ANCIENT INDIAN MATHEMATICS
ANCIENT INDIAN MATHEMATICS
Hindu mathematicians were basically number theorists, and their mathematics an appendage to Astronomy. This was because of the need to advice on the right time to sow seeds in farms or for performing of religious rituals. Naturally most of them were priests, and all mathematical principles were enunciated in the form of verse. Being purely functional, no rigorous proofs were given unlike Greek mathematics.
The difference between Hindu and Greek mathematics was also greater in its focus. While Hindu mathematics was more arithmetic oriented, Greek mathematics was geometric centered. This accounts for the refinement of numerical symbols and great advancement in algebra. The flip side was Hindu geometry was devoid of proofs and was mensuration oriented.
The growth of Hindu mathematics falls into two periods:
- The SULVASUTRA period starting roughly from 800 BC to 200 AD
- Astronomical and Mathematical period starting from 400 AD to 1200 AD
SULVASUTRAS, which means "Rules of the Cord" are supplements of KALPASUTRAS which explain the construction of sacrificial alters.
The main SULVASUTRAS were composed by Baudhayana (about 800 BC), Manava (about 750 BC), Apastamba (about 600 BC), and Katyayana (about 200 BC). The aims very mostly religious, but the contents of the manuals were about geometric shapes such as squares, circles, rectangles an example of which is shown below.
The rope which is stretched across the diagonal of a square produces an area double the size of the original square.
The Indian Mathematicians of the ancient era excelled in Diophantine Equations (probably erroneously credited to Diophantus , because the Vedics were adept at it long before Diophantus), algebraic equations in which only solutions in integers are permitted.
These ancient Indian mathematicians were interested in very practical aspects of mathematics, such as determining the positions of the stars, developing a PANCHANGA (calendar/almanac), and ordinary mathematics for everyday use, like measurements of land and weights etc
Some of the prominent ancient Mathematicians were:
Aryabhata wrote Aryabhatiya when he was only twenty-three years of age, and it was due to his efforts Astronomy was delinked from Mathematics. He was able to calculate the value of pi to 3.1416 and the length of the solar year to 365.3586805 days.Aryabhata wrote Aryabhatiya in Kusumapura near Pataliputra (situated in modern Bihar) which was the capital of the Gupta Empire. He was far sighted in many of his opinions. He believed that the earth was a sphere and rotated on its axis, that the shadow of the earth falling on the moon was the cause of eclipse. In fact Aryabhata was the most scientific of ancient Indian astronomers. Significantly he had another illustrious contemporary by name Varahamihira who wrote Panchasiddhantika(Five schools of Astronomy). Varahamihira focused on three different branches of astronomy as studied during the period and they were; astronomy and mathematics, astrology and horoscopy. There was however one major difference, while Varahamihira gave importance to astrology, Aryabhata emphasized astronomy.
Brahamagupta was the foremost Indian mathematician of his time. He was born possibly in 598 in Ujjain, India. Brahmagupta'smagnum opus was Brahmasphutasiddhanta (The Opening of the Universe) which he wrote in 628. The book has 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty. Brahmagupta became the head of an astronomical observatory at Ujjain and his second work on mathematics and astronomy was the Khandakhadyaka
Brahmagupta's understanding of the number systems went far beyond that of others of the period. He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equation.
In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows:-
When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.
He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):-
A debt minus zero is a debt.
A fortune minus zero is a fortune.
Zero minus zero is a zero.
A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product of zero multiplied by a debt or fortune is zero.
The product of zero multipliedby zero is zero.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.
He wrote about quadratic equations, lunar eclipses, planetary conjunctions, and the determination of the positions of the planets. He died in the year 670 AD
Mahavira (or Mahaviracharya)
Mahavira was another great mathematician of yore. He was born around 800 AD possibly in Mysore, South India. He was a Jain and was familiar with Jaina mathematics. He wrote the book Ganita Sara Samgraha, in 850 AD, which was basically an updating of Brahmagupta's works
The nine chapters of the Ganita Sara Samgraha are:
2. Arithmetical operations
3. Operations involving fractions
4. Miscellaneous operations
5. Operations involving the rule of three
6. Mixed operations
7. Operations relating to the calculations of areas
8. Operations relating to excavations
9. Operations relating to shadows
He died around the year 870 AD
He was born around 870 AD possibly in Bengal, India. Sridhara is known as the author of two mathematical treatises, namely the Trisatika (sometimes called the Patiganitasara ) and the Patiganita. Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation. Most of the original manuscripts are no longer available, and the only source of information about him is from the works of Bhaskara II:- One rule for solving a quadratic equation is shown below
Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.
He died around the year 930 AD
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