466 16. THE CALCULUS
FIGURE 3. RobervaFs quadrature of the cycloid.volume was known. The first figure was then asserted to be equal or proportional
to the second. The principle was stated in 1635 by Bonaventura Cavalieri (1598-
1647), a Jesuit priest and a student of Galileo. At the time it was customary
for professors to prove their worthiness for a chair of mathematics by a learned
dissertation. Cavalieri proved certain figures equal by pairing off congruent sections
of them, in a manner similar to that of Archimedes' Method and the method by
which Zu Chongzhi and Zu Geng found the volume of a sphere. This method implied
that figures in a plane lying between two parallel lines and such that all sections
parallel to those lines have the same length must have equal area. This principle is
now called Cavalieri's principle. The idea of regarding a two-dimensional figure as
a sum of lines or a three-dimensional figure as a sum of plane figures was extended
by Cavalieri to consideration of the squares on the lines in a plane figure, then to
the cubes on the lines in a figure, and so on.
The cycloid. Cavalieri's principle was soon applied to find the area of the cycloid.
Roberval, who found the tangent to the cycloid, also found the area beneath it by
a clever use of the method of indivisibles. He considered along with half an arch of
the cycloid itself a curve he called the companion to the cycloid. This companion
curve is generated by a point that is always directly below or above the center of
the generating circle as it rolls along and at the same height as the point on the rim
that is generating the cycloid. As the circle makes half a revolution (see Fig. 3), the
cycloid and its companion first diverge from the ground level, then meet again at
the top. Symmetry considerations show that the area under the companion curve
is exactly one-half of the rectangle whose vertical sides are the initial and final
positions of the diameter of the generating circle through the point generating the
cycloid. But by definition of the two curves their generating points are always at
the same height, and the horizontal distance between them at any instant is half of
the corresponding horizontal section of the generating circle. Hence by Cavalieri's
principle the area between the two curves is exactly half the area of the circle.
Rectangular approximations and the method of exhaustion. Besides the method of
indivisibles (Cavalieri's principle), mathematicians of the time also applied the
method of polygonal approximation to find areas. In 1640 Fermat wrote a pa-
per on quadratures in which he found the areas under certain figures by a method
that he saw could easily be generalized. He considered a "general hyperbola," as
in Fig. 4, a curve referred to asymptotes AR and AC and defined by the property