230 SUCCESSORS OF THE GREAT GEOMETERS
What follows is actually that both EQ and AB (or EB) and r
are parallel to LV, and Geminus assumes that EQ, AB As A]
are coincident (in other words, that through a given point fact c
only one parallel can be drawn to a given straight line, an was h
assumption known as Playfair’s Axiom, though it is actually En<
stated in Proclus on Eucl. I. 31). inf on
The proof therefore, apparently ingenious as it is, breaks write:
down. Indeed the method is unsound from the beginning, In
since (as Saccheri pointed out), before even the definition of Posid
parallels by Geminus can be used, it has to be 'proved that is the
‘ the geometrical locus of points equidistant from a straight Alexs
line is a straight line ’, and this cannot be proved without a his cc
postulate. But the attempt is interesting as the first which parell
has come down to us, although there must have been many Ponti
others by geometers earlier than Geminus. helio(
Coming now to the things which follow from the principles betwt
(to, /¿era ray dp\dy), we gather from Proclus that Geminus heave
carefully discussed such generalities as the nature of elements, subst
the different views which had been held of the distinction their
between theorems and problems, the nature and significance prove
of Siopur¡Mol (conditions and limits of possibility), the meaning astroi
of ‘ porism ’ in the sense in which Euclid used the word in his provi
Porisms as distinct from its other meaning of ‘ corollary ’, the tions
different sorts of problems and theorems, the two varieties of when
converses (complete and partial), topical or Zoc-us-theorems, sun a
with the classification of loci. He discussed also philosophical an d
questions, e.g. the question whether a line is made up of conm
indivisible parts (e£ dpepcov), which came up in connexion size,
with Eucl. I. 10 (the bisection of a straight line). prov<
The book was evidently not less exhaustive as regards with
higher geometry. Not only did Geminus mention the spiric certa
curves, conchoids and cissoids in his classification of curves; For
he showed how they were obtained, and gave proofs, presum- appei
ably of their principal properties. Similarly he gave the we i
proof that there are three homoeomeric or uniform lines or circh
curves, the straight line, the circle and the cylindrical helix. circle
The proof of ‘ uniformity ’ (the property that any portion of ways
the line or curve will coincide with any other portion of the abou
same length) was preceded by a proof that, if tjro straight
lines be drawn from any point to meet a uniform line or curve
