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.
[Illustration: 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.)]
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.
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 240.]
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.
[Illustration: Fig. 241.]
[Illustration: Fig. 242.]
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].
[Illustration: Fig. 243. _A_, a helicoid, _B_, a scorpioid cyme.]
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.
[Illustration: 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 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.
[Illustration: 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}
[Illustration: 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.
[Illustration: Fig. 247.]
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.
[Illustration: 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 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.
[Illustration: Fig. 249.]
[Illustration: 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 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:
· · · · · ·
· · · · · ·
· · · · · ·
· · · · · ·
· · · · · ·
· · · · · ·
[Illustration: Fig. 251.]
[Illustration: Fig. 252.]
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).
[Illustration: Fig. 253.]
[Illustration: 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 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}
[Illustration: 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).
[Illustration: Fig. 256. Logarithmic spiral derived from corresponding
points in a system of squares.]
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.
[Illustration: Fig. 257. The same in a system of hexagons.]
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}
[Illustration: 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.]
[Illustration: 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.]
[Illustration: Fig. 260. Another spiral foraminifer, _Cristellaria_.]
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.
[Illustration: 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
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.
[Illustration: Fig. 262.]
Widths of successive Terms of a geometrical progression,
whorls measured in inches whose first term is the width of
and parts of an inch the widest whorl, and whose
common ratio is 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.
[Illustration: 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.
[Illustration: 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 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.
――――――――――
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.
[Illustration: 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 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}
[Illustration: Fig. 266. 1, _Harpa_; 2, _Dolium_. The ridges on the
shell correspond in (1) to generating curves, in (2) to generating
spirals.]
In the complete mathematical formula (such as I have not ventured
to set forth[509]) 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 idea[510]. 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 concrevit[511].” {529}
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 Helicometer[512], 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.
[Illustration: Fig. 267. D’Orbigny’s Helicometer.]
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 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.
[Illustration: Fig. 268.]
――――――――――
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.
[Illustration: Fig. 269.]
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_ _d_θ/_dr_,
therefore _dr_/_r_ = _d_θ 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}.
[Illustration: 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 figures[513]: {534}
Ratio of breadth of each
whorl to the next preceding Constant angle
_r_/1 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
[Illustration: Fig. 271.]
We learn several interesting things from this short table. We see,
in the first place, that where each whorl is about three times the
breadth of its neighbour and predecessor, as is the case in Nautilus,
the constant angle is in the neighbourhood of 80°; and hence also that,
in all the ordinary Ammonitoid shells, and in all the typically spiral
shells of the Gastropods[514], 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.
[Illustration: 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.
[Illustration: 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_ = _a_ε^{θ cot α},
this, for any given spiral, is equivalent to _a_ε^{_k_θ}.
Therefore log(_r_/_a_) = _k_θ,
or, 1/_k_ = θ/log(_r_/_a_).
Then, if θ increase by 2π, while _r_ increases to _r__{1},
1/_k_ = (θ + 2π)/log(_r__{1}/_a_),
which leads, by subtraction to
1/_k_ ⋅ log(_r__{1}/_r_) = 2π.
Now, as α tends to 0, _k_ (i.e. cot α) tends to ∞, and therefore, as
_k_ → ∞, log(_r__{1}/_r_) → ∞ and also _r__{1}/_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. Blake[515]. {537}
(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.
[Illustration: Fig. 274.]
(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}
[Illustration: Fig. 275. An Ammonite, to shew corrugated
surface-pattern.]
[Illustration: Fig. 276.]
(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.
[Illustration: 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, ρ_{1} ρ_{2}. 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 = (ρ_{1} − ρ_{2})/_s_.
The following method[516], 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 _x__{1}, _x__{2}, we have
ρ_{1} = (_x__{1}^2 + μ^2)/2μ, approximately,
and cot α = (_x__{2}^2 − _x__{1}^2)/2μ_s_.
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
Naumann[517], by Sandberger[518], and by Grabau[519], 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._
Ratio of breadth of
Breadth of whorls successive whorls The angle (α)
(180° apart) (360° apart) as calculated
0·30 mm. — — —
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._
Ratio of breadth of
Breadth of whorls successive whorls The spiral angle
(180° apart) (360° apart) (α) as calculated
0·25 mm. — — —
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.