320 11. POST-EUCLIDEAN GEOMETRY
by the Stoic philosopher Geminus, whose dates are a subject of disagreement among
experts, but who probably lived sometime between 50 BCE and 50 CE. Geminus
wrote an encyclopedic work on mathematics, which has been entirely lost, except
for certain passages quoted by Proclus, Eutocius, and others. Proclus said that
the parallel postulate should be completely written out of the list of postulates,
since it is really a theorem. The asymptotes of hyperbolas provided the model on
which he reasoned that converging is not the same thing as intersecting. But still
he thought that such behavior was impossible for straight lines. He claimed that
a line that intersected one of two parallel lines must intersect the other,^1 and he
reports a proof of Geminus that assumes in many places that certain lines drawn
will intersect, not realizing that by doing so he was already assuming the parallel
postulate.
Proclus also reports an attempt by Ptolemy to prove the postulate by arguing
that a pair of lines could not be parallel on one side of a transversal "rather than"
on the other side. (Proclus did not approve of this argument.) But of course the
assumption that parallelism is two-sided is one of the properties of Euclidean geom-
etry that does not extend to hyperbolic geometry. These early attempts to prove
the parallel postulate began the process of unearthing more and more plausible
alternatives to the postulate, but of course did not lead to a proof of it.
1.3. Heron. We have noted already the limitations of the Euclidean approach
to geometry, the chief one being that lengths are simply represented as lines, not
numbers. After Apollonius, however, the metric aspects of geometry began to
resurface in the work of later writers. One of these writers was Heron (ca. 10-ca.
75), who wrote on mechanics; he probably lived in Alexandria. Pappus discusses
his work at some length in Book 8 of his Synagoge. Heron's geometry is much
more concerned with measurement than was the geometry of Euclid. The change
of interest in the direction of measurement and numerical procedures signaled by
his Metrica is shown vividly by his repeated use (130 times, to be exact) of the word
area (embadon), a word never once used by Euclid, Archimedes, or Apollonius.^2
There is a difference in point of view between saying that two plane figures are equal
and saying that they have the same area. The first statement is geometrical and is
the stronger of the two. The second is purely numerical and does not necessarily
imply the first. Heron discusses ways of finding the areas of triangles from their
sides. After giving several examples of triangles that are either integer-sided right
triangles or can be decomposed into such triangles by an altitude, such as the
triangle with sides of length 13, 14, 15, which is divided into a 5-12-13 triangle and
a 9-12-15 triangle by the altitude to the side of length 14, he gives "a direct method
by which the area of a triangle can be found without first finding its altitude." He
(^1) This assertion is an assumption equivalent to the parallel postulate and obviously equivalent to
the form of the postulate commonly used nowadays, known as Playfair's axiom: Through a given
point not on a line, only one parallel can be drawn to the line.
(^2) Reporting (in his commentary on Ptolemy's Almagest) on Archimedes' measurement of the
circle, however, Theon of Alexandria did use this word to describe what Archimedes did; but
that usage was anachronistic. In his work on the sphere, for example, Archimedes referred to its
surface (epiphdneia), not its area. On the other hand, Dijksterhuis (1956, pp. 412-413) reports the
Arabic mathematician al-Biruni as having said that "Heron's formula" is really due to Archimedes.
Considering the contrast in style between the proof and the applications, it does appear plausible
that Heron learned the proof from Archimedes. Heath (1921, Vol. 2, p. 322) endorses this assertion
unequivocally.