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101-625: The Shulba Sutras are part of the larger corpus of texts called the Shrauta Sutras , considered to be appendices to the Vedas . They are the only sources of knowledge of Indian mathematics from the Vedic period . Unique Vedi (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

202-465: A square into a rectangle , an isosceles trapezium , an isosceles triangle , a rhombus , and a circle , 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

303-503: A Pythagorean triple. The triple is primitive, that is the three triangle sides have no common factor, if p and q are coprime and not both odd. Neugebauer and Sachs propose the tablet was generated by choosing p and q to be coprime regular numbers (but both may be odd—see Row 15) and computing d = p + q , s = p − q , and l = 2 pq (so that l is also a regular number). For example, line 1 would be generated by setting p = 12 and q = 5. Buck and Robson both note that

404-488: A common factor. It is possible that 45 and 1 15 are to be understood as 3/4 and 5/4, which is consistent with the standard (0.75,1,1.25) scaling of the familiar (3,4,5) right triangle in Babylonian mathematics. In each row, the number in the second column can be interpreted as the shorter side s {\displaystyle s} of a right triangle, and the number in the third column can be interpreted as

505-614: A composition roughly during the 1st millennium BCE . 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 Pāṇini in probably

606-452: A condition which guarantees that p  −  q is the long leg and 2 pq is the short leg of the triangle and which, in modern terms, implies that the angle opposite the leg of length p  −  q is less than 45°. The ratio is least in Row 15 where p / q =9/5 for an angle of about 31.9°. Furthermore, there are exactly 15 regular ratios between 9/5 and 12/5 inclusive for which q

707-695: A corresponding series of primitive diagonal triples (wth the front, length, and the diagonal equal to integers without common factors)." Scholars still differ on how these numbers were generated. Buck (1980) and Robson (2001) both identify two main proposals for the method of construction of the table: the method of generating pairs, proposed in Neugebauer & Sachs (1945) , and the method of reciprocal pairs, proposed by Bruins and elaborated on by Voils, Schmidt (1980) , and Friberg. To use modern terminology, if p and q are natural numbers such that p > q then ( p − q , 2 pq , p + q ) forms

808-509: A mistake to see in [the altar builders'] 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 [was] consciously incorporated into ritual practice". The sutras contain statements of

909-496: 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. Kalpa (Vedanga) Divisions Sama vedic Yajur vedic Atharva vedic Vaishnava puranas Shaiva puranas Shakta puranas Kalpa ( Sanskrit : कल्प ) means "proper, fit" and

1010-451: A number of scholars have proposed that this error is much more plausibly explained as an error in the calculation leading up to the number, for example, the scribe's overlooking a medial zero (blank space representing a zero digit) when performing a multiplication. This explanation of the error is compatible with both of the main proposals for the method of construction of the table. (See below.) The remaining three errors have implications for

1111-555: A problem with a nice answer, the problem setter would simply need to choose such an x and let the initial datum c equal x − 1/ x . As a side effect, this produces a rational Pythagorean triple, with legs v 1 and 1 and hypotenuse v 4 . It should be pointed out that the problem on YBC 6967 actually solves the equation x − 1   00 x = x − 60 x = c {\textstyle x-{\tfrac {1\ 00}{x}}=x-{\tfrac {60}{x}}=c} , which entails replacing

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1212-477: A product of powers of 2, 3, and 5. It is for this reason that the numbers in the first column are exact, as dividing an integer by a regular number produces a terminating sexagesimal number. For instance, line 1 of the table can be interpreted as describing a triangle with short side 119 and hypotenuse 169, implying long side 169 2 − 119 2 = 120 {\displaystyle {\sqrt {169^{2}-119^{2}}}=120} , which

1313-430: A rational Pythagorean triple. Moreover, the three sides all have finite sexagesimal representations. Advocates of this proposal point out that regular reciprocal pairs ( x ,1/ x ) show up in a different problem from roughly the same time and place as Plimpton 322, namely the problem of finding the sides of a rectangle of area 1 whose long side exceeds its short side by a given length c (which nowadays might be computed as

1414-400: A square into a circle, [a cord of length] half the diagonal [of the square] is stretched from the centre to the east [a part of it lying outside the eastern side of the square]; with one-third [of the part lying outside] added to the remainder [of the half diagonal], the [required] circle is drawn. and the statement of squaring the circle is given as: 2.10. To transform a circle into a square,

1515-541: A table of four columns and 15 rows of numbers in the cuneiform script of the period. This table lists two of the three numbers in what are now called Pythagorean triples , i.e., integers a , b , and c satisfying a + b = c . From a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem. There has been significant scholarly debate on

1616-523: 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 Old Babylonia and India got it from a third source". Seidenberg suggests that this source might be Sumerian and may predate 1700 BC. In contrast, Pingree cautions that "it would be

1717-415: A value of π as 3.088, while the construction in 2.11 gives π as 3.004. Altar construction also led to an estimation of the square root of 2 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 [third] again by its own fourth less the thirty-fourth part [of that fourth]; this is [the value of] the diagonal of a square [whose side

1818-453: A wide variety of different mathematical operations and propose the translation "'solving number of the width (or the diagonal).'" Similarly, Friberg (1981) (p. 300) proposes the translation "root". In Column 1, the first parts of both lines of the heading are damaged. Neugebauer & Sachs (1945) reconstructed the first word as takilti (a form of takiltum ), a reading that has been accepted by most subsequent researchers. The heading

1919-487: 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

2020-551: A work tablet (or possibly from an earlier copy of the table). The error in Row 8, Column 1 (replacing the two sexagesimal digits 45 14 by their sum, 59) appears not to have been noticed in some of the early papers on the tablet. It has sometimes been regarded (for example in Robson (2001) ) as a simple mistake made by the scribe in the process of copying from a work tablet. As discussed in Britton, Proust & Shnider (2011) , however,

2121-538: Is a regular number (2 ·3·5). The number in Column 1 is either (169/120) or (119/120) . Each column has a heading, written in the Akkadian language . Some words are Sumerian logograms , which would have been understood by readers as standing for Akkadian words. These include ÍB.SI 8 , for Akkadian mithartum ("square"), MU.BI.IM, for Akkadian šumšu ("its line"), and SAG, for Akkadian pūtum ("width"). Each number in

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2222-473: Is a single-digit sexagesimal number, and these are in one-to-one correspondence with the rows of the tablet. He also points out that the even spacing of the numbers might not have been by design: it could also have arisen merely from the density of regular-number ratios in the range of numbers considered in the table. It was argued by de Solla Price that the natural lower bound for the ratio would be 1, which corresponds to an angle of 0°. He found that, maintaining

2323-499: Is different in the problem on YBC 6967 than on MS 3052 and MS 3971 (and by extension, on Plimpton 322). In the problem of YBC 6967, the members of the reciprocal pair are the lengths of the sides of a rectangle of area 1. The geometric meaning of x and 1/ x is not stated in the surviving text of the problems on MS 3052 and MS 3971. The goal appears to have been to apply a known procedure for producing rectangles with finite sexagesimal width and diagonal. It should also be pointed out that

2424-553: Is necessary to reinterpret Columns 2 and 3 as "the factor-reduced cores of the front and diagonal". The factor-reduced core of a number is the number with perfect-square regular factors removed; computing the factor-reduced core was part of the process of calculating square roots in Old Babylonian mathematics. According to Friberg, "it was never the intention of the author of Plimpton 322 to reduce his series of normalized diagonal triples (with length equal to 1 in each triple) to

2525-437: Is not stated but the calculation implies that it is taken to be 1. In modern terms, the calculation proceeds as follows: given x and 1/ x , first compute ( x +1/ x )/2, the diagonal. Then compute the width. Due to damage to the part of the tablet containing the first of the five parts, the statement of the problem for this part, apart from traces of the initial data, and the solution have been lost. The other four parts are, for

2626-618: 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 associated with Vedic ritual practice. The major texts of Kalpa Vedanga are called Kalpa Sutras in Hinduism. The scope of these texts includes Vedic rituals, rites of passage rituals associated with major life events such as birth, wedding and death in family, as well as personal conduct and proper duties in

2727-463: Is ongoing debate about their precise meaning. The headings of Columns 2 and 3 could be translated as "square of the width" and "square of the diagonal", but Robson (2001) (pp. 173–174) argues that the term ÍB.SI 8 can refer either to the area of the square or the side of the square, and that in this case it should be understood as "'square-side' or perhaps 'square root'". Similarly Britton, Proust & Shnider (2011) (p. 526) observe that

2828-412: Is similar to the explanation of the error in Row 15. An exception to the general consensus is Friberg (2007) , where, in a departure from the earlier analysis by the same author ( Friberg (1981) ), it is hypothesized that the numbers in Row 15 are not in error, but were written as intended, and that the only error in Row 2, Column 3 was miswriting 3 13 as 3 12 01. Under this hypothesis, it

2929-486: Is the measure]. which leads to the value of the square root of two as being: Indeed, an early method for calculating square roots can be found in some Sutras, the method involves the recursive formula: x ≈ x − 1 + 1 2 ⋅ x − 1 {\displaystyle {\sqrt {x}}\approx {\sqrt {x-1}}+{\frac {1}{2\cdot {\sqrt {x-1}}}}} for large values of x, which bases itself on

3030-615: The Karma kanda, or ritual parts of the Veda, in contrast to the Upanishads which are the Jnana kanda, or the knowledge part. This field of study emerged to serve the needs of priests as they officiated over domestic ceremonies such as weddings and baby naming rites of passage, so that the rituals were efficient, standardized and appeared consistent across different events. They also helped

3131-462: The Pythagorean theorem , both in the case of an isosceles right triangle and in the general case, as well as lists of Pythagorean triples . In Baudhayana, for example, the rules are given as follows: 1.9. The diagonal of a square produces double the area [of the square]. [...] 1.12. The areas [of the squares] produced separately by the lengths of the breadth of a rectangle together equal

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3232-453: The Rig . Come let us marry, let us possess offspring, united in affection, well disposed to each other, let us live for a hundred years". — Āśvalāyana Kalpa sutra, Book 1.7, Translated by Monier Monier-Williams The Dharmasūtras are texts dealing with custom, rituals, duties and law. They include the four surviving written works of the ancient Indian tradition on the subject of dharma, or

3333-482: The Schøyen Collection . Jöran Friberg translated and analyzed the two tablets and discovered that both contain examples of the calculation of the diagonal and side lengths of a rectangle using reciprocal pairs as the starting point. The two tablets are both Old Babylonian, of approximately the same age as Plimpton 322, and both are believed to come from Uruk, near Larsa. Further analysis of the two tablets

3434-494: The Smartasutras did not relate to Srauta or Vedic ceremonies, but instead focused on domestic events such as rites of passage when a baby is born and to Samayachara or "conventional everyday practices" that are part of every human being's life. However, other scholars include both. The Śrautasūtras (Shrauta-sutra) form a part of the corpus of Sanskrit sūtra literature. Their topics include instructions relating to

3535-471: The hypotenuse d {\displaystyle d} of the triangle. In all cases, the longer side l {\displaystyle l} is also an integer, making s {\displaystyle s} and d {\displaystyle d} two elements of a Pythagorean triple . The number in the first column is either the fraction s 2 / l 2 {\textstyle s^{2}/l^{2}} (if

3636-419: The "1" is not included) or d 2 l 2 = 1 + s 2 l 2 {\textstyle {\tfrac {d^{2}}{l^{2}}}\,=\,1+{\tfrac {s^{2}}{l^{2}}}} (if the "1" is included). In every case, the long side l {\displaystyle l} is a regular number , that is, an integer divisor of a power of 60 or, equivalently,

3737-502: 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

3838-566: The Kalpa Sutras text were probably composed by the 6th-century BCE, and they were attributed to famous Vedic sages out of respect for them in the Hindu traditions or to gain authority. These texts are written aphoristic sutras style, and therefore are taxonomies or terse guidebooks rather than detailed manuals or handbooks for any ceremony. Scholars such as Monier-Williams classified only Shrautasutras as part of Kalpa Vedanga, stating that

3939-417: The Old Babylonian period can account for all values of x used. Neugebauer and Sachs note that the triangle dimensions in the tablet range from those of a nearly isosceles right triangle (with short leg, 119, nearly equal to long leg, 120) to those of a right triangle with acute angles close to 30° and 60°, and that the angle decreases in a fairly uniform fashion in steps of approximately 1°. They suggest that

4040-568: The Shulba Sutras. The content of the Shulba Sutras is likely older than the works themselves. The Satapatha Brahmana and the Taittiriya Samhita , 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. The origin of

4141-465: The area [of the square] produced 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

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4242-491: The audience and the individuals integrate within customs and cultural practices, state Winternitz and Sarma, from "the moment when he is received in his mother's womb to the hour of his death", and beyond during his cremation. The rituals related ancient texts are of two kinds: (1) the Śrautasūtras , which are based on the śruti , and (2) the Smārtasūtras , or rules based on the smriti or tradition. The first versions of

4343-399: The broken-off portion of the heading takiltum may have been preceded by a-ša ("area"). There is now widespread agreement that the heading describes the relationship between the squares on the width (short side) and diagonal of a rectangle with length (long side) 1: subtracting ("tearing out") area 1 from the square on the diagonal leaves the area of the square on the width. As indicated in

4444-536: The broken-off portion of the tablet. Robson also argues that the proposal does not explain how the errors in the table could have plausibly arisen and is not in keeping with the mathematical culture of the time. In the reciprocal-pair proposal, the starting point is a single regular sexagesimal fraction x along with its reciprocal, 1/ x . "Regular sexagesimal fraction" means that x is a product of (possibly negative) powers of 2, 3, and 5. The quantities ( x −1/ x )/2, 1, and ( x +1/ x )/2 then form what would now be called

4545-437: The case where p and q are both odd. (Unfortunately, the only place where this occurs in the tablet is in Row 15, which contains an error and cannot therefore be used to distinguish between the proposals.) Proponents of the reciprocal-pair proposal differ on whether x was computed from an underlying p and q , but with only the combinations p  /  q and q  /  p used in tablet computations or whether x

4646-410: The correction in Row 15 are shown: either 53 in the third column should be replaced with twice its value, 1 46, or 56 in the second column should be replaced with half its value, 28. It is possible that additional columns were present in the broken-off part of the tablet to the left of these columns. Babylonian sexagesimal notation did not specify the power of 60 multiplying each number, which makes

4747-407: The diameter is divided into eight parts; one [such] part after being divided into twenty-nine parts is reduced by twenty-eight of them and further by the sixth [of the part left] less the eighth [of the sixth part]. 2.11. Alternatively, divide [the diameter] into fifteen parts and reduce it by two of them; this gives the approximate side of the square [desired]. The constructions in 2.9 and 2.10 give

4848-489: The end of the rectangle and to paste it to the side so as to form a gnomon 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. The Baudhayana Shulba sutra 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

4949-447: The entries in the table were the result of a deliberate selection process aimed at achieving the fairly regular decrease of the values in Column 1 within some specified bounds. Buck (1980) and Robson (2002) both mention the existence of a trigonometric explanation, which Robson attributes to the authors of various general histories and unpublished works, but which may derive from the observation in Neugebauer & Sachs (1945) that

5050-575: The expression for v 3 above with v 3 = 60 + v 2 . The side effect of obtaining a rational triple is thereby lost as the sides become v 1 , 60 {\displaystyle {\sqrt {60}}} , and v 4 . In this proposal it must be assumed that the Babylonians were familiar with both variants of the problem. Robson argues that the columns of Plimpton 322 can be interpreted as: In this interpretation, x and 1/ x (or possibly v 1 and v 4 ) would have appeared on

5151-414: 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". The four major Shulba Sutras, which are mathematically the most significant, are those attributed to Baudhayana , Manava , Apastamba and Katyayana . Their language is late Vedic Sanskrit , pointing to

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5252-502: The fourth column is preceded by the Sumerogram KI, which, according to Neugebauer & Sachs (1945) , "gives them the character of ordinal numbers." In the sexagesimal table above, italicized words and parts of words represent portions of the text that are unreadable due to damage to the tablet or illegibility, and that have been reconstructed by modern scholars. The terms ÍB.SI 8 and takiltum have been left untranslated as there

5353-417: The interpretation of these numbers ambiguous. The numbers in the second and third columns are generally taken to be integers. The numbers in the first column can only be understood as fractions, and their values all lie between 1 and 2 (assuming the initial 1 is present—they lie between 0 and 1 if it is absent). These fractions are exact, not truncations or rounded off approximations. The decimal translation of

5454-483: 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

5555-480: The life of an individual. Most Kalpasutras texts have experienced interpolation, changes and consequent corruption over their history, and Apasthamba Kalpasutra ancillary to the Yajurveda may be the best preserved text in this genre. Kalpa Sutras are also found in other Indian traditions, such as Jainism . Kalpa is a Sanskrit word that means "proper, fit, competent, sacred precept", and also refers to one of

5656-425: 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 mathematics (c. 2000 BCE–1600 BCE) as a possible origin, the c. 1800 BCE Plimpton 322 tablet containing

5757-409: The manner in which the tablet was computed. The number 7 12 1 in Row 13, Column 2, is the square of the correct value, 2 41. Assuming either that the lengths in Column 2 were computed by taking the square root of the area of the corresponding square, or that the length and the area were computed together, this error might be explained either as neglecting to take the square root, or copying

5858-482: The mathematical methodology to construct altar geometries for the Vedic rituals. The Sanskrit word "Shulba" means cord, and these texts are "rules of the cord". They provide, states Kim Plofker , what in modern mathematical terminology would be called "area preserving transformations of plane figures", tersely describing geometric formulae and constants. Five Shulba Sutras texts have survived through history, of which

5959-453: 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 to doubling and other geometric transformation problems. Seidenberg, followed by van der Waerden, sees a ritual origin for mathematics more broadly, postulating that

6060-485: The mid-fourth century BCE", but she places the Manava in the same period as the Baudhayana. 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,

6161-457: The missing digits could be; these interpretations differ only in whether or not each number starts with an additional digit equal to 1. With the differing extrapolations shown in parentheses, damaged portions of the first and fourth columns whose content is surmised shown in italics, and six presumed errors shown in boldface along with the generally proposed corrections in square brackets underneath, these numbers are Two possible alternatives for

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6262-416: The most part intact, and all contain very similar text. The reason for taking the diagonal to be half the sum of the reciprocal pair is not stated in the intact text. The computation of the width is equivalent to ( x −1/ x )/2, but that this more direct method of computation has not been used, the rule relating the square of the diagonal to the sum of the squares of the sides having been preferred. The text of

6363-576: The nature and purpose of the tablet. For readable popular treatments of this tablet see Robson (2002) recipient of the Lester R. Ford Award for expository excellence in mathematics or, more briefly, Conway & Guy (1996) . Robson (2001) is a more detailed and technical discussion of the interpretation of the tablet's numbers, with an extensive bibliography. Plimpton 322 is partly broken, approximately 13 cm wide, 9 cm tall, and 2 cm thick. New York publisher George Arthur Plimpton purchased

6464-448: The need to compute the square root of v 3 will, in general result in answers that do not have finite sexagesimal representations, the problem on YBC 6967 was set up—meaning the value of c was suitably chosen—to give a nice answer. This is, in fact, the origin of the specification above that x be a regular sexagesimal fraction: choosing x in this way ensures that both x and 1/ x have finite sexagesimal representations. To engineer

6565-445: The non-recursive identity a 2 + r ≈ a + r 2 ⋅ a {\displaystyle {\sqrt {a^{2}+r}}\approx a+{\frac {r}{2\cdot a}}} 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 . In his translation of Euclid's Elements , Heath outlines

6666-472: The number in column two, the result will be the length of the long side of the triangle. Consequently, the square root of the number (minus the one) in the first column is what we would today call the tangent of the angle opposite the short side. If the (1) is included, the square root of that number is the secant . In contraposition with these earlier explanations of the tablet, Robson (2002) claims that historical, cultural and linguistic evidence all reveal

6767-477: The numbers in Columns 2 and 3 an improper number of times in one of the columns. The number in Row 2, Column 3 has no obvious relationship to the correct number, and all explanations of how this number was obtained postulate multiple errors. Bruins (1957) observed that 3 12 01 might have been a simple miscopying of 3 13. If this were the case, then the explanation for the incorrect number 3 13

6868-425: The oldest dharmasūtras. Āpastamba Dharmasūtra Hārīta Dharmasūtra Hiraṇyakeśi Dharmasūtra Vaikhānasa Dharmasūtra Viṣṇu Dharmasūtra Plimpton 322 Plimpton 322 is a Babylonian clay tablet , notable as containing an example of Babylonian mathematics . It has number 322 in the G.A. Plimpton Collection at Columbia University . This tablet, believed to have been written around 1800 BC, has

6969-482: The oldest surviving is likely the Baudhayana Shulba Sutra (800-500 BCE), while the one by Katyayana may be chronologically the youngest (~300 BCE). The Gṛhyasūtras "domestic sūtras" are a category of Sanskrit texts prescribing Vedic rituals, mainly relating to rites of passage such as rituals of wedding , birth celebration, namegiving and coming of age (puberty). Their language is late Vedic Sanskrit , and they date to around roughly 500 BCE, contemporary with

7070-535: The pairs p , q were chosen deliberately with this goal in mind. It was observed by de Solla Price (1964) , working within the generating-pair framework, that every row of the table is generated by a q that satisfies 1 ≤  q <60, that is, that q is always a single-digit sexagesimal number. The ratio p / q takes its greatest value, 12/5=2.4, in Row 1 of the table, and is therefore always less than 2 + 1 ≈ 2.414 {\displaystyle {\sqrt {2}}+1\approx 2.414} ,

7171-413: The parameters in the problems from MS 3971 match any of the rows of Plimpton 322. As discussed below, all of the rows of Plimpton 322 have x ≥9/5, while all the problems on MS 3971 have x <9/5. The parameters of MS 3971 do, however, all correspond to rows of de Solla Price's proposed extension of the table of Plimpton 322, also discussed below. It must be emphasized that the role of the reciprocal pair

7272-402: 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, 1957-59, excavations by G. R. Sharma at Kausambi , but this altar does not conform to the dimensions prescribed by

7373-406: The presence of Column 1 is mysterious in this proposal, as it plays no role in the construction, and that the proposal does not explain why the rows of the table are ordered as they are, rather than, say, according to the value of p {\displaystyle p} or q {\displaystyle q} , which, under this hypothesis, might have been listed on columns to the left in

7474-399: The reciprocal of any regular factor common to the last sexagesimal digits of both, until no such common factor remains. As discussed above, the errors in the tablet all have natural explanations in the reciprocal-pair proposal. On the other, Robson points out that the role of Columns 2 and 3 and the need for the multiplier a remain unexplained by this proposal, and suggests that the goal of

7575-516: The reciprocal-pair proposal have also advocated this scheme. Robson (2001) does not directly address this proposal, but does agree that the table was not "full". She notes that in the reciprocal-pair proposal, every x represented in the tablet is at most a four-place sexagesimal number with at most a four-place reciprocal, and that the total number of places in x and 1/ x together is never more than 7. If these properties are taken as requirements, there are exactly three values of x "missing" from

7676-421: The requirement that q be a single-digit sexagesimal number, there are 23 pairs in addition to the ones represented by the tablet, for a total of 38 pairs. He notes that the vertical scoring between columns on the tablet has been continued onto the back, suggesting that the scribe might have intended to extend the table. He claims that the available space would correctly accommodate 23 additional rows. Proponents of

7777-415: The rules of behavior recognized by a community. Unlike the later dharmaśāstras , the dharmasūtras are composed in prose. The oldest dharmasūtra is generally believed to have been that of Apastamba , followed by the dharmasūtras of Gautama , Baudhayana , and an early version of Vashistha . It is difficult to determine exact dates for these texts, but the dates between 500 and 300 BCE have been suggested for

7878-455: The sake of good fortune"; the fingers alone, if he wishes only for daughters; the hairy side of the hand along with the thumbs if wishes for both (sons and daughters). Then, whilst he leads her towards the right three times around the fire, and round the water jar, he says in a low tone, "I am he, thou are she; thou art she, I am he, I am the heaven, thou art the earth; I am the Saman , thou art

7979-476: The same format as other administrative, rather than mathematical, documents of the period. The main content of Plimpton 322 is a table of numbers, with four columns and fifteen rows, in Babylonian sexagesimal notation. The fourth column is just a row number, in order from 1 to 15. The second and third columns are completely visible in the surviving tablet. However, the edge of the first column has been broken off, and there are two consistent extrapolations for what

8080-430: The second problem of MS 3052 has also been badly damaged, but what remains is structured similarly to the five parts of MS 3971, Problem 3. The problem contains a figure, which, according to Friberg, is likely a "rectangle without any diagonals". Britton, Proust & Shnider (2011) emphasize that the preserved portions of the text explicitly state the length to be 1 and explicitly compute the 1 that gets subtracted from

8181-764: The six Vedanga fields of study. In Vedanga context, the German Indologist Max Muller translates it as "the Ceremonial". The word is widely used in other contexts, such as "cosmic time" (one day for Brahma, 4.32 billion human years), as well as "formal procedures" in medicine or other secular contexts. The Kalpa field of study traces its roots to the Brahmana layer of texts in the Vedas, however its texts are more focussed, clear, short and practical for ceremonies. Kalpa Sutras are related to

8282-466: The solutions to the quadratic equation x − 1 x = c {\textstyle x-{\tfrac {1}{x}}=c} ). Robson (2002) analyzes the tablet, YBC 6967, in which such a problem is solved by calculating a sequence of intermediate values v 1 = c /2, v 2 = v 1 , v 3 = 1 + v 2 , and v 4 = v 3 , from which one can calculate x = v 4 + v 1 and 1/ x = v 4 − v 1 . While

8383-448: The square of the diagonal in the process of calculating the width as the square of the length. The initial data and computed width and diagonal for the six problems on the two tablets are given in the table below. The parameters of MS 3971 § 3a are uncertain due to damage to the tablet. The parameters of the problem from MS 3052 correspond to a rescaling of the standard (3,4,5) right triangle, which appears as Row 11 of Plimpton 322. None of

8484-406: The style of handwriting used for its cuneiform script : Robson (2002) writes that this handwriting "is typical of documents from southern Iraq of 4000–3500 years ago." More specifically, based on formatting similarities with other tablets from Larsa that have explicit dates written on them, Plimpton 322 might well be from the period 1822–1784 BC. Robson points out that Plimpton 322 was written in

8585-570: The table above, most scholars believe that the tablet contains six errors, and, with the exception of the two possible corrections in Row 15, there is widespread agreement as to what the correct values should be. There is less agreement about how the errors occurred and what they imply with regard to the method of the tablet's computation. A summary of the errors follows. The errors in Row 2, Column 1 (neglecting to leave spaces between 50 and 6 for absent 1s and 10s) and Row 9, Column 2 (writing 9 for 8) are universally regarded as minor errors in copying from

8686-403: The tablet from an archaeological dealer, Edgar J. Banks , in about 1922, and bequeathed it with the rest of his collection to Columbia University in the mid-1930s. According to Banks, the tablet came from Senkereh, a site in southern Iraq corresponding to the ancient city of Larsa . The tablet is believed to have been written around 1800 BC (using the middle chronology ), based in part on

8787-466: The tablet in the broken-off portion to the left of the first column. The presence of Column 1 is therefore explained as an intermediate step in the calculation, and the ordering of rows is by descending values of x (or v 1 ). The multiplier a used to compute the values in columns 2 and 3, which can be thought of as a rescaling of the side lengths, arises from application of the "trailing part algorithm", in which both values are repeatedly multiplied by

8888-451: The tablet to be more likely constructed from "a list of regular reciprocal pairs ." Robson argues on linguistic grounds that the trigonometric theory is "conceptually anachronistic": it depends on too many other ideas not present in the record of Babylonian mathematics from that time. In 2003, the MAA awarded Robson with the Lester R. Ford Award for her work, stating it is "unlikely that

8989-434: The tablet under these assumptions is shown below. Most of the exact sexagesimal fractions in the first column do not have terminating decimal expansions and have been rounded to seven decimal places. As before, an alternative possible correction to Row 15 has 28 in the second column and 53 in the third column. The entries in the second and third columns of Row 11, unlike those of all other rows except possibly Row 15, contain

9090-438: The tablet's author was to provide parameters not for quadratic problems of the type solved on YBC 6967, but rather "for some sort of right-triangle problems." She also notes that the method used to generate the table and the use for which it was intended need not be the same. Strong additional support for the idea that the numbers on the tablet were generated using reciprocal pairs comes from two tablets, MS 3052 and MS 3971, from

9191-402: The tablet, which she argues might have been omitted because they are unappealing in various ways. She admits the "shockingly ad hoc " nature of this scheme, which serves mainly as a rhetorical device for criticizing all attempts at divining the selection criteria of the tablet's author. Otto E. Neugebauer  ( 1957 ) argued for a number-theoretic interpretation, but also believed that

9292-417: The term often appears in the problems where completing the square is used to solve what would now be understood as quadratic equations, in which context it refers to the side of the completed square, but that it might also serve to indicate "that a linear dimension or line segment is meant". Neugebauer & Sachs (1945) (pp. 35, 39), on the other hand, exhibit instances where the term refers to outcomes of

9393-410: The trailing point algorithm was not used to rescale the side lengths in these problems. The quantity x in the reciprocal-pair proposal corresponds to the ratio p  /  q in the generating-pair proposal. Indeed, while the two proposals differ in calculation method, there is little mathematical difference between the results as both produce the same triples, apart from an overall factor of 2 in

9494-612: The use of the śruti corpus in ritual ('kalpa') and the correct performance of these rituals. Some early Śrautasūtras were composed in the late Brahmana period (such as the Baudhyanana and Vadhula Sūtras), but the bulk of the Śrautasūtras are roughly contemporary to the Gṛhya corpus of domestic sūtras, their language being late Vedic Sanskrit , dating to the middle of the first millennium BCE (generally predating Pāṇini ). Not verified though The Śulbasûtra (or Shulva-sutras ) deal with

9595-411: The values of the first column can be interpreted as the squared secant or tangent (depending on the missing digit) of the angle opposite the short side of the right triangle described by each row, and the rows are sorted by these angles in roughly one-degree increments. In other words, if you take the number in the first column, discounting the (1), and derive its square root, and then divide this into

9696-430: The word in Old Babylonian mathematics. While they note that, in almost all cases, it refers to the linear dimension of the auxiliary square added to a figure in the process of completing the square, and is the quantity subtracted in the last step of solving a quadratic, they agree with Robson that in this instance it is to be understood as referring to the area of a square. Friberg (2007) , on the other hand, proposes that in

9797-410: The wrong number from a work tablet. If the error in Row 15 is understood as having written 56 instead of 28 in Column 2, then the error can be explained as a result of improper application of the trailing part algorithm, which is required if the table was computed by means of reciprocal pairs as described below. This error amounts to applying an iterative procedure for removing regular factors common to

9898-438: The Śrautasūtras. They are named after Vedic shakhas . Vedic sacrifice rituals at a wedding West of the (sacred) fire, a stone (for grinding corn and condiments) is placed and northeast a water jar. The bridegroom offers an oblation, standing, looking towards the west, and taking hold of the bride's hands while she sits and looks towards the east. If he wishes only for sons, he clasps her thumbs and says, "I clasp thy hands for

9999-422: Was carried out in Britton, Proust & Shnider (2011) . MS 3971 contains a list of five problems, the third of which begins with "In order for you to see five diagonals" and concludes with "five diagonals". The given data for each of the five parts of the problem consist of a reciprocal pair. For each part the lengths of both the diagonal and the width (short side) of a rectangle are computed. The length (long side)

10100-411: Was generally regarded as untranslatable until Robson (2001) proposed inserting a 1 in the broken-off part of line 2 and succeeded in deciphering the illegible final word, producing the reading given in the table above. Based on a detailed linguistic analysis, Robson proposes translating takiltum as "holding square". Britton, Proust & Shnider (2011) survey the relatively few known occurrences of

10201-458: Was obtained directly from other sources, such as reciprocal tables. One difficulty with the latter hypothesis is that some of the needed values of x or 1/ x are four-place sexagesimal numbers, and no four-place reciprocal tables are known. Neugebauer and Sachs had, in fact, noted the possibility of using reciprocal pairs in their original work, and rejected it for this reason. Robson, however, argues that known sources and computational methods of

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