Guptas: Literature, scientific literature Part II
Astronomy
 Earliest astronomical knowledge:
 The earliest evidence of ancient Indian astronomical knowledge is contained in the Vedanga texts on jyotisha or astrology, the main focus of which was to fix the date of sacrificial rituals. selfstudyhistory.com
 Greek influence:
 The Sanskrit names of the signs of the zodiac have Greek origins, and it seems that Greek influence led to the sequence of planets being fixed in the names of the seven days of the week in Indian texts.
 A Sanskrit text known as the Yavanajataka reflects the transmission of Hellenistic astronomical ideas into India.
 However, Indian astronomers appear to have made certain major breakthroughs independently.
 Varahamihira’s Panchasiddhantika (6th century) summarizes the astronomical works and ideas of the preceding centuries, but ascribes their authorship to divine or semidivine beings.
Aryabhata I:
 Aryabhata I, (476–550 CE) was astronomer and mathematician.
 He is the earliest known historical astronomer in India and wrote two works:
 Aryabhatiya:
 deals with astronomy and mathematics
 The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry.
 It also contains continued fractions, quadratic equations, sumsofpower series, and a table of sines.
 Aryabhatasiddhanta:
 A lost work on astronomical computations, is known through the writings of Aryabhata’s contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I.
 Aryabhatiya:
 Residence:
 He seems to have been a native of Ashmaka country (on the Godavari) as the 7th century mathematician Bhaskara I calls the Aryabhatiya the Ashmakatantra and the followers of Aryabhata Ashmakiyas.
 A statement in the Aryabhatiya indicates that Aryabhata lived in Kusumapura, i.e., Pataliputra.
 Both Hindu and Buddhist tradition, as well as Bhaskara I identify Kusumapura as Paṭaliputra, modern Patna.
 He was aware of the ideas and methods of his predecessors, but struck his own course. In Aryabhatiya, he writes:
 ‘I dived deep in the ocean of astronomical theories, true and false, and rescued the precious sunken jewel of true knowledge by means of the boat of my own intellect.’
 Aryabhata had an earthcentric view of the universe—he thought that the planets moved around the earth in circular epicycles.
 Scientific explanation of eclipses:
 He was the first astronomer to give a scientific explanation of eclipses.
 He established that eclipses were not caused by the demons Rahu and Ketu, but by the moon coming between the earth and sun.
 He worked out how to ascertain which part of the moon would be obscured during an eclipse.
 Other works:
 He was also the first to discover that the earth rotated on its axis.
 Another one of his many achievements was to find out the sine functions and use them in astronomy.
 He worked out the equation for calculating the orbit of a planet, and gave an extremely accurate estimate of the length of a year (365.2586805 days).
 In Aryabhatiya, he gives the number of rotations of the earth in yuga.
 Aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds; the modern value is 23:56:4.091.
 He discovered that the Moon and planets shine by reflected sunlight.
 Influence:
 Aryabhata’s work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations.
 Some of his results are cited by AlKhwarizmi and in the 10th century AlBiruni stated that Aryabhata’s followers believed that the Earth rotated on its axis.
 Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing the Panchangam (Hindu calendar). In the Islamic world, they formed the basis of the Jalali calendar introduced in 1073 CE by a group of astronomers, versions of which are the national calendars in use in Iran and Afghanistan today.
Varahamihira:
 Varahamihira was a 6th century astrologer, astronomer, and mathematician who belonged to Avanti (in western Malwa) towards the end of the fifth century in Ujjain.
 He is considered to be one of the Navaratnas of the court of legendary ruler Yashodharman Vikramaditya of Malwa.
 He wrote several treatises on astronomy and horoscopy.
 Panchasiddhantika:
 His Panchasiddhantika deals with five schools of astronomy:
 Surya Siddhanta,
 Romaka Siddhanta:
 so called from the Rum, ie. the subjects of the Roman Empire, composed by Srishena
 It was influenced by Greek and Roman ideas.
 Paulisa Siddhanta:
 so called from Pulisa, the Greek, composed by Pulisa.
 Vasishtha Siddhanta:
 so called from one of the stars of the Great Bear, composed by Vishnucandra
 Paitamaha Siddhantas.
 Two of these (Romaka and Paulisa) reflect a close knowledge of Greek astonomy.
 The Romaka Siddhanta (“Doctrine of the Romans”) and the Paulisa Siddhanta (“Doctrine of Paul”) were two works of Western origin which influenced Varahamihira’s thought.
 Indian astronomers valued the work of Greek astronomers with which they were familiar, but they arrived at their results independently, which were usually more correct.
 He declared that Suryasiddhanta was the best of all five extant siddhantas available to him.
 His Panchasiddhantika deals with five schools of astronomy:
 He was also an astrologer. He wrote on all the three main branches of Jyotisha astrology:
 LaghuJataka,
 BrihatJataka,
 Brihat Samhita.
 Brihatsamhita:
 His Brihatsamhita is an encyclopaedic work dealing with diverse topics including
 how to sharpen swords,
 how to ascertain the value of precious metals and stones,
 how to make trees bear fruit out of season,
 how to distinguish the good breeds of animals, and
 how to divine the location of water.
 It also discusses the nature and structure of temples, palaces, and houses.
 It gives an explanation of seasons and discusses meteorological issues such as the correlation between the clouds, winds, and amount of rainfall.
 His Brihatsamhita is an encyclopaedic work dealing with diverse topics including
Brahmagupta:
 Brahmagupta (598–670 CE) was an mathematician and astronomer who wrote two works on Mathematics and Astronomy:
 Brahmasphutasiddhanta (Extensive Treatise of Brahma), a theoretical treatise in 628 CE, and
 Khandakhadyaka (665 CE), a more practical text.
 These texts became very influential within India, and their Arab translations and adaptations introduced Indian astronomy to the Arabs.
 Brahmasputasiddhanta:
 The Brahmasputasiddhanta is the first surviving Indian text containing a systematic discussion of astronomical instruments, as well as methods of computing astronomical elements from readings taken with them.
 The instruments include:
 accessories,
 astronomical instruments for measuring time and observing the celestial bodies,
 instruments that turn automatically for the duration of one day, and ones that rotate perpetually.
 The text mentions nine astronomical instruments like:
 chakra (a circular wooden plate graduated into 360º),
 dhanus (a semicircular plate),
 turyagola (a quarter plate),
 kartari (two semicircular plates joined together at different levels) etc.
 The instruments, made of wood or bamboo, are very simple in design and could not have provided much precision in measurement.
 This suggests that astronomers probably relied more on their superior computing skills.
 However, Brahmagupta also referred to complex automatic devices called svayamvaha yantras, which reflects an awareness of the idea of perpetual motion.
 Khandakhadyaka (meaning “edible bite”):
 It is an astronomical treatise.
 The treatise contains eight chapters covering such topics as:
 the longitudes of the planets,
 diurnal rotation,
 lunar and solar eclipses,
 risings and settings,
 the moon’s crescent and
 conjunctions of the planets.
 Khandakhadyaka was known in Sanskrit to AlBiruni. The treatise was written as a response to Aryabhata’s Ardharatrikapaksa.
 Brahmagupta criticized the Puranic view that the Earth was flat or hollow. Instead, he observed that the Earth and heaven were spherical.
Mathematics
 Shulvasutras:
 The roots of Indian mathematics can be traced to the Shulvasutras, appendices to the Shrautasutras.
 Shulva means measurement and the Shulvasutras are manuals for the preparation of the site where Vedic sacrificial rituals were to be performed, dealing with the construction of Vedic brick fire altars.
 These manuals contain one of the earliest expressions of the principle behind what later came to be known as Pythagora’s theorem in geometry.
 The Shulvasutras also made suggestions for squaring a circle, i.e., to construct, using only ruler and compasses, a square whose area is equal to that of a given circle.
 In later times, the term ganitashastra was the most frequently used term for mathematical science.
 Decimal system of notation and zero:
 One of the most important discoveries of ancient Indian mathematicians was the decimal system of notation, based on the place value of the first nine numbers and the use of a symbol known as bindu for zero.
 The use of this system greatly simplified arithmetical calculations.
 The oldest datable evidence of the decimal placevalue system of notation is in a 3rd century work on astrology called the Yavanajataka by Sphujidhvaja. This work does not, however, mention the zero.
 The zero symbol, a dot, was used in metrics (chhandas) by Pingala in the Chhandasutra, a pre2nd century BCE work.
 Varahamihira’s Panchasiddhantika is the earliest dateable text to give zero both as a symbol and as a number.
 The decimal system of notation was used by Varahamihira and was referred to by Aryabhata in his Aryabhatiya.
 Aryabhata’s method of extracting the square root and cube root presupposes the decimal place value of numbers.
 This shows that Indian mathematicians were using the system in the 5th century CE.
 A Gupta inscription of AD 448 from Allahabad district suggests that the decimal system was known in India at the beginning of the fifth century.
 In Europe, the old cumbersome system was followed till the 12th century, when the Europeans learnt the new system from the Arabs.
 Arab writers such as Ibn Washiya, AlMasudi, and AlBiruni give the credit for the discovery of the system to the ‘Hindus’.
 One of the most important discoveries of ancient Indian mathematicians was the decimal system of notation, based on the place value of the first nine numbers and the use of a symbol known as bindu for zero.
 Aryabhata:
 Aryabhata’s Aryabhatiya is a work mainly on astronomy and deals only incidentally with problems of mathematics. It deals with the arithmetical progression of numbers and their squares and cubes.
 Aryabhata displays an awareness of both the zero system and the decimal system.
 Geometry:
 Aryabhata describes the various properties of a circle. and gives a very accurate value for pi (π) correct to 4 decimal places at 3.1416.
 He calculated pi (π) correct to 4 decimal places at 3.1416.to, remarkably close to recent estimates.
 Aryabhata is regarded as the father of algebra. His work solves a number of complex simultaneous equations.
 Trigonometry:
 The use of the sine (jya) functions in solving problems in astronomy indicates the development of trigonometry.
 His definitions of sine (jya), cosine (kojya), versine (utkramajya), and inverse sine (otkram jya) influenced the birth of trigonometry.
 The Aryabhatiya gives tables for the trigonometric ratio sine for angles from 0 to 90 degrees at intervals of 3.75°. The same sine tables are also found in the Surya Siddhanta.
 Aryabhata also perfected the methods of solving in integers certain types of indeterminate equations.
 Later mathematicians such as Brahmagupta and Bhaskara II also made contributions in this sphere.
 Varahamihira:
 He improved the accuracy of the sine tables of Aryabhata I.
 He defined the algebraic properties of zero as well as of negative numbers.
 He was among the first mathematicians to discover a version of what is now known as the Pascal’s triangle. He used it to calculate the binomial coefficient.
 Among Varahamihira’s contribution to physics is his statement that reflection is caused by the backscattering of particles and refraction.
 Unlike Greek writers on geometry, ancient Indian mathematicians did not give proofs or demonstrations.
 Developments in later centuries:
 In the 7th century, Indian mathematics came to be divided into two main areas—arithmetic with mensuration and algebra.
 Bhaskara I (Just after Guptas, early 7th century):
 Bhaskara I wrote a commentary on the Aryabhatiya, Aryabhatiyabhasya in 629 CE where he gave an interesting geometrical treatment for algebraic formulae.
 He gave a unique and remarkable rational approximation of the sine function in his commentary.
 This commentary is the oldest known prose work in Sanskrit on mathematics and astronomy.
 He was the first to write numbers in the Hindu decimal system with a circle for the zero.
 He also wrote two astronomical works in the line of Aryabhata’s school, the Mahabhaskariya and the Laghubhaskariya.
 Bhaskara I wrote a commentary on the Aryabhatiya, Aryabhatiyabhasya in 629 CE where he gave an interesting geometrical treatment for algebraic formulae.
 Brahmagupta (7th century):
 Brahmagupta made important contributions to geometry.
 Brahmagupta’s most famous result in geometry is his formula for area of cyclic quadrilaterals.
 He was the first mathematician to discuss the method of obtaining a cyclic quadrilateral having rational sides.
 He also put forward theories on the circumdiameter of a triangle and for finding the diagonals of a cyclic quadrilateral in terms of its sides.
 Brahmasphutasiddhanta contains:

 role of zero, rules of using zero with negative and positive numbers,
 a method for computing square roots,
 methods of solving linear and quadratic equations, and
 rules for summing series.
 The book was written completely in verse and does not contain any kind of mathematical notation.
 Nevertheless, it contained the first clear description of the quadratic formula.
 It is the earliest known text to treat zero as a number in its own right.

 Brahmagupta uses 3 as a “practical” value of π, and as an “accurate” value of π.
 Brahmagupta made important contributions to geometry.
 Mahavira (9th century):
 Mahavira (9th century) was a famous mathematician of Karnataka who lived in the court of the Rashtrakuta king Amoghavarsha Nripatunga of Manyakheta.
 He wrote a book called Ganitasarasangraha which dealt with various mathematical problems.
 He also gave formulae for the area and circumference of an ellipse.
 The formula he gave for the area of an ellipse was incorrect, but the one for the circumference was correct.
 Bhaskara II (12th century):
 Bhaskara II, author of the Lilavati, was another important mathematician, whose writings contain some important ideas of calculus.