A history of Greek mathematics - Wilbourhall.org

230 SUCCESSORS OF THE GREAT GEOMETERS
What follows is actually that both EQ and AB (of EB)
are parallel to LV, and Geminus assumes that EQ, AB
are coincident (in other words, that through a given point
only one parallel can be drawn to a given straight line, an
assumption known as PJayfair's Axiom, though it is actually
stated in Proclus on Eucl. I. 31).
The proof therefore, apparently ingenious as it is, breaks
down. Indeed the method is unsound from the beginning,
since (as Saccheri pointed out), before even the definition of
parallels by Geminus can be used, it has to be proved that
'
the geometrical locus of points equidistant from* a straight
line is a straight line ', and this cannot be proved without a
postulate. But the attempt is interesting as the first which
has come down to us, although there must have. been many
others by geometers earlier than Geminus.
Coming now to the things which follow from the principles
(rd fxtTa ras dp\ds), we gather from Proclus that Geminus
carefully discussed such generalities as the nature of elements,
the different views which had been held of the distinction
between theorems and problems, the nature and significance
of Siopio-jioL (conditions and limits of possibility), the meaning
of ( porism in the sense in which Euclid used the word in his
'
Porisms as distinct from its other meaning of corollary ' ', the
different sorts of problems and theorems, the two varieties of
converses (complete and partial), topical or locus-theorems,
with the classification of loci. He discussed also philosophical
questions, e.g. the question whether a line is made up of
indivisible parts (e£ djxepcov), which came up in connexion
with Eucl. I. 10 (the bisection of a straight line).
The book was evidently not less exhaustive as regards
higher geometry. Not only did Geminus mention the spiric
curves, conchoids and cissoids in his classification of curves ;
he showed how they were obtained, and gave proofs, presumably
of their principal properties. Similarly he gave the
proof that there are three homoeomeric or uniform lines or
curves, the straight line, the circle and the cylindrical helix.
The proof of ' uniformity ' (the property that any portion of
the line or curve will coincide with any other portion of the
same length) was preceded by a proof that, if two straight
lines be drawn from any point to meet a uniform line or curve