34 university of colorado studies 



2. Point, Straight Line and Plane. 



The points of space of three dimensions may be conveniently 

 determined by the circles of a fixed plane in this space, if the center 

 of every circle is considered as the orthographic projection of a point 

 in space upon this plane, and its radius as the distance of the point 

 from the plane. In this manner every point in space is determined 

 by a certain circle in the plane. It is, however, evident that every 

 circle in the plane represents two points on opposite sides of the 

 plane (equi-distant from, and in the same perpendicular to, the plane). 

 To distinguish two points of this kind by their representative circles, 

 we may state that all circles described in the same sense represent 

 points on one side of the plane, and all circles described in a contrary 

 sense points on the other side of the plane. For graphical represen- 

 tation we assume the plane of reference as coincident with the plane 

 of the drawing. A circle described counter-clockwise shall represent 

 a point above, and one described clockwise a point below the plane. 

 Graphically this may be indicated by arrows on the circles. 



A straight line in a general position in space intersects the 

 plane of reference in a point 0. The centers of the circles represent- 

 ing such a line are situated on the orthographic projection of this 

 line upon the plane, and the radi of these circles are proportional to 

 the distances of their respective centers from the origin 0. Con- 

 versely, all circles whose centers are situated on a straight line and 

 whose radi are proportional to the distances of their respective centers 

 from a fixed point on the given straight line, represent a straight line 

 in space. Designating the radius of one of the circles by r and the 

 distance of its center from O by c/, there is 



where X is a factor of proportionality. Designating the angle of in- 

 clination of the straight line with the plane by a. 



on this subject in the VierteljahrssGhrift der Maturforschenden 

 Gesellschaft in Zurich: Ein neuer Weg zur Theorie der Kegel- 

 schnitte, Vol. 25, 1880, and Zur Geschichte und Theorie der 

 elementaren Ahhildungsmethoden^ Vol. 27, 1882. See also Fiedler's 

 Cyclographie, Leipzig, 1882. 



