History Of Maths:calculus

Calculus is a branch of mathematics consisting of two main areas; differentiation and integration. The first development of ideas around calculus and its applications can be traced back to the 17th Century. However, it was in 1684 when Leibniz published the first detailed explanation and notation of calculus in `Nova Methodus pro Maximis et Minimis`. (Britannica Encyclopaedia, 2006)

The first known reference to the ideas which underpin calculus can be traced back to Greek mathematicians. Democritus and Zeno of Elea both used the idea of infinitesimals which divide an object into an infinite number of sub sections. References to this idea can be found in around 450 BC when Zeno of Elea argued that `If a body moves from A to B then before it reaches B it passes through the mid-point, say B1 of AB. Now to move to B1 it must first reach the mid-point B2 of AB. Continue this argument to see that A must move through an infinite number of distances and so cannot move` (O`Connor and Robertson, 1996). However, these ideas were rejected and not considered further as they were seen to be contradictory. Although these ideas did not lead to a formalisation of calculus they formed the foundation for further progress.

According to David Bressoud (2004, p.1) `calculus developed over many centuries in many different parts of the world, not just western Europe but also ancient Greece, the Middle East, India, China and Japan.` Abu Ali al-Hasan ibn al-Haytham was an Arab Mathematician in the 11th Century whom authored over 90 books on subjects including astronomy and optics. According to David Bressoud (2004, p.1) he was the first person to have integrated a fourth-degree polynomial.

Similarly Rajeev (2006, pp.1-11) looks at the Kerala School of Astronomy and Mathematics which also made substantial contributions in the field of calculus; although their findings were not exported to Europe. A written account of foundational references to calculus can also be found in the Yuktibhasa which was written by Jyesthadeva in AD 1530. (Sarma and Hariharan, 1991, p.185)

According to Grattan-Guinness (1980, p.13) `in the 17th century calculus was closely bound up with the investigation of curves...inherited from the Greeks: conic sections, Hippias`s quadratics, Archimedean spiral`. The Greek ideas were carried forward in the works of prominent academics. In 1637 Descartes explored the ideas around calculus in the context of optics in `La géométrie` while Galileo explored ideas linked to motion and uniform acceleration. The common theme among these academics is the overlap between mathematics and physics. This demonstrates that in the early 17th Century there was more of an emphasis and greater interest in the practical application of mathematics. Although work around the concepts of calculus started in Ancient Greece the first detailed account of calculus was formally published in 1684 by Leibniz. Leibniz was a German mathematician and philosopher whom independently discovered calculus at the same time as Newton. Leibniz came across calculus as a way of formalising a solution to the tangent problem from a philosophical viewpoint. In contrast Newton came across calculus from a physics perspective. The Stanford Encyclopaedia of Philosophy (Look, 2007) explains that the ideas of calculus first came across Leibniz when he was reading the works of another mathematician Pascal. According to O`Connor and Robertson (2001) by 1676 Leibniz had discovered the formula d(xn) = nxn-1dx. It wasn`t until 1684 when Leibniz first published his work on calculus.

According to Gardner (2005) `it was between 1665 and 1666 that Newton first discovered calculus during the closure of his university due to an outbreak the plague`. Newton wrote about his ideas of `fluxionary calculus` in his unpublished work `De Analysi per Aequationes Numero Terminorum Infinitas` in 1669. Newton`s methods of fluxions where based on his insight that `the integration of a function is merely the inverse procedure to differentiating it.` (O`Connor and Robertson, 1996) Newton was led to the ideas of calculus from his work in physics and geometry. Newton saw calculus as a way of formalising the motion and magnitude of forces such as gravity. Despite developing his ideas Newton never formally published his worked.

The publication of Leibniz`s work led to a dispute between Newton and Leibniz over who was the first to discover calculus and led to claims of plagiarism from both sides. O`Connor and Robertson (2001) describe how the dispute centred on letters sent by Newton to Leibniz. In the first letter which was delayed Newton talked about the results of his work on calculus but did not disclose the methodology behind them. Leibniz replied promptly but unaware of the delay Newton assumed that Leibniz had worked on his reply for six weeks. However, the letter did prompt Leibniz to bring forward the publication of a detailed account of his methods. O`Connor and Robertson (1998) further describe that on 24 October 1676 Newton wrote a second letter to Leibniz `believing that Leibniz had stolen his methods`. However, Leibniz only received this letter in June 1677. In response to this Leibniz disclosed the underlying principles behind his work. According to O`Connor and Robertson (1998) Newton`s response to this was that `not a single previously unsolved problem was solved.` Following this Leibniz wrote a letter to the Royal Society calling for an investigation into who discovered calculus first.

As president of the Royal society Newton appointed an `impartial` committee to decide whether Newton or Leibniz invented calculus first. The impartiality of the investigating committee was hence severely compromised. In the capacity of President Newton also wrote the official report `Cornmercium epistolicum` on the findings of the committee. As expected The Royal Society found in favour of Newton as inventor of calculus but accredited Leibniz as the first publisher. (O`Connor and Robertson, 2000) It is now understood that both Newton and Leibniz discovered and developed calculus independently of each other.

When writing his work Newton used notation with which he was comfortable and only took into account his personal preference. However, Leibniz gave greater prominence to notation and placed a greater emphasis on finding a notation which could be used by all mathematicians. It was on the 21st November 1675 when Leibniz first used the notation ?(24&f(x); dx);??; in his manuscri pts. Leibniz`s notation is still used today.

As Anglin (1992) quotes, `Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost`. In this way the ideas of calculus have evolved over time with mathematicians developing new and extending old ideas and lines of thought. Calculus is still being developed with new ideas being applied to a variety of different subject areas such as economics and medical sciences.

References 1. Boyer, C B., (1949) The History of the Calculus and Its Conceptual Development. New York, Dover Publications, Inc. 2. Grattan-Guinness, I. (1980) From the Calculus to Set Theory, 1630-1910: An Introductory History. London, Gerald Duckworth & Co. Ltd. 3. Sarma, K and Hariharan, S. (1991) A Book of Rationales in Indian Mathematics and Astronomy; An Analytical Appraisal. Indian Journal of History of Science, 26 (2) pp 185-207. 4. Anglin, W. (1992) Mathematics and History. The Mathematical Intelligencer. Volume 14, Number 4 pp 6-12. 5. Belaval, Y and Look, B. (2006) Encyclopaedia Britannica Gottfried Wilhelm Leibniz. 6. O`Connor and Robertson. (2011) The MAC Tutor History of Mathematics Archive, University of St Andrews, Scotland. 7. Bressoud, D. (2004) Calculus Before Newton and Leibniz: Part I. 8. Rajeev, S. (2006) Neither Newton nor Leibnitz: The Pre-History of Calculus in Medieval Kerala. University of Rochester. 9. Gardner, K. (2005) A Brief History of Calculus. Acadia University. 10. Look, B. (2007) The Stanford Encyclopaedia of Philosophy: Leibniz.