The History of Mathematical Proof in Ancient Traditions

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Archimedes’ writings: through Heiberg’s veil 197


results in pure geometry. Th is is oft en done by obtaining a common
centre of gravity to all pairs, suitably defi ned, of the n -dimensional
objects; assuming that the centre of gravity is then inherited by a pair
of n +1-dimensional objects constituted by the n -dimensional objects;
and fi nally applying the results that follow from the geometrical
proportions inherent in the Law of the Balance. Th is is illustrated by
Archimedes through a variety of results arranged by Heiberg as propo-
sitions 1–11. As Archimedes clarifi es in the introduction, his intent
is to provide also ‘classical’ or purely geometrical proofs for a couple
of new results, measuring the volumes of (a) the intersection of a cyl-
inder and a triangular prism, (b) the intersection of two orthogonally
inclined cylinders. Nothing survives of the proofs for (b), but we have
considerable evidence for no fewer than three proofs for (a). Th e fi rst,
arranged by Heiberg as the two propositions 12–13, is a proof based
on both a method of indivisibles as well as geometrical mechanics. Th e
second is proposition 14, on which more below; the third – called by
Heiberg ‘proposition 15’ – survives in fragmentary form, but it is clear
beyond reasonable doubt that this forms, indeed, a ‘classical’ proof
based on standard geometrical principles applied elsewhere. Th is is in
fact a proof based on the method of exhaustion.
Proposition 14 therefore occupies a middle ground between the
special procedures of the Method , and the standard geometrical princi-
ples applied elsewhere. Indeed, it uses only one part of the procedures
of the Method. It makes no use of geometrical mechanics, based instead
on indivisibles alone. Archimedes considers a certain proportion
obtained for any arbitrary slice in the solid fi gures – so that a certain
triangle A is to another triangle B as a certain line segment C is to
another line segment D. Th e set of all triangles A constitutes the trian-
gular prism; the set of all triangles B constitutes the intersection of cyl-
inder and triangular prism that Archimedes sets out to measure; the set
of all line segments C constitutes a certain rectangle; the set of all line
segments D constitutes a parabolic segment enclosed by that rectangle.
Heiberg’s readings reached this point, and then Heiberg hit what
was, for him, a lacuna in his readings. He picked up the thread of
the argument as follows. It is assumed that, since the proportion
holds between all n -dimensional fi gures, it will also hold between all
n +1-dimensional fi gures. We therefore have the proportion: a trian-
gular prism to the intersection of a cylinder and a triangular prism,
as rectangle to parabolic segment. Since the ratio of a rectangle to the
parabolic segment it contains is known, and since the triangular prism

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