MISCELLANEOUS PAPERS. 377 



(this statement depends on a former proposition not given here). 

 The angle ECD is then acute and less than CEB, which is a right 

 angle. That is, we have CAB less than ACD, and CEB greater than 

 ECD. If the line CE revolve about the point C to the position of 

 CA, the angle at E will decrease to the angle A, and the angle at C 

 will increase to a right angle. There will be some position, say CF, 

 where these two angles become equal; that is CFB = FCD. Draw 

 MN perpendicular to CF at its middle point and revolve the figure 

 about MN as an axis. CD will fall upon the original position of AB, 

 and AB will fall upon the original of CD. Therefore CD is parallel 

 to AB. 



Another theorem : In a rectilineal triangle the sura of the three 

 angles cannot be greater than the two right angles. 



Demonstration : Suppose in the triangle ABC the sum of the 

 three angles is equal to 180°+ « ; then choose in case of the inequality 

 of the sides the smallest BC, halve it in D ; draw from A through D 

 the line AD, and make the prolongation of it DE, equal to AD ; then 

 join the point E to the point C by the straight line EC. In the con- 

 gruent triangles ADB and CDE, the angle ABD = DCE.and BAD = 

 DEC ; whence follows that in the triangle ACE the sum of the three 

 angles must be equal to 180'' 4-«; but also the smallest angle BAG 

 of the triangle ABC in passing over into the new triangle ACE has 

 been cut into the two parts EAC and AEC. Continuing this process, 

 continually halving the side opposite the smallest angle we must 

 finally attain to a triangle in which the sum of the three angles is 

 180°+ a, but wherein are two angles, each of which in absolute mag- 

 nitude is less than \a ; since now, however, the third angle cannot be 

 greater than 180^, so must a be either null or negative. 



Such are some of the methods of demonstration used in the non- 

 Euclidean geometry of Lobachevsky. On the other hand, if we take 

 the non-Euclidean geometry of Riemann, it may be shown that if a 

 straight line is determined by two points, but take the contradictory 

 of the axiom that a straight line is of infinite size ; then the straight 

 line returns into itself ; but two having intersected get back to that 

 intersection point without ever again meeting. Two intersecting 

 complete straight lines enclose a plane figure, and two such plane 

 figures are congruent if their angles are equal. All complete straight 

 lines are of the same length I. In a given plane all the perpendicu- 

 lars to a given straight line intersect in a single point, whose distance 

 from the straight line is \l. Inversely, the locus of all, the points at 

 a distance \l on straight lines passing through a given point and lying 

 in a given plane, is a straight line perpendicular to all the radiating 

 lines. 



The total volume of the universe, therefore, is equal to l^/tr. The 



