WILSON AND LP:\V1S. — UELATIVITV. 403 



XII. The tanijent to a pseudo-circle lies between the curve and 

 its center, and the portion of the tangent intercepted between the 

 two fixed lines is bisected at the point of tangency. 



11. In a pseudo-circle the radius and the tangent at its extremity 

 are said to be perpendicular. Or in virtue of XII we may say that the 

 perpendicular from any point to any non-singular line is the line 

 from to the middle point of that segment of the line which is inter- 

 cepted by the fixed lines through 0. The construction of a perpendic- 

 ular to any line of class (7) or (6) at a point of the line is equally simple. 



By the aid of propositions concerning similar triangles, the follow- 

 ing theorems concerning perpendiculars are readily proved. 



XIII. If a line a is perpendicular to a line b, then b is perpendicular 

 to a. 



XTV. Through any point one and only one perpendicular can be 

 drawn to any line. 



XV. All lines perpendicular to the same line are parallel. 



XVI. The singular line of one class 

 which is drawn through the intersection 

 of any two perpendicular lines will bisect 

 the segment intercepted by these lines 

 upon any singular line of the other class 



(Figure 10). 



15 '■■' J 



XVII. The perpendicular to a (7)-line Figure 10. 



is a (5) -line, and vice versa. 



Intervals along lines of class (5) cannot be compared by congruence 

 with intervals along lines of the (7) class. We may, therefore, arbi- 

 trarily define equality of intervals between the two classes. // two 

 viutually perpendicular lines are drawn from any point and terminate 

 on a singular line, the intervals of these lines will be said to be equal}^ 

 The consistency of this definition is readily proved. 



The definition of perpendicularity is such that if two lines are per- 

 pendicular they must remain perpendicular after a translation or 

 rotation. The former case is obvious, and the latter becomes so 

 when the lines are considered as radius and tangent in a pseudo-circle 

 generated by the rotation; the more general case in which neither of 

 the perpendicular lines passes through the center of rotation then 

 follows with the aid of XV. It is important to observe one peculiar 



15 In the figure BO and OC are equal, and AB and AC are perpendicular. 



16 In Figure 10, the intervals AC and AB are therefore equal by thi.s 

 definition. 



