On Growth and Form

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*, 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

* For the correction. to be applied in the case of the helicoid, or “‘turbinate” 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 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 = ae’ Ob, this, for any given spiral, is equivalent to ae? Therefore log (r/a) = ké, or, I/k = =a : which leads, by subtraction to L/k . log (7,/r) = 27. Now, as a tends to 0, k (i.e. cot a) tends to #, and therefore, as k —> o, log (r,/r) —~> and also 7r,/r —> o. 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*.

(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 #; then draw to the inner whorl a tangent parallel to GEH, touching the curve in some point Ff. The straight line joining the points of contact, HF, must evidently pass through the pole: and, accordingly, the angle GHF 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 Fig. 274. boundaries of the whorl. But it is obvious that this method will be apt to fail us when the angle a is very small; and when, consequently, the points H 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 radi. The angle (?) between two of them, as AC, BD, is therefore equal to the angle @ between the polar radii from A and B, or from C and D; and therefore BD/AC = e°°°**, which gives us the angle a in terms of known quantities. (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 im 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, a, required. Or we may determine, graphically, at two points, the radii of | curvature, p,p2. 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)

The following method*, 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 wp; the arc AB, s; and AC, BE, respectively x,, x,, we have 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

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*, by Sandbergert, and by Grabaut, 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) Ratio of breadth of Breadth of whorls successive whorls ‘The angle (a)

t Grabau, A. H.. Ueber die Nawmannsche Conchospirale, etc. Inauguraldiss. Leipzig, 1872: Die Spiralen von Conchylien, etc. Programm, Nr. 502, Leipzig, 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 a, 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 imphes 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 increase as we pass from the centre of the coil outwards, and consequently for the values of the angle a 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 shght 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 agical 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 A be the ratio of growth between the outer and the inner curve, then, the outer curve being represented by and 6 may then be called the angle of retardation, to which the inner curve is subject by virtue of its slower rate of growth. Dispensing with mathematical formulae, the several conditions may be illustrated as follows: In the diagrams (Fig. 278), OP,P,Ps;, etc. represents a radius, on which P,, P,, P;, are the points attained by the outer border of the tubular shell after as many entire consecutive revolutions. And P,’, P,’, P,', are the points similarly intersected by the inner border; OP/OP’ being always = 4, which is the ratio of growth, or “cutting-down factor.” Then, obviously, when OP, is less than OP,’ the whorls will be separated by an interspace (qa); (2) when OP, = OP,’ they will be in contact (6b), and (3) when OP, is greater than OP,’ 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 A be very large, that is to say if OP, be greater, not only than OP,’ but also than OP,', OP,', 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 NV. wmbilicatus; and the difference between these two forms, or “species,” 1s constituted accordingly by a difference in the value of A. (5) There is also a final case, not easily distinguishable externally from (4), where P’ lies on

the opposite side of the radius vector to P, and is therefore imaginary. This final condition is exhibited in Argonauta. 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 27. Let r and r’ refer to the inner, and R, R’ to the outer sides of the two whorls. Then, if we consider Now in the three cases (a, b, c) represented in Fig. 278, it is plain that 7’ 2 R, respectively. That is to say,

the case represented in Fig. 278, b: that is to say, the particular case, for each value of a, where the consecutive whorls just touch, without interspace or overlap. For such cases, then, we may tabulate the values of A, as follows: Constant angle a Ratio (A) of rate of growth of inner border of tube, We see, accordingly, that in plane spirals whose constant angle les, 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 2 to “2.

If on the other hand we have, for any given value of a, a value of A 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 ¢. And, just as we have constructed this table of values-of A 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). In this case, let us call OA=R, OC =R,, and OB=r. We then have R, = OA = acct, R= OG = ae? ten) cate’ RR, pe ae? (8t7) cota = 72 oe And ich aa OV tae epee whence, equating, 1/A = e708, The corresponding values of 4 are as follows: Ratio (A) 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 (a) breadths of the whorls themselves 90° 1-00 (imaginary) * It has been pointed out to me that it does not follow at once and obviously that, because the interspace AB is a mean proportional between the breadths of the adjacent whorls, therefore the whole distance OB is a mean proportional between OA and OC. This is a corollary which requires to be proved; but the proof is easy. it is evident that when £ = 27, that will mean that A=1. In other words, the outer and inner borders of the tube are identical, and the tube is constituted by one continuous line.

When A 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 6 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 A nearly approaches unity in the one case, and is still large in the other, 8 being correspondingly small; while we can easily find cases where f is very large, and A 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 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 wmbilicus, 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 277) on the same radius vector, the locus of intersection will follow a spiral line upon the shell, which is called the “suture” by conchologists. Itis 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 hes internally, winding round the open cone of the umbilicus*; 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 a is too small ever to give us a complete revolution of the spire.

* A beautiful construction: stwpendum Naturae artificcwm, Linnaeus. 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*: ‘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 shellt. T'0o such phylogenetic hypotheses the mathematical or dynamical study of the forms of shells lends no valid support. It 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.

* English edition, p. 537, 1900. The chapter is revised by Prof. Alpheus Hyatt, to whom the nomenclature is largely due. For a more copious terminology, see Hyatt, Phylogeny of an Acquired Characteristic, p. 422 seq., 1894. + This latter conclusion is adopted by Willey, Zoological Results, p. 747, 1902. 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.