Chapter XI: The Logarithmic Spiral (2)

It is plain that the entire resulting shell may now be looked upon in either of two ways. It is, on the one hand, an _ensemble of similar closed curves_ spirally arranged in space, gradually increasing in dimensions, in proportion to the increase of their vectorial angle from the pole. In other words, we can imagine our shell cut up into a system of rings, following one another in continuous spiral succession from that terminal and largest one, which constitutes the lip of the orifice of the shell. Or, on the other hand, we may figure to ourselves the whole shell as made up of an _ensemble of spiral lines_ in space, each spiral having been {526} traced out by the gradual growth and revolution of a radius vector from the pole to a given point of the generating curve.

Both systems of lines, the _generating spirals_ (as these latter may be called), and the closed _generating curves_ corresponding to successive margins or lips of the shell, may be easily traced in a great variety of cases. Thus, for example, in Dolium, Eburnea, and a host of others, the generating spirals are beautifully marked out by ridges, tubercles or bands of colour. In Trophon, Scalaria, and (among countless others) in the Ammonites, it is the successive generating curves which more conspicuously leave their impress on the shell. And in not a few cases, as in Harpa, _Dolium perdix_, etc., both alike are conspicuous, ridges and colour-bands intersecting one another in a beautiful isogonal system. {527}

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

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

――――――――――

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

For some years after the appearance of Moseley’s paper, a number of writers followed in his footsteps, and attempted, in various ways, to put his conclusions to practical use. For instance, D’Orbigny devised a very simple protractor, which he called a Helicometer[512], and which is represented in Fig. 267. By means of this little instrument, the apical angle of the turbinate shell was immediately read off, and could then be used as a specific and diagnostic character. By keeping one limb of the protractor parallel to the side of the cone while the other was brought into line with the suture between two adjacent whorls, another specific angle, the “sutural angle,” could in like manner be recorded. And, by the linear scale upon the instrument, the relative breadths of the consecutive whorls, and that of the terminal chamber to the rest of the shell, might also, though somewhat roughly, be determined. For instance, in _Terebra dimidiata_, the apical angle was found to be 13°, the sutural angle 109°, and so forth.

It was at once obvious that, in such a shell as is represented in Fig. 267 the entire outline of the shell (always excepting that of the immediate neighbourhood of {530} the mouth) could be restored from a broken fragment. For if we draw our tangents to the cone, it follows from the symmetry of the figure that we can continue the projection of the sutural line, and so mark off the successive whorls, by simply drawing a series of consecutive parallels, and by then filling into the quadrilaterals so marked off a series of curves similar to one another, and to the whorls which are still intact in the broken shell.

But the use of the helicometer soon shewed that it was by no means universally the case that one and the same right cone was tangent to all the turbinate whorls; in other words, there was not always one specific apical angle which held good for the entire system. In the great majority of cases, it is true, the same tangent touches all the whorls, and is a straight line. But in others, as in the large _Cerithium nodosum_, such a line is slightly convex to the axis of the shell; and in the short spire of Dolium, for instance, the convexity is marked, and the apex of the spire is a distinct cusp. On the other hand, in Pupa and Clausilia, the common tangent is concave to the axis of the shell.

So also is it, as we shall presently see, among the Ammonites: where there are some species in which the ratio of whorl to whorl remains, to all appearance, perfectly constant; others in which it gradually, though only slightly increases; and others again in which it slightly and gradually falls away. It is obvious that, among the manifold possibilities of growth, such conditions as these are very easily conceivable. It is much more remarkable that, among these shells, the relative velocities of growth in various dimensions should be as constant as it is, than that there should be an occasional departure from perfect regularity. In such cases as these latter, the logarithmic law of growth is only approximately true. The shell is no longer to be represented as a _right_ cone which has been rolled up, but as a cone which had grown trumpet-shaped, or conversely whose mouth had narrowed in, and which in section is a curvilinear instead of a rectilinear triangle. But all that has happened is that a new factor, usually of small or all but imperceptible magnitude, has been introduced into the case; so that the ratio, log _r_ = θ log α, is no longer constant, but varies slightly, and in accordance with some simple law. {531}

Some writers, such as Naumann and Grabau, maintained that the molluscan spiral was no true logarithmic spiral, but differed from it specifically, and they gave to it the name of _Conchospiral_. They pointed out that the logarithmic spiral originates in a mathematical point, while the molluscan shell starts with a little embryonic shell, or central chamber (the “protoconch” of the conchologists), around which the spiral is subsequently wrapped. It is plain that this undoubted and obvious fact need not affect the logarithmic law of the shell as a whole; we have only to add a small constant to our equation, which becomes _r_ = _m_ + _a_^θ.

There would seem, by the way, to be considerable confusion in the books with regard to the so-called “protoconch.” In many cases it is a definite structure, of simple form, representing the more or less globular embryonic shell before it began to elongate into its conical or spiral form. But in many cases what is described as the “protoconch” is merely an empty space in the middle of the spiral coil, resulting from the fact that the actual spiral shell has a definite magnitude to begin with, and that we cannot follow it down to its vanishing point in infinity. For instance, in the accompanying figure, the large space _a_ is styled the protoconch, but it is the little bulbous or hemispherical chamber within it, at the end of the spire, which is the real beginning of the tubular shell. The form and magnitude of the space _a_ are determined by the “angle of retardation,” or ratio of rate of growth between the inner and outer curves of the spiral shell. They are independent of the shape and size of the embryo, and depend only (as we shall see better presently) on the direction and relative rate of growth of the double contour of the shell.

――――――――――

Now that we have dealt, in a very general way, with some of the more obvious properties of the logarithmic spiral, let us consider certain of them a little more particularly, keeping in {532} view as our chief object the investigation (on elementary lines) of the possible manner and range of variation of the molluscan shell.

There is yet another equation to the logarithmic spiral, very commonly employed, and without the help of which we shall find that we cannot get far. It is as follows:

_r_ = ε^{θ cot α}.

This follows directly from the fact that the angle α (the angle between the radius vector and the tangent to the curve) is constant.

For, then,

tan α (= tan ϕ) = _r_ _d_θ/_dr_,

therefore _dr_/_r_ = _d_θ cot α,

and, integrating,

log _r_ = θ cot α,

or _r_ = ε^{θ cot α}.

――――――――――

As we have seen throughout our preliminary discussion, the two most important constants (or chief “specific characters,” as the naturalist would say) in any given logarithmic spiral, are (1) the magnitude of the angle of the spiral, or “constant angle,” α, and (2) the rate of increase of the radius vector for any given angle of revolution, θ. Of this latter, the simplest case is when θ = 2π, or 360°; that is to say when we compare the breadths, along the same radius vector, of two successive whorls. As our two magnitudes, that of the constant angle, and that of the ratio of the radii or breadths of whorl, are related to one another, we may determine either of them by actual measurement and proceed to calculate the other.

In any complete spiral, such as that of Nautilus, it is (as we have seen) easy to measure any two radii (_r_), or the breadths in {533} a radial direction of any two whorls (_W_). We have then merely to apply the formula

_r__{_n_ + 1}/_r__{_n_} = _e_^{θ cot α}, or _W__{_n_ + 1}/_W__{_n_}
= _e_^{θ cot α},

which we may simply write _r_ = _e_^{θ cot α}, etc.; since our first radius or whorl is regarded, for the purpose of comparison, as being equal to unity.

Thus, in the diagram, _OC_/_OE_, or _EF_/_BD_, or _DC_/_EF_, being in each case radii, or diameters, at right angles to one another, are all equal to _e_^{π/2 cot α}. While in like manner, _EO_/_OF_, _EG_/_FH_, or _GO_/_HO_, all equal _e_^{π cot α}; and _BC_/_BA_, or _CO_/_OB_ = _e_^{2π cot α}.

As soon, then, as we have prepared tables for these values, the determination of the constant angle α in a particular shell becomes a very simple matter.

A complete table would be cumbrous, and it will be sufficient to deal with the simple case of the ratio between the breadths of adjacent, or immediately succeeding, whorls.

Here we have _r_ = _e_^{2π cot α}, or log _r_ = log _e_ × 2π × cot α, from which we obtain the following figures[513]: {534}

Ratio of breadth of each
whorl to the next preceding Constant angle
_r_/1 α
1·1 89° 8′
1·25 87 58
1·5 86 18
2·0 83 42
2·5 81 42
3·0 80 5
3·5 78 43
4·0 77 34
4·5 76 32
5·0 75 38
10·0 69 53
20·0 64 31
50·0 58 5
100·0 53 46
1,000·0 42 17
10,000 34 19
100,000 28 37
1,000,000 24 28
10,000,000 21 18
100,000,000 18 50
1,000,000,000 16 52

We learn several interesting things from this short table. We see, in the first place, that where each whorl is about three times the breadth of its neighbour and predecessor, as is the case in Nautilus, the constant angle is in the neighbourhood of 80°; and hence also that, in all the ordinary Ammonitoid shells, and in all the typically spiral shells of the Gastropods[514], the constant angle is also a large one, being very seldom less than 80°, and usually between 80° and 85°. In the next place, we see that with smaller angles the apparent form of the spiral is greatly altered, and the very fact of its being a spiral soon ceases to be apparent (Figs. 271, 272). Suppose one whorl to be an inch in breadth, then, if the angle of the spiral were 80°, the {535} next whorl would (as we have just seen) be about three inches broad; if it were 70°, the next whorl would be nearly ten inches, and if it were 60°, the next whorl would be nearly four feet broad. If the angle were 28°, the next whorl would be a mile and a half in breadth; and if it were 17°, the next would be some 15,000 miles broad.

In other words, the spiral shells of gentle curvature, or of small constant angle, such as Dentalium or Nodosaria, are true logarithmic spirals, just as are those of Nautilus or Rotalia: from which they differ only in degree, in the magnitude of an angular constant. But this diminished magnitude of the angle causes the spiral to dilate with such immense rapidity that, so to speak, “it never comes round”; and so, in such a shell as Dentalium, we never see but a small portion of the initial whorl.

We might perhaps be inclined to suppose that, in such a shell as Dentalium, the lack of a visible spiral convolution was only due to our seeing but a small portion of the curve, at a distance from the pole, and when, therefore, its {536} curvature had already greatly diminished. That is to say we might suppose that, however small the angle a, and however rapidly the whorls accordingly increased, there would nevertheless be a manifest spiral convolution in the immediate neighbourhood of the pole, as the starting point of the curve. But it may be shewn that this is not so.

For, taking the formula _r_ = _a_ε^{θ cot α},

this, for any given spiral, is equivalent to _a_ε^{_k_θ}.

Therefore log(_r_/_a_) = _k_θ,

or, 1/_k_ = θ/log(_r_/_a_).

Then, if θ increase by 2π, while _r_ increases to _r__{1},

1/_k_ = (θ + 2π)/log(_r__{1}/_a_),

which leads, by subtraction to

1/_k_ ⋅ log(_r__{1}/_r_) = 2π.

Now, as α tends to 0, _k_ (i.e. cot α) tends to ∞, and therefore, as _k_ → ∞, log(_r__{1}/_r_) → ∞ and also _r__{1}/_r_ → ∞.

Therefore if one whorl exists, the radius vector of the other is infinite; in other words, there is nowhere, even in the near neighbourhood of the pole, a complete revolution of the spire. Our spiral shells of small constant angle, such as Dentalium, may accordingly be considered to represent sufficiently well the true commencement of their respective spirals.

Let us return to the problem of how to ascertain, by direct measurement, the spiral angle of any particular shell. The method already employed is only applicable to complete spirals, that is to say to those in which the angle of the spiral is large, and furthermore it is inapplicable to portions, or broken fragments, of a shell. In the case of the broken fragment, it is plain that the determination of the angle is not merely of theoretic interest, but may be of great practical use to the conchologist as being the one and only way by which he may restore the outline of the missing portions. We have a considerable choice of methods, which have been summarised by, and are partly due to, a very careful student of the Cephalopoda, the late Rev. J. F. Blake[515]. {537}

(1) The following method is useful and easy when we have a portion of a single whorl, such as to shew both its inner and its outer edge. A broken whorl of an Ammonite, a curved shell such as Dentalium, or a horn of similar form to the latter, will fall under this head. We have merely to draw a tangent, _GEH_, to the outer whorl at any point _E_; then draw to the inner whorl a tangent parallel to _GEH_, touching the curve in some point _F_. The straight line joining the points of contact, _EF_, must evidently pass through the pole: and, accordingly, the angle _GEF_ is the angle required. In shells which bear _longitudinal_ striae or other ornaments, any pair of these will suffice for our purpose, instead of the actual boundaries of the whorl. But it is obvious that this method will be apt to fail us when the angle α is very small; and when, consequently, the points _E_ and _F_ are very remote.

(2) In shells (or horns) shewing rings, or other _transverse_ ornamentation, we may take it that these ornaments are set at a constant angle to the spire, and therefore to the radii. The angle (θ) between two of them, as _AC_, _BD_, is therefore equal to the angle θ between the polar radii from _A_ and _B_, or from _C_ and _D_; and therefore _BD_/_AC_ = _e_^{θ cot α}, which gives us the angle α in terms of known quantities. {538}

(3) If only the outer edge be available, we have the ordinary geometrical problem,—given an arc of an equiangular spiral, to find its pole and spiral angle. The methods we may employ depend (1) on determining directly the position of the pole, and (2) on determining the radius of curvature.

The first method is theoretically simple, but difficult in practice; for it requires great accuracy in determining the points. Let _AD_, _DB_, be two tangents drawn to the curve. Then a circle drawn through the points _ABD_ will pass through the pole _O_; since the angles _OAD_, _OBE_ (the supplement of _OBD_), are equal. The point _O_ may be determined by the intersection of two such circles; and the angle _DBO_ is then the angle, α, required.

Or we may determine, graphically, at two points, the radii of curvature, ρ_{1} ρ_{2}. Then, if _s_ be the length of the arc between them (which may be determined with fair accuracy by rolling the margin of the shell along a ruler)

cot α = (ρ_{1} − ρ_{2})/_s_.

The following method[516], given by Blake, will save actual determination of the radii of curvature.

Measure along a tangent to the curve, the distance, _AC_, at which a certain small offset, _CD_, is made by the curve; and from another point _B_, measure the distance at which the curve makes an equal offset. Then, calling the offset μ; the arc _AB_, _s_; and _AC_, _BE_, respectively _x__{1}, _x__{2}, we have

ρ_{1} = (_x__{1}^2 + μ^2)/2μ, approximately,

and cot α = (_x__{2}^2 − _x__{1}^2)/2μ_s_.

Of all these methods by which the mathematical constants, or specific characters, of a given spiral shell may be determined, the only one of which much use has been made is that which Moseley first employed, namely, the simple method of determining {539} the relative breadths of the whorl at distances separated by some convenient vectorial angle (such as 90°, 180°, or 360°).

Very elaborate measurements of a number of Ammonites have been made by Naumann[517], by Sandberger[518], and by Grabau[519], among which we may choose a couple of cases for consideration. In the following table I have taken a portion of Grabau’s determinations of the breadth of the whorls in _Ammonites_ (_Arcestes_)

_Ammonites intuslabiatus._

Ratio of breadth of
Breadth of whorls successive whorls The angle (α)
(180° apart) (360° apart) as calculated

0·30 mm. — — —
0·30 1·333 87° 23′
0·40 1·500 86 19
0·45 1·500 86 19
0·60 1·444 86 39
0·65 1·417 86 49
0·85 1·692 85 13
1·10 1·588 85 47
1·35 1·545 86 2
1·70 1·630 85 33
2·20 1·441 86 40
2·45 1·432 86 43
3·15 1·735 85 0
4·25 1·683 85 16
5·30 1·482 86 25
6·30 1·519 86 12
8·05 1·635 85 32
10·30 1·416 86 50
11·40 1·252 87 57
12·90 — — —
──────
Mean 86° 15′

{540}

_intuslabiatus_; these measurements Grabau gives for every 45° of arc, but I have only set forth one quarter of these measurements, that is to say, the breadths of successive whorls measured along one diameter on both sides of the pole. The ratio between _alternate_ measurements is therefore the same ratio as Moseley adopted, namely the ratio of breadth between _contiguous whorls_ along a radius vector. I have then added to these observed values the corresponding calculated values of the angle α, as obtained from our usual formula.

There is considerable irregularity in the ratios derived from these measurements, but it will be seen that this irregularity only implies a variation of the angle of the spiral between about 85° and 87°; and the values fluctuate pretty regularly about the mean, which is 86° 15′. Considering the difficulty of measuring the whorls, especially towards the centre, and in particular the difficulty of determining with precise accuracy the position of the pole, it is clear that in such a case as this we are scarcely justified in asserting that the law of the logarithmic spiral is departed from.

In some cases, however, it is undoubtedly departed from. Here for instance is another table from Grabau, shewing the corresponding ratios in an Ammonite of the group of _Arcestes tornatus_. In this case we see a distinct tendency of the ratios to

_Ammonites tornatus._

Ratio of breadth of
Breadth of whorls successive whorls The spiral angle
(180° apart) (360° apart) (α) as calculated
0·25 mm. — — —
0·30 1·400 86° 56′
0·35 1·667 85 21
0·50 2·000 83 42
0·70 2·000 83 42
1·00 2·000 83 42
1·40 2·100 83 16
2·10 2·179 82 56
3·05 2·238 82 42
4·70 2·492 81 44
7·60 2·574 81 27
12·10 2·546 81 33
19·35 — — —
──────
Mean 83° 22′

{541}

increase as we pass from the centre of the coil outwards, and consequently for the values of the angle α to diminish. The case is precisely comparable to that of a cone with slightly curving sides: in which, that is to say, there is a slight acceleration of growth in a transverse as compared with the longitudinal direction.

――――――――――

In a tubular spiral, whether plane or helicoid, the consecutive whorls may either be (1) isolated and remote from one another; or (2) they may precisely meet, so that the outer border of one and the inner border of the next just coincide; or (3) they may overlap, the vector plane of each outer whorl cutting that of its immediate predecessor or predecessors.

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:

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.

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.

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.

――――――――――

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.

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.

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.

(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.