- ARCHIMEDES 303
FIGURE 19. Volumes of sphere, cone, and cylinder.
1906 and 1908 he journeyed to Constantinople and established the text, as far as
was possible. Attempts had been made to wash off the mathematical text during
the Middle Ages so that the parchment could be used to write a book of prayers.
The 177 pages of this manuscript contain parts of the works just discussed and a
work called Method. The existence of such a work had been known because of the
writings of commentators on Archimedes.
There are quotations from the Method in a work of the mathematician Heron
called the Metrica, which was discovered in 1903. The Method had been sent to
the astronomer Eratosthenes as a follow-up to a previous letter that had contained
the statements of two theorems without proofs and a challenge to discover the
proofs. Both of the theorems involve the volume and surface of solids of revolution.
In contrast to his other work on this subject, however, Archimedes here makes
free use of the principle now commonly known as Cavalieri's principle, which we
mentioned in connection with the Chinese computation of the volume of a sphere.
Archimedes' Method is a refinement of this principle, obtained by imagining the
sections of a region balanced about a fulcrum. The reasoning is that if each pair
of corresponding sections balance at distances a and b, then the bodies themselves
will balance at these distances, and therefore, by Archimedes' principle of the lever,
the area or volume of the two bodies must be have the ratio b : a.
The volume of a sphere is four times the volume of the cone with base equal to
a great circle of the sphere and height equal to its radius, and the cylinder with base
equal to a great circle of the sphere and height equal to the diameter is half again
as large as the sphere.
Archimedes' proof is based on Fig. 19. If this figure is revolved about the line
CAH, the circle with center at Ê generates a sphere, the triangle AEF generates
parabola...". If these were the original words, it appears that the nomenclature for conic sections
was changing in Archimedes' time.