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Bhāskara II

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Bhāskara II was a preeminent Indian polymath, mathematician, and astronomer of the 12th century. Often called the greatest mathematician of medieval India, he made massive contributions to scientific knowledge. He served as the leader of a cosmic observatory at Ujjain. This location was the primary mathematical center of ancient India. His life and work represent a peak of mathematical achievement during his era.

Page from Lilavati, the first volume of Siddhānta Śiromaṇī. Use of the Pythagorean theorem in the corner
Page from Lilavati, the first volume of Siddhānta Śiromaṇī. Use of the Pythagorean theorem in the corner

Bhāskara was born in 1114 CE in a family of scholars. His family belonged to the Hindu Deśastha Brahmin community. Scholars debate his exact birthplace, but many place it in the Maharashtra region. Some suggest he lived near the Satpura mountain ranges or the Godavari river. His father, Maheśvara, was an astronomer and mathematician who taught him. Bhāskara continued this tradition by teaching his son, Lokasamudra. His descendants even helped establish a school for his writings in 1207.

Page from Lilavati, the first volume of Siddhānta Śiromaṇī. Use of the Pythagorean theorem in the corner
Page from Lilavati, the first volume of Siddhānta Śiromaṇī. Use of the Pythagorean theorem in the corner

His most significant work is titled Siddhānta Śiromaṇi, meaning "Crown of Treatises." He completed this major work in 1150 CE when he was 36 years old. The text is divided into four distinct sections. The first is Līlāvatī, named after his daughter, which covers arithmetic and geometry. The second is Bījagaṇita, a treatise on algebra. The third is Grahagaṇita, which focuses on the mathematics of the planets. The fourth is Golādhyāya, which deals with the study of spheres.

Page from Lilavati, the first volume of Siddhānta Śiromaṇī. Use of the Pythagorean theorem in the corner
Page from Lilavati, the first volume of Siddhānta Śiromaṇī. Use of the Pythagorean theorem in the corner

In the realm of arithmetic, the Līlāvatī section is incredibly thorough. It includes 277 verses covering many complex topics. These include permutations, measurement, and arithmetical progressions. Bhāskara explored the properties of zero, including its rules in division. He also worked with negative numbers and surds. The text provides methods for multiplication, squaring, and estimating the value of pi. It even includes problems regarding interest computation.

Bhāskara's work in algebra, or Bījagaṇita, was revolutionary for its time. This section contains 213 verses and introduces several advanced concepts. He was the first to recognize that a positive number has two square roots. These are the positive and negative square roots. He also studied indeterminate equations, such as the one known as Pell's equation. He developed a cyclic method called chakravala to solve these equations. This method was more efficient than later methods used by European mathematicians in the 17th century.

Bhaskaracharya proof of pythagorean Theorem
Bhaskaracharya proof of pythagorean Theorem

One of his most famous achievements was a proof of the Pythagorean theorem. He proved the relationship between the sides of a triangle by calculating areas in two different ways.

Teorema de Pitágoras.Bhaskara
Teorema de Pitágoras.Bhaskara

Bhāskara also made surprising connections to what we now call calculus. In his astronomical studies, he looked at the instantaneous speeds of planets. He suggested that a planet's speed is zero at its highest point. This idea is related to the concept of a derivative. He even showed that the cosine is the derivative of the sine function. While he did not develop the formal notion of a derivative, these were early precursors to differential calculus. His work includes early forms of Rolle's theorem and the mean value theorem.

Teorema de Pitágoras.Bhaskara
Teorema de Pitágoras.Bhaskara

His influence reached far beyond his own lifetime. His mathematical rules were similar to those used by Renaissance mathematicians centuries later. He solved problems, such as those involving Diophantine equations, that eluded European mathematicians like Pierre de Fermat. His work in trigonometry was also highly advanced. He developed spherical trigonometry and calculated sines for specific angles. Bhāskara's legacy remains a vital part of the history of global mathematics.

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File:Page from Lilavati, the first volume of Siddhānta Śiromaṇī. Use of the Pythagorean theorem in the corner.jpg
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