The eight tropical spines lie between the eight polar and the four equatorial spines, four in each hemisphere; their distal points fall in two parallel circles, which correspond exactly to the two tropics of the globe. Therefore the four northern tropical spines may be called "canceral spines" (as their ends fall in the Tropic of Cancer) and the four southern correspondingly "capricornal spines" (as their points lie in the Tropic of the Capricorn). In the figures of the Pls. [131]-[140] (as well as in my Monograph, 1862, Taf. xv.-xxii.) the four northern or canceral spines are marked by the characters b1 to b4, and the four southern or capricornal spines by the characters d1 to d4. Also the eight tropical spines lie (crossed in pairs) in two meridian planes; they do not lie, however, in those perradial planes, in which are placed the twelve other spines; but in two different meridian planes, crossing the former at angles of 45°; we call these the "secondary" or "interradial" meridian planes. Each of these planes is determined by the spineless axis and by two crossed interradial or secondary axes; in each of the latter lie two opposite tropical spines. In the first interradial meridian plane lie b1 and b3, d1 and d3, in the second b2 and b4, d2 and d4.
It is a most interesting and important fact, that in all Icosacantha (Acanthonida and Acanthophracta) this regular disposition of the twenty spines (in five parallel zones and four meridian planes) becomes constantly preserved by heredity, whilst the form and size of the different spines are extremely varied by adaptation.
Only in a minority of the Icosacantha are all twenty spines perfectly equal or nearly equal in size and form; and then it is often very difficult to distinguish the different zones in their disposition. But in far the greater part the size or the form of the twenty spines becomes different in different zones; and then we can commonly distinguish easily the five different zones. Firstly, in all Quadrilonchida and Dorataspida, the four equatorial are distinguished from the sixteen other spines either by form or by size, and often in a very remarkable degree. As soon as these four principal spines are recognised, it is easy to determine also the sixteen others; for the eight polar spines lie in the same two (perradial) meridian planes as the former, whilst the eight tropical spines lie in two different (interradial) meridian planes, intersecting the two former at angles of 45°. Commonly, therefore, this distinction is rather easy.
In the majority of the Icosacantha all four equatorial spines are exactly of the same form and size. But in four families the two opposite spines of one equatorial axis are much larger, or of another form, than those of the crossing axis. This is the case in the Amphilonchida, Belonaspida, Hexalaspida, and Diploconida. Therefore we here call the major equatorial axis (with larger spines) the "hydrotomical axis," and the minor axis (with smaller spines) the "geotomical axis." Correspondingly, the meridian plane, in which the two larger equatorial spines are placed (c1, c3) and the appertaining four polar spines (a1, a3, e1, e3) may be called the "hydrotomical plane"; in the remarkable family of Hexalaspida (Pl. [139]) all six spines of this hydrotomical plane are much larger than the other fourteen. Perpendicular to this plane is the second perradial meridian plane, which we call the "geotomical plane"; in it lie the two smaller equatorial spines (c2, c4) and the corresponding four polar spines (a2, a4, e2, e4). In some Hexalaspida (Hexonaspis and Hexacolpus) the six spines of the hydrotomical plane become so preponderant that the other fourteen spines appear rudimentary; and in some of them the two equatorial spines of the hydrotomical plane are much larger than the four polar spines of the same plane. This curious relation reaches its maximum in the Diploconida (Pl. [140]).
The different development of the two equatorial axes (of the larger hydrotomical and the smaller geotomical axis) is the first and most important cause of the peculiar forms, which are produced in the four cited families. We derive these terms also from the metaphor of the terrestrial globe. The hydrotomical plane is that meridian plane of the globe which intersects almost only the water-hemisphere (the island of Ferro in the Atlantic, the island of Pandora in the Pacific). Perpendicular to this is the geotomical plane, the meridian of which intersects great land-masses in both hemispheres (Bombay in India, Athabasca in Canada). Both poles of the smaller geotomical axis are everywhere equal (the East Indian and the Western American). However, both poles of the larger hydrotomical axis (the eastern Atlantic and the western Pacific) are in some genera very different, e.g., in Amphibelone among the Amphilonchida, and in Zygostaurus among the Quadrilonchida. In this case we call the anterior (commonly more developed) pole of the hydrotomical axis the frontal pole, the opposite posterior (commonly smaller) the caudal pole (Pl. [131], figs. 7, 8; Pl. [132], figs. 9, 10). On both sides of these (right and left) lie symmetrically the two equal poles of the geotomical lateral axis.
The promorphology of the Acantharia demonstrates that the geometrical fundamental form in those groups is different. In the majority of the Acantharia, where the two equatorial axes are equal, that form is a double square-pyramid or a "quadrate octahedron"; the four equal equatorial spines indicate the two diagonals of the square, which is the common base of the united regular four-sided pyramids; their common axis is the spineless axis of the body; the ends of the polar spines fall on the edges of the pyramids, while the ends of the tropical spines fall on the halving lines of their faces. However, in those Acantharia in which the two equatorial axes become different, the square double pyramid becomes changed into a rhombic double pyramid; the common base of the united pyramids is thus a rhombus; the hydrotomical axis is the larger, the geotomical axis the smaller diagonal of the rhombus.
Opposed to the Icosacantha, under the name "Adelacantha," is the small group of Actinelida, in which the number and disposition of the radial spines is variable, not determined by the Müllerian law. Probably this group is the common ancestral stock, from which the Icosacantha have been derived by gradual development of their peculiar disposition. Probably the oldest and most primitive form of all Acantharia is Actinelius, in which a variable and undetermined (often very large) number of radial spines is united in one common central point, and therefore forms a needle-sphere. Whilst here all spines (often more than a hundred) are of equal size and form, in the nearly allied Astrolophus large and small spines are intermingled. Both genera together form the small ancestral family of Astrolophida. In the strange family of Litholophida the radial spines do not radiate within a spherical space (equally disposed in all directions), but within a quadrant or even an octant, forming a conical brush or pencil.
One very remarkable form of Actinelida is Actinastrum, forming the transition from these Adelacantha to the common regular Icosacantha. In the two observed species of Actinastrum we find thirty-two radial spines, twenty of which are disposed after the Müllerian law, as in the Icosacantha. The other twelve are four interradial equatorial spines (lying in the two secondary meridian planes) and eight perradial tropical spines (lying in the two primary meridian planes). Therefore here in each primary meridian plane are placed ten spines (two equatorial, four tropical, and four polar spines), whereas in each secondary meridian plane are placed six spines (two equatorial and four tropical). But here also all thirty-two spines are so regularly placed that their free distal ends fall into five parallel zones, four in each polar zone, eight in each tropical zone, and eight in the equatorial zone.
The Central Junction of the radial spines in the Acantharia becomes effected in four different ways:—(1) by simple apposition of the pyramidal central ends or bases; (2) by a basal leaf-cross, or by broad wings, four on each spine, supported one upon the other; (3) by a central concrescence of the meeting bases of all the twenty spines, growing perfectly together; and (4) by a concrescence in pairs of every two opposite spines. The most common and probably the original mode of junction is the first—by pyramidal apposition; the spines at the central base are pointed in the form of a pyramid, and the triangular faces of the neighbouring pyramids are simply placed upon one another. Often the small basal pyramids are imperfectly separated from the spines by an annular constriction. Commonly the basal pyramids of the four equatorial spines are six-sided, those of the sixteen other spines five-sided.
The second mode of junction, by a basal leaf-cross, is developed from the first and appears as a strengthening or a mechanical elaboration of it. Immediately above the basal pyramid arise from its radial edges four thin and broad triangular leaves or wings, and the meeting edges of the neighbouring wings are in apposition one with the other, so that between the bases of every three or four neighbouring spines a hollow pyramidal space remains open. The apex of such a pyramidal space is directed towards the centre of the body, but separated from it by the small basal pyramid; its open base is directed outwards. The twenty-two hollow pyramidal spaces are disposed regularly in four different groups:—(A) Four equatorial spaces, four-sided, each limited by two equatorial and two tropical spines (one canceral and one capricornal); (B) eight perizonal spaces (four northern and four southern), four-sided, each limited by one equatorial, two tropical, and one polar spine; (C) eight peripolar spaces (four northern and four southern), three-sided, each limited by one tropical and two polar spines; (D) two polar spaces (one northern and one southern), four-sided, each limited by four neighbouring polar spines.