164 Mr. Davies Gilbert on the Regular or Platonic Solids. 

 If therefore z — any number from infinity to 2. 



y may be 2. Indicating that the sphere is divided 

 into any number of parts by great cir- 

 cles (meridians) passing through the 

 same two poles. 



z = 6 y at its maximum 3, x infinite. Hence the 

 sphere may be covered by an infinite number 

 of equilateral triangles meeting with 6 angles 

 at each point. 



% = 5, y at its maximum = 3 x = 20. Hence the 

 sphere may be covered by 20 equilateral tri- 

 angles meeting 5 in a point. 



% = 4, y at its maximum = 4, x infinite. Hence the 

 sphere maybe covered by an infinite of squares 

 or tetragons meeting 4 in a point. 



y = 3, x = 8. Hence the sphere may be co- 

 vered by 8 equilateral triangles meeting 3 in 

 a point. 



* = 3, y at its maximum = 6, x infinite. Hence the 

 sphere may be covered by an infinite number 

 of hexagons meeting 3 in a point. 



y = 5, x = 12. The spheres may therefore be 

 covered by 12 pentagons meeting 3 in a 

 point. 



y = 4 . x = 6. The sphere may therefore be 

 covered by 6 squares or tetragons meeting 

 3 in a point. 



y = 3, .r = 4. The sphere may therefore be 

 covered by 4 equilateral triangles meeting 

 3 in a point. 



z = 2 ^ at its maximum infinite x any number what- 

 ever. Appearing to indicate that a sphere 

 being divided by a mathematical plane into 

 two equal parts, may have that plane bounded 

 by any regular polygon, when 2 angles only 

 will meet in a point. 



y = 2 The plane appears to contract itself into a right 

 line. 



The 



