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

similar, not equal;
3) was the first to discover that if two straight lines cut one another, the vertical
angles are equal (I, 15), whereas the scientific proof for this theorem was given
later by Euclid, as Proclus adds; and 4) knew the theorem about the equality of
the triangles that have one side and two angles equal (I, 26), which he must have
used to determine the distances of ships from the shore.
110
Did Thales really
need to go Egypt to learn there that the diameter divides the circle in half? On
the other hand, his theorems concerning angles and triangles are not in the least
related to Egyptian mathematics, which was never preoccupied with comparing
angles and establishing the similarity of triangles.
111
In fact, Egyptian geometry
lacked the notion of an angle as a measurable quantity – it was, in this respect,
‘linear’, unlike the ‘angle’ geometry of the Greeks, in which angles first be-
came objects of measurement.
112
The statement that the diameter divides the circle in half is not proved in Eu-
clid, but accepted as a definition. Thales, in his demonstration, must have re-
sorted to the method of superposition,
113
which in early Greek geometry was
employed much more often than in the time of Eudemus and Euclid. As von
Fritz observed,
All theorems ascribed to Thales are either directly related to the problems of sym-
metry and can be ‘demonstrated’ by the method of superposition, or such that the
first step of the demonstration is evidently based on considerations of symmetry
while the second, which brings the argument to conclusion, is simply an addition
or subtraction.
114
In the fourth century, mathematicians tried to avoid this visual and overly em-
pirical method. It is still to be found, however, in some theorems from Euclid’s
book I (4, 8). When saying that Thales treated certain things in geometry more
empirically (aısqhtikøteron), Eudemus could well have meant the method of
superposition. It does not follow, however, that Thales appealed in his demon-
strations to nothing but the visualizability of the geometrical drawing. In the
case of I, 5 we are able to verify this claim: there is a proof of this theorem in
Aristotle that is different from the one found in Euclid and might well go back
to Thales.
115
It is based on the equality of mixed angles – the angles of a semi-
circle and the angles of a circular segment, in particular – which, in turn, could
only be demonstrated by superposition or follow from the definition of such
angles. The proof proceeds in the following way:
110
Fr. 134–135, Procl.
In Eucl., 157.10f., 250.20f.
111
Vogel, K.
Vorgriechische Mathematik, Hannover 1958–1959, Pt. 1, 72; Pt. 2, 23 n. 2,
39 n. 4.
112
Gands, S. The origin of angle-geometry,
Isis 12 (1929) 452–482.
113
Heath.
Elements 1, 225; von Fritz. Grundprobleme, 401ff., 477f.
114
Von Fritz.
Grundprobleme, 568 n. 79.
115
APr 41b 13–22. See Heath. Elements 1, 252f.; von Fritz. Grundprobleme, 475f.;
Neuenschwander. VB, 358f.

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4. Early Greek geometry according to Eudemus
193
ABC is an isosceles triangle with the apex in the center of
a circle. Prove that the angles at the base are equal.
∠ 1
= ∠ 2 since both of them are angles of the semicircle.
∠
3 = ∠ 4 since two angles of any segment are equal to
each other. Subtracting equal angles 3 and 4 from equal
angles 1 and 2, we obtain that angles CAB and CBA are
equal to each other.
The proof in Aristotle derives most probably from the
Elements by Leon or
Theudius, whereas Eudemus’ source must have retained Thales’ archaic termi-
nology (Ômoioi instead of Ísoi).
The theorem on the equality of vertical angles (I, 15) could also have been
proved by the method of superposition. Euclid gives a different demonstration
of this theorem, based on I, 13. According to Neuenschwander’s reconstruc-
tion, I, 13–15 in their present form entered the
Elements either in the time of Eu-
clid or shortly before him.
116
That fully agrees with Proclus’ remark that the
scientific proof of I, 15 belongs to Euclid. Proclus (or his source) could have
come to this conclusion by comparing the proof of Thales cited by Eudemus
with Euclid’s proof, which was, naturally, more rigorous.
Unlike the first three, Thales’ fourth theorem is ascribed to him by Eudemus
on the force of an indirect argument: the method Thales used to determine the
distances of ships from the shore presupposes the use of this theorem. Without
going into the details of different reconstructions of this method,
117
we note that
Eudemus used his sources with discrimination and was perfectly able to distin-
guish between what was passed on directly by the tradition, on the one hand,
and his personal conjectures and hypotheses, on the other. The fact that here we
might be dealing with a mere reconstruction, possibly a fallacious one,
118
does
not cast doubt on other things we learn from Eudemus about Thales – no history
of mathematics can dispense with reconstructions. Numerous attempts to cast
doubt on the facts Eudemus reports, and, along with them, Thales’ place in the
history of Greek geometry have so far all proved futile.
119
The information re-
116
Neuenschwander. VB, 361f.
117
The most convincing one is suggested by Heath.
History 1, 131f., cf. van der
Waerden.
EW, 144f.
118
Gigon.
Ursprung, 55, supposed that the method of calculating the distance to a ship
at sea derives from the
Nautical Astronomy, ascribed to Thales (11 B 1). But this
work was written in verse and so hardly suitable for the exposition of geometrical
proofs. The practical value of this method for navigation is doubtful as well.
119
See von Fritz’s detailed answer (
Grundprobleme, 337ff.) on doubts expressed by
K. Reidemeister (
Das exakte Denken der Griechen, Hamburg 1949, 18ff.) and
Neugebauer (
ES, 142). The hypercritical position of D. R. Dicks (Thales, CQ 9
[1959] 294–309, esp. 301f.) is not convincing; cf. Zaicev.
Griechisches Wunder,
210f.

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Chapter 5: The history of geometry
194
ported by Eudemus is rich and accurate and includes details that could not have
been invented; the theorems he attributes to Thales show close mutual intercon-
nection.
120
Even if we cannot identify the source of Eudemus’ information on
Thales’ geometry, this only means that he had at his disposal certain texts that
are inaccessible to us.
121
About Mamercus, the next geometer after Thales, Eudemus must have only
known the little he learned from Hippias (
In Eucl., 64.11f.). Judging by the
general context in Hippias’
Collection, investigated by A. Patzer, this reference
was hardly isolated. Most likely, Mamercus was named along with the other fa-
mous geometers of his time, Thales and Pythagoras. This suggestion seems the
more probable since, in the early doxography, Hippias played an important role
as a transmitter of evidence about Thales. Though Patzer himself believed that
geometry also belongs to the circle of themes examined by Hippias,
122
his own
reconstruction of the
Collection leaves little room to suppose that it contained