Page:Popular Science Monthly Volume 68.djvu/25

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THE FOUNDATIONS OF GEOMETRY

 THE FOUNDATIONS OF GEOMETRY

an historical sketch and a simple example

By Dr. OSWALD VEBLEN,

PRINCETON UNIVERSITY

GEOMETRY 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 contrasting 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.
5. 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