460 THE POPULAR SCIENCE MONTHLY 
face. Wentworth, on the other hand, says it is the ratio to a standard 
unit times the unit of measure. Certainly these definitions do not give 
the word the same meaning. Which is right? The writer is inclined 
to think the Wentworth definition correct, for the reason that if one 
is asked for the area of a field, he does not say, e. g., 11, but 11 acres. 
There is, of course, the same distinction made in the definitions of 
volume, and the same distinction could be made in defining contents, 
weight, length, ete. 
Perhaps the most striking thing in connection with mathematical 
definitions is the weakness of their statement in our dictionaries. 
These definitions are often stated in synonyms when the real thing 
could just as well be given; they are stated obscurely when it is just as 
easy to give clear definitions; and they are stated in Latin terms, the 
language of the schools two or three centuries ago, though the vast 
body of users of the dictionary do not have the least idea of the mean- 
ing of the Latin roots. Thus in defining number, Webster’s Interna- 
tional says “ it is a unit or an aggregate of units; a numerable aggregate 
or collection of individuals; an assemblage made up of distinct things 
expressible by figures; that which admits of being counted.” Compare 
these definitions with this: a number is one or more units, or ones. 
They all have this meaning. Only one of the Webster definitions is 
simple, and it could be simplified still further to advantage by saying 
that a number is that which can be counted. Notice that number has 
virtually been defined by a synonym in this definition, since counting 
would have to be defined. However, counting is a familiar act to every 
one. 
The Old International, latest edition, defines a perpendicular as a 
line that makes right angles with another, and then a right angle as 
one that is formed by a perpendicular! The Standard Dictionary does 
the same thing. If a pupil in a geometry class were to do this, the 
teacher, metaphorically at least, would box his ears. There is no oc- 
casion for defining one of these terms by the other, and then the latter 
by the former. It would be all right to define either by the other, pro- 
viding an essential definition were given for the other. In this case the 
geometries say right angles (or a perpendicular) are formed when one 
line meets another so as to make the adjacent angles formed equal. 
The dictionary should do the same. The Century Dictionary says a 
perpendicular is the shortest distance from a point to a line, and then 
defines right angle as one formed by a perpendicular. This definition 
for perpendicular is open to the same objection as the definition of a 
straight line, which says it is the shortest distance between two points. 
But the Century evidently does not fall into the silly course of Webster 
and the Standard. Worcester says a right angle is one of 90°, and 
defines one degree as one three-hundred-and-sixtieth of a circumference. 
