MOORE. — MINIMUM GEOMETRY. 209 



ab = ac (where xy denotes the angle between the lines x and y) if 

 the line joining the point of intersection of b and c to F meets a on /, 

 that is the envelope of lines making equal angles with a given line is a 

 point. 



The angle which a line a, not through F makes with a line through F 

 is infinite if a does not meet it in /. If this is the case the angle is 

 indeterminate. 



Angle is a directed quantity, that is if a positive rotation is assigned 

 around one point of the plane it is assigned around each point of the 

 plane. A construction was given for measuring about any point, 

 beginning at a fixed line, an angle equal to a given angle. 



The relations connecting distance and angle are the expressions for 

 the angles of a triangle in terms of the sides. These are, 



. a+b+c a+b+c a+b+c 



A = -, > Jt> = — » ks — 1 • 



be ac ab 



Solved for a, b, c, these relations become, 



A+B + C , A+B+C A+B + C 

 a = BC ' b= AC ' C= AB 



From these can be derived, 



a be 

 A~ B~ C' 



which corresponds to the sine law of ordinary trigonometry. It 

 should be kept in mind that distance is a directed quantity and there- 

 fore a direction around the triangle must be assumed. 



Coordinate systems. 



Having these triangle relations it is easy to set up a coordinate 

 system. A very simple one is an ordinary polar system consisting of 

 a point from which distances are measured and a line /, through it 

 from which angles are measured. The coordinates are then the dis- 

 tance from and the direction 6, measured from I. This system does 

 not have the indeterminate character of polar coordinates in the 

 euclidean plane since a point will have just one set of coordinates. 

 It does have the disadvantage, however, of having an indeterminate 

 character for the points of the line OF. The coordinates of any point 

 on this line are (0, oo), since the distance from any point to F is zero 

 and the angle which any line makes with a line through F is infinite. 



