Varahamihira is the Brihat-Samhita

Brihat – Samhita

Another important contribution of Varahamihira is the Brihat-Samhita. It covers wide ranging subjects of human interest, including astrology, planetary movements, eclipses, rainfall, clouds, architecture, growth of crops, manufacture of perfume, matrimony, domestic relations, gems, pearls, and rituals. The volume expounds on gemstone evaluation criterion found in the Garuda Purana, and elaborates on the sacred Nine Pearls from the same text. It contains 106 chapters and is known as the “great compilation”.

On Astrology

He was also an astrologer. He wrote on all the three main branches of Jyotisha astrology:

  • Brihat Jataka – is considered as one of the five main treatises on Hindu astrology on horoscopy.
  • Laghu Jataka – also known as ‘Swalpa Jataka’
  • Samasa Samhita – also known as ‘Lagu Samhita’ or ‘Swalpa Samhita’
  • Brihat Yogayatra – also known as ‘Mahayatra’ or ‘Yakshaswamedhiya yatra’
  • Yoga Yatra – also known as ‘Swalpa yatra’
  • Tikkani Yatra
  • Brihat Vivaha Patal
  • Lagu Vivaha Patal – also known as ‘Swalpa Vivaha Patal’
  • Lagna Varahi
  • Kutuhala Manjari
  • Daivajna Vallabha (apocryphal)

His son Prithuyasas also contributed in the Hindu astrology; his book Hora Sara is a famous book on horoscopy. Khana (also named Lilavati elsewhere) the medieval Bengali poetess astrologer is believed to be the daughter-in-law of Varahamihir.

Influences

The Romaka Siddhanta (“Doctrine of the Romans”) and the varahawere two works of Western origin which influenced Varahamihira’s thought, though this view is controversial as there is much evidence to suggest that it was actually Vedic thought indigenous to India which first influenced Western astrologers and subsequently came back to India reformulated. Number of his writings share similarities with with the earlier texts like Vedanga Jyotisha .

A comment in the Brihat-Samhita by Varahamihira says: “The Greeks, though Barbarians, must be honored since they have shown tremendous interest in our science…..” (“mleccha hi yavanah tesu samyak shastram kdamsthitam/ rsivat te ‘p i pujyante kim punar daivavid dvijah” (Brihat-Samhita 2.15)).

Place values system and Zero interduce to world by Aryabhata

Mathematics

Place value system and zero

The palce values system, first seen in the 3rd-century Bakhshali was clearly in place in his work. While he did not use a symbol for Zero, the French mathematician Georges Ifrah   argues that knowledge of zero was implicit in Aryabhata’s  Palce – Values System as a place holder for the powers of ten with Null -Coefficient

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic  tradition from  Vedic Times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sine’s in a mnemonic form.

Place values Chart

 

 Pi ( π ) values

Aryabhata worked on the approximation for pi, and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (Ganitapada 10), he writes:

Caturadhikam satama stagunam devasa stistatha sahasranam

Ayutadayavi siambhasyasanno vittaparinahah


“Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.”

This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figure.

OR

Aryabhatta

chaturadhikaM shatamaShTaguNaM dvAShaShTistathA sahasrANAm AyutadvayaviShkambhasyAsanno vr^ttapariNahaH.
[gaNita pAda, 10] Aryabhatiyam (499 CE)
“Add 4 to 100, multiply by 8 and add to 62,000. This is approximately the circumference of a circle whose diameter is 20,000.”
i.e.
correct to four places.

 

It is speculated that Aryabhata used the word ‘asana’ (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by “Lambert”.

After Aryabhatiya was translated into Arabic  (c. 820 CE) this approximation was mentioned in Al-Khwarizmi’s  book on algebra.

he value of pi is being used in India from ancient times. It gives us an insight about how evolved our past was. There is a shloka, a hymn to Lord Krishna or Shiva, which gives the value of pi upto 31 decimal places.

Its amazing that our forefathers used an encryption technique to easily remember it. What is more astonishing is that they needed pi upto 31 places.

Importance of Pi
Pi deals with circles and circles are very important in many fields. Pi is a very important number in the fields of :

  • Geometry and trigonometry,
  • Complex number and analysis,
  • Number theory,
  • Nrobability and statistic physics,
  • Engineering and geology.
  • Computers and many more
  • Katapayadi Encryption
  • gopiibhaagya madhuvraataH shruMgashodadhi saMdhigaH .
    khalajiivitakhaataava galahaalaa rasaMdharaH
  • This shloka, a hymn to Lord Krishna or Shiva, gives the value of pi upto 31 decimal places.
Pi using Encryption
  • • Katapayadi system is used to encode numbers in many shlokas

    ga – 3 pii – 1 bhaa – 4 gya – 1 ma – 5 dhu – 9 ra – 2 ta -6 shru – 5 ga – 3 sho – 5 da – 8 dhi – 9 sa – 7 dha – 9 ga – 3 kha – 2 la – 3 jii – 8 vi – 4 ta – 6 kha – 2 ta – 6 va – 4 ga – 3 la – 3 ra – 2 sa – 7 dha – 9 ra – 2

    pi = 3.1415926535897932384626433832792

 

to be continue …….2

Brahmagupta uses 3 as a “practical” value of π, and as an “accurate” value of π

Brahmagupta

Brahmagupta (598–.670 CE)Pi

 In verse 40, he gives values of

12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.

So Brahmagupta uses 3 as a “practical” value of π, and as an “accurate” value of π.

Measurements and constructions

In some of the verses before verse 40, Brahmagupta gives constructions of various figures with arbitrary sides. He essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and a scalene cyclic quadrilateral.

After giving the value of pi, he deals with the geometry of plane figures and solids, such as finding volumes and surface areas (or empty spaces dug out of solids). He finds the volume of rectangular prisms, pyramids, and the frustum of a square pyramid. He further finds the average depth of a series of pits. For the volume of a frustum of a pyramid, he gives the “pragmatic” value as the depth times the square of the mean of the edges of the top and bottom faces, and he gives the “superficial” volume as the depth times their mean area.

 

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Brahmagupta’s Theorem

Brahmagupta

Brahmagupta (598–.670 CE) : Brahmagupta’s Theorem

Brahmagupta’s theorem states that AF = FD

 bbs11

Brahmagupta continues,

12.23. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes].

So, in a “non-unequal” cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is .

He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral. This leads up to Brahmagupta’s Famous theorem,

12.30-31. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals. The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle]. Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular.

 

 

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Indian Mathematician and astronomer Brahmagupta

BrahmaguptaBrahmagupta (598–.670 CE)Triangles

Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle’s base is divided into by its altitude:

12.22. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment. Thus the lengths of the two segments are    sq1.

He further gives a theorem on rational triangles. A triangle with rational sides a, b, c and rational area is of the form:

sq2

 

for some rational numbers u, v, and w.

 

 

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Indian Mathematician and astronomer Brahmagupta

BrahmaguptaBrahmagupta (598–.670 CE) Bramhgupta Formla

 

circle

Main article: Bramhagupta’s formual

Brahmagupta’s most famous result in geometry is his format for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure’s area,

12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.

So given the lengths p, q, r and s of a cyclic quadrilateral, the approximate area

 

 

p1is while, p2    the exact area is

p2 p3

 

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. Heron’s formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.

 

 

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Brahmagupta generating soluation ot certain instances of Diophantine

BrahmaguptaPell’s Equation

Brahmagupta (598–.670 CE),Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as (called Pell’s Equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the “pulverizer” since it breaks numbers down into ever smaller pieces.

The nature of squares:
18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.

The key to his solution was the identity,

which is a generalization of an identity that was discovered by Diophantus,

Using his identity and the fact that if and are solutions to the equations and , respectively, then is a solution to , he was able to find integral solutions to the Pell’s equation through a series of equations of the form . Unfortunately, Brahmagupta was not able to apply his solution uniformly for all possible values of N, rather he was only able to show that if has an integer solution for k = ±1, ±2, or ±4, then has a solution. The solution of the general Pell’s equation would have to wait for Bhaskara II in c. 1150 CE.

 

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