164. _Genealogical Tree of the Sphæroidea._--_Cenosphæra_, the simplest form of the spherical lattice-shell, may be unhesitatingly regarded as the common stem-form of all the #Sphæroidea# (pp. 50-284, Pls. 5-30). _Cenosphæra_ (p. 61, Pl. 12) arose directly from _Actissa_ simply by the silicification of the spherical exoplasmatic network of the sarcodictyum around the central capsule, on the surface of the concentric calymma. From this simple siliceous extracapsular lattice-sphere all other forms of #Sphæroidea# have arisen, in the main by the manifold combination of two simple processes, first by the formation of radial spines on the surface of the lattice-sphere, and second, the addition of concentric spherical lattice-shells. Both processes may be utilised as the foundation for a systematic treatment of the #Sphæroidea# (compare pp. 52-58).
If in the #Sphæroidea# the characteristic number and disposition of the _radial spines_ be regarded as the most important heritable peculiarity of the different families, then we have the following natural arrangement:--(1) Liosphærida, without radial spines; (2) Cubosphærida, with six radial spines (opposite in pairs in three axes perpendicular to each other); (3) Staurosphærida, with four radial spines (in two axes crossed at right angles); (4) Stylosphærida, with two opposite radial spines (in the vertical main axis); and (5) Astrosphærida, with numerous regularly or irregularly distributed radial spines (eight to twenty or more). If, on the contrary, more stress be laid upon the number of the concentric lattice-shells, then we have the following artificial grouping:--(1) Monosphærida, with one simple lattice-sphere; (2) Dyosphærida, with two concentric lattice-spheres; (3) Triosphærida, with three; (4) Tetrasphærida, with four; (5) Polysphærida, with numerous (five to twenty or more) concentric lattice-shells; (6) Spongosphærida, with a spongy spherical shell. In general the former arrangement appears more natural than the latter, since the number of primary radial spines, which grow out from the primary lattice-sphere, determines their ground-form from the outset, whatever may be the number of secondarily added shells. Strictly speaking, according to the view adopted, these Liosphærida which have several shells, on the outer surface of which there are no radial spines, ought to be classified according to the number and arrangement of their internal radial connecting beams and distributed among the other families. The practical application of this correct principle meets, however, with great difficulties. Also in many cases the phylogenetic relations of the different #Sphæroidea# are more complicated than would appear from both these classificatory principles. In general their phylogeny will quite correspond with their ontogeny, since from the innermost first formed { cix}lattice-shell (primary medullary shell) a number of radial spines arises, and upon these the secondary shells are formed from within outwards.