1. Definition of the Radiolaria.—RADIOLARIA are marine Rhizopoda, whose unicellular body always consists of two main portions, separated by a membrane; an inner Central capsule (with one or more nuclei) and an Extracapsulum (the external calymma, which has no nucleus, and the pseudopodia); the endoplasm of the former and the exoplasm of the latter are connected by openings in the capsule-membrane, The central capsule is partly the general central organ of the Radiolarian cell, partly the special organ of reproduction, since its intracapsular protoplasm, along with the nuclei embedded in it, serves for the formation of flagellate spores. The extracapsulum is partly the general organ for intercourse with the outer world (by means of the pseudopodia), partly the special organ of protection (calymma) and nutrition (sarcomatrix). The majority of Radiolaria develop also a skeleton for support and protection, which presents the utmost variety of form, and is generally composed of silica, sometimes of an organic substance (acanthin). The Radiolarian cell usually leads an isolated existence (Monozoa vel Monocyttaria); only in a small minority (of one legion) are the unicellular organisms united in colonies or coenobia (Polyzoa vel Polycyttaria).

The extent of the Radiolaria, as limited by the above definition, which I have made as compact as possible, differs in several important respects from that allowed to the group by all previous diagnoses. The shortest expression of its scope might perhaps be :—Rhizopoda with central capsule and calymma; for the most important character of the Radiolaria, and that by which they are distinguished from all other Rhizopoda, is the differentiation of the unicellular body into two principal parts of equal importance and their separation by a constant capsule-membrane.

2. The Two Subclasses of the Radiolaria.—The systematic catalogue of the Radiolaria, which forms the second part of this Report, and is brought up to the year 1884, contains 20 orders, 85 families, 739 genera, and 4318 species. The consideration that but a small proportion of the ocean has yet been investigated renders it likely, however, that even this large number does not include the half of the recent species. The great progress which our knowledge of the organisation of the Radiolaria has made, by means of comparative study, renders it possible to arrange this enormous mass of forms in four main divisions or legions, and these are again related in pairs, so that two divisions of the highest rank or subclasses are constituted, the Porulosa (or Holotrypasta) and Osculosa (or Merotrypasta).

The division of the Radiolaria into two subclasses and four legions (or principal orders), I sought to establish in 1883 in a communication, on the Orders of the Radiolaria (Sitzb. Jena Gesellsch. Med. u. Naturwiss., February 16, 1883). As a believer in the theory of descent, I regard all the systematic arrangements of specialists as artificial, and all their divisions as subjective abstractions, and hence I shall be guided in the establishment of such groups as subclasses, legions, orders, &c., by purely practical considerations, especially by the desire to give as ready a survey as possible of the complex multitude of forms (compare §§ 154 to 156).

3. Porulosa or Holotrypasta.—The subclass Porulosa or Holotrypasta includes the two legions, PERIPYLEA or SPUMELLARIA, and ACTIPYLEA or ACANTHARIA, which agree in the following constant and important characters :—( 1) The Central Capsule is primitively a sphere, and retains this homaxon form in the majority of the species. (2) The Membrane of the central capsule is everywhere perforated by very numerous minute pores, but possesses no larger principal aperture (osculum). (3) The Pseudopodia radiate in all directions and in great numbers from the central capsule, passing through its pores. (4) The Equilibrium of the floating unicellular body is in most Porulosa pantostatic (indifferent) or polystatic (plural-stable), since a vertical axis is either absent, or, if present, has its two poles similarly constituted. (5) The Ground-forms of the skeleton are therefore almost always either spherotypic or isopolar-monaxon, very rarely zygotypic. The two legions of the Porulosa are distinguished mainly by the skeleton of the SPUMELLARIA (or PERIPYLEA) being siliceous, never centrogenous, nor composed of acanthin, whilst in the ACANTHARIA (or ACTIPYLEA) it is always centrogenous and made up of acanthin; hence in the former the nucleus is always central, in the latter always excentric.

4. Osculosa or Merotrypasta.—The subclass Osculosa or Merotrypasta includes the two legions MONOPYLEA or NASSELLARIA, and CANNOPYLEA or PHAEODARIA, which agree in the following constant and important characters :—1) The Central Capsule is originally monaxon (ovoid or spheroidal) and retains this ground-form in most of the species. (2) The Membrane of the central capsule possesses a single large principal aperture (osculum) at the basal pole of the vertical main axis. (3) The Pseudopodia radiate from a stream of sarcode which passes out from the central capsule only on one side, namely, through the principal aperture. (4) The Equilibrium of the floating body is monostatic or unistable, since the two poles of the principal axis are always more or less different from each other. (5) The Ground-forms of the skeleton are, therefore, for the most part grammotypic (centraxon) or zygotypic (centroplan), rarely spherotypic. The two legions of the Osculosa are distinguished chiefly by the principal opening (osculum) being closed by a porous plate (porochora with its podoconus) in the NASSELLARIA (or MONOPYLEA), and by a radiate cover (operculum with its astropyle) in the PHAEODARIA (or CANNOPYLEA).

5. The four Legions of Radiolaria.—The four principal groups of Radiolaria, to which we have given the name "legions," are natural units, since the most important peculiarities in the structure of the central capsule are quite constant within the limits of the same legion, and since all the forms in the same legion may be traced without violence to the same phylogenetic stem. The four legions are, however, related to each other, in so far as they all exhibit those characters which distinguish the Radiolaria from other Protista. The two which compose the Porulosa (§ 3) seem somewhat more nearly related to each other than to the two which make up the Osculosa (§ 4). When, however, the attempt is made to bring them all into a phylogenetic relationship, it undoubtedly appears that the SPUMELLARLA (or PERIPYLEA) are the primitive stem, out of which the other three have been developed as independent branches. All three have been derived, probably independently, from the most ancient stem-form of the SPUMELLARIA, the spherical Actissa.

6. Peripylea or Spumellaria.—Those Radiolaria which we call "PERIPYLEA" on account of the constitution of their central capsule, or "SPUMELLARIA" on account of the nature of their skeleton, are separated from the other three legions of the class by the combination of the following constant characters :—(1) The Membrane of the central capsule is single and evenly perforated all over by innumerable fine pore-canals, but without any larger principal opening (osculum). (2) The Nucleus always lies centrally in the SPUMELLARIA monozoa and is serotinous, for it divides only at a later period into the nuclei of the spores; in the SPUMELLARIA polyzoa it is precocious, and divides early into many small nuclei. (3) The Pseudopodia are exceedingly numerous and distributed evenly over the whole surface of the central capsule. (4) The Calymma contains no phaeodium. (5) The Skeleton is seldom wanting, is never centrogenous, and is always siliceous. (6) The Ground-form of the central capsule is originally spherical (often modified); that of the skeleton is also spherical or, in the majority of cases, derived in different ways from the sphere.

7. Actipylea orAcantharia—Those Radioloria which we call "ACTIPYLEA" on account of the constitution of their central capsule, or "ACANTHARLA" from the formation of their skeleton, are separated from the other three legions by the combination of the following constant characters :—(1) The Membrane of the central capsule is single and perforated by numerous fine pore-canals, which are regularly distributed in series or groups, but without a larger principal opening (osculum). (2) The Nucleus is always excentric and generally precocious, since it divides early by a peculiar process of budding into numerous small nuclei. (3) The Pseudopodia are very numerous and distributed regularly in groups (or series united into a network). (4) The Calymma contains no phaeodium. (5) The Skeleton is generally present, always centrogenous, and composed of acanthin. (6) The Ground-form of the central capsule is originally spherical (often modified), that of the skeleton polyaxon (often modified).

8. Monopylea or Nassellaria.—Those Radiolaria wliich we call "MONOPYLEA" from the formation of their central capsule, or "NASSELLARIA" from the nature of their skeleton, are distinguished from the other three legions of the class by the combination of the following constant characters:—(1) The Membrane of the central capsule is single, and has only one large principal opening (osculum) at the basal pole of the vertical main axis; this osculum is closed by a perforated lid (porochora or operculum porosum) from which there arises within the central capsule a peculiar cone of threads or pseudopodia (podoconus). (2) The Nucleus is usually excentric and is always serotinous, since it only divides at a comparatively late period into spore-nuclei. (3) The Pseudopodia are not very numerous and arise by division of a single stem or bundle of threads of sarcode, which issues from the porochora. (4) The Calymma contains no phaeodium. (5) The Skeleton (very rarely absent) is never centrogenous, but always extracapsular and siliceous. (6) The Ground-form of the central capsule is always monaxon (with a vertical allopolar main axis), originally ovoid, often modified; that of the skeleton is also generally monaxon, often modified (triradial or bilateral).

9. Cannopylea or Phaeodaria.—Radiolaria which we call "CANNOPYLEA" from the constitution of their central capsule, or "PHAEODARIA" on account of their peculiar phaeodium, are distinguished from the other three legions by the combination of the following characters :—(1) The Membrane of the central capsule is double, consisting of a strong outer and delicate inner capsule, and has only one principal opening (osculum) at the basal pole of the vertical main axis; this osculum is closed by a radiate cover (astropyle or operculum radiatum), from the centre of which arises an external tubular spout (proboscis). Occasionally a few small accessory openings parapylae) are present besides the principal opening. (2) The Nucleus lies centrally or subcentrally in the capsule (in the vertical main axis), and is serotinous, inasmuch as it only divides at a late period into spore-nuclei. (3) The Pseudopodia are usually very numerous and arise from a thick sarcomatrix, formed by the spreading out of a thick stem of sarcode, which issues from the astropyle. (4) The Calymma always contains a phaeodium or peculiar voluminous excentric mass of pigment. (5) The Skeleton (very rarely absent) is never centrogenous, always extracapsular and formed of a silicate of carbon. (6) The Ground-form of the central capsule is always monaxon (with a vertical allopolar main axis) and generally spheroidal; that of the skeleton is very varied.

10. Synopsis of the Subclasses and Legions:

11. Individuality of the Radiolaria.—Like other Protozoa the Radiolaria are unicellular organisms, the whole fully developed organisation of which falls under the category of a single cell, both morphologically and physiologically. Since this view is based upon the composition of the individual body out of two different morphological elements, nucleus and protoplasm, it is at once justified in the case of the majority of Radiolaria, in which the plasmatic body encloses only a single nucleus (the so-called "Binnen-Bläschen"); such is the case in all the SPUMELLARIA monozoa, NASSELLARIA and PHAEODARIA. This aspect of the case might appear doubtful in those Radiolaria in which the simple primary cell-nucleus divides early into numerous small secondary nuclei, as is the case in the SPUMELLARIA polyzoa and most ACANTHARIA. Strictly speaking, the multinucleate central capsule should in such cases be regarded as a syncytium; but since the individual unity of the unicellular organism is as clearly defined in these precocious multinuclear Radiolaria as in the ordinary serotinous forms, the former must be considered unicellular Rhizopods just as are the latter. This mode of regarding the case is the more necessary, inasmuch as the early division of the nucleus has no further influence upon the organisation. Just as in many other classes of the Protista there are monozootic (solitary) and polyzootic (social) forms, so also in the Radiolaria there are in addition to the ordinary monozootic or monobious forms certain families in which colonies or coenobia are formed by the association of individuals; this distinction may be expressed by the terms "Monocyttaria" and "Polycyttaria."

The unicellular nature of the Radiolaria was first established by Richard Hertwig in 1879 (L. N. 33),¹ and brought into conformity with our present histiological knowledge and the new reform of the cell-theory. Huxley, however, who was in 1851 the first to examine living Radiolaria accurately, declared Thalassicolla nucleata to be a unicellular Protozoon, and the individual central capsules of Sphaerozoum  punctatum to be cells, but, owing to the then condition of the cell-theory, he was unable to give a conclusive demonstration of this view. Later, when Johannes Müller in 1858 and myself in 1862 recognised the peculiar "yellow cells" which occur in large numbers in many Radiolaria as true nucleated cells, it appeared impossible any longer to maintain the unicellular nature of the Radliolaria; also the great complication which I showed to exist in the structure of Thalassicolla appeared to contradict it. Only after Cienkowski (1871) and Brandt (1881) had shown that the "yellow cells" do not belong to the Radiolarian organism, but are symbiotic unicellular aIgae, was it possible to revive and demonstrate anew the unicellular nature of the Radiolaria.

12. Morphological Individuality.—From the morphological standpoint the individuality of the unicellular elementary organism is obvious in the ordinary solitary Radiolaria (Monobia), and is to be so regarded that the whole body with all its constituent parts, and not merely the central capsule, is to be regarded as a cell. Naturally the xanthellae or yellow cells (§§ 76, 90), which as independent algae live in symbiosis with many Radiolaria, must be excluded. The unicellular organisation of the Radiolaria is further to be distinguished from that of the other Protista, inasmuch as an internal membrane (capsule-membrane) separates the central (medullary) from the peripheral (cortical) portion. In the coenobia of the social Radiolaria (or Polycyttaria), the morphological individuality persists only as regards the medullary portions of the aggregated cells (the individual central capsules), while the cortical portions fuse completely to form a common extracapsulum. Hence in these SPUMELLARLA polyzoa two different stages of morphological individuality must be distinguished, the Cell as a Morphon of the first stage, and the Coenobium as a Morphon of the second stage.

13. Physiological Individuality.—From the physiological standpoint also the individuality of the unicellular organism is immediately obvious in the case of the ordinary solitary Radiolaria (Monobia); as in other Protista it fulfils all the functions of life by itself alone. This physiological individuality of the monobious Radiolarian cell is further more not influenced by the xanthellae, which live as independent algae in symbiosis with many Radiolaria; even though these often by the production of starch assist in the nourishment of the Radiolaria, yet they are by no means indispensable to them. On the other hand, the physiological individuality offers more complicated relations in the social Radiolaria (Polycyttaria) which live united in colonies or coenobia.  Here the actual Bion (or the fully developed physiological individual) is not represented by the individual cells, but by the whole multicellular coenobium, which in each species has a definite form and size. In these coenobia, which are usually spherical or cylindrical jelly-like masses, several millimeters in diameter, numerous cells are so intimately united that only their medullary portions (the central capsule with the endoplasm) remain independent; the cortical portions (calymma and exoplasm) on the contrary uniting into a common extra capsulum. This discharges, as a whole, the functions of locomotion, sensation, and inception of nutriment, while the separate central capsules act in the main only as reproductive organs (forming spores) and partly also as the central organs of metastasis (digestion). Each coenobium may also be regarded as a polycyttarium, i.e., a "multicellular Radiolarian," whose numerous central capsules represent so many sporangia or spore-capsules.

    On this head compare the section in my monograph of 1862 (L. N. 16), entitled Die Organisation der Radiolarien-Colonien; Polyzoen oder Polycyttarien? (pp. 116 to 126); and also R. Hertwig, Zur Histologie der Radiolarien, 1876 (L. N. 26, p. 23).

14. Monocyttaria and Polycyttaria.—In the majority of the Radiolaria each unicellular organism passes its individual life in an isolated condition (as a Monocyttarium). Only in a part of the SPUMELLARIA numerous unicellular individuals are united into societies which are regarded as coenobia or colonies (Polycyttaria). This is the case in three different families belonging to the PERIPYLEA, in the Collozoida (without a skeleton, Pl. 3), the Sphaerozoida (with a Beloid skeleton, Pl. 4), and the Collosphaerida (with a Sphaeroid skeleton, Pls. 5, 6, 7, 8). All three families of Polycyttaria (or social Radiolaria), agree in their mode of forming colonies, since the central capsules of the social individuals remain separate and lie in a common jelly-like mass, which is formed by the fusion of their extracapsulum. The chief part of the voluminous colonies, which attain a diameter of several millimetres (sometimes more than 1 cm.), and are generally spherical, ellipsoidal or cylindrical, consists therefore of the jelly-like calymma, and this is penetrated by a sarcoplegma, to whose meshes all the individual organisms contribute by means of the pseudopodia, which radiate from their sarcomatrix. A further peculiarity in which the social SPUMELLARIA differ from the solitary consists in the fact that the former are precocious and the latter serotinous in the division of the nucleus (§ 64). Whilst in the solitary or monozootic SPUMELLARIA the middle of the central capsule is occupied by the simple nucleus, and this divides only at a late period (immediately before the formation of spores) into the numerous spore nuclei, in the colonial or polyzootic SPUMELLARIA this division takes place very early, and the middle of each central capsule is usually occupied by an oil-globule.

    The colonial Radiolaria were described as early as the year 1834 by Meyen, the first investigator of the class, under the name Sphaerozoum, and, as Palmellaria, compared with the gelatinous colonies of the Nostochineae. The first accurate observations upon their structure were, however, made in 1851 by Huxley, who described examples of all three families under the name Thalassicolla punctata. More extended, however, were the investigations of Johannes Müller, who in his fundamental work (1858) divided the whole class Radiolaria into Solitaria and Polyzoa. The Radiolaria solitaria he divided into Thalassicolla, Polycystina and Acanthometra, the Radiolaria polyzoa into Sphaerozoa (without a shell) and Collosphaera (with a shell). The most accurate delineation of the Polycyttaria was given by Hertwig in his beautiful memoir, Zur Histologie der Radiolarien (1876). Quite recently, however (1886), since the completion of my manuscript upon the Challenger Radiolaria, a very complete Monograph of the Polycyttaria has appeared by Karl Brandt, Die colonie-bildenden Radiolarien (Sphaerozoen) des Golfes von Neapel und der angrenzenden Meeres-Abschnitte (276 pp., 8 pls., Berlin). It contains in particular most valuable contributions to the physiology and histology.

15. The Central Capsule and Extracapsulum.—The special peculiarity of the unicellular Radiolarian organism, by which it is clearly distinguished from all other Rhizopoda (and indeed from most other Protista), is its differentiation into two separate chief constituents, the central capsule and extracapsulum, and the formation of a special membrane which separates them. This, the capsule-membrane, is not to be compared with an ordinary cell-membrane, as an external layer, but rather to be regarded as an internal differentiated product. The extracapsulum or external (cortical) portion of the body is in most Radiolaria more voluminous than the central capsule or inner (medullary) portion. The exoplasm of the former (the cortical or extracapsular protoplasm) is emphatically different from the endoplasm of the latter (the medullary or intracapsular protoplasm). Besides the most important vital processes are distributed by division of labour so completely between them that they appear most distinctly co-ordinated. The central capsule is on the one hand the general central organ of the "cell-soul" for the discharge of its sensory and motor functions (comparable to a ganglion-cell), on the other hand the special organ of reproduction (sporangium). The extracapsulum, also, is not less significant, since on the one hand its calymma acts as a protecting envelope to the central capsule, as a support to the pseudopodia, and a foundation for the skeleton or a matrix for the development of the shell, and on the other hand its pseudopodia are of the utmost importance as peripheral organs of movement and sensation as well as of nutrition and respiration. The central capsule and the extracapsulum are therefore to be regarded both morphologically and physiologically as the two characteristic co-ordinated principal parts of the unicellular Radiolarian organism.

    In most of the more modern delineations of the Radiolaria the central capsule is regarded as the "cell proper" and its membrane as the "cell-wall." The, following facts are opposed to the correctness of this interpretation :—1. In most Radiolaria the exoplasm is clearly different from the endoplasm, and the former is more voluminous than the latter. 2. In all Radiolaria the division of labour is so carried out between the central capsule and the extracapsulum, that the physiological significance and independence of both principal parts of the cell is almost equally great. 3. It is only in the ACANTHARIA that the formation of the skeleton takes place within the central capsule; in all the other three legions it is quite independent of it.

16. The Malacoma and Skeleton.—Whilst the division of the unicellular organism into central capsule and extracapsulum is undoubtedly the most important character of the Radiolarian organism, the development of a skeleton of peculiar and most varied form is of very striking significance. This skeleton is always a secondary product of the cell, but is always anatomically so independent, and so clearly marked off from the soft parts or malacoma, that it seems advisable to regard both separately in a general morphological survey. The skeleton stands in a different relation to each of the two principal constituents of the malacoma. Only in the ACANTHARIA is it centrogenous and developed from the central capsule outwards. In the other three legions the skeleton never arises in the centre of the capsule; in the NASSELLARIA and PHAEODARIA it is always extracapsular; in the SPUMELLARIA it is also outside the central capsule originally, but afterwards becomes often surrounded by it, and finally lies in most cases partly within and partly without the central capsule. The chemical basis of the skeleton in the ACANTHARIA is the curious acanthin (an organic substance allied to chitin), in the PHAEODARIA a silicate of carbon, and in the NASSELLARIA and SPUMELLARIA silica.

17. Ground-Forms of the Radiolaria (Promorphology).—The ground-forms of the the Radiolaria exhibit a greater variety than those of any other class in the organic world, greater indeed than is to be found in all the remaining groups together. For every conceivable ground-form which can be defined in the system of promorphology is actually present in the Radiolaria; their skeleton exhibits, as it were, in material existence, certain geometrical ground-forms which are found in no other organisms. The cause of this unexampled richness in different forms lies chiefly in the static relations of the Radiolaria, which swim freely in the sea, partly also in the peculiar plasticity of their protoplasm and the material of their skeletons.

Regarding the general system of ground-forms compare my Generelle Morphologie (1866, Bd. i. pp. 375-552; Bd. iv., Aligemeine 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 Sphaerotypic, 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 Sphaerotypic Ground-Forms.—The first group of geometrical ground-forms, here distinguished as sphaerotypic 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 sphaerotypic 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 sphaerotypic 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 Sphaeroidea 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, sub-regular 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, jn 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 orAtypic 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 ameboid central capsule of some Colloidea (Collodastrum, p. 27, Pl. 3, figs. 4, 5) among the SPUMELLARIA, the irregular shells of many Collosphaerida (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 sphaerotypic) 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 PHAEODARIA (in most Phaeosphaeria); 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 Sphaeroidea (Pls. 11-80) as well as of 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 Sphaeroidea, but is also found among the ACANTHARLA (especially in the Astrolophida and Sphaerophracta), as well as among the Phaeosphaeria (in most genera of the Orosphaerida, Sagosphaerida, and Aulosphaerida). 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 Sphaeroidea), 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.)

26. The Regular Icosahedral Ground-Form.—The ground-form whose geometrical type is the regular icosahedron (bounded by twenty equilateral triangles) is rarely exemplified, but it occurs among the PHAEODARIA in the Circoporid genus Circogonia (Pl. 117, fig. 1), and also in certain Aulosphaerida, but, apparently, only as an accidental variation (e.g., Aulosphaera icosahedra). Furthermore, this ground-form may also be assumed to occur in those Sphaeroidea whose spherical lattice-shells bear twelve equidistant radial spines (e.g., many species of Acanthosphaera, Heliosphaera, and other Astrosphaerida); the basal points of these spines indicate the twelve angles of the regular icosahedron. (See on this head Gener. Morphol., Bd. i. p. 411.)

27. The Regular Dodecahedral Ground-Form.—The ground-form whose geometrical type is the regular dodecahedron (or pentagonal dodecahedron), bounded by twelve equilateral and equiangular pentagons, is very rarely found perfectly developed, as in Circorrhegma dodecahedra (Pl. 117, fig. 2). This form is by no means so common among the Radiolaria as in the pollen grains of plants (e.g., Buchholzia maritima, Fumaria spicata, Polygonum amphibium, &c.). It can, however, be regarded as present in all those Sphaeroidea whose spherical lattice-shells bear twenty equal and equidistant radial spines (e.g., many species of Acanthosphaera, Heliosphaera. and other Astrosphaerida); the basal points of these spines mark out the twenty angles of the regular pentagonal dodecahedron. (See Gener. Morphol., Bd. i. p. 412.)

28. The Regular Octahedral Ground-Form.—The ground-form whose geometrical type is the regular octahedron (bounded by eight equilateral triangles), commonly appears among the SPUMELLARIA in the family Cubosphaerida (p. 169, Pls. 21, 22, 23, 24, 25). In these Sphaeroidea the typical ground-form is usually indicated by six equal radial spines, which are opposed to each other in pairs, and lie in three similar axes perpendicular to each other; these are the three axes of the tesseral crystallographic system; one of them is vertical, whilst the other two cross each other at right angles in its centre. Occasionally, too, the spherical form of the lattice-shell passes over into that of the regular octahedron (Pl. 22, figs. 8, 10). The same form recurs in Circoporus (Pl. 117, fig. 6) among the PHAEODARIA. In the vegetable kingdom it is exhibited by the antheridia of Chara. It is not found in the NASSELLARLA and ACANTHARIA. (See Gener. Morphol., Bd. i. p. 412.)

29. The Regular Cubic Ground-Form.—The ground-form whose geometrical type is that of a die or cube, is actually presented in a very striking manner by various Radiolaria. Among the SPUMELLARIA it occurs in certain Sphaeroidea, e.g., in the Astrosphaerid genera Centrocubus and Octodendrum (Pl. 18, figs. 1-3); in these the central medullary shell is a complete cube, bounded by six equal squares, from the eight angles of which eight equal radial spines project. This form can also be regarded as present in those Sphaeroidea whose spherical lattice-shell bears eight equal and equidistant radial spines (many Astrosphaerida). Besides these the cubic ground-form is to be seen in certain NASSELLARIA of the family Tympanida, especially in Lithocubus (Pl. 82, fig. 12; Pl. 94, fig. 13), in many species of Acrocubus, Microcubus, &c.; the twelve bars of its lattice-skeleton correspond often exactly to the edges of the cube. (See Gener. Morphol., Bd. i. p. 413.)

30. The Regular Tetrahedral Ground-Form.—The ground-form whose geometrical type is the regular tetrahedron, bounded by four equilateral triangles, occurs less frequently in the Radiolaria than the other four regular polyhedra. Among the SPUMELLARIA it is found in the Beloidea, and especially in those members of the Thalassosphaerida and Sphaerozoida whose spicules bear four equal branches, diverging at equal angles from a common centre. Precisely the same structure is seen also among the NASSELLARIA in some Plectoidea, as in Tetraplagia among the Plagonida, and Tetraplecta among the Plectanida. The skeleton of both these genera consists of four equal rods, which radiate at equal angles from a common centre, just as do the axes of the regular tetrahedron. The tetrahedral form of these Plectoidea is the more important and interesting since on the one hand it is related to the similar spicular form of the Beloidea, and on the other perhaps furnishes the starting point from which Cortina among the NASSELLARLA may be derived (Plagoniscus, Plectaniscus). (See Gener. Morphol., Bd. i. p. 415.)

31. The Isopolar-Monaxon or Phacotypic Ground-Form.—The isopolar uniaxial or phacotypic ground-form is characterised by the possession of a vertical main axis with equal poles, whilst no transverse axes are differentiated. All horizontal planes which cut the axis at right angles are circles, and increase in size from the poles towards the equator. The most important ground-forms of this group are the phacoid (the lens or oblate spheroid) and the ellipsoid (or prolate spheroid). Phacoids (or geometrical lenses with blunt margins) are very often presented by the central capsules of the Discoidea and of many ACANTHARLA (Quadrilonchida and Hexalaspida), but the lattice-shells of many SPUMELLARIA and ACANTHARIA exhibit the same form, as also do a few PHAEODARIA (e.g., Aulophacus). True geometrical ellipsoids are seen in the central capsules of many Prunoidea among the SPUMELLARIA, and of many Amphilonchida and Belonaspida among the ACANTHARIA. Furthermore, the lattice-shells of many species of these groups retain the same essential form, e.g., many Ellipsida, Druppulida, and Spongurida (Pls. 13-17, and 39), as well as most Belonaspida. (See Gener. Morphol., Bd. i. p. 422.)

32. Allopolar-Monaxon or Conotypic Ground-Form.—The allopolar uniaxial or conotypic ground-form is characterised by the possession of a vertical main axis whose two poles are unlike, while no transverse axes are differentiated. All horizontal planes cutting the main axis at right angles are circles, and decrease more rapidly from the largest plane towards the basal than towards the apical pole. The most important ground-forms of this group are the ovoid, the cone, and the hemisphere. They often occur (and in geometrical perfection) in the egg-shaped central capsule and podoconus of the NASSELLARIA, as well as in the shells of several groups of this legion, particularly in the Cyrtocalpida or Monocyrtida eradiata (Pl. 51, figs. 10-13), and in many Stichocyrtida eradiata; furthermore, they are also seen among the PHAEODARIA, e.g., certain Challengerida (Pl. 99, figs. 19-22). (See Gener. Morphol., Bd. i. p. 426.)

33. The Regular Dipyramidal or Quadrilonchial Ground-Form.—The ground-forms whose geometrical type is the regular double pyramid are characterised by a vertical main axis which possesses equal poles, and which is crossed at its centre by several equal transverse axes. The horizontal equatorial plane is therefore a regular polygons and divides the body into two equal regular pyramids. The simplest and commonest form of this group is the quadratic octahedron, the ground-form of the quadratic crystal lographic system; its equatorial plane is a square. This regular dipyramidal ground- form occurs among the SPUMELLARLA in the shells of the Staurosphaerida as well as of many Discoidea, in which several equidistant radial spines or arms lie in the quadratic equatorial plane of the body, and project from the margin of the lenticular disc (e.g., Sethostaurus, Pl. 31; Histiastrum, Pl. 46, &c.). It is, however, among the ACANTHARIA that the most important part is played by this ground-form (and especially by the quadratic octahedron); it forms the basis of all those Acanthometra and Acanthophracta in which twenty radial spines are disposed according to the Müllerian Law, and in which the four equatorial spines are of equal dimensions (Icosacantha). (See Gener. Morphol., Bd. i. p. 436-446.)

34. The Amphithect Dipyramidal or Lentelliptical Ground-Forms.—The ground-forms whose geometrical type is the lenticular or "triaxial" ellipsoid, may also be designated amphithect double pyramids; they are characterised by the possession of a vertical main axis which has similar poles, and is crossed at its middle by two transverse axes, unequal but isopolar. The horizontal equatorial plane of the body is therefore an amphithect or elongated polygon (a rhombus in the simplest case possible), and divides the whole body into two equal amphithect pyramids. The simplest and commonest form of this group is the rhombic octahedron, which is also the ground-form of the rhombic crystallographic system. It plays an important part in those ACANTHARIA in which twenty radial spines are disposed according to the Müllerian Law, but in which the two pairs of equatorial spines are unequal (different geotomical and hydrotomical axes, see p. 719); to this category belong the Amphilonchida (Pl. 132), Belonaspida (Pl. 136), Hexalaspida (Pl. 139), and Diploconida (Pl. 140). A form essentially identical obtains also among the SPUMELLARIA in the majority of the Larcoidea, both in their triaxial lattice-shells, and in their lentelliptical central capsules, which present geometrically accurate triaxial ellipsoids, with three unequal isopolar axes at right angles to each other. (See Gener. Morphol., Bd. i. p. 446-452.)

35. The Regular Pyramidal Ground-Forms.—The ground-forms whose geometrical type is the regular pyramid, and which are the most conspicuous in the Medusae, Polyps, Corals, and regular Echinoderms (the Radiata of earlier authors), are almost confined among the Radiolaria to the legion NASSELLARIA; they occur, however, in the great majority of these, and especially in those families which may be classed together as "Cyrtoidea triradiata et multiradiata." Strictly speaking, however, almost all these NASSELLARIA, at all events in their origin, are bilateral or dipleuric, since the primary sagittal ring with its characteristic apophyses marks out the sagittal median plane, and further, since the three feet of the basal tripod are usually divided into an unpaired dorsal (pes caudalis) and two paired ventral or lateral (pedes pectorales, dexter et sinister). On the other hand, it is noteworthy, firstly, that among the primitive Plectoidea there are perfectly regular radial forms, without any indication of an original bilateral symmetry, and secondly, that similar forms are also very common among the Cyrtoidea, probably as secondary radial forms, developed from primitive bilateral ones. Similar cases also occur in certain PHAEODARIA (e.g., the Medusettida and Tuscarorida, Pls. 100, 120), but they.are entirely wanting among the ACANTHARIA and SPUMELLARIA. The multiradial NASSELLARIA have arisen from the triradial by the interpolation of three, six, nine, or more interradial and adradial secondary apophyses between the three primary perradial ones. (See Gener. Morphol., Bd. i. pp. 459-874.)

36. The Amphithect Pyramidal Ground-Forrns.—The ground-forms whose geometrical type is the amphithect pyramid, are distinguished from the regular pyramidal forms, just discussed, chiefly by the form of the basal plane, which is not a regular, but an amphithect or elongated polygon (in the simplest case a rhombus). Hence in this case the alloplar main axis of the body is crossed by two transverse axes which are isopolar and at right angles, but are unequal; they cannot, however, be distinguished as sagittal and frontal axes as is the case in the zeugites. In the animal as well as in the vegetable kingdom, an important part is played by this ground-form, e.g., in the Ctenophora, where it is the rhombic pyramid. Among the Radiolaria it is not common, though it is clearly expressed among the NASSELLARIA in a number of Stephoidea (Stephanida and Tympanida), as well as in many Spyroidea (e.g., the bipedal Zygospirida). It is very accurately developed among the PHAEODARIA in the bivalved Phaeoconchia (Pls. 121-128), where the two valves of the shell (dorsal and ventral) are generally exactly alike, their median keels corresponding to the poles of the sagittal axis. In the slit between the two valves lie the two secondary openings (right and left) of the tripylean central capsule, corresponding to the two poles of the frontal axis, and the main axis stands perpendicularly to both these, its oral pole being indicated by the astropyle, or principal aperture. (See Gener. Morphol., Bd. i. pp. 479-494.)

37. The Amphipleural Ground-Forms.—By the term amphipleural ground-forms are to be understood those usually defined as "bilaterally radial"; their geometrical type is a half amphithect pyramid. The best known examples of this form in the animal kingdom are the bilateral five-rayed Echinoderms (Spatangus, Clypeaster), in the vegetable kingdom the symmetrical five-rayed flowers (Viola, Trifolium). The three dimensive axes have the same relation as in the zygopleura, to be next discussed, and which also resemble them in being divisible only by one plane (the sagittal median plane) into two equal halves. They differ, however, the amphipleural body not being made up of two antimeres, but of at least three pairs of antimeres (or three parameres), being therefore primitively radial. Hence each of the symmetrical halves of the body contains more than one antimere. Among the Radiolaria this form does not occur in the SPUMELLARLA, ACANTHARIA, or PHAEODARIA; it is very common, however, among the NASSELLARIA; many Cyrtoidea multiradiata and Spyroidea multiradiata show this bilaterally radial ground-form, inasmuch as the body consists of two symmetrical halves, and is also composed of numerous (usually three, six, nine, or more) radial parameres. In the multi-radiate Dicyrtida and Tricyrtida the cephalis (the first joint) is usually bilateral, whilst the thorax (the second joint) is multiradial. (See Gener. Morphol., Bd. i. pp. 495-506.)

38. The Zygopleural Ground-Forms.—As zygopleural or dipleural ground-forms, as opposed to the amphipleural, are classed those zeugites or centroplana which are known as "bilaterally symmetrical" in the strictest sense of the term. This is the most important ground-form in the animal kingdom, inasmuch as it obtains almost exclusively among the higher animals (Vertebrata, Articulata, Mollusca, Vermes). The body consists of only two antimeres, which correspond to the two symmetrical halves of the body. Of the three dimensive axes two are allopolar, one isopolar; the oral pole of the longitudinal main axis is different from the aboral; the dorsal pole of the sagittal axis is different from the ventral; but the right pole of the frontal axis is equal to the left. The right antimere is usually precisely similar to the left (Eudipleura), more rarely it is slightly dissimilar or asymmetrical (Dysdipleura). Among the Radiolaria this ground-form is entirely wanting in the Porulosa or Holotrypasta (SPUMELLARIA and ACANTHARIA), but on the contrary it is very common in the Osculosa or Merotrypasta (NASSELLARIA and PHAEODARIA). In the NASSELLARIA it is of special importance, for the typical Cortina (the combination of the primary sagittal ring with the basal tripod) exhibits the zygopleural ground-form clearly sketched out; indeed it is usually clearly seen even in the sagittal ring itself, for its ventral segment is more strongly curved than the dorsal; its basal (or oral) pole is always different from the apical (or aboral). Of the three feet of the basal tripod the unpaired (caudal) one is directed dorsally and backwards, the two paired (pectoral) ones ventrally and forwards. The majority of the NASSELLARIA may be regarded as modifications of this original ground-form. Its relation to the primitively triradiate tripod presents a still unsolved problem, and the numerous relations of the zygopleural to the multiradiate ground-forms in the NASSELLARIA are exceedingly complicated. The zygopleural ground-form is less widely distributed among the PHAEODARIA, though it is very characteristically developed in the rich and varied group of Challengerida (Pl. 99). (See Gener. Morphol., Bd. i. pp. 507-527.)
 

39. Synopsis of the Geometrical Ground-Forms:—

Principal Groups of Ground-Forms.	Subsidiary Groups of Ground-Forms.	Geometrical Type.	Examples.
I. CENTROSTIGMA. The geometrical centre of the body is a point. Main axis wanting.	I. Homaxonia all axes equal. 1. Sphere Central capsule of the Sphaeroidea and of many ACANTHARIA. II. Polyaxonia Endospherical polyhedra.    All the angles of the    body lie on the surface    of a sphere. Numerous    isopolar axes 2. Endospherical polyhedron Lattice-spheres of the Sphaeroidea, Sphaerophracta, and Phaeosphaeria. 3. Icosahedron, Circogonia. 4. Dodecahedron, Circorrhegma. 5. Octahedron, Cubosphaerida, Circoporus. 6. Cube, Centrocubus, Lithocubus, &c. 7. Tetrahedron, Tetraplagia, Tetraplecta, &c.
II CENTRAXONIA. The geomeetrical centre of     the body is a straight     line (the vertical main axis). Constant transverse axes     (perpendicular to the main     axis) are wanting in the    Monaxonia (which have    circular transverse sections);     on the contrary they are     differentiated in the    Stauraxonia (which have    polygonal transverse     sections).	III. Monaxonia Uniaxial ground-forms or    centraxonia without    transverse axes.  The    transverse planes     (perpendicular to the main    axis) are circles. 8. Monaxonia isopola. (Spheroids and ellipsoids; both poles of the main axis similar.) Central capsule and lattice-shell of many Discoidea (lenses) and Prunoidea (ellipsoids), Belonaspida, &c. 9. Monaxonia allopola. (Cone, ovoid and hemisphere; the two poles of the axis dissimilar.) Central capsule and lattice-shell of many NASSELLARIA, especially the Cyrtoidea eradiata (Cyrtocalpida, &c.). IV. Stauraxonia. Pyramidal ground-forms    or centraxonia with    transverse axes.  The    transverse planes    (perpendicular to the main    axis) are either regular or    amphithect polygons. 10. Dipyramides regulares. (Quadratic octahedron, or quadrilonchial forms and regular double pyramids.) ACANTHARIA with twenty radial spines, the four equatorial being equal. Multiradial Discoidea and Staurospherida. 11. Dipyramides amphithectae. (Rhomboid octahedron, lentellipsoid, and amphithect double pyramids.) ACANTHARIA with twenty radial spines, the four equatorial being unequal but paired. Many Larcoidea. 12. Pyramides regulares. (Regular pyramids.) Many NASSELLARIA (triradial and multiradial). Medusettida and Tuscarorida. 13. Pyramides amphithectae. (Rhombic pyramids.) Phaeoconchia. Bipedal Spyroidea and Stephoidea.
III CENTROPLANA. The Geometrical centre of the body is a plane (the sagittal plane).	V. Bilateralia     (or Zeugita). Bilateral forms in the general sense, with right and left halves. 14. Amphipleura. (Bilaterally radial ground-form) Many Cyrtoidea and Spyroidea multiradiata. 15. Zygopleura. (Bilaterally symmetrical ground-form) Most NASSELLARIA (primitively at least), many Challengerida.
IV. ACENTRA. There is no geometrical centre.	VI. Anaxonia. No definitet axes can be determined. 16. Irregularia. (Absolutely irregular ground-forms) Colladastrum, Collosphaera, Phorticida, Soreumida.

40. Mechanical Causes of the Geometrical Ground-Forms.—The great variety of ground-forms exhibited by the Radiolaria is of special interest, since in most instances their causes admit of recognition, and since they are so intimately related to each other that even in the remaining cases the assumption that they have arisen by purely mechanical causae efficientes seems justified. In this respect the first rank is taken by statical conditions, especially the indifferent or stable equilibrium of the whole organism, which floats freely in the water. With regard to these fundamental statical relations, three principal groups of ground-forms may be distinguished, pantostatic, polystatic, and monostatic.

41. Pantostatic Ground-Forms.—By pantostatic or indifferently stable ground-forms are meant those in which the centre of gravity coincides with the centre of the body, so that they are in equilibrium in any given position. Strictly speaking, the only form which possesses perfectly indifferent equilibrium is the sphere, that being the only truly homaxon and perfectly regular form. Nevertheless, in a somewhat wider sense many Polyaxonia, especially the endospherical polyhedra with very numerous sides, may be included in this category. Such indifferently stable bodies are found among the SPUMELLARIA in many Collodaria and Sphaeroidea, as well as in the Astrolophida among the ACANTHARIA. On the contrary they are entirely wanting among the NASSELLARIA and PHAEODARIA, since their central capsule constantly presents a main axis with a differentiated basal pole, and determines the position of stable equilibrium.

42. Polystatic Ground-Forms.—Those ground-forms are defined as polystatic or multistable in which the body is in equilibrium in several different positions (though not in an infinite number). The number of these positions is usually twice as many as that of the constant equal isopolar axes exhibited by the form. Hence the regular polyhedra have as many positions of equilibrium as they have angles or sides, the icosahedron twenty, dodecahedron twelve, octahedron eight, cube six, tetrahedron four. The isopolar monaxon ground-forms (lens, ellipsoid, cylinder), and the diplopyramidal ground-forms (quadrilonchial. and lentelliptical) have two positions of stable equilibrium, since the two poles of the vertical axis are equal and similar and the body is divided into equal halves by the equatorial plane. This is the case in many SPUMELLARIA (especially Discoidea, Prunoidea, and Larcoidea), as well as in the great majority of ACANTHARIA. Perhaps the same holds good also in certain NASSELLARIA (e.g., isopolar Tympanida) and PHAEODARIA (e.g., isopolar Phaeosphaeria), though here unistable equilibrium appears to be necessitated by the constant main axis of the central capsule and the differentiated basal pole of the main axis.

43. Monostatic Ground-Forms.—Those ground-forms are classed as monostatic or unistable in which the body is in equilibrium only in one position, since the centre of gravity of the body lies in a constant vertical axis below its centre. This fixed position is only rarely and exceptionally found among the SPUMELLARIA (e.g., in Xiphostylus, Sphaerostylus, Lithomespilus, Lithapium) and among the ACANTHARIA (e.g., in Zygostaurus and Amphibelone). On the contrary it is quite usual among the NASSELLARIA and PHAEODARIA (with but few exceptions); for here a vertical main axis, with a differentiated basal pole, is determined even by the formation of the central capsule, and usually also by the corresponding structure of the skeleton. Among the NASSELLARLA this basal pole, with the porochora of the central capsule, appears always to be the lower; as also in most Phaeogromia among the PHAEODARIA. In the peculiar bivalved Phaeoconchia, on the other hand, the basal pole with the cannopyle is directed upwards; as also in the Challengerida and Tuscarorida. The Phaeosphaeria and Phaeocystina are probably to a large extent polystatic.  In general unistable equilibrium may be assumed in the following categories of ground-forms :—( 1) Allopolar monaxon (conical and ovoid); (2) pyramidal (regular and amphithect); (3) Centroplana (amphipleura and zygopleura); (4) Anaxonia.

44. Principal Axes.— From the foregoing consideration of the statical conditions and their direct causal connection with the geometrical ground-forms of the Radiolaria, the great mechanical significance of the differentiation of definite axes in these unicellular free-swimming organisms will be manifest. The most important of these is the primary main axis (axis principalis, or protaxon), which in all cases has a vertical direction. It is wanting in the Centrostigma (spheres. and endospherical polyhedra), and in the Anaxonia (acentra). It is isopolar in the phacotypic forms (Monaxonia isopola), and in the double pyramids (Stauraxonia isopola). It is allopolar in all monastatic ground-forms, in the conotypic forms (Monaxonia allopola), pyramids (Stauraxonia allopola), and the Centroplana (or bilateral forms).

45. Secondary or Transverse Axes—In contrast to the vertical main axis all the other constant axes differentiated in the body may be called "secondary axes," or "transverse axes," since they cross the former at definite points. All ground-forms whose vertical axis is crossed by a fixed number of such axes at definite angles may be called "Stauraxonia." They are divided into two groups, double pyramids and single pyramids; in the former the two poles of the main axis (or the two halves of the body separated by the equatorial plane) are similar (Stauraxonia homopola), in the latter dissimilar (Stauraxonia heteropola). If all the secondary axes be equal, the stauraxon ground-form is regularly radial. If some of them be unequal they are arranged in certain relations towards two primary transverse axes, perpendicular to each other, to which all the other secondary axes are subsidiary; the ground-forms are then either ampliithect or bilateral. The two primary transverse axes, which may also be designated "ideal transverse axes" (euthyni), divide the vertical main axis in its centre; one of them is the sagittal, the other the frontal. These three dimensive axes give the factors which accurately determine the ground-form and the dimensions in most Radiolaria; the vertical main axis determines the length (principal axis); one horizontal transverse axis determines the thickness (sagittal axis), and the other the breadth (frontal axis). Those ground-forms in which the transverse axes are isopolar are termed "amphithect," and those in which the one (frontal or lateral) is isopolar and the other (sagittal or dorso-ventral) is allopolar, are termed "bilateral," or better "zeugitic."

46. Primary and Secondary Gronnd-Forms.—The geometrical sphere must be regarded as the original ground-form of the Radiolaria; it being understood that its monophyletic derivation from a single stem-form, Actissa, is correct. The simplest forms of Actissa (Procyttarium,Pl. 1, fig. 1) are in fact geometrically perfect spheres; indeed even the individual parts which compose their unicellular bodies (nucleolus, nucleus, central capsule and calymma) are concentric spheres. But in addition the central capsules of most other SPUMELLARLA, especially the Sphaeroidea, as well as of many ACANTHARLA are true spheres. Furthermore the simple or concentrically composed lattice-spheres of Sphaeroidea, Sphaerophracta, and Phaeosphaeria may be regarded as spheres, although strictly speaking they are endospherical polyhedra. From the primary spherical form of the Radiolaria all other secondary forms may be derived in the following order :—1. By the development of a main axis the Monaxonia arise. 2. By the development of transverse axes the Stauraxonia arise. 3. In both groups (Monaxonia and Stauraxonia) the two poles (or upper and lower halves of the body) are at first similar (Isopola). 4. By differentiation in the two poles or halves of the body (distinction between the basal pole and the apical) the forms with different poles (Allopola) arise. 5. The transverse axes of the Stauraxonia are at first equal (regular pyramids and double pyramids). 6. By differentiation in the transverse axes (distinction between the sagittal and the frontal axis) the amphithect pyramids and double pyramids arise. 7. From the amphithect pyramids the Amphipleura arise by differentiation of both poles of the sagittal axis. 8. The zygopleural ground-form appears last, as the simplest form of the Amphipleura.

47. The Ground-Forms of the Spumellaria.—The SPUMELLARIA, being the oldest and most primitive Radiolaria, have for the most part either indifferent or multistable equilibrium; e.g., all Colloidea and Beloidea which have a spherical central capsule, and also most Spaeroidea. Among these primitive Centrostigma true spheres and endospherical polyhedra are represented in the utmost variety, and the regular polyhedra in particular. By the development of a vertical main axis these Centrostigma have also given rise to very numerous Centraxonia, which are usually isopolar, very rarely allopolar. Sometimes they are Monaxonia (circular in transverse section), sometimes Stauraxonia (polygonal in transverse section). The vertical main axis is longer in the Prunoidea, shorter in the Discoidea than any of the other axes. The Larcoidea are distinguished by their lentelliptical or triaxial ellipsoid form; the three different but isopolar axes corresponding with those of the rombic octahedron; but even among the Sphaeroidea, Prunoidea, and Discoidea, this form is sometimes produced by the differentiation of two different transverse axes at right angles to each other. Whilst these ground-forms (Centraxonia and Centrostigma) occur in the utmost variety among the SPUMELLARIA, the centroplanar (or true bilateral) ground form is entirely wanting.

48. The Ground-Forms of Acantharia.—In the small family Astrolophida, which contains the most archaic forms of the legion (Actinelius, Astrolophus), the ACANTHARLA show a direct relation to the most primitive SPUMELLARIA (Actissa), and like these have indifferent equilibrium; their central capsule is a sphere, their calymma an endospherical polyhedron, whose angles are indicated by the distal ends of the numerous equal radial spines. In the great majority of ACANTHARIA, however (all Acanthonida and Aecanthophracta), twenty radial spines are present, regularly distributed, according to Müller's icosacanthan law, in five parallel circles, each containing four crossed spines (p. 717). Usually the twenty spines are equal, and the ground-form is the quadratic octahedron, or a regular double pyramid with sixteen sides. But in some groups (the Amphilonchida and Prunophracta) two opposite equatorial spines are much more strongly developed than the other eighteen, and therefore the hydrotomical axis ins the equatorial plane is larger than the geotomical axis (p. 719); the isopolar stauraxonian form passes over into the allopolar, and the ground-form becomes the rhombic octahedron or the amphithect double pyramid (compare §§ 33 and 34, and p. 720). The centroplanar ground-form is entirely wanting in the ACANTHARIA.

49. The Ground-Forms of the Nassellaria.—The NASSELLARIA all possess monostatic ground-forms, inasmuch as by the very structure of their monopylean central capsule a vertical main axis is necessitated, whose basal pole occupies the porochora. The same arrangement is also for the most part clearly recognisable in the corresponding structure of the skeleton, which is generally either centraxon or centroplanar. Among their manifold skeletal forms different larger groups of ground-forms may be recognised according as the vertical allopolar main axis is crossed by differentiated transverse axes or not (Stauraxonia or Monaxonia) ; the former are either triradial or multiradial. The triradial, with three lateral or terminal radial apophyses, constitute the greater part of the NASSELLARIA, and have probably been derived originally from the triradial Plectoidea (Triplagia, Triplecta); a more careful examination, however (especially with reference to the structure of the cortinar septum), reveals the tact that the ground-form is not strictly regularly pyramidal (with three equal radii), but amphipleural (with two paired ventral and one unpaired dorsal radius), and that it usually passes over into a distinctly zygopleural form. The same holds true of the multiradial NASSELLARIA, where for the most part three interradial or six adradial (sometimes more) apophyses are intercalated between the three primary perradial ones; sometimes here also the ground-form is a quite regular hexagonal or nonagonal pyramid, but usually it is more or less amphithect or amphipleural. Among the eradial NASSELLARIA, which have no radial apophyses, the ground-form is sometimes allopolar monaxon (conical, ovoid, hemispherical, &c.), sometimes amphithect pyramidal (even in the simplest Stephanida, Archicircus, &c.), or sometimes distinctly zygopleural or bilateral (many Plectellaria).

50. The Ground-Forms of the Phaeodaria.—The PHAEODARIA agree with the NASSELARIA in the possession of a primitively centraxon ground-form, and like them are monostatic, since a vertical main axis whose basal pole passes through the astropyle is present, owing to the characteristic structure of their cannopylean central capsule. In the great majority of PHAEODARIA the spheroidal central capsule also possesses a pair of parapyke near the opposite apical pole of the main axis (Tripylea), and these determine (as the right and left secondary openings) an isopolar frontal axis. Hence, strictly speaking, in most PHAEODARIA the central capsule has the geometrical ground-form of the amphithect pyramid (as in the Ctenophora), with an allopolar vertical main axis, and two unequal, but isopolar, horizontal transverse axes. In many PHAEODARIA the skeleton also has this amphithect pyramidal ground-form, e.g., the bivalved Phaeoconchia and part of the Phaeogromia. On the contrary, in the rest of the PHAEODARIA the skeleton exhibits very various geometrical ground-forms, independent of that of the central capsule. In the Phaeosphaeria it forms preferably spheres or endospherical polyhedra, as also in the Castanellida and Circoporida among the Phaeogromia; among the Circoporida there are also seen with remarkable distinctness the regular polyhedra (especially the dodecahedron and icosahedron). Isopolar monaxonia are found among the Aulosphaerida (Aulatractus) and Orosphaerida; allopolar monaxonia among the Challengerida (Lithogromia). The Medusettida and Tuscarorida show various forms of regular pyramids (allopolar stauraxonia); and finally, the Challengerida are for the most part centroplanar or bilateral. Thus the PHAEODARIA present a great wealth of different geometrical ground-forms in the development of their skeleton, not in that of their central capsule.
 

¹ The numbers preceded by L. N. refer to the list of names of authors in the Bibliography on p. clxxvi.