B. The radial structure, indicated by radial apophyses arising from the shell, offers three principal differences, according to which the whole group of Cyrtoidea may be divided into three large groups or sections, viz., (1) Pilocyrtida, or Cyrtoidea triradiata, with three radial apophyses; (2) Astrocyrtida, or Cyrtoidea multiradiata, with numerous radial apophyses (four to nine or more); and (3) Corocyrtida, or Cyrtoidea eradiata, without external radial apophyses. The majority of Cyrtoidea are Pilocyrtida, with three radial apophyses, which are probably homologous to the three primary feet of the Plectoidea and of Cortina (therefore "cortinar feet"). The Astrocyrtida, or the Cyrtoidea with a variable number of radial apophyses (at least four to six) may be derived from the Pilocyrtida by interpolation of secondary or interradial apophyses between the three primary or perradial apophyses. The Corocyrtida, however, or the Cyrtoidea without external radial apophyses, may have originated by reduction and loss of the latter, either from the Pilocyrtida or from the Astrocyrtida.
C. The shape of the basal mouth in the Cyrtoidea exhibits two essential differences only, viz., (1) the terminal mouth of the shell is a simple wide opening in the Stomocyrtida, or (2) the terminal mouth is closed by a lattice-plate, in the Clistocyrtida. As these two different cases occur in all the twelve families, which we have distinguished according to the differences in the number of joints and in the radial structure, we get altogether twenty-four subfamilies which are synoptically arranged in the following table:—
| Synopsis of the four sections, twelve families and twenty-four subfamilies of CYRTOIDEA. | PILOCYRTIDA. Cyrtoidea triradiata. (Three radial apophyses.) |
ASTROCYRTIDA. Cyrtoidea multiradiata. (Four to nine or more apophyses.) |
COROCYRTIDA. Cyrtoidea eradiata. (No radial apophyses.) | |||
|---|---|---|---|---|---|---|
| Mouth of the shell. | Aperta. | Clausa. | Aperta. | Clausa. | Aperta. | Clausa. |
| MONOCYRTIDA. (Cyrtoidea monothalamia). |
Tripocalpida.    
|
Phænocalpida.
|
Cyrtocalpida.
| |||
| Archipilida. | Archiperida. | Archiphormida. | Archiphænida. | Archicorida. | Archicapsida. | |
| DICYRTIDA. (Cyrtoidea dithalamia). |
Tripocyrtida.
|
Anthocyrtida.
|
Sethocyrtida.
| |||
| Sethopilida. | Sethoperida. | Sethophormida. | Sethophænida. | Sethocorida. | Sethocapsida. | |
| TRICYRTIDA. (Cyrtoidea trithalamia). |
Podocyrtida.
|
Phormocyrtida.
|
Theocyrtida.
| |||
| Theopilida. | Theoperida. | Theophormida. | Theophænida. | Theocorida. | Theocapsida. | |
| STICHOCYRTIDA. (Cyrtoidea polythalamia). |
Podocampida.
|
Phormocampida.
|
Lithocampida.
| |||
| Stichopilida. | Stichoperida. | Stichophormida. | Stichophænida. | Stichocorida. | Stichocapsida. | |
