Bhaskara II: The Genius Mathematician And Astronomer Of Medieval India – OpEd

Bhaskara II, also known as Bhaskaracharya, was a renowned Indian mathematician and astronomer who lived in the 12th century. He is considered one of the greatest mathematicians of medieval India, whose contributions to mathematics and astronomy have had a significant impact on the development of these fields. His works, particularly Siddhanta Shiromani and Lilavati, continue to be appreciated for their depth and clarity.

Bhaskara II was born in Saka 1036 (A.D. 1114) in Biddur, Bijapur, Karnataka, India. He wrote Siddhanta Shiromani in Saka 1072 (A.D. 1150) at the age of 36. At the end of the chapter on problems in Goladhyaya, he mentioned details about his family. He belonged to the Sandilya lineage and lived in Vijjalavida. His father, Mahesvara, was a great astrologer who wrote two books, Karana-Grantha and Jataka-Tika-Grantha. Under the guidance of his father, Bhaskara II gained mastery over various fields of knowledge.

Bhaskara II as a Mathematician and Astronomer

Bhaskara II was not only a mathematician but also an excellent astronomer and scientist. He became the head of the astronomical observatory at Ujjain, which was the leading mathematical center of ancient India. He was the successor of eminent mathematicians like Varahamihira (505-587 A.D.) and Brahmagupta. His works demonstrated a level of mathematical sophistication that was not achieved in Europe for several centuries.

As an astronomer, he made significant contributions to understanding planetary positions, eclipses, and celestial mechanics. His mathematical works, particularly Lilavati and Bijaganita, introduced and developed the systematic use of the decimal number system. He also compiled and expanded upon earlier problems introduced by Brahmagupta and other mathematicians.

Contributions to Mathematics

Bhaskara II’s mathematical works were extensive and groundbreaking. Some of his most significant contributions include:

Decimal Number System

One of Bhaskara II’s key contributions was the systematic use of the decimal number system, including the decimal point, which was a significant advancement over numeral systems used in other parts of the world at the time. His arithmetic text, “Lilavati,” covered topics such as arithmetical and geometrical progressions, plane and solid geometry, the shadow of the gnomon, and methods for solving indeterminate equations. He introduced the rules of operations with zero, used negative numbers and surds, and refined the estimation of π. His work also provided solutions to problems involving interest computation and combination theory. His approach to solving indeterminate equations was particularly significant, as the methods he described were later rediscovered by European mathematicians in the 17th century. His systematic approach and improvements on earlier methods established his legacy as a brilliant mathematical thinker.

Arithmetic and Lilavati

His work Lilavati covers a wide range of topics in arithmetic, including fractions, progressions, and equations. It is divided into 13 chapters and includes:

• Definitions and arithmetical terms
• Properties of zero, including division and operations with zero
• Use of negative numbers and surds
• Approximation of π (3.141666)
• Methods of multiplication and squaring
• Indeterminate equations (Kuttaka method)

The Lilavati also includes excellent problems that demonstrate practical applications of arithmetic. Bhaskara II’s work in this field was far ahead of the mathematical developments in Europe.

Algebra and Bijaganita

His algebraic work, “Bijaganita,” consisted of twelve chapters and was the first to acknowledge that a positive number has both a positive and negative square root. His book detailed the properties of positive and negative numbers, surds, and the methods for solving equations with multiple unknowns. He introduced the Chakravala method, a cyclic algorithm for solving indeterminate quadratic equations of the form ax² + bx + c = y². His contributions to the Pell equation, an important concept in number theory, remain significant to this day.

Contributions to Astronomy

In the field of astronomy, Bhaskara II built upon the work of Brahmagupta. He accurately defined many astronomical quantities, such as the length of the sidereal year, estimating it to be approximately 365.2588 days—remarkably close to the modern measurement of 365.25636 days. His astronomical text, “Siddhanta Shiromani,” is divided into two parts: mathematical astronomy and the study of celestial spheres. The first part covers topics such as planetary longitudes, diurnal motion, syzygies, solar and lunar eclipses, planetary latitudes, and conjunctions of planets. He also examined the crescent shape of the moon, sunrise equations, and the paths of the sun and moon. His understanding of celestial mechanics and his accurate astronomical calculations demonstrated a deep knowledge of planetary motion, contributing significantly to the field of mathematical astronomy.

Bhaskara II’s work laid the foundation for future mathematical and astronomical developments. His discoveries influenced both Indian and global mathematical traditions, and his systematic approach to problem-solving remains an inspiration. His works, particularly “Lilavati” and “Bijaganita,” continue to be studied for their mathematical depth, and his contributions to algebra, arithmetic, and celestial mechanics established him as one of the greatest scholars of medieval India.

Legacy and Influence

Bhaskara II’s influence extended far beyond his lifetime. His mathematical techniques, particularly his work on number systems, algebra, and astronomy, were studied and used by later scholars. His work influenced the mathematical developments of the Renaissance and provided a foundation for many modern mathematical principles.

Bhaskara II remains one of the most celebrated mathematicians of India. His ability to blend mathematical precision with astronomical insights makes him a towering figure in the history of science. His works, particularly Siddhanta Shiromani, continue to be studied and appreciated for their remarkable depth and accuracy.

Conclusion

Bhaskara II was a visionary mathematician and astronomer whose contributions to mathematics and astronomy were centuries ahead of his time. His works not only built upon the foundations laid by earlier mathematicians but also introduced new concepts that influenced later developments in these fields. His legacy endures as a testament to the rich mathematical heritage of India, inspiring scholars and mathematicians even today.

References

1. Datta, B., & Singh, A. N. (1935). History of Hindu Mathematics: A Source Book. Asia Publishing House.
2. Plofker, K. (2009). Mathematics in India. Princeton University Press.
3. Gupta, R. C. (1997). “The Chakravala Method of Bhaskara II,” Indian Journal of History of Science, 32(3), 161-177.
4. Joseph, G. G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics. Princeton University Press.
5. Yadav, B. S. (2011). Ancient Indian Leaps into Mathematics. Birkhäuser.

Authors:

• Dr. Baljinder Kour, Assistant Professor, Department of Mathematics, Akal University, Talwandi Sabo, Bathinda, Punjab, India.
• Prof. (Dr.) Ashish Arora, Professor, Department of Mathematics, IK Gujral Punjab Technical University, Kapurthala, Punjab, India.