378 KANSAS ACADEMY OF SCIENCE. 



sum of the angles of a plane triangle is greater than a straight angle 

 by an excess proportional to its area. 



The greater the area of a triangle, the greater the excess or differ- 

 ence of the angle from 180^ 



In conclusion, it may be said that the geometry of Euclid is in no 

 danger of being superseded by either the hyperbolic or the elliptic 

 geometry. The Euclidean geometry, ever since it was given to the 

 world by its author, has been accepted as a geometrical bible, from 

 which the truth, the whole truth and nothing but the truth could be 

 obtained. But as the years rolled on, and the nineteenth century un- 

 folded its years, exceptions began to be taken to Euclid's axioms. The 

 Euclidean axioms are all accepted by the non-Euclidean, excepting 

 •the last, which the latter denies and replaces "by its contradictory — 

 that the sum of the angles made on the same side of a transversal by 

 two straight lines may be less than a straight angle without the lines 

 meeting." All of the foregoing is given to illustrate the subject of 

 non- Euclidean geometry, and to call the attention of teachers of ge- 

 ometry to the wonderful method of overcoming the objectionable 

 axiom of Euclid. 



Quotations have been made from various writers on the subject, 

 some unchanged and others modified to suit the views of the writer 

 of this paper. 



