by the claim, King Hiero asked him to prove it.
Archimedes had, at this time, discovered the principles
of the levers and pulleys, and set about constructing a
mechanical device that allowed him, single-handedly, to
launch a ship from the harbor that was too large and
heavy for a large group of men to dislodge.
Dubbed a master of invention, Archimedes also
devised a water-pumping device, now known as the
Archimedes screw and still used in many parts of the
world today, along with many innovative machines of
war that were used in the defense of Sicily during the not-
infrequent Roman invasions. (These devices included
parabolic mirrors to focus the rays of the sun to burn
advancing ships from shore, catapult devices, and spring-
loaded cannons.) But despite the fame he received for his
mechanical inventions, Archimedes believed that pure
mathematics was the only worthy pursuit. His accom-
plishments in mathematics were considerable.
By bounding a circle between two regular polygons
and calculating the ratio the perimeter to diameter of
each, Archimedes found one of the earliest estimates for
the value of π, bounding it between the values 3 10/71
and 3 1/7. (This latter estimate, usually written as 22/7,
is still widely used today.) Archimedes realized that by
using polygons with increasingly higher numbers of
sides yielded better and better approximations, and that
by “exhausting” all the finite possibilities, the true value
of πwould be obtained. Archimedes also used this
method of exhaustion to demonstrate that the length of
any segment of a parabola is 4/3 times the area of the
triangle with the same base and same height.
By comparing the cross-sectional areas of parallel
slices of a sphere with the slices of a cylinder that
encloses the sphere, Archimedes demonstrated that the
volume of a sphere is 2/3 that of the cylinder. The vol-
ume of the sphere then follows: V= (2/3)(2r×πr2) =
(4/3)πr3. (Here ris the radius of the sphere.) Archimedes
regarded this his greatest mathematical achievement,
and in his honor, the figures of a cylinder and an
inscribed sphere were drawn on his tombstone.
Archimedes also computed the surface area of a
sphere as four times the area of a circle of the same
radius of the sphere. He did this again via a method of
exhaustion, by imagining the sphere as well approxi-
mated by a covering of flat tiny triangles. By drawing
lines connecting each vertex of a triangle to the center
of the sphere, the volume of the figure is thus divided
into a collection of triangular pyramids. Each pyramid
has volume one-third its base times it height (essentially
the radius of the sphere), and the sum of all the base
areas represents the surface area of the sphere. From
the formula for the volume of the sphere, the formula
for its surface area follows.
One cannot overstate the influence Archimedes has
had on the development of mathematics, mechanics,
and science. His computations of the surface areas and
volumes of curved figures provided insights for the
development of 17th-century calculus. His understand-
ing of Euclidean geometry allowed him to formulate
several axioms that further refined the logical under-
pinnings of the subject, and his work on fluids and
mechanics founded the field of hydrostatics. Scholars
and noblemen of his time recognized both the theoreti-
cal and practical importance of his work. Sadly,
Archimedes died unnecessarily in the year 212
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During the conquest of Syracuse by the Romans, it is
20 Archimedes of Syracuse
Archimedes of Syracuse, regarded as one of the greatest
scientists of all time, pioneered work in planar and solid geo-
metry, mechanics, and hydrostatics. (Photo courtesy of the
Science Museum, London/Topham-HIP/The Image Works)