:x
1020:x
988:=
975:=
963:4
957:3
953:1
942:4
936:3
932:1
927:+
922:3
919:1
914:+
911:1
903:2
864:.
808:e
801:t
794:v
780:π
772:π
506:π
463:)
457:,
451:,
445:(
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418:,
412:,
409:8
406:(
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379:,
373:,
370:5
367:(
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343:5
340:,
337:4
334:,
331:3
328:(
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