ON LEAF-ARRANGEMENT, OR PHYLLOTAXIS
The beautiful configurations produced by the orderly arrangement of
leaves or florets on a stem have long been an object of admiration and
curiosity. Leonardo da Vinci would seem, as Sir Theodore Cook tells
us, to have been the first to record his thoughts upon this subject;
but the old Greek and Egyptian geometers are not likely to have
left unstudied or unobserved the spiral traces of the leaves upon a
palm-stem, or the spiral curves of the petals of a lotus or the florets
in a sunflower.
The spiral leaf-order has been regarded by many learned botanists
as involving a fundamental law of growth, of the deepest and most
far-reaching importance; while others, such as Sachs, have looked
upon the whole doctrine of “phyllotaxis” as “a sort of geometrical or
arithmetical playing with ideas,” and “the spiral theory as a mode
of view gratuitously introduced into the plant.” Sachs even goes so
far as to declare this doctrine “in direct opposition to scientific
investigation, and based upon the idealistic direction of the
Naturphilosophie,”—the mystical biology of Oken and his school.
The essential facts of the case are not difficult to understand;
but the theories built upon them are so varied, so conflicting, and
sometimes so obscure, that we must not attempt to submit them to
detailed analysis and criticism. There are two chief ways by which we
may approach the question, according to whether we regard, as the more
fundamental and typical, one or other of the two chief modes in which
the phenomenon presents itself. That is to say, we may hold that the
phenomenon is displayed in its essential simplicity by the corkscrew
spirals, or helices, which mark the position of the leaves upon a
cylindrical stem or on an {636} elongated fir-cone; or, on the other
hand, we may be more attracted by, and regard as of greater importance,
the logarithmic spirals which we trace in the curving rows of florets
in the discoidal inflorescence of a sunflower. Whether one way or the
other be the better, or even whether one be not positively correct and
the other radically wrong, has been vehemently debated. In my judgment
they are, both mathematically and biologically, to be regarded as
inseparable and correlative phenomena.
The helical arrangement (as in the fir-cone) was carefully studied in
the middle of the eighteenth century by the celebrated Bonnet, with
the help of Calandrini, the mathematician. Memoirs published about
1835, by Schimper and Braun, greatly amplified Bonnet’s investigations,
and introduced a nomenclature which still holds its own in botanical
textbooks. Naumann and the brothers Bravais are among those who
continued the investigation in the years immediately following, and
Hofmeister, in 1868, gave an admirable account and summary of the work
of these and many other writers[573].
Starting from some given level and proceeding upwards, let us mark
the position of some one leaf (_A_) upon a cylindrical stem. Another,
and a younger leaf (_B_) will be found standing at a certain distance
_around_ the stem, and a certain distance _along_ the stem, {637}
from the first. The former distance may be expressed as a fractional
“divergence” (such as two-fifths of the circumference of the stem)
as the botanists describe it, or by an “angle of azimuth” (such as
ϕ = 144°) as the mathematician would be more likely to state it. The
position of _B_ relatively to _A_ must be determined, not only by
this angle ϕ, in the horizontal plane, but also by an angle (θ) in
the vertical plane; for the height of _B_ above the level of _A_, in
comparison with the diameter of the cylinder, will obviously make a
great difference in the appearance of the whole system, in short the
position of each leaf must be expressed by _F_(ϕ ⋅ sin θ). But this
matter botanical students have not concerned themselves with; in other
words, their studies have been limited (or mainly limited) to the
relation of the leaves to one another in _azimuth_.
Whatever relation we have found between _A_ and _B_, let precisely the
same relation subsist between _B_ and _C_: and so on. Let the growth
of the system, that is to say, be continuous and uniform; it is then
evident that we have the elementary conditions for the development of a
simple cylindrical helix; and this “primary helix” or “genetic spiral”
we can now trace, winding round and round the stem, through _A_, _B_,
_C_, etc. But if we can trace such a helix through _A_, _B_, _C_, it
follows from the symmetry of the system, that we have only to join _A_
to some other leaf to trace another spiral helix, such, for instance,
as _A_, _C_, _E_, etc.; parallel to which will run another and similar
one, namely in this case _B_, _D_, _F_, etc. And these spirals will run
in the opposite direction to the spiral _ABC_.
In short, the existence of one helical arrangement of points implies
and involves the existence of another and then another helical pattern,
just as, in the pattern of a wall-paper, our eye travels from one
linear series to another.
A modification of the helical system will be introduced when, instead
of the leaves appearing, or standing, in singular succession, we get
two or more appearing simultaneously upon the same level. If there
be two such, then we shall have two generating spirals precisely
equivalent to one another; and we may call them _A_, _B_, _C_, etc.,
and _A′_, _B′_, _C′_, and so on. These are the cases which we call
“whorled” leaves, or in the simplest case, where {638} the whorl
consists of two opposite leaves only, we call them decussate.
――――――――――
Among the phenomena of phyllotaxis, two points in particular have been
found difficult of explanation, and have aroused discussion. These are
(1), the presence of the logarithmic spirals such as we have already
spoken of in the sunflower; and (2) the fact that, as regards the
number of the helical or spiral rows, certain numerical coincidences
are apt to recur again and again, to the exclusion of others, and so to
become characteristic features of the phenomenon.
The first of these appears to me to present no difficulty. It is a
mere matter of strictly mathematical “deformation.” The stem which
we have begun to speak of as a cylinder is not strictly so, inasmuch
as it tapers off towards its summit. The curve which winds evenly
around this stem is, accordingly, not a true helix, for that term is
confined to the curve which winds evenly around the _cylinder_: it
is a curve in space which (like the spiral curve we have studied in
our turbinate shells) partakes of the characters of a helix and of a
logarithmic spiral, and which is in fact a logarithmic spiral with
its pole drawn out of its original plane by a force acting in the
direction of the axis. If we imagine a tapering cylinder, or cone,
projected, by vertical projection, on a plane, it becomes a circular
disc; and a helix described about the cone necessarily becomes in the
disc a logarithmic spiral described about a focus which corresponds
to the apex of our cone. In like manner we may project an identical
spiral in space upon such surfaces as (for instance) a portion of a
sphere or of an ellipsoid; and in all these cases we preserve the
spiral configuration, which is the more clearly brought into view the
more we reduce the vertical component by which it was accompanied.
The converse is, of course, equally true, and equally obvious,
namely that any logarithmic spiral traced upon a circular disc or
spheroidal surface will be transformed into a corresponding spiral
helix when the plane or spheroidal disc is extended into an elongated
cone approximating to a cylinder. This mathematical conception is
translated, in botany, into actual fact. The fir-cone may be looked
upon as a cylindrical axis contracted at both ends, until {639} it
becomes approximately an ellipsoidal solid of revolution, generated
about the long axis of the ellipse; and the semi-ellipsoidal capitulum
of the teasel, the more or less hemispherical one of the thistle, and
the flattened but still convex one of the sunflower, are all beautiful
and successive deformations of what is typically a long, conical, and
all but cylindrical stem. On the other hand, every stem as it grows
out into its long cylindrical shape is but a deformation of the little
spheroidal or ellipsoidal surface, or cone, which was its forerunner in
the bud.
This identity of the helical spirals around the stem with spirals
projected on a plane was clearly recognised by Hofmeister, who was
accustomed to represent his diagrams of leaf-arrangement either in one
way or the other, though not in a strictly geometrical projection[574].
――――――――――
According to Mr A. H. Church[575], who has dealt very carefully and
elaborately with the whole question of phyllotaxis, the logarithmic
spirals such as we see in the disc of the sunflower have a far greater
importance and a far deeper meaning than this brief treatment of mine
would accord to them: and Sir Theodore Cook, in his book on the _Curves
of Life_, has adopted and has helped to expound and popularise Mr
Church’s investigations.
Mr Church, regarding the problem as one of “uniform growth,” easily
arrives at the conclusion that, _if_ this growth can be conceived as
taking place symmetrically about a central point or “pole,” the uniform
growth would then manifest itself in logarithmic spirals, including
of course the limiting cases of the circle and straight line. With
this statement I have little fault to find; it is in essence identical
with much that I have said in a previous chapter. But other statements
of Mr Church’s, and many theories woven about them by Sir T. Cook
and himself, I am less able to follow. Mr Church tells us that the
essential phenomenon in the sunflower disc is a series of orthogonally
intersecting logarithmic spirals. Unless I wholly misapprehend Mr
Church’s meaning, I should say that this is very far from essential.
The spirals {640} intersect isogonally, but orthogonal intersection
would be only one particular case, and in all probability a very
infrequent one, in the intersection of logarithmic spirals developed
about a common pole. Again on the analogy of the hydrodynamic lines
of force in certain vortex movements, and of similar lines of force
in certain magnetic phenomena, Mr Church proceeds to argue that the
energies of life follow lines comparable to those of electric energy,
and that the logarithmic spirals of the sunflower are, so to speak,
lines of equipotential[576]. And Sir T. Cook remarks that this “theory,
if correct, would be fundamental for all forms of growth, though it
would be more easily observed in plant construction than in animals.”
The parallel I am not able to follow.
Mr Church sees in phyllotaxis an organic mystery, a something for which
we are unable to suggest any precise cause: a phenomenon which is to
be referred, somehow, to waves of growth emanating from a centre, but
on the other hand not to be explained by the division of an apical
cell, or any other histological factor. As Sir T. Cook puts it, “at the
growing point of a plant where the new members are being formed, there
is simply _nothing to see_.”
But it is impossible to deal satisfactorily, in brief space, either
with Mr Church’s theories, or my own objections to them[577]. Let
it suffice to say that I, for my part, see no subtle mystery in the
matter, other than what lies in the steady production of similar
growing parts, similarly situated, at similar successive intervals
of time. If such be the case, then we are bound to have in {641}
consequence a series of symmetrical patterns, whose nature will
depend upon the form of the entire surface. If the surface be that of
a cylinder we shall have a system, or systems, of spiral helices: if
it be a plane, with an infinitely distant focus, such as we obtain by
“unwrapping” our cylindrical surface, we shall have straight lines;
if it be a plane containing the focus within itself, or if it be any
other symmetrical surface containing the focus, then we shall have
a system of logarithmic spirals. The appearance of these spirals is
sometimes spoken of as a “subjective” phenomenon, but the description
is inaccurate: it is a purely mathematical phenomenon, an inseparable
secondary result of other arrangements which we, for the time being,
regard as primary. When the bricklayer builds a factory chimney, he
lays his bricks in a certain steady, orderly way, with no thought of
the spiral patterns to which this orderly sequence inevitably leads,
and which spiral patterns are by no means “subjective.” The designer
of a wall-paper not only has no intention of producing a pattern of
criss-cross lines, but on the contrary he does his best to avoid
them; nevertheless, so long as his design is a symmetrical one, the
criss-cross intersections inevitably come.
Let us, however, leave this discussion, and return to the facts of the
case.
――――――――――
Our second question, which relates to the numerical coincidences so
familiar to all students of phyllotaxis, is not to be set and answered
in a word.
Let us, for simplicity’s sake, avoid consideration of simultaneous or
whorled leaf origins, and consider only the more frequent cases where a
single “genetic spiral” can be traced throughout the entire system.
It is seldom that this primary, genetic spiral catches the eye, for
the leaves which immediately succeed one another in this genetic
order are usually far apart on the circumference of the stem, and it
is only in close-packed arrangements that the eye readily apprehends
the continuous series. Accordingly in such a case as a fir-cone, for
instance, it is certain of the secondary spirals or “parastichies”
which catch the eye; and among fir-cones, we can easily count these,
and we find them to be {642} on the whole very constant in number,
according to the species.
Thus in many cones, such as those of the Norway spruce, we can trace
five rows of scales winding steeply up the cone in one direction,
and three rows winding less steeply the other way; in certain other
species, such as the common larch, the normal number is eight rows
in the one direction and five in the other; while in the American
larch we have again three in the one direction and five in the other.
It not seldom happens that two arrangements grade into one another
on different parts of one and the same cone. Among other cases in
which such spiral series are readily visible we have, for instance,
the crowded leaves of the stone-crops and mesembryanthemums, and (as
we have said) the crowded florets of the composites. Among these we
may find plenty of examples in which the numbers of the serial rows
are similar to those of the fir-cones; but in some cases, as in the
daisy and others of the smaller composites, we shall be able to trace
thirteen rows in one direction and twenty-one in the other, or perhaps
twenty-one and thirty-four; while in a great big sunflower we may find
(in one and the same species) thirty-four and fifty-five, fifty-five
and eighty-nine, or even as many as eighty-nine and one hundred and
forty-four. On the other hand, in an ordinary “pentamerous” flower,
such as a ranunculus, we may be able to trace, in the arrangement of
its sepals, petals and stamens, shorter spiral series, three in one
direction and two in the other. It will be at once observed that these
arrangements manifest themselves in connection with very different
things, in the orderly interspacing of single leaves and of entire
florets, and among all kinds of leaf-like structures, foliage-leaves,
bracts, cone-scales, and the various parts or members of the flower.
Again we must be careful to note that, while the above numerical
characters are by much the most common, so much so as to be deemed
“normal,” many other combinations are known to occur.
The arrangement, as we have seen, is apt to vary when the entire
structure varies greatly in size, as in the disc of the sunflower.
It is also subject to less regular variation within one and the same
species, as can always be discovered when we examine a sufficiently
large sample of fir-cones. For instance, out of 505 {643} cones of
the Norway spruce, Beal[578] found 92 per cent. in which the spirals
were in five and eight rows; in 6 per cent. the rows were four and
seven, and in 4 per cent. they were four and six. In each case they
were nearly equally divided as regards direction; for instance of the
467 cones shewing the five-eight arrangement, the five-series ran in
right-handed spirals in 224 cases, and in left-handed spirals in 243.
Omitting the “abnormal” cases, such as we have seen to occur in a small
percentage of our cones of the spruce, the arrangements which we have
just mentioned may be set forth as follows, (the fractional number
used being simply an abbreviated symbol for the number of associated
helices or parastichies which we can count running in the opposite
directions): 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144.
Now these numbers form a very interesting series, which happens to have
a number of curious mathematical properties[579]. We see, for instance,
that the denominator of each fraction is the numerator of the next; and
further, that each successive numerator, or denominator, is the sum
of the preceding two. Our immediate problem, then, is to determine,
if possible, how these numerical coincidences come about, and why
these particular numbers should be so commonly met with {644} as to
be considered “normal” and characteristic features of the general
phenomenon of phyllotaxis. The following account is based on a short
paper by Professor P. G. Tait[580].
[Illustration: Fig. 325.]
Of the two following diagrams, Fig. 325 represents the general case,
and Fig. 326 a particular one, for the sake of possibly greater
simplicity. Both diagrams represent a portion of a branch, or fir-cone,
regarded as cylindrical, and unwrapped to form a plane surface. _A_,
_a_, at the two ends of the base-line, represent the same initial leaf
or scale: _O_ is a leaf which can be reached from _A_ by _m_ steps in a
right-hand spiral (developed into the straight line _AO_), and by _n_
steps from _a_ in a left-handed spiral _aO_. Now it is obvious in our
fir-cone, that we can include _all_ the scales upon the cone by taking
so many spirals in the one direction, and again include them all by
so many in the other. Accordingly, in our diagrammatic construction,
the spirals _AO_ and _aO_ _must_, and always _can_, be so taken that
_m_ spirals parallel to _aO_, and _n_ spirals parallel to _AO_, shall
separately include all the leaves upon the stem or cone.
If _m_ and _n_ have a common factor, _l_, it can easily be shewn that
the arrangement is composite, and that there are _l_ fundamental, or
genetic spirals, and _l_ leaves (including _A_) which are situated
exactly on the line _Aa_. That is to say, we have here a _whorled_
arrangement, which we have agreed to leave unconsidered in favour of
the simpler case. We restrict ourselves, accordingly, to the cases
where there is but one genetic spiral, and when _therefore_ _m_ and _n_
are prime to one another.
Our fundamental, or genetic, spiral, as we have seen, is that which
passes from _A_ (or _a_) to the leaf which is situated nearest to
the base-line _Aa_. The fundamental spiral will thus be right-handed
(_A_, _P_, etc.) if _P_, which is nearer to _A_ than to _a_, be this
leaf—left-handed if it be _p_. That is to say, we make it a convention
that we shall always, for our fundamental spiral, run {645} round the
system, from one leaf to the next, _by the shortest way_.
[Illustration: Fig. 326.]
Now it is obvious, from the symmetry of the figure (as further shewn
in Fig. 326), that, besides the spirals running along _AO_ and _aO_,
we have a series running _from the steps on_ _aO_ to the steps on
_AO_. In other words we can find a leaf (_S_) upon _AO_, which, like
the leaf _O_, is reached directly by a spiral series from _A_ and from
_a_, such that _aS_ includes _n_ steps, and _AS_ (being part of the
old spiral line _AO_) now includes _m_ − _n_ steps. And, since _m_ and
_n_ are prime to one another (for otherwise the system would have been
a composite or whorled one), it is evident that we can continue this
process of convergence until we come down to a 1, 1 arrangement, that
is to say to a leaf which is reached by a single step, in opposite
directions from _A_ and from _a_, which leaf is therefore the first
leaf, next to _A_, of the fundamental or generating spiral. {646}
If our original lines along _AO_ and _aO_ contain, for instance, 13 and
8 steps respectively (i.e. _m_ = 13, _n_ = 8), then our next series,
observable in the same cone, will be 8 and (13 − 8) or 5; the next 5
and (8 − 5) or 3; the next 3, 2; and the next 2, 1; leading to the
ultimate condition of 1, 1. These are the very series which we have
found to be common, or normal; and so far as our investigation has yet
gone, it has proved to us that, if one of these exists, it entails,
_ipso facto_, the presence of the rest.
In following down our series, according to the above construction, we
have seen that at every step we have changed direction, the longer
and the shorter sides of our triangle changing places every time. Let
us stop for a moment, when we come to the 1, 2 series, or _AT_, _aT_
of Fig. 326. It is obvious that there is nothing to prevent us making
a new 1, 3 series if we please, by continuing the generating spiral
through three leaves, and connecting the leaf so reached directly with
our initial one. But in the case represented in Fig. 326, it is obvious
that these two series (_A_, 1, 2, 3, etc., and _a_, 3, 6, etc.) will be
running in the same direction; i.e. they will both be right-handed, or
both left-handed spirals. The simple meaning of this is that the third
leaf of the generating spiral was distant from our initial leaf by
_more than the circumference_ of the cylindrical stem; in other words,
that there were more than two, but _less than three_ leaves in a single
turn of the fundamental spiral.
Less than two there can obviously never be. When there are exactly
two, we have the simplest of all possible arrangements, namely that
in which the leaves are placed alternately on opposite sides of the
stem. When there are more than two, but less than three, we have the
elementary condition for the production of the series which we have
been considering, namely 1, 2; 2, 3; 3, 5, etc. To put the latter
part of this argument in more precise language, let us say that: If,
in our descending series, we come to steps 1 and _t_, where _t_ is
determined by the condition that 1 and _t_ + 1 would give spirals both
right-handed, or both left-handed; it follows that there are less
than _t_ + 1 leaves in a single turn of the fundamental spiral. And,
determined in this manner, it is found in the great majority of cases,
in fir-cones and a host of other examples of phyllotaxis, that _t_
= 2. In other words, in the {647} great majority of cases, we have
what corresponds to an arrangement next in order of simplicity to the
simplest case of all: next, that is to say, to the arrangement which
consists of opposite and alternate leaves.
“These simple considerations,” as Tait says, “explain completely the
so-called mysterious appearance of terms of the recurring series
1, 2, 3, 5, 8, 13, etc.[581] The other natural series, usually but
misleadingly represented by convergents to an infinitely extended
continuous fraction, are easily explained, as above, by taking _t_ = 3,
4, 5, etc., etc.” Many examples of these latter series have been given
by Dickson[582] and other writers.
――――――――――
We have now learned, among other elementary facts, that wherever any
one system of helical spirals is present, certain others invariably and
of necessity accompany it, and are definitely related to it. In any
diagram, such as Fig. 326, in which we represent our leaf-arrangement
by means of uniform and regularly interspaced dots, we can draw one
series of spirals after another, and one as easily as another. But
in our fir-cone, for instance, one particular series, or rather two
conjugate series, are always conspicuous, while the others are sought
and found with comparative difficulty.
[Illustration: Fig. 327.]
The phenomenon is illustrated by Fig. 327, _a_–_d_. The ground-plan of
all these diagrams is identically the same. The generating spiral in
each case represents a divergence of 3/8, or 135° of azimuth; and the
points succeed one another at the same successional distances parallel
to the axis. The rectangular outlines, which correspond to the exposed
surface of the leaves or cone-scales, are of equal area, and of equal
number. Nevertheless the appearances presented by these diagrams
are very different; for in one the eye catches a 5/8 arrangement,
in another a 3/5; and so on, down to an arrangement of 1/1. The
mathematical side of this very curious phenomenon I have not attempted
to investigate. But it is quite obvious that, in a system within {648}
which various spirals are implicitly contained, the conspicuousness
of one set or another does not depend upon angular divergence. It
depends on the relative proportions in length and breadth of the leaves
themselves; or, more strictly speaking, on the ratio of the diagonals
of the rhomboidal figure by which each leaf-area is circumscribed.
When, as in the fir-cone, the scales by mutual compression conform to
these rhomboidal outlines, their inclined edges at once guide the eye
in the direction of some one particular spiral; and we shall not fail
to notice that in such cases the usual {649} result is to give us
arrangements corresponding to the middle diagrams in Fig. 327, which
are the configurations in which the quadrilateral outlines approach
most nearly to a rectangular form, and give us accordingly the least
possible ratio (under the given conditions) of sectional boundary-wall
to surface area.
The manner in which one system of spirals may be caused to slide, so to
speak, into another, has been ingeniously demonstrated by Schwendener
on a mechanical model, consisting essentially of a framework which can
be opened or closed to correspond with one after another of the above
series of diagrams[583].
The determination of the precise angle of divergence of two consecutive
leaves of the generating spiral does not enter into the above general
investigation (though Tait gives, in the same paper, a method by which
it may be easily determined); and the very fact that it does not
so enter shews it to be essentially unimportant. The determination
of so-called “orthostichies,” or precisely vertical successions of
leaves, is also unimportant. We have no means, other than observation,
of determining that one leaf is vertically above another, and spiral
series such as we have been dealing with will appear, whether such
orthostichies exist, whether they be near or remote, or whether the
angle of divergence be such that no precise vertical superposition ever
occurs. And lastly, the fact that the successional numbers, expressed
as fractions, 1/2, 2/3, 3/5, represent a convergent series, whose final
term is equal to 0·61803..., the _sectio aurea_ or “golden mean” of
unity, is seen to be a mathematical coincidence, devoid of biological
significance; it is but a particular case of Lagrange’s theorem that
the roots of every numerical equation of the second degree can be
expressed by a periodic continued fraction. The same number has a
multitude of curious arithmetical properties. It is the final term of
all similar series to that with which we have been dealing, such for
instance as 1/3, 3/4, 4/7, etc., or 1/4, 4/5, 5/9, etc. It is a number
beloved of the circle-squarer, and of all those who seek to find, and
then to penetrate, the secrets of the Great Pyramid. It is deep-set in
Pythagorean as well as in Euclidean geometry. It enters (as the chord
of an angle of 36°), {650} into the thrice-isosceles triangle of
which we have spoken on p. 511; it is a number which becomes (by the
addition of unity) its own reciprocal; its properties never end. To
Kepler (as Naber tells us) it was a symbol of Creation, or Generation.
Its recent application to biology and art-criticism by Sir Theodore
Cook and others is not new. Naber’s book, already quoted, is full of
it. Zeising, in 1854, found in it the key to all morphology, and the
same writer, later on[584], declared it to dominate both architecture
and music. But indeed, to use Sir Thomas Browne’s words (though it
was of another number that he spoke): “To enlarge this contemplation
into all the mysteries and secrets accommodable unto this number, were
inexcusable Pythagorisme.”
If this number has any serious claim at all to enter into the
biological question of phyllotaxis, this must depend on the fact,
first emphasized by Chauncey Wright[585], that, if the successive
leaves of the fundamental spiral be placed at the particular azimuth
which divides the circle in this “sectio aurea,” then no two leaves
will ever be superposed; and thus we are said to have “the most
thorough and rapid distribution of the leaves round the stem, each
new or higher leaf falling over the angular space between the two
older ones which are nearest in direction, so as to divide it in
the same ratio (_K_), in which the first two or any two successive
ones divide the circumference. Now 5/8 and all successive fractions
differ inappreciably from _K_.” To this view there are many simple
objections. In the first place, even 5/8, or ·625, is but a moderately
close approximation to the “golden mean”; in the second place the
arrangements by which a better approximation is got, such as 8/13,
13/21, and the very close approximations such as 34/55, 55/89, 89/144,
etc., are comparatively rare, while the much less close approximations
of 3/5 or 2/3, or even 1/2, are extremely common. Again, the general
type of argument such as that which asserts that the plant is “aiming
at” something which we may call an “ideal angle” is one that cannot
commend itself to a plain student of physical science: nor is the
hypothesis rendered more acceptably when Sir T. Cook qualifies it by
telling us that “all that a plant can do {651} is to vary, to make
blind shots at constructions, or to ‘mutate’ as it is now termed;
and the most suitable of these constructions will in the long run be
isolated by the action of Natural Selection.” Finally, and this is the
most concrete objection of all, the supposed isolation of the leaves,
or their most complete “distribution to the action of the surrounding
atmosphere” is manifestly very little affected by any conditions which
are confined to the angle of azimuth. If we could imagine a case in
which all the leaves of the stem, or all the scales of a fir-cone, were
crushed down to one and the same level, into a simple ring or whorl of
leaves, then indeed they would have their most equable distribution
under the condition of the “ideal angle,” that is to say of the “golden
mean.” But if it be (so to speak) Nature’s object to set them further
apart than they actually are, to give them freer exposure to the air
than they actually have, then it is surely manifest that the simple way
to do so is to elongate the axis, and to set the leaves further apart,
lengthways on the stem. This has at once a far more potent effect than
any nice manipulation of the “angle of divergence.” For it is obvious
that in _F_(ϕ ⋅ sin θ) we have a greater range of variation by altering
θ than by altering ϕ. We come then, without more ado, to the conclusion
that the “Fibonacci series,” and its supposed usefulness, and the
hypothesis of its introduction into plant-structure through natural
selection, are all matters which deserve no place in the plain study of
botanical phenomena. As Sachs shrewdly recognised years ago, all such
speculations as these hark back to a school of mystical idealism.
{652}