Regarding the general system of ground-forms compare my Generelle Morphologie (1866, Bd. i. pp. 375-552; Bd. iv., Allgemeine Grundformenlehre). The ground-forms there proposed and systematically defined have, however, found but little acceptance (chiefly, no doubt, owing to the difficult and complicated nomenclature); but having now, twenty years after their publication, anew carefully revised and critically studied them, I can find no sufficient reason for abandoning the principles there adopted. On the contrary the study of the Challenger Radiolaria during the last ten years, with its incomparable wealth of forms, has only confirmed the accuracy of my system of ground-forms. The customary treatment of these in zoological and botanical handbooks (such as those of Claus and Sachs) is quite insufficient.
18. The Principal Groups of Geometrical Ground-Forms.—The great variety of the geometrical ground-forms which are actually realised in the variously shaped bodies of the Radiolaria, renders it desirable to classify these in as small a number as possible of principal groups and a larger number of subdivisions. As extensive principal groups four at least must be distinguished; the Centrostigma or Sphærotypic, the Centraxonia or Grammotypic, the Centroplana or Zygotypic, and the Acentrica or Atypic. The natural centre of the body, about which all its parts are regularly arranged, is in the first group a point (stigma), in the second a straight line (principal axis), in the third a plane (sagittal plane), in the fourth a centre is of course wanting.
19. The Centrostigma or Sphærotypic Ground-Forms.—The first group of geometrical ground-forms, here distinguished as sphærotypic or the centrostigma, is undoubtedly the most important among the Radiolaria, inasmuch as if these be considered monophyletic, it must be the original one from which all the other ground-forms have been derived. The common character of all these sphærotypic ground-forms is that their natural centre is a point (stigma); thus there is no single principal axis (or protaxon) such as is characteristic of the two following groups. The sphærotypic ground-forms are subdivided into two important smaller groups, the spheres (Homaxonia) and the endospherical polyhedra (Polyaxonia). The spherical ground-forms, fully developed in the central capsule and calymma of Actissa and the Sphæroidea as well as in many Acantharia, present no different axes; all possible axes passing through the centre of the body are equal (Homaxonia). In the endospherical polyhedra, on the contrary, numerous axes (three at least) may be distinguished, which are precisely equal to each other and different from all the remaining axes (Polyaxonia). If the extremities of these axes, or the poles, which are all equidistant from the common centre, be united by straight lines, a polyhedral figure is produced whose angles all lie in the surface of the sphere. According as the poles of the axes are at equal, subequal, or at different distances from each other, we may divide the endospherical polyhedra into regular, subregular and irregular. (See Gener. Morphol., Bd. i. pp. 404-416.)
20. The Centraxonia or Grammotypic Ground-Forms.—The second principal group of organic ground-forms, here called grammotypic or centraxonia, is characterised by the fact that a straight line (gramma) or a single principal axis (protaxon) forms the natural centre of the body. This important and extensive group is divided into two subgroups, those with one axis (Monaxonia) and those with crossed axes (Stauraxonia); in the latter different secondary transverse or cross-axes may be distinguished, but not in the former. In the Monaxonia, therefore, every transverse section of the body perpendicular to the principal axis is a circle, in the Stauraxonia, on the contrary, a polygon. The Monaxonia are further subdivided into two groups, in one of which the two poles of the principal axis are equal and similar (Isopolar), in the other of which they are different (Allopolar); in the former the two halves of the body, which are separated by the equatorial plane (or the largest transverse plane, perpendicular to the principal axis), are equal, in the latter unequal. Among the isopolar uniaxial ground-forms (Monaxonia isopola) may be mentioned the ellipsoidal, spheroidal, lenticular, &c.; to the allopolar uniaxial forms (Monaxonia allopola) belong the conical, hemispherical, ovoid, &c. In the same way the pyramidal ground-forms with crossed axes are divisible into two groups, according as the two poles of the principal axis are equal or not. The ground-form of the former is the double pyramid, that of the latter the single pyramid. Both the double and the single pyramids may again be subdivided, each into two important lesser groups, the regular and the amphithect. In the first division the equatorial plane of the double and the basal plane of the single pyramid is a regular polygon (square, &c.), whilst in the other division it is an elongated or amphithect polygon (rhombus, &c.); the crossed axes are equal in the former, unequal in the latter. (See Gener. Morphol., Bd. i. pp. 416-494.)
21. The Centroplana or Zygotypic Ground-Forms.—The third principal group of ground-forms includes those which are bilaterally symmetrical in the ordinary sense, or zeugitic or zygotypic; the natural centre of their body is a plane. These forms are the only ones in which the distinction between right and left is possible, since their body is divided by the median plane (planum sagittale) into two symmetrical halves (right and left). In all these zeugites the position of every part is determined by three axes perpendicular to each other, and of these three dimensive axes two are allopolar, one is isopolar. The two unlike poles of the principal (or longitudinal) axis are the oral and aboral, the two unlike poles of the sagittal (or vertical) axis are the dorsal and ventral; the two similar poles of the frontal (or transverse) axis, however, are the right and left. This important group of zeugitic or bilateral forms may also be divided into two clearly distinct lesser groups, the Amphipleura and the Zygopleura. In the Amphipleura (or bilaterally radial ground-forms) the "radial two-sided" body is produced by modification of a regular pyramid (as Spatangus from Echinus), and hence is composed of several (not less than three) antimeres. In the Zygopleura (or bilaterally symmetrical ground-forms) on the other hand, the bodies consist of two antimeres (as in all the higher animals, Vertebrata, Arthropoda, &c.). (See Gener. Morphol., Bd. i. pp. 495-527.)
22. The Acentrica or Atypic Ground-Forms.—Among the acentrica or anaxonia are included all those ground-forms which are absolutely irregular, and in which neither a definite centre nor constant axes can be distinguished (e.g., most Sponges). These quite irregular ground-forms are very rare among the Radiolaria, but nevertheless there may be referred to them the amœboid central capsule of some Colloidea (Collodastrum, p. [27], Pl. [3], figs. 4, 5) among the Spumellaria, the irregular shells of many Collosphærida (Pl. [8], fig. 2), and the absolutely irregular shells of the Phorticida and Soreumida among the Larcoidea. (See Gener. Morphol., Bd. i. p. 400.)
23. The Subsidiary Groups of Geometrical Ground-Forms.—The four natural principal groups of ground-forms, which have just been defined according to the nature of the centre of their bodies, may be divided again into numerous subsidiary groups, defined by the relations of the constant axes and the two poles of each axis, as well as by the number of the axes and the differentiation of the secondary with respect to the principal axis. The most important of these subsidiary groups into which the principal ones are immediately divided are the following:—(1) The Centrostigma (or sphærotypic) are divided into spheres (Homaxonia) and endospherical polyhedra (Polyaxonia). (2) The Centraxonia (or grammotypic) into uniaxial (Monaxonia) and those with crossed axes (Stauraxonia); among the former of these may be distinguished the isopolar (phacotypic) and the allopolar (conotypic); among the latter the double and single pyramids. (3) The Centroplana (or bilaterals) are divided into amphipleura (or bilaterally radial) and zygopleura (or bilaterally symmetrical). (4) The Acentrica (or Anaxonia) or absolutely irregular ground-forms, present no special subdivisions.
For a complete system of the geometrical ground-forms and their relation to promorphological classification, see Gener. Morphol., Bd. i. pp. 555-558.
24. The Spherical or Homaxon Ground-Form.—The spherical is the only absolutely regular ground-form, since only in it are all axes which pass through the centre equal; it is very often realised among the Radiolaria, especially in the Spumellaria and Acantharia, where it furnishes the common original ground-form, but it is often to be seen in the shells of many Phæodaria (in most Phæosphæria); on the other hand, it is never found among the Nassellaria. Geometrical spheres, in the strict sense of the term, are only to be found among the Spumellaria and Acantharia, namely, in the central capsule of many Collodaria (Pls. [1], [2]) and all Sphæroidea (Pls. [11]-[30]) as well as many Acanthometra and Acanthophracta (Pls. [128]-[138]). Nevertheless, speaking generally, one includes those central capsules and skeletons which have been distinguished here as endospherical polyhedra. (On these ground-forms see Gener. Morphol., Bd. i. pp. 404-406.)
25. The Endospherical Polyhedral Ground-Form.—The endospherical polyhedron or polyaxon ground-form naturally follows the spherical or homaxon. Under it are included all polyhedra whose angles fall in the surface of a sphere; this ground-form is especially common among the Spumellaria, especially in the shells of Sphæroidea, but is also found among the Acantharia (especially in the Astrolophida and Sphærophracta), as well as among the Phæosphæria (in most genera of the Orosphærida, Sagosphærida, and Aulosphærida). Strictly speaking, all those lattice-shells which have been incorrectly called "spherical" belong to this category, for they are none of them true spheres in the geometrical sense (like the central capsules of the Sphæroidea), but rather endospherical polyhedra, whose angles are indicated by the nodal points of the lattice shell, or the radial spines which spring from them. These endospherical polyhedra may be divided into three groups, the regular, subregular, and irregular. Of regular polyhedra, properly so-called, it may be shown geometrically that only five can exist, namely, the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron. All these are actually manifested among the Radiolaria, although but seldom. Much more common are the subregular endospherical polyhedra, e.g., spherical lattice-shells with regular hexagonal meshes of equal size; they are never exactly equal nor perfectly regular, but the divergences are so insignificant that they escape superficial observation (Pl. [20], figs. 3, 4; Pl. [26], figs. 1-3). On the contrary in the irregular endospherical polyhedra the meshes of the lattice-sphere are more or less different in size and often in form also (Pl. [28], figs. 4, 8; Pl. [30], figs. 4, 6). The five truly regular polyhedra require separate notice on account of their importance. (See Gener. Morphol., Bd. i. p. 406.)