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Apastamba (600 BCE - 540 BCE) was the author of a Sulbasutra which is certainly later than the Sulbasutra of Baudhayana. Apastamba's Sulbasutra is the most interesting from a mathematical point of view. Apastamba was neither a mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like Ahmes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and to improve and expand on the rules which had been given by his predecessors. It is clear from the writing that Apastamba, as well as being a priest and a teacher of religious practices, would have been a skilled craftsman. He must have been himself skilled in the practical use of the mathematics he described as a craftsman who himself constructed sacrificial altars of the highest quality. This work is an expanded version of that of Baudhayana. Apastamba's work consisted of six chapters while the earlier work by Baudhayana contained only three. The general linear equation was solved in the Apastamba's Sulbasutra. He also gives a remarkably accurate value for v2 namely 1 + 1/3 + 1/(34) - 1/(3434). This gives an answer correct to five decimal places. A possible way that Apastamba might have reached this remarkable result is described in the article Indian Sulbasutras. As well as the problem of squaring the circle, Apastamba considers the problem of dividing a segment into 7 equal parts.

 

Aryabhata (476550 CE) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499 CE, when he was 23 years old) and the Arya-siddhanta. Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. 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 sinesHis works and achievements are given below.

 

 His works in Mathematics

       Place value system and zero

       Pi as irrational

       Mensuration and trigonometry

       Indeterminate equations

       Algebra

 

 His works in Astronomy

       Motions of the solar system

       Eclipses

       Sidereal periods

       Heliocentrism

 
 

Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (ca. 820 CE), was particularly influential. Some of his results are cited by Al-Khwarizmi and in the 10th century Al-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.

 

Aryabhata II (c. 920 c. 1000 CE) was an Indian mathematician and astronomer, and the author of the Maha-Siddhanta. The treatise consists of eighteen chapters and was written in the form of verse in Sanskrit. The initial twelve chapters deal with topics related to mathematical astronomy and covers the topics that Indian mathematicians of that period had already worked on. The various topics that have been included in these twelve chapters are: the longitudes of the planets, lunar and solar eclipses, the estimation of eclipses, the lunar crescent, the rising and setting of the planets, association of the planets with each other and with the stars. The next six chapters of the book includes topics such as geometry, geography and algebra, which were applied to calculate the longitudes of the planets. In about twenty verses in the treatise, he gives elaborate rules to solve the indeterminate equation: by = ax + c. These rules have been applied to a number of different cases such as when c has a positive value, when c has a negative value, when the number of the quotients is a even number, when this number of quotients is an odd number, etc. Aryabhata II played a vital role in it by constructing a sine table, which was accurate up to five decimal places.