1 The Birth of Science - MSRI

2.7 An Application of the “Method of Exhaustion” 45
But this inequality is false, because it contradicts (∗). Thus we have ex-
cluded the case S > 4
3 A0.
Suppose instead that S < 4
3 A0. Using the postulate again, we see that if n
is a sufficiently large integer the area 1
3 An must be less than the difference
4
3 A0 − S. Using (∗) we deduce that
that is,
4
3 A0 < A0 + A1 + A2 + · · · + An + 4
3 A0 − S,
S < A0 + A1 + A2 + · · · + An.
This inequality, too, is clearly false, since the right-hand side represents
the area of a portion of the segment of parabola of area S. This concludes
the proof of the theorem.
We note (and it will be clearer to those who read the Appendix) that page 76
the proof depends crucially on the study of triangles, which don’t appear
at all in the formulation of the problem. They are used merely as a tool.
This example makes it clear why Hellenistic mathematicians laid out with
great care such simple theories as that of triangles, presented in the Elements:
they were useful tools for tackling even problems whose original
statement had no connection whatsoever with the auxiliary theory. Triangles,
in particular, were studied so that figures could be “triangulated”.
We will encounter a similar use of circles as a tool in the study of planetary
orbits.
Every real number different from zero has a multiple that is greater
than an arbitrary fixed real number. In modern axiomatizations of the real
numbers, this property is assumed true, and is called the “Archimedean
postulate”. The postulate that Archimedes actually stated is different: the
“magnitudes” that he (and Hellenistic mathematicians in general) considered
in fact form a “non-Archimedean” set (in the modern terminology),
in that the magnitude of a segment, though nonzero, has no multiple that
exceeds the magnitude of a square. According to Hellenistic mathematicians,
two magnitudes “have a ratio”, and are called homogeneous, if each
has a multiple that exceeds the other. 60 Archimedes explicitly postulates
that the difference between any two inequivalent surfaces is homogeneous
with (has a ratio with) any other surface. 61
Archimedes’ surviving writings may give the impression that the level
of scientific works transmitted through late antiquity and the Middle Ages
was not as low as we have maintained (see page 4). In fact, the selection
60 Euclid, Elements, V, Def. 4.
61 A more general version of the postulate, applying not only to surfaces but also to lines and
solids, appears in Archimedes, De sphaera et cylindro, 11, 16–20 (ed. Mugler, vol. I).
Revision: 1.12 Date: 2003/01/10 06:11:21