Ancient-education/Philosophers and Scholars/Bhaskara II

Bhaskara II

Bhaskara II, often called Bhāskarāchārya, was a great Indian mathematician and astronomer of the 12th century. He lived in a time when learned people wrote in Sanskrit and worked at centres of learning such as Ujjain. Bhaskara wrote clear, lively books that taught methods for arithmetic, algebra, and the motions of the planets. His writings mix rules, work examples, and short verses so learners can remember them. Many later scholars used his books and wrote notes on them. His ideas travelled beyond India and helped the development of mathematics and astronomy in later centuries.

Birth and early life

Bhaskara II was born around 1114 CE, in a place called Vijjadavida (or Vijjalavida). He lived in the Satpura mountain ranges of the Western Ghats. Scholars believe this place is the town of Patana in Chalisgaon, which is in the Khandesh region of Maharashtra today. His father, Maheshvara, was an astronomer and taught him the basics of calculations and sky work. (Wikipedia contributors, n.d.)

Bhaskara grew into a teacher and scholar himself. He lived and worked at Ujjain, a well-known centre for astronomy and mathematics in medieval India. There he had access to earlier works by mathematicians such as Brahmagupta. Bhaskara studied these earlier books, corrected some ideas, and wrote longer, clearer textbooks that students could use. His reputation spread quickly thanks to the practical quality of his methods. (Bhāskara II, 2015)

Siddhanta Shiromani

Siddhanta Shiromani is the most important work of Bhaskara II. The book was written in 1150 when he was 36 years old. It is composed in the Sanskrit language in 1450 verses. The work is written in four main parts, which are:

• Lilavati (arithmetic)
• Bijaganita (algebra)
• Ganitadhyaya (planetary mathematics)
• Goladhyaya (sphere and sky calculations)

Each part is written in short Sanskrit verses that explain rules, methods, and examples in a clear, organised manner.

Bhaskara created his work as a teaching resource for both new and experienced students. The new students could learn simple processes like adding, subtracting, figuring fractions, and determining geometrical shapes, while the skilled students learnt to use it as a tool to investigate more complex topics, including eclipses and the motions of planets. The book presents a solid balance of theory and practical methods, which made it popular across India for many centuries. Siddhanta Shiromani became a standard textbook and influenced many later mathematicians and astronomers. (Bhaskara II, 2015)

Lilavati

Lilavati is the first part of Siddhanta Shiromani and is named after his daughter. The book teaches arithmetic, mensuration and simple geometry. The book consists of 13 chapters and 278 verses. It is written as short verses, and many problems are framed as little puzzles or stories to make learning pleasant. Topics include basic operations, fractions, proportions, progressions, and rules for areas and volumes.

Bhaskara proposes methods that are direct and useful for everyday calculations. Because of its clear style and many examples, Lilāvati was widely taught; translations and commentaries appear in many languages. It is known for a neat blend of education and gentle humour. (Bhāskara II, 1908).

Bijagnita

Bijagaṇita is the second part of Siddhānta Śiromaṇi. The book is divided into six parts and contains 213 verses. It mainly deals with Bhaskara’s algebra section. In it, Bhaskara treats positive and negative numbers, zero, simple and higher algebraic equations, as well as methods to solve linear and quadratic equations. He also explains rules for square roots and gives solutions to indeterminate equations of certain types. Bhaskara built on earlier algebraic work but added clearer solution methods and many worked examples so students could follow step by step. The book helped shape later algebra in India and beyond. (Bhāskara II, 2015).

Grahaganita

Grahaganita, the third section, contains 468 verses on how to calculate the positions of the Sun, Moon, and the five visible planets. Bhaskara II used the idea of epicycles, or small circles on larger circles, to explain planetary motion.

For eclipses, he provided precise guidelines, sine tables for angles, and a close value for the year's duration. His methods helped predict conjunctions, risings, and daily motions, making astronomy useful for calendars, farming, and religious work.

He defined rules for the timing of the occurrences of an eclipse, provided sine tables for angles, and established an approximate length of the year. Also, he was able to accurately calculate the predictions for near conjunctions, when stars would rise, and daily motion, making astronomy useful in the preparation of calendars, crop planting, and the performance of religious activities. (Bhāskara II, 2015).

The Goladhyaya

The Goladhyaya, the fourth section, consists of 472 verses and provides an extensive description of star patterns and explains the spherical shape of Earth. Bhaskara stated that Earth rotates on its axis, which causes day and night. He also explained ocean tides using the force of attraction from the Moon and effectively used cones and nodes to explain the occurrences of solar and lunar eclipses. Spherical trigonometry enabled the calculation of distances and angles in relation to the celestial sphere. He also explained the use of early instruments for timekeeping, like the ghati yantra. His ideas made astronomical observation clearer and more scientific. (Bhāskara II, 2015)

Karana-kutuhala

The Karana-kutuhala is a short, practical handbook Bhaskara wrote later. It condenses many of the computational rules from Siddhanta Shiromani into an easy, ready-to-use form. The purpose was to provide a quick set of formulae and procedures for daily astronomical work, for example, for making eclipse predictions. Because it is short and handy, many teachers and practitioners prefer it for applied work. The text became very popular in west and northwest India and survives in a great many copies and commentaries.  (Bhāskara II, n.d.).

Other works and commentaries

Besides these main books, Bhaskara wrote smaller pieces, notes and explanations for older works. He composed commentaries and smaller treatises that helped readers follow older texts and apply rules in practice. Scholars later wrote commentaries on Bhaskara’s books, and teachers used his examples in classrooms. Because Bhaskara mixed proofs with worked examples and clear rules, his pages were useful both to scholars who sought profound understanding and to learners who wanted solid calculation methods. Over time, his influence reached Persian and Arabic readers as well, and some ideas travelled westwards through translations. Collections of his works have been preserved on palm-leaf manuscripts and later printed editions. (Bhāskara II, n.d.)

Legacy

Bhaskara II stands out because he wrote books that explain both why a rule works and how to use it. His clear language, lively examples and step-by-step methods made difficult topics easier for students. He treated algebra with care, worked on indeterminate problems, and gave practical rules for the motion of planets, all at a high level of skill.  Later generations of mathematicians and astronomers used his techniques, resulting in even greater breakthroughs. Modern scholars recognize Bhaskara as one of India’s finest mediaeval mathematicians. (Wikipedia contributors, n.d.)

References

Wikipedia contributors. (n.d.) https://en.wikipedia.org/wiki/Bh%C4%81skara_II

Bhāskara II. (2015). The Siddhānta Śiromaṇi of Bhāskarāchārya https://dn721909.ca.archive.org/0/items/in.ernet.dli.2015.405727/2015.405727.The-Siddhant-siromani.pdf

Bhāskara II. (1908). Lilāvati of Bhāskarāchārya https://dn790008.ca.archive.org/0/items/EtgU_lilavati-of-bhaskaracharya-1908-khemraj/Lilavati%20of%20Bhaskaracharya%201908%20-%20Khemraj.pdf

Bhāskara II. (n.d.) https://archive.org/details/dli.ernet.496170

Bhāskara II. (n.d.). Bhāskara II treatises https://archive.org/details/in.ernet.dli.2015.312414