Aryabhata Wasinee Siewsrichol Aryabhata's life Around 1,200 years 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 astronomy which began the Zij period. Mansur wanted more accurate astronomical tables and a better approximation of the circumference of the Earth. This conference was published in the newspaper where a Hindu astronomer named Kankah translated the starting point of the Zij astronomy in India which was the translation of the first Sanskrit 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, arithmetic 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 mathematics and as well as astronomy. He was very influential in the Arabic 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 Patna. Aryabhata was a mathematician 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 Brahmagupta who said, Since Aryabhata knows nothing of mathematics, celestial sphere or time, I 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 universe 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 yugas (of the current yuga) 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, Aryabhatiya 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 mathematicians 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 Iran. 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 Pythagorean theorem, he obtained one of the two methods for making his table of sines. Other topics in the Ganita also include geometric measurements, mathematical series, 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 calendar because the periods of the repetition of certain relative positions of planets such as the Sun and Moon 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 Euclidean algorithm. 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 area of a triangle 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 algebra 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 Indian mathematics 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, Indias 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.