458 THH POPULAR SCIENCE MONTHLY 
lines as lines that are everywhere the same distance apart (Webster) ; 
others that they are lines in the same plane that will not meet however 
far produced; and others still that they are lines having the same 
direction. In one sense these definitions mean the same thing, but 
in the development of geometry they give rise to very different series 
of propositions. 
Perhaps the most far-reaching of differences in definitions are 
found in those for parallel lines. Wentworth, following Chauvenet, 
says parallel lines are those having the same direction throughout their 
whole extent. This definition is very objectionable for two reasons; 
first, because the meaning of the word direction is ambiguous, the word 
being used to signify either one way or the exactly opposite, or in the 
sense of the angle a line makes with a standard line; second, because 
the idea of direction in the sense intended is difficult to explain. The 
Century Dictionary gives this definition: “The direction of point A 
from point B is or is not the same as that of another point C’ from 
point D, according as a straight line drawn from B to A and continued 
to infinity would or would not cut the celestial sphere at the same point 
as the straight line from D to C continued to infinity.” Chauvenet 
and Wentworth thought they had found a way to simplify the defini- 
tion of parallelism. It is clear from the preceding that what they did 
was to slur over a very complex concept. As a matter of fact the use 
of the word direction in trying to define parallels was not new. Thus 
Dr. Johnson defines parallels “as lines extended in the same direction, 
and preserving always the same distance.” The definition used by 
Euclid, viz., lines in the same plane that will not meet, however far 
produced, is practically the best, and has the merit of preparing the 
student for the non-Euclidean geometry. 
Not only have parallel lines been defined differently by different 
authors, but other important terms have met the same fate. The con- 
cept angle has been presented in three or four ways: (1) As the figure 
formed by two lines meeting, which is essentially a description, not a 
definition. (2) As the difference in direction of two lines. (3) As the 
inclination of one line to another. (4) As the amount of divergence of 
two lines that meet. The objection to the first is that it does not call 
attention to an angle as a magnitude, but rather as a shape. A recent 
author gives this definition and then asks on the next line whether 
increasing the lengths of the lines would increase the size of the angle? 
Of course it would if the pupil judged by the definition given. The 
use of the term direction to define angle is as objectionable as for 
parallels. The third definition, Euclid’s, is better than the others, but 
not as clear as it might be on account of the meaning of the word in- 
clination. Thus we are led to the fourth definition, which is objection- 
able chiefly on the ground that, following the usual custom in English, 
