Both systems of lines, the generating spirals (as these latter may be called), and the closed generating curves cor­re­spon­ding to successive margins or lips of the shell, may be easily traced in a great variety of cases. Thus, for example, in Dolium, Eburnea, and a host of others, the generating spirals are beautifully marked out

Fig. 266. 1, Harpa; 2, Dolium. The ridges on the shell correspond in (1) to generating curves, in (2) to generating spirals.

by ridges, tubercles or bands of colour. In Trophon, Scalaria, and (among countless others) in the Ammonites, it is the successive generating curves which more conspicuously leave their impress on the shell. And in not a few cases, as in Harpa, Dolium perdix, etc., both alike are conspicuous, ridges and colour-bands intersecting one another in a beautiful isogonal system. {527}

In the complete math­e­mat­i­cal formula (such as I have not ventured to set forth[509]) for any given turbinate shell, we should have, accordingly, to include factors for at least the following elements: (1) for the specific form of the section of the tube, which we have called the generating curve; (2) for the specific rate of growth of this generating curve; (3) for its specific rate of angular rotation about the pole, perpendicular to the axis; (4) in turbinate (as opposed to nautiloid) shells, for its rate of shear, or screw-translation parallel to the axis. There are also other factors of which we should have to take account, and which would help to make our whole expression a very complicated one. We should find, for instance, (5) that in very many cases our generating curve was not a plane curve, but a sinuous curve in three dimensions; and we should also have to take account (6) of the inclination of the plane of this generating curve to the axis, a factor which will have a very important influence on the form and appearance of the shell. For instance in Haliotis it is obvious that the generating curve lies in a plane very oblique to the axis of the shell. Lastly, we at once perceive that the ratios which happen to exist between these various factors, the ratio for instance between the growth-factor and the rate of angular revolution, will give us endless possibilities of permutation of form. For instance (7) with a given velocity of vectorial rotation, a certain rate of growth in the generating curve will give us a spiral shell of which each successive whorl will just touch its predecessor and no more; with a slower growth-factor, the whorls will stand asunder, as in a ram’s horn; with a quicker growth-factor, each whorl will cut or intersect its predecessor, as in an Ammonite or the majority of gastropods, and so on (cf. p. [541]).

In like manner (8) the ratio between the growth-factor and the rate of screw-translation parallel to the axis will determine the apical angle of the resulting conical structure: will give us the difference, for example, between the sharp, pointed cone of Turritella, the less acute one of Fusus or Buccinum, and the {528} obtuse one of Harpa or Dolium. In short it is obvious that all the differences of form which we observe between one shell and another are referable to matters of degree, depending, one and all, upon the relative magnitudes of the various factors in the complex equation to the curve.

The paper in which, nearly eighty years ago, Canon Moseley thus gave a simple math­e­mat­i­cal expression to the spiral forms of univalve shells, is one of the classics of Natural History. But other students before him had come very near to recognising this math­e­mat­i­cal simplicity of form and structure. About the year 1818, Reinecke had suggested that the relative breadths of the adjacent whorls in an Ammonite formed a constant and diagnostic character; and Leopold von Buch accepted and developed the idea[510]. But long before, Swammerdam, with a deeper insight, had grasped the root of the whole matter: for, taking a few diverse examples, such as Helix and Spirula, he shewed that they and all other spiral shells whatsoever were referable to one common type, namely to that of a simple tube, variously curved according to definite math­e­mat­i­cal laws; that all manner of ornamentation, in the way of spines, tuberosities, colour-bands and so forth, might be superposed upon them, but the type was one throughout, and specific differences were of a geometrical kind. “Omnis enim quae inter eas animad­verti­tur dif­fer­en­tia ex sola nascitur di­versi­tate gyra­tionum: qui­bus si in­super ex­terna quae­dam adjun­gun­tur orna­menta pin­narum, sinuum, an­fractuum, plan­i­tierum, eminen­tiarum, pro­fun­di­ta­tum, ex­ten­sionum, impres­sionum, cir­cum­volu­tionum, colo­rum­que: ... tunc dein­ceps facile est, quarum­cum­que Coch­learum figuras geo­met­ricas, cur­vosque, ob­liquos atque rec­tos angulos, ad uni­cam om­nes speciem re­digere: ad ob­longum videlicet tubulum, qui vario modo curvatus, crispatus, extrorsum et introrsum flexus, ita concrevit[511].” {529}

Fig. 267. D’Orbigny’s Helicometer.