History of the Calculus and its Development
Introduction
Calculus is historically referred to as infinitesimal calculus, and is defined as a discipline whose main focus is derivates, limits, integrals, functions and infinite series. Ideas that lead up to the notions of derivative, function and integral developed in the entire phase of the 17th century. The history of this discipline and its development will be presented.
Discussion
Greek mathematicians are popularly known for their wide utilization of infinitesimals. The first person who is recorded to have considered the division of objects into infinite number of cross-sections seriously, but failed to accept the idea due to his inability to rationalize discrete cross-sections with a cone’s smooth slope was known as Democritus. These infinitesimals were further discredited by Zeno Elea on the grounds of the paradoxes created (Boyer, 110).
Heuristic methods which look like today’s calculus concepts were developed by Syracuse and Archimedes. The methods were alter incorporates into a general outline of integral calculus at the time of Newton. The first person to find the tangent to a curve, other than a circle was Archimedes, who used a method known as differential calculus. Diligent students of Archimedes include calculus pioneers like Johann Bernoulli and Isaac Barrow (Dauben and Scriba, 25).
The first essential advance of Archimedes was to prove the fact that the area of parabola segment equaled to four thirds, the area of the triangle with a similar vertex and base and two thirds the circumscribed parallelogram area. He formed a construction of an infinite series of triangles beginning with a triangle with area A and then adding more continually as shown below:
A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , …
He concluded that the area of the parabola was:
A(1 + 1/4 + 1/42 + 1/43 + ….) = (4/3)A. , which represents the first known sample showing summation of infinite series (Cahan, 69).
Other integrations made by Archimedes include surface area and volume of sphere, area and volume of a cone, volume of parabola segment and the surface area of an ellipse.
Major contributions were also made by three mathematicians, namely; Cavalieri, Roberval and Fermat. Cavalieri invented the method of indivisibles. It seems that he held the view that an area consisted of component lines which were summed up to an infinite number of indivisibles. Using the method, Cavalieri showed that the integral of xn starting from zero to infinity was an+1/(n + 1) through revealing the outcomes of the number of nvalues and making an inference of the general result.
Though Roberval considered similar problems, he was more rigorous compared to Cavalieri. He considered the are between a line and a curve to consist of an infinite number of rectangular strips that were infinitely narrow. According to Roberval, the integral of xmstarting from zero to one was approximated as: (0m + 1m + 2m + … + (n-1)m)/nm+1. He then calculated the area by asserting that the equations tended to 1/(m + 1) as n tended to infinity (Dauben, 173).
Fermat was known for generalizing the hyperbola and parabola. For a parabola, he generalized; y/a = (x/b)2 to (y/a)n = (x/b)m while for a hyperbola, he generalized; y/a = b/x to (y/a)n = (b/x)m. Fermat computed the sum of rp from r = 1 to r = n during the examination of y/a = (x/b)p. Another contribution made by Fermat was the investigation of minima and maxima through considering a situation when the tangent to the curve and the x-axis are in parallel (Boyer, 128).
Descartes contribution was a method sued to determine normal, which is based on double intersection. The translation of double intersection into double roots was later advanced by De Beaune. A simpler method known as the Hudde’s Rule, which typically uses a derivative, was discovered by Hudde. Newton was influenced through Descartes method and the Hudde’s Rule.
Newton discussed the converse problem when the relationship between x and y‘/x‘ is given and one is supposed to determine y. Newton found out that the problem could be solved through anti-differentiation. The series expansion for sin x and cos x were also calculated by Newton, as well as the expansion for the exponential function, which was established following introduction of the present notation by Euler (Stewart, 35).
The method of infinite series through which x flowing becomes X+0 and then xnbecomes (x+o)n was also determined by Newton. This is shown by, xn + noxn-1 + (nn–n)/2 ooxn-2 + . . . at the very end, the increment o disappears by way of taking limits. According to Newton’s pint of view, variables changed with time. Leibniz on the other hand held the view that the x and y variables ranged over sequences of values that were infinitely close. Additionally, Newton considered integration to include finding of fluents for a particular fluxion, implying differentiation and integration. Leibniz made use of integration as a sum, in a way that was the same as that used by Cavalieri (Edwards, 98).
Hence, calculus traces its root back to the Greek mathematicians in the 17thcentury. Its development is also owed to people like Newton, Hudde, Leibniz, Cavalieri and others.
References
Boyer, C. B. The history of the calculus and its conceptual development. Courier Dover Publication, 1949
Cahan, D. From natural philosophy to the sciences: writing the history of the nineteenth-century science. University of Chicago Press, 2003
Dauben, J., & Scriba, C. J. Writing the history of mathematics: its historical development. Birkhauser, 2002
Edwards, C. H. The historical development of the calculus. Springer, 1994
Stewart, J. Calculus: concepts and contexts. Cengage Learning, 2009
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