The calculus had its origin in the logical difficulties encountered by the ancient Greek mathematicians in their attempt to express their intuitive ideas on the ratios or proportionalities of lines, which they vaguely recognized as continuous, in terms of numbers, which they regarded as discrete.
Archimedes (287 BC – 212 BC) developed a significant part of the calculus. The fundamental notion of the calculus - that of a limit - was well understood by Archimedes, although he did not call it by name.
The first known definition of continuity according to Aristotle (383-322 BC): A thing is continuous when of any two successive parts the limits at which they touch are one and the same and are, as the word implies, held together.
Indian mathematicians produced a number of works with some ideas of calculus. The formula for sum of the curve was first written by Aryabhata in 500 AD, order to find the volume of a cube, which was an important step in the development of integral calculus.
Indian astronomers also came very close to creating what now call calculus. They had advanced to the point where they could apply ideas from both integral and deferential calculus to derive the infinites series expansion of the sine, cosine and arctangent functions.
Early applications for calculus included the study of gravity and planetary motion, fluid flow and ship design and geometric curves and bridge engineering.
Ancient history of calculus
