Q)Briefly discuss the developments in mathematics and astronomy during Gupta period.
Ans)MATHEMATICS
The Jainas attached great importance to mathematical proficiency in their religious teaching. Various works deal at great lengths with mensuration, surds, fractions, permutations and combinations, geometry , law of indices, classification of numbers etc. In mensuration, they worked out the relations between the diameter, circumference and arc and chord of a circle. Whey also found out approximations to find approximate value of a surd. Jainas used large and complicated arithmetical factors. They frequently resorted to approximation in dealing with large numbers. When ever the fractional part was greater them 1/2 it was replaced by unitywhen less than 1/2 it was neglected. Jainas speculated about the possible nembers of combinations out of n fundamental categories, taken one at a time, two at a time, three at a time or more at a time. In all the cases they succeeded in finding permutation and combination formulas known to us today.
As a distinct branch of mathematics, algebra appeared from about the time of Bramagupta(about 598 A.D.). Indian algebraists, psiible for the first time in history, used abbreviations of names and colours or gems or symbolsof unknown quantities, and operations like powers, roots etc.
They had classified algebriac equations into three groups :
i) equations in one unknown,
ii)equations in several unknwons
iii)equations in products of unknowns,
Solutions of linear and quadratic equations were known to them. They also knew how to find solutions for intermediate equations of first and second degree with more than one unknown. Solutions of higher dgree equations were attemted by them.
Numerals
Apart from algebra, possibly the greatest contribution of ancient Indian civilization was the invention of numerals. The necessity for numerals and numerical notation by words and letters had arisen when human beings started dealing with very large and very small numbers such as astronomy and in precision measurement of precious metals. We have already pointed out that the expansion of trade and navigation in this age promoted these areas of science. Human beings could no longer express their trading exchanges, number and distance of stars or even number of days denoting the periodicity of a star by vertical strokes one after the other as was the custom since the time of Harappan civilization. The decimal or the zero system was first found in a Gujara grant plate inscription of 595 A.D. and later in other stone inscription from Gwalior, Mahipala, Bauka etc. It seems to have been used by the contemporary and later astronomers who constantly improved it. Arabs adopted this system and greatly improved upon the earlier numerals, so much so that today these numerals are called Arab numerals or numbers.
ASTRONOMY
The origin of Indian astronomy is dated to the Vedic times. These have been described in texts called Siddhantas. The Indian astronomy of those times recorded accurate observations of the sun, moon and the planets. It could not build a rational and convincing theory of how the planetary system worked. Aryabhatta born in 476 A.D., was the greatest astronomer of the Gupta period. It was his firm belief that the earth was rotating and the heavens resting.He also gave a scientific explanation for the occurrence of eclipses as opposed to the prevailing ideas that Rahu and Ketu caused eclipses.Another great achievement to Aryabhatta's credit was the construction of trigonometric tables.He computed trigonometric tables geometrically and used the values of "sine" and "cosine" in his astronomical calculations. Besides these he developed formulas for the sum of arithmetic and geometric series.Aryabhatta was followed by Varahamihira who recorded the works of Arysbhatta and older astronomical findings in his classical work Brihatasamhita.
Ans)MATHEMATICS
The Jainas attached great importance to mathematical proficiency in their religious teaching. Various works deal at great lengths with mensuration, surds, fractions, permutations and combinations, geometry , law of indices, classification of numbers etc. In mensuration, they worked out the relations between the diameter, circumference and arc and chord of a circle. Whey also found out approximations to find approximate value of a surd. Jainas used large and complicated arithmetical factors. They frequently resorted to approximation in dealing with large numbers. When ever the fractional part was greater them 1/2 it was replaced by unitywhen less than 1/2 it was neglected. Jainas speculated about the possible nembers of combinations out of n fundamental categories, taken one at a time, two at a time, three at a time or more at a time. In all the cases they succeeded in finding permutation and combination formulas known to us today.
The jainas could find formulas like
nC1=n,
nC2=n(n-1)/1X2
nC3=n(n-1)(n-2)/1X2X3
nP1=n
nP1=n(n-1)
nP1=n(n-1)(n-2)
Algebra in India
An implicit basic formulation used in all Jaina work on numbers is the modern law of indices:
amxan=a(m+n),and (am)n=amn
where m,n may be integral or fractional. As a distinct branch of mathematics, algebra appeared from about the time of Bramagupta(about 598 A.D.). Indian algebraists, psiible for the first time in history, used abbreviations of names and colours or gems or symbolsof unknown quantities, and operations like powers, roots etc.
They had classified algebriac equations into three groups :
i) equations in one unknown,
ii)equations in several unknwons
iii)equations in products of unknowns,
Solutions of linear and quadratic equations were known to them. They also knew how to find solutions for intermediate equations of first and second degree with more than one unknown. Solutions of higher dgree equations were attemted by them.
Numerals
Apart from algebra, possibly the greatest contribution of ancient Indian civilization was the invention of numerals. The necessity for numerals and numerical notation by words and letters had arisen when human beings started dealing with very large and very small numbers such as astronomy and in precision measurement of precious metals. We have already pointed out that the expansion of trade and navigation in this age promoted these areas of science. Human beings could no longer express their trading exchanges, number and distance of stars or even number of days denoting the periodicity of a star by vertical strokes one after the other as was the custom since the time of Harappan civilization. The decimal or the zero system was first found in a Gujara grant plate inscription of 595 A.D. and later in other stone inscription from Gwalior, Mahipala, Bauka etc. It seems to have been used by the contemporary and later astronomers who constantly improved it. Arabs adopted this system and greatly improved upon the earlier numerals, so much so that today these numerals are called Arab numerals or numbers.
ASTRONOMY
The origin of Indian astronomy is dated to the Vedic times. These have been described in texts called Siddhantas. The Indian astronomy of those times recorded accurate observations of the sun, moon and the planets. It could not build a rational and convincing theory of how the planetary system worked. Aryabhatta born in 476 A.D., was the greatest astronomer of the Gupta period. It was his firm belief that the earth was rotating and the heavens resting.He also gave a scientific explanation for the occurrence of eclipses as opposed to the prevailing ideas that Rahu and Ketu caused eclipses.Another great achievement to Aryabhatta's credit was the construction of trigonometric tables.He computed trigonometric tables geometrically and used the values of "sine" and "cosine" in his astronomical calculations. Besides these he developed formulas for the sum of arithmetic and geometric series.Aryabhatta was followed by Varahamihira who recorded the works of Arysbhatta and older astronomical findings in his classical work Brihatasamhita.
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