Chapter XI: The Logarithmic Spiral (1)

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 lack[495]. 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.

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.

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.

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 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 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 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 confusion[496].

In Fig. 243_B_ (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.

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 spiral[497].” 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_ = _a_θ.

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_ = _b_θ. 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′_ + _a_θ cot α. The outer whorls resemble those of an Archimedean spiral, because of the other term _a_θ 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.

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}

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 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, spiral[498].”

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.

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 learning[499]; and its bearing on our biological problem of the shell, though apparently indirect, is yet so close that it deserves our further consideration.

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 parallelogram[500], 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 = 1^2,

1^2 + 3 = 2^2,

2^2 + 5 = 3^2,

3^2 + 7 = 4^2 _etc._,

which relation we may illustrate graphically σχηματογραφεῖν by the successive numbers of dots which keep the annexed figure a perfect square[501]: as follows:

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

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

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 gnomon[502]. 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 mathematicians[503]. {512}

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 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 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 structure[504]. {515}

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 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.

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 mollusca[505].

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 vector[506]. 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 duplicatus_[507] 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.

Widths of successive Terms of a geometrical progression,
whorls measured in inches whose first term is the width of
and parts of an inch the widest whorl, and whose
common ratio is 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 ──────────────────────── Six whorls Three whorls Ratio µ

5·37 2·03 2·645
4·55 1·72 2·645

Four whorls Two whorls

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)^½ = 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 Logarithms of Difference of
successive whorls successive whorls 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.

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.

In a number of such opercula, Moseley measured the breadths of the successive whorls along a radius vector[508], 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.

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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.