Proofs of Fermat's theorem on sums of two squares

The Shulba Sutras or Śulbasūtras (Sanskrit Template:IAST: "string, cord, rope") are sutra texts belonging to the Śrauta 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-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 composed by Baudhayana, Manava, Apastamba and Katyayana, about whom very little is known. The texts are dated by comparing their grammar and vocabulary with that of other Vedic texts. The texts have been dated from around 800 BCE to 200 CE, with the oldest being a sutra attributed to Baudhayana around 800 BCE to 600 BCE.

There are competing theories about the origins of the geometrical material found in the Shulba sutras. 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 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.

The Satapatha Brahmana and the Taittiriya Samhita were probably also aware of the Pythagoras theorem. 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". Seidenberg suggested that this source might be Sumerian and may predate 1700 BC. Template:Harvnb illustrates an application of the Pythagorean Theorem in the Shulba Sutra to convert a rectangle to a square of equal area.

Pythagorean triples

Pythagorean triples are found in Apastamba's rules for altar construction. They were used for the construction of right angles. The complete list is:

• $(3,4,5)$
• $(5,12,13)$
• $(8,15,17)$
• $(12,35,37).$

However, since these triples are easily derived from an old Babylonian rule, Mesopotamian influence is not unlikely.

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. 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.

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].

which leads to the value of the square root of two as being:

${\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}={\frac {577}{408}}=1.4142...$

One conjecture about how such an approximation was obtained is that it was taken by the formula:

${\sqrt {a^{2}+r}}\approx a+{\frac {r}{2a}}-{\frac {(r/2a)^{2}}{2(a+{\frac {r}{2a}})}},$ with $a=4/3$ and $r=2/9$

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.

This formula is also similar in structure to the formula found on a Mesopotamian tablet from the Old Babylonian period (1900-1600 BCE):

${\sqrt {2}}=1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}=1.41421297.$

which expresses ${\sqrt {2}}$ in the sexagesimal system, and which too is accurate up to 5 decimal places (after rounding).

Indeed an early method for calculating square roots can be found in some Sutras, the method involves the recursive formula: ${\sqrt {x}}\approx {\sqrt {x-1}}+{\frac {1}{2\cdot {\sqrt {x-1}}}}$ for large values of x, which bases itself on the non-recursive identity ${\sqrt {a^{2}+r}}\approx a+{\frac {r}{2\cdot a}}$ 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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Certain rules given by the famous Indian grammarian Pāṇini (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 and irrational numbers.

List of Shulba Sutras

The following Shulba Sutras exist in print or manuscript

1. Apastamba
2. Baudhayana
3. Manava
4. Katyayana
5. Maitrayaniya (somewhat similar to Manava text)
6. Varaha (in manuscript)
7. Vadhula (in manuscript)
8. Hiranyakeshin (similar to Apastamba Shulba Sutras)

Further reading

• Parameswaran Moorthiyedath, "Sulbasutra"
• 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

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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Citations and footnotes

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