A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

230 SUCCESSORS OF THE GREAT GEOMETERS
GEMINUS 229
Lastly, by the triangles FLS, QVS, in which the sides FS,
curves, the straight line, the circle and the cylindrical helix.
SQ are equal and two angles are respectively equal, Q V = FL.
The proof of uniformity ' ' (the property that any portion of
Therefore QV = LE.
the line or curve will coincide with any other portion of the
Since then EL, QV are equal and parallel, so are EQ, LV,
same length) was preceded by a proof that, if two straight
and (says Geminus) it follows that AB passes through Q. lines be drawn from any point to meet a uniform line or curve
EF, and let the interior angles BEF, EFD be together less
What follows is actually that both EQ and AB (of EB)
than two right angles.
are parallel to LV, and Geminus assumes that EQ, AB
Take any point H on FD and draw HK parallel to AB
are coincident (in other words, that through a given point
meeting EF in K. Then, if we bisect EF at L, LF at M, MF
only one parallel can be drawn to a given straight line, an
at X, and so on, we shall at last have a length, as FN, less
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
than FK. Draw FG, NOP parallel to AB. Produce FO to Q,
of Siopio-jioL (conditions and limits of possibility), the meaning
and let i^Q be the same multiple of FO that FE is of i^iY
of ( porism in the sense in which Euclid used the word in his
'
then shall AB, CD meet in Q.
Porisms as distinct from its other meaning of corollary ' ', the
Let $ be the middle point of FQ and R the middle point of
different sorts of problems and theorems, the two varieties of
FS. Draw through R, S, Q respectively the straight lines
converses (complete and partial), topical or locus-theorems,
RPG, STU, QV parallel to EF. Join MR, LS and produce
with the classification of loci. He discussed also philosophical
them to T, V Produce FG to U.
questions, e.g. the question whether a line is made up of
Then, in the triangles FON, ROP, two angles are equal
indivisible parts (e£ djxepcov), which came up in connexion
respectively, the vertically opposite angles FON, ROP and
with Eucl. I. 10 (the bisection of a straight line).
the alternate angles NFO, PRO
;
and FO = OR
; therefore
The book was evidently not less exhaustive as regards
RP = FK
higher geometry. Not only did Geminus mention the spiric
And FN, PG in the parallelogram FNPG are equal ; therefore
RG = 2FN= FM (whence MR is parallel to FG or AB)
he showed how they were obtained, and gave proofs, presum-
curves, conchoids and cissoids in his classification of curves ;
Similarly we prove that SU = 2 FM = FL, and LS is
ably of their principal properties. Similarly he gave the
parallel to FG or AB.
proof that there are three homoeomeric or uniform lines or