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 arise merely, as Jäger expresses it, from that concentric lamination, or, with Perrier, from that polymorphism of the members of the colony, which is associated with organic and social existence. The idea of the antimere is omitted, as being essentially a promorphological conception (for a medusoid or a starfish, though of widely distinct order of individuality, is equally so divisible); that of the metamere is convenient to denote the secondary units of a linear tertiary individual; the term persona, however, seems unlikely to survive, not only on account of its inseparable psychological connotations, but because it has been somewhat vaguely applied alike to aggregates of the second and third order; and the term colony, corm or deme may indifferently be applied to those aggregates of primary, secondary, tertiary or quaternary order which are not, however, integrated into a whole, and do not reach the full individuality of the next higher order. The term zooid is also objectionable as involving the idea of individualized organs, a view natural while the medusoid gonophores of a hydrozoon were looked at as evolved of its homologue in Hydra, whereas the latter may be a degenerate form of the former. Passing to the vegetable world, here, as before, the cell is the unit of the first order, while aggregates representing almost every stage in the insensible evolution of a secondary unit are far more abundant than among animals. Complete unity of the second order can hardly be allowed to the thallus, which Spencer proposes to compound and integrate into tertiary aggregates—the higher plants; as in animals, the embryological method is preferable, both as avoiding gratuitous hypothesis and as leading to direct results. Such a unit is clearly presented by the embryo of higher plants in which the cell-aggregate is at once differentiated into parts and integrated into a whole. Such an embryo possesses axis and appendages as when fully developed (fig. 2). The latter, however, being as organs mere lateral expansions of the concentric layers into which the plant embryo, like the animal, is differentiated, and so neither stages of evolution nor capable of separate existence, are not entitled to individual rank. The embryo, the bud, shoot or uniaxial plant, all thus belong to the second order of individuality, like the hydroid they resemble. Like the lower coelenterates, too, aggregates of such axes are formed by branching out from their low degree of integration. Such colonies can hardly be termed individuals of the third, much less of higher, order, at least without somewhat abandoning that unity of treatment of plants and animals without which philosophical biology disappears. Individuality of the second order is most fully reached by the flower—the most highly differentiated and integrated form of axes and appendages. Such a simple inflorescence as a raceme or umbel approximates to unity of the third order, to which a composite flower-head must be admitted to have attained while a compound inflorescence is on the way to a yet higher stage. If, as seems probable, a nomenclature be indispensable for clear expression, it may be simply arranged in conformity with this view. Starting from the unit of the first order, the plastid or monad, and terming any undifferentiated aggregate a deme, we have a monad-deme integrating into a secondary unit or dyad, this rising through dyad-demes, into a triad, this forming triad-demes, and these when differentiated becoming tetrads, the botryllus-colony with which the evolution of compound individuality terminates being a tetrad-deme. The separate living form, whether monad, dyad, triad, or tetrad, requires also some distinguishing name, for which persona will probably ultimately be found most appropriate, since such usage is most in harmony with its inevitable physiological and psychological connotations, while the genealogical individual of Gallesio and Huxley, common also to all the categories, may be designated with Haeckel the ovum-product or ovum-cycle, the complete series of forms needed to represent the species being the species-cycle (though this coincides with the former save in cases where the sexes are separate, or polymorphism occurs). For such a peculiar case as Diplozoon paradoxum, where two separate forms of the same species coalesce, and still more for such heterogeneous individuality as that of a lichen, where a composite unit arises from the union of two altogether distinct forms—fungus and alga—yet additional categories and terms are required.

Promorphology.—Just as the physiologist constantly seeks to interpret the phenomena of function in terms of mechanical, physical, and chemical laws, so the morphologist is tempted to inquire whether organic as well as mineral forms are not alike reducible to simple mathematical law. And just as the crystallographer constructs an ideally perfect mathematical form from an imperfect or fragmentary crystal, so the morphologist has frequently attempted to reduce the complex-curved surfaces of organic beings to definite mathematical expression. Canon Moseley (Phil. Trans., 1838) succeeded in showing, by a combination of measurement and mathematical analysis, that the curved surface of any turbinated or discoid shell might be considered as generated by the revolution, about the axis of the shell, of a curve, which continually varied its dimensions

according to the law of the logarithmic spiral. For Goodsir this logarithmic spiral, now carved on his tomb, seemed a fundamental expression of organic curvature and the dawn of a new epoch in natural science—that of the mathematical investigation of organic form—and his own elaborate measurements of the body, its organs, and even its component cells seemed to yield, now the triangle, and again the tetrahedron, as the fundamental form. But such supposed results, savouring more of the Naturphilosophie than of sober mathematics, could only serve to discourage further inquiry and interest in that direction. Thus we find that even the best treatises on botany and zoology abandon the subject, satisfied with merely contrasting the simple geometrical ground-forms of crystals with the highly curved and hopelessly complicated lines and surfaces of the organism.

But there are other considerations which lead up to a mathematical conception of organic form, those namely of symmetry and regularity. These, however, are usually but little developed, botanists since Schleiden contenting themselves with throwing organisms into three groups—first, absolute or regular; second, regular and radiate; third, symmetrical bilaterally or zygomorphic—the last being capable of division into two halves only in a single plane, the second in two or more planes, the first in none at all. H. C. C. Burmeister, and more fully H. G. Bronn, introduced the fundamental improvement of defining the mathematical forms they sought not by the surfaces but by axes and their poles; and Haeckel has developed the subject with an elaborateness of detail and nomenclature which seems unfortunately to have impeded its study and acceptance, but of which the main results may, with slight variations chiefly due to Jäger (Lehrb. d. Zool. i. 283), be briefly outlined.

A. ANAXONIA: Forms destitute of axes, and consequently wholly irregular in form, e.g. Amoebae and many sponges.

B. AXONIA: Forms with definite axes.

I., all axes equal.

(a) Spheres, where an indefinite number of equal axes can be drawn through the middle point, e.g. Sphaerozoum.

(b) Polyhedra, with a definite number of like axes.

Of these a considerable number occur in nature, for example, many radiolarians (fig. 3), pollen-grains, &c., and they are again classifiable by the number and regularity of their faces. II. , where all the parts are arranged round a main axis, and of these we distinguish—

1. Monaxonia, with not more than one definite axis. Here are distinguished (a) those with similar poles, spheroid (Coccodiscus) and cylinder (Pyrosoma) and (b) those with dissimilar poles, cone (Conulina).

2. Stauraxonia, where, besides the main axes, a definite number of secondary axes are placed at right angles, and the stereometric ground-form becomes a pyramid. Here, again, may be distinguished (a) those with poles similar, Stauraxonia homopola, where the stereometric form is the double pyramid (fig. 4), and (b) those with poles dissimilar, Stauraxonia heteropola, where the stereometric form is the single pyramid, and where we distinguish a basal, usually oral, pole from an apical, aboral or anal pole. The bases of these may be either regular or irregular polygons, and thus a new classification into Homostaura and Heterastaura naturally arises.

The simpler group, the Homostaura, may have either an even or an odd number of sides, and thus among the Homostaura we have even-sided and odd-sided, single and double pyramids. In those Homostaura with an even number of sides, such as medusae, the radial and inter-radial axes have similar poles; but in the series with an odd number of sides, like most echinoderms, each of the transverse axes is half radial and half semi-radial (fig. 5). Of the group of regular double pyramids the twelve-sided pollen-grain of Passiflora (fig. 4) may be taken as an example, having the ground-form of the hexagonal system, the hexagonal dodecahedron. Of the equal even-sided single pyramids (Heteropola homostaura), Alcyonium, Geryonia, Aurelia may be taken as examples of the eight-sided, six-sided, and four-sided, pyramids While those with an odd number of sides may be illustrated by Ophiura or Primula with five sides, and the flower of lily or rush with three sides.

In the highest and most complicated group, the Heterostaura,