The thorax, or the second shell-joint of the Cyrtoidea, is in the majority homologous with the thorax of the Phormospyrida and Androspyrida, and therefore developed by apophyses, which arise from the base of the cephalis and become united by transverse branches forming a lattice-plate. Its size is generally in inverse proportion to that of the cephalis. The more the cephalis becomes reduced, the more the thorax is developed. Its form is very variable, usually three-sided pyramidal or prismatic in the triradiate, polyhedral in the multiradiate, and conical or cylindrical in the eradiate Cyrtoidea. Its terminal mouth is either a simple wide opening, or closed by a lattice-plate. In the majority of Cyrtoidea the thorax is separated from the cephalis not only by the external collar constriction, but also by the internal cortinar septum, a horizontal lattice-plate which exhibits the typical basal pores of the Semantida (usually two smaller jugular and two larger cardinal pores). But this septum is often reduced or perfectly lost, and then the external collar constriction alone indicates the separation of the cephalis and the thorax.
The abdomen, or the third shell-joint of the Cyrtoidea, absent in the Monocyrtida and Dicyrtida (as also in all Spyroidea), occurs constantly in all Tricyrtida and Stichocyrtida. It is a simple large chamber in the Tricyrtida, but forms an annulated body, composed of a variable number of successive joints, in the Stichocyrtida. The constrictions between these joints, and also the lumbar constriction, between abdomen and thorax, are usually provided with a lattice-girdle, projecting into the cavity of the shell, like a diaphragm. Usually this horizontal girdle bears only a single circle of pores, rarely two or more. In many Cyrtoidea it is replaced by a solid horizontal ring of silex, and often it is wanting. It originates by the insertion of the following shell-joint, which takes place not on the terminal mouth of the preceding joint, but somewhat above it.
The annular joints of the Stichocyrtida succeeding the third joint, and very variable in number, may be regarded either as a series of new postabdominal chambers, succeeding the true abdomen, or as secondary joints of the annulated abdomen itself. The latter view may be sustained by the fact that these joints are usually of an indifferent shape, and do not possess the characteristic features which we find in the first three joints, the abdomen, the thorax and the cephalis.
The lattice-work of the shell exhibits in the Cyrtoidea an extraordinary variety, similar to that of the Sphæroidea; it serves in the first place for the distinction of species. The three first joints of the shell are often distinguished by the different character of the lattice-work. The cephalis has usually very small and simple pores. The lattice-work of the thorax is often characterised by radial structures. The pores of the abdomen are usually very numerous and regular. The numerous joints in the annulated abdomen of the Stichocyrtida commonly exhibit little variety.
The closure of the mouth, effected by a convex or horizontal terminal lattice-plate, has a different signification in the Monocyrtida and in the jointed Cyrtoidea. In the Monocyrtida clausa this closing plate is the original cortinar plate or the basal plate of the cephalis. In the jointed Cyrtoidea, however, the lattice-plate which closes the terminal mouth of the thorax or of the abdomen (of the last annular joint in the Stichocyrtida), is produced by central union of the convergent edges, which grow centripetally from the margin of the mouth of the last joint towards its centre.
The radial apophyses arising from the shell of the Cyrtoidea may probably be always derived from that tripodal structure which is found in all Plectoidea, in Cortina and Cortiniscus among the Stephoidea, and in the majority of Spyroidea. Therefore the prototype of this radial structure would be Plagoniscus and Cortina, with four radial spines united in a common point, the cortinar centrum; an ascending apical horn and three descending basal feet. The odd posterior or caudal foot is usually similar in shape to the two paired anterior or pectoral feet, but may be distinguished from these latter by its relation to the apical horn. Very frequently an internal vertical free columella arises in the cephalis, or instead of it an ascending rib in the dorsal wall of the cephalis, which connects the base of the apical horn with the origin of the caudal foot. This is probably the remaining part of the sagittal ring. More rarely also a part of the ventral rod of the latter is preserved, or on the anterior pole of the basal rod of the cephalis an ascending procolumella arises which is inserted on the frontal face of the cephalis, and sometimes prolonged into a nasal horn (the rod, C, of Bütschli). These two odd horns, the posterior apical horn and the anterior nasal horn, are usually different and divergent. In some genera a variable number of accessory radial horns is developed on the convex face of the cephalis. In many hornless genera the free apical horn is lost, but not unfrequently the columella is preserved which connects the caudal foot with that point of the cephalis, in which formerly the apical horn was inserted.
The three primary radial beams, corresponding to the three basal feet of Plectaniscus and Cortina, exhibit in the Cyrtoidea the greatest variety in form and size, and chiefly in their relation to the shell, the latter serving mainly for the distinction of genera. Originally these three cortinar beams arise from the basal plate of the cephalis, the odd caudal foot appearing as a prolongation of the basal rod of that plate, and the paired pectoral feet as prolongations of its coracal rods (between the jugular and cardinal pores). The lattice-work of the thorax is developed usually between the three cortinar feet, more rarely inside or outside of them. Therefore the three beams appear commonly as three divergent ribs in the wall of the thorax, and continue over its basal mouth as three free terminal feet. With the increasing length of the shell and the number of its joints the three radial ribs are also prolonged, and their free distal ends may be prominent at very different points, either as three lateral wings or as three terminal feet. These are either solid spines or lattice-plates, sometimes more or less ramified.
The three radial apophyses are prevalent in the majority of the Cyrtoidea, which we call "Pilocyrtida" (or Cyrtoidea triradiata). Their number increases in the Astrocyrtida (or Cyrtoidea multiradiata). The most frequent cases of multiplication are here caused by the development of six or nine radial apophyses; these may be enclosed ribs, or lateral wings, or terminal feet. In the sexradial Cyrtoidea there are three secondary or interradial apophyses interpolated between the three primary or perradial; in the nine-radial Cyrtoidea, however, there are six adradial apophyses interpolated.
A third and last great group is formed by the Corocyrtida or Cyrtoidea eradiata. These exhibit no radial apophyses, neither enclosed ribs, nor free lateral wings, nor terminal feet. But in a great number of them internal traces of an original triradiate structure are visible, mainly in the cortinar septum between cephalis and thorax; this often exhibits three or four, and sometimes six cortinar or collar pores, of the same typical shape as in the triradial Spyroidea. Sometimes even an internal columella with three radial branches is preserved, as in Axocorys. It is therefore very probable that a great part of these Cyrtoidea eradiata (if not all) may be derived from triradiate or multiradiate ancestral forms, by reduction and loss of the radial apophyses. In another part of this group, mainly in the Monocyrtida eradiata (Cyrtocalpida) it is possible, or even probable, that their eradiate shell has originated independently from Nassellida, and that they have no true relation to radial Cyrtoidea.
The Central Capsule of the Cyrtoidea, first observed by J. Müller (1858), and more fully described in my Monograph (1862), was very accurately examined by Richard Hertwig (1879). His observations were confirmed by numerous new forms, which I was able to examine in well-preserved preparations of the Challenger. The central capsule, according to these, exhibits the same typical shape, which is characteristic of all Monopylea (with porochora and podoconus), and may be derived with the latter from the common ancestral forms, Cystidium and Nassella (= the skeletonless Nassellida). In the majority, however, of Cyrtoidea, the capsule develops on its basal face a number of depending lobes, as were also found in some Spyroidea (and probably also Botryodea). In this respect we may distinguish two main forms of the capsule in the Cyrtellaria, viz., the primary simple, not lobate form, and the secondary lobate form. The central capsule is originally always enclosed in the cephalis, and has there a simple, subspherical, ellipsoidal or ovate form. As soon as their growth increases, and the enclosing cephalis becomes too narrow, it sends out prolongations in the form of basal lobes, which depend from its base, and proceed through the pores of the basal lattice of the cephalis, or the cortinar pores. In the great majority of Cyrtoidea in which the capsule was observed, either three or four such lobes were seen (already described by J. Müller). Of course this number depends upon the number of cortinar pores, which is either three or four; therefore in the Cyrtoidea with three pores in the cortinar plate, we find three lobes of the central capsule (an odd posterior and two paired anterior); in the Cyrtoidea, however, with four pores in the cortinar plate (the majority) we find four lobes of the central capsule (two smaller anterior jugular and two larger posterior cardinal lobes). Usually each lobe is ovate or pear-shaped and encloses a large oil-globule, and often also an apophysis of the cell-nucleus.