1 The Birth of Science - MSRI

10.14 The First Few Definitions in Euclid’s Elements 285
A straight [] line is a line that equally with respect to [all] points
on itself lies straight [] and maximally taut between its extremities.
220
The origin of this characterization can be traced fairly easily. Already the
Stoics had defined a straight line as a line taut between its endpoints. 221
Archimedes took it as a postulate that among lines sharing the same extremities
the straight line was the shortest. 222 In the imperial period a
“straight line” (which in Euclid, unless otherwise specified, means a segment)
came to mean one that extends endlessly in both directions. 223 A
person wishing to use the Stoic definition or the Archimedean postulate
to characterize the “new” infinite straight line of course could not base
a statement on one pair of endpoints only, but must instead say that the
straight line has this tautness property “equally with respect to all points
on itself”. The statement found in the Definitions is thus clear in this post-
Euclidean context.
In view of the difficulty in memorizing long chunks of text and of the page 355
type of teaching prevailing in the imperial age, one can imagine the poor
students being encouraged to work from crib sheets abridged from the
Definitions, where each entry was truncated as soon as the syntax allowed
it. 224 Such a crib sheet, from being copied together with the Elements, might
easily have merged eventually with the Euclidean text. The fact that by
this procedure we obtain exactly the definitions found in the Elements —
even “A straight line is a line that lies equally with respect to the points on
itself”, which no mathematician has ever been able to make proper sense
of — demonstrates the plausibility of our conjecture.
We should every so often pause to consider the legions of students
who throughout the centuries were forced to memorize a half-sentence
by teachers who did not know the second half that could have made it
meaningful.
Proclus seems to have preserved, through channels not easily identifiable,
a memory of the relationship between Archimedes’ shortest-line
postulate and the definition of a line given in the Elements. He struggles to
show, using strange arguments whose origin remains mysterious, that the
Elements statement is just a reformulation of the shortest-line postulate. 225
220 ¨ ¡
(Heronis Definitiones, 16, 22–24, ed. Heiberg).
221 Simplicius, In Aristotelis Categorias commentarium ([CAG], vol. VIII), 264, 33–36 = [SVF], II, 456.
222 Archimedes, De sphaera et cylindro, I, 10, 23–25 (ed. Mugler).
223 The term is used for one class of infinite lines in the classification by Geminus reported by
Proclus ([Proclus/Friedlein], 111, 1–12).
224 Papyrus Michigan iii, 143, already mentioned, may have been a late examplar.
225 [Proclus/Friedlein], 109–110.
Revision: 1.11 Date: 2003/01/06 02:20:46