Leonid Zhmud The Origin of the History of Science in Classical Antiquity

Elements, and several theorems of book VI
(27–29), can be reformulated into algebraic identities and quadratic equations
and, for this reason, were often termed ‘geometrical algebra’.
150
For example,
the application of areas with a deficiency means the construction on a given
line
a of the rectangle ax, so that by subtracting from it the square x
2
, the given
square
b
2
is obtained (
ax–x
2
= b
2
). This does not mean, however, that the appli-
cation of areas really stemmed from the solution of quadratic equations, let
alone that they are of Babylonian origin.
151
The Pythagorean character of this
theory is obvious: the area of a rectilinear figure (II, 14) is determined by find-
ing the geometric mean
x between lines a and b; i.e., a square with the side x
equals a rectangle
ab (x
2
=
ab). Since Hippocrates of Chios was familiar with
this theory and developed it, the application of areas can be dated in the first
half of the fifth century.
It seems to follow from Eudemus’ words that book II, related entirely to the
application of areas, along with book IV, was created by the Pythagoreans.
148
Neuenschwander. VB, 374f.; van der Waerden.
Pythagoreer, 341ff.
149
In general form this problem is formulated as the application to a given straight line a
rectangle equal to a given rectilinear figure and exceeding or falling short by a square
(Heath.
History 1, 151).
150
E.g. proposition II, 2 can be reformulated as the identity (
a + b) c = ac + bc, and II, 14
as the equation
x
2
= ab. See Heath. Elements 1, 343f.; idem. History 1, 150ff.; Bek-
ker.
Denken, 60f.; van der Waerden. Pythagoreer, 341ff.
151
The algebraic and Babylonian origin of book II, which used to be almost unani-
mously accepted, was subjected to shattering criticism in recent decades. For refer-
ences, see Zhmud.
Wissenschaft, 149 n. 37.

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Chapter 5: The history of geometry
200
Why, then, did Eudemus refer in this case to the application of areas, rather than
to book II directly? First, this theory is expounded not only in book II, but also
in books I and VI. Proclus, in particular, quotes Eudemus in his commentary to
I, 44, whereas the terms ûperbol2 and Élleiyi~, which (unlike parabol2)
are missing in book II, occur only in VI, 27–29. Second, not long before Euclid,
book II underwent some mainly stylistic changes, and a new notion, the paral-
lelogram, was introduced into it.
152
That was why Eudemus could have pre-
ferred to speak of the theory whose antiquity was warranted by its authentic
form, which was different from the
Elements of his own time, as well as by its
provenance from the Pythagorean compendium.
Eudemus tells much less about Ionian geometers before Hippocrates than
about the Pythagoreans. This is easily accounted for by the fact that, till the
mid-fifth century, the Pythagorean school was foremost in geometry. But one
cannot ignore the selective approach of Eudemus himself, let alone those who
subsequently used his
History of Geometry. Anaxagoras is mentioned in the
Catalogue as a man who touched upon many geometrical problems. What his
discoveries in geometry actually were, remains unclear; nor can we be sure that
Eudemus knew anything about them.
153
Eudemus regarded Oenopides, the founder of the Chian school of math-
ematicians, as having been “a little younger” than Anaxagoras. The traditional
date of Anaxagoras’ birth is ca. 500; ölígœ neøtero~ refers here, as in the
other places in the