Re: [HM] Archimedes: What Did He Do Besides Cry Eureka?
Barnabas Hughes (barnabas.hughes@csun.edu)
Thu, 19 Aug 1999 13:22:25 +0000
What you see on the cover of Sherman Stein's book, "Archimedes - What
Did He Do Besides Cry Eureka?" is a digitally enhanced photo of a folio
page from the tenth century Archimedean codex. The text is from "On
floating bodies". Sometime later a monk separated the pages of the
original codex, cut each in two, and made out of each part two quarto
pages half the size of the folio page. If Stein's book is turned
sideways, you can see two such pages with the thread fixing the sheet to
its quire. With the consent of the new owner of the codex, specialists
at The Walters Gallery will separate all the pages, repair and
strengthen each, and extract the Archimedean text for the use of
scholars and the benefit of the history of mathematics.
Meanwhile Sherman Stein has provided us with a provocative commentary
on specific Archimedean works: the law of the lever, the mechanical
method, summation of two series, the area within a parabolic section,
floating bodies, the spiral, the sphere, and pi. Four appendices are
attached the most notable of which develops properties of the parabola
by means of affine geometry. Stein thought it better to justify
Archimedes' geometry by mapping because such proofs "are short[er]" than
what Archimedes or Euclid used. He wrote the book for the mathematical
community; that is, "anyone who recognizes the equation of a
parabola." Stein's avowed goal in preparing his commentary was to
discover and then elucidate "... [the] mathematics that lies at the
core of [each] argument... It is the core that I want to expose in
such a way that the busiest reader could easily follow and appreciate
the reasoning" (p. x). While for the most part Stein approached his
goal as closely as most readers would wish, I think that his discussion
of the mechanical method needs some adjustment for smooth reading.
A preliminary remark will assist in the adjustment. Archimedes
based his method on the law of the lever; specifically, to bring at
least two weights at opposite ends of a balance beam into equilibrium
about a fulcrum. This requires the lever to be horizontal. I stress
this obvious point about the lever because none of the figures in Heath's
translation nor most of those in Stein's commentary show a horizontal
beam. Each beam is either vertical or at an angle. While I fully
expect the elucidation of the Archimedean codex to display vertical
beams in its figures (both Tischendorf and Heiberg copied them thusly),
I seriously doubt that Archimedes drew them so in his letter to
Eratosthenes in which he explained his method.
THE MECHANICAL METHOD displays how Archimedes found answers to many
of his questions. He used the law of the lever and centers of gravity to
balance corresponding lines of plane figures or corresponding sections
of solids. With the parts in equilibrium he concluded that all such
lines or sections were in equilibrium; hence, the planes or solids were
in equilibrium. Thereby, the solution to the problem was evident. Once
the desired conclusion was found, Archimedes proved it geometrically
according to Euclidean norms.
Having taught Archimedes' method for the past 30 years to prospective
and actual high school teachers of mathematics, I certainly agree with
Stein that the place to start is with Proposition IV, to find the volume
of a paraboloid. This is simpler than finding the area of a segment of
a parabola, as Archimedes begins his exposition. Proposition V, to find
its [the paraboloid's] center of gravity, as Stein continues, makes for
good reinforcement. Further, he does well to display the balance been
in the proper horizontal position. Stein creates the paraboloid, as
would Archimedes, by rotating a parabola ab out its axis. Seeking a
modern analogy he refers to "the equation of the parabola ... y = x^2
relative to suitable axes" (p. 34). Since the beam becomes the suitable
axis, the equation x = y^2 would have been more appropriate. This
change does not affect the geometric argument that follows Archimedes'
way of thinking.
The figures Stein offers, however, are not so helpful. Figure 2
(p.35) could have been improved by showing the sections of the cylinder
and parabola opposite each other and in equilibrium about the fulcrum.
Figure 3 (p. 36) is quite difficult to interpret: two illustrations are
separated by the words "top view" (the top of which figure??).
Further, in no way would Archimedes have stood the section of a
paraboloid atop the end of the balance beam--it should be hung by its
vertex from the end of the beam. Thereafter in the finding of the
volume of the sphere, the center of gravity of a hemisphere, and the
area of a parabolic segment, Stein might have made the balance beams
horizontal.
Stein's book is a useful companion to anyone who wishes to study
the works of Archimedes in depth. Most of his explanations and
illustrations are quite helpful. The Mathematical Association of
America, despite its rush at getting the book into print, has performed
a valuable service to those who desire more than a nodding acquaintance
with "What [Archimedes Did] Besides Cry Eureka".
Barney Hughes