On Growth and Form

For, in the logarithmic spiral, when a tends to 90°, the expression 7 = qi cote tends to r= a(1+ @cot a); while the equation to the Archimedean spiral is r=b@. 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 q@’ in the expression 7 = a’ + a cot a. The outer whorls resemble those of an Archimedean spiral, because of the other term aé cot a 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’ may be written in the form log r = @ log a, or 6 = log r/log a, or (since @ is a constant), 0=klogr. 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. Tn 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 Roe constancy of the angle between tangent Fig. 245. 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.

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 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

B most frequent case, we have, as aye 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; TZ, 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 Z 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.

Furthermore we see how any slight change in any one of the forces P, T, L will tend to modify the angle a, 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*.”

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 1s 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 sym-

* See an interesting paper by Whitworth, W. A., “The Equiangular Spiral, its chief properties proved geometrically,” in the Messenger of Mathematics (1), 1, p. 5, 1862. metrical 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 6); 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

on two of its sides, as inc. 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* ; 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 Fig. 249. Fig. 250. which we have added is called, by Aristotle (Phys. m1, 4), a “gnomon.” Kuclid extends the term to include the case of any parallelogram}, 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 eiéntvxol apOuot, which can be translated into form, by means of rows of dots or other signs (cf. Arist. Metaph. 1092b12), or in the pattern of a tiled floor: all according to “the mystical way of

* T am well aware that the debt of Greek science to Egypt and the East is vigorously denied by many scholars, some of whom go so far as to believe that the Egyptians never had any science, save only some “rough rules of thumb for measur- Pythagoras, and the secret magick of numbers.” Thus for example, the odd numbers are “gnomonic numbers,” because which relation we may illustrate graphically (cynwatoypadetv) by the successive numbers of dots which keep the annexed figure

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:V. 9, it is obvious that, on doubling it, we obtain a precisely similar figure; for 1:V2::V2:2; and 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: }(V 5 Lyon 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*. There is good reason to believe that this triangle was especially studied by the Pythagoreans ; for it les 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 7.

* This is the so-called Dreifachgleichschenkelige Dreieck; cf. Naber, op. infra cit. The ratio 1: 0-618 is again not hard to find in this construction. + See, on the mathematical history of the Gnomon, Heath’s Euclid, 1, passim, 1908; Zeuthen, Theoréme de Pythagore, Genéve, 1904; also a curious and interesting book, Das Theorem des Pythagoras, by Dr H. A. Naber, Haarlem, 1908. 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 The. 256. Logarithmic spiral derived from corresponding points in 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, ete.); (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

mathematical accident, to the nature or mode of development of the actual structure*. 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 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. * For many beautiful geometrical constructions based on the molluscan shell, see Colman, S. and Coan, C. A., Nature’s Harmonic Unity (ch. ix, Conchology), New York, 1912.

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 “timeelement” 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 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.

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 Q 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 MOP, 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 SBS ae a complete revolution, and when, therefore P, Q Fig. 261. and R all he upon the same radius.

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*. 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}. Thus (in Fig. 262), ab is one-third of bce, de of ef, gh of hi, and kl of Im. The curve is therefore a logarithmic spiral.”