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