The Univalve Shell: a summary

The surface of any shell, whether discoid or turbinate, may be imagined to be generated by the revolution about a fixed axis of a closed curve, which, remaining always geometrically similar to itself, increases continually its dimensions: and, since the rate of growth of the generating curve and its velocity of rotation follow the same law, the curve traced in space by cor­re­spon­ding points {554} in the generating curve is, in all cases, a logarithmic spiral. In discoid shells, the generating figure revolves in a plane perpendicular to the axis, as in Nautilus, the Argonaut and the Ammonite. In turbinate shells, it slides continually along the axis of revolution, and the curve in space generated by any given point partakes, therefore, of the character of a helix, as well as of a logarithmic spiral; it may be strictly entitled a helico-spiral. Such turbinate or helico-spiral shells include the snail, the periwinkle and all the common typical Gastropods.

The generating figure, as represented by the mouth of the shell, is sometimes a plane curve, of simple form; in other and more numerous cases, it becomes more complicated in form and its boundaries do not lie in one plane: but in such cases as these we

Fig. 283. Section of a spiral, or turbinate, univalve, Triton corrugatus, Lam. (From Woodward.)

may replace it by its “trace,” on a plane at some definite angle to the direction of growth, for instance by its form as it appears in a section through the axis of the heli­coid shell. The gen­er­ating curve is of very various shapes. It is circular in Scalaria or Cyclostoma, and in Spirula; it may be con­si­dered as a seg­ment of a circle in Natica or in Plan­orbis. It is ap­prox­i­mate­ly tri­an­gular in Conus, and rhom­boidal in Solarium or Potam­ides. It is very com­monly more or less elliptical: the long axis of the el­lipse being parallel to the axis of the shell in Oliva and Cypraea; all but per­pen­di­cu­lar to it in many Trochi; and oblique to it in many well-marked cases, such as Sto­ma­tella, La­mel­laria, Si­gar­etus hal­io­toides (Fig. ) and Haliotis. In Nautilus pom­pi­lius it is ap­prox­i­mate­ly a semi-ellipse, and in N. um­bil­i­catus rather more than a semi-ellipse, the long axis lying in both cases per­pen­di­cu­lar to the axis of the shell. Its {555} form is seldom open to easy math­e­mat­i­cal ex­pres­sion, save when it is an actual circle or ellipse; but an exception to this rule may be found in certain Am­mo­nites, for­ming the group “Cordati,” where (as Blake points out) the curve is very nearly rep­re­sent­ed by a cardioid, whose equation is r = a(1 + cos θ).

The generating curve may grow slowly or quickly; its growth-factor is very slow in Dentalium or Turritella, very rapid in Nerita, or Pileopsis, or Haliotis or the Limpet. It may contain the axis in its plane, as in Nautilus; it may be parallel to the axis, as in the majority of Gastropods; or it may be inclined to the axis, as it is in a very marked degree in Haliotis. In fact, in Haliotis the generating curve is so oblique to the axis of the shell that the latter appears to grow by additions to one margin only (cf. Fig. ), as in the case of the opercula of Turbo and Nerita referred to on p. ; and this is what Moseley supposed it to do.

Fig. 284. A, Lamellaria perspicua; B, Sigaretus haliotoides.
(After Woodward.)

The general appearance of the entire shell is determined (apart from the form of its generating curve) by the magnitude of three angles; and these in turn are determined, as has been sufficiently explained, by the ratios of certain velocities of growth. These angles are (1) the constant angle of the logarithmic spiral (α); (2) in turbinate shells, the enveloping angle of the cone, or (taking half that angle) the angle (θ) which a tangent to the whorls makes with the axis of the shell; and (3) an angle called the “angle of retardation” (β), which expresses the retardation in growth of {556} the inner as compared with the outer part of each whorl, and therefore measures the extent to which one whorl overlaps, or the extent to which it is separated from, another.

The spiral angle (α) is very small in a limpet, where it is usually taken as = 0°; but it is evidently of a significant amount, though obscured by the shortness of the tubular shell. In Dentalium it is still small, but sufficient to give the appearance of a regular curve; it amounts here probably to about 30° to 40°. In Haliotis it is from about 70° to 75°; in Nautilus about 80°; and it lies between 80° and 85°, or even more, in the majority of Gastropods.

The case of Fissurella is curious. Here we have, apparently, a conical shell with no trace of spiral curvature, or (in other words) with a spiral angle which approximates to 0°; but in the minute embryonic shell (as in that of the limpet) a spiral convolution is distinctly to be seen. It would seem, then, that what we have to do with here is an unusually large growth-factor in the generating curve, which causes the shell to dilate into a cone of very wide angle, the apical portion of which has become lost or absorbed, and the remaining part of which is too short to show clearly its intrinsic curvature. In the closely allied Emarginula, there is likewise a well-marked spiral in the embryo, which however is still manifested in the curvature of the adult, nearly conical, shell. In both cases we have to do with a very wide-angled cone, and with a high retardation-factor for its inner, or posterior, border. The series is continued, from the apparently simple cone to the complete spiral, through such forms as Calyptraea.

The angle α, as we have seen, is not always, nor rigorously, a constant angle. In some Ammonites it may increase with age, the whorls becoming closer and closer; in others it may decrease rapidly, and even fall to zero, the coiled shell then straightening out, as in Lituites and similar forms. It diminishes somewhat, also, in many Orthocerata, which are slightly curved in youth, but straight in age. It tends to increase notably in some common land-shells, the Pupae and Bulimi; and it decreases in Succinea.

Directly related to the angle α is the ratio which subsists between the breadths of successive whorls. The following table gives a few illustrations of this ratio in particular cases, in addition to those which we have already studied. {557}

Ratio of breadth of consecutive whorls.

Pointed Turbinates Obtuse Turbinates and Discoids

Telescopium fuscum 1·14 Conus virgo 1·25

Acus subulatus 1·16 Conus litteratus 1·40

*Turritella terebellata 1·18 Conus betulina 1·43

*Turritella imbricata 1·20 *Helix nemoralis 1·50

Cerithium palustre 1·22 *Solarium perspectivum 1·50

Turritella duplicata 1·23 Solarium trochleare 1·62

Melanopsis terebralis 1·23 Solarium magnificum 1·75

Cerithium nodulosum 1·24 *Natica aperta 2·00

*Turritella carinata 1·25 Euomphalus pentangulatus 2·00

Acus crenulatus 1·25 Planorbis corneas 2·00

Terebra maculata (Fig. ) 1·25 Solaropsis pellis-serpentis 2·00

*Cerithium lignitarum 1·26 Dolium zonatum 2·10

Acus dimidiatus 1·28 *Natica glaucina 3·00

Cerithium sulcatum 1·32 Nautilus pompilius 3·00

Fusus longissimus 1·34 Haliotis excavatus 4·20

*Pleurotomaria conoidea 1·34 Haliotis parvus 6·00

Trochus niloticus (Fig. ) 1·41 Delphinula atrata 6·00

Mitra episcopalis 1·43 Haliotis rugoso-plicata 9·30

Fusus antiquus 1·50 Haliotis viridis 10·00

Scalaria pretiosa 1·56

Fusus colosseus 1·71

Phasianella bulloides 1·80

Helicostyla polychroa 2·00

Those marked * from Naumann; the rest from Macalister .

In the case of turbinate shells, we require to take into account the angle θ, in order to determine the spiral angle α from the ratio of the breadths of consecutive whorls; for the short table given on p. is only applicable to discoid shells, in which the angle θ is an angle of 90°. Our formula, as mentioned on p. now becomes

R = ε 2π sin θ cot α .

For this formula I have worked out the following table. {558}

Table shewing values of the spiral angle α cor­re­spon­ding to certain ratios of breadth of successive whorls of the shell, for various values of the apical semi-angle θ.

Ratio
R ⁄ 1 θ = 5° 10° 15° 20° 30° 40° 50° 60° 70° 80° 90°

1·1 80° 8 ′ 85° 0 ′ 86° 44 ′ 87° 28 ′ 88° 16 ′ 88° 39 ′ 88° 52 ′ 89° 0 ′ 89° 4 ′ 89° 7 ′ 89° 8 ′

1·25 67 51 78 27 82 11 84 5 85 56 86 50 87 21 87 39 87 50 87 56 87 58

1·5 53 30 69 37 76 0 79 21 82 39 84 16 85 13 85 44 86 4 86 15 86 18

2·0 38 20 57 35 66 55 73 11 77 34 80 16 81 52 82 45 83 18 83 37 83 42

2·5 30 53 50 0 60 35 67 0 73 45 77 13 79 19 80 26 81 11 81 35 81 42

3·0 26 32 44 50 56 0 63 0 70 45 74 45 77 17 78 35 79 28 79 56 80 5

3·5 23 37 41 5 52 25 59 50 68 15 72 45 75 35 77 2 78 1 78 33 78 43

4·0 21 35 38 10 49 35 57 15 66 10 71 3 74 9 75 42 76 47 77 22 77 34

4·5 20 0 36 0 47 15 55 5 64 25 69 35 72 54 74 33 75 43 76 20 76 35

5·0 18 45 34 10 45 20 53 15 62 55 68 15 71 48 73 31 74 45 75 25 75 38

10·0 13 25 25 20 35 15 43 5 53 45 60 20 64 57 67 4 68 42 69 35 69 53

20·0 10 25 20 0 28 30 35 45 46 25 53 25 58 52 61 10 63 6 64 10 64 31

50·0 8 0 15 35 22 35 28 50 38 45 45 55 52 1 54 18 56 28 57 42 58 6

100·0 6 50 13 20 19 30 25 5 34 20 41 15 47 35 49 45 52 3 53 20 53 46

{559}

From this table, by interpolation, we may easily fill in the ap­prox­i­mate values of α, as soon as we have determined the apical angle θ and measured the ratio R; as follows:

R θ α

Turritella sp. 1·12 7° 81°

Cerithium nodulosum 1·24 15 82

Conus virgo 1·25 70 88

Mitra episcopalis 1·43 16 78

Scalaria pretiosa 1·56 26 81

Phasianella bulloides 1·80 26 80

Solarium perspectivum 1·50 53 85

Natica aperta 2·00 70 83

Planorbis corneus 2·00 90 84

Euomphalus pentangulatus 2·00 90 84

We see from this that shells so different in appearance as Cerithium, Solarium, Natica and Planorbis differ very little indeed in the magnitude of the spiral angle α, that is to say in the relative velocities of radial and tangential growth. It is upon the angle θ

Fig. 285. Terebra maculata, L.

that the difference in their form mainly depends: that is to say the amount of longitudinal shearing, or displacement parallel to the axis of the shell.

The enveloping angle, or rather semi-angle (θ), of the cone may be taken as 90° in the discoid shells, such as Nautilus and Planorbis. It is still a large angle, of 70° or 75°, in Conus or in Cymba, somewhat less in Cassis, Harpa, Dolium or Natica; it is about 50° to 55° in the various species of Solarium, about 35° in the typical Trochi, such as T. niloticus or T. zizyphinus, and about 25° or 26° in Scalaria pretiosa and Phasianella bulloides; it becomes a very acute angle, of 15°, 10°, or even less, in Eulima, Turritella or Cerithium. The costly Conus gloria-maris, one of the {560} great treasures of the conchologist, differs from its congeners in no important particular save in the somewhat “produced” spire, that is to say in the comparatively low value of the angle θ.

A variation with advancing age of θ is common, but (as Blake points out) it is often not to be distinguished or disentangled from an alteration of α. Whether alone, or combined with a change in α, we find it in all those many Gastropods whose whorls cannot all be touched by the same enveloping cone, and whose spire is accordingly described as concave or convex. The former condition, as we have it in Cerithium, and in the cusp-like spire of Cassis,

Fig. 286. Trochus niloticus, L.

Dolium and some Cones, is much the commoner of the two. And such tendency to decrease may lead to θ becoming a negative angle; in which case we have a depressed spire, as in the Cypraeae.

When we find a “reversed shell,” a whelk or a snail for instance whose spire winds to the left instead of to the right, we may describe it math­e­mat­i­cally by the simple statement that the angle θ has changed sign. In the genus Ampullaria, or Apple-snails, inhabiting tropical or sub-tropical rivers, we have a remarkable condition; for in certain “species” the spiral turns to the right, in others to the left, and in others again we have a flattened {561} “discoid” shell; and furthermore we have numerous intermediate stages, on either side, shewing right and left-handed spirals of varying degrees of acuteness. In this case, the angle θ may be said to vary, within the limits of a genus, from somewhere about 35° to somewhere about 125°.

The angle of retardation (β) is very small in Dentalium and Patella; it is very large in Haliotis. It becomes infinite in Argonauta and in Cypraea. Connected with the angle of retardation are the various possibilities of contact or separation, in various degrees, between adjacent whorls in the discoid, and between both adjacent and opposite whorls in the turbinated shell. But with these phenomena we have already dealt sufficiently.