25. _The Endospherical Polyhedral Ground-Form._--The endospherical polyhedron or polyaxon ground-form naturally follows the spherical or homaxon. Under it are included all polyhedra whose angles fall in the surface of a sphere; this ground-form is especially common among the SPUMELLARIA, especially in the shells of #Sphæroidea#, but is also found among the ACANTHARIA (especially in the Astrolophida and #Sphærophracta#), as well as among the #Phæosphæria# (in most genera of the Orosphærida, Sagosphærida, and Aulosphærida). Strictly speaking, all those lattice-shells which have {xiii}been incorrectly called "spherical" belong to this category, for they are none of them true spheres in the geometrical sense (like the central capsules of the #Sphæroidea#), but rather endospherical polyhedra, whose angles are indicated by the nodal points of the lattice shell, or the radial spines which spring from them. These endospherical polyhedra may be divided into three groups, the regular, subregular, and irregular. Of _regular polyhedra_, properly so-called, it may be shown geometrically that only five can exist, namely, the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron. All these are actually manifested among the Radiolaria, although but seldom. Much more common are the _subregular endospherical polyhedra_, _e.g._, spherical lattice-shells with regular hexagonal meshes of equal size; they are never exactly equal nor perfectly regular, but the divergences are so insignificant that they escape superficial observation (Pl. 20, figs. 3, 4; Pl. 26, figs. 1-3). On the contrary in the _irregular endospherical polyhedra_ the meshes of the lattice-sphere are more or less different in size and often in form also (Pl. 28, figs. 4, 8; Pl. 30, figs. 4, 6). The five truly regular polyhedra require separate notice on account of their importance. (See Gener. Morphol., Bd. i. p. 406.) 26. _The Regular Icosahedral Ground-Form._--The ground-form whose geometrical type is the regular icosahedron (bounded by twenty equilateral triangles) is rarely exemplified, but it occurs among the PHÆODARIA in the Circoporid genus _Circogonia_ (Pl. 117, fig. 1), and also in certain Aulosphærida, but, apparently, only as an accidental variation (_e.g._, _Aulosphæra icosahedra_). Furthermore, this ground-form may also be assumed to occur in those #Sphæroidea# whose spherical lattice-shells bear twelve equidistant radial spines (_e.g._, many species of _Acanthosphæra_, _Heliosphæra_, and other Astrosphærida); the basal points of these spines indicate the twelve angles of the regular icosahedron. (See on this head Gener. Morphol., Bd. i. p. 411.)