The distinction between protists and histons is much more important than the familiar division of organisms into plants and animals, in respect of their fundamental forms and their configuration. For the protists, the unicellular organisms (without tissue) exhibit a much greater freedom and variety in the development of their fundamental forms than the histons, the multicellular tissue-forming organisms. In the protists (both protophyta and protozoa) the constructive force of the elementary organism, the individual cell, determines the symmetry of the typical form and the special form of its supplementation; but in the histons (both metaphyta and metazoa) it is the plasticity of the tissue, made up of a number of socially combined cells, that determines this. On the ground of this tectological distinction we may divide the whole organic world into four kingdoms (or sub-kingdoms), as the morphological system in the seventh table shows.
In respect of the general science of fundamental forms (promorphology), the most interesting and varied group of living things is the class of the radiolaria. All the various fundamental forms that can be distinguished and defined mathematically are found realized in the graceful flinty skeletons of these unicellular sea-dwelling protozoa. I have distinguished more than four thousand forms of them, and illustrated by one hundred and forty plates, in my monograph on the Challenger radiolaria [translated].
Only a very few organic forms seem to be quite irregular, without any trace of symmetry, or constantly changing their formless shape, as we find, for instance, in the amœbæ and the similar amœboid cells of the plasmodia. The great majority of organic bodies show a certain regularity both in their outer configuration and the construction of their various parts, which we may call "symmetry" in the wider sense of the word. The regularity of this symmetrical construction often expresses itself at first sight in the arrangement side by side of similar parts in a certain number and of a certain size, and in the possibility of distinguishing certain ideal axes and planes cutting each other at measurable angles. In this respect many organic forms are like inorganic crystals. The important branch of mineralogy that describes these crystalline forms, and gives them mathematical formulæ, is called crystallography. There is a parallel branch of the science of biological forms, promorphology, which has been greatly neglected. These two branches of investigation have the common aim of detecting an ideal law of symmetry in the bodies they deal with and expressing this in a definite mathematical formula.
The number of ideal fundamental forms, to which we may reduce the symmetries of the innumerable living organisms, is comparatively small. Formerly it was thought sufficient to distinguish two or three chief groups: (1) radial (or actinomorphic) types, (2) bilateral (or zygomorphic) types, and (3) irregular (or amorphic) types. But when we study the distinctive marks and differences of these types more closely, and take due account of the relations of the ideal axes and their poles, we are led to distinguish the nine groups or types which are found in the sixth table. In this promorphological system the determining factor is the disposition of the parts to the natural middle of the body. On this basis we make a first distinction into four classes or types: (1) the centrostigma have a point as the natural middle of the body; (2) the centraxonia a straight line (axis); (3) the centroplana a plane (median plane); and (4) the centraporia (acentra or anaxonia), the wholly irregular forms, have no distinguishable middle or symmetry.
I. Centrostigmatic Types.—The natural middle of the body is a mathematical point. Properly speaking, only one form is of this type, and that is the most regular of all, the sphere or ball. We may, however, distinguish two sub-classes, the smooth sphere and the flattened sphere. The smooth sphere (holospœra) is a mathematically pure sphere, in which all points at the surface are equally distant from the centre, and all axes drawn through the centre are of equal length. We find this realized in its purity in the ovum of many animals (for instance, that of man and the mammals) and the pollen cells of many plants; also cells that develop freely floating in a liquid, the simplest forms of the radiolaria (actissa), the spherical cœnobia of the volvocina and catallacta, and the corresponding pure embryonic form of the blastula. The smooth sphere is particularly important, because it is the only absolutely regular type, the sole form with a perfectly stable equilibrium, and at the same time the sole organic form which is susceptible of direct physical explanation. Inorganic fluids (drops of quicksilver, water, etc.) similarly assume the purely spherical form, as drops of oil do, for instance, when put in a watery fluid of the same specific weight (as a mixture of alcohol and water).
The flattened sphere, or facetted sphere (platnosphæra), is known as an endospherical polyhedron; that is to say, a many-surfaced body, all the corners of which fall in the surface of a sphere. The axes or the diameters, which are drawn through the angles and the centre, are all unequal, and larger than all other axes (drawn through the facets). These facetted spheres are frequently found in the globular silicious skeletons of many of the radiolaria; the globular central capsule of many spheroidea is enclosed in a concentric gelatine envelope, on the round surface of which we find a net-work of fine silicious threads. The meshes of this net are sometimes regular (generally triangular or hexagonal), sometimes irregular; frequently starlike silicious needles rise from the knots of the net-work (A-f, 1, 51, 91). The pollen bodies in the flower-dust of many flowering plants also often assume the form of facetted spheres.
II. Centraxonia Types.—The natural middle of the body is a straight line, the principal axis. This large group of fundamental forms consists of two classes, according as each axis is the sole fixed ideal axis of the body, or other fixed transverse axes may also be distinguished, cutting the first at right angles. We call the former uniaxial (monaxonia), and the latter transverse-axial (stauraxonia). The horizontal section (vertically to the chief axis) is round in the uniaxials and polygonal in the transverse-axial.
In the monaxonia the form is determined by a single fixed axis, the principle axis; the two poles may be either equal (isopola) or unequal (allopola). To the isopola belong the familiar simple forms which are distinguished in geometry as spheroids, biconvex, ellipsoids, double cones, cylinders, etc. A horizontal section, passing through the middle of the vertical chief axis, divides the body into two corresponding halves. On the other hand, many of the parts are unequal in size and shape in the allopola. The upper pole or vertex differs from the basal pole or ground surface; as we find in the oval form, the planoconvex lens, the hemisphere, the cone, etc. Both sub-classes of the monaxonia, the allopola (conoidal) and the isopola (spheroidal), are found realized frequently in organic forms, both in the tissue-cells of the histona and the independently living protists (A-f, 4, 84).
In the stauraxonia the vertical imaginary principal axis is cut by two or more horizontal cross-axes or radial-axes. This is the case in the forms which were formerly generally classed as regular or radial. Here also, as with the monaxonia, we may distinguish two sub-classes, isopola and allopola, according as the poles of the principal axis are equal or unequal.
Of the stauraxonia isopola we have, for instance, the double pyramids, one of the simplest forms of the octahedron. This form is exhibited very typically by most of the acantharia, the radiolaria in which twenty radial needles (consisting of silicated chalk) shoot out from the centre of the vertical chief axis. These twenty rays are (if we imagine the figure of the earth with its vertical axis) distributed in five horizontal zones, with four needles each, in this wise: two pairs cross at right angles in the equatorial zone, but on each side (in north and south hemispheres) the points of four needles fall in the tropical zone, and the points of four polar needles in the polar circles; twelve needles (the four equatorial and eight polar) lie in two meridian planes that are vertical to each other; and the eight tropical needles lie in two other meridian planes which cross the former at an angle of forty-five degrees. In most of the acantharia (the radial acanthometra and the mailed acanthophracta)—there are few exceptions—this remarkable structural law of twenty radial needles is faithfully maintained by heredity. Its origin is explained by adaptation to a regular attitude which the sea-dwelling unicellular body assumes in a certain stage of equilibrium (A-f, 21, 41). If the points of the real needles are connected by imaginary lines, we get a polyhedrical body, which may be reduced to the form of a regular double pyramid. This typical form of the equipolar stauraxonia is also found in other protists with a plastic skeleton, as in many diatomes and desmidiacea (A-f, 24). It is more rarely found embodied in the tissue-cells of the histona.