What did Fermat contribute to the field of calculus?

Answer 1

Fermat contributed to the development of calculus. His study of curves and equations prompted him to generalize the equation for the ordinary parabola ay=x2, and that for the rectangular hyperbola xy=a2, to the form an-1y=xn. The curves determined by this equation are known as the parabolas or hyperbolas of Fermat according as n is positive or negative (Kolata). He similarly generalized the Archimedean spiral, r=aQ. In the 1630s, these curves then directed him to an algorithm, or rule of mathematical procedure, that was equivalent to differentiation. This procedure enabled hi m to find tangents to curves and locate maximum, minimum, and inflection points of polynomials (Kolata). His main contribution was finding the tangents of a curve as well as its points of extrema. He believed that his tangent-finding method was an extension of his method for locating extrema (Rosenthal, page 79). For any equation, Fermat 's method for finding the tangent at a given point actually finds the subtangent for that specific point (Eves, page 326). Fermat found the areas bounded by these curves through a summation process. "The creators of calculus, including Fermat, reli ed on geometric and physical (mostly kinematical and dynamical) intuition to get them ahead: they looked at what passed in their imaginations for the graph of a continuous curve..." (Bell, page 59). This process is now called integral calculus. Fermat founded formulas for areas bounded by these curves through a summation process that is now used for the same purpose in integral calculus. Such a formula is: A= xndx = an+1 / (n + 1) It is not known whether or not Fermat noticed that differentiation of xn, leading to nan-1, is the inverse of integrating xn. Through skillful transformations, he handled problems involving more general algebraic curves. Fermat applied his analysis of infinitesimal quantities to a variety of other problems, including the calculation of centers of gravity and finding the length of curves (Mahoney, pages 47, 156, 204-205). Fermat was unable to notice what is now considered the Fundamental Theorem of Calculus, however, his work on this subject aided in the development of differential calculus (Parker, page 304). Additionally, he contributed to the law of refraction by disagreeing with his contemporary, the philosopher and amateur mathematician, René Descartes. Fermat claimed that Descartes had incorrectly deduced his law of refraction since it was deep-seated in his assumptions. As a result, Desc artes was irritated and attacked Fermat's method of maxima, minima, and tangents (Mahoney, pages 170-195). Fermat differed with Cartesian views concerning the law of refraction, published by Descartes in 1637 in La Dioptrique. Descartes attempted to justify the sine law through an assumption that light travels more rapidly in the denser of the two media involved in the refraction. (Mahoney, page 65). Twenty years later, Fermat noted th at this appeared to be in conflict with the view of the Aristotelians that nature always chooses the shortest path. "According to [Fermat's] principle, if a ray of light passes from a point A to another point B, being reflected and refracted in any manner during the passage, the path which it must take can be calculated...th e time taken to pass from A to B shall be an extreme" (Bell, page 63). Applying his method of maxima and minima, Fermat made the assumption that light travels less rapidly in the denser medium and showed that the law of refraction is concordant with his "principle of least time." "From this principle, Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence (in refraction) is a constant number times the sine of the angle of refraction in passing from one medium to anot her" (Bell, page 63). His argument concerning the speed of light was found later to be in agreement with the wave theory of the 17th-century Dutch scientist Huygens, and was verified experimentally in 1869 by Fizeau. In addition to the law of refraction, Fermat obtained the subtangent to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x - e. There is nothing to indicate that he was aware that the process was general, and it is likely that he never separated it his method from the context of the particular problems he was considering (Coolidge, page 458). The first definite statement of the method was due to Barrow, and was published in 1669. Fermat also obtained the areas of parabolas and hyperbolas of any order, and determined the centers of mass of a few simple laminae and of a paraboloid of revolution (Ball, pages 49, 77 , 108). Fermat was also strongly influenced by Viète, who revived interest in Greek analysis. The ancient Greeks divided their geometric arguments into two categories: analysis and synthesis. While analysis meant "assuming the pro position in question and deducing from it something already known," synthesis is what we now call "proof" (Mahoney, page 30). Fermat recognized the need for synthesis, but he would often give an analysis of a theorem. He would then state that it could easily be converted to a synthesis. Source:http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html