As an example of the method of treatment adopted we may take Oken's matured account of the composition of the cranial vertebræ, as given in the English translation of his Lehrbuch. "To a perfect vertebra," he says, "belong at least five pieces, namely, the body, in front the two ribs, behind the two arches or spinous processes" (p. 370). In the cervical vertebræ the transverse processes represent the ribs. The skull consists of four vertebræ, the occipital, the parietal, the frontal and the nasal, or, named after the sense with which each is associated, the auditory, the lingual, the ocular and the olfactory. The "bodies" of these vertebræ are the body of the occipital (basioccipital), the two bodies of the sphenoid (basi- and pre-sphenoid), and the vomer. The transverse processes of each are the condyles of the occipitals (exoccipitals), the alæ of the two sphenoids (alisphenoids and orbitosphenoids) and the lateral surfaces of the vomer. The arches or spinous processes are the occipital crest, the parietals, the frontals, and the nasals.

The cranium is thus composed of four rings of bone, each composed of the typical elements of a vertebra.

The arbitrary nature of the comparison is obvious enough. As Cuvier pointed out in the posthumous edition of his Leçons, it is only the occipital segment that shows any real analogy with a vertebra—an analogy which Cuvier ascribed to similarity of function. He admitted a faint resemblance of the parietal segment to a vertebra:—"The body of the sphenoid does indeed look like a repetition of the basioccipital, but having a different function it takes on another form, especially above, by reason of its posterior clinoid apophyses."[157] He denied the resemblance of the frontal and nasal "vertebræ" to true vertebræ, pointing out that both parietals and frontals are bones specially developed for the purpose of roofing over and protecting the cerebrum.

A very curious development was given to the vertebral theory by K. G. Carus, who seems to have taken as his text a saying of Oken's, that the whole skeleton is only a repeated vertebra.[158] His system is worthy of some consideration, for he tries to work out a geometry of the skeleton.[159]

His method of deduction is a good example of pure Naturphilosophie. Life, he says, is the development of something determinate from something indeterminate. A finite indeterminate thing, that is, a liquid, must take a spherical form if it is to exist as an individual. Hence the sphere is the prototype of every organic body. Development takes place by antagonism, by polarity, typically by the division and multiplication of the sphere. In the course of development the sphere may change, by expansion into an egg-shaped body, or by contraction into a crystalline form, the changes due to expansion being typical of living things, those due to contraction being typical of dead. At the surface of the primitive living sphere is developed the protective dermatoskeleton, which naturally takes the shape of a hollow sphere; round the digestive cavity which is formed in the living sphere is developed the splanchnoskeleton; round the nervous system (which is, as it were, the animal within the animal) is developed the neuroskeleton. All skeletal formations belong to one or other of these systems.

Carus defines his aim to be the discovery of the inner law which presides over the formation of the skeleton throughout the animal kingdom; he desires to know "how such and such a formation is realised in virtue of the eternal laws of reason" (iii., p. 93). Here we touch the kernel of Naturphilosophie—the search for rational laws which are active in Nature; the discontent with merely empirical laws.

The thesis which Carus sustains is that all forms of skeleton, whether of dermatoskeleton, splanchnoskeleton, or neuroskeleton, can be deduced from the hollow sphere, which is the primary form of any skeleton whatsoever (p. 95). That means, put empirically, that every skeleton can be represented schematically by a number of hollow spheres, suitably modified in shape, and suitably arranged. The chief modification in shape exhibited by bones is one which is intermediate between the organic and the crystalline series of modifications of the sphere. The organic modifications are bounded by curved lines, the crystalline by straight; the intermediate partly by curved and partly by straight lines. They are the dicone (the shape of a diabolo) and the cylinder. These forms must necessarily be of importance for the skeleton, which is intermediate between the organic and the inorganic. "The dicone embodies the real significance of the bone," writes Carus. Each dicone and cylinder composing the skeleton is called by Carus a vertebra.

We may expect then all skeletons to be composed of spheres, cylinders and dicones in diverse arrangements. Nature being infinite, all the possible types of arrangement of these elements must exist in the test or skeleton of some animal, living, fossil, or to come (p. 127). One conceives easily what the main types of skeleton must be. In some animals, e.g., sea-urchins, the skeleton is a simple sphere; in others, e.g., starfish, secondary rows of spheres radiate out from a central sphere or ring; in annulate animals the skeleton consists of a row of partially fused spheres.

In Vertebrates the arrangement is more complex. There are first the protovertebral rings of the dermatoskeleton, these being principally the ribs, limb-girdles, and jaws. Round the central nervous system are developed the deutovertebral rings of the neuroskeleton (vertebræ in the ordinary sense). The apophyses and bodies of the vertebræ, and the bones of the members[160] are composed of columns of tritovertebræ, or vertebræ of the third order. Thus the whole vertebrate skeleton is a particular arrangement of vertebræ, which in their turn are modifications of the primary hollow sphere.

The German transcendentalists were more or less contemporary with E. Geoffroy, and no doubt influenced him, especially in his later years, as they certainly did his follower Serres. Oken indeed wrote, in a note[161] appended to Geoffroy's paper on the vertebral column of insects, that "Mr Geoffroy [sic] is without a doubt the first to introduce in France Naturphilosophie into comparative anatomy, that is to say, that philosophy one of whose doctrines it is to seek after the signification of organs in the scale of organised beings." This is, however, an exaggeration, for Geoffroy was primarily a morphologist, whereas the morphology of the German transcendentalists was only a side-issue of their Naturphilosophie.