Looking, as we have done, upon the spiral shell as being essentially

a cone rolled up, it is plain that, for a given spiral angle,

intersection or non-intersection of the successive whorls will depend

upon _the apical angle_ of the original cone. For the wider the cone,

the more rapidly will its inner border tend to encroach on the outer

border of the preceding whorl.

But it is also plain that the greater be the apical angle of the

cone, and the broader, consequently, the cone itself be, the greater

difference will there be between the total _lengths_ of its inner

and outer border, under given conditions of flexure. And, since the

inner and outer borders are describing precisely the same spiral about

the pole, it is plain that we may consider the inner border as being

_retarded_ in growth as compared with the outer, and as being always

identical with a smaller and earlier part of the latter.

If λ be the ratio of growth between the outer and the inner curve,

then, the outer curve being represented by

_r_ = _a_ _e_^{θ cot α},

the equation to the inner one will be

_r′_ = _a_λ_e_^{θ cot α},

or _r′_ = _a_ _e_^{(θ − β)cot α},

and β may then be called the angle of retardation, to which the inner

curve is subject by virtue of its slower rate of growth. {542}

Dispensing with mathematical formulae, the several conditions may be

illustrated as follows:

[Illustration: Fig. 278.]

In the diagrams (Fig. 278), _O_ _P__{1} _P__{2} _P__{3}, etc.

represents a radius, on which _P__{1}, _P__{2}, _P__{3}, are the points

attained by the outer border of the tubular shell after as many entire

consecutive revolutions. And _P__{1}′, _P__{2}′, _P__{3}′, are the

points similarly intersected by the inner border; _OP_/_OP′_ being

always = λ, which is the ratio of growth, or “cutting-down factor.”

Then, obviously, when _O_ _P__{1} is less than _O_ _P__{2}′ the

whorls will be separated by an interspace (_a_); (2) when _O_ _P__{1}

= _O_ _P__{2}′ they will be in contact (_b_), and (3) when _O_ _P__{1}

is greater than _O_ _P__{2}′ there will a greater or less extent

of overlapping, that is to say of concealment of the surfaces of

the earlier by the later whorls (_c_). And as a further case (4),

it is plain that if λ be very large, that is to say if _O_ _P__{1}

be greater, not only than _O_ _P__{2}′ but also than _O_ _P__{3}′,

_O_ _P__{4}′, etc., we shall have complete, or all but complete

concealment by the last formed whorl, of the whole of its predecessors.

This latter condition is completely attained in _Nautilus pompilius_,

and approached, though not quite attained, in _N. umbilicatus_; and

the difference between these two forms, or “species,” is constituted

accordingly by a difference in the value of λ. (5) There is also a

final case, not easily distinguishable externally from (4), where _P′_

lies on {543} the opposite side of the radius vector to _P_, and is

therefore imaginary. This final condition is exhibited in Argonauta.

The limiting values of λ are easily ascertained.

[Illustration: Fig 279]

In Fig. 279 we have portions of two successive whorls, whose

corresponding points on the same radius vector (as _R_ and _R′_) are,

therefore, at a distance apart corresponding to 2π. Let _r_ and _r′_

refer to the inner, and _R_, _R′_ to the outer sides of the two whorls.

Then, if we consider

_R_ = _a_ _e_^{θ cot α},

it follows that _R′_ = _a_ _e_^{(θ + 2π)cot α},

_r_ = λ_a_ _e_^{θ cot α} = _a_ _e_^{(θ − β)cot α},

and _r′_ = λ_a_ _e_^{(θ + 2π)cot α} = _a_ _e_^{(θ + 2π − β)cot α}.

Now in the three cases (_a_, _b_, _c_) represented in Fig. 278, it is

plain that _r′_ ⪌ _R_, respectively. That is to say,

λ_a_ _e_^{(θ + 2π)cot α} ⪌ _a_ _e_^{θ cot α},

and λ_e_^{2π cot α} ⪌ 1.

The case in which λ_e_^{2π cot α} = 1, or −log λ = 2π cot α log ε, is

the case represented in Fig. 278, _b_: that is to say, the particular

case, for each value of α, where the consecutive whorls just touch,

without interspace or overlap. For such cases, then, we may tabulate

the values of λ, as follows:

Constant angle Ratio (λ) of rate of growth of inner border of tube,

α of spiral as compared with that of the outer border

89° ·896

88 ·803

87 ·720

86 ·645

85 ·577

80 ·330

75 ·234

70 ·1016

65 ·0534

{544}

We see, accordingly, that in plane spirals whose constant angle

lies, say, between 65° and 70°, we can only obtain contact between

consecutive whorls if the rate of growth of the inner border of the

tube be a small fraction,—a tenth or a twentieth—of that of the outer

border. In spirals whose constant angle is 80°, contact is attained

when the respective rates of growth are, approximately, as 3 to 1;

while in spirals of constant angle from about 85° to 89°, contact is

attained when the rates of growth are in the ratio of from about 3/5 to

9/10.

[Illustration: Fig. 280.]

If on the other hand we have, for any given value of α, a value of λ

greater or less than the value given in the above table, then we have,

respectively, the conditions of separation or of overlap which are

exemplified in Fig. 278, _a_ and _c_. And, just as we have constructed

this table of values of λ for the particular case of simple contact

between the whorls, so we could construct similar tables for various

degrees of separation, or degrees of overlap.

For instance, a case which admits of simple solution is that in which

the interspace between the whorls is everywhere a mean proportional

between the breadths of the whorls themselves (Fig. 280). {545}

In this case, let us call _OA_ = _R_, _OC_ = _R__{1} and _OB_ = _r_. We

then have

_R__{1} = _OA_ = _a_ _e_^{θ cot α},

_R__{2} = _OC_ = _a_ _e_^{(θ + 2π) cot α},

_R__{1} _R__{2} = _a_ _e_^{2(θ + π) cot α} = _r_^2 [520].

And _r_^2 = (1/λ)^2 ⋅ ε^{2θ cot α},

whence, equating, 1/λ = _e_^{π cot α}.

The corresponding values of λ are as follows:

Ratio (λ) of rates of growth of outer and inner

border, such as to produce a spiral with interspaces

between the whorls, the breadth of which

interspaces is a mean proportional between the

Constant angle (α) breadths of the whorls themselves

90° 1·00 (imaginary)

89 ·95

88 ·89

87 ·85

86 ·81

85 ·76

80 ·57

75 ·43

70 ·32

65 ·23

60 ·18

55 ·13

50 ·090

45 ·063

40 ·042

35 ·026

30 ·016

As regards the angle of retardation, β, in the formula

_r′_ = λ_e_^{θ cot α}, or _r′_ = _e_^{(θ − β)cot α},

and in the case

_r′_ = _e_^{(2π − β)cot α}, or −log λ = (2π − β)cot α,

{546}

it is evident that when β = 2π, that will mean that λ = 1. In other

words, the outer and inner borders of the tube are identical, and the

tube is constituted by one continuous line.

When λ is a very small fraction, that is to say when the rates of

growth of the two borders of the tube are very diverse, then β will

tend towards infinity—tend that is to say towards a condition in which

the inner border of the tube never grows at all. This condition is not

infrequently approached in nature. The nearly parallel-sided cone of

Dentalium, or the widely separated whorls of Lituites, are evidently

cases where λ nearly approaches unity in the one case, and is still

large in the other, β being correspondingly small; while we can easily

find cases where β is very large, and λ is a small fraction, for

instance in Haliotis, or in Gryphaea.

For the purposes of the morphologist, then, the main result of this

last general investigation is to shew that all the various types of

“open” and “closed” spirals, all the various degrees of separation or

overlap of the successive whorls, are simply the outward expression of

a varying ratio in the _rate of growth_ of the outer as compared with

the inner border of the tubular shell.

――――――――――

The foregoing problem of contact, or intersection, of the successive

whorls, is a very simple one in the case of the discoid shell but a

more complex one in the turbinate. For in the discoid shell contact

will evidently take place when the retardation of the inner as compared

with the outer whorl is just 360°, and the shape of the whorls need not

be considered.

As the angle of retardation diminishes from 360°, the whorls will stand

further and further apart in an open coil; as it increases beyond 360°,

they will more and more overlap; and when the angle of retardation is

infinite, that is to say when the true inner edge of the whorl does not

grow at all, then the shell is said to be completely involute. Of this

latter condition we have a striking example in Argonauta, and one a

little more obscure in _Nautilus pompilius_.

In the turbinate shell, the problem of contact is twofold, for we have

to deal with the possibilities of contact on the _same_ side of the

axis (which is what we have dealt with in the discoid) and {547} also

with the new possibility of contact or intersection on the _opposite_

side; it is this latter case which will determine the presence or

absence of an _umbilicus_, and whether, if present, it will be an

open conical space or a twisted cone. It is further obvious that, in

the case of the turbinate, the question of contact or no contact will

depend on the shape of the generating curve; and if we take the simple

case where this generating curve may be considered as an ellipse, then

contact will be found to depend on the angle which the major axis of

this ellipse makes with the axis of the shell. The question becomes a

complicated one, and the student will find it treated in Blake’s paper

already referred to.

When one whorl overlaps another, so that the generating curve cuts

its predecessor (at a distance of 2π) on the same radius vector, the

locus of intersection will follow a spiral line upon the shell, which

is called the “suture” by conchologists. It is evidently one of that

_ensemble_ of spiral lines in space of which, as we have seen, the

whole shell may be conceived to be constituted; and we might call it

a “contact-spiral,” or “spiral of intersection.” In discoid shells,

such as an Ammonite or a Planorbis, or in _Nautilus umbilicatus_,

there are obviously two such contact-spirals, one on each side of the

shell, that is to say one on each side of a plane perpendicular to the

axis. In turbinate shells such a condition is also possible, but is

somewhat rare. We have it for instance, in _Solarium perspectivum_,

where the one contact-spiral is visible on the exterior of the cone,

and the other lies internally, winding round the open cone of the

umbilicus[521]; but this second contact-spiral is usually imaginary,

or concealed within the whorls of the turbinated shell. Again, in

Haliotis, one of the contact-spirals is non-existent, because of the

extreme obliquity of the plane of the generating curve. In _Scalaria

pretiosa_ and in Spirula there is no contact-spiral, because the growth

of the generating curve has been too slow, in comparison with the

vector rotation of its plane. In Argonauta and in Cypraea, there is no

contact-spiral, because the growth of the generating curve has been too

quick. Nor, of course, is there any contact-spiral in Patella or in

Dentalium, because the angle α is too small ever to give us a complete

revolution of the spire. {548}

The various forms of straight or spiral shells among the Cephalopods,

which we have seen to be capable of complete definition by the help of

elementary mathematics, have received a very complicated descriptive

nomenclature from the palaeontologists. For instance, the straight

cones are spoken of as _orthoceracones_ or _bactriticones_, the loosely

coiled forms as _gyroceracones_ or _mimoceracones_, the more closely

coiled shells, in which one whorl overlaps the other, as _nautilicones_

or _ammoniticones_, and so forth. In such a succession of forms the

biologist sees undoubted and unquestioned evidence of ancestral

descent. For instance we read in Zittel’s _Palaeontology_[522]: “The

bactriticone obviously represents the primitive or primary radical of

the Ammonoidea, and the mimoceracone the next or secondary radical

of this order”; while precisely the opposite conclusion was drawn by

Owen, who supposed that the straight chambered shells of such fossil

cephalopods as Orthoceras had been produced by the gradual unwinding

of a coiled nautiloid shell[523]. _To such phylogenetic hypotheses

the mathematical or dynamical study of the forms of shells lends no

valid support._ If we have two shells in which the constant angle of

the spire be respectively 80° and 60°, that fact in itself does not at

all justify an assertion that the one is more primitive, more ancient,

or more “ancestral” than the other. Nor, if we find a third in which

the angle happens to be 70°, does that fact entitle us to say that

this shell is intermediate between the other two, in time, or in blood

relationship, or in any other sense whatsoever save only the strictly

formal and mathematical one. For it is evident that, though these

particular arithmetical constants manifest themselves in visible and

recognisable differences of form, yet they are not necessarily more

deep-seated or significant than are those which manifest themselves

only in difference of magnitude; and the student of phylogeny scarcely

ventures to draw conclusions as to the relative antiquity of two allied

organisms on the ground that one happens to be bigger or less, or

longer or shorter, than the other. {549}

At the same time, while it is obviously unsafe to rest conclusions

upon such features as these, unless they be strongly supported

and corroborated in other ways,—for the simple reason that there

is unlimited room for _coincidence_, or separate and independent

attainment of this or that magnitude or numerical ratio,—yet on the

other hand it is certain that, in particular cases, the evolution of

a race has actually involved gradual increase or decrease in some one

or more numerical factors, magnitude itself included,—that is to say

increase or decrease in some one or more of the actual and relative

velocities of growth. When we do meet with a clear and unmistakable

series of such progressive magnitudes or ratios, manifesting themselves

in a progressive series of “allied” forms, then we have the phenomenon

of “_orthogenesis_.” For orthogenesis is simply that phenomenon

of continuous lines or series of form (and also of functional or

physiological capacity), which was the foundation of the Theory of

Evolution, alike to Lamarck and to Darwin and Wallace; and which we

see to exist whatever be our ideas of the “origin of species,” or of

the nature and origin of “functional adaptations.” And to my mind,

the mathematical (as distinguished from the purely physical) study

of morphology bids fair to help us to recognise this phenomenon of

orthogenesis in many cases where it is not at once patent to the eye;

and also, on the other hand, to warn us, in many other cases, that

even strong and apparently complex resemblances in form may be capable

of arising independently, and may sometimes signify no more than the

equally accidental numerical coincidences which are manifested in

identity of length or weight, or any other simple magnitudes.

――――――――――

[Illustration: Fig. 281. An ammonitoid shell (_Macroscaphites_) to shew

change of curvature.]

I have already referred to the fact that, while in general a very great

and remarkable regularity of form is characteristic of the molluscan

shell, that complete regularity is apt to be departed from. We have

clear cases of such a departure in Pupa, Clausilia, and various Bulimi,

where the enveloping cone of the spire is not a right cone but a cone

whose sides are curved. It is plain that this condition may arise

in two ways: either by a gradual change in the ratio of growth of

the whorls, that is to say in the logarithmic spiral itself, or by a

change in the velocity of {550} translation along the axis, that is

to say in the helicoid which, in all turbinate shells, is superposed

upon the spiral. Very careful measurements will be necessary to

determine to which of these factors, or in what proportions to each,

the phenomenon is due. But in many Ammonitoidea where the helicoid

factor does not enter into the case, we have a clear illustration of

gradual and marked changes in the spiral angle itself, that is to say

of the ratio of growth corresponding to increase of vectorial angle.

We have seen from some of Naumann’s and Grabau’s measurements that

such a tendency to vary, such an acceleration or retardation, may be

detected even in Ammonites which present nothing abnormal to the eye.

But let us suppose that the spiral angle increases somewhat rapidly;

we shall then get a spiral with gradually narrowing whorls, and this

condition is characteristic of Oekotraustes, a subgenus of Ammonites.

If on the other hand, the angle α gradually diminishes, and even falls

away to zero, we shall have the spiral curve opening out, as it does in

Scaphites, Ancyloceras and Lituites, until the spiral coil is replaced

by a spiral curve so gentle as to seem all but straight. Lastly, there

are a few cases, such as _Bellerophon expansus_ and some Goniatites,

where the outer spiral does not perceptibly change, but the whorls

become more “embracing” or the whole shell more involute. Here it is

the angle of retardation, the ratio of growth between the outer and

inner parts of the whorl, which undergoes a gradual change.

――――――――――

In order to understand the relation of a close-coiled shell to one of

its straighter congeners, to compare (for example) an {551} Ammonite

with an Orthoceras, it is necessary to estimate the length of the right

cone which has, so to speak, been coiled up into the spiral shell. Our

problem then is, To find the length of a plane logarithmic spiral, in

terms of the radius and the constant angle α. In the annexed diagram,

if _OP_ be a radius vector, _OQ_ a line of reference perpendicular to

_OP_, and _PQ_ a tangent to the curve, _PQ_, or sec α, is equal in

length to the spiral arc _OP_. And this is practically obvious: for

_PP′_/_PR′_ = _ds_/_dr_ = sec α, and therefore sec α = _s_/_r_, or the

ratio of arc to radius vector.

[Illustration: Fig. 282.]

Accordingly, the ratio of _l_, the total length, to _r_, the radius

vector up to which the total length is to be measured, is expressed by

a simple table of secants; as follows:

α _l_/_r_

5° 1·004

10 1·015

20 1·064

30 1·165

40 1·305

50 1·56

60 2·0

70 2·9

75 3·9

80 5·8

85 11·5

86 14·3

87 19·1

88 28·7

89 57·3

89° 10′ 68·8

20 85·9

30 114·6

40 171·9

50 343·8

55 687·5

59 3437·7

90 Infinite

Putting the same table inversely, so as to shew the total {552} length

in whole numbers, in terms of the radius, we have as follows:

Total length (in terms

of the radius) Constant angle

2 60°

3 70 31′

4 75 32

5 78 28

10 84 16

20 87 8

30 88 6

40 88 34

50 88 51

100 89 26

1000 89 56′ 36″

10,000 89 59 30

Accordingly, we see that (1), when the constant angle of the spiral

is small, the spiral itself is scarcely distinguishable from a

straight line, and its length is but very little greater than that

of its own radius vector. This remains pretty much the case for a

considerable increase of angle, say from 0° to 20° or more; (2) for a

very considerably greater increase of the constant angle, say to 50°

or more, the shell would only have the appearance of a gentle curve;

(3) the characteristic close coils of the Nautilus or Ammonite would

be typically represented only when the constant angle lies within a

few degrees on either side of about 80°. The coiled up spiral of a

Nautilus, with a constant angle of about 80°, is about six times the

length of its radius vector, or rather more than three times its own

diameter; while that of an Ammonite, with a constant angle of, say,

from 85° to 88°, is from about six to fifteen times as long as its own

diameter. And (4) as we approach an angle of 90° (at which point the

spiral vanishes in a circle), the length of the coil increases with

enormous rapidity. Our spiral would soon assume the appearance of the

close coils of a Nummulite, and the successive increments of breadth

in the successive whorls would become inappreciable to the eye. The

logarithmic spiral of high constant angle would, as we have already

seen, tend to become indistinguishable, without the most careful

measurement, from an Archimedean spiral. And it is obvious, moreover,

that our ordinary methods of {553} determining the constant angle of

the spiral would not in these cases be accurate enough to enable us to

measure the length of the coil: we should have to devise a new method,

based on the measurement of radii or diameters over a large number of

whorls.

The geometrical form of the shell involves many other beautiful

properties, of great interest to the mathematician, but which it is

not possible to reduce to such simple expressions as we have been

content to use. For instance, we may obtain an equation which shall

express completely the surface of any shell, in terms of polar or of

rectangular coordinates (as has been done by Moseley and by Blake),

or in Hamiltonian vector notation. It is likewise possible (though

of little interest to the naturalist) to determine the area of a

conchoidal surface, or the volume of a conchoidal solid, and to find

the centre of gravity of either surface or solid[524]. And Blake has

further shewn, with considerable elaboration, how we may deal with

the symmetrical distortion, due to pressure, which fossil shells

are often found to have undergone, and how we may reconstitute by

calculation their original undistorted form,—a problem which, were the

available methods only a little easier, would be very helpful to the

palaeontologist; for, as Blake himself has shewn, it is easy to mistake

a symmetrically distorted specimen of (for instance) an Ammonite,

for a new and distinct species of the same genus. But it is evident

that to deal fully with the mathematical problems contained in, or

suggested by, the spiral shell, would require a whole treatise, rather

than a single chapter of this elementary book. Let us then, leaving

mathematics aside, attempt to summarise, and perhaps to extend, what

has been said about the general possibilities of form in this class of

organisms.

_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.

[Illustration: Fig. 283. Section of a spiral, or turbinate, univalve,

_Triton corrugatus_, Lam. (From Woodward.)]

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 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. 284) 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[525]. 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. 258), as in the case

of the opercula of Turbo and Nerita referred to on p. 522; and this is

what Moseley supposed it to do.

[Illustration: 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

_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.

[Illustration: Fig. 285. _Terebra maculata_, L.]

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.

[Illustration: Fig. 286. _Trochus niloticus_, L.]

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_.

[Illustration: Fig. 287.]

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

[Illustration: Fig. 288.]

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.

[Illustration: Fig. 289.]

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.

[Illustration: Fig. 290. _Caprinella adversa._ (After Woodward.)]

[Illustration: Fig. 291. Section of _Productus_ (_Strophomena_) sp.

(From Woods.)]

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.

――――――――――

[Illustration: Fig. 292. Skeletal loop of _Terebratula_. (From Woods.)]

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

[Illustration: Fig. 293. Spiral arms of _Spirifer_. (From Woods.)]

[Illustration: Fig. 294. Inwardly directed spiral arms of _Atrypa_.]

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].

[Illustration: Fig. 295. Pteropod shells: (1) _Cuvierina columnella_;

(2) _Cleodora chierchiae_; (3) _C. pygmaea_. (After Boas.)]

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.

[Illustration: Fig. 296. Diagrammatic transverse sections, or outlines

of the mouth, in certain Pteropod shells: A, B, _Cleodora australis_;

C, _C. pyramidalis_; D, _C. balantium_; E, _C. cuspidata_. (After

Boas.)]

[Illustration: Fig. 297. Shells of thecosome Pteropods (after Boas).

(1) _Cleodora cuspidata_; (2) _Hyalaea trispinosa_; (3) _H. globulosa_;

(4) _H. uncinata_; (5) _H. inflexa_.]

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.

[Illustration: Fig. 298. _Cleodora cuspidata._]

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.

[Illustration: Fig. 299. Curves obtained by transforming radial

ordinates, as in Fig. 298, into vertical equidistant ordinates. 1,

_Hyalaea trispinosa_; 2, _Cleodora cuspidata_.]

[Illustration: Fig. 300. Development of the shell of _Hyalaea_

(_Cavolinia_) _tridentata_, Forskal: the earlier stages being the

“_Pleuropus longifilis_” of Troschel. (After Tesch.)]

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.

[Illustration: Fig. 301. Pteropod shells, from the side: (1) _Cleodora

cuspidata_; (2) _Hyalaea longirostris_; (3) _H. trispinosa_. (After

Boas.)]

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.

[Illustration: Fig. 302.]

[Illustration: Fig. 303.]

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].

[Illustration: Fig. 304. Section of _Nautilus_, shewing the contour

of the septa in the median plane: the septa being (in this plane)

logarithmic spirals, of which the shell-spiral is the evolute.]

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

[Illustration: Fig. 305. Cast of the interior of _Nautilus_: to shew

the contours of the septa at their junction with the shell-wall.]

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_.

[Illustration: Fig. 306. _Ammonites_ (_Sonninia_) _Sowerbyi_. (From

Zittel, after Steinmann and Döderlein.)]

[Illustration: Fig. 307. Suture-line of a Triassic Ammonite

(_Pinacoceras_). (From Zittel, after Hauer.)]

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.

{587}