Lagadha
लगध
Lagadha is the name attached to the Vedanga Jyotisha, the oldest surviving Indian text devoted to astronomy and time-keeping. It survives in two recensions: a Rigvedic version of 36 verses and a Yajurvedic version of 43 or 44 verses. The text describes the winter solstice as occurring when the Sun and Moon are at the beginning of the Shravishtha (Dhanishtha) nakshatra; from this, some scholars have dated the underlying observations to the late second millennium BCE, while others, on linguistic grounds, place the composition of the text several centuries later.
The Vedanga Jyotisha sets out a five-year cycle (yuga) of 1,830 civil days containing 62 synodic lunar months, two of which are intercalary, and 67 sidereal months. It uses tithis, nakshatras and the muhurta of 48 minutes, and gives rules for the changing length of daylight through the year. Its purpose was practical: to fix the days for Vedic sacrifices tied to the new and full moon and the solstices.
Five-year luni-solar cycleIntercalary monthsNakshatra-based time-keepingDaylight-length rule
Aryabhata
आर्यभट
Aryabhata tells us in his own work that he was 23 years old when 3,600 years of the Kali Yuga had elapsed, which corresponds to 499 CE, giving his birth year as 476 CE. His Aryabhatiya is a compact work of 121 verses in four sections: Gitikapada (astronomical constants), Ganitapada (mathematics), Kalakriyapada (time-reckoning and planetary motion) and Golapada (the celestial sphere). It introduced an alphabetic notation for writing very large numbers in verse.
Aryabhata stated that the apparent daily rotation of the stars is caused by the rotation of the Earth on its axis, an idea that many later Indian astronomers rejected. He explained solar and lunar eclipses in terms of the shadows of the Earth and Moon rather than the demon Rahu, gave a table of sine differences (jya) at intervals of 3°45′, and gave π as 62832/20000 = 3.1416, which he described as an approximation (asanna). He also presented the kuttaka ("pulveriser") method for solving linear indeterminate equations. His sidereal year differs from the modern value by only a few minutes.
Rotation of the EarthEclipse explanationSine tableπ ≈ 3.1416Kuttaka algorithm
Varahamihira
वराहमिहिर
Varahamihira was the great compiler of Indian astronomy and its related sciences. His Pancha-siddhantika ("Five Treatises") summarises five astronomical systems current in his day — the Paitamaha, Vasishtha, Romaka, Paulisha and Surya (Saura) Siddhantas — and uses an epoch in 505 CE. It is our main source for several of these early systems, two of which show Hellenistic influence.
His Brihat Samhita is an encyclopaedic work covering the Samhita branch of Jyotisha: planetary motions and omens, weather and rainfall prediction, earthquakes, architecture and temple-building, the examination of gems, agriculture, water-finding and much more. His Brihat Jataka became one of the standard textbooks of horoscopy (Hora). Varahamihira is the classic source for the description of Jyotisha as having three branches — Siddhanta, Samhita and Hora.
Compilation of five SiddhantasBrihat Samhita encyclopaediaTrigonometric relationsClassification of Jyotisha
Brahmagupta
ब्रह्मगुप्त
Brahmagupta composed the Brahmasphuta-siddhanta at the age of 30, in 628 CE. Besides astronomy, it contains chapters on mathematics in which zero is treated as a number in its own right, with rules for adding, subtracting and multiplying with zero and with negative quantities ("debts") and positive ones ("fortunes"). He gave the formula for the area of a cyclic quadrilateral that bears his name, solutions of quadratic equations, and a composition rule (bhavana) for generating solutions of the equation Nx² + 1 = y², later called Pell's equation.
His later handbook, the Khandakhadyaka, includes a second-order interpolation formula for computing sines. Brahmagupta's work was translated into Arabic in Baghdad in the eighth century and helped transmit Indian astronomy and numerals to the Islamic world. He was a sharp critic of some of his predecessors, including Aryabhata.
Arithmetic of zeroNegative numbersCyclic quadrilateral formulaBhavana for Pell's equationInterpolation
Bhaskaracharya (Bhaskara II)
भास्कराचार्य
Bhaskara II, known as Bhaskaracharya, headed the astronomical tradition associated with Ujjain. His Siddhanta Shiromani, completed in 1150 CE, has four parts: Lilavati (arithmetic and measurement), Bijaganita (algebra), Grahaganita (planetary mathematics) and Goladhyaya (the sphere). Lilavati, written in elegant verse with problems addressed to a girl named Lilavati, was used as a textbook in India for centuries.
In algebra he gave a refined form of the chakravala ("cyclic") method for solving Nx² + 1 = y², solving difficult cases such as N = 61. In astronomy he worked with the instantaneous velocity (tatkalika gati) of planets and used the idea that the change in the sine is proportional to the cosine — results that anticipate concepts of differential calculus. His Karanakutuhala (1183) is a practical handbook for computing almanacs.
Lilavati and BijaganitaChakravala methodInstantaneous motionSpherical astronomy
Madhava of Sangamagrama
सङ्गमग्राम माधव
Madhava is regarded as the founder of the Kerala school of astronomy and mathematics. Few of his own writings survive, and most of his mathematical results are known through later members of the school — notably Nilakantha Somayaji and Jyeshthadeva, whose Malayalam text Yuktibhasha gives proofs.
The Kerala school attributes to Madhava infinite power series for the sine, cosine and arctangent functions, and the series π/4 = 1 − 1/3 + 1/5 − 1/7 + …, together with correction terms that make the series converge much faster. Using such methods he computed π correctly to eleven decimal places. These results predate their rediscovery in Europe by James Gregory, Isaac Newton and Gottfried Leibniz in the seventeenth century, which is why the arctangent series is now often called the Madhava–Leibniz or Madhava–Gregory series.
Infinite series for πSine and cosine seriesArctangent seriesCorrection termsKerala school
Nilakantha Somayaji
नीलकण्ठ सोमयाजी
Nilakantha Somayaji was a leading figure of the Kerala school in its later phase. His Tantrasangraha, completed in 1501, revised the planetary models of earlier Indian astronomy, and his long commentary on the Aryabhatiya preserves and discusses many results of Madhava and his successors.
In Nilakantha's revised model, Mercury, Venus, Mars, Jupiter and Saturn move around the mean Sun, which in turn moves around the Earth. This gave a much better treatment of the inner planets Mercury and Venus than earlier Indian models and is geometrically similar to the system that Tycho Brahe proposed later in the sixteenth century. He also wrote on the importance of checking computations against observation.
TantrasangrahaRevised planetary modelAryabhatiya commentaryEmphasis on observation
A continuous tradition
These astronomers did not work in isolation. Varahamihira summarised the systems that preceded him; Brahmagupta
argued with Aryabhata; Bhaskara II built on Brahmagupta; and the Kerala astronomers wrote extensive commentaries on
the Aryabhatiya a thousand years after it was composed. Their parameters and methods also passed into the
astronomical handbooks (karanas) used by almanac-makers, so that the Panchangam consulted today is a direct
descendant of their work, even when its positions are now computed with a modern ephemeris.
To see what this astronomy is used for, explore the Vedic units of time, the
calendar systems of India, and today's Panchangam.