Transcript Slides for Lecture 11

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.
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 (15961650). 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.
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.
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.