228 SUCCESSORS OF THE GREAT GEOMETERS 
straight lines which happens in the case of the lines referred 
to h 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.’ 1 
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 eqv/idistance. An-Nairîzî 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 ahsurdum). 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 con 
verse of I. 28, which is proved by reductio ad ahsurdum, 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, 
whicli is to the following effect. 
Let AB, CD be two straight lines met by the straight line 
1 Proclus on Eucl. I, pp. 192. 5-193. 3.