MOORE. — MINIMUM GEOMETRY. 



205 



is a conic tangent to the line/ at the point F. In the minimum plane 

 this is the curve which Study calls the parabolic circle. 



Area. The line area of the triangle whose vertices are the points 

 (*i, Vi), O2, 2/2), (*b, yz), is 



2/1 - 2/2 y\ 



2/3 



Oi — .r 2 ) (.r 3 — .Ti) = 

 - .r 2 a* — U' 3 _ 



(2/30-1 - 2/1*3) + (2/1*2 — 2/2-Tl) + (2/2*3 - 2/3*2) = 



1 2/1*1 



1 2/2*2 

 1 2/3 *3 



which agrees with the ordinary formula for area. The angle area is 



1 



(*i — *a) (*2 — *s) (*3 - a-i) 



1 2/i*i 

 1 ?/ 2 *2 

 1 2/3*3 



That is, 



Line area = angle area multiplied by the product of the three sides. 

 The curvature of the parabolic circle circumscribing the above 

 triangle is 



1 2/i *i 

 1 2/2 *2 



1 2/3*3 



— . = 2 X Angle area = 2 X line area -£- abc. 



(*1 — *2) (*2 — X Z ) (*3 — *l) 



This agrees with the curvature of a circle in euclidean geometry except 

 for a factor 2. This is due to the definition of area but there seems to 

 be no need to complicate the formula just to make a closer agreement. 

 The condition for a point of inflection is the same here as in ordinary 

 geometry, 



d 2 y 



dx 2 



= 0. 



Collineations of the plane. We -saw from the definition of distance 

 that the sides and angles of a triangle are in a measure independent. 

 In fact if the vertices are moved along the lines which join them to the 

 point F the sides are not changed in length while the angles can be 

 made to vary from zero to infinity. We should then expect to find 

 collineations which leave distance invariant and change angle and 

 vice versa. If distance is to be preserved both the point F and the 



