Too large to write
But here is about just aryabhatta an other indians
The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BCE) and Vedic civilization (1500-500 BCE) to modern India .
Indian mathematicians have made major contributions to the development of mathematics as we know it today. One of the bigest contributions of Indian mathematics is the modern arithmetic and decimal notation of numbers used universally throughout the world (known as the Hindu-Arabic numerals). John Playfair, the famous scottish mathematician published a dissertation titled "Remarks on the astronomy of Brahmins" in 1790.His following quotation shows the appreciation of the then European Scientific community on the achievements of ancient Indian mathematicians and scientists.
"The Constructions and these tables imply a great knowledge of geometry,arithmetic and even of the theoretical part of astronomy.But what, without doubt is to be accounted,the greatest refinement in this system, is the hypothesis employed in calculating the equation of the centre for the Sun,Moon and the planets that of a circular orbit having a double eccentricity or having its centre in the middle between the earth and the point about which the angular motion is uniform.If to this we add the great extent of the geometrical knowledge required to combine this and the other principles of their astronomy together and to deduce from them the just conclusion;the possession of a calculus equivalent to trigonometry and lastly their approximation to the quadrature of the circle, we shall be astonished at the magnitude of that body of science which must have enlightened the inhabitants of India in some remote age and which whatever it may have communicated to the Western nations appears to have received another from them...."
Albert Einstein in the 20th century also comments on the importance of Indian arithmetic: "We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made."
Said Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of antiquity, Archimedes and Apollonius."
Other examples include zero, negative numbers, and the trigonometric functions of sine and cosine, which have all provided some of the biggest impetuses to advances in the field. Concepts from ancient and medieval India were carried to China and the Middle East, where they were studied extensively. From there they made their way to Europe and other parts of the world.
Indian mathematicians have made outstanding contributions to the development of mathematics as we know it today. The Indian decimal notation of numbers, concept of zero have probably provided some of the biggest impetus' to advances in the field. Concepts from India were carried to the Middle East, where they studied extensively. From there they made their way to Europe.
Unfortunately, some concepts of Indian origin have not been given due acknowledgement in modern history, with some discoveries/inventions by Indian mathematicians now attributed to their western counterparts. Some historians have argued that much of modern mathematics was either originally developed by Indian mathematicians or most probably known to them during periods of rapid mathematical development during the 19th and 20th century.
This remains an open question, and much more research still needs to be done. There exist significant bodies of mathematically fluent scientists and engineers within India that are testimony to the general mathematical educational standards within that nation.
Here is the list some important Indian mathematicians :-
BC
•Yajnavalkya, the author of the altar mathematics of the Shatapatha Brahmana.
•Lagadha - Author of a 1350 BC text on Vedic astronomy
•Baudhayana, 800 BC
•Manava, 750 BC
•Apastamba, 700 BC
•Aksapada Gautama, 550 BC, Logician
•Katyayana, 400 BC
•Panini, 400 BC, Algebraic grammarian
•Pingala, 5th century BC
•Bharata Muni, 4th century BC, combinatorics in music
AD 1-1000
•Aryabhata - Astronomer who gave very accurate calculations for astronomical constants, 476-520
•Varahamihira
•Bhaskara I, 620
•Brahmagupta - Helped bring the concept of zero into arithmetic
•Matanga Muni - Combinatorics in music
•Virahanka (8th century) - Gave explicit rules for the Fibonacci series.
•Shridhara (between 650-850) - Gave a good rule for finding the volume of a sphere.
•Lalla, 720-790
•Govindsvamin (9th century)
•Virasena
•Mahavira (9th century)
•Jayadeva (9th century)
•Prithudaka, 9th century
•Halayudha, 10th century
•Aryabhata II, 920-1000
•Vateshvara (10th century)
•Manjula, 930
AD 1000-1800
•Brahmadeva, 1060-1130
•Sripati, 1019-1066
•Gopala - Studied Fibonacci numbers before Fibonacci
•Hemachandra - Also studied Fibonacci numbers before Fibonacci
•Bhaskara II - Conceived of Differential Calculus
•Gangesha Upadhyaya, 13th century, Logician, Mithila school
•Pakshadhara, sone of Gangehsa, Logician, Mithila school
•Shankara Mishra, Logician, Mithila school
•Narayana Pandit
•Madhava - Considered the father of mathematical analysis, Founded some concepts of Calculus
•Parameshvara (1360-1455), discovered drk-ganita, a mathematical model of astronomy based on observations, Madhava's Kerala school
•Nilakantha Somayaji,1444-1545 - Mathematician and Astronomer, Madhava's Kerala school
•Mahendra Suri (14th century)
•Shankara Variyar (c. 1530)
•Vasudeva Sarvabhauma, 1450-1525, Logician, Navadvipa school
•Raghunatha Shiromani, (1475-1550), Logician, Navadvipa school
•Jyeshtadeva , 1500-1610, Author of Yuktibhasa, the world's first calculus text, Madhava's Kerala school
•Achyuta Pisharati, 1550-1621, Astronomer/mathematician, Madhava's Kerala school
•Mathuranatha Tarkavagisha, c. 1575, Logician, Navadvipa school
•Jagadisha Tarkalankara, c. 1625, Logician, Navadvipa school
•Gadadhara Bhattacharya, c. 1650, Logician, Navadvipa school
•Munishvara (17th century)
•Kamalakara (1657)
•Jagannatha Samrat (1730)
Born in 1800s
•Srinivasa Ramanujan (1887-1920)
•A. A. Krishnaswami Ayyangar (1892-1953)
•Prasanta Chandra Mahalanobis (1893-1972
•Satyendra Nath Bose (1894-1974)
•Sanjeev Shah (1803- 1896)
•Raghunath Purushottam Paranjape
Born in 1900s
•Raj Chandra Bose (1901-1987)
•S. N. Roy (1906-1966)
•Sarvadaman Chowla (1907-1995)
•Subrahmanyan Chandrasekhar (1910-1995)
•D.K. Ray-Chaudhuri
•Harish-Chandra (1923-1983)
•C. R. Rao
•Shreeram Shankar Abhyankar (1930-)
•Vijay Kumar Patodi (1945-1976)
•Narendra Karmarkar (1957-)
•Manjul Bhargava (1975 - )
•M.V. Subbarao (1921-2006)
•Divakar Viswanath
•Dhananjay P. Mehendale
•Bhama Srinivasan (1935-)
Born : 476 in Kusumapura (now Patna), India
Died : 550 in India
There were at least two mathematicians who lived by the name Aryabhata. This is a biography about Aryabhata from Kusumapura, now Patna. Aryabhata belonged to the Kusumapura School, but was probably a native of Kerala since his tradition is still in vogue there.
We know the year of Aryabhata's birth date since he tells us that he was twenty-three years of age when he wrote Aryabhatiya which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As Parameswaran writes :-
... No final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.
We do know that Aryabhata wrote Aryabhatiya in Kusumapura at the time when Pataliputra was the capital of the Gupta Empire and a major centre of learning, but there have been numerous other places proposed by historians as his birthplace. Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In it is claimed that Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although the author accepted that he lived most of his life in Kusumapura in the Gupta Empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the Aryabhatiya.
We should note that Kusumapura became one of the two major mathematical centers of India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close to Pataliputra) is on the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to reach it easily, and also allowed the mathematical and astronomical advances made by Aryabhata and his school to reach across India and also eventually into the Islamic world.
As to the texts written by Aryabhata only one has survived. However Jha claims that:-
... Aryabhata was an author of at least three astronomical texts and wrote some free stanzas as well.
At age 23 he wrote his small but famous work on astronomy and mathematics , the Aryabhatiya. In it he organized and combined existing knowledge of astronomy and mathematics. He says, "I delved deep in the astronomical theories, true and false, and rescued the precious sunken (hollow) jewel of the knowledge by means of the best of my intellect and by the grace of God". Aryabhata considered his work as a whole, but Brahmagupta divides the work into two parts in his Brahma Sphuta Siddhanta. It consists of 121 slokas-three slokas (verses) forming the introduction and the conclusion, ten slokas written in the Geetika metre, followed by 108 slokas in the Aryavrata metre. Brahmagupta called the Geetika metre and the Aryavrata metre, Dasgeetika and Aryashtasatam respectively. He then divided the Aryashtasatam into three parts:
Ganita (mathematics),
Kala-kriya (calculation of time),
Gola (sphere).
The Ganita deals with pure mathematics and therefore addresses topics such as: the methods of determining square and cube roots, geometrical problems, the progression, problems involving quadratic equations and indeterminate equations of the first degree. The method of solving these equations has been called Kuttaka by later mathematicians.
In the chapter “Gola”, Aryabhata defines all the circles given in the armillary sphere together with the small circles representing the diurnal motion of the sun. He was the first astronomer to mention that the diurnal motion of the heavens is due to the rotation of the earth about its axis.
Other contributions Aryabhata made towards pure mathematics were his approximation of pi and his sine tables.
Aryabhata's name spread rapidly and disciples gathered from all over to see him. Among his disciples were Bhaskara I, Sankaranarayana, Surya-deva, and Nilkantha.
The surviving text is Aryabhata's masterpiece the Aryabhatiya which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time. Its mathematical section contains 33 verses giving 66 mathematical rules without proof. The Aryabhatiya contains an introduction of 10 verses, followed by a section on mathematics with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of 50 verses being on the sphere and eclipses.
The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines. Let us examine some of these in a little more detail.
First we look at the system for representing numbers which Aryabhata invented and used in the Aryabhatiya. It consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, .... In fact the system allows numbers up to 1018to be represented with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols and the place-value system. He writes in [3]:-
... It is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.
Next we look briefly at some algebra contained in the Aryabhatiya. This work is the first we are aware of which examines integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers. The problem arose from studying the problem in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to solve problems of this type. The word kuttaka means "to pulverise" and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. The method here is essentially the use of the Euclidean algorithm to find the highest common factor of a and b but is also related to continued fractions.
Aryabhata gave an accurate approximation for π. He wrote in the Aryabhatiya the following:-
“Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.”
This gives π = 62832/20000 = 3.1416 which is a surprisingly accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, it is perhaps even more surprising that Aryabhata does not use his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but, for example, Ahmad considers this value as an approximation to half the perimeter of a regular polygon of 256 sides inscribed in the unit circle. However, Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of π by Aryabhata is [22] where Jha writes:-
“Aryabhata I's value of π is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that Aryabhata devised a particular method for finding this value. It is shown with sufficient grounds that Aryabhata himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered this value independently and also realized that π is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be ascribed to the celebrated mathematician, Aryabhata I.”
We now look at the trigonometry contained in Aryabhata's treatise. He gave a table of sines calculating the approximate values at intervals of 90 /24 = 3 45'. In order to do this he used a formula for sin(n+1)x - sin nx in terms of sin nx and sin (n-1)x. He also introduced the versine (versin = 1 - cosine) into trigonometry.
Other rules given by Aryabhata include that for summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of a circle which are correct, but the formulae for the volumes of a sphere and of a pyramid are claimed to be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V = Ah/2 for the volume of a pyramid with height ‘h’ and triangular base of area ‘A’. He also appears to give an incorrect expression for the volume of a sphere. However, as is often the case, nothing is as straightforward as it appears and Elfering (see for example [13]) argues that this is not an error but rather the result of an incorrect translation.
This relates to verses 6, 7, and 10 of the second section of the Aryabhatiya and in [13] Elfering produces a translation which yields the correct answer for both the volume of a pyramid and for a sphere. However, in his translation Elfering translates two technical terms in a different way to the meaning which they usually have. Without some supporting evidence that these technical terms have been used with these different meanings in other places it would still appear that Aryabhata did indeed give the incorrect formulae for these volumes.
We have looked at the mathematics contained in the Aryabhatiya but this is an astronomy text so we should say a little regarding the astronomy which it contains. Aryabhata gives a systematic treatment of the position of the planets in space. He gave the circumference of the earth as 4967 yojanas and its diameter as 15811/24 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24 835 miles, which is an excellent approximation to the currently accepted value of 24 902 miles. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. This is a quite remarkable view of the nature of the solar system which later commentators could not bring themselves to follow and most changed the text to save Aryabhata from what they thought were stupid errors!
Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.
It is in stanza 5 of Aryabhatiya that aryabhata tells of this method:
One should divide the second aghana by three times the square of the cube roots of the preceeding ghana. The square (of the quotient) multiplied by three times the purva (that part of the cube root already found) is to be subtracted from the first aghana and the cube (of the quotient of the above division) is to be subtracted from the ghana.
Certain steps have been left out in Aryabhata's method for claculating the cube root. This may have been due to limitations of the Sanscrit language. It was common at this time to pass on teachings orally, hence, it is understandable that some written methods may be vague.
Here is an example of how Aryabhata solved the cube, taken from
Find the cube of 1860867. Counting from right to left, the first, fourth, seventh and so on places are named ghana(cubic), the second, fifth, eighth and so on are called the first aghana (noncubic), while the third, sixth, ninth places and so on are called the second aghana.
So in this example we start by taking the cube root of 1=1
1-1=0, and you bring down the 8.
08
Three times the square of the root= 3((1)^2)=3
8=3(2)+2, so the first 2=the quotient or next digit of the root. So we have a remainder of 2 and now we bring down the 6.
26
Square of the quotient multiplied by three times the purva=((2)^2)(3)(1)=12.
26-12=14, now bring down the 0
140
The cube of the quotient is (2)^3=8
140-8=132, now bring down the 8.
1328
Three times the square of the root=3((12)^2)=432
1328= 432(3) + 32, so 3 is the quotient or next digit of the root.
We have a remainder of 32, and we bring down the 6.
326
Square of the quotient multiplied by 3 times the purva =((3)^2)(3)(12)=324
326-324=2, bring down the 7.
27
cube of the quotient =(3)^3=27
27-27=0, therefore the cube root of 1860867 is 123.
The following identities occur for the first time in Aryabhata's work, the Aryabhatiya.
Stanza 22:
The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one and twice the number of terms plus one, is the sum of squares. The square of the sums of the original series is the sum of the cubes.
This can be written as:
Sn^2 = 1^2 + 2^2 + 3^2 +...+ n^2 = (n(n+1)(2n+1))/6
Sn^3 = 1^3 + 2^3 + 3^3 +...+ n^3 = (1 + 2 +...+ n)^2
How Aryabhata arrived at these identities is unknown.