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 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". The four major Shulba Sutras, which are mathematically the most significant, are those attributed to Baudhayana , Manava , Apastamba and Katyayana.
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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 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". The four major Shulba Sutras, which are mathematically the most significant, are those attributed to Baudhayana , Manava , Apastamba and Katyayana.
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.
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. Sharma at Kausambi , 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 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. Several Mathematicians and Historians mention that the earliest of the texts were written beginning in BCE by Vedic Hindus based on compilations of an oral tradition dating back to BCE.
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 BC.
The sutras contain statements of the Pythagorean theorem , both in the case of an isosceles right triangle and in the general case, as well as lists of Pythagorean triples. The diagonal of a square produces double the area [of the square]. The areas [of the squares] produced separately by the lengths of the breadth of a rectangle together equal the area [of the square] produced by the diagonal.
This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 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 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. These include transforming a square into a rectangle , an isosceles trapezium , an isosceles triangle , a rhombus , and a circle , and transforming a circle into a square.
As an example, the statement of circling the square is given in Baudhayana as:. If it is desired to transform 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.
To transform a circle into a square, 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]. 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.
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:. 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 is the measure]. 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.
From Wikipedia, the free encyclopedia. Part of a series on Hinduism Hindus History Origins. Main traditions. Vaishnavism Shaivism Shaktism Smartism. Rites of passage. Philosophical schools. Gurus, saints, philosophers. Other texts. Text classification. Other topics. Retrieved The American Mathematical Monthly. Geometry and Algebra in Ancient Civilizations. Springer Verlag. Mathematics in School. However all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the Sulvasutras 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 Sulbasutras 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. Origin of Vedas, Chapter 5. Notion Press. Princeton University Press. 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.
Glossary of Hinduism terms Hinduism portal. Part of a series of articles on the. Area of a circle Circumference Use in other formulae. Irrationality Transcendence. Chronology Book. Legislation Pi Day. Wikiquote has quotations related to: Shulba Sutras.
Square Roots in the Sulbasutra
They belong to the Taittiriya branch of the Krishna Yajurveda school and are among the earliest texts of the genre, perhaps compiled in the 8th to 6th centuries BCE. Moreover, the text has undergone alterations in the form of additions and explanations over a period of time. The date of the commentary is uncertain but according to Olivelle it is not very ancient. Olivelle states that Book One and the first sixteen chapters of Book Two are the 'Proto-Baudhayana'  even though this section has undergone alteration. Chapter 17 and 18 in Book Two lays emphasis on various types of ascetics and acetic practices. The first book is primarily devoted to the student and deals in topics related to studentship. It also refers to social classes, the role of the king, marriage, and suspension of Vedic recitation.
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