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Home , Aryabhata, Aryabhatiya, Jalali calendar, Sridhara, Yojana

Wasinee Siewsrichol

Aryabhata’s life

Around 1,200 ago, Abdullah Al Mansur, the second Abbasid Caliph, founded his new capital Baghdad, where he conducted a scientific conference. The Greek, Nestorian, Byzantine, Jewish, and Hindu scholars were invited.

This was a monumental scientific conference because the theme of the conference was observational which began the period. Mansur wanted more accurate astronomical tables and a better approximation of the of the . This conference was published in the newspaper where a Hindu astronomer named Kankah translated the starting of the Zij astronomy in which was the translation of the first text.

Then in 1881, archeologists found a 70-page manuscript in the village of Bakshali about 70 kilometers from a famous archeological site, Takshila. The manuscript contains mathematical discoveries of quadratic equations, finding square roots of numbers that are not perfect squares, and geometric progressions.

In the next era which was the 5th century AD, Aryabhata was born. Aryabhatta is considered one of very few people who changed the history of and as well as astronomy. He was very influential in the science world where he is referred to as Arjehir. Some historians believed that he was born in south India, either in Kerala, Tamil Nadu, or Andhra Pradesh. Others believe that he was born in Bengal, India which is the northeastern part of India. However, most historians can agree that he lived most of his life in Kusumapura or present day .

Aryabhata was a who was ahead of his time. People in the modern era celebrated this achieve- ment, however, his peers and astronomers of later generations ridiculed his ideas such as 7th century astronomer who said, Since Aryabhata knows nothing of mathematics, or time, have not mentioned separately his demerits. It was very rare for an Astronomer to think that the Earth was spinning which was different from the long accepted geocentric view. This geocentric view is that the Earth was the center of the and everything else spun around the Earth, not the other way around like what Aryab- hata thought. Aryabhatas views remained long forgotten until the heliocentric view of Nicolaus Copernicus became accepted.

Aryabhata’s mathematical works

Aryabhatiya and Aryabhatasiddhanta

2 Aryabhata wrote, ”When sixty times sixty years and three quarter (of the current ) had elapsed, twenty-three years had then passed since my birth. Historians do not know much about his biography except for his name and that he wrote Aryabhata when he was 23. However, marked the start of a new science in India and it was studied and analyzed over centuries. So far, there have been 12 commentaries to the work on record. The earliest commentary dated the first quarter of the 6th century and the latest one dated to the mid-19th century. The commentators include famous Indian and astronomers such a Bhaskara I (7th century), Paramesvara (15th century), and Nilakantha (15th-16th century). These preserved commentaries have shown that Aryabhatas work had been studied extensively. The original Aryabhatiya was written in Sanskrit but it has been translated into Hindi, Telugu, and Malayalam. Aryabhatiya was written in the traditional Indian form of verses.

At the end of the 8th century, the book was translated into Arabic and was renamed Zij al-Aryabhar. Around the same time, two books by Brahmagupta were also translated which had some of Aryabhatas mathematical and astronomical innovations. The Arabic version of Aryabhatas work were then translated into Latin, which made Aryabhatas work spread across West European scientists.

One of the few of his surviving books, Aryabhatiya, was split into four parts. His other famous work was Aryabhatasiddhanta. Aryabhatasiddhanta was never found, but it circulated mainly in the northwest of India and, through the Ssnian dynasty (224651) of . It is one of the earliest astronomical works to determine the start of each day to midnight. On the other hand, Aryabhatiya was a famous work in South India, and the work layed the foundation for astronomy. One historian claimed that Aryabhata was an author of at least three astronomical texts and wrote some free stanzas. Aryabhatiya is split into an introduction and three other parts: Ganita (Mathematics), Kala-kriya (Time Calculations), and Gola (Sphere). The introduction includes astronomical tables and Aryabhatas system. In Ganita, it uses the decimal number system to 10 decimal places and gives algorithms for obtaining the square and cubic roots. By using the , he obtained one of the two methods for making his table of . Other topics in the Ganita also include geometric measurements, mathematical , quadratic equations, compound interest, ratios, and solutions to linear equations. Aryabhatiya is written in 118 verses. This masterpiece starts with a 10 verse introduction, a 33 verse mathematical section, and 66 mathematical rules. However, there are no proofs given. The next section, the Kala-kriya, consists of 25 verses. And the final section, the Gola, has 50 verses.

The Phonemic Number Notation

3 Aryabhata does not use numbers but he instead uses letters or a phonemic number notation represented by a consonant-vowel monosyllable. There are 33 consonants of the Indian alphabet to represent the numbers 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are made of these consonants followed by a vowel to obtain 100, 10000,..., 10 to the 18. These alphabetic monosyllables were made so that there is not a long string of words when numbers are written in verbal form.

Zero and the Place Value System

It is also extremely likely that he knew about zero because of two things. First, his alphabetical counting system would be impossible without zero or a place value system. Second, the calculations for the square and cubic roots which were also impossible without the place value system and zero.

Linear Indeterminate

Linear indeterminate was enormously important in India because of the need to set a standard because the periods of the repetition of certain relative positions of such as the and have different revolution periods.

In the 3rd century AD, a Greek mathematician Diophantus was also concerned with linear indeterminate equations but he was only searching for rational solutions. However, Aryabhata tried to solve these equations in positive integers. There is a low possibility that there was any direct Greek influence, but they arrived at this problem because of different needs and using different methods. Historians have speculated that Indians got into contact with linear indeterminate from the Chinese. However, most scholars have agreed that Aryabhatas contribution was very valuable and he was the first person in the entire world to develop very elegant methods of integer solution.

In order to solve linear indeterminate, Aryabhata essentially used the . He used the kuttaka method and kuttaka means to pulverise and this method consisted of breaking the problem down into new problems. The coefficients will in turn become smaller and smaller. In his works, he did not use the number place value system or did he use the symbol zero. He instead used the letters if the alphabet to show powers of tens and null coefficient. He was also able to give the of a and solve for indeterminate equations which are equations for which there is more than one solution; for example, 2x = y is a simple indeterminate equation. The portion of Aryabhatiya describes integer solutions to equations. Forms such as by=ax+c and by=ax-c where a, b, c are integers. He talks about the rule of three which is to find the value of x when three numbers a, b and c are given.

There is a central part in all of which was the Rule of Three. The Rule of Three teaches people how to find the number x by knowing three given numbers a, b, c such that ab=cx. The Indian Rule of Three was passed over into the Arab and then into West Europe. The Rule of Three was also known elsewhere such as in China, Greece, and Egypt. However, rule was the most prominent since it was translated into problem solving methods. The Rule of Three was extended into five, seven, and etc. The article in ProQuest states an example, The treatise considers several problems which reduce to solving a linear equation in one unknown. One problem, set forth in part II, rule 30, is to calculate the value of an object if it is known that two men having equal wealth possess a different number of objects, a1 a2 and different pieces of money remaining after the purchase, b1, b2. The problem reduces itself to solving the equation a1x + b1 = a2x + b2. Then Aryabhata solves the equation and writes, Aryabhata formulates the rule of solving the linear equation in this manner: ”Divide the difference between therupakas with two persons by the difference between their gulikas. The quotient is the value of one gulika, if the possessions of the two persons are of equal value” (See ref. 1, part II, rule 30). That is to say, x=b2b1a1a2.x=b2b1a1a2.

In a commentary on Aryabhata, Bhaskara I wrote, ”Here Acarya Aryabhata has described the rule of three only. How the well-known rules of five, etc. are to be obtained? I say thus: The Acarya has described only

4 the fundamental ofanupata (proportion). All others such as the rule of five, etc. follow from the fundamental rule of proportion. How? The rule of five, etc. consists of combinations of the rule of three. In the rule of five there are two rules of three, in the rule of seven, three rules of three, and so on.”

There is a famous problem called the Problem of Messengers. This problem needs a person to calculate the time of meeting between two planets which are moving in opposite directions or in the same direction. Aryabhata writes, ”Divide the distance between the two bodies moving in the opposite directions by the sum of their speeds, and the distance between the two bodies moving in the same direction by the difference of their speeds; the two quotients will give the time elapsed since the two bodies met or to elapse before they will meet” (See ref. 1, part II, rule 31). Thus, if the distance S between the two bodies and their velocities V1 and V2 are known, the time of meeting is found as t=SV1+V2t=SV1+V2 when they are moving in opposite directions or as t=SV3V2t=SV3V2 when they are moving in the same directions. Aryabhata makes the solution in a way that avoids negative numbers.

Other Rules

Aryabhatas problems lead to quadratic equations in which a commentator states, capital A yields an unknown monthly profit x, which is then itself lent for interest for T months. The initial profit added together with the new interest is equal to B. Find the initial interest rate. Aryabhata gives the solution of the equation Tx2 + Ax = AB in verbal form corresponding to this expression: x=BAT+(A2)2A2T.x=BAT+(A2)2A2T.

Aryabhata set rules for the summation of arithmetical progression and states, of an arithmetical progression are set forth by Aryabhata in the part II, rule 19: ”Diminish the given number of terms by one, then divide by two, then increase by the number of the proceeding terms (if any), then multiply by common difference, and then increase by the first term of the (whole) series: the result is the arithmetic mean (of the given number of terms) This multiplied by the given number of terms is the sum of the given terms. Alternatively, multiply the sum of the first and last terms (of the series or partial series which is to be summed up) by half the number of terms” The first part of the rule finds the sum S of the terms of an arithmetical progression from the term p+1 to the term p+n: S=n[a+(n12+p)d].S=n[a+(n12+p)d]. Aryabhata also develops rules for finding the number of terms of an arithmetical progession: n=12[8ds+(d2a)22ad+1].n=12[8ds+(d2a)22ad+1].

Aryabhata gives the pertinent rule in part II, rule 32-33 for the Solution of this problem: find a number N, which, when divided by given numbers a, c yields two known remainders p, q. The problem leads to these indeterminate equations of the first degree: ax + b = cy, if p ¿ q (b = p q) ax b = cy, if p ¡ q Incidentally, the latter equation can be reduced to the former by substitution of the unknown.

Aryabhata’s geometrical rules include several verbal formulas. For example, he defines the area of a triangle as the product of the height multiplied by a half of the base as a half of the ’s multiplied by a half of the .

The area of any plane figure, writes Aryabhata in part II, rule 9, can be found if we single out two sides and then multiply one by the other. The commentator Paramesvara explains that what is meant here is the mean length and width.

According to ProQuest, Aryabhata determines the volume of a pyramid as base area multiplied by half the height. This, rather rough approximation is refined by other mathematicians, and in particular by , who finds the volume as the base area multiplied by a third of the height. Aryabhata calculates the volume of a sphere by the formula r2r2, which is equal to 147r3. This is rather approximative as compared with the exact formula for the volume of the sphere, 43r343r3 given in Bhaskara II.

An essential mathematical constant, which also had a great practical value, was the number estimating the ratio of the length of a circle of its diameter. For his time, Aryabhata’s estimation was rather accurate. The

5 value which was given by Aryabhata is correct to four decimal places: 3.1416. In part II, rule 14, Aryabhata gives the Pythagorean theorem: Add the square of the height of the to the square of its shadow. The of that sum is the semi-diameter of the circle of shadow.

In part II, rule 13, the scholar gives several geometrical definitions which are rather rare in Indian mathematical literature: A circle should be constructed by means of a pair of compasses; a triangle and quadrilateral by means of the two hypotenuses. The level of ground should be tested by means of water; and verticality by means of plumb.

Trigonometry

Arybhatiya contains topics on arithmetic, algebra, plane and spherical trigonometry, , quadratic equations, sums of power, series, and a table of sines. He did some work in trigonometry as well. He was the first mathematician to specify and for 1-cosx tables,with a 3.75 interval from 0 to 90 to an accuracy of 4 decimal places. He also defined cosine (kojya), versine (ukramajya), and inverse sine (otkram jya). Aryabhatas table of sines consisted of approximating the values at intervals of 90 degrees/24=3 degrees and 45 minutes. He achieved this by using the formula for sin(n+1)x-sin(nx) in terms of sin(nx) and sin(n-1)x. He gave the formulas for the of a triangle as well as the . He worked out the area of a triangle. His exact words were, ribhujasya phalashariram samadalakoti bhujardhasamvargah which translates for a triangle, the result of a with the half side is the area. But the formulas for the volumes of a sphere and pyramids were wrong as stated by most historians. For the volume of the pyramid, he gave the formula V=Ah/2 where V is the volume, A is the area, and h is the height. However, the very small portion of historians believe that it is not an error but a result of an incorrect translation. Historians such as Elfering claims that in verses 6, 7, and 10 in Aryabhatiya, he translated the verses in which the correct answer for both the formula for the volume of a pyramid and a sphere were present. This is because Elfering translated two terms in a different way. These terms have been used in different ways in the past, however, most historians concluded that he gave the wrong formulas for the pyramid and sphere.

By looking at Aryabhatas work, historians have concluded that Aryabhata knew about basic properties of sim- ilar and proportions, had idea about derived proprotions, relations of the segments of two intersecting chords, and the properties of the diameter perpendicular to a chord. In the book Aryabhatiya, the Indians have gotten some of their background information about trigonometry from the early Hellinistic astronomers who developed trigonometry chords. The Indians did not use chords, but used sines instead, which allowed them to work into problems relating to the sides and angles of right angled triangles.

Pi

6 He may or may not have realized that was an but he was able to correctly approximate pi to five digits, 3.1416. In Aryabhatiya, he was able to give an accurate approximation for pi and in the article it states, 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 essentially means that pi=62832/20000=3.1416. Aryabhata actually prefers to use the square root of 10 instead of 62832/20000 and square root of 10 is 3.1622. Many scholars criticize him because he does not provide proofs for these accurate values. In another interesting paper, it states, Aryabhata I’s value of 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 is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered this value independently and also realised that is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating . Thus the credit of discovering this exact value of may be ascribed to the celebrated mathematician, Aryabhata I.

Astronomy

Kala-kriya is about astronomy, but it focuses on planetary motion along the . In Gola, he predicted solar and lunar eclipses and explicitly states that the westward motion of the stars is due to the rotation about its axis. He also said that the brightness of the Moon and planets is due to the light that comes from the Sun. The reflection of the Sun is what humans see when they look at the moon or other planets. Scholars concluded that Aryabhata believed that the earth rotated about its axis and in his works, and he says that the Sun and Moon are each carried by epicycles, a circle in which a moves and which has a center that is itself carried around at the same time on the circumference of a larger circle, and they revolve around the Earth. In the Paitmahasiddhnta, it says the the motion of the planets are controlled by two epicycles. There is a smaller and larger epicycle which he calls manda and ghra. Manda means slow and ghra means fast. He also said that the Moon was the closest to the Earth followed by , , Sun, , , , and asterisms or other small groups of stars. In the astronomy portion of Aryabhatiya, he gave the circumference of the Earth as 4,967 and its diameter as 1,581 and 1/24 yojanas. One is five . Therefore, the circumference is 24,835 miles and the actual circumference of the earth is 24902 miles. He was only off by 67 miles.

7 Aryabhata and other Indian mathematicians used the shadow casted by a vertical pole and the gnomon to determine heights and distances. Then he created a number of rules and problems by the information he got from the shadows.

Eclipses

He was also able to explain the phenomena of eclipses by the shadows cast by the Earth and the shadows that other things cast on the Earth. His methods allowed future Indian astronomers to predict the length of the and they were short by a mere 41 seconds. He was one of very few people in his time to come up with scientific explanations for one of Earths great mysteries which was the eclipses. The popular Indian belief at the time was that the eclipses were caused by the demon Rahu.

Sidereal Rotation

Aryabhata was also able to calculate the length of the sidereal rotation which is the length of the earths rotation to the fixed stars. It was 23 hours, 56 minutes, and 41 seconds. The time that the modern scientists calculated was 23 hours, 56 minutes, and 4.091 seconds. He was also able to calculate the length of the sidereal to 365 days, 6 hours, 12 minutes, and 30 seconds with an error of 3 minutes and 20 seconds.This calculation was probably the most accurate in his period.

Collaboration with other scholars

Since there is only one surviving works of Aryabhata, he did not mention in his works any collaboration with other scholars.

Historical events that marked Aryabhata’s life.

Aryabhata traveled to Kusumapura which was a major communications network as well as one of the two major mathematical centers in all of India. The other mathematical center was . He lived there during the , also known as the Golden Years of India. There are four notable periods in Indias history which they called yugas. These yugas were the Golden Age, the Silver Age, the Bronze Age, and the Iron Age. In Kusumapura, it was easy to facilitate scientific knowledge and so his works spread across India and the Islamic world quite easily. He has also been speculated to have worked as the head of the University.

Prosperity in the Gupta Empire initiated a period known as the Golden Age of India, marked by extensive inventions and discoveries in science, technology, engineering, art, dialectic, literature, logic, mathematics, astronomy, religion, and philosophy. The Gupta empire was founded by Maharaja Sri-Gupta. The decimal system, chess, and the concept of zero were discovered during this time period. Many mathematicians, scien- tists, and scholars made huge contributions during this time. The philosophers of this time period discovered that the Earth was not flat, but that it was round and rotated on its axis. This rotation in turn caused the lunar eclipse. People discovered gravity and that there were a system of stars outside the Earth. This became known as planets and the solar system. Famous scholars of this time were Aryabhatta, Kalidasa, and .

There were developments in other areas as well. Literature became popular in India. This includes famous story tales of Panchatantra, Kam , and epics of Ramanyana and were also written during the Gupta Empire. The rulers during the Gupta empire promoted peace and prosperity which made the scientific and artisitic movements possible. Most of the rulers of the Gupta Empire were strong leaders, administrators, and traders. The rulers thought that trade was important and so they developed strong ties

8 with their neighboring countries. These countries include Burma, Malay Archipelago, Sri Lanka, and Indochina. Major rulers of Gupta Empire who contributed towards the Golden Age of India are Chandragupta (319 335 A.D), Samudragupta (335 375 A.D), Chandragupta II (375 414 A.D), Kumaragupta I (415 455 AD), and Skandagupta (455 467 A.D).

There were also some magnificent and beautiful architectural buildings created during this time. Many of the scultures were related to religion and spiritual realm, such as standing the Buddha of Mathura and the sitting Buddha in Sarnath, and famous caves of rock-cut monasteries in Ajanta. There were also paintings of Badami and Bagh. The style of Gandhara School of art that was originally developed in Mathura had gained much prominence during this period. The architectures started to change the way temples look by installing statues of Gods in the temples. During the Golden Age, temples became bigger, more extravagant, and fully decorated with carvings. However, sadly and unfortunately, invaders destroyed most of the temples that were created during this period.

Significant historical events around the world during Aryabhata’s life

According to History Central,

”500 AD Arthur’s Victory Over Saxons-The legendary Arthur won a battle against the Saxons at Mound Badon in Dorset, in Southern England. This slowed the Saxon conquest of England.

500 AD Svealand- The First Swedish State- Svealand, the first Swedish state was founded around 500 A.D. The Goths inhabited the Southern part of the Swedish peninsula. Much of what is known about early Sweden has been taken from the epic ”Beowulf”, written in 700 A.D.

503 - 557 AD Persian-Roman Wars- Between 503 and 557 A.D.,three successive wars – interrupted by periods of peace – are fought between the Persian Empire and the Eastern Roman Empire. All have the same basic cause – an inability to define the borders and the relationship between the two empires. In 567 a ’definitive’ peace was reached. Under its terms, Rome agreed to pay the 30,000 pieces of gold annually. The borders between the empire were reaffirmed, Christian worship was to be protected in the Persian Empire, and regulation of trade and diplomatic relations were laid out.

507 AD Kingdom Of Franks - The Franks’ Clovis defeated the Visigoths under Alaric II at the Battle of Vouille. The Visigoths retreated into Spain, where they retained their Empire.

532 AD Nika Revolt- A popular uprising took place in Constantinople against Justinian. Constantinople was nearly destroyed by fire. The insurrection quelled with great cruelty by Belisarius. Thirty thousand people were slain.

537 AD Hagia Sophia Cathedral Built- The Hagia Sophia Cathedral in Constantinople was completed. The cathedral represented the culmination of Byzantine architecture, with a large domed basilica.

552 AD Battle at Taginae- The Byzantine army invaded Italy and defeated the Ostrogoths at the Battle of Taginae. The Byzantines, using a combination of pikes and bows, decimated the Ostrogothic Kingdom of Italy.

558 - 650 AD The Avars- The Avars, a Turkish Mongolian group, formed an Empire that extended from the Volga to the Hungarian plains. In 626 A.D., they laid siege to Constantinople, but were forced to withdraw.

565 AD Justinian Great- Justinian the Great died in 565 A.D. bringing to end 38 years of rule as leader of the Byzantine Empire. Under his stewardship, the Empire expanded to include all of North Africa and parts of the Middle East as well as Italy and Greece. Under Justinian, the first comprehensive compilation of Roman

9 Law was published.

572 AD Leovigild King Of Visigoths- Leovigild the King of Visigoths set off to reinvigorate the empire. He extended the Vistigoth dominance to all parts of the Iberian Pennisular.

577 AD Battle At Deorham- At the Battle of Deorham in southwestern England, the Saxons defeated the Welsh. This victory virtually completed the Saxon conquest of England.

581 AD Sui Dynasty Reunites China- After nearly four centuries of internal divisions and strife, China was reunited under the leadership of Yang Jian. A member of a respected aristocratic family, Yang Jian founded the Sui Dynasty. Yang Jian used Buddhism to help unite the kingdom.

598 AD Pope Greogory Obtains 30 Year Truce- Gregory the Great was the first monk to become Pope. For many, he was a model for the future papacy. Gregory controlled the civil affairs of Rome and expanded the power of the Church. Gregory also negotiated a 30-year truce with Lombards, insuring the independence of Rome.”

Significant mathematical progress during the Aryabhata’s lifetime

Chinese traveler Fa Xian visited India from 399-405 CE during the reign of Emperor Chandragupta II who promoted the synthesis of science, art, philosophy, and religion during the Golden Age of India. He took detailed observations and wrote them in his journal which he later published. In his journal, he wrote that the Golden Age was the height of the Gupta Empire because this period marked the extensive inventions in science, technology, engineering, art, dialectic, literature, logic, mathematics, astronomy, religion, and philosophy that contributed to Hindu culture. Ayurvedic was established, and ayuverdic was a form of alternative medicine. Vavartna, also known as the nine jewels, was a group of nine scholars in the court of Chandragupta II who contributed many advancements in their academic fields. Aryabhata was one of these scholars who believed in the concept of zero and knew the approximation for the value of pi. Aryabhata is also believed to be the first of the Indian mathematician-astronomers who postulated the theory that the Earth moves round the Sun and is not flat, but instead is round and rotates about its own axis. He also may have discovered that the Moon and planets shine by reflected . Varahamihira was an astronomer and mathematician like Aryabhata. Sushruta was a famous physician who wrote the Samhita, a Sanskrit text on all of the major concepts of ayurvedic medicine with innovative chapters on surgery.

Connections between history and the development of mathematics

He made calculations for the calendar, and they are still being used today in India for fixing the panchanga or the . The Islamic world also used these calculations as a basis for the which was introduced in 1703. The modified versions which includes and they are the national for Iran and today.

Remarks

There were some historic events and objects that were named in his honor which includes Indias first satellite launched by the Indian ISRO (Indian Space Research Organization), the lunar crater, and the interschool Aryabhata maths Competition.

10 A research establishment has been set up in Nainital, called the Aryabhatta Research Institute of Observational Sciences (ARIOS) to honor his contribution to the field of science. There is also a lunar crater and a species of bacteria discovered by ISRO named after Aryabhatta. Bhaskara I who wrote a commentary on the Aryabhatiya about 100 years later wrote of Aryabhata, Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.

References

www.boundless.com/world-history/textbooks/boundless-world-history-i-ancient-civilizations-enlightenment-textbook/ early-civilizations-in-the-indian-subcontinent-4/the-gupta-empire-29/the-golden-age-of-india-119-13221/ www.newworldencyclopedia.org/entry/Aryabhata www-history.mcs.st-and.ac.uk/Biographies/Aryabhata_I.html www.britannica.com/biography/Aryabhata-I search.proquest.com/docview/362773868/C5D9F4C2028A4391PQ/1?accountid=14552 www.shalusharma.com/aryabhatta-the-indian-mathematician www.math10.com/en/maths-history/math-history-in-india/mathematical_achievements_of_aryabhatta. html http://www.historycentral.com/dates/500ad.html

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