P which does not cut' the given Line. 'As soon as we turn this Line about P it would meet it to the right or to the left.'
Min. Certainly. And what then? Do you expect me to admit that, because case (ε) would lead to a consequence not obviously absurd, therefore it is the case which always happens, to the exclusion of cases (γ) and (α)?
Nie. (hesitatingly) Well, I think that is what we expect. But we first deduce the real existence of Parallels. 'Thus we are led to the conclusion that there exist Lines in a Plane which, though both be unlimited, do not meet. Such Lines are called parallel.'
Min. Oh most lame and impotent conclusion! After all these magnificent Catherine-wheels of revolving half-rays, to deduce Euc. I. 27! And even this wretched result you have no right to. Just consider what your argument has been. There are five conceivable cases, (α), (β), (γ), (δ), and (ε). If (α) or (β) were true, no Line could be drawn, through P, parallel to the given Line: if (γ), many such Lines could be drawn: if (δ), two such Lines: if (ε), one such Line. Now what have you proved? Positively nothing whatever but this—that case (α) would lead to an absurd result. You leave me perfectly free to range about among the other four cases, one of which, (β), denies the real existence of Parallels, which existence you tell me you have proved! And so, for the 'long course of logical reasoning' which you object to so much in Euclid, you substitute a short course of illogical reasoning! But you deduce another conclusion, do you not?
