1. Calculating area and volumes Early Greek Geometry by Thales (600 B.C.) and the Pythagorean school (6 th century B.C) Hippocrates of Chios mid-5 th 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 Cauchy’s notion of integral. His theory of integration is usually taught in the calculus classes. Henri Léon Lebesgue (1875-1941) gave a generalization of Riemann’s integral which is more powerful and is the theory of integration used today. His integration is based on a theory of measures.