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, etc.
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 meaning of the Latin roots. Thus in defining number, Webster's International 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 occasion 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, providing 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.
