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PROCEEDINGS OF THE AMERICAN ACADEMY. 



considered a branch of a hyperbola.^* Since, however, this curve is 

 here generated by the rotation of a Hne OP about its terminus Q, 

 we shall call this locus (taken Avith the other branch Q Q' Q" sym- 

 metrically situated with respect to 0) the pseudo-circle. 



By means of such a rotation we are able to compare intervals upon 

 any line with intervals upon any other line of the same class. For 

 the intervals of the congruent radii OP, OP', OP" will be called equal. 

 When we consider the fixed lines we observe that the effect of 

 rotation is to carry the segment OA into OA' or OA". It is therefore 

 evident that segments are congruent by rotation which are incongru- 

 ent by transla,tion. This source of ambiguity exists only in the case 

 of singular lines, for in no other case is it possible to compare two 

 segments both by rotation and by translation. We may remove this 

 ambiguity at once by stating that intervals along singular lines, al- 

 though metrically comparable with intervals on other singular lines 



of the same class by translation, are 

 all of zero magnitude when compared 

 with intervals on any non-singular 

 line. This will become more evident 

 later. 



Consider next (Figure 9) the inter- 

 cept AB terminating on the fixed lines 

 corresponding to a rotation with cen- 

 ter at 0. Let P be the middle point 

 of the line, and C any other point. 

 Through C draw a line parallel to OB, 

 and on this line mark the point P' 

 such that the area ODP'G equals the 

 area OFPH. The area OECG is less 

 than each of these by X. Hence 

 P' lies on the further side of AB 

 from 0. But P' is a point on the pseudo-circle through P concentric 

 with 0, as we have just seen. Since C was any point of AB, it follows 

 that P' may be any point of the pseudo-circle. Hence as the line 

 AB meets the pseudo-circle at P and only at P, it is tangent to the 

 curve. As a species of converse, we may state the theorem : 



Figure 9. 



14 There is no special significance in the fact that a rectangular hyperbola is 

 drawn in the figure and that the fixed line.s «, are perpendicular in the 

 Euclidean sense; in subsequent figures the singular lines are often oblique. 

 From the non-Euclidean viewpoint the question of perpendicularity or 

 obliquity of the singular lines is of course naeaningless. 



