Chapter XI: The Logarithmic Spiral (3)

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

_Telescopium fuscum_ 1·14
_Acus subulatus_ 1·16
*_Turritella terebellata_ 1·18
*_Turritella imbricata_ 1·20
_Cerithium palustre_ 1·22
_Turritella duplicata_ 1·23
_Melanopsis terebralis_ 1·23
_Cerithium nodulosum_ 1·24
*_Turritella carinata_ 1·25
_Acus crenulatus_ 1·25
_Terebra maculata_ (Fig. 285) 1·25
*_Cerithium lignitarum_ 1·26
_Acus dimidiatus_ 1·28
_Cerithium sulcatum_ 1·32
_Fusus longissimus_ 1·34
*_Pleurotomaria conoidea_ 1·34
_Trochus niloticus_ (Fig. 286) 1·41
_Mitra episcopalis_ 1·43
_Fusus antiquus_ 1·50
_Scalaria pretiosa_ 1·56
_Fusus colosseus_ 1·71
_Phasianella bulloides_ 1·80
_Helicostyla polychroa_ 2·00

Obtuse Turbinates and Discoids

_Conus virgo_ 1·25
_Conus litteratus_ 1·40
_Conus betulina_ 1·43
*_Helix nemoralis_ 1·50
*_Solarium perspectivum_ 1·50
_Solarium trochleare_ 1·62
_Solarium magnificum_ 1·75
*_Natica aperta_ 2·00
_Euomphalus pentangulatus_ 2·00
_Planorbis corneas_ 2·00
_Solaropsis pellis-serpentis_ 2·00
_Dolium zonatum_ 2·10
*_Natica glaucina_ 3·00
_Nautilus pompilius_ 3·00
_Haliotis excavatus_ 4·20
_Haliotis parvus_ 6·00
_Delphinula atrata_ 6·00
_Haliotis rugoso-plicata_ 9·30
_Haliotis viridis_ 10·00

Those marked * from Naumann; the rest from Macalister[526].

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. 534 is only applicable to discoid shells, in which the angle θ is an angle of 90°. Our formula, as mentioned on p. 518 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 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, 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[527]. 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.

_Of Bivalve Shells._

Hitherto we have dealt only with univalve shells, and it is in these that all the mathematical problems connected with the spiral, or helico-spiral, are best illustrated. But the case of the bivalve shell, of Lamellibranchs or of Brachiopods, presents no essential difference, save only that we have here to do with two conjugate spirals, whose two axes have a definite relation to one another, and some freedom of rotatory movement relatively to one another.

The generating curve is particularly well seen in the bivalve, where it simply constitutes what we call “the outline of the shell.” It is for the most part a plane curve, but not always; for there are forms, such as Hippopus, Tridacna and many Cockles, or Rhynchonella and Spirifer among the Brachiopods, in which the edges of the two valves interlock, and others, such as Pholas, Mya, etc., where in part they fail to meet. In such cases as these the generating curves are conjugate, having a similar relation, but of opposite sign, to a median plane of reference. A great variety of form is exhibited by these generating curves among the bivalves. In a good many cases the curve is approximately circular, as in Anomia, Cyclas, Artemis, Isocardia; it is nearly semi-circular in Argiope. It is approximately elliptical in Orthis and in Anodon; it may be called semi-elliptical in Spirifer. It is a nearly rectilinear {562} triangle in Lithocardium, and a curvilinear triangle in Mactra. Many apparently diverse but more or less related forms may be shewn to be deformations of a common type, by a simple application of the mathematical theory of “Transformations,” which we shall have to study in a later chapter. In such a series as is furnished, for instance, by Gervillea, Perna, Avicula, Modiola, Mytilus, etc., a “simple shear” accounts for most, if not all, of the apparent differences.

Upon the surface of the bivalve shell we usually see with great clearness the “lines of growth” which represent the successive margins of the shell, or in other words the successive positions assumed during growth by the growing generating curve; and we have a good illustration, accordingly, of how it is characteristic of the generating curve that it should constantly increase, while never altering its geometric similarity.

Underlying these “lines of growth,” which are so characteristic of a molluscan shell (and of not a few other organic formations), there is, then, a “law of growth” which we may attempt to enquire into and which may be illustrated in various ways. The simplest cases are those in which we can study the lines of growth on a more or less flattened shell, such as the one valve of an oyster, a Pecten or a Tellina, or some such bivalve mollusc. Here around an origin, the so-called “umbo” of the shell, we have a series of curves, sometimes nearly circular, sometimes elliptical, and often asymmetrical; and such curves are obviously not “concentric,” though we are often apt to call them so, but are always “co-axial.” This manner of arrangement may be illustrated by various analogies. We might for instance compare it to a series of waves, radiating outwards from a point, through a medium which offered a resistance increasing, with the angle of divergence, according to some simple law. We may find another, and perhaps a simpler illustration as follows:

In a very simple and beautiful theorem, Galileo shewed that, if we imagine a number of inclined planes, or gutters, sloping downwards (in a vertical plane) at various angles from a common starting-point, and if we imagine a number of balls rolling each down its own gutter under the influence of gravity (and without hindrance from friction), then, at any given instant, the locus of {563} all these moving bodies is a circle passing through the point of origin. For the acceleration along any one of the sloping paths, for instance _AB_ (Fig. 287), is such that

_AB_ = ½_g_ cos θ ⋅ t^2
= ½_g_ ⋅ _AB_/_AC_ ⋅ t^2.

Therefore

_t_^2 = 2/_g_ ⋅ _AC_.

That is to say, all the balls reach the circumference of the circle at the same moment as the ball which drops vertically from _A_ to _C_.

Where, then, as often happens, the generating curve of the shell is approximately a circle passing through the point of origin, we may consider the acceleration of growth along various radiants to be governed by a simple mathematical law, closely akin to that simple law of acceleration which governs the movements of a falling body. And, _mutatis mutandis_, a similar definite law underlies the cases where the generating curve is continually elliptical, or where it assumes some more complex, but still regular and constant form.

It is easy to extend the proposition to the particular case where the lines of growth may be considered elliptical. In such a case we have _x_^2/_a_^2 + _y_^2/_b_^2 = 1, where _a_ and _b_ are the major and minor axes of the ellipse.

Or, changing the origin to the vertex of the figure

_x_^2/_a_^2 − 2_x_/_a_ + _y_^2/_b_^2 = 0,

giving (_x_ − _a_)^2/_a_^2 + _y_^2/_b_^2 = 1.

Then, transferring to polar coordinates, where _r_ ⋅ cos θ = _x_, _r_ ⋅ sin θ = _y_, we have

(_r_ ⋅ cos^2 θ)/_a_^2 − (2 cos θ)/_a_ + (_r_ ⋅ sin θ)/_b_^2 = 0,

{564}

which is equivalent to

_r_ = (2_a_ _b_^2 cos θ)/((_b_^2 cos^2 θ) + (_a_^2 sin^2 θ)),

or, eliminating the sine-function,

_r_ = (2_a_ _b_^2 cos θ)/((_b_^2 − _a_^2) cos^2 θ + _a_^2).

Obviously, in the case when _a_ = _b_, this gives us the circular system which we have already considered. For other values, or ratios, of _a_ and _b_, and for all values of θ, we can easily construct a table, of which the following is a sample:

_Chords of an ellipse, whose major and minor axes (a, b) are in certain given ratios._

θ _a_/_b_ = 1/3 1/2 2/3 1/1 3/2 2/1 3/1 0° 1·0 1·0 1·0 1·0 1·0 1·0 1·0 10 1·01 1·01 1·002 ·985 ·948 ·902 ·793 20 1·05 1·03 1·005 ·940 ·820 ·695 ·485 30 1·115 1·065 1·005 ·866 ·666 ·495 ·289 40 1·21 1·11 ·995 ·766 ·505 ·342 ·178 50 1·34 1·145 ·952 ·643 ·372 ·232 ·113 60 1·50 1·142 ·857 ·500 ·258 ·152 ·071 70 1·59 1·015 ·670 ·342 ·163 ·092 ·042 80 1·235 ·635 ·375 ·174 ·078 ·045 ·020 90 0·0 0·0 0·0 0·0 0·0 0·0 0·0

The coaxial ellipses which we then draw, from the values given in the table, are such as are shewn in Fig. 288 for the ratio _a_/_b_ = 3/1, and in Fig. 289 for the ratio _a_/_b_ = ½; these are fair approximations to the actual outlines, and to the actual arrangement of the lines of growth, in such forms as Solecurtus or Cultellus, and in Tellina or Psammobia. It is not difficult to introduce a constant into our equation to meet the case of a shell which is somewhat unsymmetrical on either side of the median axis. It is a somewhat more troublesome matter, however, to bring these configurations into relation with a “law of growth,” as was so easily done in the case of the circular figure: in other words, to {565} formulate a law of acceleration according to which points starting from the origin _O_, and moving along radial lines, would all lie, at any future epoch, on an ellipse passing through _O_; and this calculation we need not enter into.

All that we are immediately concerned with is the simple fact that where a velocity, such as our rate of growth, varies with its direction,—varies that is to say as a function of the angular divergence from a certain axis,—then, in a certain simple case, we get lines of growth laid down as a system of coaxial circles, and, when the function is a more complex one, as a system of ellipses or of other more complicated coaxial figures, which figures may or may not be symmetrical on either side of the axis. Among our bivalve mollusca we shall find the lines of growth to be approximately circular in, for instance, Anomia; in Lima (e.g. _L. subauriculata_) we have a system of nearly symmetrical ellipses with the vertical axis about twice the transverse; in _Solen pellucidus_, we have again a system of lines of growth which are not far from being symmetrical ellipses, in which however the transverse is between three and four times as great as the vertical axis. In the great majority of cases, we have a similar phenomenon with the further complication of slight, but occasionally very considerable, lateral asymmetry.

In certain little Crustacea (of the genus Estheria) the carapace takes the form of a bivalve shell, closely simulating that of a {566} lamellibranchiate mollusc, and bearing lines of growth in all respects analogous to or even identical with those of the latter. The explanation is very curious and interesting. In ordinary Crustacea the carapace, like the rest of the chitinised and calcified integument, is shed off in successive moults, and is restored again as a whole. But in Estheria (and one or two other small crustacea) the moult is incomplete: the old carapace is retained, and the new, growing up underneath it, adheres to it like a lining, and projects beyond its edge: so that in course of time the margins of successive old carapaces appear as “lines of growth” upon the surface of the shell. In this mode of formation, then (but not in the usual one), we obtain a structure which “is partly old and partly new,” and whose successive increments are all similar, similarly situated, and enlarged in a continued progression. We have, in short, all the conditions appropriate and necessary for the development of a logarithmic spiral; and this logarithmic spiral (though it is one of small angle) gives its own character to the structure, and causes the little carapace to partake of the characteristic conformation of the molluscan shell.

The essential simplicity, as well as the great regularity of the “curves of growth” which result in the familiar configurations of our bivalve shells, sufficiently explain, in a general way, the ease with which they may be imitated, as for instance in the so-called “artificial shells” which Kappers has produced from the conchoidal form and lamination of lumps of melted and quickly cooled paraffin[528].

In the above account of the mathematical form of the bivalve shell, we have supposed, for simplicity’s sake, that the pole or origin of the system is at a point where all the successive curves touch one another. But such an arrangement is neither theoretically probable, nor is it actually the case; for it would mean that in a certain direction growth fell, not merely to a minimum, but to zero. As a matter of fact, the centre of the system (the “umbo” of the conchologists) lies not at the edge of the system, but very near to it; in other words, there is a certain amount of growth all round. But to take account of this condition would involve more troublesome mathematics, and it is obvious that the foregoing illustrations are a sufficiently near approximation to the actual case. {567}

Among the bivalves the spiral angle (α) is very small in the flattened shells, such as Orthis, Lingula or Anomia. It is larger, as a rule, in the Lamellibranchs than in the Brachiopods, but in the latter it is of considerable magnitude among the Pentameri. Among the Lamellibranchs it is largest in such forms as Isocardia and Diceras, and in the very curious genus Caprinella; in all of these last-named genera its magnitude leads to the production of a spiral shell of several whorls, precisely as in the univalves. The angle is usually equal, but of opposite sign, in the two valves of the Lamellibranch, and usually of opposite sign but unequal in the two valves of the Brachiopod. It is very unequal in many Ostreidae, and especially in such forms as Gryphaea, or in Caprinella, which is a kind of exaggerated Gryphaea. Occasionally it is of the same sign in both valves (that is to say, both valves curve the same way) as we see sometimes in Anomia, and much better in Productus or Strophomena.

Owing to the large growth-factor of the generating curve, and the comparatively small angle of the spiral, the whole shell seldom assumes a spiral form so conspicuous as to manifest in a typical way the helical twist or shear which is so conspicuous in the {568} majority of univalves, or to let us measure or estimate the magnitude of the apical angle (θ) of the enveloping cone. This however we can do in forms like Isocardia and Diceras; while in Caprinella we see that the whorls lie in a plane perpendicular to the axis, forming a discoidal spire. As in the latter shell, so also universally among the Brachiopods, there is no lateral asymmetry in the plane of the generating curve such as to lead to the development of a helix; but in the majority of the Lamellibranchiata it is obvious, from the obliquity of the lines of growth, that the angle θ is significant in amount.

――――――――――

The so-called “spiral arms” of Spirifer and many other Brachiopods are not difficult to explain. They begin as a single structure, in the form of a loop of shelly substance, attached to the dorsal valve of the shell, in the neighbourhood of the hinge. This loop has a curvature of its own, similar to but not necessarily identical with that of the valve to which it is attached; and this curvature will tend to be developed, by continuous and symmetrical growth, into a fully formed logarithmic spiral, so far as it is permitted to do so under the constraint of the shell in which it is contained. In various Terebratulae we see the spiral growth of the loop, more or less flattened and distorted by the restraining pressure of the ventral valve. In a number of cases the loop remains small, but gives off two nearly parallel branches or offshoots, which continue to grow. And these, starting with just such a slight curvature as the loop itself possessed, grow on and on till they may form close-wound spirals, always provided that the “spiral angle” of the curve is such that the resulting spire can be freely contained within the cavity of the shell. Owing to the bilateral symmetry of the whole system, the case will be rare, and unlikely to occur, in which each separate arm will coil strictly _in a plane_, so as to constitute a discoid spiral; for the original {569} direction of each of the two branches, parallel to the valve (or nearly so) and outwards from the middle line, will tend to constitute a curve of double curvature, and so, on further growth, to develop into a helicoid. This is what actually occurs, in the great majority of cases. But the curvature may be such that the helicoid grows outwards from the middle line, or inwards towards the middle line, a _very_ slight difference in the initial curvature being sufficient to direct the spire the one way or the other; the middle course of an undeviating discoid spire will be rare, from the usual lack of any obvious controlling force to prevent its deviation. The cases in which the helicoid spires point towards, or point away from, the middle line are ascribed, in zoological classification, to particular “families” of Brachiopods, the former

condition defining (or helping to define) the Atrypidae and the latter the Spiriferidae and Athyridae. It is obvious that the incipient curvature of the arms, and consequently the form and direction of the spirals, will be influenced by the surrounding pressures, and these in turn by the general shape of the shell. We shall expect, accordingly, to find the long outwardly directed spirals associated with shells which are transversely elongated, as Spirifer is; while the more rounded Atrypas will tend to the opposite condition. In a few cases, as in Cyrtina or Reticularia, where the shell is comparatively narrow but long, and where the uncoiled basal support of the arms is long also, the spiral coils into which the latter grow are turned backwards, in the direction where there is room for them. And in the few cases where the shell is very considerably flattened, the spirals (if they find room {570} to grow at all) will be constrained to do so in a discoid or nearly discoid fashion, and this is actually the case in such flattened forms as Koninckina or Thecidium.

_The Shells of Pteropods._

While mathematically speaking we are entitled to look upon the bivalve shell of the Lamellibranch as consisting of two distinct elements, each comparable to the entire shell of the univalve, we have no biological grounds for such a statement; for the shell arises from a single embryonic origin, and afterwards becomes split into portions which constitute the two separate valves. We can perhaps throw some indirect light upon this phenomenon, and upon several other phenomena connected with shell-growth, by a consideration of the simple conical or tubular shells of the Pteropods. The shells of the latter are in few cases suitable for simple mathematical investigation, but nevertheless they are of very considerable interest in connection with our general problem.

The morphology of the Pteropods is by no means well understood, and in speaking of them I will assume that there are still grounds for believing (in spite of Boas’ and Pelseneer’s arguments) that they are directly related to, or may at least be directly compared with, the Cephalopoda[529].

The simplest shells among the Pteropods have the form of a tube, more or less cylindrical (Cuvierina), more often conical (Creseis, Clio); and this tubular shell (as we have already had occasion to remark, on p. 258), frequently tends, when it is very small and delicate, to assume the character of an unduloid. (In such a case it is more than likely that the tiny shell, or that portion of it which constitutes the unduloid, has not grown by successive {571} increments or “rings of growth,” but has developed as a whole.) A thickened “rib” is often, perhaps generally, present on the dorsal side of the little conical shell. In a few cases (Limacina, Peraclis) the tube becomes spirally coiled, in a normal logarithmic spiral or helico-spiral.

In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is curiously modified. In the first place, its cross-section, originally {572} circular or nearly so, becomes flattened or compressed dorso-ventrally; and the angle, or rather edge, where dorsal and ventral walls meet, becomes more and more drawn out into a ridge or keel. Along the free margin, both of the dorsal and the ventral portion of the shell, growth proceeds with a regularly varying velocity, so that these margins, or lips, of the shell become regularly curved or markedly sinuous. At the same time, growth in a transverse direction proceeds with an acceleration which manifests itself in a curvature of the sides, replacing the straight borders of the original cone. In other words, the cross-section of the cone, or what we have been calling the generating curve, increases its dimensions more rapidly than its distance from the pole.

In the above figures, for instance in that of _Cleodora cuspidata_, the markings of the shell which represent the successive edges of the lip at former stages of growth, furnish us at once with a “graph” of the varying velocities of growth as measured, radially, from the apex. We can reveal more clearly the nature of these variations in the following way which is simply tantamount to converting our radial into rectangular coordinates. Neglecting curvature (if any) of the sides and treating the shell (for simplicity’s sake) as a right cone, we lay off equal angles from the apex _O_, along the radii _Oa_, _Ob_, etc. If we then plot, as vertical equidistant ordinates, the magnitudes _Oa_, _Ob_ ... _OY_, and again on to _Oa′_, we obtain a diagram such as the following (Fig. 299); by {573} help of which we not only see more clearly the way in which the growth-rate varies from point to point, but we also recognise much better than before, the similar nature of the law which governs this variation in the different species.

Furthermore, the young shell having become differentiated into a dorsal and a ventral part, marked off from one another by a lateral edge or keel, and the inequality of growth being such as to cause each portion to increase most rapidly in the median line, it follows that the entire shell will appear to have been split into a dorsal and a ventral plate, both connected with, and projecting from, {574} what remains of the original undivided cone. Putting the same thing in other words, we may say that the generating figure, which lay at first in a plane perpendicular to the axis of the cone, has now, by unequal growth, been sharply bent or folded, so as to lie approximately in two planes, parallel to the anterior and posterior faces of the cone. We have only to imagine the apical connecting portion to be further reduced, and finally to disappear or rupture, and we should have a _bivalve shell_ developed out of the original simple cone.

In its outer and growing portion, the shell of our Pteropod now consists of two parts which, though still connected together at the apex, may be treated as growing practically independently. The shell is no longer a simple tube, or simple cone, in which regular inequalities of growth will lead to the development of a spiral; and this for the simple reason that we have now two opposite maxima of growth, instead of a maximum on the one side and a minimum on the other side of our tubular shell. As a matter of fact, the dorsal and the ventral plate tend to curve in opposite directions, towards the middle line, the dorsal curving ventrally and the ventral curving towards the dorsal side.

In the case of the Lamellibranch or the Brachiopod, it is quite possible for both valves to grow into more or less pronounced spirals, for the simple reason that they are _hinged_ upon one another; and each growing edge, instead of being brought to a standstill by the growth of its opposite neighbour, is free to move out of the way, by the rotation about the hinge of the plane in which it lies.

But where, as in the Pteropod, there is no such hinge, the dorsal and ventral halves of the shell (or dorsal and ventral valves, if we may call them so), if they curved towards one another (as they do in a cockle), would soon interfere with one another’s progress, and the development of a pair of conjugate spirals would become impossible. Nevertheless, there is obviously, in both dorsal and ventral valve, a _tendency_ to the development of a spiral curve, that of the ventral valve being more marked than that of the larger and overlapping dorsal one, exactly as in the two unequal valves of Terebratula. In many cases (e.g. _Cleodora cuspidata_), the dorsal valve or plate, {575} strengthened and stiffened by its midrib, is nearly straight, while the curvature of the other is well displayed. But the case will be materially altered and simplified if growth be arrested or retarded in either half of the shell. Suppose for instance that the dorsal valve grew so slowly that after a while, in comparison with the other, we might speak of it as being absent altogether: or suppose that it merely became so reduced in relative size as to form no impediment to the continued growth of the ventral one; the latter would continue to grow in the direction of its natural curvature, and would end by forming a complete and coiled logarithmic spiral. It would be precisely analogous to the spiral shell of Nautilus, and, in regard to its ventral position, concave towards the dorsal side, it would even deserve to be called directly homologous with it. Suppose, on the other hand, that the ventral valve were to be greatly reduced, and even to disappear, the dorsal valve would then pursue its unopposed growth; and, were it to be markedly curved, it would come to form a logarithmic spiral, concave towards the ventral side, as is the case in the shell of Spirula[530]. Were the dorsal valve to be destitute of any marked curvature (or in other words, to have but a low spiral angle), it would form a simple plate, as in the shells of Sepia or Loligo. Indeed, in the shells of these latter, and especially in that of Sepia, we seem to recognise a manifest resemblance to the dorsal plate of the Pteropod shell, as we have it (e.g.) in Cleodora or Hyalaea; {576} the little “rostrum” of Sepia is but the apex of the primitive cone, and the rounded anterior extremity has grown according to a law precisely such as that which has produced the curved margin of the dorsal valve in the Pteropod. The ventral portion of the original cone is nearly, but not wholly, wanting. It is represented by the so-called posterior wall of the “siphuncular space.” In many decapod cuttle-fishes also (e.g. Todarodes, Illex, etc.) we still see at the posterior end of the “pen,” a vestige of the primitive cone, whose dorsal margin only has continued to grow; and the same phenomenon, on an exaggerated scale, is represented in the Belemnites.

It is not at all impossible that we may explain on the same lines the development of the curious “operculum” of the Ammonites. This consists of a single horny plate (_Anaptychus_), or of a thicker, more calcified plate divided into two symmetrical halves (_Aptychi_), often found inside the terminal chamber of the Ammonite, and occasionally to be seen lying _in situ_, as an operculum which partially closes the mouth of the shell; this structure is known to exist even in connection with the early embryonic shell. In form the Anaptychus, or the pair of conjoined Aptychi, shew an upper and a lower border, the latter strongly convex, the former sometimes slightly concave, sometimes slightly convex, and usually shewing a median projection or slightly developed rostrum. From this “rostral” border the curves of growth start, and course round parallel to, finally constituting, the convex border. It is this convex border which fits into the free margin of the mouth of the Ammonite’s shell, while the other is applied to and overlaps the preceding whorl of the spire. Now this relationship is precisely what we should expect, were we to imagine as our starting-point a shell similar to that of Hyalaea, in which however the dorsal part of the split cone had become separate from the ventral half, had remained flat, and had grown comparatively slowly, while at the same time it kept slipping forward over the growing and coiling spire into which the ventral half of the original shell develops[531]. In short, I think there is reason to believe, or at least to suspect, that we {577} have in the shell and Aptychus of the Ammonites, two portions of a once united structure; of which other Cephalopods retain not both parts but only one or other, one as the ventrally situated shell of Nautilus, the other as the dorsally placed shell for example of Sepia or of Spirula.

In the case of the bivalve shells of the Lamellibranchs or of the Brachiopods, we have to deal with a phenomenon precisely analogous to the split and flattened cone of our Pteropods, save only that the primitive cone has been split into two portions, not incompletely as in the Pteropod (Hyalaea), but completely, so as to form two separate valves. Though somewhat greater freedom is given to growth now that the two valves are separate and hinged, yet still the two valves oppose and hamper one another, so that in the longitudinal direction each is capable of only a moderate curvature. This curvature, as we have seen, is recognisable as a logarithmic spiral, but only now and then does the growth of the spiral continue so far as to develop successive coils: as it does in a few symmetrical forms such as _Isocardia cor_; and as it does still more conspicuously in a few others, such as Gryphaea and Caprinella, where one of the two valves is stunted, and the growth of the other is (relatively speaking) unopposed.

_Of Septa._

Before we leave the subject of the molluscan shell, we have still another problem to deal with, in regard to the form and arrangement of the septa which divide up the tubular shell into chambers, in the Nautilus, the Ammonite and their allies (Fig. 304, etc.).

The existence of septa in a Nautiloid shell may probably be accounted for as follows. We have seen that it is a property of a cone that, while growing by increments at one end only, it conserves its original shape: therefore the animal within, which (though growing by a different law) also conserves its shape, will continue to fill the shell if it actually fills it to begin with: as does a snail or other Gastropod. But suppose that our mollusc fills a part only of a conical shell (as it does in the case of Nautilus); then, unless it alter its shape, it must move upward as it grows in the growing cone, until it come to occupy a space similar in form {578} to that which it occupied before: just, indeed, as a little ball drops far down into the cone, but a big one must stay farther up. Then, when the animal after a period of growth has moved farther up in the shell, the mantle-surface continues its normal secretory activity, and that portion which had been in contact with the former septum secretes a septum anew. In short, at any given epoch, the creature is not secreting a tube and a septum by separate operations, but is secreting a shelly case about its rounded body, of which case one part appears to us as the continuation of the tube, and the other part, merging with it by indistinguishable boundaries, appears to us as the septum[532].

The various forms assumed by the septa in spiral shells[533] present us with a number of problems of great beauty, simple in their essence, but whose full investigation would soon lead us into mathematics of a very high order.

We do not know in great detail how these septa are laid down; but the essential facts are clear[534]. The septum begins as a very thin cuticular membrane (composed apparently of a substance called conchyolin), which is secreted by the skin, or mantle-surface, of the animal; and upon this membrane nacreous matter is gradually laid down on the mantle-side (that is to say between the animal’s body and the cuticular membrane which has been thrown off from it), so that the membrane remains as a thin pellicle over the _hinder_ surface of the septum, and so that, to begin with, the membranous septum is moulded on the flexible and elastic surface of the animal, within which the fluids of the body must exercise a uniform, or nearly uniform pressure.

Let us think, then, of the septa as they would appear in their uncalcified condition, formed of, or at least superposed upon, an {579} elastic membrane. They must then follow the general law, applicable to all elastic membranes under uniform pressure, that the tension varies inversely as the radius of curvature; and we come back once more to our old equation of Laplace, that

_P_ = _T_(1/_r_ + 1/_r′_).

Moreover, since the cavity below the septum is practically closed, and is filled either with air or with water, _P_ will be constant over the whole area of the septum. And further, we must assume, at least to begin with, that the membrane constituting the incipient septum is homogeneous or isotropic.

Let us take first the case of a straight cone, of circular section, more or less like an Orthoceras; and let us suppose that the septum is attached to the shell in a plane perpendicular to its axis. The septum itself must then obviously be spherical. Moreover the extent of the spherical surface is constant, and easily determined. For obviously, in Fig. 302, the angle _LCL′_ equals the supplement of the angle (_LOL′_) of the cone; that is to say, the circle of contact subtends an angle at the centre of the spherical surface, which is constant, and which is equal to π − 2θ. The case is not excluded where, owing to an asymmetry of tensions, the septum meets the side walls of the cone at other than a right angle, as in Fig. 303; and here, while the septa still remain portions of spheres, the geometrical construction for the position of their centres is equally easy.

If, on the other hand, the attachment of the septum to the inner walls of the cone be in a plane oblique to the axis, then it is evident that the outline of the septum will be an ellipse, and its surface an {580} ellipsoid. If the attachment of the septum be not in one plane, but form a sinuous line of contact with the cone, then the septum will be a saddle-shaped surface, of great complexity and beauty. In all cases, provided only that the membrane be isotropic, the form assumed will be precisely that of a soap-bubble under similar conditions of attachment: that is to say, it will be (with the usual limitations or conditions) a surface of minimal area.

If our cone be no longer straight, but curved, then the septa will be symmetrically deformed in consequence. A beautiful and interesting case is afforded us by Nautilus itself. Here the outline of the septum, referred to a plane, is approximately bounded by two elliptic curves, similar and similarly situated, whose areas are to one another in a definite ratio, namely as

_A__{1}/_A__{2} = (_r__{1} _r′__{1})/(_r__{2} _r′__{2}) = ε^{−4π cot α},

and a similar ratio exists in Ammonites and all other close-whorled spirals, in which however we cannot always make the simple assumption of elliptical form. In a median section of Nautilus, we see each septum forming a tangent to the inner and to the outer wall, just as it did in a section of the straight Orthoceras; but the curvatures in the neighbourhood of these two points of contact are not identical, for they now vary inversely as the radii, drawn from the pole of the spiral shell. The contour of the septum in this median plane is a spiral curve identical with the original logarithmic spiral. Of this it is the “invert,” and the fact that the original curve and its invert are both identical is one of the most beautiful properties of the logarithmic spiral[535].

But while the outline of the septum in median section is simple and easy to determine, the curved surface of the septum in its entirety is a very complicated matter, even in Nautilus which is one of the simplest of actual cases. For, in the first place, since the form of the septum, as seen in median section, is that of a logarithmic spiral, and as therefore its curvature is constantly altering, it follows that, in successive _transverse_ sections, the {581} curvature is also constantly altering. But in the case of Nautilus, there are other aspects of the phenomenon, which we can illustrate, but only in part, in the following simple manner. Let us imagine a pack of cards, in which we have cut out of each card a similar concave arc of a logarithmic spiral, such as we actually see in the median section of the septum of a Nautilus. Then, while we hold the cards together, foursquare, in the ordinary position of the {582} pack, we have a simple “ruled” surface, which in any longitudinal section has the form of a logarithmic spiral but in any transverse section is a straight horizontal line. If we shear or slide the cards upon one another, thrusting the middle cards of the pack forward in advance of the others, till the one end of the pack is a convex, and the other a concave, ellipse, the cut edges which combine to represent our septum will now form a curved surface

of much greater complexity; and this is part, but not by any means all, of the deformation produced as a direct consequence of the form in Nautilus of the section of the tube within which the septum has to lie. And the complex curvature of the surface will be manifested in a sinuous outline of the edge, or line of attachment of the septum to the tube, and will vary according to the configuration of the latter. In the case of Nautilus, it is easy to shew empirically (though not perhaps easy to demonstrate {583} mathematically) that the sinuous or saddle-shaped form of the “suture” (or line of attachment of the septum to the tube) is such as can be precisely accounted for in this manner. It is also easy to see that, when the section of the tube (or “generating curve”) is more complicated in form, when it is flattened, grooved, or otherwise ornamented, the curvature of the septum and the outline of its sutural attachment will become very complicated indeed[536]; but it will be comparatively simple in the case of the first few sutures of the young shell, laid down before any overlapping of whorls has taken place, and this comparative simplicity of the first-formed sutures is a marked feature among Ammonites[537].

We have other sources of complication, besides those which are at once introduced by the sectional form of the tube. For instance, the siphuncle, or little inner tube which perforates the septa, exercises a certain amount of tension, sometimes evidently considerable, upon the latter; so that we can no longer consider each septum as an isotropic surface, under uniform pressure; and there may be other structural modifications, or inequalities, in that portion of the animal’s body with which the septum is in contact, and by which it is conformed. It is hardly likely, for all these reasons, that we shall ever attain to a full and particular explanation of the septal surfaces and their sutural outlines throughout the whole range of Cephalopod shells; but in general terms, the problem is probably not beyond the reach of mathematical analysis. The problem might be approached experimentally, after the manner of Plateau’s experiments, by bending {584} a wire into the complicated form of the suture-line, and studying the form of the liquid film which constitutes the corresponding surface _minimae areae_.

In certain Ammonites the septal outline is further complicated in another way. Superposed upon the usual sinuous outline, with its “lobes” and “saddles,” we have here a minutely ramified, or arborescent outline, in which all the branches terminate in wavy, more or less circular arcs,—looking just like the ‘landscape marble’ from the Bristol Rhaetic. We have no difficulty in recognising in this a surface-tension phenomenon. The figures are precisely such as we can imitate (for instance) by pouring a {585} few drops of milk upon a greasy plate, or of oil upon an alkaline solution.

We have very far from exhausted, we have perhaps little more than begun, the study of the logarithmic spiral and the associated curves which find exemplification in the multitudinous diversities of molluscan shells. But, with a closing word or two, we must now bring this chapter to an end.

In the spiral shell we have a problem, or a phenomenon, of growth, immensely simplified by the fact that each successive increment is irrevocably fixed in regard to magnitude and position, instead, of remaining in a state of flux and sharing in the further changes which the organism undergoes. In such a structure, then, we have certain primary phenomena of growth manifested in their original simplicity, undisturbed by secondary and conflicting phenomena. What actually _grows_ is merely the lip of an orifice, where there is produced a ring of solid material, whose form we have treated of under the name of the generating curve; and this generating curve grows in magnitude without alteration of its form. Besides its increase in areal magnitude, the growing curve has certain strictly limited degrees of freedom, which define its motions in space: that is to say, it has a vector motion at right angles to the axis of the shell; and it has a sliding motion along that axis. And, though we may know nothing whatsoever about the actual velocities of any of these motions, we do know that they are so correlated together that their _relative_ velocities remain constant, and accordingly the form and symmetry of the whole system remain in general unchanged.

But there is a vast range of possibilities in regard to every one of these factors: the generating curve may be of various forms, and even when of simple form, such as an ellipse, its axes may be set at various angles to the system; the plane also in which it lies may vary, almost indefinitely, in its angle relatively to that of any plane of reference in the system; and in the several velocities of growth, of rotation and of translation, and therefore in the ratios between all these, we have again a vast range of possibilities. We have then a certain definite type, or group of forms, mathematically isomorphous, but presenting infinite diversities of outward appearance: which diversities, as Swammerdam {586} said, _ex sola nascuntur diversitate gyrationum_; and which accordingly are seen to have their origin in differences of rate, or of magnitude, and so to be, essentially, neither more nor less than _differences of degree_.

In nature, we find these forms presenting themselves with but little relation to the character of the creature by which they are produced. Spiral forms of certain particular kinds are common to Gastropods and to Cephalopods, and to diverse families of each; while outside the class of molluscs altogether, among the Foraminifera and among the worms (as in Spirorbis, Spirographis, and in the Dentalium-like shell of Ditrupa), we again meet with similar and corresponding forms.

Again, we find the same forms, or forms which (save for external ornament) are mathematically identical, repeating themselves in all periods of the world’s geological history; and, irrespective of climate or local conditions, we see them mixed up, one with another, in the depths and on the shores of every sea. It is hard indeed (to my mind) to see where Natural Selection necessarily enters in, or to admit that it has had any share whatsoever in the production of these varied conformations. Unless indeed we use the term Natural Selection in a sense so wide as to deprive it of any purely biological significance; and so recognise as a sort of natural selection whatsoever nexus of causes suffices to differentiate between the likely and the unlikely, the scarce and the frequent, the easy and the hard: and leads accordingly, under the peculiar conditions, limitations and restraints which we call “ordinary circumstances,” one type of crystal, one form of cloud, one chemical compound, to be of frequent occurrence and another to be rare.

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