Arya Bhata was born about 476 AD in India, probably in central India, though nobody knows for sure. When Arya Bhata was a teenager, he left his hometown and went to Kusumapura (modern Patna) in north-eastern India, on the banks of the Ganges river, to study at Nalanda university there. At this time, Nalanda University was a very good college; it was one of the earliest universities anywhere. Nalanda was a Buddhist school, and King Ashoka paid for some of its buildings. One of the buildings was an astronomical observatory. Students came from far away, even from China, to study at Nalanda.
Arya Bhata must have worked hard in college, because by the time he was 23 years old, in 499 AD, he was already writing an important book about math and astronomy of his own. It is a very short book, summarizing important new ideas in 108 short poems.
For the astronomy part of the book, Arya Bhata seems to have thought that the earth spun around on its axis (as it really does). But he also thought that the stars moved around the sky, not realizing that they only seem to move because of the movement of the earth.
Because Arya Bhata believed that the earth was at the center of the universe and the sun and the planets and the stars all moved around the earth in different orbits, he saw astronomy as a process of calculating distances and movements from the earth to these orbits. To calculate these distances, you needed trigonometry, and Arya Bhata used the trigonometry developed by Hipparchus and Ptolemy for this purpose. Arya Bhata defined the concept of the sine and cosine, which he called jya and kojya, meaning "chord" and "perpendicular chord", and wrote down a table of sines. He took trigonometry further than Ptolemy had by working with half-chords.
At the same time, Arya Bhata was also working on defining tangents and cotangents.
In another math section of his book, Arya Bhata was inspired by recent work on using place values to help add and multiply large numbers. Zero had not yet been invented, but Arya Bhata took some more steps along the path from place values towards the idea of zero.
Arya Bhata also wrote several other books about math and astronomy, but we no longer have copies of some of them. One of these lost books is the Arya-siddhanta. In this book, Arya Bhata described ways of measuring time, like sundials and water clocks. Although we may not think of sundials and clocks and sines and tangents as all being related now, to Arya Bhata and other scientists of his time, all of these were things that you needed in order to understand the movements of the stars and the planets around the Earth.
To find out more about Indian science, check out these books from your local library or from Amazon:
Science in Ancient India, by Melissa Stewart (2002). Written for kids.
Eyewitness India, by Manini Chatterjee (2002). Written for kids.
Ancient India, by Virginia Schomp (2005). Written for middle schoolers. Very good for reports.