WILSON AND LEWIS. — RELATH ITV. 



395 



9°. If a sufHcient nunil)er of equal inter\iils he laid ofT on a line, 

 any point of the line may l)e surpassed. 



Now the whole theory of conunensural)ility or incommensurability 

 of two intervals along the same line or parallel lines may be treated 

 by the usual methods. Thus the intervals along a line, starting from 

 any origin upon the line, may be brought into one-to-one correspond- 

 ence with the series of real numbers. It is, however, to be especially 

 emphasized that we have not established, and cannot establish by the 

 translation alone, any comparison between intervals on non-parallel 

 lines. 



III. The diagonals of a parallelogram bisect each other.'' 

 For let (Figure 1) the parallelogram ABCD, of which the diagonals 

 intersect at E, be translated into the 

 position BB' C C (by translating A to 

 iS), in which the diagonals intersect at 

 E'. Now BE' is parallel to EC, and EB 

 to CE'. Hence BE' which is congruent 

 to AE, is congruent to EC by II. Con- 

 sequently AE is congruent to EC by 8°. 



l\\ If two triangles have the sides of one respectively parallel 

 to the sides of the other, and if one side of one is congruent to one side 



of the other, then the remaining sides of the 

 one are respectively congruent to the remain- 

 ing sides of the other. 



For if the two congruent sides are brought 

 into coincidence by translation, the two tri- 

 angles will either coin- 

 cide throughout, or will 

 together (Figure 2) form 

 a parallelogram (II). 



Two triangles with the 

 sides of one respectively 

 parallel to the sides of the 

 other will be called similar. 



V. In two similar triangles the sides of 

 the one are respectively proportional to the 

 sides of the other. 



For if ABC and A'B'C are the triangles, the vertex A' may be 

 made to coincide with A by a translation (Figure 3). Suppose, now, 



7 Theorems like this and the preceding and some which are to follow are 

 proved in elementary geometries by the aid of propositions (on congruence of 

 triangles) not deducible from translations alone. 



o^a' 



Figure 2. 



Figure 3. 



