The family Liosphærida comprises all those solitary Sphæroidea in which the surface of the spherical shell is not armed with radial spines. Nearly the half of this large group is formed by the Ethmosphærida, in which the carapace is a quite simple, spherical lattice-shell; this subfamily is probably the common ancestral group from which all other Sphæroidea, or even all Sphærellaria, can be derived in a phylogenetical as well as in a morphological sense. The central capsule in this first subfamily is constantly enclosed by the fenestrated shell, and separated from it by the jelly-veil. The shell is therefore an extracapsular or medullary shell.
To these simple Ethmosphærida all other subfamilies can be opposed as "Liosphærida concentrica," as their carapace is composed of two or more concentric lattice-shells; two in the Carposphærida, three in the Thecosphærida, four in the Cromyosphærida, five or more in the Caryosphærida. In all these four subfamilies the concentric shells are simple (not spongy) fenestrated shells. In a sixth subfamily, in the Plegmosphærida, the shell is wholly or partially composed of spongy wicker-work, with or without a latticed medullary shell in the centre.
The internal radial beams, in the "Liosphærida composita" connecting the concentric spheres, exhibit in their number and disposition similar important differences, such as the external radial spines in the Astrosphærida. The following eight different cases of regular disposition were observed:—(A) two opposite beams in one axis; (B) four beams, opposite in pairs in two axes perpendicular one to another; (C) six beams, opposite in pairs in the three dimensive axes; (D) eight beams, opposite in pairs in the four diagonals of the regular cube; (E) twelve beams corresponding to the twelve axes of the regular icosahedron; (F) fourteen beams quite regularly disposed (six corresponding to the three axes of the regular octahedron, eight to the central points of its faces); (G) twenty beams (probably corresponding to the twenty corners of a regular dodecahedron); (H) thirty-two beams, regularly disposed. Rarely the number of the radial beams is intermediate between these eight cases, and rarely it is higher; then commonly the disposition is irregular. The regularity of their disposition in the great majority of cases is very remarkable and evident.
Synopsis of the Genera of Liosphærida.
I. Subfamily Ethmosphærida. (Shell one single latticed sphere.) |  | Pores of the shell simple, not prolonged into free tubuli. |  | Shell cavity simple, | 15. Cenosphæra. |
| Shell cavity with radial beams united in the centre, | 16. Stigmosphæra. | ||||
| Pores prolonged into free conical or cylindrical tubuli. |  | Tubuli external, centrifugal, | 17. Ethmosphæra. | ||
| Tubuli internal, centripetal, | 18. Sethosphæra. | ||||
II. Subfamily Carposphærida. (Two concentric spheres.) |  | One shell medullary (intracapsular), the other cortical (extracapsular), | 19. Carposphæra. | ||
| Both shells cortical (near together), | 20. Liosphæra. | ||||
III. Subfamily Thecosphærida. (Three concentric spheres.) | | Two shells medullary (intracapsular), one shell cortical (extracapsular), | 21. Thecosphæra. | ||
| One shell medullary (intracapsular), two shells cortical (extracapsular), | 22. Rhodosphæra. | ||||
IV. Subfamily Cromyosphærida. (Four concentric spheres.) |  | Two inner medullary shells (intracapsular), and two outer cortical shells (extracapsular), | 23. Cromyosphæra. | ||
V. Subfamily Caryosphærida. (Five or more concentric spheres.) |  | Two inner medullary shells, and three or more outer cortical shells, | 24. Caryosphæra. | ||
VI. Subfamily Plegmosphærida. (Spherical shell wholly or partially of spongy structure.) |  | Spongy sphere without latticed medullary shell in the centre. | | Sphere solid, | 25. Styptosphæra. |
| Sphere with a central cavity, | 26. Plegmosphæra. | ||||
| Spongy sphere with one or two latticed medullary shells in the centre. | | One single medullary shell, | 27. Spongoplegma. | ||
| Two concentric medullary shells, | 28. Spongodictyon. | ||||
| I. Subfamily Ethmosphærida. (Shell one single latticed sphere.) | ||||||
| Pores of the shell simple, not prolonged into free tubuli. | ||||||
| Shell cavity simple, | ||||||
| 15. Cenosphæra. | ||||||
| Shell cavity with radial beams united in the centre, | ||||||
| 16. Stigmosphæra. | ||||||
| Pores prolonged into free conical or cylindrical tubuli. | ||||||
| Tubuli external, centrifugal, | ||||||
| 17. Ethmosphæra. | ||||||
| Tubuli internal, centripetal, | ||||||
| 18. Sethosphæra. | ||||||
| II. Subfamily Carposphærida. (Two concentric spheres.) | ||||||
| One shell medullary (intracapsular), the other cortical (extracapsular), | ||||||
| 19. Carposphæra. | ||||||
| Both shells cortical (near together), | ||||||
| 20. Liosphæra. | ||||||
| III. Subfamily Thecosphærida. (Three concentric spheres.) | ||||||
| Two shells medullary (intracapsular), one shell cortical (extracapsular), | ||||||
| 21. Thecosphæra. | ||||||
| One shell medullary (intracapsular), two shells cortical (extracapsular), | ||||||
| 22. Rhodosphæra. | ||||||
| IV. Subfamily Cromyosphærida. (Four concentric spheres.) | ||||||
| Two inner medullary shells (intracapsular), and two outer cortical shells (extracapsular), | ||||||
| 23. Cromyosphæra. | ||||||
| V. Subfamily Caryosphærida. (Five or more concentric spheres.) | ||||||
| Two inner medullary shells, and three or more outer cortical shells, | ||||||
| 24. Caryosphæra. | ||||||
| VI. Subfamily Plegmosphærida. (Spherical shell wholly or partially of spongy structure.) | ||||||
| Spongy sphere without latticed medullary shell in the centre. | ||||||
| Sphere solid, | ||||||
| 25. Styptosphæra. | ||||||
| Sphere with a central cavity, | ||||||
| 26. Plegmosphæra. | ||||||
| Spongy sphere with one or two latticed medullary shells in the centre. | ||||||
| One single medullary shell, | ||||||
| 27. Spongoplegma. | ||||||
| Two concentric medullary shells, | ||||||
| 28. Spongodictyon. | ||||||
Subfamily Ethmosphærida,[[25]] Haeckel, 1862, Monogr. d. Radiol., p. 348 (sensu restricto).
Definition.—Liosphærida with one single spherical lattice-shell; living solitary, not aggregated in colonies.
Genus 15. Cenosphæra,[[26]] Ehrenberg, 1854, Monatsber. d. k. preuss. Akad. d. Wiss. Berlin, p. 237.
Definition.—Liosphærida with one single latticed sphere, with simple shell-pores (not prolonged into free tubuli) and with simple shell-cavity (without internal radial beams).