462 THE POPULAR SCIENCE MONTHLY 
“volume is the space occupied as measured by cubic units, i. e., cu. ft., 
cu. in., ete. | 
The new dictionary, unlike the old, defines a right angle, not by a 
synonym, but by saying it is an angle included between two radi 
subtended by a quarter circle. Under angle, “right angle” is not 
given. Thus the new dictionary can not be criticized as the old was. 
But it does seem a pity that the dictionaries can not give the simple, 
plain essential definition found in almost all geometry text-books. 
Under “straight line,” the first definition, a line having an in- 
variable direction, is credited to Newcomb, thus retaining the old weak 
idea of direction, used in defining parallel lines. Next, Euclid’s defi- 
nition is given, and then the Hilbert axiom as a definition. Legendre’s 
definition is criticized. Of course the reader of the dictionary will not 
learn the meaning of a straight line from the Newcomb definition, but 
will learn the meaning “of the same direction” from his knowledge 
of a straight line. It would have been far wiser to have told what 
Euclid meant by his definition. 
In defining angle the same discredited definition of the difference 
in direction of two lines appears again. Curiously enough the gener- 
alized, or trigonometrical, definition of an angle is found not under 
the word angle at all, but from a cross reference to “ Mathematical 
angle.” It is only at this place that the essential quality of an angle 
as a magnitude is given. 
The word congruent, for some unknown reason is not given the 
meaning applied to it in foreign and recent American text-books on 
elementary geometry. 
Under “ Parallel lines” we find “lying evenly everywhere in the 
same direction, however far extended; in all parts equally distant.” 
This is said to be the Euclid idea of the term! Then there is given the 
modern geometry conception of a point and line at infinity in which 
parallel lines and parallel planes meet. 
Under “Parallel Postulate” is presented a good idea of the dif- 
ference between Huclidean, Lobachevskian and Elliptical space. Such 
features as this go far to show that the dictionary is up-to-date in deal- 
ing with important ideas of mathematics which the general public has 
not had a chance to know about heretofore. While this explanation of 
the parallel postulate deals with one of the most abstruse matters in 
modern mathematics it is still reasonably intelligible to the ordinary 
reader. But there have been introduced into the New International 
the definitions of numerous highly technical mathematical terms whose 
meaning is quite beyond the ken of all except a very limited number 
of technical school graduates. Thus, under Dirichlet’s theorem is 
found a triple integral involving perhaps a dozen elements. Similar 
technical matter will be found under numerous headings. 
