The three dimensive axes, which determine the typical form of Larcoidea, are commonly differentiated in such a way, that the first, the longitudinal or principal axis, is the longest; both its poles, oral and aboral (or anterior and posterior) are equal. The second, the lateral or transverse axis, is commonly less than the first, greater than the third axis; both its poles are the equal lateral poles (right and left not differentiated). The third dimensive axis, the equatorial or sagittal axis, is commonly the shortest; both its equal poles are the sagittal poles (dorsal and ventral poles not different). The relative size of the three dimensive axes in the human body exhibits similar relations.
The three dimensive planes of the Larcoidea, the sagittal, lateral, and transverse planes, are elliptical, all three of different sizes. The first plane, the median or sagittal plane, is commonly as regards size between the two others; its major axis is the principal, its minor the sagittal axis; it separates the right half of the body from the left. The second plane or lateral plane, is commonly larger than the two others; its major axis the principal, its minor the transverse axis; it separates the dorsal half of the body from the ventral. The third plane, the equatorial or zonal plane, is commonly less than the two others; its major axis the lateral, its minor the sagittal axis; it separates the two principal halves of the body, the oral and aboral halves.
In my Monograph (1862) only very few forms of Larcoidea are described, Tetrapyle and Lithelius (the latter representing a peculiar family, Lithelida). In my Prodromus (1881, pp. 463, 464) I disposed all observed forms of Larcoidea in two different families, the Pylonida and Lithelida. The rich materials of the Challenger collection have since offered an astonishing number of new and interesting forms of this section, so that I can enumerate here fifty-one genera and two hundred and sixty-five species. I dispose them here in four subsections and nine families. Three of these have regular lentelliptical shells, which are not articulate, and without annular constrictions (Larcarida, Larnacida, Pylonida); these form the subsection Pylolarcida. Two other families (Tholonida and Zonarida) are distinguished by annular constrictions, which divide the regular lentelliptical shell into a number of dome-shaped chambers or cupolas; we call these Thololarcida. A third group, Spirolarcida, comprises the Larcoidea with spiral growth; the two families of Lithelida and Streblemida. Finally a fourth group the Sorolarcida is formed by the Larcoidea with irregular shells, also two families, the Phorticida and Soreumida.
The first family of Larcoidea, the Larcarida, contains the most simple forms, beginning with Cenolarcus, a quite simple lentelliptical latticed shell. In Coccolarcus we find already two concentric shells, connected by radial beams, an inner medullary and an outer cortical shell. In Spongolarcus the lentelliptical shell becomes spongy.
The second family, Larnacida, is very similar to the Larcarida, and seems to diverge only by the different mode of connection between the two concentric lentelliptical shells. But in truth this slight difference is of great morphological importance, as it depends on a quite different and peculiar mode of growth. In the foregoing Larcarida (Coccolarcus, &c.), the concentric shells originate in the same manner as in the concentric Prunoidea and Sphæroidea, by radial beams, which arise from the surface of the inner (medullary) shell and become connected by a network to form the outer (cortical) shell. Here, in the Larnacida, a quite similar shell originates in a quite different way, first arrived at in the Pylonida (Trizonium). Both concentric shells become here connected by peculiar lattice girdles, which are developed in the perimeter of the three elliptical dimensive planes. Firstly, on both sides of a simple, spherical, or lentelliptical central chamber, arise two lateral wings (on the poles of the transverse axis), and build around the former a transverse girdle. This is crossed by a larger lateral girdle, the minor axis of which is the major of the former, and perpendicular to both girdles is yet developed a third, the sagittal girdle. If the open fissures or "gates" between these three girdles become closed by network, we obtain Larnacilla, the probable ancestral form of all Larnacida.
Whilst in Larnacilla and Larnacidium this typical trizonal lentelliptical shell constitutes by itself alone the whole skeleton, in the other Larnacida it becomes overgrown by outer envelops, and so becomes enclosed in the interior of the central capsule as a "Larnacilla-shaped medullary shell." If the enclosing external envelops be simply latticed, we get the subfamily Larnacalpida; if they be spongy, we get the Larnacospongida.
The third family, Pylonida, is the most important of all Larcoidea, as not only the largest and most interesting number of species belong to it, but also many other genera (far the greater part of all Larcoidea) may be derived from it. The peculiar character of the Pylonida is determined by the imperfect fenestration of the lentelliptical shell growing in the three dimensive axes in a quite different manner. Each elliptical dimensive plane becomes circumscribed by an elliptical latticed girdle (or fenestrated ring), and between these three girdles (perpendicular one to another) remain wide open fissures of the shell or "gates" (Pylæ). The beginning of the shell-building is the same as in Larnacilla, the most simple form of Larnacida. From a quite simple medullary shell, a spherical, subspherical, ellipsoidal, or lentelliptical central chamber, arise two latticed wings, opposite on the poles of the transverse axis (Monozonium). Both wings are short and wide hollow fenestrated tubes, the axes of which are parallel to the principal axis. Therefore they form together with the central chamber an elliptical transverse girdle. This first girdle becomes crossed by a second lateral girdle; from both poles of the transverse axis arise latticed wings, which unite on the poles of the principal axis, therefore the minor axis of this second larger ring is the major axis of the first smaller ring (Dizonium). Between the two crossed rings remain four wide open gates. Now follows the development of a third sagittal girdle, arising from both poles of the principal axis and overgrowing the four gates. But as this third girdle is larger than the second, four other larger gates arise between the two (in planes perpendicular to the former four gates). Now we have the characteristic and most important trizonal shell (Trizonium), composed of three elliptical lattice-girdles of different size, perpendicular one to another, and enclosing a simple central chamber. If the four gates of this Trizonium become closed by lattice-work, it passes over into Larnacilla, the most important ancestral form of the Larnacida.
This most significant "trizonal shell," either incompletely latticed in Trizonium (with four open gates), or completely latticed by fenestration of the four gates, in Larnacilla, is to be found in far the greater part of all Larcoidea, representing the medullary shell, which is overgrown by an outer cortical shell. In many Larcoidea, in which this "Larnacilla-shell" is absent, it is perhaps lost by phylogenetic reduction, or retrograde metamorphosis.
The same process of triple girdle-building, by which the typical Trizonium-shell or Larnacilla-shell is produced (Haplozonaria), is repeated once or twice in the larger forms of Pylonida. The first system of three girdles (perpendicular one to another) becomes overgrown by a second system of the same formulation in the Diplozonaria, and this becomes overgrown by a third system in the Triplozonaria; in the highest genus of this group, Pylozonium, we find not less than nine girdles (three systems, each of three girdles). Till now only one genus of the whole polymorphous family was well known, Tetrapyle (with five girdles, three of the medullary, two of the cortical shell). If the gates between the girdles remain open, all these forms must be regarded as Pylonida; if the gates afterwards become closed by a network, they pass over into other families.
The fourth family of the Larcoidea is the Tholonida, distinguished by the polythalamous shell being composed of a certain number of roundish or hemispherical chambers (domes or cupolas), which surround a primordial central chamber in quite regular disposition, lying opposite in pairs on the poles of the three dimensive axes. If we imagine that each "wing" (or open half-girdle) of the Pylonida becomes closed by a lattice-work, and so transformed into a hemispherical or roundish cupola, we obtain the characteristic shell of the Tholonida. Indeed every girdle of the former corresponds to a pair of opposite domes of the latter. The axis of each pair of domes is one of the three dimensive axes.