(Redirected from Sulba Sutras)The '''Shulba Sutras''' (
Sanskrit '': "string, cord, rope") are
sutra texts belonging to the
Åšrauta ritual and containing geometry related to fire-altar construction.
Purpose and origins
The Shulba Sutras are part of the larger corpus of texts called the
Shrauta Sutras, considered to be appendices to the
Vedas. They are the only sources of knowledge of
Indian mathematics from the
Vedic period. Unique fire-alter shapes were associated with unique gifts from the Gods. For instance, "he who desires heaven is to construct a fire-alter in the form of a falcon"; "a fire-alter 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-alter in the form of a rhombus".
[1]
The four major Shulba Sutras, which are mathematically the most significant, are those composed by
Baudhayana,
Manava,
Apastamba and
Katyayana, about who very little is known.
The texts are dated by comparing their grammar and vocabulary with the grammar and vocabulary of other Vedic texts.
The texts have been dated from around
800 BCE to
200 AD,
with the oldest being the sutra that was written by Baudhayana around 800 BCE to
600 BCE.
There are competing theories about the origin of the geometry that is found in the Shulba sutras, and of geometry in general. According to the theory of the ritual origins of geometry, different shapes symbolized different religious ideas, and the need to manipulate these shapes lead to the creation of the pertinent mathematics. Another theory is that the mystical properties of numbers and geometry were considered spiritually powerful and consequently, led to their incorporation into religious texts.
Mathematics
Pythagorean theorem
The sutras contain discussion and non-axiomatic demonstrations of cases of the
Pythagorean theorem and
Pythagorean triples. It is also implied and cases presented in the earlier work of
Apastamba[2] and
Baudhayana. Although there is no consensus on whether or not Apastamba's rule is derived from Mesopotamia. In Baudhayana, 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 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.[3]
The ''
Satapatha Brahmana'' and the ''
Taittiriya Samhita'' were probably also aware of the Pythagoras theorem.
[4] Seidenberg (1983) argued that either "Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source".
[5] Seidenberg suggested that this source might be
Sumerian and may predate
1700 BC.
Pythagorean triples
Pythagorean triples are found in
Apastamba's rules for altar construction.
[ Joseph, G. G. 2000. ''The Crest of the Peacock: The Non-European Roots of Mathematics''. Princeton University Press. 416 pages. ISBN 0691006598. page 229.] They were used for the construction of right angles.
The complete list is:
★
★
★
★
★
However, since these triples are easily derived from an old Babylonian rule, Mesopotamian influence is not unlikely.
[6]
Geometry
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 a
square into a
rectangle, an
isosceles trapezium, an isosceles
triangle, a
rhombus, and a
circle, and transforming a circle into a square.
[7] In these texts approximations, such as the transformation of a circle into a square, appear side by side with more accurate statements. As an example, the statement of circling the square is given in Baudhayana as:
2.9. If it is desired to transform a square into a circle, [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, 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 a value of π as 3.088, while the construction in 2.11 gives π as 3.004.
[8]
Square roots
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 is the measure].[9]
which leads to the value of the square root of two as being:
One conjecture about how such an approximation was obtained is that it was taken by the formula:
:
with
and
[10]
which is an approximation that follows a rule given by the twelfth century Muslim mathematician
Al-Hassar.
The result is correct to 5 decimal places.
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''.
Numerals
Before the period of the Sulbasutras was at an end, the
Brahmi numerals had definitely begun to appear (c. 300BCE) and the similarity with modern day numerals is clear to see. More importantly even still was the development of the concept of decimal place value. Certain rules given by the famous
Indian grammarian
Panini (c. 500 BCE) add a zero suffix (a suffix with no phonemes in it) to a base to form words, and this can be said somehow to imply the concept of the mathematical
zero.
Incommensurables
It has sometimes been suggested the sutras contain knowledge of irrationality, but such claims are not well substantiated and unlikely to be true.
[11]
List of Shulba Sutras
The following Shulba Sutras exist in print or manuscript
# Apastamba
# Baudhayana
# Manava
# Katyayana
# Maitrayaniya (somewhat similar to Manava text)
# Varaha (in manuscript)
# Vadhula (in manuscript)
# Hiranyakeshin (similar to Apastamba Shulba Sutras)
Further reading
★ Seidenberg, A. 1983. "The Geometry of the Vedic Rituals." In The Vedic Ritual of the Fire Altar. Ed. Frits Staal. Berkeley: Asian Humanities Press.
★ Sen, S.N., and A.K. Bag. 1983. The Sulbasutras. New Delhi: Indian National Science Academy.
References
★
The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, , Kim, Plofker, Princeton University Press, 2007,
★
A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc, 1991,
★
The History of Mathematics: A Brief Course, , Roger, Cooke, Wiley-Interscience, 1997,
Citations and footnotes
1. , , Kim, Plofker, , 2007,
2. The rule in the Apastamba cannot be derived from Old Babylon (Cf. Bryant 2001:263)
3. , , Kim, Plofker, , 2007,
4. Seidenberg 1983. Bryant 2001:262
5. Seidenberg 1983, 121
6. , , , Boyer, , 1991,
7. , , Kim, Plofker, , 2007,
8. {{cite book | first = Kim | last = Plofker | year = 2007 | pages = 392 | title = | quote = 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 s = 2r·13/15), which is already considered only "approximate." In 2.12, the ratio of a square's diagonal to its side (our is considered to be 1 + 1/3 + 1/(3·4) - 1/(3·4·34) = 1.4142.]}}
9. , , Kim, Plofker, , 2007,
10. {{cite book |last=Cooke |authorlink=Roger Cooke |title= |year=1997 |chapter=The Mathematics of the Hindus |pages=200 | quote = The Hindus had a very good system of approximating irrational square roots. Three of the ''Sulva Sutras'' contain the approximation for the diagonal of a square of side 1 (that is ). [...] We can only conjecture how such an approximation was obtained. One guess is the approximation
with a = 4/3 and r = 2/9. This approximation follows a rule given by the twelfth century Muslim mathematician Al-Hassar.}}
11. , , , Boyer, , 1991,
External links
★
Sulba Sutras
★
The Indian Sulbasutras
العربية
ä¸å›½
Français
Deutsch
Ελληνική
हिनà¥à¤¦à¥€
Italiano
日本語
Português
РуÑÑкий
Español
