16 
Journal of the Mitchell Society. 
[May 
“All these demonstrations, some ingenious, are without exception 
false, defective in their foundations and without the necessary rigor 
of deduction.” 
Lobatchevsky classifies all the co-planar lines through a 
given point A with reference to another co-planar line BC 
not passing through A, under two heads — cutting and non- 
cutting (Fig. 3). The transition from the non-cutting lines, 
FIG 3. 
such as EA and GA, to the cutting lines, such as FA, is 
marked by one line HA — the boundary line between the two 
classes; this he entitles the parallel line . From the assumpt- 
ions, there arises the necessity of making a distinction of 
sides in parallelism, and hence there must be two parallels, 
so-called, one on each side. One logical consequence of this 
is that “if in any rectilineal triangle the sum of the three 
angles is equal to two right angles, this is also the case for 
every triangle” — one instance is the criterion for all. 
