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

Elements.
187
As for the method of diorism,
though the Academic work relates its discovery to the mid-fourth century, we
have good reason to suppose that it had been used earlier.
188
It is possible that
Leon formulated the method clearly or improved it, rather than invented it.
Much more detailed is the information Eudemus provides on Eudoxus. Ac-
cording to the
Catalogue, Eudoxus 1) “was the first to increase the number of
the so-called general theorems”, 2) added three new proportionals to the three
already known, and 3) “multiplied the theorems concerning the section …,
183
See Heath.
History 1, 246ff.; Becker. Denken, 76f.; van der Waerden. EW, 150f.;
Böker,
op. cit., 1203f.; Knorr. AT, 50f.
184
Neuenschwander, E. Zur Überlieferung der Archytas-Lösung des delischen Prob-
lems,
Centaurus 18 (1974) 1–5; Knorr. AT, 50f.; idem. TS, 100ff. It does not mean
that the
History of Geometry was inaccessible to Eutocius: he refers to it in another
work (
In Archim. De dimens. circ., 228.20 = Eud. fr. 139). Like Simplicius, he could
have quoted Eudemus at second hand as well, if it was convenient for him. Knorr.
TS,
100ff., suggested Geminus or Sporus as possible intermediaries between them, but
there is no evidence that Geminus used Eudemus (see above, 184 f.).
185
See above, 176f.
186
See above, 173ff.
187
Lasserre.
Birth, 18.
188
Heath.
 History 1, 319f.; Lasserre. Léodamas, 516f. This method can be clearly
traced, in particular, in Plato’s
Meno, written ca. 385. See Knorr. AT, 73f.; Menn, S.
Plato and the method of analysis,
Phronesis 47 (2002) 193–223.

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4. Early Greek geometry according to Eudemus
207
employing the method of analysis for their solution”.
189
To this one should
add: 4) the information from the scholia that Eudoxus was the author of book
V of the
Elements, containing the general theory of proportions, 5) Eratos-
thenes’ and Eutocius’ evidence on Eudoxus’ solution to the problem of doubl-
ing the cube, and, finally, 6) the words of Archimedes that Eudoxus proved the
theorem that each cone equals one-third of the cylinder and every pyramid
equals one-third of the prism with the same base and height.
190
Nos. 1 and 4
refer, it seems, to one and the same discovery, namely, to the general theory of
proportions. This, in any case, is how “the so-called general theorems” (tà
kaloúmena kaqólou qewr2mata) are usually understood.
191
Similar ex-
pressions occurring in Aristotle (1 kaqólou maqhmatik2 and tà kaqólou
ën ta$~ maq2masin), are normally interpreted as a reference to Eudoxus’ the-
ory of proportions applied equally to all magnitudes (numbers, lines, figures,
etc.).
192
By contrast, the “theorems concerning the section” (tà perì t3n
tom3n), whose number Eudoxus multiplied, applying to them the method of
analysis, remain obscure. Lasserre, following Bretschneider, was inclined to
relate them to the section of line in extreme and mean ratio (golden section).
193
More probable, however, is the interpretation preferred by Tannery and Heath
that ‘section’ means the section of solids,
194
which might correspond to no. 6
on our list.
Also unclear remains the question of Eudoxus’ solution to the problem of
doubling the cube. Eratosthenes’ epigram and his letter to King Ptolemy III
mention ‘curved lines’ (kampúlai grammaí); Eutocius, however, who quotes
both these texts, leaves Eudoxus’ solution out. As he explains, the ‘curved
lines’ Eudoxus mentions in the ‘introduction’ are not found in the text of the
proof and a discrete proportion is used here as if it were continuous (