Шүлба Сутралары: Веда дәуірі геометриясы және құрбандық алаптары

The Shulva Sutras or Śulbasūtras (Sanskrit: शुल्बसूत्र; śulba: "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire altar construction.

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. Their language is late Vedic Sanskrit, pointing to a composition roughly during the 1st millennium BCE. According to Plofker, the Katyayana was composed after "the great grammatical codification of Sanskrit by Pāṇini in probably 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, a word later applied to mean a rule or algorithm in general) or verse, particularly in the Classical period. Naturally, ease of memorization sometimes interfered with ease of comprehension. As a result, most treatises were supplemented by one or more prose commentaries " There are multiple commentaries for each of the Shulba Sutras, but these were written long after the original works. The commentary of Sundararāja on the Apastamba, for example, comes from the late 15th century CE and the commentary of Dvārakãnātha on the Baudhayana appears to borrow from Sundararāja. According to Staal, certain aspects of the tradition described in the Shulba Sutras would have been "transmitted orally", and he points to places in southern India where the fire altar ritual is still practiced and an oral tradition preserved. The fire altar tradition largely died out in India, however, and Plofker warns that those pockets where the practice remains may reflect a later Vedic revival rather than an unbroken tradition. 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 800 BCE by Vedic Hindus based on compilations of an oral tradition dating back to 2000 BCE. It is possible, as proposed by Gupta, that the geometry was developed to meet the needs of ritual. Some scholars go farther: Staal hypothesizes a common ritual origin for Indian and Greek geometry, citing similar interest and approach to doubling and other geometric transformation problems. Seidenberg, followed by van der Waerden, sees a ritual origin for mathematics more broadly, postulating that the major advances, such as discovery of the Pythagorean theorem, occurred in only one place, and diffused from there to the rest of the world. Van der Waerden mentions that author of Sulbha sutras existed before 600 BCE and could not have been influenced by Greek geometry. While Boyer mentions Old Babylonian mathematics (c. 2000 BCE–1600 BCE) as a possible origin, however also states that Shulba sutras contain a formula not found in Babylon sources. 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 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".

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:
<blockquote>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 is the measure]. Indeed, an early method for calculating square roots can be found in some Sutras, the method involves the recursive formula: for large values of x, which bases itself on the non recursive identity for values of r extremely small relative to a. It has also been suggested, for example by Bürk that this approximation of √2 implies knowledge that √2 is irrational. 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.