Chapter XI · THE LOGARITHMIC SPIRAL

The very numerous examples of spiral conformation which we meet with in our studies of organic form are peculiarly adapted to math­e­mat­i­cal methods of in­ves­ti­ga­tion. But ere we begin to study them, we must take care to define our terms, and we had better also attempt some rough preliminary clas­si­fi­ca­tion of the objects with which we shall have to deal.

In general terms, a Spiral Curve is a line which, starting from a point of origin, continually diminishes in curvature as it recedes from that point; or, in other words, whose radius of curvature continually increases. This definition is wide enough to include a number of different curves, but on the other hand it excludes at least one which in popular speech we are apt to confuse with a true spiral. This latter curve is the simple Screw, or cylindrical Helix, which curve, as is very evident, neither starts from a definite origin, nor varies in its curvature as it proceeds. The “spiral” thickening of a woody plant-cell, the “spiral” thread within an insect’s tracheal tube, or the “spiral” twist and twine of a climbing stem are not, math­e­mat­i­cally speaking, spirals at all, but screws or helices. They belong to a distinct, though by no means very remote, family of curves. Some of these helical forms we have just now treated of, briefly and parenthetically, under the subject of Geodetics.

Of true organic spirals we have no lack495. We think at once of the beautiful spiral curves of the horns of ruminants, and of the still more varied, if not more beautiful, spirals of molluscan shells. Closely related spirals may be traced in the arrangement {494} of the florets in the sunflower; a true spiral, though not, by the way, so easy of in­ves­ti­ga­tion, is presented to us by the outline of a cordate leaf; and yet again, we can recognise typical though transitory spirals in the coil of an elephant’s trunk, in the “circling {495} spires” of a snake, in the coils of a cuttle-fish’s arm, or of a monkey’s or a chameleon’s tail.

Fig. 237. The shell of Nautilus pompilius, from a radiograph: to shew the logarithmic spiral of the shell, together with the arrangement of the internal septa. (From Messrs Green and Gardiner, in Proc. Malacol. Soc. II, 1897.)

Among such forms as these, and the many others which we might easily add to them, it is obvious that we have to do with things which, though math­e­mat­i­cally similar, are biologically speaking fundamentally different. And not only are they biologically remote, but they are also physically different, in regard to the nature of the forces to which they are severally due. For in the first place, the spiral coil of the elephant’s trunk or of the chameleon’s tail is, as we have said, but a transitory configuration, and is plainly the result of certain muscular forces acting upon a structure of a definite, and normally an essentially different, form. It is rather a position, or an attitude, than a form, in the sense in which we have been using this latter term; and, unlike most of the forms which we have been studying, it has little or no direct relation to the phenomenon of Growth.

Fig. 238. A Foraminiferal shell (Globigerina).

Again, there is a manifest and not unimportant difference between such a spiral conformation as is built up by the separate and successive florets in the sunflower, and that which, in the snail or Nautilus shell, is apparently a single and indivisible unit. And a similar, if not identical difference is apparent between the Nautilus shell and the minute shells of the Foraminifera, which so closely simulate it; inasmuch as the spiral shells of these latter are essentially composite structures, combined out of successive and separate chambers, while the molluscan shell, though it may (as in Nautilus) become secondarily subdivided, has grown as one continuous tube. It follows from all this that there cannot {496} possibly be a physical or dynamical, though there may well be a math­e­mat­i­cal Law of Growth, which is common to, and which defines, the spiral form in the Nautilus, in the Globigerina, in the ram’s horn, and in the disc of the sunflower.

Of the spiral forms which we have now mentioned, every one (with the single exception of the outline of the cordate leaf) is an example of the remarkable curve known as the Logarithmic Spiral. But before we enter upon the mathematics of the logarithmic spiral, let us carefully observe that the whole of the organic forms in which it is clearly and permanently exhibited, however different they may be from one another in outward appearance, in nature and in origin, nevertheless all belong, in a certain sense, to one particular class of conformations. In the great majority of cases, when we consider an organism in part or whole, when we look (for instance) at our own hand or foot, or contemplate an insect or a worm, we have no reason (or very little) to consider one part of the existing structure as older than another; through and through, the newer particles have been merged and commingled, by intussusception, among the old; the whole outline, such as it is, is due to forces which for the most part are still at work to shape it, and which in shaping it have shaped it as a whole. But the horn, or the snail-shell, is curiously different; for in each of these, the presently existing structure is, so to speak, partly old and partly new; it has been conformed by successive and continuous increments; and each successive stage of growth, starting from the origin, remains as an integral and unchanging portion of the still growing structure, and so continues to represent what at some earlier epoch constituted for the time being the structure in its entirety.

In a slightly different, but closely cognate way, the same is true of the spirally arranged florets of the sunflower. For here again we are regarding serially arranged portions of a composite structure, which portions, similar to one another in form, differ in age; and they differ also in magnitude in a strict ratio according to their age. Somehow or other, in the logarithmic spiral the time-element always enters in; and to this important fact, full of curious biological as well as math­e­mat­i­cal significance, we shall afterwards return. {497}

It is, as we have so often seen, an essential part of our whole problem, to try to understand what distribution of forces is capable of producing this or that organic form,—to give, in short, a dynamical expression to our descriptive morphology. Now the general distribution of forces which lead to the formation of a spiral (whether logarithmic or other) is very easily understood; and need not carry us beyond the use of very elementary mathematics.

Fig. 239.

If we imagine growth to act in a perpendicular direction, as for example the upward force of growth in a growing stem (OA), then, in the absence of other forces, elongation will as a matter of course proceed in an unchanging direction, that is to say the stem will grow straight upwards. Suppose now that there be some constant external force, such as the wind, impinging on the growing stem; and suppose (for simplicity’s sake) that this external force be in a constant direction (AB) perpendicular to the intrinsic force of growth. The direction of actual growth will be in the line of the resultant of the two forces: and, since the external force is (by hypothesis) constant in direction, while the internal force tends always to act in the line of actual growth, it is obvious that our growing organism will tend to be bent into a curve, to which, for the time being, {498} the actual force of growth will be acting at a tangent. So long as the two forces continue to act, the curve will approach, but will never attain, the direction of AB, perpendicular to the original direction OA. If the external force be constant in amount the curve will ap­prox­i­mate to the form of a hyperbola; and, at any rate, it is obvious that it will never tend to assume a spiral form.

In like manner, if we consider a horizontal beam, fixed at one end, the imposition of a weight at the other will bend the beam into a curve, which, as the beam elongates or the weight increases, will bring the weighted end nearer and nearer to the vertical. But such a force, constant in direction, will obviously never curve the beam into a spiral,—a fact so patent and obvious that it would be superfluous to state it, were it not that some naturalists have been in the habit of invoking gravity as the force to which may be attributed the spiral flexure of the shell.

But if, on the other hand, the deflecting force be inherent in the growing body, or so connected with it in a system that its direction (instead of being constant, as in the former case) changes with the direction of growth, and is perpendicular (or inclined at some constant angle) to this changing direction of the growing force, then it is plain that there is no such limit to the deflection from the normal, but the growing curve will tend to wind round and round its point of origin. In the typical case of the snail-shell, such an intrinsic force is manifestly present in the action of the columellar muscle.

Many other simple illustrations can be given of a spiral course being impressed upon what is primarily rectilinear motion, by any steady deflecting force which the moving body carries, so to speak, along with it, and which continually gives a lop-sided tendency to its forward movement. For instance, we have been told that a man or a horse, travelling over a great prairie, is very apt to find himself, after a long day’s journey, back again near to his starting point. Here some small and imperceptible bias, such as might for instance be caused by one leg being in a minute degree longer or stronger than the other, has steadily deflected the forward movement to one side; and has gradually brought the traveller back, perhaps in a circle to the very point from which he set out, {499} or else by a spiral curve, somewhere within reach and recognition of it.

We come to a similar result when we consider, for instance, a cylindrical body in which forces of growth are at work tending to its elongation, but these forces are unsymmetrically distributed. Let the tendency to elongation along AB be of a magnitude proportional to BB′, and that along CD be of a magnitude proportional to DD′; and in each element parallel to AB and CD, let a parallel force of growth, proportionately intermediate in magnitude, be at work: and let EFF′ be the middle line. Then at any cross-section BFD, if we deduct the mean force FF′, we have a certain positive force at B, equal to Bb, and an equal and opposite force at D, equal to Dd. But AB and CD are not separate

Fig. 240.

structures, but are connected together, either by a solid core, or by the walls of a tubular shell; and the forces which tend to separate B and D are opposed, accordingly, by a tension in BD. It follows therefore, that there will be a resultant force BG, acting in a direction intermediate between Bb and BD, and also a resultant, DH, acting at D in an opposite direction; and accordingly, after a small increment of growth, the growing end of the cylinder will come to lie, not in the direction BD, but in the direction GH. The problem is therefore analogous to that of a beam to which we apply a bending moment; and it is plain that the unequal force of growth is equivalent to a “couple” which will impart to our structure a curved form. For, if we regard the part ABDC as practically rigid, and the part BB′D′D as pliable, this couple {500} will tend to turn strips such as B′D′ about an axis perpendicular to the plane of the diagram, and passing through an intermediate point F′. It is plain, also, since all the forces under consideration are intrinsic to the system, that this tendency will be continuous, and that as growth proceeds the curving body will assume either a circular or a spiral form. But the tension which we have here assumed to exist in the direction BD will obviously disappear if we suppose a sufficiently rapid rate of growth in that direction. For if we may regard the mouth of our tubular shell as perfectly extensible in its own plane, so that it exerts no traction whatsoever on the sides, then it will be drawn out into more and more elongated ellipses, forming the more and more oblique orifices of a straight tube. In other words, in such a structure as we have presupposed, the existence or

Fig. 241.

maintenance of a constant ratio between the rates of extension or growth in the vertical and transverse directions will lead, in general, to the development of a logarithmic spiral; the magnitude of that ratio will determine the character (that is to say, the constant angle) of the spiral; and the spirals so produced will include, as special or limiting cases, the circle and the straight line.

We may dispense with the hypothesis of bending moments, if we simply presuppose that the increments of growth take place at a constant angle to the growing surface (as AB), but more rapidly at A (which we shall call the “outer edge”) than at B, and that this difference of velocity maintains a constant ratio. Let us also assume that the whole structure is rigid, the new accretions solidifying as soon as they are laid on. For example, {501} let Fig. 242 represent in section the early growth of a Nautilus-shell, and let the part ARB represent the earliest stage of all, which in Nautilus is nearly semicircular. We have to find a law governing the growth of the shell, such that each edge shall develop into an equiangular spiral; and this law, accordingly, must be the same for each edge, namely that at each instant the direction of growth makes a constant angle with a line drawn from a fixed point (called the pole of the spiral) to the point at which growth is taking place. This growth, we now find, may be considered as effected by the continuous addition of similar quadrilaterals. Thus, in Fig. 241, AEDB is a quadrilateral with AE, DB parallel, and with the angle EAB of a certain definite

Fig. 242.

magnitude, = γ. Let AB and ED meet, when produced, in C; and call the angle ACE (or xCy) = β. Make the angle yCz = angle xCy, = β. Draw EG, so that the angle yEG = γ, meeting Cz in G; and draw DF parallel to EG. It is then easy to show that AEDB and EGFD are similar quadrilaterals. And, when we consider the quadrilateral AEDB as having infinitesimal sides, AE and BD, the angle γ tends to α, the constant angle of an equiangular spiral which passes through the points AEG, and of a similar spiral which passes through the points BDF; and the point C is the pole of both of these spirals. In a particular limiting case, when our quadrilaterals are all equal as well as similar,—which will be the case when the angle γ (or the angles EAC, etc.) is a {502} right angle,—the “spiral” curve will be a circular arc, C being the centre of the circle.

Another, and a very simple illustration may be drawn from the “cymose inflorescences” of the botanists, though the actual mode of development of some of these structures is open to dispute, and their nomenclature is involved in extraordinary historical confusion496.

Fig. 243. A, a helicoid, B, a scorpioid cyme.

In Fig. 243B (which represents the Cicinnus of Schimper, or cyme unipare scorpioide of Bravais, as seen in the Borage), we begin with a primary shoot from which is given off, at a certain definite angle, a secondary shoot: and from that in turn, on the same side and at the same angle, another shoot, and so on. The deflection, or curvature, is continuous and progressive, for it is caused by no external force but only by causes intrinsic in the system. And the whole system is symmetrical: the angles at which the successive shoots are given off being all equal, and the lengths of the shoots diminishing in constant ratio. The result is that the successive shoots, or successive increments of growth, are tangents to a curve, and this curve is a true logarithmic spiral. But while, in this simple case, the successive shoots are depicted as lying in a plane, it may also happen that, in addition to their successive angular divergence from one another within that plane, they also tend to diverge by successive equal angles from that plane of reference; and by this means, there will be superposed upon the logarithmic spiral a helicoid twist or screw. And, in the particular case where this latter angle of divergence is just equal to 180°, or two right angles, the successive shoots will once more come to lie in a plane, but they will appear to come off from one another on alternate sides, as in Fig. 243 A. This is the Schraubel or Bostryx of Schimper, the cyme unipare hélicoide of Bravais. The logarithmic spiral is still latent in it, as in the other; but is concealed from view by the deformation resulting from the helicoid. The confusion of nomenclature would seem to have arisen from the fact that many botanists did not recognise (as the brothers Bravais did) the math­e­mat­i­cal significance of the latter case; but were led, by the snail-like spiral of the scorpioid cyme, to transfer the name “helicoid” to it.

In the study of such curves as these, then, we speak of the point of origin as the pole (O); a straight line having its extremity in the pole and revolving about it, is called the radius vector; {503} and a point (P) which is conceived as travelling along the radius vector under definite conditions of velocity, will then describe our spiral curve.

Of several math­e­mat­i­cal curves whose form and development may be so conceived, the two most important (and the only two with which we need deal), are those which are known as (1) the equable spiral, or spiral of Archimedes, and (2) the logarithmic, or equiangular spiral.

Fig. 244.

The former may be illustrated by the spiral coil in which a sailor coils a rope upon the deck; as the rope is of uniform thickness, so in the whole spiral coil is each whorl of the same breadth as that which precedes and as that which follows it. Using its ancient definition, we may define it by saying, that “If a straight line revolve uniformly about its extremity, a point which likewise travels uniformly along it will describe the equable spiral497.” Or, putting the same thing into our more modern words, “If, while the radius vector revolve uniformly about the pole, a point (P) travel with uniform velocity along it, the curve described will be that called the equable spiral, or spiral of Archimedes.” {504}

It is plain that the spiral of Archimedes may be compared to a cylinder coiled up. And it is plain also that a radius (r = OP), made up of the successive and equal whorls, will increase in arithmetical progression: and will equal a certain constant quantity (a) multiplied by the whole number of whorls, or (more strictly speaking) multiplied by the whole angle (θ) through which it has revolved: so that r = aθ.

But, in contrast to this, in the logarithmic spiral of the Nautilus or the snail-shell, the whorls gradually increase in breadth, and do so in a steady and unchanging ratio. Our definition is as follows: “If, instead of travelling with a uniform velocity, our point move along the radius vector with a velocity increasing as its distance from the pole, then the path described is called a logarithmic spiral.” Each whorl which the radius vector intersects will be broader than its predecessor in a definite ratio; the radius vector will increase in length in geometrical progression, as it sweeps through successive equal angles; and the equation to the spiral will be r = a^θ . As the spiral of Archimedes, in our example of the coiled rope, might be looked upon as a coiled cylinder, so may the logarithmic spiral, in the case of the shell, be pictured as a cone coiled upon itself.

Now it is obvious that if the whorls increase very slowly indeed, the logarithmic spiral will come to look like a spiral of Archimedes, with which however it never becomes identical; for it is incorrect to say, as is sometimes done, that the Archimedean spiral is a “limiting case” of the logarithmic spiral. The Nummulite is a case in point. Here we have a large number of whorls, very narrow, very close together, and apparently of equal breadth, which give rise to an appearance similar to that of our coiled rope. And, in a case of this kind, we might actually find that the whorls were of equal breadth, being produced (as is apparently the case in the Nummulite) not by any very slow and gradual growth in thickness of a continuous tube, but by a succession of similar cells or chambers laid on, round and round, determined as to their size by constant surface-tension conditions and therefore of unvarying dimensions. But even in this case we should have no Archimedean spiral, but only a logarithmic spiral in which the constant angle approximated to 90°. {505}

For, in the logarithmic spiral, when α tends to 90°, the expression r = a^{θ cot α} tends to r = a(1 + θ cot α); while the equation to the Archimedean spiral is r = bθ. The nummulite must always have a central core, or initial cell, around which the coil is not only wrapped, but out of which it springs; and this initial chamber corresponds to our a′ in the expression r = a′ + aθ cot α. The outer whorls resemble those of an Archimedean spiral, because of the other term aθ cot α in the same expression. It follows from this that in all such cases the whorls must be of excessively small breadth.

There are many other specific properties of the logarithmic spiral, so interrelated to one another that we may choose pretty well any one of them as the basis of our definition, and deduce the others from it either by analytical methods or by the methods of elementary geometry. For instance, the equation r = a^θ may be written in the form log r = θ log a, or θ = (log r) ⁄ (log a), or (since a is a constant), θ = k log r. Which is as much as to say that the vector angles about the pole are proportional to the logarithms of the successive radii; from which circumstance the name of the “logarithmic spiral” is derived.

Fig. 245.

Let us next regard our logarithmic spiral from the dynamical point of view, as when we consider the forces concerned in the growth of a material, concrete spiral. In a growing structure, let the forces of growth exerted at any point P be a force F acting along the line joining P to a pole O and a force T acting in a direction perpendicular to OP; and let the magnitude of these forces be in the same constant ratio at all points. It follows that the resultant of the forces F and T (as PQ) makes a constant angle with the radius vector. But the constancy of the angle between tangent and radius vector at any point is a fundamental property of the logarithmic spiral, and may be shewn to follow from our definition of the curve: it gives to the curve its alternative name of equiangular spiral. Hence in a structure growing under the above conditions the form of the boundary will be a logarithmic spiral. {506}

Fig. 246.

In such a spiral, radial growth and growth in the direction of the curve bear a constant ratio to one another. For, if we consider a consecutive radius vector, OP′, whose increment as compared with OP is dr, while ds is the small arc PP′, then

dr ⁄ ds = cos α = constant.

In the concrete case of the shell, the distribution of forces will be, originally, a little more complicated than this, though by resolving the forces in question, the system may be reduced to this simple form. And furthermore, the actual distribution of forces will not always be identical; for example, there is a distinct difference between the cases (as in the snail) where a columellar muscle exerts a definite traction in the direction of the pole, and those (such as Nautilus) where there is no columellar muscle, and where some other force must be discovered, or postulated, to account for the flexure. In the most frequent case, we have, as in Fig. 247, three forces to deal with, acting at a point, p : L, acting

Fig. 247.

in the direction of the tangent to the curve, and representing the force of longitudinal growth; T, perpendicular to L, and representing the organism’s tendency to grow in breadth; and P, the traction exercised, in the direction of the pole, by the columellar muscle. Let us resolve L and T into components along P (namely A′, B′), and perpendicular to P (namely A, B); we have now only two forces to consider, viz. P − A′ − B′, and A − B. And these two latter we can again resolve, if we please, so as to deal only with forces in the direction of P and T. Now, the ratio of these forces remaining constant, the locus of the point p is an equiangular spiral. {507}

Furthermore we see how any slight change in any one of the forces P, T, L will tend to modify the angle α, and produce a slight departure from the absolute regularity of the logarithmic spiral. Such slight departures from the absolute simplicity and uniformity of the theoretic law we shall not be surprised to find, more or less frequently, in Nature, in the complex system of forces presented by the living organism.

In the growth of a shell, we can conceive no simpler law than this, namely, that it shall widen and lengthen in the same unvarying proportions: and this simplest of laws is that which Nature tends to follow. The shell, like the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence, and may be made the basis of a definition, of the logarithmic spiral.

Such a definition, though not commonly used by mathematicians, has been occasionally employed; and it is one from which the other properties of the curve can be deduced with great ease and simplicity. In math­e­mat­i­cal language it would run as follows: “Any [plane] curve proceeding from a fixed point (which is called the pole), and such that the arc intercepted between this point and any other whatsoever on the curve is always similar to itself, is called an equiangular, or logarithmic, spiral498.”

In this definition, we have what is probably the most fundamental and “intrinsic” property of the curve, namely the property of continual similarity: and this is indeed the very property by reason of which it is peculiarly associated with organic growth in such structures as the horn or the shell, or the scorpioid cyme which is described on p. 502. For it is peculiarly char­ac­ter­is­tic of the spiral of a shell, for instance, that (under all normal circumstances) it does not alter its shape as it grows; each increment is geometrically similar to its predecessor, and the whole, at any epoch, is similar to what constituted the whole at another and an earlier epoch. We feel no surprise when the animal which secretes the shell, or any other animal whatsoever, grows by such {508} symmetrical expansion as to preserve its form unchanged; though even there, as we have already seen, the unchanging form denotes a nice balance between the rates of growth in various directions, which is but seldom accurately maintained for long. But the shell retains its unchanging form in spite of its asymmetrical growth; it grows at one end only, and so does the horn. And this remarkable property of increasing by terminal growth, but nevertheless retaining unchanged the form of the entire figure, is char­ac­ter­is­tic of the logarithmic spiral, and of no other math­e­mat­i­cal curve.

Fig. 248.

We may at once illustrate this curious phenomenon by drawing the outline of a little Nautilus shell within a big one. We know, or we may see at once, that they are of precisely the same shape; so that, if we look at the little shell through a magnifying glass, it becomes identical with the big one. But we know, on the other hand, that the little Nautilus shell grows into the big one, not by uniform growth or magnification in all directions, as is (though only ap­prox­i­mate­ly) the case when the boy grows into the man, but by growing at one end only.

Though of all curves, this property of continued similarity is found only in the logarithmic spiral, there are very many rectilinear figures in which it may be observed. For instance, as we may easily see, it holds good of any right cone; for evidently, in Fig. 248, the little inner cone (represented in its triangular section) may become identical with the larger one either by magnification all round (as in a), or simply by an increment at one end (as in b); indeed, in the case of the cone, we have yet a third possibility, for the same result is attained when it increases all round, save only at the base, that is to say when the triangular section increases {509} on two of its sides, as in c. All this is closely associated with the fact, which we have already noted, that the Nautilus shell is but a cone rolled up; in other words, the cone is but a particular variety, or “limiting case,” of the spiral shell.

This property, which we so easily recognise in the cone, would seem to have engaged the particular attention of the most ancient mathematicians even from the days of Pythagoras, and so, with little doubt, from the more ancient days of that Egyptian school whence he derived the foundations of his learning499; and its bearing on our biological problem of the shell, though apparently indirect, is yet so close that it deserves our further consideration.

Fig. 249. Fig. 250.

If, as in Fig. 249, we add to two sides of a square a symmetrical L-shaped portion, similar in shape to what we call a “carpenter’s square,” the resulting figure is still a square; and the portion which we have added is called, by Aristotle (Phys. III, 4), a “gnomon.” Euclid extends the term to include the case of any parallelogram500, whether rectangular or not (Fig. 250); and Hero of Alexandria specifically defines a “gnomon” (as indeed Aristotle implicitly defines it), as any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original. Included in this important definition is the case of numbers, considered geometrically; that is to say, the εἰδητικοὶ ἀριθμοί, which can be translated into form, by means of rows of dots or other signs (cf. Arist. Metaph. 1092 b 12), or in the pattern of a tiled floor: all according to “the mystical way of {510} Pythagoras, and the secret magick of numbers.” Thus for example, the odd numbers are “gnomonic numbers,” because

0 + 1 = 1² , 1² + 3 = 2² , 2² + 5 = 3² , 3² + 7 = 4² etc.,

which relation we may illustrate graphically σχηματογραφεῖν by the successive numbers of dots which keep the annexed figure a perfect square501: as follows:

There are other gnomonic figures more curious still. For instance, if we make a rectangle (Fig. 251) such that the two sides

Fig. 251. Fig. 252.

are in the ratio of 1 : √2, it is obvious that, on doubling it, we obtain a precisely similar figure; for 1 : √2 :: √2 : 2; and {511} each half of the figure, accordingly, is now a gnomon to the other. Another elegant example is when we start with a rectangle (A) whose sides are in the proportion of 1 : ½(√5 − 1), or, ap­prox­i­mate­ly, 1 : 0·618. The gnomon to this figure is a square (B) erected on its longer side, and so on successively (Fig. 252).

Fig. 253. Fig. 254.

In any triangle, as Aristotle tells us, one part is always a gnomon to the other part. For instance, in the triangle ABC (Fig. 253), let us draw CD, so as to make the angle BCD equal to the angle A. Then the part BCD is a triangle similar to the whole triangle ABC, and ADC is a gnomon to BCD. A very elegant case is when the original triangle ABC is an isosceles triangle having one angle of 36°, and the other two angles, therefore, each equal to 72° (Fig. 254). Then, by bisecting one of the angles of the base, we subdivide the large isosceles triangle into two isosceles triangles, of which one is similar to the whole figure and the other is its gnomon502. There is good reason to believe that this triangle was especially studied by the Pythagoreans; for it lies at the root of many interesting geometrical constructions, such as the regular pentagon, and the mystical “pentalpha,” and a whole range of other curious figures beloved of the ancient mathematicians503. {512}

Fig. 255.

If we take any one of these figures, for instance the isosceles triangle which we have just described, and add to it (or subtract from it) in succession a series of gnomons, so converting it into larger and larger (or smaller and smaller) triangles all similar to the first, we find that the apices (or other cor­re­spon­ding points) of all these triangles have their locus upon a logarithmic spiral: a result which follows directly from that alternative definition of the logarithmic spiral which I have quoted from Whitworth (p. 507).

Again, we may build up a series of right-angled triangles, each of which is a gnomon to the preceding figure; and here again, a logarithmic spiral is the locus of cor­re­spon­ding points in these successive triangles. And lastly, whensoever we fill up space with

Fig. 256. Logarithmic spiral derived from cor­re­spon­ding points in a system of squares.

a {513} collection of either equal or similar figures, similarly situated, as in Figs. 256, 257, there we can always discover a series of inscribed or escribed logarithmic spirals.

Once more, then, we may modify our definition, and say that: “Any plane curve proceeding from a fixed point (or pole), and such that the vectorial area of any sector is always a gnomon to the whole preceding figure, is called an equiangular, or logarithmic, spiral.” And we may now introduce this new concept and nomenclature into our description of the Nautilus shell and other related organic forms, by saying that: (1) if a growing

Fig. 257. The same in a system of hexagons.

structure be built up of successive parts, similar and similarly situated, we can always trace through cor­re­spon­ding points a series of logarithmic spirals (Figs. 258, 259, etc.); (2) it is char­ac­ter­is­tic of the growth of the horn, of the shell, and of all other organic forms in which a logarithmic spiral can be recognised, that each successive increment of growth is a gnomon to the entire pre-existing structure. And conversely (3) it follows obviously, that in the logarithmic spiral outline of the shell or of the horn we can always inscribe an endless variety of other gnomonic figures, having no necessary relation, save as a {514} math­e­mat­i­cal accident, to the nature or mode of development of the actual structure504. {515}

Fig. 258. A shell of Haliotis, with two of the many lines of growth, or generating curves, marked out in black: the areas bounded by these lines of growth being in all cases “gnomons” to the pre-existing shell.

Fig. 259. A spiral foraminifer (Pulvinulina), to show how each successive chamber continues the symmetry of, or constitutes a gnomon to, the rest of the structure.

Of these three propositions, the second is of very great use and advantage for our easy understanding and simple description of the molluscan shell, and of a great variety of other structures whose mode of growth is analogous, and whose math­e­mat­i­cal properties are therefore identical. We see at once that the successive chambers of a spiral Nautilus (Fig. 237) or of a straight Orthoceras (Fig. 300), each whorl or part of a whorl of a periwinkle or other gastropod (Fig. 258), each new increment of the operculum of a gastropod (Fig. 263), each additional increment of

Fig. 260. Another spiral foraminifer, Cristellaria.

an elephant’s tusk, or each new chamber of a spiral foraminifer (Figs. 259 and 260), has its leading char­ac­ter­is­tic at once described and its form so far explained by the simple statement that it constitutes a gnomon to the whole previously existing structure. And herein lies the explanation of that “time-element” in the development of organic spirals of which we have spoken already, in a preliminary and empirical way. For it follows as a simple corollary to this theorem of gnomons that we must not expect to find the logarithmic spiral manifested in a structure whose parts are simultaneously produced, as for instance in the margin of a leaf, or among the many curves that make the contour of a fish. But we must rather look for it wherever the organism retains for us, and still presents to us at a single view, the successive phases of preceding growth, the successive magnitudes attained, the successive outlines occupied, as the organism or a part thereof pursued the even tenour of its growth, year by year and day by day. And it easily follows from this, that it is in the hard parts of organisms, and not the soft, fleshy, actively growing parts, that this spiral is commonly and char­ac­teris­ti­cally found; not in the fresh mobile tissues whose form is constrained merely by the active forces of the moment; but in things like shell and tusk, and horn and claw, where the object is visibly composed of parts {516} successively, and permanently, laid down. In the main, the logarithmic spiral is char­ac­ter­is­tic, not of the living tissues, but of the dead. And for the same reason, it will always or nearly always be accompanied, and adorned, by a pattern formed of “lines of growth,” the lasting record of earlier and successive stages of form and magnitude.

It is evident that the spiral curve of the shell is, in a sense, a vector diagram of its own growth; for it shews at each instant of time, the direction, radial and tangential, of growth, and the unchanging ratio of velocities in these directions. Regarding the actual velocity of growth in the shell, we know very little (or practically nothing), by way of experimental measurement; but if we make a certain simple assumption, then we may go a good deal further in our description of the logarithmic spiral as it appears in this concrete case.

Let us make the assumption that similar increments are added to the shell in equal times; that is to say, that the amount of growth in unit time is measured by the areas subtended by equal angles. Thus, in the outer whorl of a spiral shell a definite area marked out by ridges, tubercles, etc., has very different linear dimensions to the cor­re­spon­ding areas of the inner whorl, but the symmetry of the figure implies that it subtends an equal angle with these; and it is reasonable to suppose that the successive regions, marked out in this way by successive natural boundaries or patterns, are produced in equal intervals of time.

If this be so, the radii measured from the pole to the boundary of the shell will in each case be proportional to the velocity of growth at this point upon the circumference, and at the time when it corresponded with the outer lip, or region of active growth; and while the direction of the radius vector corresponds with the direction of growth in thickness of the animal, so does the tangent to the curve correspond with the direction, for the time being, of the animal’s growth in length. The successive radii are a measure of the acceleration of growth, and the spiral curve of the shell itself is no other than the hodograph of the growth of the contained organism. {517}

So far as we have now gone, we have studied the elementary properties of the logarithmic spiral, including its fundamental property of continued similarity; and we have accordingly learned that the shell or the horn tends necessarily to assume the form of this math­e­mat­i­cal figure, because in these structures growth proceeds by successive increments, which are always similar in form, similarly situated, and of constant relative magnitude one to another. Our chief objects in enquiring further into the math­e­mat­i­cal properties of the logarithmic spiral will be: (1) to find means of confirming and verifying the fact that the shell (or other organic curve) is actually a logarithmic spiral; (2) to learn how, by the properties of the curve, we may further extend our knowledge or simplify our descriptions of the shell; and (3) to understand the factors by which the char­ac­ter­is­tic form of any particular logarithmic spiral is determined, and so to comprehend the nature of the specific or generic characters by which one spiral shell is found to differ from another.

Of the elementary properties of the logarithmic spiral, so far as we have now enumerated them, the following are those which we may most easily in­ves­ti­gate in the concrete case, such as we have to do with in the molluscan shell: (1) that the polar radii of points whose vectorial angles are in arithmetical progression, are themselves in geometrical progression; and (2) that the tangent at any point of a logarithmic spiral makes a constant angle (called the angle of the spiral) with the polar radius vector.

Fig. 261.

The former of these two propositions may be written in what is, perhaps, a simpler form, as follows: radii which form equal angles about the pole of the logarithmic spiral, are themselves continued proportionals. That is to say, in Fig. 261, when the angle ROQ is equal to the angle QOP, then OR : OQ :: OQ : OP.

A particular case of this proposition is when the equal angles are each angles of 360°: that is to say when in each case the radius vector makes a complete revolution, and when, therefore P, Q and R all lie upon the same radius. {518}

It was by observing, with the help of very careful measurement, this continued proportionality, that Moseley was enabled to verify his first assumption, based on the general appearance of the shell, that the shell of Nautilus was actually a logarithmic spiral, and this demonstration he was immediately afterwards in a position to generalise by extending it to all the spiral Ammonitoid and Gastropod mollusca505.

For, taking a median transverse section of a Nautilus pompilius, and carefully measuring the successive breadths of the whorls (from the dark line which marks what was originally the outer surface, before it was covered up by fresh deposits on the part of the growing and advancing shell), Moseley found that “the distance of any two of its whorls measured upon a radius vector is one-third that of the two next whorls measured upon the same radius vector506. Thus (in Fig. 262), ab is one-third of bc, de of ef, gh of hi, and kl of lm. The curve is therefore a logarithmic spiral.”

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.

From the apex of a large specimen of Turbo duplicatus507 a {519} 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 1·1804.

Fig. 262.

Widths of successive whorls measured in inches and parts of an inch Terms of a geometrical progression, whose first term is the width of the widest whorl, and whose common ratio is 1·1804

1·31 1·31

1·12 1·1098

·94 ·94018

·80 ·79651

·67 ·67476

·57 ·57164

·48 ·48427

·41 ·41026

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 {520} 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 µ 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 the formula

r = (µ − 1)^{2 ⁄ m} .”

Accordingly, Moseley made the following measurements, beginning from the second and third whorls respectively:

Width of Ratio µ

Six whorls Three whorls

5·37 2·03 2·645

4·55 1·72 2·645

Four whorls Two whorls Ratio µ

4·15 1·74 2·385

3·52 1·47 2·394

“By the ratios of the two first admeasurements, the formula gives

r = (1·645)^{1 ⁄ 3} = 1·1804.

By the mean of the ratios deduced from the second two admeasurements, it gives

r = (1·389)^{1 ⁄ 2} = 1·1806.

“It 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. {521}

Turbo duplicatus.

Relative widths of successive whorls Logarithms of successive whorls Difference of successive logarithms

131 2·11727 —

112 2·04922 ·06805

94 1·97313 ·07609

80 1·90309 ·07004

67 1·82607 ·07702

57 1·75587 ·07020

48 1·68124 ·07463

41 1·161278 ·06846

Mean difference ·07207

And ·07207 is the logarithm of 1·1805.

Fig. 263. Operculum of Turbo.

The logarithmic spiral is not only very beautifully manifested in the molluscan shell, but also, in certain cases, in the little 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 {522} (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.

Fig. 264. Opercula of (1) Turbo, (2) Nerita. (After Moseley.)

In a number of such opercula, Moseley measured the breadths of the successive whorls along a radius vector508, 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 (in inches) of successive whorls, measured from the pole.

Distance Ratio Distance Ratio Distance Ratio Distance Ratio

·24 ·16 ·2 ·18

2·28 2·31 2·30 2·30

·55 ·37 ·6 ·42

2·32 2·30 2·30 2·24

1·28 ·85 1·38 ·94

{523}

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 the 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 {524} 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 limiting 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 {525} 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.

Fig. 265. Melo ethiopicus, L.

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

Fig. 266. 1, Harpa; 2, Dolium. The ridges on the shell correspond in (1) to generating curves, in (2) to generating spirals.

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 forth509) 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 idea510. 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 concrevit511.” {529}

Fig. 267. D’Orbigny’s Helicometer.

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 Hel­i­co­meter512, 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

Fig. 268.

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.

Fig. 269.

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 dθ ⁄ dr,

therefore

dr ⁄ r = dθ 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} .

Fig. 270.

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 figures513: {534}

Ratio of breadth of each whorl to the next preceding r ⁄ 1 Constant angle 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,

Fig. 271.

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 Gastropods514, 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.

Fig. 272.

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.

Fig. 273.

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 = aε^{θ cot α} ,

this, for any given spiral, is equivalent to aε^kθ .

Therefore

log(r ⁄ a) = kθ,

or,

1 ⁄ k = θ ⁄ log(r ⁄ a).

Then, if θ increase by 2π, while r increases to r₁ ,

1 ⁄ k = (θ + 2π) ⁄ log(r₁ ⁄ a),

which leads, by subtraction to

1 ⁄ k · log(r₁ ⁄ r) = 2π.

Now, as α tends to 0, k (i.e. cot α) tends to ∞, and therefore, as k → ∞, log(r₁ ⁄ r) → ∞ and also r₁ ⁄ 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. Blake515. {537}

Fig. 274. (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.

Fig. 275. An Ammonite, to shew corrugated surface-pattern. Fig. 276. (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.

Fig. 277.

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, ρ₁ρ₂ . 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 = (ρ₁ − ρ₂) ⁄ s.

The following method516, 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 x₁ , x₂ , we have

ρ₁ = (x₁² + μ²) ⁄ 2μ , ap­prox­i­mate­ly, and

cot α = (x₂² − x₁²) ⁄ 2μs .

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 Naumann517, by Sandberger518, and by Grabau519, 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.

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

0·30 — — —

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.

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

0·25 — — —

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′ = aλe^{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:

Fig. 278.

In the diagrams (Fig. 278), O P₁ P₂ P₃ , etc. represents a radius, on which P₁ , P₂ , P₃ , are the points attained by the outer border of the tubular shell after as many entire consecutive revolutions. And P₁′, P₂′, P₃′, 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 P₁ is less than O P₂′ the whorls will be separated by an interspace (a); (2) when O P₁ = O P₂′ they will be in contact (b), and (3) when O P₁ is greater than O P₂′ 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 P₁ be greater, not only than O P₂′ but also than O P₃′, O P₄′, 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.

Fig 279.

In Fig. 279 we have portions of two successive whorls, whose cor­re­spon­ding points on the same radius vector (as R and R′) are, therefore, at a distance apart cor­re­spon­ding to 2π. Let r and r′ refer to the inner, and R, R′ to the outer sides of the two whorls. Then, if we consider

R = a e^{θ cot α} ,

it follows that

R′ = a e^{(θ + 2π)cot α} ,

r = λa e^{θ cot α} = a e^{(θ − β)cot α} ,

and

r′ = λa 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,

λa e^{(θ + 2π)cot α} a e^{θ cot α} ,

and

λe^{2π cot α} 1.

The case in which λe^{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 α of spiral Ratio (λ) of rate of growth of inner border of tube, 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.

Fig. 280.

If on the other hand we have, for any given value of α, a value of λ greater or less than the value given in the above table, then we have, respectively, the conditions of separation or of overlap which are exemplified in Fig. 278, a and c. And, just as we have constructed this table of values of λ for the particular case of simple contact between the whorls, so we could construct similar tables for various degrees of separation, or degrees of overlap.

For instance, a case which admits of simple solution is that in which the interspace between the whorls is everywhere a mean proportional between the breadths of the whorls themselves (Fig. 280). {545}

In this case, let us call OA = R, OC = R₁ and OB = r. We then have

R₁ = OA = a e^{θ cot α} ,

R₂ = OC = a e^{(θ + 2π) cot α} ,

R₁ R₂ = a e^{2(θ + π) cot α} = r² †.

And

r² = (1 ⁄ λ)² · ε^{2θ cot α} ,

whence, equating,

1 ⁄ λ = e^{π cot α} .

The cor­re­spon­ding values of λ are as follows:

Constant angle (α) 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 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′ = λe^{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 umbilicus521; 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 Palaeontology522: “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 shell523. 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.