OF SPACE. 



43 



beoome parallel, the distance of the eye, and consequently that cf 

 the vanishing point, becoming infinite. 



186. Definition. Tlie subcontrary section of a sca- 

 lene cone is that which is perpendicular to the triangular 

 section of the cone, passing through the axis, and perpen- 

 dicular to the base, and which cuts off from it a triangle 

 similar to the whole, but in a contrary position. 



187. Theorem. The subcontrary section of a scalene 

 cone is a circle. 



Through any point A of the section, let a 

 plane be d^a^vn parallel to the base ; then its 

 section will be a circle, as is easily shown by 

 the properties of similar triangles ; and the 

 common section of the planes wiil be per- 

 pendicular to the triangular section of the 

 cone to which they are both perpendicular ; 

 consequently, ABqnCB.BD ; but since the 

 triangles CBE, FBD are equiangular and similar, CBi;BE:tBF : 

 BD, and CB.BD=:BE.BF=:ABq ; tlierefore EAF is also a circle. 



188. Theorem. The stereographic projection of any 

 circle of a sphere, seen from a point in its surface, on a 

 plane perpendicular to the diameter passing through that 

 point, is a circle. 



Let ABC be a great circle of the sphere pas- 

 sing through tlie point A and the centre of the 

 circle to be projected, then the angle ACB~ 

 BADrrBEF, and ABC=CAG=:CHI, and the 

 triangle AHE is similar to ABC, and the plane 

 ABC is perpendicular to the plane BC and the 

 plane HE, therefore HE is a subcontrary sec- q" 

 tion of the cone ABC, and is consequently a circle. 



