Calculus or ‘infinitesimal calculus’ is a mathematical discipline which focuses on limits, continuity, derivatives, integrals and infinite series. The origin of Calculus can be traced back to ancient Greece, China, medieval Europe and India. In the late 17th century, Isaac Newton and Gottfried Wilhelm Leibniz developed their own independent versions of ‘Infinitesimal Calculus’. The great mathematicians argued with each other and the Leibniz-Newton calculus controversy continued until the death of Leibniz in 1716. ‘Calculus’ continues to play a major role in the field of science ever since its inception.

‘Calculus’ primarily deals with the study of functions and limits. The word ‘calculus’ means a small pebble in greek. ‘Calculus’ finds its usage across different streams. This includes propositional calculus in logic, process calculus in computing, and felicific calculus in philosophy.

Early precursors of ancient calculus

Egypt and Babylonia

Several ideas that led to integral calculus seem to have been introduced in the Egyptian and Babylonian civilizations.

Greece

Archimedes was the first mathematician to find the tangent to a curve other than a circle, using a method similar to differential calculus. Isaac Barrow and Johann Bernoulli who are considered as the pioneers of calculus, were students of Archimedes.

China

The method of exhaustion was developed in China. In the 5th century, the Cavelieri’s principle for finding the volume of a sphere was discovered by Zu Chongzhi.

Early precursors of medieval calculus

Middle East

The formula for the sum of fourth powers was derived by Hasan Ibn al-Haytham (Al hazan). The results were used to carry out integration.

India

Ideas on calculus appeared in Indian mathematics too! Most of the credits go to the Kerala school of astronomy and mathematics. Components of calculus such as Taylor series and infinite series of approximations were first stated by Madhava of Sangamagrama in the 14th century. Followed by the Kerala school of mathematicians. But they did not really convert calculus into a powerful problem-solving tool as it is today.

Europe

In the 14th century, Oxford calculators and French collaborators such as Nicole Oresme revived mathematical study of continuity.

Modern precursors

Integrals

‘Stereometrica Doliorum’ by Johannes Kepler published in 1615 forms the basis for integral calculus. He developed a method for calculating the area of an ellipse by adding up the lengths of many radii drawn from a focus.

Derivatives

In the 17th century, European mathematicians including Isaac Barrow and Blaise Pascal discussed the idea of a derivative.

Going ahead, Isaac Newton shared his early ideas about calculus.

Fundamental theorem of calculus

English mathematician Isaac Barrow gave the first full proof of the fundamental theorem of calculus. James Gregory was successful in proving a restricted version of the second fundamental theorem of calculus.

Other developments

In 1691, making use of the methods developed by the Dutch mathematician Johann van Waveren Hudde, Michel Rolle gave the first proof of Rolle’s theorem. Bernard Bolzano and Augusrin-Louis Cauchy stated the mean value theorem in its modern form.

Newton vs Leibniz

Isaac Newton and Gottfried Leibniz are two key men behind the discovery of Calculus. They independently developed the foundations for calculus. They had major differences between them. Newton considered variables changing with time.On the other hand, Leibniz assumed variables x and y as sequences of infinitely close values. He used dx and dy as differences between successive values of such sequences. Despite knowing the fact that dy/dx gives the tangent, he did not use it as a defining property. Newton, on the other hand, used finite velocities x’ and y’ to compute the tangent. While Newton thought Calculus was geometrical, Leibniz took it towards analysis.

The development of calculus can be mainly divided into three periods. Anticipation, Development and Rigorization. In the anticipation stage, mathematicians used techniques which involved infinite processes to find area under curves. Newton and Leibniz laid the foundation for calculus in the development stage, bringing all the techniques under the umbrella of derivative and integral. But it took a lot of time for mathematicians to justify their methods and put calculus on a sound mathematical foundation. This happened primarily during the Rigorization stage.

In short, calculus is the mathematical study of continuous change. The article describes the development of ‘calculus’ over the years.