Uranus, Neptune, Pluto mentioned in Mahabharata >30000 yrs ago

Uranus and Neptune were discovered through telescope in 781 AD by Herschel, but their calculated position never matched with original one.
It was corrected by German Astronomers in 1846 AD.
But Sage Veda Vyas mentioned Uranus, Neptune and Pluto in his epic poem Mahabharata (written more than 5000 years back in india).

Vyas has named them as Sweta, Syama and Teekshana.

Uranus or Sweta (Greenish White planet)

Vishesheena hi Vaarshneya Chitraam Pidayate Grahah….[10-Udyog.143]
Swetograhastatha Chitraam Samitikryamya Tishthati….[12-Bheeshma.3]

Here Vyas states that some greenish white (Sweta) planet has crossed Chitra Nakshatra.
Neelakantha of 17th century also had the knowledge of Uranus or Sweta.
Sweta means greenish white, which was later discovered to be the color of Uranus.

Neelakantha writes in his commentary on Mahabharat (Udyog 143) that Shveta, or Mahapata(one which has greater orbit) was a famous planet in the Astronomical science of India.
Neelakantha calls this “Mahapata” which means having greater orbit and it indicates a planet beyond Saturn.

Neptune or Syama (Bluish White planet)

Shukrahah Prosthapade Poorve Samaruhya Virochate Uttare tu Parikramya Sahitah Samudikshyate….[15-Bheeshma.3] Syamograhah Prajwalitah Sadhooma iva Pavakah Aaindram Tejaswi Naksha- tram Jyesthaam Aakramya Tishthati…[16-Bheeshma.3]

Vyas mentions that a bluish white (Syama) planet was in Jyeshtha and it was smoky (Sadhoom).
Neelkantha calls it “Parigha” (circumference) in his commentary on Mahabharat.
He could mean that its orbit was almost of the circumference of our solar system.

How did Sage Vyas see color of these planets ?
Mirrors and Microscopic Vision are mentioned in Mahabharata (Shanti A. 15,308).
So, lenses and telescopes must also be present at that time.
In ancient literature, Durbini (device used to see objects at far off distance, similar to binoculars) were mentioned.

Pluto or Teekshana/Teevra

Pluto was discovered to the modern world in 1930.

Krittikaam Peedayan Teekshnaihi Nakshatram…[30-Bheeshma.3]

Here Vyas states that some immobile liminary troubling Krittika (Pleides) with its sharp rays.
This was mentioned as Nakshatra because it was stationary at one place for long period, so it must be a planet in outer orbit.

It gets mentioned again as

Krittikasu Grahasteevro Nakshatre Prathame Jvalan…… [26- Bhishma.3]

Mathematical calculations make it clear that Krittika and Pluto were conjunct during mahabharata period.

Vyas has mentioned ‘seven Great planets‘, three times in Mahabharat.

Deepyamanascha Sampetuhu Divi Sapta Mahagrahah…[2-Bhishma.17]

It means that the seven great planets were brilliant and shining. In traditional indian astrology, Rahu and Ketu are nodes/shadows and do not shine like stars or planets.

Nissaranto Vyadrushanta Suryaat Sapta Mahagrahah…[4-Karna 37]

This line states that these seven great planets were ‘seen‘ moving away from the Sun.
Since Rahu & Ketu cannot be ‘seen’, they can be ruled out.
This statement is made on sixteenth day of Kurukshetra War, so the Moon has moved away from Sun.
Hence we can assume Mars, Mercury, Jupiter, Venus, Saturn, Uranus and Neptune are the seven great planets mentioned by Vyas.
Pluto was neglected due to its lesser impact on earth.

Moon was excluded because,

Praja Samharane Rajan Somam Sapta grahah Iva….[22-Drona 37]

Here again seven planets are mentioned by Vyas, excluding the Moon.

Though they were described 5100 years back, we forgot about Uranus, Neptune and Pluto because in traditional indian astrology because they were not used to predict future.

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Pythagorean (Pythagoras) Theorem in Baudhayana Sulba Sutra (800 BC)

In mathematics, the Pythagorean (Pythagoras) theorem (written around 400 BC) is a relation among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:

“In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).”

But in reality, this was written much earlier in ancient india by sage Baudhayana (around 800 BC).
He is noted as the author of the earliest Sulba Sūtra—appendices to the Vedas giving rules for the construction of altars—called the Baudhāyana Śulbasûtra, which contained several important mathematical results.
He is accredited with calculating the value of pi (π) before Pythagoras.

Solka in Baudhāyana Śulbasûtra that describes Pythagoras theorem is given below :

dīrghasyākṣaṇayā rajjuH pārśvamānī, tiryaDaM mānī, cha yatpṛthagbhUte kurutastadubhayāṅ karoti.

Baudhāyana used a rope as an example in the above sloka.
Its translation means : A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.

Proof of Pythagoras theorem has been provided by both Baudhāyana and Āpastamba in their Sulba Sutras.
Though, Baudhāyana was not the only Indian mathematician to have provided Pythagorean triplets and proof. Āpastamba also provided the proof for Pythagoras theorem, which is numerical in nature and unfortunately, Pythagoras was wrongly credited by Cicero and early Greek mathematicians for this theorem.

Also, another ancient Indian mathematician called Bhaskara later provided a unique geometrical proof as well as numerical which is known for the fact that it’s truly generalized and works for all sorts of triangles and is not incongruent.

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Pingala, Inventor of Binary Numbers in 2nd Century BCE

Pingala (Devanagari: पिङ्गल) is the author of Chandaḥśāstra (Chandaḥsūtra), the earliest known Sanskrit treatise on prosody.
Very less historical knowledge is available about Pingala, though his works are retained till date.
He is identified either as the younger brother of Pāṇini (4th century BCE), or of Patañjali, the author of the Mahabhashya (2nd century BCE).

His work, Chandaḥśāstra means science of meters, is a treatise on music and can be dated back to 2nd century BCE.
Main commentaries on ‘Chandaḥśāstra‘ are ‘Vrittaratnakara‘ by Kedara in 8th century AD, ‘Tatparyatika‘ by Trivikrama in 12th century AD and ‘Mritasanjivani‘ by Halayudha in 13th century AD. The complete significance of Pingala’s work can be understood by the explanations found in these three commentaries.

Pingala (in Chandaḥśāstra 8.23) has assigned the following combinations of zero and one to represent various numbers, much in the same way as the present day computer programming procedures.

0 0 0 0 numerical value = 1
1 0 0 0 numerical value = 2
0 1 0 0 numerical value = 3
1 1 0 0 numerical value = 4
0 0 1 0 numerical value = 5
1 0 1 0 numerical value = 6
0 1 1 0 numerical value = 7
1 1 1 0 numerical value = 8
0 0 0 1 numerical value = 9
1 0 0 1 numerical value = 10
0 1 0 1 numerical value = 11
1 1 0 1 numerical value = 12
0 0 1 1 numerical value = 13
1 0 1 1 numerical value = 14
0 1 1 1 numerical value = 15
1 1 1 1 numerical value = 16

Other numbers have also been assigned zero and one combinations likewise.
Pingala’s system of binary numbers starts with number one (and not zero). The numerical value is obtained by adding one to the sum of place values. In this system, the place value increases to the right, as against the modern notation in which it increases towards the left.

The procedure of Pingala system is as follows:

• Divide the number by 2. If divisible write 1, otherwise write 0.
• If first division yields 1 as remainder, add 1 and divide again by 2. If fully divisible, write 1, otherwise write 0 to the right of first 1.
• If first division yields 0 as remainder that is, it is fully divisible, add 1 to the remaining number and divide by 2. If divisible, write 1, otherwise write 0 to the right of first 0.
• This procedure is continued until 0 as final remainder is obtained.

Example to understand Pingala System of Binary Numbers :

Find Binary equivalent of 122 in Pingala System :

Divide 122 by 2. Divisible, so write 1 and remainder is 61.

1

Divide 61 by 2. Not Divisible and remainder is 30. So write 0 right to 1.

10

Add 1 to 61 and divide by 2 = 31.
Divide 31 by 2. Not Divisible and remainder is 16. So write 0 to the right.

100

Divide 16 by 2. Divisible and remainder is 8. So write 1 to right.

1001

Divide 8 by 2. Divisible and remainder is 4. So write 1 to right.

10011

Divide 4 by 2. Divisible and remainder is 2. So write 1 to right.

100111

Divide 2 by 2. Divisible. So place 1 to right.

1001111

Now we have 122 equivalent to 1001111.

Verify this by place value system : 1×1 + 0x2 + 0x4 + 1×8 + 1×16 + 1×32 + 1×64 = 64+32+16+8+1 = 121
By adding 1(which we added while dividing 61) to 121 = 122, which is our desired number.
In Pingala system, 122 can be written as 1001111.

Though this system is not exact equivalent of today’s binary system used, it is very much similar with its place value system having 20, 20, 21, 22, 22, 23, 24, 25, 26 etc used to multiple binary numbers sequence and obtain equivalent decimal number.

Reference : Chandaḥśāstra (8.24-25) describes above method of obtaining binary equivalent of any decimal number in detail.
These were used 1600 years before westeners invented binary system.

We now use zero and one (0 and 1) in representing binary numbers, but it is not known if the concept of zero was known to Pingala— as a number without value and as a positional location.
Pingala’s work also contains the Fibonacci number, called mātrāmeru, and now known as the Gopala–Hemachandra number. Pingala also knew the special case of the binomial theorem for the index 2, i.e. for (a + b) 2, as did his Greek contemporary Euclid.

Halayudha (10th century AD) who wrote a commentary on Pingala’s work understood and used zero in the modern sense but by then it was commonplace in India and had also begun to make its way to West Asia as well to countries like Indonesia, Cambodia and others in East and Southeast Asia. It took several centuries more before being accepted in Europe. It was Leonardo of Pisa, better known as Fibonacci who seems to have introduced it in Europe in the 13th century. (He learnt it from the Arabs but noted that it came from India. His successors were not so careful, and for centuries they were known as Arabic numerals.)

Halayudha was himself a mathematician no mean order. His discussion of combinatorics of poetic meters led him to a general version of the binomial theorem centuries before Newton. (This was the integer version only and not the full general version with arbitrary index given by Newton.) This too traveled east and west with the Persian mathematician and poet using the results in the 13th century.
Halāyudha’s commentary includes a presentation of the Pascal’s triangle for binomial coefficients (called meruprastāra).

Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables (Short = 0, Long = 1).

Use of zero is sometimes mistakenly ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, while Pingala used short and long syllables.
As Pingala’s system ranks binary patterns starting at one (four short syllables—binary “0000”—is the first pattern), the nth pattern corresponds to the binary representation of n-1, written backwards. Positional use of zero dates from later centuries and would have been known to Halāyudha but not to Pingala.

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BrahmaGupta, Ancient Mathematician-concept of ‘Negative Numbers’ & Theorem on Cyclic Quadrilaterals

Brahmagupta (Sanskrit: ब्रह्मगुप्त) was an Indian mathematician and astronomer who lived between 597–668 AD and wrote two important works on mathematics and astronomy: The Brāhmasphuṭasiddhānta in 628 AD (Correctly Established Doctrine of Brahma) which is a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text.
He is believed to be born in Bhinmal (in Hindi भीनमाल, which was originall known as Bhillamala in ancient days) which is in present day Rajasthan and he was known as Bhillamalacarya (the teacher from Bhillamala) and later went on to become the head of the astronomical observatory at Ujjain in central India.
Brahmagupta was the first to give rules to compute with zero.

Most of his works are composed in elliptic verse, a common practice in Indian mathematics at the time, and consequently have something of a poetic ring to them. But since no proofs are given(found), it is not known how Brahmagupta’s mathematics was derived.

The historian al-Biruni (c. 1050) in his book Tariq al-Hindstates that the Abbasid caliph al-Ma’mun had an embassy in India and from India a book was brought to Baghdad which was translated into Arabic as Sindhind. It is generally presumed that Sindhind is none other than Brahmagupta’s Brahmasphuta-siddhanta. That is how an important link between Indian mathematics, Astronomy and the nascent upsurge in science and mathematics in the Islamic world formed.

Brahmagupta’s work in Mathematics

Arithmetic :

In his work on arithmetic, Brahmagupta explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots. He also gave rules for dealing with five types of combinations of fractions. He gave the sum of the squares of the first n natural numbers as n(n + 1)(2n + 1)⁄ 6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)².

Usage of ZERO :

Brahmagupta stated that ‘When ZERO is added to a number or subtracted from a number, the number remains unchanged.
A number multiplied by ZERO becomes ZERO.

Positive and Negative numbers usage :

His statements about debt(negative numbers) and fortune(positive numbers) are :
A debt minus ZERO is a debt.
A fortune minus ZERO is a fotune.
Zero minus Zero is a Zero.
A debt subtracted from Zero is a fortune.
A fortune subtracted from Zero is a debt.
Zero multiplied by debt or fortune is a Zero.
Zero multipled by Zero is a Zero.
Product(multiplication) or Quotient(division) of two debts is a fortune.
Product of Quotient of two fortunes is a fortune.
Product of Quotient of a debt and a fortune is a debt.

Brahmagupta established the basic mathematical rules for dealing with zero (1 + 0 = 1; 1 – 0 = 1; and 1 x 0 = 0), although his understanding of division by zero was incomplete (he thought that 1 ÷ 0 = 0). Almost 500 years later, in the 12th Century, another Indian mathematician, Bhaskara II, showed that the answer should be infinity, not zero (on the grounds that 1 can be divided into an infinite number of pieces of size zero), an answer that was considered correct for centuries. However, this logic does not explain why 2 ÷ 0, 7 ÷ 0, etc, should also be zero – the modern view is that a number divided by zero is actually “undefined” (i.e. it doesn’t make sense)

Before Brahmagupta, the result of 3 – 4 was considered to have no answer or at the most as ‘0’. But he introduced the idea of debt(negative numbers) and showed how to borrow and subtract to attain a negative number.

Furthermore, he pointed out, quadratic equations (of the type x² + 2 = 11, for example) could in theory have two possible solutions, one of which could be negative, because 3² = 9 and -3² = 9. In addition to his work on solutions to general linear equations and quadratic equations, Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables), and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657.

Brahmagupta’s Theorem on cyclic quadrilaterals:

Brahmagupta dedicated a substantial portion of his work to geometry and trigonometry. He established √10 (3.162277) as a good practical approximation for π (3.141593), and gave a formula, now known as Brahmagupta’s Formula, for the area of a cyclic quadrilateral, as well as a celebrated theorem on the diagonals of a cyclic quadrilateral, usually referred to as Brahmagupta’s Theorem.

Astronomy :

Brahmagupta taught Arabs about Astronomy.
The famous Abbasid caliph Al-Mansur (712–775) founded Baghdad, which is situated on the banks of the Tigris, and made it a center of learning. The caliph invited a scholar of Ujjain by the name of Kankah in 770 A.D. Kankah used the Brahmasphutasiddhanta to explain the Hindu system of arithmetic astronomy. Muhammad al-Fazari translated Brahmugupta’s work into Arabic upon the request of the caliph.
In chapter 7 of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun, an idea which is maintained in scriptures. He does this by explaining the illumination of the Moon by the Sun.

• 7.1. If the moon were above the sun, how would the power of waxing and waning, etc., be produced from calculation of the [longitude of the] moon? the near half [would be] always bright.
• 7.2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun.
• 7.3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] half-month, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation.

Some of the important contributions made by Brahmagupta in astronomy are: methods for calculating the position of heavenly bodies over time (ephemerides), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses.[26] Brahmagupta criticized the Puranic view that the Earth was flat or hollow. Instead, he observed that the Earth and heaven were spherical and that the Earth is moving. In 1030, the Muslim astronomer Abu al-Rayhan al-Biruni, in his Ta’rikh al-Hind, later translated into Latin as Indica, commented on Brahmagupta’s work and wrote that critics argued:
“If such were the case, stones would and trees would fall from the earth.”

According to al-Biruni, Brahmagupta responded to these criticisms with the following argument on gravitation:
“On the contrary, if that were the case, the earth would not vie in keeping an even and uniform pace with the minutes of heaven, the pranas of the times. […] All heavy things are attracted towards the center of the earth. […] The earth on all its sides is the same; all people on earth stand upright, and all heavy things fall down to the earth by a law of nature, for it is the nature of the earth to attract and to keep things, as it is the nature of water to flow, that of fire to burn, and that of wind to set in motion… The earth is the only low thing, and seeds always return to it, in whatever direction you may throw them away, and never rise upwards from the earth.”

About the Earth’s gravity he said: “Bodies fall towards the earth as it is in the nature of the earth to attract bodies, just as it is in the nature of water to flow.“

Aryabhata contributes ‘ZERO, Pi’ etc to Mathematics and calculates Eclipses in Astronomy

Aryabhata, born in 476 CE, was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.
There is a general tendency to misspell his name as “Aryabhatta” by analogy with other names having the “bhatta” suffix, but all his astronomical text spells his name as Aryabhata.
He mentions in his work Aryabhatiya that it was composed 3,630 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476

Though his birthplace is uncertain, he went to Kusumapura (Pataliputra or modern day Patna) for advanced studies and lived there for sometime as the head of an institution (kulapati).
Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.

He wrote many books on mathematics, astronomy etc but most of them are lost today.
His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.
And Arabic translation of Aryabhata’s work is Al ntf or Al-nanf and it claims that it is a translation by Aryabhata, but the original Sanskrit name of this work is not known.

Place Value System and ZERO

The place-value system, first seen in the 3rd-century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah explains that knowledge of zero was implicit in Aryabhata’s place-value system as a place holder for the powers of ten with null coefficients.
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 sines in a mnemonic form.

Approximation of π

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

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

Translation : “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 calculates to 3.1416 close to the actual value Pi (3.14159).
Aryabhata used the word āsanna (approaching / approximating), to mean that not only is this an approximation but that the value is incommensurable (or irrational).
This is quite a sophisticated insight, because the irrationality of pi(π) was proved only in 1761 by Johann Heinrich Lambert.
After Aryabhatiya was translated into Arabic (during 820 CE) this approximation was mentioned in Al-Khwarizmi‘s book on algebra.

Contributions in Trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as :

tribhujasya phalashariram samadalakoti bhujardhasamvargah

Translation : “for a triangle, the result of a perpendicular with the half-side is the area.”

Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means “half-chord (half-wave)“. For simplicity, people started calling it jya.
When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba.
However, in Arabic writings, vowels are omitted, and it was abbreviated as jb.
Later writers substituted it with jaib, meaning “pocket” or “fold (in a garment)“.
Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means “cove” or “bay“; thence comes the English SINE.
Alphabetic code has been used by him to define a set of increments. If we use Aryabhata’s table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.

Indeterminate or Diophantine Equations

An example from Bhāskara’s commentary on Aryabhatiya :

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7
That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85.
They were discussed extensively in ancient Vedic text Baudhayana Sulba Sutras, which date to 800 BCE.
Aryabhata’s method of solving such problems is called the kuṭṭaka (कुट्टक) method.
Kuttaka means “pulverizing” or “breaking into small pieces“, and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm.
The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulbasutras.

In his contribution towards Algebra, Aryabhata provided elegant results for the summation of series of squares and cubes in his book Aryabhatiya.

Aryabhata’s contributions in Astronomy

Aryabhata’s system of astronomy was called the audAyaka system, in which days are reckoned from sunrise, dawn at lanka or “equator“.
Some of his later writings on astronomy, which apparently proposed a second model (or ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in Brahmagupta’s khanDakhAdyaka.
In some texts, he seems to ascribe the apparent motions of the heavens to the Earth’s rotation and he may have believed that the planet’s orbits as elliptical rather than circular.

Motions of the Solar System

In the first chapter of his book Aryabhatia, he insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view in other parts of the world, that the sky rotated.
Here, he gives the number of rotations of the earth in a yuga, and made more explicit in his gola chapter.

Eclipses

Lunar and Solar eclipses were scientifically explained by Aryabhata by stating that the Moon and planets shine by reflected sunlight.
Instead of the prevailing cosmogony in which eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth’s shadow and solar eclipse occurs when Moon intersects Sunrays from falling on Earth.
He discussed the size and extent of the Earth’s shadow (verses gola.38–48) and then provides the computation and the size of the eclipsed part during an eclipse. Later Indian astronomers improved on the calculations, but Aryabhata’s methods provided the core.
His computational paradigm was so accurate that 18th-century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.

Sidereal Periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds; whereas the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days) is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).

Heliocentrism

Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms of the mean speed of the sun.
Aryabhata’s calculations were based on an underlying heliocentric model, in which the planets orbit the Sun, though this has been rebutted.

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Bhaskaracharya Bhāskara II -introduced concept of ‘Infinity’

Bhaskaracharya (Bhāskara the teacher) was an Indian mathematician and astronomer of 12th century AD.
He is refered as Bhāskara II to avoid confusion with Bhāskara I (of 7th century AD).
He was born near Vijjadavida (Bijapur in modern Karnataka) and lived between 1114-1185 AD.

He represented the peaks of mathematical knowledge in the 12th century and was the head of the astronomical observatory at Ujjain, the leading mathematical centre of ancient India.
Bhaskara II’s family belonged to Deshastha Brahmin community, which served as court scholars at Kings forts.
He learned Mathematics from his father Maheswara, an astrologer.
He imparted his knowledge of mathematics to his son Lokasamudra, whose son had started a school to study the works of his grand father in 1207 AD.

His main work Siddhānta Shiromani, (Sanskrit for “Crown of treatises,“) is divided into four parts called Lilāvati(beautiful woman, named after his daughter Lilavati), Bijaganita, Grahaganita (mathematics of planets) and Golādhyāya (study of sphere/earth).
These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karna Kautoohala.
Bhāskara’s work on calculus predates Newton and Leibniz by over half a millennium.
He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.

Lilavati (meaning a beautiful woman) is based on Arithmetic. It is believed that Bhaskara named this book after his daughter Lilavati. Many of the problems in this book are addressed to his daughter. For example “Oh Lilavati, intelligent girl, if you understand addition & subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10 & 100, as well as [the remainder of] those when subtracted from 10000.” The book contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical & geometric progressions. Many of the methods in the book on computing numbers such as multiplications, squares & progressions were based on common objects like kings & elephants, which a common man could understand.

Bijaganita is on Algebra & contains 12 chapters.
“A positive number has two square-roots (a negative root & a positive root)“. This was published in this text for the very first time. It contains concepts of positive & negative numbers, zero, the ‘unknown‘ (includes determining unknown quantities), surds, simple equations & quadratic equations.

Bhaskara was the first to introduce the concept of Infinity : If any finite number is divided by zero, the result is infinity.
Also the fact that if any finite number is added to infinity then the sum is infinity. He developed a proof of the Pythogorean theorem by calculating the same area in two different ways & then cancelling out two terms to get a² + b² = c².
He is also known for his calculation of the time required (365.2588 days) by the Earth to orbit the Sun which differs from the modern day calculation of 365.2563 days, by just 3.5 minutes!
The law of Gravitation had been proved by Bhaskara 500 years before it was rediscovered by Newton.

Bhaskaracharya’s contributions

Mathematics
Some of Bhaskara’s contributions to mathematics include the following:
A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².
In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century
A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.
The first general method for finding the solutions of the problem x² − ny² = 1 (so-called “Pell’s equation“) was given by Bhaskara II.
Solutions of Diophantine equations of the second order, such as 61x² + 1 = y². This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.
Solved quadratic equations with more than one unknown, and found negative and irrational solutions.
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.
Conceived differential calculus, after discovering the derivative and differential coefficient.
Stated Rolle’s theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)

Arithmetic
Bhaskara’s arithmetic text Leelavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.
Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:
Definitions.
Properties of zero (including division, and rules of operations with zero).
Further extensive numerical work, including use of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods of multiplication, and squaring.
Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
Problems involving interest and interest computation.
Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara’s method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.
His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore the Lilavati contained excellent recreative problems and it is thought that Bhaskara’s intention may have be.

Algebra
His Bijaganita (“Algebra”) was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root).
His work Bijaganita is effectively a treatise on algebra and contains the following topics:
Positive and negative numbers.
Zero.
The ‘unknown’ (includes determining unknown quantities).
Determining unknown quantities.
Surds (includes evaluating surds).
Kuttaka (for solving indeterminate equations and Diophantine equations).
Simple equations (indeterminate of second, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the type ax² + b = y²).
Solutions of indeterminate equations of the second, third and fourth degree.
Quadratic equations.
Quadratic equations with more than one unknown.
Operations with products of several unknowns.
Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax² + bx + c = y.
Bhaskara’s method for finding the solutions of the problem Nx² + 1 = y² (the so-called “Pell’s equation“) is of considerable importance.

Trigonometry
The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara’s knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include the now well known results for sin(a + b) and sin(a – b)

Calculus
His work, the Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.
Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement. Bhaskara also goes deeper into the ‘differential calculus‘ and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ‘infinitesimals‘.
There is evidence of an early form of Rolle’s theorem in his work

• If  f (a) = f (b) = 0, then f ‘ (x) = 0 for some x with a<x<b then
• He gave the result that if x =(approx) y then sin(y) – sin(x) =(approx) (y-x) cos(y), thereby finding the derivative of sine, although he never developed the notion of derivatives.

Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.
In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
He was aware that when a variable attains the maximum value, its differential vanishes.
He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.
In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle’s theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara’s Lilavati.
Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara’s work and further advanced the development of calculus in India.

Astronomy

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is same as in Suryasiddhanta. The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes !
His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.
The twelve chapters of the first part cover topics such as:
Mean longitudes of the planets.
True longitudes of the planets.
The three problems of diurnal rotation.
Syzygies.
Lunar eclipses.
Solar eclipses.
Latitudes of the planets.
Sunrise equation
The Moon’s crescent.
Conjunctions of the planets with each other.
Conjunctions of the planets with the fixed stars.
The paths of the Sun and Moon.
The second part contains thirteen chapters on the sphere. It covers topics such as:
Praise of study of the sphere.
Nature of the sphere.
Cosmography and geography.
Planetary mean motion.
Eccentric epicyclic model of the planets.
The armillary sphere.
Spherical trigonometry.
Ellipse calculations.
First visibilities of the planets.
Calculating the lunar crescent.
Astronomical instruments.
The seasons.
Problems of astronomical calculations.

Engineering

The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever.
Bhāskara II used a measuring device known as Yasti-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.

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