1 The Birth of Science - MSRI

282 10. Lost Science
school child, contains the first ten definitions almost exactly as they have
come down to us, but it makes no reference to Euclid. 203 Thus it shows that
at the time of its writing — the third century A.D. — the definitions of fundamental
geometrical entities were taught in the form that we know from
the Elements, but not that they appeared in the Elements or that they were
associated with Euclid. The second papyrus, chronologically more interesting,
comes from Herculaneum 204 and does not contain “definitions” of
fundamental entities, but only one for a circle. This latter is quite correct,
but does not coincide with the one present in the known versions of the
Elements: in particular, the term circumference () is used in the
papyrus without being explicitly defined, whereas in the extant Elements
a definition of this term is included in the course of the definition of a circle.
205 This shows that in at least one case a definition of a term originally
left undefined was interpolated in the Elements.
Sextus Empiricus wrote before Euclid’s work took the form that has
come down to us. He discusses definitions of geometric entitites several
times. The importance of his testimony is increased by the fact that he
seems to report the definition of a circle not in the form present in today’s
Elements, but in that found in the Herculaneum papyrus; 206 this leads us page 351
to think that he had an edition of the Elements that was if nothing else less
corrupt than ours. 207
When Sextus reports Platonist-essentialist definitions of fundamental
geometrical objects similar to the first few in Book I of the Elements, they
usually differ from the latter in telling ways. Let’s examine, for example,
his passage about the definition of a point. He writes that mathematicians,
in describing () geometrical entities, say that
a point [stigme] is a “sign” [semeion] having no parts and no extension,
or the extremity of a line[.] 208
203 P. Michigan III, 143.
204 P. Herculaneum 1061.
205 The text transmitted by all manuscripts of the Elements is: “A circle is a plane figure contained
by one line, called the circumference, such that all straight lines emanating from one point inside
the figure and falling upon it — upon the circumference of the circle — are equal to one another”
(
¨ ¡ ¡
¡ ). Note the gauche amplification “upon the circumference”, especially
pointless in Greek since the pronoun cannot be misunderstood in view of its position and
gender. (Heath omits from his translation the two references to “circumference” because they were
declared spurious by Heiberg.)
206 Sextus Empiricus, Adversus mathematicos, III, 107.
207 Heiberg, for a variety of reasons, concluded that Sextus still had access to the original edition
of Euclid: see the Prolegomena to the critical edition of the Elements in [Euclid: OO], vol. V.
208 ¨ (Sextus Empiricus, Adversus
mathematicos, III, 20). For the terms stigme and semeion see page 157.
Revision: 1.11 Date: 2003/01/06 02:20:46