[FOM] Understanding Euclid

Alasdair Urquhart urquhart at cs.toronto.edu
Fri Dec 5 13:53:42 EST 2008

The fact that Euclid states the Parallel Postulate as an implication,
not an equivalence is very likely related to the fact that
the converse of the Fifth Postulate is proved as a theorem
(Euclid I.27), deduced as a consequence of I.16 without the
use of the Fifth Postulate.  It is well known that Euclid
tries to go as long as possible without using the Parallel
Postulate; it is in line with this procedure that he would
state the postulate in as weak a form as possible.

Proclus called attention to this:

 	This [Postulate 5] ought even to be struck out of the
 	Postulates altogether; for it is a theorem involving many
 	difficulties  ... and the converse of it is actually
 	proved by Euclid himself as a theorem.  [Heath's edition
 	of the Elements, Volume 1, p. 202]

Alasdair Urquhart

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