Bhaskaracharya – The Crown Jewel of Mathematics and AstronomyA Glance at the Astronomical Achievements of Bhaskaracharya…1. The Earth is not flat, has no support and has a power of attraction.

2. The north and south poles of the Earth experience six months of day and six months of night.

3. One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.

4. Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars. There is slight difference between the orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

5. Earth’s atmosphere extends to 96 kilometers and has seven parts.

6. There is a vacuum beyond the Earth’s atmosphere.

There have been several unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara’s work, but this once again seems like an attempt by European scholars to claim European influence on (all) the great works of mathematics. These claims should be ignored. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.

The period between 500 and 1200 AD was the golden age of Indian Astronomy. In this long span of time Indian Astronomy flourished mainly due to emi…nent astronomers like Aryabhat, Lallacharya, Varahamihir, Brahmagupta, Bhaskaracharya and others. Bhaskaracharya’s Siddhanta Shiromani is considered as the pinnacle of all the astronomical works of those 700 hundred years. It can be aptly called the “essence” of ancient Indian Astronomy and mathematics. In the ninth century Brahmagupta’s Brahmasphutasiddhanta was translated in Arabic. The title of the translation was ‘Sind Hind’. This translation proved to be a watershed event in the history of numbers. The Arabs quickly grasped the importance of the Indian decimal system of numbers. They played a key role in transmitting this system of numbers to Europeans. For a long time Europeans were using Roman Numerals, which were very tedious to handle. After accepting the decimal system of numbers, European mathematicians made a remarkable progress in mathematics, but that was about 500 years after Bhaskaracharya.

From 750 AD Onwards India was engulfed in waves of foreign attacks. In 1205 AD Bakhtiyar Khilji destroyed the magnificent Nalanda University, which was a renowned center of knowledge for about 800 years. India was in utter chaotic state till the country was colonized by British. All universities and learning centers in India were destroyed, knowledge was lost and hardly any progress was made in mathematics and astronomy. A few scholars like Keshav Daivadnya, Ganesh Daivadnya Madhav, Sawai Jai Singh and others tried to keep the flame of knowledge burning in that dark period.

Birth and Education of Bhaskaracharya :

Ganesh Daivadnya has bestowed a very apt title on Bhaskaracharya. He has called him ‘Ganakchakrachudamani’, which means, ‘a gem among all the calculators of astronomical phenomena.’ Bhaskaracharya himself has written about his birth, his place of residence, his teacher and his education, in Siddhantashiromani as follows,

‘ A place called ‘Vijjadveed’, which is surrounded by Sahyadri ranges, where there are scholars of three Vedas, where all branches of knowledge are studied, and where all kinds of noble people reside, a brahmin called Maheshwar was staying, who was born in Shandilya Gotra (in Hindu religion, Gotra is similar to lineage from a particular person, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ or ‘Vedas’) and ‘Smart’ (originated from ‘Smut’) Dharma, respected by all and who was authority in all the branches of knowledge. I acquired knowledge at his feet’.

From this verse it is clear that Bhaskaracharya was a resident of Vijjadveed and his father Maheshwar taught him mathematics and astronomy. Unfortunately today we have no idea where Vijjadveed was located. It is necessary to ardently search this place which was surrounded by the hills of Sahyadri and which was the center of learning at the time of Bhaskaracharya. He writes about his year of birth as follows,

‘I was born in Shake 1036 (1114 AD) and I wrote Siddhanta Shiromani when I was 36 years old.’

Bhaskaracharya has also written about his education. Looking at the knowledge, which he acquired in a span of 36 years, it seems impossible for any modern student to achieve that feat in his entire life. See what Bhaskaracharya writes about his education,

‘I have studied eight books of grammar, six texts of medicine, six books on logic, five books of mathematics, four Vedas, five books on Bharat Shastras, and two Mimansas’.

Bhaskaracharya calls himself a poet and most probably he was Vedanti, since he has mentioned ‘Parambrahman’ in that verse.

SIDDHANTASHIROMANI

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Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD when he was 36 years old. This is a mammoth work containing about 1450 verses. It is divided into four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact each part can be considered as separate book. The numbers of verses in each part are as follows,

Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses.

One of the most important characteristic of Siddhanta Shiromani is, it consists of simple methods of calculations from Arithmetic to Astronomy. Essential knowledge of ancient Indian Astronomy can be acquired by reading only this book. Siddhanta Shiromani has surpassed all the ancient books on astronomy in India. After Bhaskaracharya nobody could write excellent books on mathematics and astronomy in lucid language in India. In India, Siddhanta works used to give no proofs of any theorem. Bhaskaracharya has also followed the same tradition.

Lilawati is an excellent example of how a difficult subject like mathematics can be written in poetic language. Lilawati has been translated in many languages throughout the world. When British Empire became paramount in India, they established three universities in 1857, at Bombay, Calcutta and Madras. Till then, for about 700 years, mathematics was taught in India from Bhaskaracharya’s Lilawati and Beejaganit. No other textbook has enjoyed such long lifespan.

BHASKAR’S MATHEMATICS

Lilawati and Beejaganit together consist of about 500 verses. A few important highlights of Bhaskar’s mathematics are as follows,

Terms for numbers

In English, cardinal numbers are only in multiples of 1000. They have terms such as thousand, million, billion, trillion, quadrillion etc. Most of these have been named recently. However, Bhaskaracharya has given the terms for numbers in multiples of ten and he says that these terms were coined by ancients for the sake

of positional values.

Bhaskar’s terms for numbers are as follows:

eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).

Kuttak

Kuttak is nothing but the modern indeterminate equation of first order. The method of solution of such equations was called as ‘pulverizer’ in the western world. Kuttak means to crush to fine particles or to pulverize. There are many kinds of Kuttaks. Let us consider one example.

In the equation, ax + b = cy, a and b are known positive integers. We want to also find out the values of x and y in integers. A particular example is,

100x +90 = 63y

Bhaskaracharya gives the solution of this example as, x = 18, 81, 144, 207… And y=30, 130, 230, 330…

Indian Astronomers used such kinds of equations to solve astronomical problems. It is not easy to find solutions of these equations but Bhaskara has given a generalized solution to get multiple answers.

Chakrawaal

Chakrawaal is the “indeterminate equation of second order” in western mathematics. This type of equation is also called Pell’s equation. Though the equation is recognized by his name Pell had never solved the equation. Much before Pell, the equation was solved by an ancient and eminent Indian mathematician, Brahmagupta (628 AD). The solution is given in his Brahmasphutasiddhanta. Bhaskara modified the method and gave a general solution of this equation. For example, consider the equation 61×2 + 1 = y2. Bhaskara gives the values of x = 22615398 and y = 1766319049

There is an interesting history behind this very equation. The Famous French mathematician Pierre de Fermat (1601-1664) asked his friend Bessy to solve this very equation. Bessy used to solve the problems in his head like present day Shakuntaladevi. Bessy failed to solve the problem. After about 100 years another famous French mathematician solved this problem. But his method is lengthy and could find a particular solution only, while Bhaskara gave the solution for five cases. In his book ‘History of mathematics’, see what Carl Boyer says about this equation,

‘In connection with the Pell’s equation ax2 + 1 = y2, Bhaskara gave particular solutions for five cases, a = 8, 11, 32, 61, and 67, for 61×2 + 1 = y2, for example he gave the solutions, x = 226153980 and y = 1766319049, this is an impressive feat in calculations and its verifications alone will tax the efforts of the reader’

Henceforth the so-called Pell’s equation should be recognized as ‘Brahmagupta-Bhaskaracharya equation’.

Simple mathematical methods

Bhaskara has given simple methods to find the squares, square roots, cube, and cube roots of big numbers. He has proved the Pythagoras theorem in only two lines. The famous Pascal Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on that number triangle. Pascal was born 500 years after Bhaskara. Several problems on permutations and combinations are given in Lilawati. Bhaskar. He has called the method ‘ankapaash’. Bhaskara has given an approximate value of PI as 22/7 and more accurate value as 3.1416. He knew the concept of infinity and called it as ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara had not notions about calculus, One of his equations in modern notation can be written as, d(sin (w)) = cos dw.

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