MOORE. — MINIMUM GEOMETRY. 



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thing, however, happens in euclidean geometry as there is no way of 

 comparing intervals on minimum lines or on the line at infinity. 



Case I. 

 The fundamental facts of the first kind of geometry are : 



Distance is a directed quantity, 

 that is when a positive direction is 

 fixed on one line of the plane it is 

 fixed for each line of the plane. 



The locus of points equidistant 

 from a fixed point A is a line 

 passing through F. 



The distance between two 

 points A and B is the same as the 

 distance between any point on 

 AF and any point on BF. 



The sum of the sides of a tri- 

 angle is zero. 



The distance from a point, not 

 on /, to a point (not F) on / is 

 infinite. The distance from a 

 point not on F is indeterminate. 

 The distance from a point A to 

 any point on AF is zero. 



Angle is a directed quantity, 

 that is when a positive rotation 

 is fixed about one point of the 

 plane it is fixed about each point 

 of the plane. 



Lines making a fixed angle with 

 a given line a pass through the 

 same point on /. 



The angle between two lines 

 a and b is the same as the angle 

 between any line passing through 

 (a,/) and any line passing through 

 (b, /). 



The sum of the angles of a 

 triangle is zero. 



The angle between a line, not 

 through F, and a line (not /) 

 through F is infinite. The angle 

 between / and a line through F 

 is indeterminate. The angle be- 

 tween two lines intersecting on / 

 is zero. 



If a, b, c, denote the sides of a triangle and A, B, C, the angles oppo- 

 site, the relations connecting them were found to be, 



A + B + C = 0, 



a 

 A 



a + b + c = 0, 



6 c 

 B == C' 



Defining the line area of a triangle to be such a scalar function of the 

 sides taken in a definite order around the triangle, that triangles having 

 the same vertex and bases on the same line shall have areas propor- 

 tional to their bases, the following formula for area was derived, 



Line area = abC = acB = bcA. 



