Page:Mind (New Series) Volume 15.djvu/522

 508 HUGH MACCOLL : numberless succession of increasing real positive infinite values, Hj, H 2, H 3 , etc., that is to say, positive values all too large to be expressible solely in terms of any finite values (see 3). When the angle BCM becomes exactly a right angle, the point P, the infinite straight line MP, and the MP infinite ratio r ;, which represents tan PCM, all vanish, and the revolving line becomes parallel to the fixed line. Then, the revolution still continuing, another and different point Q, representing the intersection of the other branch (the branch CA produced) of the revolving line with the fixed line, springs into existence at an infinite distance to the left of M r and the straight line QM, after passing through a number- less series of really infinite but diminishing values, eventually becomes finite and continues to diminish till it finally van- ishes, or becomes zero, just as the variable point Q coincides with M. 7. The mistake made by non-Euclideans, when they appeal to geometrical examples like the preceding, is that they wrongly identify the variable point P which moves always- through contiguous positions to the right of M, and farther and farther away from M till it finally vanishes, with the point Q which immediately after springs into existence at an infinite distance to the left of M, and then moves through contiguous positions nearer and nearer to M till it finally coincides with it. When this coincidence takes place, the infinite branch containing both the fixed point C and the revolving point B will be pointing perpendicularly upwards, with B above C, and will contain neither the point P, which it lost when parallel to the fixed line and never recovered after,. nor the point Q, which had never belonged to it but to the other infinite branch containing the revolving point A. The fallacy of the non-Euclideans is analogous to that of the lawyers who assert that "the king never dies," because the instant the king P dies, his successor Q becomes, ipso facto, king in his place. To both we may return analogous answers : to the lawyers we reply that the dead king P is nevertheless not the living king Q ; and to the non-Euclideans we reply that the vanished point P is nevertheless not the new point Q. When the angle BCM is a right angle there is no point P, and there is no point Q, so that in this position of the revolving line (a position parallel to the fixed line) the distances MP and QM are pseudo-infinities which have only symbolic existence* Suppose the revolving line to remain for a moment stationary in this parallel position. If it then revolves through an infinitesimal angle in the screwing direction, we get the