52 A SHORT HISTORY OF SCIENCE 



build up with four others a regular octahedron, or starting with five, 

 an icosahedron with 20 faces. Six triangles, however, will fill the 

 angular space about a point, and thus not permit the formation of 

 a regular polyhedron. Using squares instead of triangles, we 

 obtain only the cube; using pentagons (angle 108), the regular 

 dodecahedron - - 12 faces, 3 at each vertex. The Egyptians must 

 have been familiar with the cube, the regular tetrahedron, and the 

 octahedron. To these, with the icosahedron, the Pythagoreans as- 

 sociated the four cosmical elements earth, air, fire, and water. 

 Their discovery of an additional body, the regular dodecahedron, 

 formed by 12 pentagons, made a break in the correspondence, and 

 the need was met by the addition of the universe, or, according 

 to others, the ether, as a fifth term in the cosmical series. This 

 correspondence was not merely symbolical, but physical, the 

 earth being supposed to consist of cubical particles, etc. We 

 cannot infer that the impossibility of a sixth regular polyhedron 

 was known. That only these five regular polyhedra are pos- 

 sible was in fact first proved by Euclid. There is a tradition 

 that the Pythagorean discoverer of the dodecahedron was 

 drowned at sea on account of the sacrilege of announcing his 

 discovery publicly. A later commentator records a similar 

 tradition that the discoverer of the irrational perished by 

 shipwreck, since the inexpressible should remain forever con- 

 cealed, and that he who touched and opened up this picture 

 of life was transported to the place of creation and there washed 

 in eternal floods. 



The regular polygons were naturally studied, and in particu- 

 lar the decomposition of them into right triangles of 45 and 

 30-60. With the pentagon the attempt naturally failed, but 

 the five-pointed star formed by drawing diagonals was a special 

 emblem of the Pythagoreans. With the inscribed pentagon 

 connects itself naturally the division of a line in extreme and 

 mean ratio, or, as it was later characterized, the "golden sec- 

 tion." This division, by which the square on the greater segment 

 of a line is equivalent to the rectangle whose sides are the other 

 segment and the whole line, occurs repeatedly in Greek archi- 



