A history of Greek mathematics - Wilbourhall.org

228 SUCCESSORS OF THE GREAT GEOMETERS
straight lines which happens in the case of the lines referred
to? Indeed, until the statement in the postulate is clinched
by proof, the facts shown in the case of the other lines may
direct our imagination the opposite way. And, though the
controversial arguments against the meeting of the straight
lines should contain much that is surprising, is there not all
the more reason why we should expel from our body of
doctrine this merely plausible and unreasoned (hypothesis) ?
It is clear from this that we must seek a proof of the present
theorem, and that it is alien to the special character of
postulates.' l
Much of this might have been written by a modern
geometer. Geminus's attempted remedy was to substitute
a definition of parallels like that of Posidonius, based on the
notion of eqvAdistance. An-Nairizi gives the definition as
follows :
'
Parallel straight lines are straight lines situated in
the same plane and such that the distance between them, if
they are produced without limit in both directions at the same
time, is everywhere the same ', to which Geminus adds the
statement that the said distance is
the shortest straight line
that can be drawn between them. Starting from this,
Geminus proved to his own satisfaction the propositions of
Euclid regarding parallels and finally the parallel-postulate.
He first gave the propositions (1) that the 'distance ' between
the two lines as defined is perpendicular to both, and (2) that,
if a straight line is perpendicular to each of two straight lines
and meets both, the two straight lines are parallel, and the
'
distance ' is the intercept on the perpendicular (proved by
reductio ad absurdum). Next come (3) Euclid's propositions
I. 27, 28 that, if two lines are parallel, the alternate angles
made by any transversal are equal, &c. (easily proved by
drawing the two equal ' distances ' through the points of
intersection with the transversal), and (4) Eucl. I. 29, the converse
of I. 28, which is proved lyy reductio ad absurdum, by
means of (2) and (3). Geminus still needs Eucl. I. 30, 31
(about parallels) and I. 33, 34 (the first two propositions
relating to parallelograms) for his final proof of the postulate,
which is to the following effect.
Let A B, CD be two straight lines met by the straight line
1
Proclus on Eucl. I, pp. 192. 5-193. 3.