The Tetrasphærida, or the Sphæroidea with four concentric shells, are in general not frequent, and not rich in different forms. In most of the observed species two inner shells are medullary, two outer cortical shells, the former within, the latter without, the central capsule; and the wall of the capsule, pierced by the connecting radial beams, lies between the two middle shells. But there are some Tetrasphærida in which all four shells seem to be external or cortical shells.
The Polysphærida, or the Sphæroidea with five or more concentric shells, seem of course to offer the greatest possibility for the development of very different forms; but in reality this group is the poorest and smallest of all; and only one part of it, the Arachnosphærida, is rather common. In this peculiar division the shell is composed of five to ten or more, very delicate, cobweb-like concentric shells, which are connected by radial beams; all are cortical shells, and lie outside the central capsule. Much more rare are those Polysphærida, in which both innermost shells, as true medullary shells, lie within the central capsule, all others being outside it. The total number of concentric shells in this group is commonly between five and ten, rarely more.
The Spongosphærida are distinguished from all other Sphæroidea by the spongy structure of the spherical shell, which is composed wholly or partially of an irregular spongy framework. The relation of this group to the other groups of Sphæroidea is probably rather complicated, for in some Spongosphærida the whole shell is composed of massive spongy reticulation, whilst in others it contains a spherical central cavity, and in a third group this cavity is filled up by one or two concentric lattice-shells, connected by radial beams. Many of these Spongosphærida are very common, and of considerable size.
The Collosphærida form a peculiar separate group of Sphæroidea, distinguished from all others by their social life or aggregation in colonies (cœnobia). They represent the only group of Sphærellaria in which this association of numerous individual capsules or cells is realised. The shell is almost constantly simple, without regularly disposed radial spines; therefore they may be called "social Monosphærida," or better "polyzoic Ethmosphærida." Only in one small group (Clathrosphærida) the shell, enveloping every central capsule, is double or surrounded by an external mantle; these may be compared to the Diplosphærida (or better to a part of the Carposphærida, Liosphæra, p. [76]). In most of the Collosphærida the lattice-shell is more or less irregular in form and structure.
The Lattice Work of the fenestrated shells is in the Sphæroidea of the greatest variability, and its innumerable modifications serve mainly for the distinction of species. In general we can distinguish as the most important modifications a regular network (with equal size, form, and distance of the pores or meshes) and an irregular network (with differences in the size, form, or distance of the meshes or pores). In both groups the pores may be either angular or round; so that there may exist together four different main forms of network—(A) regular lattice with equal hexagonal pores; (B) regular lattice with equal circular pores; (C) irregular lattice with unequal polygonal pores; (D) irregular lattice with unequal roundish pores. Besides these modifications, the pores may be prolonged into tubules which are directed radially towards the outside (rarely towards the inside) of the sphere. In other cases they are surrounded by elevated or honeycomb-like frames.
The Radial Spines exhibit in the Sphæroidea the greatest variety in form, size, disposition, &c., and their numerous modifications serve mainly for the distinction of genera, their peculiar formation and size also for the distinction of species. In general we may distinguish as the most important modifications primary and secondary spines. The primary spines or "main spines" are commonly direct outward prolongations of the internal radial beams, connecting the concentric shells. The secondary or "by-spines" arise only from the surface of the lattice-shell, without reference to the internal beams. The by-spines are commonly smaller, and much more numerous than the main spines. Regarding the form, the radial spines are either roundish (cylindrical or conical, often also club-shaped, rarely spindle-shaped) or angular (commonly three-sided, prismatic or pyramidal). The spines are constantly solid, never hollow; the "internal canals," described by some authors, are only microscopic views of the transparent edges. In many cases the spines are branched or forked. The most important difference in the variable shape of the spines is their regular or irregular number and disposition, which afford characters for the distinction of our five families.
The Three Dimensive Axes—or the three diameters of the sphere, perpendicular one to another—are in the great majority of the Sphæroidea significant in the promorphological consideration of the body, and are indicated either by the position of the external radial spines, or at least of the internal radial beams, connecting the concentric spheres. Commonly two radial spines are placed opposite in each axis. The most perfect group in this respect seems to be that of the Cubosphærida, in which the three axes are represented by three pairs of spines. Next come the Staurosphærida, in which two axes in cross-form are exhibited by two pairs of spines. The most simple group are the Stylosphærida, in which only one pair of spines is developed, indicating one single axis. These three families form together a continuous natural series,—the Sphæroidea with real dimensive axes,—and exhibit at the same time relations to the three other suborders of Sphærellaria, the Larcoidea, Discoidea, and Prunoidea respectively. At both ends of this series stand two other families, on one side the Liosphærida, without any radial spines on the surface of the sphere, on the other side the Astrosphærida, in which the radial spines are developed in great and variable numbers, at least eight to twelve, commonly twenty to forty, often more than a hundred or even a thousand.
The Liosphærida comprise all those Sphæroidea in which the surface of the shell is smooth, without radial spines (Pls. [12], [20]). The simplest of these are the Ethmosphærida, with one single lattice-shell, enveloping the spherical central capsule. Cenosphæra, the most simple form of the Ethmosphærida, may be regarded as the common ancestral form of all Sphæroidea, in an ontogenetical as well as in a phylogenetical and morphological sense. From this simple lattice sphere all other Sphæroidea can be derived either by radial or by tangential growth. If the radial beams, arising from the surface of the simple fenestrated sphere, become connected (at equal distances from the centre) by tangential beams, we get the compound shells of the "Liosphærida concentrica" (with two, three, four, or more concentric spheres). The radial beams connecting these exhibit in many Liosphærida the same regular disposition and number as the external radial spines in the Astrosphærida. Perhaps these forms in a "natural system" would be better united (e.g., Liosphærida with twelve or twenty internal radial beams, and Astrosphærida with twelve or twenty external radial spines); but in many cases (mainly for higher numbers) the certain determination of their number and disposition is very difficult or quite impossible.
The Cubosphærida (Pls. [21]-[25]) represent the large and very important family of Sphæroidea, in which all three dimensive axes are equally distinguished by pairs of spines, corresponding to three axes of a cube or of a regular octahedron, agreeing therefore also with the three axes of the cubic or regular crystalline system. In the majority of the Cubosphærida the six radial spines are accurately opposite each other in pairs in three axes, perpendicular one to another, and commonly they are of equal size and form; but in some genera the three pairs of spines become differentiated, whilst both spines of each pair remain equal. Either one pair is larger than the two others (which are equal), corresponding to the axes of the quadratic crystalline system; or all three pairs are different (corresponding to the three unequal axes of the rhombic crystalline system); the former nearer to the Discoidea, the latter to the Larcoidea. We may suppose with some probability, that the Cubosphærida are for the most part the common ancestral group of those Sphæroidea, in which a certain number of radial spines or beams is disposed in a regular order; the Staurosphærida may be derived from them by loss of one pair of spines, the Stylosphærida by loss of two pairs of spines, and most Astrosphærida by multiplying the radial spines, six to fourteen or more secondary spines being added to the six primary "dimensive spines." However, in many Astrosphærida (e.g., in those with eight spines, Centrocubus, Octodendron, &c.) the regular geometrical disposition of the radial spines seems to follow another mathematical order, quite independent of the Cubosphærida.
The Staurosphærida (Pl. [15]) are distinguished by the possession of four radial spines, opposite in pairs in two axes, perpendicular one to another. This rectangular cross determines a certain plane, the "equatorial plane," and this brings the Staurosphærida near to the Discoidea, mainly to those which also bear on the periphery of the circular equatorial plane four crossed spines (such as Staurodisculus, Stethostaurus, Staurodictya, &c.). But in these cruciform Discoidea the shell and the enclosed central capsule are discoidal or lenticular, whilst in the Staurosphærida they remain spherical. Commonly the cross is quite regular, with four right angles and four equal beams; but often also it becomes more or less irregular. In some genera one pair of equal opposite spines is larger than the other pair. These forms represent the three different axes of the rhombic crystal system, whilst the common regular Staurosphærida represent those of the quadratic crystal system. The latter can be derived from the Cubosphærida (representing the regular crystal system) by reduction of one axis and loss of its pair of spines. In general the number of species (and particularly of the individuals) is much smaller in the Staurosphærida than in all other families of Sphæroidea.