THE FOUNDATIONS OF GEOMETRY 21 



THE FOUNDATIONS OF GEOMETRY 



AN HISTORICAL SKETCH AND A SIMPLE EXAMPLE 

 BY Db. OSWALD VEBLEN, 



PRINCETON UNIVERSITY 



GEOMETEY as a logical system took its first definite form in the 

 mind of Euclid (about 330-275 B.C.) ; and since the edifice 

 constructed by the grandfather of geometry has justly retained the 

 admiration of all succeeding students, one can perhaps exhibit the 

 modern researches on the same subject in no better way than by con- 

 trasting them with some of Euclid's fundamental statements. The 

 propositions which Euclid placed at the foundation of his work have 

 come to us classified under three heads : definitions, postulates, axioms. 

 As examples of the first we may quote (from Todhunter's edition). 



1. A point is that which has no parts, or which has no magnitude. 



2. A line is length without breadth. 



3. The extremities of a line are points. 



4. A straight line is that which lies evenly between its extreme points. 



6. A superficies is that which has only length and breadth. 



7. A plane superficies is that in which, any two points being taken, the 

 straight line between them lies wholly in that superficies. 



15. A circle is a plane figure contained by one line, which is called the 

 circumference and is such that all lines drawn from a certain point within the 

 figure to the circumference are equal to one another: 



16. And this point is called the center of the circle. 



It is evident that in the first of these statements, if ' point ' is 

 defined, ' magnitude ' or ' parts ' is not ; in the second, if ' line ' is 

 defined, 'length' and 'breadth' are not; and so on. A partial list of 

 the terms undefined in the above definitions would include magnitude, 

 length, breadth, extremities, lie in, lie evenly, equal to. It is in fact 

 a commonplace among teachers and schoolboys that to any one who 

 did not already know what the terms meant, these definitions would 

 be entirely meaningless. Another way of stating the same proposition, 

 and the way upon which modern mathematicians insist, is that in 

 every process of definition there must be at least one term undefined. 

 A thing which is not defined in terms of other things we may call an 

 element. 



It is also to be observed that in the above list of undefined terms 

 there are at least two classes to be distinguished. The first four terms 

 are nouns and correspond to the notion element. The last three are 



