A Brief History of Calculus

Calculus is a mathematical discipline that mainly focuses on functions, limits derivatives, and infinite series. Isaac Newton and Gottfried Leibniz who each independently discovered the subject in the mid seventeenth century discovered the subject (Goldstine, 1980). However, each of the two disagreed as each claimed that the other had stolen the discovery from him. This bitter dispute ensued to the end of their lives with each claiming ownership over calculus. 

In the 17th century, some of the European mathematicians like Descartes, Pierre de Fermat, John Wallis, and Isaac Barrow came into constant discussions of developing a derivative. Consequently, Fermat came up with an adequality method, which would determine maxima and minima as well as tangents to curves, and this was closely related to differentiation (Goldstine, 1980).. However, Isaac Newton later indicated that his ideas about calculus were independent but they were drawn from the way Fermat drew tangents. On the other integral side, Cavalerri came up with the method of indivisibles in the period between 1630 and 1640, and this was more modern way of computing than the previous methods. Case in point, the area under the curve had previously only been calculated by Archimedes for the parabola. However this was later extended to other curves, including the cycloid by Torricelli. Later the formula used was generalized to negative powers and fractional by Wallis in the year 1656 (Goldstine, 1980). Three years later, Fermat came up with an ingenious trick that would be used for evaluating the integral of power functions directly. He also developed techniques that would help in finding centers of gravity, solid figures, and these inspired further development in quadrature. James Gregory was then influenced by Fermat's developments in tangency and quadrature provided a second theorem of calculus in the mod 17th century. However, Isaac Barrow gave the first full proof of calculus. 

Building From the previous works Newton and Leibniz each on his own developed the theory of infinitesimal calculus in the 17th century. Moreover Leibniz developed notations and concepts that were useful and consistent. On the other hand, Newton came up with vital applications in physics particularly in terms of integral calculus. Michel Rolles then came up with Rolles theorem in 1691 by the use of methods that were previously developed by Johan Wavaren the Dutch mathematician. Other important contributions were made by other many others like Huygens and Barrow. Newton's development of calculus began with the determination of the area under the curve which was followed by a reasoning of the area of a triangle whose area is a function of x and y. Newton realized that if the abscissa infinitesimal increased in abscissa, it would create a new formula and then he recalculated the areas using the binomial theorem he had discovered (Shell-Gellasch, Jardine, & Mathematical Association of America, 2005). This helped him understand inversion. In later years, Leibniz strove to develop calculus, and his main interests were by then politics, law, logic, metaphysics, and economics. However he met a mathematician Huygens in 1672, who played a pivotal role in convincing him to study mathematics. Although just like Newton he viewed the tangent as a ratio, he described it in simple terms as the ratio between abscissas and ordinates. The development later led to the invention of calculus (Goldstine, 1980). 

Before the developments by Newton and Leibniz were made the term calculus was only used to refer to the general body of mathematics. However, after the two men developed the concept, the terms were set a part to refer to the insights developed by the two. By the middles of seventieth century, the mathematical dimensions in Europe had drastically changed as compared to the previous century that depended on Hellenistic mathematics. However, they formed the beginning of the research as Newton and the contemporaries researched more on the works of modern mathematicians. Although there was need for this development in Europe, it was also no formalism but there was a lot of disordered mass of methods in form of paradoxes, notations, and theories. Consequently, Newton decided to conduct investigations on calculus in physics and geometry. In his description he described calculus as a scientific method of generating magnitude and motion. On the other hand, Leibniz laid his focus on the tangent problem and defined calculus as the metaphysical component or explanation of change. Despite these differences in opinion, they both focused on forming the inverse properties that existed between the differential and integral function as anticipated by the forerunners. The two were the first to view calculus as the way in which the new descriptive and rhetoric terms and their imagination and creativity helped them come up with the new mathematical system(Shell-Gellasch,Jardine, & Mathematical Association of America, 2005).. 

Many of Newton's discoveries were transmitted through correspondence and smaller papers and as the selected heir of Isaac Barrow at Cambridge University; he was admitted to pursue the degree. While there, he developed many formulas with the first one being the binomial theorem, which he extended and included negative and fractional exponents. He also expanded his analysis of the infinite series. One notable thing about him in the university is that he was a fast learner who understood theories and quickly expanded them into larger theories. He made most of his discoveries during the plague years of 1665 and 1666. The plague is associated with the Black Death and is marked in history as one of the most devastating pandemics in London's history. In total, it killed 75 to 200 million people in Eurasia and later peaked in Europe. The plague was associated with the bacterium caused by fleas (Woodhouse, American Mathematical Society, 2004). Moreover, the plague was responsible for many deaths, which were spread over the years of its occurrence. The land was overcrowded and hygiene could hardly be maintained, especially in the poorer sections of the city. Sanitation was also unheard of and dirty drainages flowed along the streets (Shell-Gellasch, Jardine, & Mathematical Association of America, 2005). Moreover, rubbish, animal dung, and the slops filled the streets, making them muddy and slippery. The place was also replete with flies' especially in summer and sewage was common in winter. Whenever garbage was collected, it would be thrown outside the city. making the entrance filthy. This, in turn, caused people to walk around with handkerchiefs on their noses, as they could not stand the stench. The much confusion in the city and immigrants turned the city into a slum. Notably, the plague mostly affected the poor people in London, as the rich were able to cross over to other countries or even go to their estates. However the property owners as well as the merchants lost most of their businesses, making the period a difficult economic time. This could explain why scientists and mathematicians were more eager to determine the solutions to the problems. London needed immediate rebuilding, and this was done as the government law initiated the process. The streets were made wider and the drainage systems were rebuilt. The houses were also reconstructed using stones and bricks as a mandatory measure. The growing of the economy could explain why Newton was more eager to make his development in the changing economy. In this period he posited that he was induced into isolations as he tried to evade the impact. 

References 

Goldstine, H. H. (1980).  A History of the Calculus of Variations from the 17th through the 19th Century . New York, NY: Springer New York. 

Shell-Gellasch, A., Jardine, D., & Mathematical Association of America. (2005).  From Calculus to computers: Using 200 years of mathematics history in the teaching of mathematics . Washington, D.C: Mathematical Association of America. 

Woodhouse, R., & American Mathematical Society. (2004).  A history of the calculus of variations in the eighteenth century . Providence, Rhode Island: AMS Chelsea.