394 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and propositions relating to this type of transformation, and common 

 to the two geometries. We therefore postulate for translation: 



5°. Any point P can be carried into any point P' , and any two 

 translations which carry P into P' are identical. 



6°. Any line is carried into a parallel line. 



7°. Any line parallel to PP' remains unchanged. 



8°. The succession of two translations is a translation. 



These postulates determine the characteristics of a group of geome- 

 tries of which the two most important are Euclidean geometry and 

 that non-Euclidean geometry with which we are here concerned. 

 Another non-Euclidean geometry belonging to this same group will be 

 discussed briefly in § 31. This group excludes such geometries as the 

 Lobachewskian and the Riemannian in which a parallel to a given 

 line at a given point is not uniquely defined. We shall first proceed 

 to develop some of those general theorems which are true in this 

 whole group of geometries. 



I. If two intersecting lines are parallel respectively to two other 

 intersecting lines, the corresponding angles ^ are congruent. 



For by translation the points of intersection may be made to coin- 

 cide, and the lines of the first pair, remaining parallel with the lines 

 of the other pair (6°), must come into coincidence with them, by 

 postulate 3°. 



II. The opposite sides of a parallelogram are congruent. 



For if ABCD is a parallelogram and if A be translated to B, the line 

 of DC remains unchanged, by 7°, and the line of AD falls along the line 

 of'^C by I. Hence D falls on C by 2°. 



Cor. If two points P, P' are carried by a translation into Q, Q', 

 the figure PP' Q' Q is a, parallelogram. 



7. We may now set up a system of measurement along any line 

 and hence along the whole set of parallel lines. Consider the segment 

 PP'. By the translation which carries P into P', the point P' is 

 carried into a point P" of the same line. The measure of the separa- 

 tion of P and P' we will call the interval ^ PP'. And since the segment 

 PP' is congruent to the segment P' P", the intervals PP' and P' P" 

 are said to be equal. We may thus rnark off any number of equal 

 intervals along the line. We shall assume further the Archimedean 

 postulate. 



5 The word angle here refers to a geometrical figure only, and does not as yet 

 imply any measure of angle. 



6 We use the word interval to avoid all ambiguity. The notion of distance 

 will be separately considered in Appendix, §65. 



