376 KANSAS ACADEMY OF SCIENCE. 



From these statements it must not be concluded that because the 

 Euclidean geometry is rigorous that the non-Euclidean is less so. 

 The difficulty with Euclid is in the assumption of axioms or state- 

 ments that were not capable of proof. Of the non-Euclidean geome- 

 tries, there are now two well-defined ones: First, that discovered by 

 Lobachevsky, sometimes called the hyperbolic geometry, and second, 

 the elliptic geometry, discovered by Kiemann. Riemann studied the 

 foundations of geometry "from a very different point of view, an 

 abstract algebraic point of view, considering not our space and geo- 

 metrical figures, but a system of variables." He investigated the 

 question, "What is the nature of a function of these variables which 

 can be called element of length or distance ? and found that in the 

 simplest cases it must be the square root of a quadratic function of 

 the differentials of the variables whose coeflScients may themselves be 

 functions of the variables." To Clifford "we owe the theory of paral- 

 lels in elliptic space." The lesson to be learned from non-Euclidean 

 geometry is, "that the axioms of geometry are only deductions from 

 our experience, like the theories of physical science." "The assump- 

 tions which distinguish the three kinds of geometry, Euclidean, the 

 hyperbolic, and the elliptic, may be expressed in different forms. 

 We may say that one, or two, or no parallels can be drawn through a 

 point ; or, that the sum of the angles of a triangle is equal to, less 

 than, or greater than two right angles ; or, that a straight line has 

 two real points, one real point, or no real point at infinity ; or, that in 

 a plane we can have similar figures, and that a straight line is of finite 

 or infinite length, etc." Any of these forms points out the nature of 

 the geometry. 



To illustrate the non-Euclidean geometry of Lobachevsky, let us 

 take his theorem : A straight line maintains its parallelism at all 

 points. 



Let AB be parallel to CD at E, and let F be any other point of 

 AB on either side of E, to prove that AB is parallel to CD at F. 



Demonstration : To H, on CD, draw EH and FH, If H move 

 off indefinitely on CD, these two lines will approach positions of 

 parallelism with CD. But the limiting position of EH is the line 

 passing through F, and if the limiting position of FH were some 

 other line, FK, F would be the limiting position of H, the intersec- 

 tion of EH and FH. 



Another theorem is : If one line is parallel to another, the second 

 is parallel to the first. 



Given AB parallel to CD, to prove that CD is parallel to AB. 



Demonstration: Draw AC perpendicular to CD. The angle CAB 

 will be acute ; therefore, the perpendicular CE from C to AB must 

 fall on that side of A towards which the line AB is j^arallel to CD 



