374 KANSAS ACADEMY OF SCIENCE. 



THE NON-EUCLIDEAN GEOMETRY. 



By E. Miller, University of Kansas, Lawrence. 

 Read (by title) before the Academy, at Topeka, December 31, 1904. 



TN the third century before Christ there lived the three greatest 

 -^ mathematicians of antiquity — Euclid, Archimedes, and Apol- 

 lonius. The earliest of the three was Euclid, who was born in the 

 city of Tyre, about 330 b. c, and died about 275 b. c. He was the 

 first professor of mathematics, and at the same time one of the most 

 famous professors in the University of Alexandria, Egypt. There 

 was current in his day an old saying that has appeared almost every- 

 where and in every age since. Among the French it has taken the 

 form of question and answer. In a discussion of the merits of ge- 

 ometry, one Frenchman puts the question: "Who the devil can 

 learn this?" to which the other replies: "It is the devil who has 

 patience." Euclid's "Elements of Geometry" has been for nearly 

 2100 years the chief text-book among mathematical teachers. His 

 axioms and propositions have been accepted without question or criti- 

 cism. But during the last 100 years there has gradually crept into 

 the minds of the greatest thinkers the thought that Euclid is lame in 

 spots. One writer says: "The defects of Euclid as a text-book of 

 geometry have been often stated, and are summed up in de Morgan's 

 article in the "Dictionary of Greek and Roman Biography." The most 

 prominent defects are these : (1) The definitions and axioms contain 

 many assumptions which are not obvious, and in particular the so- 

 called axiom about parallel lines is not self-evident. (2) No expla- 

 nation is given as to the reason why the proofs take the form in 

 which they are presented ; ^. e., the synthetical proof is given, but not 

 the analysis by which it was obtained. ( 3 ) There is no attempt 

 made to generalize the results arrived at ; e. g., the idea of an angle is 

 never extended so as to cover the case where it is equal to or greater 

 than two right angles. ( 4 ) The sparing use of superposition as a 

 method of proof. ( 5 ) The classification is very imperfect. ( 6 ) 

 The work is unnecessarily long and verbose." Hence there has grown 

 up during the last 100 years a demand for axioms and demonstrations 

 that shall be free from objection. The efiPort has been crowned with 

 success. 



It has been conceded in the past that the axioms of geometry could 

 neither be denied nor investigated. Men everywhere found them to 

 agree with their experience, believing that the most rigid reasoning 

 would fail to show any of them untrue. There is no question now 



