“Calculus is the mathematics of change. Differential calculus, or differentiation, determines varying rates of change. Differentiation helps solve problems involving acceleration of moving objects from a flywheel to the space shuttle, as well as rates of growth and decay, optimal values, graphs of curves, and other issues. Integration is the ‘inverse’ (or opposite) of differentiation. It measures accumulations over periods of change. Integration can find volumes and lengths of curves, measure forces and work, and solve other problems. It is used in the day-to-day work of space scientists, architectural engineers, and theoretical physicists (Kowalski, 2011.).”

In the Western world, calculus was considered a mathematical breakthrough, because it dealt with continuously varying quantities, which until mid-17th century, in western mathematics, this concept had not been recognized. England’s Sir Isaac Newton (1642-1727) and Germany’s Baron Gottfried Wilhelm Leibnitz (1646-1716) are the 17th century European mathematicians who are generally credited with inventing/discovering calculus. It is certain that both Newton and Leibnitz created calculus methods independently of one another, and they argued until their deaths about who developed calculus first (Calculus: Maths in Flux, 1999). We should recognize, however, that this important work was built upon a foundation laid by many others over the centuries. As Newton famously acknowledged, “If I have seen further it is by standing on the shoulders of giants.”

Other contributors to the field of calculus are many. For example, the Greek Archimedes (approx. 287 – 212 BC), tackled problems of finding areas under parabolas and inside spirals, and he solved how to find the volume of the sphere, spherical segments, and the paraboloid. Archimedes also showed how to compute the slope of a line tangent to a spiral, the beginnings of differential calculus. He also used a clever ‘method of exhaustion’ to approximate the area inside a spiral. Archimedes attributed this method to Eudoxus of Cnidus (408-355BC) who used it to find the volume of a pyramid.

Additionally, in the tenth century, Thabit ibn Qurra (826-901) of southern Turkey and Abu Sahl al-Kuhi (940-1000) of northern Iran had discovered their own proofs for the volume of a paraboloid. Abu Ali al-Hasan ibn al-Haytham (Latinized Alhazen) (965-1040) born in Basra, Persia (now southeastern Iraq) read the work of Qurra and al-Kuhi and almost 700 years before the formulas for integrals would be known, he found the volume of a paraboloid by stacking disks.

Likewise, there were two Indian astronomer/mathematicians who are noteworthy for their contributions to calculus. Interestingly, they share the same name. Bhaskara I (n.d.), (600-680) authored the *Mahabhaskariya*, an eight chapter work on mathematical astronomy which included topics such as the longitudes of the planets, conjunctions of the planets with each other and with bright stars, eclipses of the sun and the moon, risings and settings, and the lunar crescent. Bhaskara II (n.d.) (1114-1185) is known for his writings in* Siddhanta Shiromani*, that work with differential calculus and its application to astronomical problems and computations.

Although there is evidence of all of this work (and more) in calculus in the ancient world, with records as early as Eudoxus of Cnidus (408-355BC), it is interesting to note that it is generally understood that centuries later, during the 1600’s and in Europe, an important ‘breakthrough’ in mathematics was occurring. This breakthrough was that European mathematicians began to use the algebraic structure of real numbers and they were imposing on it the notion of geometrical ‘closeness’. This is considered the birth of Analysis, a new field of mathematics at that time, which includes calculus. This is when the French philosopher, mathematician and writer, René Descartes developed the Cartesian plane (analytic geometry). It is claimed that this is what paved the way for the development of calculus in Europe. Yet, it is clear that many of the foundations of calculus were known well before Newton and Leibnitz. Therefore, it should be stated the history of the development of calculus, as we know it today, is the result of a long evolution of mathematical thinking, not a revolution in mathematical thinking, though Newton and Leibnitz are certainly the central figures to be credited. Newton and Leibniz were the first to state, understand, and effectively use the Fundamental Theorem of Calculus.

Considering the controversial origins of calculus, another more recent argument has arisen. That is, Dr. C. K. Raju presented a paper, *The Infinitesimal Calculus: How and Why it Was Imported into Europe *, at the International Conference on Indian History, Civilization and Geopolitics (ICIH 2009) that was held at New Delhi (How Jesuits Took Calculus, n.d.). This paper claimed that development of calculus is not a European invention but rather it is an Indian invention that had been appropriated by Europeans. The claim is that in the 1600’s, Jesuit priests took trigonometric tables and planetary models from the Kerala mathematicians of the Aryabhata school to Europe to assist with Europe’s foremost problem, that of navigation. The problem of navigation was that at that time, the European calendar was off by 10 days, and this led to inaccurate measurements of latitude. By using calculation methods described by the Indian astronomer/mathematicians Bhaskara I and Bhaskara II, the navigation problem was solved. It is believed that the Indian mathematics was the source from which both Newton and Leibnitz drew their insights. They did not attribute where they obtained their knowledge, but instead credited the discovery of their new calculating methods to themselves.

Therefore, in light of this more detailed understanding of the development of calculus, we can state that both Newton and Leibniz added to a vast body of knowledge in the areas of both differential and integral calculus that began long before them. The problems that are studied in calculus: areas, volumes, related rates, position/velocity/acceleration, infinite series, and differential equations had been solved before Newton and Leibniz, but the expression of these solutions was awkward and slow. Newton and Leibniz were the first to state, understand, and effectively use the Fundamental Theorem of Calculus. The field of calculus continues to evolve and develop today. Calculus is useful in solving problems in physics, biology, chemistry, economics, business, and other disciplines. On the Western front, the calculus quarrel between Newton and Leibnitz encouraged both men to publish their work so that later mathematicians could apply and expand it. The calculus quarrel illustrates the importance of publishing scientific work, and how important discoveries should be shared and also why they should be properly credited.

In conclusion, the Western world, calculus was considered a mathematical breakthrough. Mathematicians including René Descartes, Sir Isaac Newton, and Baron Gottfried Wilhelm are all are commonly given credit for its development. Yet, it should be recognized that this important work was built upon a foundation laid by many people in many lands and over many centuries. Contributors to the field of calculus outside of Western Europe of the Middle Ages included the Greek mathematician, Archimedes (approx. 287 – 212 BC); the Turkish, Thabit ibn Qurra (826-901); Abu Sahl al-Kuhi (940-1000), Abu Ali al-Hasan ibn al-Haytham (Latinized Alhazen) (965-1040) and Qurra and al-Kuhi (each from regions that we know today as Iran); Bhaskara I (600-680) and Bhaskara II (1114-1185), both from India; and Eudoxus of Cnidus (today’s Turkey) (408-355BC), who all contributed to the foundations and development of calculus. Therefore, it is clear that the development of calculus, as we know it today, is the result of many contributions to a long evolution of mathematical thinking, adding to and building upon the knowledge of many individuals and cultures.

References:

“Bhaskara I.” http://www-history.mcs.st-and.ac.uk/Biographies/Bhaskara_I.html (accessed November 27, 2012).

“Bhaskara.” http://www-history.mcs.st-and.ac.uk/Mathematicians/Bhaskara_II.html (accessed November 27, 2012).

Calculus: Maths in flux. (1999, December 25). *The Economist*, *353*(8151), 99.

Calculus, A P, and D Bressoud. “Calculus Before Newton and Leibniz: Part I, II, III.

“How Jesuits Took Calculus from India to Europe.” http://indianrealist.wordpress.com/2009/01/26/how-jesuits-took-calculus-from-india-to-europe/ (accessed May 2e, 2014).

Kowalski, K. M. (2011, July-August). A quarrel: Who invented calculus? Odyssey, 20(6), 17+. Retrieved from http://go.galegroup.com.library.esc.edu/ps/i.do?id=GALE%7CA267032619&v=2.1&u=esc&it=r&p=ITOF&sw=w

Raju C. K.. “The Infinitesimal Calculus: How and Why it Was Imported into Europe (Abstract).” Nehru Memorial Museum and Library, Teen Murti House, New Delhi 110 011 & Centre for Studies in Civilizations, 36 Tuglaquabad Institutional Area, New Delhi 110 062. http://www.indianscience.org/essays/31-%20E–Infinitesimal%20Calculus.PDF (accessed May 2e, 2014).

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