MATHEMATICAL DEFINITIONS 459 
a Latin word is introduced to possess a technical signification, which 
has little or no meaning for most young persons. The word divergence 
should be explained etymologically in the definition as turning apart. 
Some writers introduce the simple geometrical concepts without 
attempting to give any definition of them, these concepts being thought 
of as fundamental, and not needing or capable of definition, just as 
certain truths are taken as axiomatic. Thus Hilbert, in his “ Founda- 
tions of Geometry,” so regards the concepts of point, straight line, 
plane and angle. Among his axioms he includes one which says a 
straight line is “ determined” when two of its points are given, and 
another which says a plane is determined when three of its points are 
given, the (virtual) definition for a plane thus corresponding to that 
for a straight line. Nearly all American authors say a plane is a sur- 
face such that if any two of its points be joined by a straight line, the 
line will lie wholly in the surface. To be consistent they should de- 
fine a straight line as Euclid does, viz., as a line that lies evenly between 
any two of its points. Thus a plane surface is tested by laying a 
straight edge on it, and a straight line is tested by sighting between its 
points to see if they are in the line of sight. 
There are four principal definitions for straight line, all different 
in character. Three of these have already been referred to. One of 
them, Legendre’s (that a straight line is the shortest distance between 
two points), the Century Dictionary criticizes. The fourth definition, 
that of the English Society for the Improvement of Geometrical Teach- 
ing, viz., A straight line is such a line that any part of it, however 
turned, will coincide with any other part, if its extremities lie in that 
other part, has merit as a practical description, but pedagogically it is 
not very satisfactory on account of the difficulty students have in under- 
standing it, and because the definition is not used after it is made. 
Theoretically the Hilbert plan of going at the matter is far superior 
to the association definition. 
Some terms in geometry are used ambiguously, notably circle, 
straight line and equals. By circle is meant either an area or a circum- 
ference, the latter being the usual meaning of circle in higher mathe- 
matics. By straight line is meant either an indefinite line, or a line 
segment or sect. By equal is meant either equal in area or volume, 
?. €., numerically equal, or equal in all respects, often called congruent. 
These ambiguities lead to much confusion in the minds of learners. 
It is probable that the words straight for indefinite straight line, ray 
for pi indefinite straight line, sect for line segment, and congruent 
for “equal in all respects,” will soon be generally adopted. 
A difference in the definitions of area and volume as given by 
various authors and by the dictionaries is of considerable interest. 
Wells says the area of a surface is its ratio to a standard unit of sur- 
