On Growth and Form

The numerical ratio in the case of the Nautilus happens to be one of unusual simplicity. Let us take, with Moseley, a somewhat more complicated example. + It will be observed that here Moseley, speaking a8 a mathematician and considering the linear spiral, speaks of whorls when he means the linear boundaries, or lines traced by the revolving radius vector; while the conchologist usually applies the term whorl to the whole space between the two boundaries. As conchologists, therefore, we call the breadth of a whorl what Moseley looked upon as the distance between two consecutive whorls. But this latter nomenclature Moseley himself often uses.

t In the case of Turbo, and all other “turbinate” shells, we are dealing not with a plane logarithmic spiral, as in Nautilus, but with a ‘‘ gauche” spiral, such that the radius vector no longer revolves in a plane perpendicular to the axis of the system, but is inclined to that axis at some constant angle (@). The figure still preserves its continued similarity, and may with strict accuracy be called a logarithmic spiral in space. It is evident that its envelope will be a right circular cone; and ind:ed it is commonly spoken of as a logarithmic spiral wrapped wpon a cone, its pole coinciding with the apex of the cone. It follows that the distances of successive whorls of the spiral measured on the same straight line passing through

the apex of the cone, are in geometrical pregression, and conversely: just as in the former case. But the ratio between any two consecutive interspaces (i.e. line was drawn across its whorls, and their widths were measured upon it in succession, beginning with the last but one. The measurements were, as before, made with a fine pair of compasses and a diagonal scale. The sight was assisted by a magnifying glass. In a parallel column to the following admeasurements are the terms of a geometric progression, whose first term is the width of the widest whorl measured, and whose common ratio is

Fig. 262. Widths of successive Terms of a geometrical progression, whorls measured in inches whose first term is the width of and parts of an inch the widest whorl, and whose common ratio is }-1804 The close coincidence between the observed and the calculated figures is very remarkable, and is amply sufficient to justify the conclusion that we are here dealing with a true logarithmic spiral. Nevertheless, in order to verify his conclusion still further, and to get partially rid of the inaccuracies due to successive small

measurements, Moseley proceeded to investigate the same shell, measuring not single whorls, but groups of whorls, taken several at a time: making use of the following property of a geometrical progression, that “if yz represent the ratio of the sum of every even number (m) of its terms to the sum of half that number of terms, then the common ratio (r) of the series is represented by Accordingly, Moseley made the following measurements, beginning from the second and third whorls respectively :

“By the ratios of the two first admeasurements, the formula gives By the mean of the ratios deduced from the second two admeasurements, it gives “Tt is scarcely possible to imagine a more accurate verification than is deduced from these larger admeasurements, and we may with safety annex to the species Turbo duplicatus the characteristic number 1-18.” By similar and equally concordant observations, Moseley found for Turbo phasianus the characteristic ratio, 1-75: and for Buccinum subulatum that of 1-13.

From the table referring to Turbo duplicatus, on page 519, it is perhaps worth while to illustrate the logarithmic statement of the same facts: that is to say, the elementary corollary to the fact that the successive radii are in geometric progression, that their logarithms differ from one another by a constant amount. Relative widths of Logarithms of Difference of successive whorls successive whorls —_ successive logarithms The logarithmic spiral is not only very beautifully manifested in the molluscan shell, but also, in certain cases, in the httle lid or “operculum” by which the entrance to the tubular shell is closed after the animal has withdrawn itself within. In the spiral shell of Turbo, for instance, the operculum is a thick calcareous structure, with a beautifully curved outline, which grows by successive increments applied to one portion of its edge, and shews, accordingly, a spiral line of growth upon its surface. The successive increments leave their traces on the surface of the operculum

(Fig. 264, 1), which traces have the form of curved lines in Turbo, and of straight lines in (e.g.) Nerita (Fig. 264, 2); that is to say, apart from the side constituting the outer edge of the operculum (which side is always and of necessity curved) the successive increments constitute curvilinear triangles in the one case, and rectilinear triangles in the other. The sides of these triangles are tangents to the spiral line of the operculum, and may be supposed to generate it by their consecutive intersections.

In a number of such opercula, Moseley measured the breadths of the successive whorls along a radius vector*, just in the same way as he did with the entire shell in the foregoing cases; and here is one example of his results. Operculum of Turbo sp.; breadth (an inches) of successive whorls, measured from the pole. * As the successive increments evidently constitute similar figures, similarly related to the pole (P), it follows that their linear dimensions are to one another as the radii vectores drawn to similar points in them: for instance as PP,, PP,, which (in Fig. 264, 1) are radii vectores drawn to the points where they meet the common boundary.

The ratio is approximately constant, and this spiral also is, | therefore, a logarithmic spiral. But here comes in a very beautiful illustration of that property of the logarithmic spiral which causes its whole shape to remain unchanged, in spite of its apparently unsymmetrical, or unilateral, mode of growth. For the mouth of the tubular shell, into which the operculum has to fit, is growing or widening on all sides: while the operculum is increasing, not by additions made at the same time all round its margin, but by additions made only on one side of it at each successive stage. One edge of the operculum thus remains unaltered as it is advanced into each new position, and as it is placed in a newly formed section of the tube, similar to but greater than the last. Nevertheless, the two apposed structures, the chamber and its plug, at all times fit one another to perfection. The mechanical problem (by no means an easy one), is thus solved: “How to shape a tube of a variable section, so that a piston driven along it shall, by one side of its margin, coincide continually with its surface as it advances, provided only that the piston be made at the same time continually to revolve in its own plane.”

As Moseley puts it: “That the same edge which fitted a portion of the first less section should be capable of adjustment, so as to fit a portion of the next similar but greater section, supposes a geometrical provision in the curved form of the chamber of great apparent complication and difficulty. But God hath bestowed upon this humble architect the practical skill of a learned geometrician, and he makes this provision with admirable precision in that curvature of the logarithmic spiral which he gives to the section of fhe shell. This curvature obtaining, he has only to turn his operculum slightly round in its own plane as he advances it into each newly formed portion of his chamber, to adapt one margin of it to a new and larger surface and a different curvature, leaving the space to be filled up by increasing the operculum wholly on the other margin.”

But in many, and indeed more numerous Gastropod mollusca, the operculum does not grow in this remarkable spiral fashion, but by the apparently much simpler process of accretion by concentric rings. This suggests to us another mathematical feature of the logarithmic spiral. We have already seen that the logarithmic spiral has a number of “limiting cases,” apparently very diverse from one another. Thus the right cone is a logarithmic spiral in which the revolution of the radius vector is infinitely slow; and, in the same sense, the straight line itself is a hmiting case of the logarithmic spiral. The spiral of Archimedes, though not a limiting case of the logarithmic spiral, closely resembles one in which the angle of the spiral is very near to 90°, and the spiral is coiled around a central core. But if the angle of the spiral were actually 90°, the radius vector would describe a circle, identical with the “core” of which we have just spoken; and accordingly it may be said that the circle is, in this sense, a true limiting case of the logarithmic spiral. In this sense, then, the circular concentric operculum, for instance of Turritella or Littorina, does not represent a breach of continuity, but a “limiting case”’ of the spiral operculum of Turbo; the successive “ gnomons”’ are now not lateral or terminal additions, but complete concentric rings.

Viewed in regard to its own fundamental properties and to those of its limiting cases, the logarithmic spiral is the simplest of all known curves; and the rigid uniformity of the simple laws, or forces, by which it is developed sufficiently account for its frequent manifestation in the structures built up by the slow and steady growth of organisms. . In order to translate into precise terms the whole form and growth of a spiral shell, we should have to employ a mathematical notation, considerably more complicated than any that I have attempted to make use of in this book. But, in the most elementary language, we may now at least attempt to describe the general method, and some of the variations, of the mathematical development of the shell. .

Let us imagine a closed curve in space, whether circular or elliptical or of some other and more complex specific form, not necessarily in a plane: such a curve as we see before us when we consider the mouth, or terminal orifice, of our tubular shell; and let us imagine some one characteristic point within this closed curve, such as its centre of gravity. Then, starting from a fixed origin, let this centre of gravity describe an equiangular spiral in space, about a fixed axis (namely the axis of the shell), while at the same time the generating curve grows, with each angular increment of rotation, in such a way as to preserve the symmetry of the entire figure, with or without a simultaneous movement of translation along the axis.

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

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 Fig. 266. 1, Harpa; 2, Dolium. The ridges on the shell correspond

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 In the complete mathematical formula (such as I have not ventured to set forth*) 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 growthfactor, 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 * The equation to the surface of a turbinate shell is discussed by Moseley (Phil. Trans. tom. cit. p. 370), both in terms of polar coordinates and of the rectangular coordinates +, y, z. A more elegant representation can be given in vector notation, by the method of quaternions.

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*. 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:... tune 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 Tt.”

number of writers followed in _ his footsteps, and attempted, in various ways, to put his conclusions to practical use. Forinstance, D’Orbigny devised a very simple protractor, which he called a Helicometer*, 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 lhmb 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 * Alcide D’Orbigny, Bull. de la soc. géol. Fr. xm, p. 200, 1842; Cours élém. de Paléontologie, 1, p. 5, 1851. A somewhat similar instrument was described by Boubée. in Bull. soc. géol. 1, p. 232, 1831. Naumann’s Conchyliometer (Poggend. Ann. LIV, p. 544, 1845) was an application of the screw-micrometer; it was provided also with a rotating stage, for angular measurement. It was adapted for the study of a discoid or ammonitoid shell, while D’Orbigny’s instrument was meant

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 Dohum, 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 trumpetshaped, 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 = @ log a, is no longer constant, but varies slightly, and in accordance with some simple law.

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 Conchosyiral. 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’.

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®tructure, of simple form, representing the more or less globular embryonic shell before it began to elongate into its conical or spiralform. 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 stvled 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 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:

This follows directly from the fact that the angle a (the angle between the radius vector and the tangent to the curve) is constant. 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,” a, and (2) the rate of increase of the radius vector for any given angle of revolution, 6. Of this latter, the simplest case is when

= 27, 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 radu 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 radu (r), or the breadths in

a radial direction of any two whorls (W). We have then merely to apply the formula which we may simply write 7 = e°°°**, 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, As soon, then,-as we have prepared tables for these values, the determination of the constant angle a 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 =e7""*, or logr = loge x 27 x cota, from which we obtain the following figures * : * It is obvious that the ratios of opposite whorls, or of radii 180° apart, are represented by the square roots of these values; and the ratios of whorls or radii 90° apart, by the square roots of these again.