34 TERMINOLOGY. . 38. 



Fig. 1. ; on the other all the equilateral triangles, as RST, 

 Fig. 5. But the equiangular hexagons, as R/'R / S"S"F / T', 

 produced by sections parallel to the triangles, are homolo- 

 gous with them ; for triangles of this kind can be inscribed 

 in them, or they are transformed into such triangles, if 

 brought to the same distance from the centre of the form. 



The sections of the hexahedron, which appear as oblongs 

 or rectangles, as HSU'S' Fig. 6., are likewise homologous 

 to each other ; for they are similar, if made equidistant from 

 the centre. Let the edge of the hexahedron AD be = 1, 

 and AS that part of it through the end of which the sec- 

 tion passes, = j ; the figure of this section will be a 



y 2 



square. A section of that kind, however, if made equidis- 

 tant from the centre with a rectangle, is likewise transform- 

 ed into a rectangle, and therefore homologous with these 

 figures, and not with the squares above mentioned. 



In every oblong, a rhomb can be inscribed, if we join 

 the centres of its sides by straight lines. Hence sections 

 of a rhombic figure are homologous with sections of an ob- 

 long or rectangular figure. 



Besides the sections described in the hexahedron, there 

 are none to be met with in any other solid whatever ; or 

 those which may be met with in other solids, can always be 

 traced to one of these. The different kinds of sections are, 

 therefore : 



1. Such as are either equilateral Triangles themselves, or 

 in which equilateral triangles may be inscribed ; as regu- 

 lar hexagons, or equiangular hexagons, whose alternate 

 sides, or equilateral hexagons, whose alternate angles, are 

 equal ; dodecagons of the same description, &c. 



2. Such as are either Squares themselves, or into which 

 squares may be inscribed, as regular octagons, or equiangu- 

 lar octagons, whose alternate sides are equal, or equilateral 

 ones, whose alternate angles are equal, &c. 



3. Such as are Rectangles or Rhombs, or in which rectangles 

 or rhombs may be inscribed. It must be remarked here, 

 that if among the rectangular sections, there is only one, or 

 two squares, as in the tetrahedron and in the hexahedron, 



