82 LAMBERT— THE STRAIGHT LINE CONCEPT. [April i 4 , 



THE STRAIGHT LINE CONCEPT. 

 BY P. A. LAMBERT. 



{Read April 14, /<?PJ.) 

 INTRODUCTION. 



The foundation of a science is the system of assumptions which 

 gives precision to the concepts with which the science deals. It is 

 essential that the system of assumptions together with the results 

 obtained by applying the processes of logic to the concepts shall be 

 free from contradiction. This freedom from contradiction is gen- 

 erally established by showing that the system of assumptions gives 

 precision to some complete number system of arithmetic. 



It is an important problem in any science to reduce the system 

 of assumptions to a minimum. This problem is solved by exclud- 

 ing all assumptions which are logical consequences of other assump- 

 tions. When the system of assumptions of a science is reduced to 

 a minimum the omission of any one assumption or the change of 

 any one assumption will either lead to a contradiction or change 

 the concepts of the science. 



In order that a science shall not become a mere exercise in men- 

 tal gymnastics the results obtained by applying the processes of 

 logic to the concepts of the science must agree with observed results 

 in the processes of the physical world. 



The assumptions of a science are also called the axioms of a 

 science. The assumptions of geometry are called axioms by Hil- 

 bert in the Grundlagen der Geometric 



THE STRAIGHT LINE. 



The logical entities with which rational geometry deals are the 

 concepts named the poirrt, the straight line and the plane. Pre- 

 cision is given to these concepts by axioms which have been ar- 

 ranged by Hilbert in five groups, called axioms of relation, axioms 

 of order, axioms of congruence, axioms of parallels and axioms of 

 continuity. 



The importance of the straight line, whether by straight line we 

 understand the intuitive entity of experience or the logical entity 



