Page:Popular Science Monthly Volume 77.djvu/463

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HE word definition is defined as "fixing the bounds of," or "determining the precise signification of." It may be distinguished from the word description by saying that the latter merely makes its object 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 definitions 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 boundaries 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 altogether. To the writer the distinction between axiom and postulate in Euclid is valuable and should be retained. Fortunately, most American authors follow Euclid in regarding the postulates as the fundamental 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 distance 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