Calculating area and volumes

1
Calculating area and volumes

• Early Greek Geometry by Thales (600 B.C.) and the
  Pythagorean school (6th century B.C)
• Hippocrates of Chios mid-5th century B.C. a first
  result on areas of curved shapes.
  (Squaring/quadrature of the lune) Tried the
  quadrature of the circle.
• 5th century B.C. Democritus discovered the volume
  of the cone is 1/3 of the encompassing cylinder
  using indivisibles.
• Archimedes (287 -212 B.C).
• Used the method of exhaustion invented by Euxodus
  (408-355 BC) to calculate area. This method is in
  book XII of Euclid.
• In On the sphere and cylinder he calculated the
  area of a sphere relative to a cylinder.
• In Quadrature of the parabola Archimedes finds
  the area of a segment of a parabola cut off by
  any chord.
• In The method (lost until 1899) he gives a
  physical motivation for his geometric results
  using infinitesimals, but does not consider them
  as rigorous.

2
Calculating area and volumes

• Johannes Kepler (1571-1630) worked on planetary
  motions and worked on integration in order to
  find the area of a segment of an ellipse. Also
  derived a formula to measure the volume of wine
  casks.
• Pierre de Fermat (1601-1665), Gilles Personne de
  Roberval (1602-1675) and Bonaventura Cavalieri
  (1598-1647). Used indivisibles to obtain new
  results for integration. Cavalieri wrote
  Geometria indivisibilis continuorum nova (1635)
• Roberval wrote Traité des indivisibles. He
  computed the definite integral of sin x, worked
  on the cycloid and computed the arc length of a
  spiral. He is important for his discoveries on
  plane curves and for his method for drawing the
  tangent to a curve
• Fermat also worked on tangents as did Rene
  Descartes (1596-1650). Fermat also gave criteria
  to find maxima and minima.
• Also Evangelista Torricelli (1608-1647), Blaise
  Pascal (1623-1662), René Descartes (1596-1650)
  and John Wallis (1616-1703) contributed to the
  beginning of analysis.

3
Calculating area and volumes

• Gottfried Leibniz (1646-1719) and Sir Isaac
  Newton (1643-1727)
• Independently gave a foundation of calculus with
  infinitesimals.
• We still use Leibniz notation today.
• Newton used physical intuition of moving
  particles, fluctuations and fluxes. De Methodis
  Serierum et Fluxionum was written in 1671 but
  Newton failed to get it published and it did not
  appear in print 1736.
• Leibniz used infinitely close variables dx, dy.
  In 1684 Leibniz published details of his
  differential calculus in Nova Methodus pro
  Maximis et Minimis, itemque Tangentibus... in
  Acta Eruditorum.
• There was a big controversy over priority, which
  Leibniz, who actually had published first lost.
• In 1711 Keill accused Leibniz of plagiarism in
  the Transactions of the Royal Society of London.
• The Royal Society set up a committee to pronounce
  on the priority dispute. It was totally biased,
  not asking Leibniz to give his version of the
  events. The report of the committee, finding in
  favour of Newton, was written by Newton himself
  1713 but not seen by Leibniz until the autumn of
  1714.

4
Calculating area and volumes

• Augustin-Louis Cauchy (1789-1857) gave a good
  definition of limit and integrals without
  infinitesimals. He was able to integrate
  continuous functions.
• Bernhard Riemann(1826-1866) corrected and
  expanded the Cauchys notion of integral. His
  theory of integration is usually taught in the
  calculus classes.
• Henri Léon Lebesgue (1875-1941) gave a
  generalization of Riemanns integral which is
  more powerful and is the theory of integration
  used today. His integration is based on a theory
  of measures.