Date: Thu, 23 May 1996 20:17:53 -0400
To: Athena Discuss 
Subject: Re: Archimedes & Diop (reply to S F Thomas)

FISHERGM@jmu.edu wrote:
> 
> 
> >FISHERGM@jmu.edu wrote:
> >
(( cuts ))
> I'm unclear about what is meant by "rediscovery" of "an
> established theorem" here.  Thomas Heath says in his edition of
> the works of Archimedes:  Archimedes is said to have requested
> his friends and relatives to place upon his tomb a representation
> of a cylinder circumscribing a sphere within it, together with
> an inscription giving the ratio which the cylinder bears to
> the sphere [Plutarch, *Marcellus*]; from which we may infer
> that he himself regarded the discovery of this ratio [*On
> the Sphere and Cylinder*, I. 33, 34] as his greatest achievement.
> Is Diop referring to the discovery of this ratio?  I assume
> (perhaps wrongly) that Diop bases his attribution of this
> theorem to Egyptians two millenia before Archimedes
> on figures showing a sphere inscribed in a right cylinder,
> and not in a verbal rendition of some kind?  Could
> Diop perhaps have in mind the discovery that a sphere *can*
> be inscribed in a right cylinder with height equal to
> the diameter of the sphere, which would be a different
> matter altogether, and perhaps even pre-historic, made in
> a non-mathematical context, of the sort that (say) the Egyptians
> others introduced long ago?

No.  Diop cites Struve on the Papyrus of Moscow, and Peet
on the Rhind Papyrus, to establish that the Egyptians knew
the formulae in question.

> As indicated within the quotation from Heath, the propositions
> (or theorem) in question is, in a sense, contained in #33 and
> #34 of the first book of this work by Archimedes.  Proposition
> 33 reads, in Heath's translation:  "The surface of any sphere is
> equal to four times the greatest circle in it."  By hindsight,
> we can recognize that this sort of thing is replaced today
> by the formula for the surface area of a sphere, namely
> A = 4 (pi) r^2.  However, Archimedes didn't have this kind
> of formulation available, and so far as I know, neither did
> anyone else before him.  

Apparently not.  Diop says:

 "Since V. V. Struve published the Papyrus of Moscow...
  the international scientific community knows with
  certainty that two thousand years before Archimedes,
  the Egyptians had already established the rigorous
  formula for the area of the sphere: S = 4 pi R^2."
 

> More relevant here, though, is the
> question of whether or not Archimedes was the first to
> establish, by a chain of rigorous reasoning (which Archimedes
> based on five explicitly stated assumptions) this relationship
> which allows a comparison between surface area of a sphere
> and area of a circle, in terms of a ratio, which avoids
> explicit use of pi, although in applications an approximation
> to pi can be chosen to calculate the area of a circle, and
> then this Proposition can be used to calculate the corresponding
> area of the sphere in which this is a "greatest circle".
> 
> What, you may say, does this have to do with inscribing
> a sphere in a right cylinder?  Well, Proposition 34 reads:
> "Any sphere is equal to four times the cone [not cylinder!]
> which has its base equal to the greatest circle in the
> sphere and its height equal to the radius of the sphere."
> After a demonstration of this, Archimedes presents
> the following corollary, derived from Propositions 33
> and 34:  "From what has been proved it follows that every
> cylinder whose base is the greatest circle in a sphere and
> whose height is equal to the diameter of the sphere is
> 3/2 the [area of] the sphere, and its surface together with
> bases is 3/2 of the surface of a sphere."
> 
> A figure which conveniently illustrates these theorems, or
> conclusions, is a sphere inscribed in a right cylinder with
> height equal to the diameter of the sphere, and also a
> cone occupying the position described in the statement of
> Proposition 34.  I expect the diagram on Archimedes' tomb
> had the cone as well as the cylinder and sphere, although
> Plutarch doesn't say so.  I don't know if this has ancient
> documentary confirmation or not, but this is the diagram
> customarily given in relatively recent discussions of
> the work of Archimedes.

No cone in the figure reproduced by Diop, but the relationship
is the equality between the sphere and the cylinder (not
including bases).

(( cuts ))

> How do you think this compares with Diop's attribution
> of something involving a sphere inscribed in a suitable
> right cylinder to Egyptians 2 millenia before Archimedes?

If your main question is whether the Egyptian theorems
were formulaic, the short answer is yes.

> Gordon Fisher     fishergm@jmu.edu

Regards,

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