MATHEMATICAL DEFINITIONS 457 
MATHEMATICAL DEFINITIONS IN TEXT-BOOKS AND 
DICTIONARIES 
By Dr. JOS. VY. COLLINS 
STATE NORMAL SCHOOL, STEVENS POINT, WIS. 
HE word definition is defined as “ fixing the bounds of,” or “ de- 
termining the precise signification of.” It may be distinguished 
from the word description by saying that the latter merely makes its ob- 
ject known by words or signs, very often by some non-essential quality, 
as a lady by her dress. Young, in his “ Teaching of Mathematics,” says 
a definition is simply an agreement making clear the precise meaning 
of the word defined. As mathematics is an exact science, its definitions 
are important and play a significant part in the development of the 
subject. Formerly the tendency was to give a large number of defini- 
tions at the beginning of a study, but latterly only essential ones are 
furnished, and others are introduced as needed. 
The distinction between definitions, axioms and postulates is often 
not clear, though it would appear that there should be definite boun- 
daries between them. Doubtless so far as their etymological meanings 
go the words postulate and axiom could be used interchangeably. A 
few late geometries class axioms and postulates together and call them 
all postulates. German texts usually avoid the use of these terms alto- 
gether. To the writer the distinction between axiom and postulate 
in Euclid is valuable and should be retained. Fortunately, most Amer- 
ican authors follow Euclid in regarding the postulates as the funda- 
mental propositions of constructions, one the straight-line postulate, 
and the other the circle postulate. Similarly, some writers do not 
distinguish clearly between axioms and definitions. For instance, it is 
usually given as an axiom that quantities that can be made to coincide 
are equal. This, on the face of things, simply defines the meaning of 
the term equal. Again, some writers following the lead of the popular 
French geometer, Legendre, define a straight line as the shortest dis- 
tance between two points, whereas Euclid gives this property as an 
axiom. This test for a straight line implies measurement, and hence 
the idea of measurement of lines would have to be developed before a 
straight line could be defined. Evidently Euclid’s view of the matter 
is much preferable to Legendre’s. 
The definitions of the fundamental concepts by different authors 
should amount to the same, however differently they are expressed. 
But it turns out that definitions apparently meaning the same thing 
are really very different. Thus some authors have defined parallel 
