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 people learnt to write in Sanskrit and worked at learning centres, 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.

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. ^[1]

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 clearly organised manner.

Bhaskara created this work to teach both basic and advanced ideas. Students could use it to learn simple operations like addition, subtraction, fractions, and geometry. While skilled scholars used it to study complex subjects such as eclipses and planetary motion, this involved calculations related to time. 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. ^[2]

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 as a neat blend of education and gentle humour. ^[3]

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, and simple and higher algebraic equations, as well as methods to solve linear and quadratic equations. He also discusses rules for surds (square roots) and proposes 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.

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. He gave clear rules for eclipses, sine tables for angles, and a close value for the length of the year. His methods helped predict conjunctions, risings, and daily motions, making astronomy useful for calendars, farming, and religious work.

Goladhyaya

Goladhyaya, the fourth section, explains the Earth as a sphere and describes star patterns in 472 verses. Bhaskara II stated that Earth rotates on its axis, causing day and night. He explained tides through the Moon’s pull and used cones and nodes to describe eclipses. Spherical trigonometry helped measure distances and angles in the sky. He also described early instruments, like the ghati yantra for timekeeping. His ideas made astronomical observation clearer and more scientific. ^[4]

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 and 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. _[5]

Other works and commentaries

Besides these main books, Bhaskara wrote smaller pieces, notes and explanations. 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. ^[2]

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 mathematicians and astronomers used his methods and made further advances. Today, Bhaskara is remembered as one of India’s finest medieval mathematicians, and students of the history of science study his books to learn how mathematics and astronomy were taught in pre-modern times. Many editions and translations of his works are freely available online, which helps modern readers and teachers access his clear teaching.

References

1. https://en.wikipedia.org/wiki/Bh%C4%81skara_II
2. https://dn721909.ca.archive.org/0/items/in.ernet.dli.2015.405727/2015.405727.The-Siddhant-siromani.pdf
3. https://dn790008.ca.archive.org/0/items/EtgU_lilavati-of-bhaskaracharya-1908-khemraj/Lilavati%20of%20Bhaskaracharya%201908%20-%20Khemraj.pdf
4. https://archive.org/details/dli.ernet.496170
5. https://archive.org/details/in.ernet.dli.2015.312414