1 The Birth of Science - MSRI

10.14 The First Few Definitions in Euclid’s Elements 283
Since the sentence “a point is a ‘sign’ having no parts” is very similar
to Definition 1 in the Elements, and the characterization of a point as the
extremity of a line coincides with Definition 3, it is generally thought that
Sextus is citing Euclid in this passage. But a third property of points is
mentioned, namely their extensionlessness, and further, none of the three
properties is called a definition (), but rather a description. By contrast,
when Sextus reports the definition of a circle (which we can probably
assume to go back to Euclid, because of the testimonium from the Herculaneum
papyrus), he talks instead of mathematicians defining () a
circle. 209 Now, in the Elements many statements are called definitions, but
nothing is called a description. This indicates that in the case of a point
Sextus Empiricus is alluding not to Euclid but to someone else.
What Sextus says about points is much closer to a passage from a work
titled Definitions of terms in geometry, with which Heron’s name has long page 352
been associated. This text, after a dedication where the author states his
purpose of describing the technical vocabulary of geometry, consists of a
hundred or so paragraph-length sections, each illustrating and characterizing
one geometric concept. Little of this material qualifies as definitions,
belying the work’s traditional name. 210 The first section starts:
A point is that which has no part, or an extremity without extension,
or the extremity of a line. 211
Thus we see here the two telltale differences present in the Sextus Empiricus
passage but not in the Elements: the use of the verb “describe”
() and the characterization of the point as devoid of extension.
Getting back to the Sextus passage quoted on the preceding page, here SL: check if previous page
or this page
is how it continues:
A line is a length without width or the extremity of a surface; a surface
is the extremity of a body, or a width without depth.
All of this appears both in the Definitions and in the Elements, except for
“a surface is the extremity of a body”, which is not in the latter, showing
again that Sextus was not relying on the Elements. In discussing the notion
of a straight line, too, Sextus reports both the definition that appears in the
209 Sextus Empiricus, Adversus mathematicos, III, 107.
210 Heron’s Definitions of terms in geometry ( ) is the title
it bore in the Byzantine compilation where it was preserved. Though the work is still generally
attributed to Heron, Knorr gives good reasons to think that it belongs to Diophantus ([Knorr: AS]).
It first appeared in print in 1570 as an adjunct to Dasypodius’ edition of Book I of the Elements; our
references are to Heiberg’s edition in [Heron: OO], vol. IV.
211 ¨ (Heronis Definitiones, 14,
11–12, ed. Heiberg).
Revision: 1.11 Date: 2003/01/06 02:20:46