## Introduction

One of the branches of mathematics, calculus deals with integration and derivatives to study change. “Calculus is used in many disciplines such as in science and engineering and can solve problems for which algebra alone can not” (Robertson & O’Connor). It has two main branches namely integral and differential calculus and uses the concepts of algebra, geometry and trigonometry. The ideas of calculus were discovered over long period with the first steps to development of calculus being taken by the Greek mathematicians.

Some ideals of integral calculus were introduced in ancient period which were not systematically developed. The integral calculus ideas were used to calculate volumes and areas of objects by Egyptian papyrus around 1800 BC. Greek mathematician by the name Eudoxus implemented the exhaustion method which he used to find areas and volumes of regions and solids respectively in 408 BC. “Between 287 and 212 BC, Archimedes improved the exhaustion method and invented heuristics methods which were similar to integral calculus” (Gel’fand et al). The differential calculus development was first initiated by Archimedes when he calculated the tangent to a curve. The method of exhaustion was further used by Liu Hui in 3^{rd} century and Zu Chongzhi in fifth century AD to calculate area of square and volume of sphere respectively. In 1000 AD an Islamic mathematician, Alhachen, working in Egypt derived the formula of calculating sum of 4^{th} powers in geometric progression using a method which was later popularized to finding sum of any integral powers. “He nearly developed the formula of calculating polynomial integral but he was not concerned with polynomials with powers of more than 4 degrees.” (Gel’fand et al)

Baskare II, an Indian mathematician derived derivatives representing change for the first time in 12^{th} century and developed the Rolle’s Theorem. In the same century (12^{th}) a Persian mathematician “initiated cubic polynomials derivatives and the idea of function.” (James & Thomas) He also developed differential calculus concept such as minima and maxima of curves and derivative function to help him solve equations of quadratic nature.

The modern calculus discoveries were made during the 1600s. In Japan a mathematician by the name Seki Kowe expounded on the exhaustion method and come up with methods of finding are using integral calculus. In late 17^{th} century major invention in the field of calculus were made in Europe. These major inventions on calculus made it possible to solve problem which for long time were not solvable in mathematical physics. Mathematicians such as john Wallis and Isaac Barrow made major contribution during the 17^{th} century calculus invention in Europe. They both come with idea of derivative.

Sir Isaac Newton and Gottfried Leibniz pulled these new ideas of calculus together, and are usually credited for discovery of the modern calculus. Their major contribution was the fundamental theorem of calculus by culminating all “techniques which were used in past under one umbrella of differential and integral” (Kline). They both used the “infinitesimals” quantities in development of calculus. Each individual developed the foundations of calculus independently resulting to varying thoughts on the foundations though, their contributions were instrumental. Leibniz made development on notation and concepts while Newton was the first to make application of integral calculus. They changed the way people viewed calculus from traditional perspective to modern thinking. In mid 17^{th} century a new mathematical community had cropped in Europe that had inconsistence mass of formulas, techniques, notation and theories. Isaac and Leibniz wanted to bring uniformity in these techniques and formulas. Newton contribution on calculus was through investigations in geometry and physics. He analyzed changes of variables over time as basis to understand how motion and magnitude were generated, whiles Leibniz was concerned with tangent and deduced that calculus could be used to explain change in metaphysics. Although they differed in their approaches, their differences revolved around integral and differential calculus. Isaac came into the world of calculus long before Leibniz although much of his work were no published until 1690 whiles by 1684 Leibniz has published his worker formally.

**Isaac contribution**

Newton started his mathematical education as an assistant to Isaac Barrow in Oxford. He did not publish most of his work in formal publications instead much of findings were transmitted through small papers. Newton made his first contribution in calculus in 1664 when he advanced on binomial theorem which could be used in both negative and positive exponents. “By further expanding on the binomial theorem, he used finite algebraic quantities to make deduction on infinite series” (Robertson & O’Connor). Between 1665 and 1666, Isaac published his worker on Fluxional calculus. In his work he elaborated on how to calculate area under curve: finding the rate of change and then calculating the total area: by use of the binomial theorem he calculated the area. He used x’ and y’ quantities to calculate tangent. At this time he developed an expression to calculate the area bounded by a curve by using changes in a point” (James & Young, p. 31).

**Leibniz contributions**

He started his creation on calculus in 1666 the period when Isaac developed the fluxional calculus. “In his mind he wanted to generate a general method in which the entire world’s truth would be reduced to a kind of calculation.” Leibniz was the first mathematician in 1692 to use function to explain curve concept. He also saw that coefficient of equation could be arranged in a matrix to reach a solution of the equation. “In 1675 he did his major contribution to calculus by calculating area under a function y=f(x)” (James & Young). He also introduced the integral function, ∫ used in modern mathematics. Leibniz thought that the variables x and y could take many values to form a sequence and introduced dy and dx to denote the difference between two values of x and y. He understood that dy/dx gave the tangent. He placed much importance on the notation and also emphasis on the symbols he used which later helped him in his developments. As result much of symbols used in calculus today were those developed by Leibniz.

However, their techniques were much criticized by other mathematician who came after them. Their techniques were late advanced by other mathematicians to justify them and arrange calculus in a distinct mathematical form. It took more than a hundred years before calculus was developed to what it is today. Mathematicians Cauchy, Weierstrass, and Riemann approached calculus in term of limits rather than the infinitesimals. They believed that infinitesimal quantities never existed. “The notion of quantities being close to each other replaced the notion of infinitely small quantities advanced by Isaac and Leibniz in calculus” (Kline).

## Works cited

- Gel’fand I. M., Fomin, S. V, and Richard A. Silverman,
*Calculus of Variations*. Dover Publications, 2000. - James Clerk Maxwell and Thomas Young, “Science Progress: A Review Journal of Current Scientific Advance” Michigan: Blackwell, v.48 1960: 31
- Kline Morris,
*Mathematical Thought from Ancient to Modern Times,*Oxford university press, 2005. - Robertson, E F. and O’Connor J. J., A history of the calculus. 2008.