Another argument against the interpretation of the sections as spirals is that in any one line the individual spots are of roughly uniform size. This means of course that the spiral has been cut by a plane parallel to the tangent plane. This might happen once, just as once Doctor Walcott cut all six segments of a single endopodite, but that it should happen repeatedly is highly improbable. Moreover, there is a limit to the diameter of the section which may be made from these slender spirals. Most of the spots have one diameter about one half greater than the other, but others are from three to six times as long as wide. These last could obviously be cut only from a very large spiral, and they are therefore interpreted by Walcott as setæ of epipodites. Yet all gradations are found among the sections, from the long setæ to the short dots. (See pl. 27, 1918.) In referring to one slice, Walcott says (1918, p. 152):

In the latter figure and in figure 13, plate 27, the setæ of several epipodites appear to have been cut across so as to give the effect of long rows of setæ. The same condition occurs in specimens of Marrella when the setæ of several exopodites are matted against each other.

Fig. 12.—A slice of Ceraurus pleurexanthemus in which the exopodite happened to be cut in such a way as to show a part of the shaft and some of the setæ in longitudinal section. Specimen 80. × 4.

This is certainly an apt comparison, and equally true if Neolenus, Triarthrus, or Cryptolithus were substituted for Marrella.

Now consider the "epipodites." They are well shown in Calymene in the specimens illustrated on plate 27, figure 11 (1918), and plate 3, figure 3 (1881), and less clearly in one or two others. Slices 22 (pl. 27, fig. 12, 1918) and 80 (our [fig. 12]) show what is called the same organ in Ceraurus. It will be noted that all of these slices are cut in the same way, that is, more or less parallel to the under surface of the head, or, at any rate, on a plane parallel to a plane which would be tangent to the axial portion of the coiled shell. The sections which show the spirals best are those which are cut by a plane perpendicular to the long axis of the body. If one were to attempt to cut an enrolled Triarthrus in such a way as to get a section showing the length of the setæ, one would not cut a section perpendicular to the axis of the animal, nor, in fact, would he cut one parallel to the ventral plane, but it is obvious that in this latter type of section he would stand a better chance of finding a part of the plane of the exopodite coincident with the plane of his section than in the former. And that seems to be what has happened in these sections of Calymene and Ceraurus. If the exopodites were preserved, transverse sections were bound to cut across many sets of fringes, and the resultant slice would show transverse sections of the setæ as a series of overlapping spots. A few fortunately located sections in a more nearly horizontal plane might cut the setæ and occasionally the shaft of one or more exopodites in the longitudinal plane, and the resulting effect would produce the so-called "epipodites." A careful study has shown that no one of these epipodites is complete, and they do not have the palmate form shown in Walcott's figures.

And the last and most important argument against the spiral appendages is that certain slices, of both Calymene and Ceraurus, show definitely exopodites of exactly the type found in other trilobites. These are discussed later in the detailed description of the various slices.

If these series of spots are interpreted on the basis of the known structure of Triarthrus, they are of course a series of sections through the setæ of the exopodites. It will be shown in Part IV that these setæ are not circular in section, but flattened, in Cryptolithus even blade-like, and that they overlap one another. A section across them would give the same general appearance as, for instance, that shown in figures 4, 6, 9, and 10 of Walcott's plate 3 (1881).

When both endopodites and the "spiral branchiæ" are present in the same section (pl. 1, fig. 4; pl. 2, figs. 1, 2), the "spiral branchiæ" are dorsal to the endopodites, as the setæ of the exopodites would be expected to be. The specimens which show the clear spots connected, and which suggest a spiral (pl. 3, fig. 5), may seem at first sight to bear evidence against this interpretation, but one has only to think of the effect of cutting a section along the edge where the setæ are attached to the shaft of the exopodite of Triarthrus to see that such a zig-zag effect is entirely possible. One would expect to cut just this position only rarely, and, in fact, the zig-zags are seen in only three or four sections. The bifurcation of the basal segment of the "spiral branchiæ" (pl. 3, fig. 10, 1881) is probably more apparent than real, if indeed these basal segments have anything to do with the succeeding one.