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http://www.archive.org/details/cu31924000658470
On the Relation of Phyllotaxts
to Mechanical Laws
On the
Relation of Phyllotaxis
to Mechanical Laws
By
Arthur Harry Church, M.A., D.Sc.
Lecturer in Natural Science, Jesus College, Oxford
Publication assisted by a grant from the Royal Society, August 1904
London
Williams & Norgate
14 Henrietta Street, Covent Garden
1904
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C56
1'78604
Contents
PART I.
CONSTRUCTION BY ORTHOGONAL TRAJECTORIES.
PAGES
I. Introduction: Historical Sketch, Fibonacci, Bonnet,
The Spiral Theory of Schimper, Bravais, Sachs. 1-16
II. General Observations: 1. Orthostichies; 2. Parastichies,
Pinus Pinea; 3. Huphorbia Wulfenii; 4. Cynara
Scolymus ; 5. Helianthus annuus. : ‘ 17-29
III. Geometrical Representation of Growth: The First
Zone of Growth ; Vortex Representation ; Geometry
of Uniform Growth Expansion ; ' . 30-44
IV. Application of Spiral-Vortex Construction ; Possible
Arrangements; Concentration-Systems; Construction
of Log. Spiral Curves; Application to Helianthus
Capitulum ; Helices and Spirals of Archimedes 45-65
V. Ideai Angles: Suggestions of Wiesner : . 66-74
VI. Asymmetry. ‘ ¢ ; ; 75-78
PART II.
ASYMMETRICAL AND SYMMETRICAL PHYLLOTAXIS.
J. Normal Fibonacci Phyllotaxis: Conception of Bulk-
Ratio 5 ; ‘ F F ‘ 83-89
TI. Constant Phyllotaxis: Araucaria, Podocarpus . 90-108
v
vil
III.
VIII.
Il.
IIT.
IV.
CONTENTS.
Rising Phyllotaxis : Helianthus Capitulum ; Fibonacci
Expansion ; Helianthus Seedling; Cyperus ; Falling
Phyllotaxis; Cynara; Asymmetrical Floral Diagrams
. The Symmetrical Concentrated Type: Hquisetwm
. Asymmetrical Least Concentrated Type: Cyperus,
Gasterta
. Symmetrical Non-concentrated Type .
. Multijugate Type: Bravais; Dipsacus; Expansion
System ; Stlphium, Cephalaria
Anomalous Series: Sedum reflexcum,; Lycopodium
Selago; Dichotomy of Lycopodium; General
Conclusions
PART III.
SECONDARY GROWTH-PHENOMENA.
. Notation
Rhythm: Theory of Growth-Centre and Lateral
Centres; Periodicity ; The Log. Spiral Theory of
Equi-Growth-Potential. Conclusions from Parts I.
and II.
Contact-Pressures: Theories of Schwendener and
Weisse; Apex of Aspidium; Reciprocal Pressures
and Quasi-Squares ; Influence of a Rigid Boundary ;
Packing ; Cedrus Bud; Pinus Pinea Seedling ;
Cynara; Helianthus; Hexagonal Faceting ; Anthurium
Eccentric Growth: Eccentric Homologues of the Centric
Growth-Centre ; Anisophylly and Dorsiventrality ;
Orientation of Eccentric Shoot-Systems; Selaginella
and Salvinia ; Eccentric Flowers ; Tropaeolum
PAGES
109-141
142-153
£30
154-162
163-165
166-195
196-211
215-219
220-235
236-266
267-289
CONTENTS. vu
PAGES
V. Bilaterality of Appendages: Structure of a Foliage-
bud; Phenomena of Sliding-Growth ; Displacements
and Readjustments; Spiral of Dorsiventrality ;
Spiral of Phyllody; Representation of Extreme
Bilaterality ; Contact-Cycles F . 290-315
VI. Varying Growth in Lateral Members: Retention or
Obliteration of the Primary Pattern; Pinus;
Sempervivum ; Production of a Normal Foliage-leaf. 316-326
MATHEMATICAL NOTES ON LOG: SPIRAL SYSTEMS AND
THEIR APPLICATION TO PHYLLOTAXIS PHENOMENA.
I. General Equation to the Quasi-Circle inscribed in a
Log. Spiral Quasi-Square Mesh: Bilaterality ;
Dorsiventrality ; Isophylly . 3829-333
II. Mathematical Orthostichies in Log. Spiral Systems . 334-335
III. The Form of the ‘‘ Ovoid” Curve j ; . 835-337
IV. Bulk-Ratio . . : F F . 338-339
V. The Oscillation Angle . : : : 339-341
VI. The Fibonacci Series. 3 ‘ . 341-344
VII. Continued Fractions . : : F 344
VIII. Sliding-Growth : ‘ . 345-347
General Conclusions . : . 3848-349
Errata AND Notes ro Parr I. ‘ : : ‘ 213
Errata and Notes ro Part IJ.: Pine-Cones; Dichotomy
of Helianthus annuus ; : 351
Note on Phyllotaxis.
BY
ARTHUR H. CHURCH, M.A., D.Sc.,
Lecturer in Natural Science, Jesus College, Oxford.
+H
With two Figures in the Text.
—+—
RITERS on Phyllotaxis are generally agreed in
accepting the series of formulae known as the
Schimper-Braun series of divergences, 2, 2, , &c., as
fundamental expressions of the primary phenomena of the
arrangement of lateral members. This series of fractional
expressions, which involves the utilization of the Fibonacci
ratio series 2, 3, 5, 8, 13, &c., has thus proved for over sixty
years the ground-work of all theories of phyllotaxis, and is
usually described in the early pages of textbooks. Taking
the ‘2’ as a type of these values, this expression implies that
in placing five members on a spiral which makes two complete
revolutions of an axis, the sixth member is mathematically
superposed to the first, and that successive members differ by
a divergence-angle of 144°. So simple are these relations and
so thoroughly well known that it is not necessary to dwell
further on the vast superstructure of morphological theory
which has been built up on this foundation. However, as
a matter of fact, taking the 2? divergence again as an example,
it is beyond doubt that observation of the actual plant shows
that these relations do not strictly hold, and various theories
(Annals of Botany, Vol. XV. No. LIX. September, rgor.]
482 Church.— Note on Phyllotaxis.
have at different times been proposed to show why this should
be so; these again agree in taking the fractional expressions
as representative of some mathematical law, all deviations
from which must be due to the action of secondary forces,
real or hypothetical. Such speculations include the original
prosenthesis theory of Schimper and Braun, various torsion
and displacement theories, culminating in the contact-pressure
theory of Schwendener. These various views have been
recently critically examined by Winkler (Pringsh. Jahrb., 1901,
Heft J).
Since the general plan of these investigations consists, how-
ever, in superimposing some new hypothesis on the original
conception of Schimper and Braun, a strict analysis of the
subject demands a preliminary investigation of the views of
Schimper and Braun and the scientific evidence underlying
these fractional expressions, which become translated into
accurate divergence-angles of degrees, minutes, and seconds.
So long have these numbers been accepted that it appears
somewhat gratuitous to point out that these generalizations
rest on no scientific basis whatever, and that what passed for
evidence in 1830 does not necessarily hold at the present day.
Thus Schimper and Braun elaborated these expressions of
divergence on the plan of the original 2 or guzucuncial system
proposed by Bonnet in 1754. The starting-point in dealing
with phyllctaxis is therefore the clucidation of the exact point
of view of Bonnet, which has determined the path along
which all subsequent investigation has proceeded. Now
Bonnet, who had the assistance of the mathematician Calan-
drini, studied adult axes only, and devised, as an expression
of the facts observed on elongated leafy shoots, a helix winding
round a cylinder and spacing out at equal angles five members
in two complete revolutions, the sixth member faling on the
same vertical line as the first ; a simple mathematical concep-
tion was thus utilized to express the observed phenomena.
The fact which Bonnet thoroughly understood, that on a plant-
shoot the sixth leaf did ot fall exactly over the first, but that
the series formed by every fifth leaf itself wound along a spiral
Church.—Note on Phyllotaxts. 483
path, was explained by an assumption which has exerted
a powerful influence on subsequent speculations, that the
plant in fact purposely destroyed the postulated mathematical
construction, in order that the assimilating members might
be given free transpiration-space without any overlapping.
Generally speaking, but little real advance has been made in
the investigation of the primary causes of phyllotaxis beyond
these original views of Bonnet published nearly 150 years ago.
It will be noticed that the fractional expressions of Schimper
and Braun repeat the hypothesis of Bonnet in a more
elaborated form; the Fibonacci series of ratios is introduced
in full, but these are so associated as to still imply helices
wound on cylindrical axes. However, as pointed out by the
brothers Bravais, axes are commonly conical, dome-shaped,
or even nearly plane, and on such surfaces the helices would
be carried up as spirals of equal screw-thread, and thus
become curves which in the last plane case are spirals of
Archimedes. That is to say, by expressing the helix-
construction in the form of a floral-diagram, the position of
leaves being marked on concentric circles whose radii are
in arithmetical progression, the genetic spiral becomes a spiral
of Archimedes, and the orthostichies are true radii vectores of
the system. Such a geometrical construction is implied in
the Schimper-Braun terminology which postulates the exis-
tence of orthostichies as straight lines. At the same time, by
drawing curves through the same points in different sequence,
other spirals appear in the construction, and these, distinguished
as parastichies, are similarly by construction spirals of
Archimedes.
Such geometrical plans are given in textbooks, and are
used for instilling a primary conception of the arrangement
of lateral members; the fact that they do not always agree
with actual observations is glossed over by the assumption of
secondary disturbing agencies, as for example forsion.
On examination, these fundamental expressions are seen to
be based on :—
1. The assumption of a special divergence-angle.
484 Church.—Note on Phyllotaxis.
2. The existence of accurate orthostichies: these latter
following from the construction as- being radii vectores of
a spiral of Archimedes, the spiral again being derived from
Bonnet’s helix with parallel screw-thread.
Since helices and spirals of Archimedes are also commonly
the result of torsion-action, the way becomes paved for the
addition of theories of lateral displacement or torsion-effects,
which are expected to produce secondary alterations in the
original simple system of Schimper and Braun.
It becomes therefore necessary to test the basis of these
generalizations, and to examine the possibility of checking by
direct observation either the divergence-angle or the ortho-
stichies themselves ; and finally to compare the plane construc-
tions by spirals of Archimedes and see how far these really do
interpret the appearances seen in a transverse section of the
developing system in the plant.
Such investigation shows that the hypotheses have no true
basis, while the construction by spirals of Archimedes is
a conspicuous failure. Thus, the divergence-angle is hope-
lessly beyond the error of actual observation on the plant,
since the points from which the angles have to be taken must
be judged by the eye; when, therefore, the divergence-angles
are expected to be true to a matter of minutes and seconds in
fairly high divergences, this becomes a matter of impossibility ;
and the Bravais showed in 1835 that it was in fact impossible
to disprove the standpoint that there was only one angular
divergence in such cases of normal Fibonacci phyllotaxis,
namely Schimper’s ‘Ideal Angle’ of 137°, 30’, 277-936.
Similarly, it is equally impossible to judge straight lines by
the eye alone, and the existence of orthostichies in spiral
phyllotaxis as mathematically straight lines thus becomes
as hypothetical as the Schimper-Braun divergence-angles.
In neither of the two methods used for the practical deter-
mination of phyllotaxis-constants is there then any possibility
of accurate mathematical demonstration. Although the
tabulation of appearances as judged by the eye may be
taken as an approximately accurate version of the real
Church.—Note on Phyllotaxts. 485
phenomena, it is clearly impossible to found any modern
scientific generalizations on angles which cannot be measured,
and lines which cannot be proved to be straight: it thus
follows that all speculations based on the assumption of the
Schimper-Braun series must rest on a purely hypothetical
foundation which may at any time be overturned. Such
expressions, as Sachs constantly pointed out, attempt to
imitate the phenomena observed without giving any reason
for such geometrical construction.
Again, taking the mathematical interpretation of the
Schimper-Braun system, that the genetic spiral and the
parastichies are represented by spirals of Archimedes, while
the orthostichies are radii vectores, a simple geometrical con-
struction in terms of these spirals should bring out either the
truth or error of this hypothetical relationship of the lateral
members.
Thus, from the equation to the Archimedean spiral (r=a8),
it is easy to construct a pair of spirals whose variable a shall
have the ratio of the parastichies observed on any given speci-
-men. ‘Take for example the 7) system, the primary contact
parastichies of which are 8 and 13; Fig. 2 shows such a system
geometrically planned for a left-hand genetic spiral: the
members along the twenty-one orthostichy lines differ by
twenty-one, and fall on the mathematically straight radii
vectores of the system. The intersections of these parastichy
spirals mark the pozzts at which the lateral members are
inserted, and the views of Schimper and Braun included only
the consideration of such points. It is clear, however, that if
the spaces between the spiral planes are regarded as contain-
ing the members pressed into close lateral contact, as seen in
the transverse section of a foliage bud, the appearance of the
progressive dorsiventrality of such lateral members is very
fairly zmztated. The construction, in fact, becomes more and
more like the appearances seen in the plant as the periphery
of the system is reached, but the central part which includes
the actual seat of development is very inadequately repre-
sented: thus, the areas become so relatively elongated in the
486 Church.—Note on Phyllotaxts.
radial direction as they approach the centre that they cannot
possibly represent any formation of primordia at the stem-
apex, on which such members are well known to arise as fairly
isodiametric protuberances. At the same time, it will be
noticed that the Archimedean spirals by construction all fall
into the centre and stop there, so that no room is left in the
Fig. 2. Theory of Schimper and Braun. Construction for Phyllotaxis 4.
OA.=Orthostichy line=radius vector passing through 1, 22, 43, &c. Members
along the contact parastichies differ by 8 and 13 respectively. Genetic spiral
winds left. Divergence-angle = of 360° = 137° 8’ 34”.
system for any subsequent growth and the addition of new
members which naturally obtains in the plant.
Again, further consideration shows that all spirals, whatever
their primary nature may have been, must necessarily pass
Church.—Note on Phyllotaxts. 487
into Archimedean spirals, which differ by a constant along
each radius vector, if they represent the limiting planes of
members which grow to a constant bulk and then remain
stationary, in the manner that lateral members do on the
plant. The appearance of Archimedean spirals on adult
shoots is thus secondary, and is merely the expression of the
attainment of uniform volume by members in spiral series; it
has nothing to do with the facts of actual development, during
which lateral members arise as similar protuberances, which
may be indefinitely produced without the possibility of the
system being closed by a terminal member,
In other words, the genetic spiral must be regarded mathe-
matically as winding to infinity, and being engaged in the
production of szmilar members. That is to say, the possibility
is at once suggested that the genetic spiral can only be repre-
sented by a logarithmic or equiangular spiral which makes
equal angles with all radii vectores.
Not only is this a mathematical fact there is no gainsaying,
but the introduction of log. spirals into the subject of Phyllo-
taxis at once opens up wide fields for speculation, in that
these spirals are thoroughly familiar to the mathematician
and physicist ; representing the laws of mathematical asym-
metrical growth around a point, they constitute in Hydro-
dynamics the curves of spiral-vortex movement, while their
application to Magnetism was fully investigated by Clerk
Maxwell. The possibility that the contact parastichies may
be also not only log. spirals but log. spirals which intersect
orthogonally, and thus plot out a field of distribution of energy
along orthogonally intersecting paths of equal action, is so
clearly suggested that it may at once be taken as the ground-
work of a theory of phyllotaxis more in accordance with
modern lines of thought (cf. Tait, ‘Least and Varying
Action,’ article Mechanics, Enc. Brit., vol. 15, p. 723).
A geometrical construction in terms of such spirals in the
ratio (8 : 13) (Fig. 3) may be taken as a representative system
corresponding to the preceding phyllotaxis-plan of Fig. 2.
It is difficult to avoid the conclusion that the log. spiral
488 Church.—Note on Phyllotaxis.
construction gives the true key to the problem, and that the
whole subject thus becomes a question of the mechanical dis-
tribution of energy within the substance of the protoplasmic
mass of the plant-apex : that phyllotaxis phenomena are the
result of inherent properties of protoplasm, the energy of life
being in fact distributed according to the laws which govern
Fig. 3. Log. spiral theory: Construction for Phyllotaxis system (8+13) in
terms of distribution of energy. Contact Parastichies = orthogonally intersecting
log. spirals in ratio (8 : 13). The curve through 1, 22, 43, &c., is alsoa log. spiral.
Genetic spiral winds left. Divergence-angle=137° 30’ 38”. Ludk-ratio of axis
to primordium=O04., AB.=1: 5 within a small error, or=Sin 402 =.204 for
the true curve.
the distribution of energy in any other form: and that the
original orthogonal planes, the relics of which survive in the
contact parastichies of the system, represent the natural
consequence of a mechanical system of energy-distribution
directly comparable with that which produces the orthogonal
intersection of cell-walls at the moment of their first formation,
Church.—Note on Phyllotaxis. 489
which was deduced by Sachs from the analogy of the ortho-
gonally intersecting planes of thickening observed in cell-
walls and starch-grains.
The readiness with which the several problems of phyllo-
taxis may be solved from this standpoint, when once the key
to the whole subject is grasped, is very remarkable, and these
views have been elaborated to considerable length in a paper
which awaits publication. The results are so varied and
striking that it is difficult to give any summary of them in
a small space: based as they are on the relative value of
the spirals of Archimedes and logarithmic spirals as inter-
preting the true developmental spiral of the plant-apex, it is
evident that the discussion of such curves is beyond the
province of the non-mathematical botanist. The object of the
present note is therefore merely to point out that the subject
of phyllotaxis thus enters entirely new ground which promises
results more fundamental than any yet obtained in the domain
of plant morphology: for example, it follows in such con-
structions that an equation may be given for the plane section
of a lateral primordium which will serve as a true mathe-
matical definition of a leaf, differentiating it from a stem:
the true divergence-angles may be calculated, and a definite
primordium which
determines any given system; while the geometrical con-
structions, on the plan of Fig. 3, have the advantage that
they do agree with the appearances observed in the plant ;
they obey and amplify Hofmeister’s law, and from the stand-
point of energy-distribution afford the clue to the subsequent
building up of the elaborate ‘ expansion-systems’ of which the
capitulum of Helianthus may be taken as a type.
It is not proposed at present to go into further detail as to
these questions which are very fully discussed in the paper
already prepared for publication ; until logarithmic spirals are
more familiar to the botanist it will be sufficient to point out
that the true key to phyllotaxis is undoubtedly to be found in
the solution of the problems of symmetrical or asymmetrical
: . F axis
numerical value can be given to the ratio
490 Church.—Note on Phyllotaxis.
distribution of energy in orthogonally intersecting planes
around an initial ‘growth-centre’; in the latter case the
whole of the spiral paths are log. spirals. The perfection of
such a construction involves uniform growth in the system ;
and owing to the obvious impairment of this uniform rate of
growth behind the plane portion of the apex, the true log-
spirals are possibly never to be observed on the plant, although
the approximation has been found in, certain cases to be
extremely close. Ultimately all these curves pass into spirals
of Archimedes as the members cease growth on the attain-
ment of constant volume, and these latter curves therefore
occur on adult axes and appeal to the eye in the macroscopic
view of the entire shoot. They were thus correctly isolated
by Bonnet, to whom the detailed construction of the growing
point was naturally unknown in 1754. The curves seen in
transverse section of an apical system of developing members
are thus probably curves transitional between log. spirals and
spirals of Archimedes.
On the other hand it will be noted that the new con-
structions are equally incapable of absolute verification by
any angular measurements on the plant; Schimper’s ortho-
stichies have vanished, as pointed out by the Bravais, for
the more general examples of phyllotaxis, and the differ-
ence between the two spiral systems is very slight to the eye:
but, while the Schimper-Braun School only sought to imitate
the appearances seen on the plant, the log. spiral theory gives
at least an equally correct summary of the facts observed, and
is in addition founded on definite mechanical laws of con-
struction by orthogonal trajectories which have already been
accepted for plant anatomy; it is so far then the logical
outcome of Sachs’ theory of the orthogonal intersection of
cell-walls, and represents therefore another special case of the
distribution of energy along planes of equal action}.
BOTANIC GARDENS, OXFORD.
May, 1901.
* Cf. Church, On the Relation of Phyllotaxis to Mechanical Laws. Part I,
Construction by Orthogonal Trajectories. Igor.
The Principles of Phyllotaxis.
BY
ARTHUR H. CHURCH, M.A., D.Sc.
Lecturer in Natural Science, Jesus College, Oxford.
With seven Figures in the Text.
N a preliminary note published some time ago}, exception was taken
to the conventional methods adopted for the description and even
interpretation of phyllotaxis phenomena, and a suggestion was made
that appeared to be not only more in accord with modern conceptions
of the phenomena of energy distribution, but it was further indicated that
such a theory when carried to its mathematical limits threw a strong light
both on the mechanism of shoot production and the inherent mathematical
properties of the lateral appendage usually described as a ‘leaf-member,’
as opposed to any secondary and subsidiary biological adaptations.
As publication of the entire paper has been delayed, and the new
standpoint has not received any special support from botanists to whom
the mathematical setting proved possibly a deterrent, the object of the
present note is to place the entire argument of the original paper in as
concise a form as possible. The preliminary discussion is sufficiently
familiar *.
The conventional account of phyllotaxis phenomena involves a system
of ‘fractional expressions’ which become interpreted into angular diver-
gences; and in practice the appearance of ‘ orthostichies’ has been taken
as a guide to the determination of the proper ‘fractional expression.’
This method, elaborated by Schimper (1830-5), has more or less held
the field to the present time; and, for want of something better, has
received the assent, though often unwilling, of such great investigators
as Hofmeister and Sachs, to say nothing of lesser lights. Although
elaborated into a system by Schimper and Braun, who added the peculiar
mathematical properties of the Fibonacci series to the academical account
1 Note on Phyllotaxis, Annals of Botany, xv, p. 481, 1901.
2 On the Relation of Phyllotaxis to Mechanical Laws. Part I, Construction by Orthogonal
Trajectories, 1901. Part II, Asymmetry and Symmetry, 1902.
3 Descriptive Morphology-Phyllotaxis. New Phytologist, i, p. 49.
[Annals of Botany, Vol. XVII. No. LXX. April, 1904.)
228 Church.—The Principles of Phyllotaxts.
of the subject, the geometry of the system is based solely on a mathematical
conception put forward by Bonnet and Calandrini in 1754; and this
mathematical conception applied only to adult shoots and adult members
of equal volume arranged in spiral sequence, and thus involved a system
of intersecting helices of equal screw-thread, or, reduced to a plane
expression, of spirals of Archimedes, also with equal screw-thread. A
system of helical mathematics was thus interpolated into botanical science,
and these helical systems were correctly tabulated by ‘ orthostichies’ and
‘ divergence angles’ obtained from simple fractional expressions themselves
deduced from the observation of orthostichies.
But in transferring the study of phyllotaxis to the ontogenetic sequence
of successively younger, and therefore gradated, primordia at the apex of
a growing plant-shoot which was not cylindrical, these mathematical
expressions were retained, although the helices originally postulated have
absolutely vanished ; and it is somewhat to the discredit of botanical science
that this simple error should have remained so long undetected and
unexpressed. As soon as one has to deal with spirals which have not an
equal screw-thread, the postulated orthostichies vanish as straight lines;
the fractional expressions therefore no longer present an accurate statement
of the facts; and the divergence angles, calculated to minutes and seconds,
are hopelessly out of the question altogether; while any contribution
to the study of phyllotaxis phenomena which continues the use of such
expressions must only serve to obscure rather than elucidate the inter- _
pretation of the phenomena observed. That the required orthostichies
were really non-existent at the growing point, a feature well known to
Bonnet himself, has thus formed the starting-point for new theories of
displacement of hypothetically perfect helical systems, as, for example,
in the contact-pressure theory of Schwendener. But once it is grasped
that the practice of applying helical mathematics to spiral curves which,
whatever they are, cannot be helices, is entirely beside the mark, it is
clear that the sooner all these views and expressions are eliminated the
better, and the subject requires to be approached without prejudice from
an entirely new standpoint.
The first thing to settle therefore is what this new standpoint is to be;
and how can such a remarkable series of phenomena be approached on
any general physical or mathematical principles?
Now in a transverse section of a leaf-producing shoot, at the level
of the growing point, the lateral appendages termed /eaves are observed
to arrange themselves in a gradated sequence as the expression of a
rhythmic production of similar protuberances, which takes the form of
a pattern in which the main construction lines appear as a grouping ;
of intersecting curves winding to the centre of the field, which is occupied
by the growing point of the shoot itself. As the mathematical properties
Churth.—The Principles of Phyllotaxis. 229
of such intersecting curve systems are not specially studied in an ordinary
school curriculum, a preliminary sketch of some of their interesting features
may be excused, since geometrical. relationships have clearly no inherent
connexion with the protoplasmic growth of the plant-shoot, but are merely
properties of lines and numbers.
Thus, by taking first, for example, a system in which spiral curves
of any nature radiate from a central point in such a manner that 5 are
1
‘
i
i
1
1
'
ee ee |
Fic. 35. Curve-system (5+8): Fibonacci series. A full contact-cycle of eight members is
represented by circular primordia.
‘turning in one direction and 8 in the other, giving points of intersection
in a uniform sequence, a system of meshes and points of intersection is
obtained, and to ‘either of these units a numerical value may be attached.
‘That is to say, if any member along the ‘5’ curves be called 1, the next
inmost member along the same series will be 6, since the whole system
is made of 5 rows, and this series will be numbered by differences of 5.
230 Church.—The Principles of Phyllotaxis.
In the same way differences of 8 along the ‘8’ curves will give a numerical
value to these members ; and by starting from 1, all the meshes, or points,
if these are taken, may be numbered up as has been done in the figure
(Fig. 35, (5 +8)
Observation of the figure now shows what is really a very remarkable
property: all the numerals have been used, and 1, 2, 3, 4, &c., taken in
order, give also a spiral sequence winding to the centre. This is merely
Fic. 36. Curve-system (6+ 8): Bijugate type. Contact-cycle as in previous figure.
a mathematical property of the system (5+ 8), in that these numbers are
only divisible by unity as a common factor; but the single spiral thus
obtained becomes in a botanical system the genetic-spiral which has been
persistently regarded as the controlling factor in the whole system, since
if such a construction be elongated sufficiently far, as on a plant-shoot,
this spiral will alone be left visible.
The first point to be ascertained in phyllotaxis is the decision as to
Church.—The Principles of Phyllotaxis. 2o4
which is to be the prime determining factor; that is to say, does the
possession by the. plant of a ‘genetic-spiral’ work out the subsidiary
pattern of the parastichies, or are the parastichies the primary feature, and
the genetic-spiral a secondary and unimportant consequence of the
construction ?
Now, other systems may quite as easily be drawn; thus take next
a system of 6 curves crossing 8. On numbering these up by differences
of 6 and 8 respectively in either series, it will be found that this time
all the numerals are zot employed, but that there are two sets of 1, 3, 5,
&c., and 1’, 3’, 5’, &c., showing that pairs of members on exactly opposite
sides of the system are of equal value. There is thus no single genetic
spiral now present, but two equal and opposite systems—a fact which
follows mathematically from the presence of a common factor (2) to the
numbers 6 and 8. The existence of such factorial systems in plants has
created much confusion, and the term diugate applied to such a construction
by the brothers Bravais may be legitimately retained as its designation
(Fig. 36, system (6 + 8)).
Again, on constructing a system of 7 curves crossing 8, and numbering
by respective differences, this time of 7 and 8; as in the first case, since
these numbers have 1 only as common factor, all the numerals are
utilized in numbering the system; the genetic-spiral may be traced even
more readily than in the first example, the adjacent members along it
being now in lateral contact, so that the resulting spiral obviously winds
round the apex. This effect is common among Cacti, and is the result
of a general property of these curve systems which may be summed up
as follows :—Given a set of intersecting curves, the same points of inter-
section (with others) will also be plotted by another system of curves
representing the diagonals of the first meshes, and the number of these
curves, and also of course the difference in numerical value of the units
along their path, will be given by the sum and difference of the numbers
which determine the system, for example, 5 and 8 have as complementary
system 3 and 13; and also other systems may be deduced by following the
addition and subtraction series, e. g. :—
5— 8
3—13
Z—21
I— 34.
Whereas the (7+8) system gives only 1 and 15; the single so-called
‘genetic-spiral, which includes all the points, being reached at the first
process. Thus a Cactus built on these principles would show an obvious
‘genetic-spiral’ winding on the apex and 15 ridges, which in the adult
state become vertical as a true helical construction is secondarily produced
as the internodes attain a uniform bulk (Fig. 37 (7 + 8)).
R
232 Church —The Principles of Phyllotaxis.
Finally, take the case of 8 curves crossing 8, and number in the
same way by differences of 8 along both series. It immediately
becomes clear that there are 8 similar series: all other spirals have
been eliminated; there is no ‘genetic-spiral’ at all, but only a system
of alternating circles of members of absolutely identical value in each
circle. We have now, that is to say, systems of true whorls, and also ~
learn in what a true whorl consists—the members must be exactly and
ie
"Ytn-~ none ee
Loon funomnn.
Fic. 37. Curve-system (7 +8): anomalous type.
mathematically equal in origin—while the expression a successive whorl -
is a contradiction in terms.
From such simple and purely geometrical considerations it thus
follows that the so-called ‘genetic-spiral’ is a property solely of inter-
secting curve-systems which only possess I as a common factor, and
is therefore only existent in one case out of three possible mathematical}
forms (Figs. 35, 36, 38). While if these four systems were subjected to
Church.—The Principles of Phyllotaxts. 233
a secondary Zone of Elongation, No. 1 would pull out as a complex
of spirals in which four distinct sets might be traced; No. 2 as two
spiral series leaving paired and opposite members at each ‘node’;
No. 3 as a spiral series with two complementary sets only; while
No. 4 would give the familiar case of alternating whorls with 8
members at each ‘node.’ Further these cases are not merely arbitrary:
they may all occur in the plant-kingdom, though the first is admittedly
Fic. 38. Curve-system (8 x 8): symmetrical type.
the most frequent; but any theory which interprets one should equally
well interpret the others. Similarly all changes of system may be discussed
with equal readiness from the standpoint of the addition or loss of certain
curves, and only from such a standpoint; since it is evident that once
it is granted that new curves may be added to or lost from the system,
the numerical relations of the members may be completely altered by
R2
Church.—The Principles of Phyllotaxis.
234
ae
Fic. 39. System (5 +8): eccentric construction in the plane of No. 2.
Church—The Principles of Phyllotaxis. 235
the addition of one curve only, as in the difference between the systems
(7+8), (8+ 8), &c, (Figs. 35-38)%. q
Thus the hypothesis of a genetic-spiral, since it entirely fails to
account for the arrangement of the members of all phyllotaxis systems
in a single spiral, may be conveniently wholly eliminated from future
discussions of these systems. It remains as a mere geometrical accident
of certain intersecting curve-systems, and the fact that such systems
may be very common in plant construction does, not affect the main
principle at all.
On the other hand, it may be urged that in these special cases one
cannot get away from the fact that it does actually represent the building-
path as seen in the visible ontogeny of the component members, and must
therefore ever remain the most important feature of these systems as
checked by actual observation apart from theoretical considerations. But
even this view is not absolute; and such a case in which the ontogenetic
sequence of development is not the single spiral obtained by numbering
the members in theoretical series would naturally confuse the observer
of direct ontogeny.
For example, in the previous cases figured the proposition of centric
- growth systems was alone considered, as being the simplest to begin
with; it is obvious that even a small amount of structural eccentricity
will produce a very different result. Thus in Fig. 39 the (5+8) system
is redrawn in an eccentric condition, the so-called ‘dorsiventrality’ of
the morphologist; on numbering the members in the same manner as
before it is clear that the series obtained is very different from any
empirical ontogenetic value which would be founded on the observation
of the relative bulk of the members at any given moment. The occurrence
of such systems in plant-shoots—and it may be stated that this figure was
originally devised to illustrate certain phenomena of floral construction
in the case of 7ropacolum—gives in fact the final proof, if such were any
longer needed, of the simple geometrical generalization that such systems
of intersecting curves are always readily interpreted in terms of the
number of curves radiating in either direction, and not in any other
manner. The presence of a circular zone (whorl) or a genetic-spiral is
a wholly secondary geometrical consequence of the properties of the
numerals concerned in constructing the system. The preference of any
individual botanist, either in the past or at present, for any particular method
‘Cf. Relation of Phyllotaxis to Mechanical Laws. Part II, p. 109, Rising and Falling
Phyllotaxis. Part IV, Cactaceae.
. Though the figures (35-38) have, as a matter of fact, been drawn by means of suitable ortho-
gonally intersecting logarithmic spirals, because these curves are easily obtained and the schemes are
subsequently held to be the representation of the true construction system of the plant-apex, the
nature of the spirals does not affect the general laws of intersection so long as this takes place
uniformly.
236 Church —The Principles of Phyllotaxis.
of interpreting any of these systems has little bearing on the case: the
subject is purely a mathematical one; and the only view which can be
acceptable is that which applies equally well to all cases, in that the
question is solely one of the geometrical properties of lines and numbers, ©
and must therefore be settled without reference to the occurrence of
such constructions in the plant.
If all phyllotaxis systems are thus to be regarded solely as cases of
intersecting curves, which are selected in varying numbers in the shoots
of different plants, and often in different shoots of the same plant, with
a tendency to a specific constancy which is one of the marvellous features
of the plant-kingdom, it remains now to discuss the possibility of attaching
a more direct significance to these curves, which in phyllotaxis construction
follow the lines of what have been termed the contact-parastichies; that
is to say, to consider
I. What is the mathematical nature of the spirals thus traced ?
II. What is the nature of the intersection? and
III. Is it possible to find any analogous construction in the domain
of purely physical science?
The suggestion of the logarithmic spiral theory is so obvious that
it would occur naturally to any physicist: the spirals are primarily of.
the nature of logarithmic spirals; the intersections are orthogonal; and
the construction is directly analogous to the representation of lines
of equipotential in a simple plane case of electrical conduction. In
opposition to this most fruitful suggestion, it must be pointed out however _
that the curves traced on a section are obviously never logarithmic spirals,
and the intersections cannot be measured as orthogonal. But then it
is again possible that in the very elaborate growth-phenomena of a plant-
shoot secondary factors come into play which tend to obliterate the —
primary construction; in fact, in dealing with the great variety of ©
secondary factors, which it only becomes possible to isolate when: the
primary construction is known, the marvel is rather that certain plants —
should yield such wonderfully approximately accurate systems. To begin.
with, logarithmic spiral constructions are zujfinite, the curves pass out to
infinity, and would wind an infinite number of times before reaching the
pole. Plant constructions on the other hand are finite, the shoot attains
a certain size only, and the pole is relatively large. The fact that similar
difficulties lie in the application of strict mathematical construction to
a vortex in water, for example, which must always possess an axial tube
of flow for a by no means perfect fluid, or to the distribution of potential
around a wire of appreciable size, does not affect the essential value of
the mathematical conception to physicists. And, though the growth of
the plant is finite, and therefore necessarily subject to retarding influences
of some kind, there is no reason why a region may not be postulated,
Church—The Principles of Phyllotaxis. 237
however small, at which such a mathematical distribution of ‘growth-
potential’ may be considered as accurate; and such a region is here
termed a ‘Growth-Centre’ Since the interpretation of all complex phe-
nomena must be first attacked from the standpoint of simple postulates,
it now remains to consider the construction and properties of as simple
a centre of growth as possible.
Thus in the simplest terms the growth may be taken as uniform
fy)
Fic. 40. Scheme for Uniform Growth Expansion: a circular meshwork of quasi-squares.
Symmetrical construction from which asymmetrical homologues are obtained by the use of logarithmic
spirals.
and centric: the fact that all plant growth is subject to a retardation
effect or may be frequently eccentric, may at present be placed wholly
on one side, since the simplest cases evidently underlie these. The case
of uniform centric growth is that of a uniformly expanding sphere; or,
238 Church.—The Principles of Phyllotaxis.
since it is more convenient to trace a solid in separate planes, it will be
illustrated by a diagram in which a system of concentric circles encloses
a series of similar figures, which represent a uniform growth increment
in equal intervals of time. Such a circular figure, in which the expanding
system is. subdivided into an indefinite number of small squares repre-
senting equal time-units, is shown in Fig. 40, and presents the general
theory of mathematical growth, in that in equal times the area represented
by one ‘square’ grows to the size of the one immediately external
to it?.
Now it is clear that while these small areas would approach true
squares if taken (sufficiently small, at present they are in part bounded °
by circular lines whe intersect the radii orthogonally; they may there-
fore be termed gwasi-squares: and while a true square would contain
a true inscribed circle, the homologous curve similarly inscribed in a quasi-
square will be a guasi-circle.
It is to this quasi-circle that future interest attaches; because, just
as the section of the whole shoot was conceived as containing a centric
growth-centre, so the lateral, i.e. secondary, appendages of such a shoot
may be also conceived as being initiated from a point and presenting a
centric growth of their own. These lateral growth-centres, however, are
component parts of a system which is growing as a whole. The con-
ception thus holds that the plane representation of the primary centric
shoot-centre is a civcular system enclosing quasi-circles as the representatives
of the initiated appendages.
To this may now be added certain mathematical and botanical facts
which are definitely established.
I. Any such growth-construction involving sémilar figures (and quasi-
circles would be similar) implies a construction by logarithmic spirals.
II. A growth-construction by intersecting logarithmic spirals, and
only by curves drawn in the manner utilized in constructing these diagrams
(Figs. 35-38), is the only possible mathematical case of continued orthogonal |
intersection ®. ;
III. The primordia of the lateral appendages of a plant only make
contact with adjacent ones in a definite manner, which is so clearly that
of the contacts exhibited by quasi-circles in a quasi-square meshwork,
that Schwendener assumed both a circular form and the orthogonal
arrangement as the basis of his Dachstuhl Theory: these two points being
here just the factors for which a rigid proof is required, since given these
the logarithmic spiral theory necessarily follows.
A construction in terms of quasi-circles would thus satisfy all theo-
1 The same figure may also be used to illustrate a simple geometrical method of drawing any
required pair of orthogonally intersecting logarithmic spirals.
2 For the formal proof of this statement I am indebted to Mr. H. Hilton.
Church.—The Principles of Phyllotaxis. 239
retical generalizations of the mathematical conception of uniform growth,
and would be at the same time in closest agreement with the facts of
observation ; while no other mathematical scheme could be drawn which
would include primordia arranged in such contact relations and at the
same time give an orthogonal construction. If, that is to say, the guast-
circle can be established as the mathematical representative of the
ptimordium of a lateral appendage, the orthogonal construction, which
is the one point most desired to be proved, will necessarily follow.
Fic. 41. Quasi-circles of the systems (a+ 2), (I+1) and (1 +2) arranged for illustration in the
plane of median symmetry. C’, C’”’, C’’’, the centres of construction of the respective curves.
(After E. H. Hayes.)
It remains therefore now to discuss the nature of the curves denoted
by the term guasz-circles; their equations may be deduced mathematically,
and the curves plotted on paper from the equations. These determinations
have been made by Mr. E. H. Hayes. Thus a general equation for the
quasi-circular curve inscribed in a mesh made by the orthogonal inter-
240 Churth.—The Principles of Phyllotaxis.
section of # spirals crossing ~, in the manner required, is given in such’
a form as,
lo r= lo Cc + I: 64 8 —_———— — .0000 0864 62,
soa S — 3 3 y) 5 ne 3
where the logarithm is the tabular logarithm, and 0 is measured in
degrees ; or where the logarithm is the natural logarithm and @ in circular
measure:
rs? a
(tog) ee ne +nt
From these equations the curve required for any phyllotaxis system
can be plotted out; and a series of three such curves is shown in Fig. 41,
grouped together for convenience of illustration, i.e. those for the lowest
systems (2+ 2), (+2) and (1+1).
It will be noticed immediately that the peculiar characters of these
curves are exaggerated as the containing spiral curves become fewer:
thus with a larger number than 3 and 5, the difference between the shape
of the curve and that of a circle would not be noticeable to the eye.
While in the kidney-shaped (1+1) curve the quasi-circle would no longer
be recognized as at all comparable in its geometrical properties with
a true centric growth-centre. But even these curves, remarkable as they
are, are zot the shape of the primordia as they first become visible at the
apex of a shoot constructing appendages in any one of these systems.
The shape of the first formed leaves of a decussate system, for example,
is never precisely that of the (2+2) curve (Fig. 41), but it is evidently of
the same general type; and it may at once be said that curves as near
as possible to those drawn from the plant may be obtained from these
quasi-circles of uniform growth by taking into consideration the necessity
of allowing for a growth-retardation.. Growth in fact has ceased to be
uniform even when the first sign of a lateral appendage becomes visible
at a growing point ; but, as already stated, this does not affect the correct-
ness of the theory in taking this mathematical construction for the
starting-point ; and, as has been insisted upon, the conception of the actual
existence of a state of uniform growth only applies to the hypothetical
‘ growth-centre.’
On the other hand, the mere resemblance of curves copied from the
plant to others plotted geometrically according to a definite plan which
is however modified to fit the facts of observation, will afford no strict
proof of the validity of the hypothesis, although it may add to its general
probability, since there is obviously no criterion possible as to the actual
nature of the growth-retardation ; that is to say, whether it may be taken
as uniform, or whether, as may be argued from analogy, it may exhibit daily
or even hourly variations. Something more than this is necessary before
the correctness of the assumption of quasi-circular leaf-homologues can
Church.—The Principles of Phyllotaxis. 241
be taken as established; and attention may now be drawn to another
feature of the mathematical proposition.
It follows from the form of the equation ascribed to the quasi-circle
that whatever value be given to m and 2, the curve itself is dzlaterally
symmetrical about a radius of the whole system drawn through its centre
of construction. That it should be so when m=z, i.e. in a symmetrical
(whorled) \eaf-arrangement, would excite no surprise ; but that the primor-
dium should be bilaterally symmetrical about a radius drawn through
its centre of construction, even when the system is wholly asymmetrical
and spiral, is little short of marvellous, since it implies that identity of
leaf-structure in both spiral and whorled systems, which is not only their
distinguishing feature, but one so usually taken for granted that it is
not considered to present any difficulty whatever. Thus, in any system of
spiral phyllotaxis, the orientation of the rhomboidal leaf-base is obviously
obligue, and as the members come into lateral contact they necessarily
‘become not only oblique but asymmetrical, since they must under mutual
pressure take the form of the full space available to each primordium,
the quasi-square area which appears in a spiral system as an oblique
unequal-sided rhomb (Fig. 35). Now the base of a leaf (in a spiral system)
is always such an oblique, azisophyllous structure, although the free appen-
dage is zsophyllous, bilaterally symmetrical, and flattened in a horizontal
plane?. The quasi-circle hypothesis thus not only explains the inherent
bilaterality of a lateral appendage, but also that peculiar additional attri-
bute which was called by Sachs its ‘dorsiventrality; or the possession
of different upper and lower sides, and what is more remarkable, since
it cannot be accounted for by any other mathematical construction, the
isophylly of the leaves produced in a spiral phyllotaxis system *.
It has been the custom so frequently to assume that a leaf-primordium
takes on these fundamental characters as a consequence of biological
adaptation to the action of such external agencies as light and gravity, that
it is even now not immaterial to point out that adaptation is not creation,
and that these fundamental features of leaf-structure must be present in the
-original primordium, however much or little the action of environment may
+ These relations are beautifully exhibited in the massive insertions of the huge succulent leaves
of large forms of Agave: the modelling of the oblique leaf-bases with tendency to rhomboid section,
as opposed to that of the horizontal symmetrical portion of the upper free region of the appendage,
“may be followed by the hand, yet only differs in bulk from the case of the leaves of Sempervivum or
the still smaller case of the bud of Przus,
2, Anisophylly is equally 2 mathematical necessity of all eccenéric shoot systems.
It will also ‘be noted that the adjustment required in the growing bud, as the free portions of
such spirally placed primordia tend to orientate their bilaterally symmetrical lamina in a radial and
not spiral plane, gives the clue to those peculiar movements in the case of spiral growth systems, which,
in that they could be with difficulty accounted for, although as facts of observation perfectly obvious,
has resulted in the partial acceptance of Schwendener’s Dachstuhl Theory. This theory was in fact
mainly based on the necessity for explaining this ‘slipping’ of the members, but in the logarithmic
spiral theory it follows as a mathematical property of the construction,
242 Churth.—The Principles of Phyllotaxis.
result in their becoming obvious to the eye. The fact that the quasi-circle
hypothesis satisfies all the demands of centric growth systems, whether
symmetrical or asymmetrical, as exhibited in the fundamental character
of foliar appendages, and that these characters may be deduced as the
mathematical consequences of the simple and straightforward hypothesis
of placing centres of lateral growth in a centric system which is also grow-
ing, may be taken as a satisfactory proof of the correctness of the original
standpoint. And it is difficult to see what further proof of the relation.
between a leaf-primordium as it is first initiated, and the geometrical .
properties of a quasi-circle growth system is required ; but it still remains °
to connect this conception with that of orthogonal construction.
This however naturally follows when it is borne in mind, firstly that no
other asymmetrical mathematical growth-construction is possible, except
the special quasi-square system which will include such quasi-circles ; and
secondly, that the contact-relations of the quasi-circles: in these figures are
identical with those presented by the primordia in the plant, and could only .
be so in orthogonal constructions. It thus follows that with the proving of
the quasi-circle hypothesis, the proof is further obtained that the intersection
of the spiral paths must be mutually orthogonal; and it becomes finally
established that in the construction-of a centric phyllotaxis system, along
logarithmic spiral lines, the segmentation of the growth system at the
hypothetical growth-centre does follow the course of paths intersecting |.
at right angles; and the principle of construction by orthogonal trajectories,
originally suggested by Sachs for the lines of cell-structure and details
of thickened walls, but never more fully proved, is now definitely estab-
lished for another special ‘case of plant-segmentation, which involves the
production of lateral appendages without any reference to the segmentation
of the body into ‘ cell’ units. wd
But even this is not all; the point still remains,—What does such
construction imply in physical terms? Nor can it be maintained that the
present position of physical science affords any special clue to. the still
deeper meaning of the phenomena. The fact that the symmetrical con-
struction in terms of logarithmic spirals agrees with the diagram for dis-
tribution of lines of equipotential and paths of current flow in a special case
of electric conduction, while the asymmetrical systems are similarly homo-
logous with lines of equal pressure and paths of flow ina vortex in a perfect
fluid, the former a static proposition, the latter a kinetic one, may be only
_an ‘accident.’ On the other hand it must always strike an unprejudiced
-observer that there may be underlying all these cases the working of some
still more fundamental law which finds expression in a similar mathematical |
form.
In conclusion, it may be noted that if the proof here given of the
principle of plant construction by orthogonal trajectories is considered satis-
. Church, —The Principles of Phyllotaxis. 243
factory, it adds considerably to the completeness of the principles of proto-
plasmic segmentation, and may be extended in several directions with
further interesting results. It is only necessary to point out that the case
of centric-growth is after all only a first step; and the most elaborate
growth forms of the plant-kingdom, as exhibited for instance in the seg-
mentation of the leaf-lamina, may be approached along similar lines, and
by means of geometrical constructions which are consequent on the more or
less perfect substitution of eccentric and ultimately wholly wxilateral growth-
extension, which again must ever be of a retarded type. The subject thus
rapidly gains in complexity ; but that the study of growth-form, which
after all is the basis of all morphology, must be primarily founded on such
simple conceptions as that of the ‘ growth-centre’ which has here been put
forward, should I think receive general assent, and in the case of the quasi-
circle, there can be little doubt as to the extreme beauty of the results
of the mathematical consideration.
On the Relation of Phyllotaxis to Mechanical
Laws.
By
Arthur H. Church, M.A., D.Sc.,
Lecturer in Natural Science, Jesus College, Oxford.
PART I.
CONSTRUCTION BY ORTHOGONAL TRAJECTORIES,
I. Introduction.
In the doctrine of Metamorphosis and the enunciation of the
Spiral Theory we have handed down to us two remarkable
generalizations which, originating in the fertile imagination of
Goéthe, have passed through the chaos of Nature Philosophy and
emerged in a modern and purified form, quite different from their
primary conception, to form the groundwork of our present views
of Plant morphology.
That leaves are usually arranged in spiral series had long been
recognized by botanists; but it was left for Goéthe, in 1831, to
connect the spiral-twining and torsion of stems, the spiral thicken-
ing of vessels, and the spirals of leaf-cycles into one ever-present
“ spiral-tendency ” of vegetation.
The Spiral Theory proper, as applied to Phyllotaxis, owes its
elaboration and geometrical completeness to Schimper and Braun
(1830-1835), by whom it was worked out with such precision, and
the ideas carried to their ultimate logical conclusion with such
uncompromising vigour, that it still forms, in the early pages of
text-books, the starting-point for our consideration of the relative
positions of the members of the plant body.
A
2 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
And in this, of all older botanical generalizations, perhaps, it is
alone worthy a place beside the Linnean system of classification,
that it first introduced methods of precise observation, record, and
geometrical representation into the interpretation of the growth of
the plant body as one whole organism, and thus paved the way for
the classic morphological researches of Wydler, Irmisch, and Eichler.
To Hofmeister and Sachs, as founders of the modern school,
the theory of Schimper and Braun, based on the observation
of matured organisms, struck on the rock of development; but,
while Hofmeister convinced himself of the utter inadequacy of the
theory, he did not substitute any more comprehensive view, and
Sachs did not investigate the matter at all deeply, regarding it as a
mere playing with figures and geometrical constructions, of little
interest except to those to whom it was practically useful.*
Further attempts at a more mechanical solution of the problems
have been made by Schwendener ; and an admirable summary by
Weisse in Goebel’s Organography of Plants presents the methods
adopted in explaining the phenomena observed by the action of the
mechanical forces of contact-pressure.
The subject can, however, by no means be regarded as placed
on a satisfactory footing. It is clear, that if mechanical agencies
come into play, they should be referable to the established laws of
mechanics, capable of resolution into their component forces, and
of diagrammatic representation in the different planes; while the
part, if any, that is not mechanical, but due to some inherent
“organizing property” of the protoplasm, requires to be clearly
isolated from the products of known mechanical laws.
From a mechanical standpoint, it is perhaps in the diagrams
that one feels most the absence of geometrical or mathematical
constructions. Thus Weisse, in using Schwendener’s not at all
* Sachs, On the Physiology of Plants, Eng. trans. p. 499: “For my part
I have from the first regarded the theory of phyllotaxis more as a sort of geo-
metrical and arithmetical playing with ideas, and have especially regarded the
spiral theory as a mode of view gratuitously introduced into the plant, as may
be read clearly enough in the four editions of my text-book.”
Sachs, Teat-book, edit. i, Eng. trans. p. 174: “The treatment of the subject
(Parastichtes) is only of value to those who are practically concerned with
phyllotaxis.”
INTRODUCTION. 5
easily grasped simile of the twist on the girders of a span-roof,
remarks that it is readily shown on a model but not on paper.
When to this is added the puzzling results of abnormal cases, the
general feeling left is that the mechanical forces are so well under
the control of the living protoplasm of the plant that they may or
may not actin any given case,*
Even if the diagrams and observations here recorded have no
permanent value, it is hoped that they may tend to revive an
interest in the methods of plotting out what may be termed
architectural studies of vegetable life.
PHYLLOTAXIS.
By the oldest botanists the arrangement of leaves in series
which formed alternating rows, when viewed horizontally or
vertically, was very aptly described by the term “ Quincuncial,”
from the analogy of the familiar method of planting vines in the
vineyard (Daubeny, Lectures on Roman Husbandry, 1857, p. 152).
Though such a diagonal pattern was produced by the indefinite
multiplication of the quincunx (V), no reference to any special
number (5) was implied, and all cases of spiral phyllotaxis and the
great majority of whorled clearly come under this wide generali-
zation + (Fuchs, De Historia Stirpium, 1542).
A more detailed classification appears to have been first. proposed
by Sauvages in 1743 (Sauvages, Mémoire sur une nouvelle Méthode
de Connottre les Plantes par les Feuilles, 1743).
* Goebel, Organography of Plants, Eng. trans., Weisse, p. 75.
Schwendener, .Mechanische Theorie der Blattstellungen, 1878, p. 12: “Die
Schumann’schen Einwande gegen meine Theorie der Blattstellungen,” Berichte
Konig. Preuss. Akad. Wiss., Berlin, 1899, p. 901.
+The view put forward by Fuchs, that the quincunx (V) was formed by
halving the X, is not endorsed by modern authorities; the 5-dot arrangement
of a dice-cube being a more possible primitive form.
This original signification of the term Quincunctal was revived by Naumann
in 1845 (“Ueber den Quincunx als Grundgesetz der Blattstellung vieler
Pflanzen”). From observations on Sigillaria, Lepidodendron, and Cactus stems,
he formulated a hypothesis of ridge and furrow construction, each ridge of a
cactus being a row of the Quincuncial system.
4 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Four types were established : the cases of opposite leaves, whorled,
alternate and scattered (jfewilles éparses) respectively; the
definition of the last named being that it included all instances in
which the members were arranged in no constant order.
Linnzus scarcely went farther than this. In his Philosophia
Botanica, 1751, the types are increased to nine; Dispositio sparsa
being extended to Conferta, Imbricata, and Fasciculata: the
definition of sparsa being again “sine ordine.”
Bonnet first determined a spiral arrangement, and his observa-
tions contain the germs of all subsequent spiral theories
(Recherches sur Cusage des Fewilles dans les plantes, 1754, p. 159).
He classified leaf arrangement according to five types:
(1) Alternating,
(2) Decussate (Paires eroissées),
(3) Whorled,
(4) Quincuncial,
(5) Multiple Spirals (Spirales redoublées) :
the last two of these being the ones which present the essential
points of interest.
Not only did Bonnet thus originate the spiral construction, but
he claimed to have discovered the “final cause” of the arrange-
ment of leaves, and his generalization, that « Transpiration which
takes place in the leaves demands that air should: circulate freely
around them, and that they should overlap as little as possible,”
has had a remarkably persistent influence on subsequent in-
vestigators.
Omitting this physiological standpoint, the morphological
generalizations of Bonnet were sufficiently striking. In this fourth
type, he included the true 2 spiral as we now understand it, in
which a spiral makes two revolutions to insert five members, thus
ultimately producing five vertical rows on the axis; and this
arrangement he checked on sixty-one species of plants. The term
quineuncial, thus defined, became limited to a special type of spiral
phyllotaxis quite apart from its original signification. He further
noted the tendency of the 2 phyllotaxis to vary to vertical rows of 3
or 8 on the same apeaibe the variation in the rise of the spiral,
INTRODUCTION. 5
right or left, in individual cases; and the correlation of the 2
arrangement with a 5-channelled stem.
The fifth type of “ Redoubled Spirals ” is of even greater interest,
in that it contains the germ not only of the parastichies of Braun,
but also of the multwugate systems of Bravais.
Only two examples were noted: Pinus, in which three parallel
spirals of seven members each resulted in a cycle of 21
members, and Abies, in which five parallel spirals of eleven
members each gave a total of 55. These latter observations are
eredited to Calandrini, who also drew the figures.
The lack of higher divergences appears to be due to Bonnet’s
preference for the longest leafy axes, and his special precautions to
avoid the terminal bud as much as possible, since this did not give
accurate results! Notwithstanding this, he saw quite clearly in
the case of the Apricot (p. 180) that successive 2 cycles were
really not vertically superposed, and that, in fact, the first members
of each successive cycle also formed a spiral, and so in practice no
leaf was vertically superposed to another on the same axis. This
he regarded, not as the expression of any fault in the theory, but
as a confirmation of his law, since such a secondary displacement
would give room for the proper function of every leaf.
Subsequently, arrangements in which eight and. thirteen parallel
spirals could be counted (the latter in the staminal cone of Cedrus)
were distinguished by De Candolle (A. P. de Candolle, Organographie
Végétale, 1827, vol. i. p. 329). -
From such a medley of observations on vertical rows and parallel
spirals, the more modern theory of phyllotaxis was evolved by the
genius of Schimper and Braun.
The vertical rows become “ orthostichies,” the parallel spirals
“parastichies,’ the number of leaves between two superposed
members becomes a “cycle,” and these are tabulated in a series :—
4 4% 8 Ys gp ete,*
* The properties of the Schimper-Braun series, 1, 2, 3, 5, 8, 13, etc., had long
been recognized by mathematicians (Gerhardt, Lamé¢), and appear to have been
first discussed by Leonardo da Pisa (Fibonacci) in the 13th century.
Kepler, in 1611, speculated on the occurrence of these numbers in the vegetable
kingdom, and went so far as to suggest that the pentamerous flower owed its
6 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
from the central commonest type (2), the quincuncial system of
Bonnet. The essence of the Schimper-Braun theory, however,
consists‘in the fact that these ratios of the numbers of members
(denominator) to the turns of the spiral (numerator) being thus
expressed in fractional form, become reduced to angular measure-
ments expressed in degrees of arc (the divergence), and that a single
genetic spiral controls the whole system.
When expressed in degrees, these divergences show an oscillation
between 4 and 4, or 180° and 120°, towards a central station of
rest, an angle to which the term “ideal angle” was applied by
Schimper.*
Thus, 3 = 144° 21 = 137° 27’ 16”:36
2=135° 4 = 187° 31’ 4112
Ps = 138° 27’ 41-54 bi = 137° 30’ 0”
S=137° 8’ 34-28 | “Ideal angle” =137° 30’ 27-936
4= 187° 38’ 49"-41
It will be noticed that the differences become extremely minute
in the higher fractions, and that beyond +; the difference is much
less than one degree of arc; an angle quite impossible of observation
on most plants or offaccurate marking on a small diagram.t+
No satisfactory attempt could be made at measuring the angles;
in fact, the brothers Bravais came to the conclusion that within the
error of observation all these higher divergences might be due to a
constant angle.t
structure to the fact that 5 was a member of the series. Cf. Ludwig, “ Weiteres
tiber Fibonacci-curven,” Bot. Centralb. lxviii. p. 7, 1896.
* It will be noted that Schimper’s formule are based on the type of the
quincuncial system (2) of Bonnet. The construction proposed by the latter, with
the co-operation of the mathematician Calandrini, was that of a helix drawn on a
cylinder. Such a system transferred to the plane representation of a floral
diagram, become a spiral of Archimedes, in which the sixth member falls on the
same radius vector as the first. The parastichies differing by two or three re-
spectively will similarly be Archimedean spirals. The truth of these systems
will therefore stand or fall acording as constructions by means of spirals of
Archimedes, derived from a consideration of adult cylindrical shoots, will explain
the facts observed’in the actual ontogeny of the members.
+ Cf. Bravais, Ann. Sct. Nat., 1837, pp. 67-71.
t Of. C. de Candolle, Théorie de Vangle unique en Phyllotacie, 1865.
INTRODUCTION. 4
This clearly formed the weakest point of the theory. It is quite
useless to take angular measurements as the basis of a theory when
they cannot be checked.
Again, in considering the common quincuncial (2) type, it is quite
easy to suppose that if five members developed in spiral series were
left isolated on a stem, they would space themselves out at equal
angles of 72° if they developed symmetrically: but it does not
follow that they were produced at exact successive angles of 144°,
although this number may have been approximated.
It is, in fact, a matter of ready observation, as Bonnet noticed,
that in none of the cases usually described as 2, and continued for
several members, does the sixth member come exactly over the first,
but rather falls a little earlier in the gap between 1 and 3. The
longer the internodes, the nearer it appears to so come, but the
range of error may clearly be very large: thus, to form the 6th
leaf of a 2 cycle the spiral should have rotated 5 x 144=720°; the
nearest 6th leaf of any other cycle is that of the 35, to form which
the spiral rotates 692°. In a given case, therefore, when it becomes
necessary to decide whether the cycle stops at 2, or is continued on
to 9s, a range of error as great as 28-—14° requires to be negotiated.
Such a range in a system which in higher values comes down to
minutes and seconds does not tend to render the original spiral
theory very acceptable.
The determination of the fractional value depends, therefore,
- since angular measurements are out of the question, on the deter-
mination of a member vertically superposed, to one taken as a
starting point. The theory of Schimper and Braun really stands
or falls, then, with the observation of “ orthostichies,” that is to say,
according as a leaf which appears to stand vertically above any
given one is actually so. Of this, again, proof is impossible: the
very fact that in going up the series to count the divergence on a
specimen, a nearer and nearer vertical point is obtained at every
rise, suggests that the one ultimately selected is only an approxima-
tion, the eye being as incapable of judging a mathematically straight
line as it is of measuring an angle to fractions of a degree.
That orthostichies tend to become ewrviserial in the higher
divergences was more fully recognised by Bravais, and very in-
8 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
genious constructions were adopted by Braun and Eichler to adapt
the “ obliquely vertical” rows of stamens in several Ranunculaceous
flowers as true orthostichies. But it is clear that no sharp line
can be drawn between parastichies and orthostichies when once the
latter become curved.
Hofmeister, who approached the subject with the most open mind,
came nearest the truth in formulating the statement that, in the
bud, a new member always arises in the widest gap between two
older ones. That the logical consequence of this would be ‘that no
member would ever be vertically superposed to another, nor again
would it be so if developed at the “ideal angle,” has beiéé duly
recognized. But such conclusions have always been slurred over by
supporters of the spiral theory: either the observations must be
imperfect, or the specimens must have suffered from torsions or
displacements; the remarkable series of mathematical fractions
could not possibly be wrong: the perfect accuracy of the “ideal
angle” could not be expected of the plant: the object to be attained
namely, the best possible distribution of assimilating surface being
sufficiently approximated at a comparatively low divergence.*
When once phyllotaxis is committed to this series of fractions,
expressing actual ratios of angular measurement, all deductions
from the mathematical properties of such a series naturally follow.
The remarkable superstructure therefore stands or falls according
to the correctness of the original series, based, as already noticed,
* Cf. Bonnet, 1754, p. 160 ; De Candolle, 1827, Organographic Végétale, vol.
i. p. 331.
Cf. Chauncey Wright, 1871. “On the uses and origin of arrangements of
leaves in plants” (Mem. Amer. Acad. ix. 387, 390). The continuation of this
theory of leaf distribution initiated by Bonnet, affords a remarkable example of
the method of biological interpretation of phenomena. Because a spiral series
gives a scattered arrangement of leaves and is very generally met with, it
does not at all follow that such a scattered arrangement is beneficial or at all an
aim on the part of the plant: nor again that the “ideal angle” would give
the ideal distribution. It is clear that in the intercalary growth of petiole-
formation the plant has a means of carrying leaves beyond their successors
whatever the phyllotaxis may be; while if the ideal angle of a spiral phyllo-
taxis becomes the ideal angle of leaf-distribution, the formation of whorled
series from primitive spirals, to say nothing of secondary dorsiventral systems
becomes curiously involved, "
INTRODUCTION. 9
on orthostichies which cannot be proved to be straight and angles
which cannot be measured.
Thus, if the angle of divergence within one cycle is constant, a
transition from one cycle to another of different value must involve
a special angle at the point of transition. To meet this difficulty
the theory of “ prosenthesis” was added to the original conception
by Schimper and Braun; a hypothesis again incapable of proof by
any actual measurements on the plant.*
Prosenthesis was also called upon to explain the alternation of
cycles in the common type of flower; and, in the same way, in
the formation of whorls of foliage leaves which usually alternate,
prosenthesis was required at every node.
Still more remarkable were the constructions adopted to explain
the “ obliquely vertical rows” of stamens in the flowers of certain
Ranunculacee, In order to bring these into line with “ortho-
stichies,” peculiar transitional divergences were adopted; a % spiral
eg. might, with a tendency to approach ;5;, give a somewhat larger
angle to every new cycle; and, owing to this special form of pro-
senthesis, the true orthostichies would take an oblique position, in
this case, along the course of the genetic spiral.t
Once, however, it is admitted that such transitional divergences
may render orthostichies oblique, the whole theory becomes con-
siderably weakened, since no clue is given to the causes which may
produce such an effect in one case and not in another; while the
fact that what it has been the custom of older writers to call ortho-
stichies should prove to be really a little curved, does not at first
strike the observer as necessarily affecting the validity of the
original hypothesis.t
On the other hand, with all its faults, the definite notation of the
Schimper-Braun theory, and the brevity and apparent simplicity
with which it sums up complicated constructions, is so closely
interwoven with our whole conception of the subject, that it becomes
* Eichler, Bliithendiagramme, i. p. 14.
+ Eichler, Bliithendiagramme, ii. p. 157.
{ Sachs, Physiology, Eng. trans., p. 497. “The theory of phyllotaxis, with its
assumption of the spiral as a fundamental law of growth, has, to the great injury
of all deeper insight into the growth of the plant, established itself so firmly
that even now it is not superfluous to show up its errors point by point.”
10 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
extremely difficult to take up an unbiassed standpoint, or recast the
matter in a new phraseology ; while to deny the actual existence of
the genetic spiral otherwise than, as Sachs has suggested, an unim-
portant accessory of the construction, savours of direct heresy.
‘The criticism of Sachs, which strikes at the root of the theory of
Schimper and Braun as applied to living organisms, applies equally
well to the work of other observers, and requires to be constantly
borne in mind.*
Because, writes Sachs, we can describe a circle by turning a
radius around one of its extremities, it does not follow that circles
are produced by this method in nature. Because we can draw a
spiral line through a series of developing members, it does not follow
that the plant is attempting to make a spiral, or that a spiral series
would be of any advantage to it. Geometrical constructions do not
give any clue to the causes which produce them, but only express
what is seen, and this subjective connection of the leaves by a
spiral does not at all imply any inherent tendency in the plant to
such a system of construction.t
Much of this, again, applies to the methods adopted by
Schwendener. Because an empirical system can be forced by
pressure into a condition resembling that obtaining in the plant,
it does not follow that a similar pressure acting on a similar
system is in operation in the plant itself.
Schwendener,} it is true, made a great advance in dealing with
solid bodies and spheres, rather than the abstract geometrical points
of the Schimper-Braun theory ; and, so far, Goebel is undoubtedly
right in stating that further research must be conducted along the
lines laid down by him. But at the base of all Schwendener’s con-
structions lies the fact that he begins by assuming the fractional
series of Schimper and Braun, and then arranges a mechanism to
convert these into systems more in accord with what is actually
observed in the plant.
* Sachs, History of Botany, Eng. trans., p. 168.
+ Mechanische Theorie der Blattstellungen, 1878,
Cf. Airy, Proceedings of the Royal Society, 1874, vol. xxii. p. 297, for a very
similar hypothesis of pressure on actual primordia without reference to the
actual structure of the growing point.
{ Goebel, Organography, Eng. trans., p. 73.
INTRODUCTION. 11
It is clear, however, that whatever subsequent alterations
are made in the system, the construction remains fundamentally
that of Schimper and Braun, and must stand or fall with the truth
of the premises which govern the original fractional series; and
these, as has been pointed out, are extremely vague, and have to
a great extent been rejected by Hofmeister and Sachs.
Contemporaneously with Schimper and Braun, the problems of
phyllotaxis were being attacked by the brothers A. and L. Bravais,
with in some respects identical results.*
Very scant justice has been done by Sachs} to the remarkable
work of these French observers. The parts in which they
appeared to agree with Schimper and Braun have been accepted,
those in which they differed have been rejected. It is not too
much to say that in the latter case they were wholly correct, and in
the former they came under the same erroneous influences as the
rival German school.
Thus, Sachs sums up by saying that their theories presented the
defects and not the merits of the Schimper-Braun system, in that
they made use of mathematical formule to an even greater extent
without paying attention to genetic conditions, and the whole was
“much inferior as regards serviceableness in the methodic descrip-
tion of plants to the simple views of Schimper.”
It is evident that Sachs’ distaste for the whole subject prevented
him from going into the matter very carefully, as the first thing
that strikes the reader is the very definite attempt made by the
Bravais to actually measure the angles and confirm their results
experimentally. It was owing to failure in this respect that they
fell back on the method of orthostichies and on this basis erected
very consistent hypotheses. When orthostichies obviously failed,
they approached the actual truth much nearer than Schimper and
Braun. They thus distinguished two kinds of spiral phyllotaxis
(1), that in which orthostichies were present and rectiserial ; (2)
that in which the so-called orthostichies were obviously curviserial.
The former applied to cylindrical structures and was so far identical
with Schimper’s theory, which was also based on mature cylindrical
* Ann. Sci. Nat., 1837, p. 42.
+ History of Botany, Eng. trans., p. 169.
12 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
organs; but, in the latter, they pointed out that the axis was often
conical or circular: in such case the straight orthostichies were
wanting and successive cycles were not accurately superposed.
More complete acquaintance with the structure of growing-points
would have shown them that the first case was wholly unnecessary,
and that the second hypothesis, based on a cone which might be
flattened to a circular disk, was alone required. Again, in common
with Schimper and Braun, they shared the view that the lateral
members were equal in bulk, or might be expressed by points, when
in point of fact they present in development a gradated series,
They, however, arrived safely at the conclusion that in such systems
the construction could not be expressed by a fractional divergence,
but only by the number of interesting parastichies (sinistrorsum
and dextrorsum), and the figure drawn for the theoretical structure
of a Composite inflorescence is very nearly correct, although its
method of construction (probably by modified Archimedean spirals)
is not described. Still more remarkable was the care with which
they worked out the multijugate types, in which the fractional
expression was divisible by a common factor (2-8), and thus clearly
pointed to the presence of two or more concurrent genetic spirals,
a case not contemplated by the spiral theory of Schimper and
Braun.
Restricted to the doubtful method of orthostichies, the Bravais
followed Schimper and Braun in the elaboration of other sets of
divergence fractions.*
Thus if 4, 4, 2, 2, etc, pointed, as stages of a continuous frac-
tion, to an ideal angle of 137° 30’ 28”, why might not there be a
complementary system 4, 4, 2, #, +4; pointing to 151° 8’ 8”? As
also 4, 4, #, 7p, etc., leading on to an ideal angle of 99° 30’ 6”, and
4, 1, 4 +5, ete, to 77° 57’ 19”!
It is clear that by such hypotheses any fraction that can be
counted may be regarded as a member of some system; and, as
Sachs has pointed out, this degenerates into mere “playing with
figures”; while no progress along such lines is possible when a
physiological reason is asked for. Still, these formule were founded
* Bravais, Ann. Sei. Nat., 1837, p. 87; Van Tieghem, Traité de Botanique,
p. 55, 1891,
INTRODUCTION. 13
on direct observations of plants, and the results are so far logically
carried out along Schimper-Braun lines of argument.
If these arrangements are regarded as the reductio ad absurdum
of the whole subject, it follows that the original premises are
possibly incorrect. It is so far only necessary to point out that
these cases are relatively much less numerous, and occur in plants
which exhibit marked adaptations to special biological environ-
ment, or, in modern phraseology, are markedly xerophytic, as for
example, Dipsacus, Sedum, Pothos, Bromelia, Cactacece.
By adopting the following
construction, and using the ep
usual terminology, a very
plausible diagram, which con-
veys a useful summary of the
Schimper-Braun theory, may
be plotted out (fig. 1). If it
be granted that, given a con-
stant type of lateral member,
the phyllotaxis would rise, as
expressed in the fractional
series, with successive increase
in the diameter of the axis, it
might also follow that it would
fall on a constant axis if the
members increased in bulk, or
rise if they were diminished,
according to the number of
members which would fill a
cycle round the stem.
Again, since members pack
more or less together, spheres to a certain degree extending into
the rows adjacent to them, while rhomboid figures each press one
half their length into adjacent cycles ; and since, to take the general
case, the plant commences growth from two symmetrically placed
cotyledons (divergence 3), it would pass on to a spiral arrangement
in the simplest manner by placing one member on one side and two
on the other (=divergence +). With no further increase in the
pthc eth Mion alsa
Fig.1.—General scheme for the orientation of
the cycles of the Schimper- Braun hypothesis.
14 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
bulk of the axis, or with increase in both axis and lateral members
definitely correlated, the phyllotaxis would remain 3. If a further
rise took place, the five gaps would be filled by the five members of
a 2 cycle, and in the same manner in successive cycles, two new
members being always added opposite the larger gaps corresponding
to the members of the last cycle but one, and thus each new
cycle would equal the sum of its two predecessors, and the rise
in divergence would be repeated ontogenetically in every in-
dividual.
The members of each cycle would have their appropriate angular
divergence (although this is only approximated in the figure), and
for a constant type of member such an ascending series would be
produced with an increased diameter in the stem; lateral branches,
proceeding from two symmetrically placed prophylls, would take on
a spiral construction according to their relative bulk.
The whole figure is orientated for the 2 position, so uniformly
present in the quincuncial calyx, and the members numbered in
this relation, so that No. 2 is median posterior.
An enormous number of facts may be collected in support of
such a construction and incorporated with it without, however,
necessarily establishing its accuracy. Thus the orientation of a 3
cycle with regard to a # is in all cases exactly as shown. For
example, in Helleborus foetidus, the flower possesses a 2 calyx with
normal orientation, and eight nectary petals of a 2 series, of which
most commonly 1-5, 6, 7 are present. The missing ones, 8, 7, 6,
as the case may be, always leave gaps in the positions marked by
these numbers with absolute constancy. The relation of two
cycles having been established, the other cycles may be regarded
as following the same plan, and may readily be numbered from
the divergence scheme—No. 1 being given by line which
zigzags through No. 1 of successive cycles to approach the “ideal
angle.”
It may be noted that the 4 spiral gives the odd member anterior,
the typical position in the case of trimerous monocotyledonous
flowers, while the } cycle falls transversely, as in the case of the
two prophylls.
Although a multitude of facts may be fitted into such a scheme,
INTRODUCTION. 15
and the relationship of members is thus readily tabulated and
placed in diagrammatic form, as in the construction of floral
diagrams, it affords-no explanation of the fact why, for example, a
2 divergence may be continued indefinitely,and then, when it does
rise, passes into a 3 or even directly into a +;, as in the construction
usually given for the nectaries of Helleborus niger. One begins to
regard with suspicion the convention which infers from five
members a # spiral, and from thirteen members a 8; spiral, while a
fall to five carpels may be interpreted as a reversion to a 2 spiral
again. The conventions do not explain anything; and it is not
clear, if angular distances cannot be checked, what criterion can
distinguish between five members of a ? spiral and the first five, for
example, of an <f; series.
Schwendener in his constructions accepts the divergences as
standard quantities, and proposes figures of transition in which the
varying bulk of the elements is taken into account. ‘The point of
view adopted here will be that, in the case of normal and phylo-
genetically primitive modes of growth, the divergences themselves
convey an erroneous impression, and that all theories which include
their acceptance must necessarily fall to the ground.
Nothing is more striking, in dealing with the subject of phyllo-
taxis, than the large number of hypotheses put forward which are
almost equally incapable of direct proof or disproof; and the
difficulty of the problem consists in determining a sure foundation
on which subsequent theories may be elaborated.
The hypothesis of Schimper and Braun does not satisfy the
demands of modern investigation, in that its premises do not admit
of strict observation and measurement, and for similar reasons
Schwendener’s views on mechanical contact-pressures are incapable
of direct proof.*
_ That such contact-pressures exist, and operate to a very consider-
able extent in producing secondary changes, is undoubted; but it
does not follow that they are so pre-eminently important and lead
to such great disturbances of the original construction, since it is
possible that by being equally distributed the disturbing effect
* Of. K. Schumann, Morphologische Studien, Heft 2, 1899, p. 312; C. de
Candolle, Considerations sur Petude de la Phyllotawie, 1881, p. 27.
16 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
would be inappreciable. The objection to Schwendener’s theory is
then, that it superimposes a second doubtful hypothesis on the
original unsatisfactory one of Schimper and Braun. On the other
hand, if the subject can be dealt with ab initio from a new stand-
point, it is not necessary to discuss the details of the conflicting
observations of Schwendener and Schumann.
In the present paper an attempt will be made to base all deduc-
tions on a single hypothesis, the mathematical proposition for uni-
form growth, as that of a mechanical system in which equal distribu-
tion of energy follows definite paths which may be studied by
means of geometrical constructions.
PLATE I.
“g1jU9d BY} MOI Lapdo UT peTaquANtt
SaAvo] OY} PUB ‘ATESTOASUB] 4d querd aures oy [—"s “OT
SU
sae as i
\
‘(g +.) satprysered snoraqo a1ou
£ JaJSWIRIP "WU OY “qAoTT ‘tung.uM979 WNAMMITMIS—"G OI
Lo
Pa
GENERAL OBSERVATIONS. 17
II. General Observations.
1, Mersop oF DETERMINING ORTHOSTICHIES.
As the difficulty of determining a vertical series increases with
the length of the internodes, and may be complicated by possible
torsion and irregular growth curvatures, a plant may be selected in
which growth and subsequent displacements are obviously at a
minimum,
A typical “rosette-plant,” such as Sempervivum, affords suitable
material: the leaves develop symmetrically and retain the positions
impressed on them in the bud.
The plant presents, as a whole, the usual curved systems of inter-
secting parastichies, the most obvious being five in one direction and
eight in the other (fig. 2 ; also fig. 4).
By a horizontal cut, the whole plant may be reduced to one
plane, and, commencing at the centre, the leaves may be readily
numbered in serial order of development from one to about fifty
(fig. 3). A vertical row of leaves, or an orthostichy, should clearly
appear in the section as a radius passing through the centre of the
leaves, differing in number by the denominator of the fractional
expression. The leaves 1, 6, 11, 16, etc., form a very obvious curve,
so that 2 is rejected ; the leaves 1, 9, 17, 25, etc., form an equally
obvious curve in the opposite direction, hence 3 is rejected. For
the same reason, 1, 14, 27, 40 form a lesser curve, but still distinct
enough to disqualify =; In 1, 22,43 the curve hardly appears,
and 1, 35 is possibly straighter, but further investigation is limited
by the number of leaves.
A comparison of the closer and closer proximity of these curves
B
18 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
to a central ideal line, suggests most clearly that no orthostichy is
really possible until the “ideal angle” is reached; that is to say,
only at infinity will a leaf be found exactly on the same radius as
1. The series in the specimen is bounded by a few leaves, and so
No. 22 is near enough for practical purposes, and the phyllotaxis
is usually given as ,; but there is no proof of its position over
No. 1; on the contrary, a very strong presumption against the
acceptance of any orthostichies at all.
2. Braun’s MetHop OF DETERMINING PARASTICHIES.
(Pinus Pine.)
The ripened carpellary cones of Pinus afford useful permanent
examples of spiral phyllotaxis. The large cone of P. Pinea, 54
inches by 3, is especially suitable for observation, and the smooth
scales are large enough to be clearly numbered.
Such a cone is observed to consist of obliquely intersecting rows of
scales (fig. 6-7), of which eight long curves intersecting thirteen
shorter ones are the most obvious.
Since the cone may be regarded as built up of a certain number
of oblique rows winding left, or again of a certain number of rows
winding right, a complete cycle may be regarded as formed by
taking one member from each of the two series. Thus in the case
figured (fig. '7), the structure may be regarded as built up of—
L, of oblique rows of the type 1, 9, 17, etc., of which eight can be
counted going all round the cone.
IL, of rows 1, 14, 27, etc., of which thirteen can be counted,
IIL, of rows of the type 1, 22, 43, etc., of thich there are twenty-
one.
In the first case, the scales will differ by eight, the number of
the curves, in the second by thirteen, and in the third by twenty-
one.
A simple method thus follows for numbering the scales in the
serial order of their development (genetic spiral). By taking any
given scale as one, the number of each one adjacent to it may be
determined by counting round the cone the number of curves in
PLATE II.
(EL4 9) sonporsvaed favopaq Woy
woas ouoo papuvdxa Kap “YU ‘won.GsnY sri —"g “OL
; ‘(13+ 81)
poyop uteqsAs aaano ‘wasopnwds wnapasoduag —"p 1h
GENERAL OBSERVATIONS. 19
the same sense with it, and by zigzagging from scale to scale the
whole cone is numbered up. From a consideration of the para-
stichies of other systems, Braun* tabulated a series of fractions 4, 2,
3, , s, #4, etc, complementary to those of Schimper’s series of
orthostichies. The method is strictly accurate, and clearly affords
the only way of numbering up the members of a complicated spiral
system when the lateral members retain their original close contact
on a condensed axis.
However, as soon as attempts are made to bring the system of
parastichies into line with that of orthostichies, difficulties arise.
The system of (8+13) parastichies corresponds in the Schimper-
Braun theory to a phyllotaxis of $,, but it is at once clear that 22
is not superposed to 1. The scale marked 35 is practically over it,
and hence the phyllotaxis would be usually given as one stage
higher, i.e. 13, to fit which divergence it would be necessary to
assume that the correct parastichies are those passing through
1, 14, 27, and 1, 22, 43 respectively. In other words, the steepest
parastichies are to be taken as a guide. As in the preceding case
of Sempervivwm, however, there is no evidence that 35 is vertically
superposed to 1; the figure, in fact, shows that it is only approxi-
mately so, and that if 56 were normally placed it might be nearer the
vertical line. Owing to the sloping off of the cone, 56 is clearly
well off the line, and 35 remains the nearest for practical purposes.
There is no proof of its accuracy; but by comparison with Semper-
vivum, the strongest presumption in favour of its being only an
approximation, owing to the limited number of members on a
cylindrical portion of the axis. There is certainly no clear justifi-
cation for assuming any secondary displacements in order to save
the theory.
In fact, there is only one mathematically accurate statement
which can be made about such a construction, and that is, that
taking four scales in contact, or making use of a rhomb of rhombs
(eg., fig. 7: 1, 9, 22, 14), the cone is composed of (8413) inter-
secting spirals, of which eight are longer and thirteen shorter.
Adopting the convention that the right-hand direction is that
marked by the hand of a watch at 12 o’clock, the cone figured
* Flora, 1835, p. 157.
20 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
(fig. 6) shows eight right- and thirteen left-handed spirals. (Fig. 5
vice versa.) Reversing this expression for the position seen when
looking down on the apex of the cone instead of up from below as
in the figure, in order to express the cone in terms of a floral
diagram, the constant becomes (8 L+13 R), this being in fact the
actual terminology proposed by the brothers Bravais. The direc-
tion of the spirals, however, may vary from cone to cone, and the
more essential point is that the lower number of the expression
gives the number of the longer curves, so that the formula (8 long
+13 short spirals) remains the only cone-constant about which
there is no mathematical doubt. Braun’s method marks a real
advance in that it replaces the long genetic spiral, quite impossible
of observation in high divergences, by short intersecting curves
which may be readily and accurately counted in the highest series.
The closer the members, the more difficult the question of
orthostichies becomes, the method of transferring a system of
pavastichies to fractional terms expressing an accurate angular
divergence becomes conventional, and one is unavoidably brought
into agreement with Bravais, that in such a case as Pinus, the
parastichies are all-important and the orthostichies imaginary.
The tapering of the Pine cone, however, militates against its being
taken as a type of a cylindrical system, and the method, as appli-
cable to the whole series of divergences, requires to be tested on
cylindrical axes.
3. PHYLLOTAXIS OF EupHoRBIA WULFENII, Hoppe.
Euphorbia Wulfenit, a handsome Spurge growing 5-6 ft. high, as
cultivated in the Oxford Botanic Garden, affords excellent cylin-
drical stems on which, owing to delayed.formation of cork, the leaf-
scars remain and are well marked.
The flowering-shoots, bearing evergreen leaves, persist for 3-4
years, terminating in a compound inflorescence, without, however,
producing a terminal flower.
Beyond the two cotyledons of the seedling, or the two prophylls
of a lateral shoot, the phyllotaxis rises to a medium elevation,
indicated roughly by .8,, and produces vegetative leaves on a
PLATE ITT.
“69 0} [ WATS & LOI Japi1o UL palaquind *(@L +8) satyorjsered-qorytva
so[vos ayy § Suoy sayour Fg ‘auoo sures oy —"y “org { aJOWBIP satpuUt g fated pasopo ‘TT ‘var
Ad —"9 “DU
GENERAL OBSERVATIONS. 21
cylindrical stem for a distance of 3-4 feet or more before the end
tapers to the inflorescence. Old flowering stems, cut in September
or October, when all the leaves have been shed, afford the best.
material. On these cylindrical stems, owing to the absence of
winter-bud formation, the alternation of the seasons is shown by the
alternating approximation and separation of the leaf-scar cycles ;
the scars being closer together in autumn and fairly scattered in
spring. The figure (fig. 8) represents a continuous portion of a
stem A, B, 0, D, E, cut into four sections for convenience of illustra-
tion. By using the method of parastichies on the spring area (A),
rhombs are readily noted formed by (3+5) intersecting curves,
pointing to a 3 phyllotaxis; but the top member of the rhomb (9)
is not on the same vertical line as the bottom one, and the fraction
may therefore be higher. On another piece (B, C) the same
rhombs are conspicuous, but another set may be marked out form-
ing (5+8) curves of a 3; phyllotaxis, but the top member of the
rhomb is again not vertically superposed to the bottom one. Ona
third piece (C, D) the same rhombs may be traced, or a steeper
series due to (8+ 13) curves of 58; type. The top member is again
not in vertical series. The stem is erect and contains little wood,
there is no sign of torsion on it, but the phyllotaxis, as defined by
the observation of orthostichies, seems ever elusive. In the same
way still steeper curves may be located, as in D, E, where 13+ 21
point to a 13 system, and, given a long cylinder, they may be made
as steep as one likes; as soon as the eye becomes accustomed to one
set, a still steeper may be seen. It becomes clear that the curves
may be carried the whole length of the stem before the series comes
to a compulsory end. This range was short in the Pine cone; in the
cylinder it becomes indefinitely prolonged, a leaf accurately super-
posed to No. 1 being, in fact, only reached at some quite indefinite
station, although a nearer approach is gained at every rise of the
phyllotaxis series. Thus the actual expression given becomes a
convention, since the ever-steepest curves pass beyond the limit of
observation. These are the facts, but what do they mean ?
It is clear that the phyllotaxis fraction, whatever numerical
value is given to it, rises in the series as the axis is telescoped,
and falls as it is lengthened. In fact, if a high phyllotaxis be
22 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
plotted out on a rubber-tube, and this greatly extended, it can be
made to fall as low as one pleases. The mechanical effect of such
tension or compression has been noted by Schwendener and Weisse.
Now, in this plant, the expansions and contractions of the
system are due to varying rates of growth in the main axis. The
spiral itself is constant and the same genetic spiral runs uniformly
through the whole shoot, including different sets of parastichy
curves, which, when marked out along the axis, present the appear-
ance of a spiral spring expanded and contracted at different points.
The actual arrangement of the members on the stem is, in fact,
here quite secondary; varying phyllotaxis phenomena are produced
by varying rates of growth, a conclusion already reached by Bravais
from the study of the rosette of Sempervivwm and its flowering
axis. The spiral arrangement is not an end at which the plant
is aiming, but the mere retention of a uniform system impressed
on it in the terminal bud.
It follows, then, that phyllotaxis is the obvious and visible
expression of more obscure phenomena in the growing apex, and
must be referred ¢o the first Zone of Growth, since in passing through
the Zone of Elongation it may be fundamentally altered in appear-
ance.
Confirmation of the view that the spiral is only the relic of
the effect of certain agencies at the growing point, and is not
directly essential to the welfare of the plant, is shown by the
general occurrence of cases in which the originally spirally arranged
leaves become secondarily dorsiventral in arrangement by torsion
of the leaf-stalks; 2.¢., the effect of the spiral becomes secondarily
corrected as soon as it becomes a distinct disadvantage to the plant.
It follows again that, for any spiral leaf arrangement that has
passed through this second zone of elongation, no expression
which is not a purely arbitrary and conventional one can be
formulated.*
Phyllotaxis is to be studied in the growing apex itself, or in
structures which have undergone so little elongation that the
* Transverse sections of the apices of shoots of Euphorbia Wulfenii (of. figs.
90, 91) show systems in which the contact parastichies are (8+13), (5+8), and
in very weak shoots (3+5).
Bap ee
PLATI
ot ‘d ‘oO ‘g SV sixe oxy Jo uornsod snonuy
“LOJIWVIP UL sayour g wN[NgIdeg -uod v uo satyoryseted jo suaqyshs suolea ! Jajawelp
“MOTI WloIJ SaTROs [BIONTOAUTT “TT ‘sn whjoag nevuhg—'6 “Ol Ul "LUM GT Weg raddoy ‘nwalingl niqeoydngy—'g "Ol
GENERAL OBSERVATIONS. 23
individual members remain in their original close lateral contact.
It is impossible to exaggerate the importance of the acceptance
of such a generalization. At one sweep it clears away all the
foundations of the Schimper-Braun theory, and all other formule
based on the study of mature cylindrical organs with isolated
members, which from the time of Bonnet have been regarded as
the types of construction, and for which expressions were required
by descriptive systematists,*
4. PHYLLOTAXIS OF CyNARA ScoLymus, Z.
The great inflorescence heads of the cultivated Globe Artichoke
produced an involucre of protective scale leaves 8-9 inches in
diameter in the expanded head. These form a definite spiral series
starting from a very low divergence on the elongated main axis
and reaching a very high one at the periphery of the disk. The
scales are large and the insertion of the lower one quite clear, so
that these may be easily numbered in rotation (fig. 9), and the series
finished off by Braun’s method. The numbering is quite definite,
and the system clearly presents a “rising phyllotaxis.” Thus if
the sixth member were over the first, it would be 2, but it is not;
nor does 9 come opposite 1, nor again 14, though 22 is so near, that
taking all the leaves in sight as in Sempervivwm, it might be con-
sidered sufficiently accurate. If the series has stopped at any one
of these points and remained constant afterwards, the transition
* Henceforward, therefore, the term phyllotaxis will be used exclusively for
the primary arrangement of lateral members at the moment of their actual
development on the growing-apex. Schwendener, following the custom of
systematists, uses the term as applicable to any leaf arrangement at any part of
the plant, as calculated, that is to say, by the very questionable method of
orthostichies. Thus the terminal bud of Pinus sylvestris would present scale-
leaves and spur-branch origins with a divergence of 23, while in the shoot
immediately below the divergence was ;;. It is clear that the arrangement at
the growing-point is the feature of primary importance ; subsequent alterations
are secondary in character, and the actual arrangement observed in adult mem-
bers may present merely the obscured relics of a primary construction ; the
hypothesis that the phenomena observed on an adult member has been from
the first the aim of the organism having no satisfactory basis.
24 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
from one cycle to the next would have been clear. In this actual
specimen the ultimate stage reached, as shown by the parastichies
of the disk-florets, would have been +; (cp. fig. 53): the scales,
however, pass on to this from the low divergence of the axis with
absolutely no visible line of demarcation between ascending
members of the series. That is to say, there is no visible ‘“‘pro-
senthesis” anywhere; the spiral, as in Sempervivwm, can be given
any fractional quality according to the point at which the observer
or the specimen stops. In fact, just as orthostichies cannot be
proved, so prosenthesis is a purely idealistic conception ; the angles
subtending the scales pass on by imperceptible gradations, so that,
whichever scale be numbered 1 as starting point, no difference in
the construction can be observed.
Clearly, then, there is only one uniform genetic spiral as indicated
by the numerical order of development, and no manipulation of
phyllotaxis-fractions will explain the system: as pointed out by
Bravais,* the error of observation is such that it is impossible to
disprove that the angle is not constant for every member, and.
might in fact be the “ideal angle” throughout.
From the preceding example of Huphorbia, the conclusion was
reached that phyllotaxis is a function of the first zone of growth
in which no elongation takes place beyond the equal growth of all
the isodiametric initial-cells isodiametrically.
All further investigations, therefore, demand the elimination of
the secondary elongation of the second zone of growth, either by
looking vertically down on the growing point, or by resolving
all forces into their transverse components. The following example
of a type in which this elimination of the longitudinal extension
has been combined with transverse extension on a large scale may
be considered.
5. PHYLLOTAXIS OF HELIANTHUS ANNUUS, L.
The most perfect examples of phyllotaxis easily obtainable are
afforded by the common sunflower, so frequently selected as a typical
* Loc. cit., p. 71,
‘
PLATE V.
19
as
+a
ae
2e
en
38
Sa
ee
ao
on
ee
2
x
Pm
ae
ey
&
|
the ovary portions of 35 abortive ray
11,— Helianthus annuus.
Fis,
Capitulum in fruit,
3 inches diameter ; contact parastichies (34 +55).
Fra. 10.—ArVianthus annuus, L.
GENERAL OBSERVATIONS. 25
Angiosperm, both in anatomical and physiological observations,
owing to the fact that it exhibits, par excellence, what is regarded
as a normal structure little modified by specialization for any
peculiar environment.
Not only is Helianthus a leading type of the Compositae which
hold the highest position among Angiosperm families, but amongst
this family it flourishes in the best stations, in which sunlight, air
and water-supply are perhaps at an optimum for modern vegetation.
The very fact that it is as near an approximation to the typical
Angiosperm as can perhaps be obtained, suggest that the phenomena
of growth exhibited by it will also be normal, and from the time
of Braun to that of Schwendener it has afforded a classical example
of spiral phyllotaxis.
As is well known, the plant forms a main axis of only annual
duration, growing 7-8 feet, or in special cases even 15 feet, during
the summer months; the leaves are typical in character, and are
borne spirally with a divergence to which, since the stem has
passed through a very active zone of elongation, the application of
any fractional value must be purely empirical, but it would be
generally said to range between 2 and ;5;. In the open, the stem
grows erect without torsion, and terminates in the main inflor-
escence. Branches are normally developed to the second degree, and
these again terminate in inflorescences similar to the terminal one,
but on a progressively smaller scale. The vegetative leaves pass
gradually by reduction into an involucre of leaf-base scales (fig. 14) ;
and contemporaneously with the formation of these, the axis
broadens out, elongation practically ceases, but lateral extension is
very considerable, so that the capitulum disk approaches a level
surface and the whole energy of growth is directed radially.
As in Cynara Scolymus, the involucral scales exhibit therefore
the phenomenon of a “rising phyllotaxis,” and it is futile to at-
tempt to give it any fractional value until the broadest diameter
of the inflorescence is reached. Here the leaf-members become.
fertile scaly bracts and subtend the florets of the disk; the
sterilized ray-florets being subtended by the innermost series of
the large scales. The fertile bracts mark out rhomboidal areas
and the enclosed flower-primordia are circular in section:
26 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
bracts and subtended flowers are developed centripetally (<¢., in
normal series), and by a wonderful correlation of growth in the
whole capitulum, the receptacle extends laterally and symmetri-
cally as new flowers are added, expand, and pass on into the
fruiting condition. The result is that the head in fruit exhibits
practically the same structure as in its early stages, but on a large
discoid area.
Such capitula admit of ready observation. By taking a head in
which the last flowers are withering, and clearing away the corolla
tubes, the developing ovaries are seen to mark out rhomboidal
facets (fig. 10), and when the fruits are ripened and have been
shed, the subtending bracts still form rhomboidal sockets (fig. 11).
These sockets, with or without fruits, form series of intersecting
curves (“parastichies” of Braun, “ contact-lines ” of Schwendener),
identical with those of the Pine cone, only reduced to a horizontal
plane.
A fairly large head, 5-6 inches in diameter in the fruiting
condition, will show exactly 55 long curves crossing 89 shorter
ones (fig. 12), A head slightly smaller, 3-5 inches across the disk,
exactly 34 long and 55 short (figs. 10,11); very large 11 inch heads
give 89 long and 144 short (fig. 13): the smallest tertiary heads
reduce to 21434, and ultimately 13+21 may be found; but these
being developed late'in the season are frequently distorted and do
not set fruit well.*
A record head grown at Oxford in 1899 measured 22 inches in
diameter, and, though it was not counted, there is every reason to
believe that its parastichies belonged to a still higher series
(144+ 233). The Sunflower is thus limited in its inflorescence to
certain set patterns (according to the strength of the inflorescence
axis,) ¢.g., $$, 34, 24, 33, 8%. These were first observed by Braun,t
and translated into terms of the Schimper-Braun series they would
correspond to divergences of 33, 24, 24, 22;, and 89, respectively.
* Cf. A. Weisse: “Die Zahl der Randbliithen an Compositenkopfchen in
ihrer Beziehung zur Blattstellung und Ernihrung (Pring. Jahrb., xxx. 453).
Complete data for 140 specimens are tabulated ; out of 61 poorly nourished
pot-cultures 27 produced (13+ 21) terminal capitula.
+ Flora, 1835, p. 157,
PLATE VI.
“(FEL + 68) satyporsered-qoryttoo
£ Loyaweip saqout TL ASI
SUNUUD SRYRMMUYIEL— ET OT
“(69 + 6g) sattporsered-jorz U0
£ apaweIp sayout Fg YSIq
‘SANUUD SYR LEO L ONT
GENERAL OBSERVATIONS. 27
Under normal circumstances of growth, the ratio of the curves is
practically constant,* and in noting the wonderful accuracy with
which these high divergences are illustrated, one naturally con-
cludes that laws which hold with such mathematical exactness in
the higher series must also be true when lower divergences are
considered. Such an apparent confirmation of the Schimper-Braun
theory is very striking; in fact, its very perfection leads one to
question the accuracy of the spiral hypothesis; if a single growth
spiral can here work at an angle correct to minutes and seconds,
and the difference between 34 and =; is only 1’ 41”, why is the
mathematical precision not equal in simple cases, and the divergence
as readily measured as the angles of a crystal ? +
* Of. Weisse. Out of 140 plants 6 only were anomalous, the error being thus
only 4 per cent.
+ A primordium of the highest system described (Helcanthus capitulum, fig.
13) subtends an angle of only 2°.5’.
On the other hand the range of error in lower systems is very considerable.
As an example a seedling of Nymphaea alba may be taken, since the leaves arise
perfectly free from one another. There are thus no lateral pressures in the bud ;
the interstices are packed with hairs among which the leaf bases slide without
meeting any resistance (fig. 94).
By making a careful camera-lucida drawing of such a plant 3 mm, in diameter,
magnified 70 diameters, restored to normal volume as far as possible by clearing
in Eau de Javelle, the centre of the vascular bundles may be taken as represent-
ing the centre of construction and the angular divergence measured on the
drawing with a considerable amount of accuracy. In such a specimen the angles
measured were—
Between 1 and 2 ; 3 : < . 147°
ie average 139°
144°
OD mw w bo
WOM!
143°
134° average 1402°
144° (Fig. 94.)
Three leaves form a complete cycle, and the error of observation diminishes
in taking the average of the three ; while the centre of each leaf may be fairly
accurately marked, the actual centre of the system is difficult to judge on an
apex which is bulging asymmetrically.
Thus on another apex (fig. 94, corner) an obvious error of § mm. in judging
the centre thus empirically, introduced in the divergence of two particular leaves,
an error of 143° to 147° ; the angle to the next leaf not so much affected by this
error being 133° or 132°, In the Schimper-Braun system, the phyllotaxis of
28 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Further observation shows again that while as in Pinus and
Euphorbia Wulfenii, a formula may be obtained from the curves,
no obvious orthostichies are present, and the genetic spiral cannot be
traced without numbering the members. An alternative hypothesis
thus immediately presents itself, as in preceding cases, namely, that
the single genetic spiral of Schimper is not a primary cause of the
formation of parastichies, but that the parastichies are the primary,
as they are the only constant feature; and that the genetic spiral
is of no real significance, but an appearance produced on elongated
axes by pulling out such a system as that here found in an arrested
condition,
Further examination of the head brings out the interesting detail
that the sterile ray-florets are closely related to the long curves, and
typically equal to them in number, while their three-angled
ovaries form the “ half-bricks” which fill out the mosaic of the disk
to the circular outline of the involucre (fig. 11).*
Other Composites, especially those with bracteate inflorescence
receptacles, are equally constant, though in usually lower series;
good examples being afforded by Aster, Chrysanthemum, and the
monstrous forms of florists’ Dahlias (fig. 16).
To sum up, the fruit-heads of the Sunflower present a persistent
phyllotaxis system in which the members still retain, as in the
Pine cone, the actual lateral contact they had when they were
formed, unmodified by longitudinal extension. Growth has
operated so symmetrically that the structure of the capitulum is
practically the same as when the flowers were being first laid down
on the apical cone. That such is the case may be readily checked
by transverse sections of the young inflorescence, in which the
circular outline of the flower primordia is clearly defined. The
primary members, it is true, are reduced to supporting frameworks
enclosing cylindrical florets; but allowing for this peculiarity, the
Sunflower-head presents on a large scale the actual conditions
this seedling would be described as 2, or 144° angular divergence. According
to the theory subsequently put forward, the angular divergence of the system
approximates 138°°5, as measured on a geometrical construction.
* Of. Weisse, loc. cit. Ludwig, Bot. Centralb., lxiv. p. 100: “ Ueber Varia-
tionskurven und Variations flachen der Pflanzen,” =
PLATE VIL
“SOAIND OY] JO WOTJOIIIp ayy AVOYsS
04 poaotuar A{pergaed szinaz ¢ (Gg + FE) saryorse. ved-3oe} 09,
ad Aq B SB payoatas WU VIL
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PLATE VIIL
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xodv a[orps ayy Jo Surmerp ‘prow weg waysas (¢ +) ‘(EL 49)
SMLOZIYL JO FulOd BuLMolyy twp pda wena pers pr
"Ge HTT samporysuivil § tutoy snoaqgsuot waprery § Y2/YNT—"OL “Dig
GENERAL OBSERVATIONS. 29
obtaining in a transverse plane at the growing apex of a stem,
and the whole forms a striking image of a growing point covered
- with primordia which tend to assume the form of spheres.
It will be further noted that the structure cannot be defined
in terms of the Schimper-Braun theory. As in the case of the
Pine cone, no formula can be given for it except that which
includes the number of intersecting parastichies. But on the other
hand, the older theory gave a method for constructing the diagram,
which, even if erroneous, was almost within the limit of the error
of observation. Clearly no advantage is to be gained by throwing
away the Schimper-Braun construction until it can be replaced by
a better. If the number of curved parastichies gives the only
strictly accurate account of the system, it becomes necessary to
examine these curves and see if they can be brought into line with
mathematical terminology and geometrical methods of construction.
30 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
III. Geometrical Representation of Growth.
STRUCTURE OF THE FIRST ZONE OF GROWTH.
IF spiral phyllotaxis is thus reducible to a function of the first zone
of growth, and is, for example, wholly independent of the circum-
nutation spiral which is a function of the second zone of growth,*
often exhibited most clearly in stems with a symmetrical arrange-
ment of leaves, it is necessary before deducing constructions for
phyllotaxis to determine as far as possible the various agencies in
operation in the former zone.
The conception of the first zone of growth, as elaborated by Sachs,
* The same observation applies to the theory proposed by Airy (Proceedings
of the Royal Society, vol. xxii., 1874, p. 297, 307), in which a strong condensing
force was supposed to act with a telescoping effect on a simple whorled series
with a tendency to lateral displacement! Such a condensing force, considered
possible as a phenomenon of arrested development of the axis, would be clearly
a property of the second zone, and though it may come under the head of
secondary changes, will have no influence whatever on the actual origin of the
new centres of growth in Zone I.
Schwendener’s theory (Mechanische Theorie der Blattstellungen, 1878) of a
vertical compressing force resembles in many points that of Airy; his pressure
is again derived from the mutual contact of the primordia at their bases after
they are formed. But such alterations, again, must obviously belong to the
second zone of growth, and will, if the pressures are unequal, induce packing of
the primordia in close hexagonal series.
Lastly, it may not be amiss to point out that the correspondence of vascular
bundles (Bonnet), the shape of the pith (Palisot de Beauvais, 1812), or the
presence of ridges on the stem (Naumann), being phenomena originated in the
third zone of differentiation, have still less importance as indicating any relation
whatever to the actual arrangement of the new centres of growth in the
embryonic protoplasm of Zone I.
GEOMETRICAL REPRESENTATION OF GROWTH. 31
includes an apical region in which the embryonic protoplasm is
engaged in the formation of new cell-units, without regard to any
marked longitudinal extension of the member as a whole, and may
be taken to imply the general increase of the protoplasmic mass,
equally in all directions in space, and uniformly throughout its
substance; the actual subdivision into units of an approximately
equal size being a secondary specialization. The mass of protoplasm
may be thus considered as a whole, without reference to the
cell-membranes of the component units, and this in its structure,
and in being supplied with new material along an axial-con-
ducting portion, presents many analogies with a jet of semi-fluid
substance. :
Sachs * also pointed out the remarkable similarity of the shape of
the growing apex of a plant to a paraboloid of revolution, and that
in a radial longitudinal section of a typical apex the periclinal
walls formed a series of confocal parabolas crossed by a coaxial
system of confocal parabolas which formed the anticlinal walls. The
mathematical fact that two such sets of confocal parabolas intersect
at all points orthogonally was of the utmost importance in enabling
him to formulate his theory of the orthogonal intersection of cell-
walls. Sachs, however, left the matter entirely a theory of
geometrical construction, although, as he himself states in deciding
against the spiral theory, geometrical constructions tell nothing of
causes but only express facts; still it is clear that these phenomena
must be based on some definite laws, probably mechanical.
Thus Errera t+ has shown that the cell-wall at the moment of its
formation has the properties of a weightless fluid film, and that the
direction taken by such cell-membranes is identical with that taken
by soap-bubble films which impinge orthogonally on previously-
formed films. A large number of cases proposed as presenting an
apparent contradiction of Sachs’ generalization have been shown
* Sachs, Lectures on the Physiology of Plants, Eng. trans., 1887, p. 448: “Many
hundreds of median longitudinal sections through growing points of shoots and
roots, drawn by very different observers without even the most distant per-
ception of the fundamental principle, accord with the construction I have given,
and demonstrate tts accuracy.”
+ Errera, “Ueber Zellenformen und Seifenblasen,” Biol, Centralb., 1887,
1888, p. 728.
32 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
by Wildeman * to really support it, so far, that is to say, as an angle
of 90° can be judged by the eye ina cleared preparation ; and the
law of the orthogonal intersection of cell-walls at a growing apex
may be taken as generally accepted.
The most remarkable deduction from this theory of orthogonal
cell-formation is that the cells at the initial group in Zone I. are
laid down in accordance with definite mechanical laws, and not as
the expression of any aim on the part of the protoplasm as to their
ultimate use in the plant economy. The fact that hexagonal
packing appears to be the more frequent case in mature organs is
therefore due to secondary alterations in the arrangement, and is
the expression, that is 6o say, of secondary forces initiated away
from the growing point itself. In other words, plants form their
cells fairly isodiametric and orthogonal in Zone I.; in the second
zone elongation and further orthogonal division takes place ; while
a third zone of differentiation is necessary to correct the errors of
Zone I, and convert the mechanically produced cells into units more
suited for the performance of their special functions. The unequal
tensions set up in this process may result in the rolling of the
cells over each other as they tend to round off and become more
independent.
On the other hand, it is quite true that the orthogonal position is
very difficult to prove absolutely in any particular cell. Angles
may look very like right angles without being exactly 90°; for
example, it is often difficult in a transverse section of the apical cell
of Pteris to say whether the curvilinear angles are 90° or nearer
120°. It is, in fact, only by bringing the great mass of facts into
line with some general mechanical principle, as Errera has done,
that the probability becomes practical proof.
Thus when the large oospheres of Fucus or the tetraspores of one
of the Florideae are discharged, they assume an apparently per-
fectly spherical form, although this is not the shape in which they
are actually developed. But regarded as masses of a dense fluid
freely suspended in a non-miscible medium of approximately
equal density, the surface tension is sufficient to reduce their surface
to a minimum, and spherical form is attained by these reproductive
* Wildeman, Etudes sur Pattache des cloisons cellulaires, 1898,
GEOMETRICAL REPRESENTATION OF GROWTH. 33
cells which are not exhibiting any phenomena of growth. The sur-
face tension is sufficiently great to warrant the neglect of the weight
of the mass of the spore, which would tend to destroy the spherical
form if the densities were slightly unequal; thus so far as observa-
tions can go, the spores are absolutely spherical; but no proof of
this exists unless the mechanical theory of surface tension can be
applied. That is to say, the absolute proof of the shape assumed can
only be determined by physical deductions and not simply by observation.
In the same way, no amount of actual measurement of a specimen
would convince a mathematician that the apparently parabolic
curves seen in sections were of the strict (y? = 4 ax) type, unless
some mechanical determining cause can be adduced in support of
such a statement ; as, for example, a hypothesis that the cells might
be regarded as homologous with projectiles discharged from the
growing apex.
The paraboloid theory of Sachs still remains a good working
hypothesis, and will stand or fall as the theories based on it can or
cannot explain other allied phenomena; its value depends on the
extent to which other facts can be deduced from it.
Thus, if the section of the growing apex is a true parabola, over
which the superficial cells may be supposed to glide until they reach
a position of rest on the cylindrical surface of the full-grown stem,
it is possible that the motion in the particles composing the fluid
mass of protoplasm might be resolved into a transverse velocity and
a longitudinal acceleration ; the former, a steady uniform movement-
due apparently to the expansion of growth ; the latter, the expression
of the constant action of some retarding force acting along the axis
of the paraboloid apex.
In a simple case in which none of the particles were discharged
above the horizontal line, it is clear that a paraboloid of revolution
would mark out the enclosing curve of the line of fall of all of
them ; while if the particles are regarded as being discharged in all
directions, as in the fall of particles of a bursting shell, the
enveloping curve would still be such a paraboloid, so that it is
immaterial whether the initial point be regarded as situated on the
surface of the apex, or at the focus of the parabola, so far as the
main outline of the curve is alone concerned.
c
34 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
But there is no reason to believe that growth can be expressed as
a uniform velocity, nor can the retarding force be in any way
measured, so that the parabola cannot be at present constructed
from physical data. This geometrical construction therefore remains
purely hypothetical; and although the approximation may appear
close to the eye, it cannot be used as a basis on which a mechanical
theory of the apex can be built.
From the apparent paraboloidal shape, Sachs deduced the ortho-
gonal intersection of cell walls. The latter may, however, still be
true and yet the curves not be parabolas; the law of the orthogonal
distribution of paths of equal action being a generalization of
which intersecting confocal parobolas is only one special case.*
* The theory of the orthogonal intersection of cell walls, built up by Sachs
and Schwendener, was elaborated by the former in one of the most suggestive
chapters of his Vegetable Physiology (Eng. trans. 1887, p. 431). Plants
exhibiting circular symmetry presented radial anticlinals intersecting circular
periclinals; in elliptical forms, the periclinal ellipses were intersected by
hyperbolic anticlinals, and in the growing apex two orthogonally intersecting
systems of confocal parabolas were assumed ; again, in the asymmetrical growth
of a tree-trunk (p. 445), a diagram constructed by eccentric circle systems
showed that the medullary rays followed approximately the paths of radiating
orthogonally intersecting curves. The completeness of the generalization is
somewhat marred by the consideration that the most remarkable feature of all
would be the fact that the plant body, out of the infinite variety of curves,
should be so prone to express its form in terms of conic sections. The fallacy
is at once suggested, that such plant-curves only approximate these conics to
the .eye, merely because the eye may be prejudiced in favour of such compara-
tively simple curves in that they are the first curves to be studied mathematically.
From such doubtful premises, Sachs deduced the law of orthogonal intersection
of cell-walls ; the latter fact may be perfectly true, and there appears to be in
fact so much physical evidence in support of the view that it may be instead
taken as the real starting point for determining the nature of the main curve.
Thus, if a section is mathematically circular, the anticlinals must be radii,
if elliptical they must be hyperbolas, if parabolic the anticlinals must be
confocal parabolas in the reverse sense, but it is first necessary to prove the
circle, ellipse, or parabola, as the case may be. There may be an infinite
number of curves which look like these much-studied conics, but it does not
follow that they exist in the plant until their mathematical equations can be
studied from physical data. Thus Sachs grasped the idea that the construction
and segmentation of the plant into layers of cells was only a form of the same
general action of forces which produces the thickening deposit of cell-walls and
the layering of starch-grains, That the orthogonal construction lines of these
GEOMETRICAL REPRESENTATION OF GROWTH. 35
Since the longitudinal section affords no clue, it is therefore
necessary to fall back on the transverse components of the growing
system.
A transverse section shows a simple concentric circular structure
in which cell-walls follow the paths of circles and radii, intersecting
therefore orthogonally. That is to say, the circles and radii re-
present reciprocal paths of equal action, and since the protoplasm
is asemi-fluid mass, such paths may be compared to the lines of equal
pressure and flow in a plane circular system.
Thus, if fluid films are laid down in connection with radial lines
of equal pressure, the periclinal walls will be established, and may
be subsequently fixed by a deposit of cellulose. In the same
manner, because the anticlinal walls follow the paths of radii, it
follows that their position results from another uniform action
along the circular paths. These orthogonal paths are interchange-
able, and what can be said of one can be inferred of the other.
The formation of anticlinal and periclinal walls in such a theoretical
apex may be considered therefore as resulting from two motions in
the tuid protoplasm, one a radiating current, the other a free circular
vortex. Main current movements of protoplasm in the whole
growing apex, apart from subsidiary currents in individual cells,
may thus be regarded as following along the general lines readily
observable in single cells, and known respectively as movements
of Circulation and Rotation. The diagram for the paths of equal
action in a transverse section of an apex would be the same as
that for the circulation and rotation of protoplasm in an isolated
spherical cell, and the mechanical law underlying the geometrical
construction of Sachs for the orthogonal formation and intersec-
tion of cell-walls would be that such orthogonal paths represent
the geometrical consequence of the fact that lines of equal
pressure and flow in a fluid medium are mutually at right angles.
bodies might be due to crystallization formed the keystone of the Micellar
Theory. With such a standpoint it is the more remarkable that Sachs did not
explain the layering of the tree trunk along the lines of an ovoid starch-grain,
and did not note that the small end of a typical starch-grain is equally indis-
tinguishable by the eye from a parabola, and presents an equally good imitation
of the construction lines of a growing apex.
36 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS
The same will hold good for the radial longitudinal section, and
since lines of equal action are here marked out by orthogonally
intersecting cell-membranes, it is possible, though not at all essential,
that these curves may be parabolas.
But although in dealing with a semi-fluid protoplasmic mass in
which movements are undoubtedly taking place one is tempted
to use the terminology of lines of equal pressure and flow, it is
clear that no definite movements, implying any considerable trans-
fer of material along any radial or circular paths, can be estab-
lished throughout the multicellular apex characteristic of vascular
plants, or even in the coenocytic apex of the Siphoneae. The
same orthogonal paths would in a plane system of electrical con-
duction mark out lines of equipotential and current flow, the two
sets of phenomena being in fact only special cases of the general
proposition of the distribution of energy along interchangeable
orthogonal paths.
The more ambiguous term, paths of equal “action,” may there-
fore be used in preference to any other terminology, although in
mapping out such systems the phenomena of the more obvious
vortex-motion of a fluid may be used metaphorically, and as a term
implying a definite geometrical construction.
Again, circular symmetry is clearly secondary ; all lower plants,
the majority of Algae, Bryophyta, and toa certain extent Vascular
Cryptogams, present asymmetrical growth at the apex, due to the
fact that new lateral members, in the form of single cells, can
only be added one at a time; this being especially well seen in the
growth of filamentous cellular algae.
A transition to a more bulky stage is accompanied by the de-
velopment of an initial cell cutting off segments in serial lateral
order, the three-sided apical cell of the Fucaceae and most Mosses and
Ferns being the most typical case.
Primarily, then, it may be said plants possess asymmetrical growth
as a necessary consequence of the limitation of new members to
serial succession of individual units, and that the symmetrical con-
dition, in which new cells are added at the apex in all directions
contemporaneously, is a secondary phenomenon, evolved.as a
distinct improvement on the older method in correlation with the
GEOMETRICAL REPRESENTATION OF GROWTH. 37
more perfect and uniform production of a radially symmetrical
axis.
In other words, the “ circular-vortex” construction of the sym-
metrical apex is secondary, and must be regarded as a special case
of a more primitive “spiral-vortex” construction, which is not,
however, necessarily a peculiar property of protoplasm, as assumed
in the original conception of the spiral theory, but the mere ex-
pression of asymmetrical growth.
In dealing with the spiral development of lateral members, it is
therefore necessary to take as a starting point the more general
case of a Spiral Vortex, rather than the circular one implied by
the typical Angiosperm apex of Sachs.
In such a spiral vortex, the stream lines are logarithmic or equi-
angular spirals,* which only reach their pole at infinity, and lines
of equal pressure and flow will be marked out by the paths of
orthogonally intersecting log. spirals. In other words, each circle
becomes a coil of a log. spiral, and the radius is represented by a
portion of that log. spiral, which cuts the other orthogonally.
Just as the circular-vortex construction is that of an ideal apex,
and is usually masked in any given specimen by secondary
phenomena of unequal growth and pressure of the component cells,
and possibly even at the theoretically initial group by subsidiary
vortices in the main stream, so the spiral vortex structure will also
be masked and almost obliterated. Beautiful examples of circular-
vortex construction persist in the loose and undifferentiated endo-
cortex of many roots (cf. Zea, Philodendron), while a more typical
root shows parenchyma more or less hexagonally packed.
The apex of the root of a Fern + affords a convenient example
* The logarithmic spiral is the curve whose polar equation is r=a’, where a
is constant. It is called logarithmic because another form of the equation is
log. r=@ log. a. The log. spiral has the property that the tangent at any point
makes a constant angle with the radius vector.
+ The tetrahedral cell of the Fern-root is here selected as an illustration, owing
to the fact that it is easily observable, fairly large, and in the large apex of a
healthy root a considerable number of segments can be obtained in a fairly level
series,
At the same time it must be clearly understood that the cell in question cuts
off a fourth segment in the sequence to form the root-cap. The exact series is
38 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
for the consideration of the spiral vortex plan and the disturbing
effect produced by secondary agencies.
As seen in transverse section, the tetrahedral apical-cell cuts off
similar members in serial order along a right or left-hand spiral,
which would clearly go on forming segments to infinity before
reaching the centre of the system.
Since by Sachs’ theory of cell-formation, and observations on
the phenomena of karyokinesis, every cell-segment halves the
initial cell, the system represents a type of dichotomy of the apex
in which successive segments are distributed throughout a spiral
system; and since the volumes of regular tetrahedra are as the
cubes of their edges, it follows that the new segment-wall would,
if the walls were plane, be formed almost exactly one-fifth * along
the side of the initial-cell, and the segments should be very
approximately five times as long as wide. Since the walls are
curved, and the exact curve unknown, it is not possible to get exact
data; but observation and measurement of the segments show
that such a ratio is very closely approximated in Péeris, as in other
thus broken at every third member in the transverse section. The segments in
order are not equally graded and do not form a true log. spiral. This can be
actually checked on a careful drawing ; the centres of successive segments do not
lie on a series of circles in G.P., but a gap is left at every third one. Measure-
ments of the relative length and breath of the segments show the same fact.
The general plan of construction is, however, sufficient for an illustration, and
for practical purposes the section might be assumed to be that of the apex of the
stem of Hquisetwm, which is unsuitable owing to its sharply conical form, The
exact shape of the tetrahedral apical-cell of the Fern-root is still doubtful. It
is clear that it cannot be contained by four confocal paraboloids of revolution,
since these curves would intersect at 120° and not 90°; and all four faces appear
identical. Nor can the section be formed by the intersection of three circles at
90°; the figure is obviously dissimilar. It is probable that, at any rate, so long
as it is actively dividing, and the asymmetrical construction follows the plan of
a spiral vortex, the three walls seen in section must be planes of equal action in
such a system, and therefore also as seen in section log. spirals intersecting
orthogonally. Such a construction would follow the lines of the diagram more
closely than any other.
*Ratio= 3/9 . 32-1
= 1:2597838 : 259783
=1 : 206.
GEOMETRICAL REPRESENTATION OF GROWTH. 39
three-sided apical cells (Aimanthalia, reproductive shoot); and
similarly, successive segments, as they continue to grow, notwith-
standing displacements of the segment ends, retain the same ratio
very approximately.
The segments cut off from the apical cell, therefore, are very
approximately, by actual measurement, similar figures, and conform
to the law of uniform growth; the spiral series of such figures is,
therefore arranged along a log. spiral; that is to say, a line drawn
through the centres of construction of the segments would also
form a log, spiral; and if the cell-walls were determined only by the
lines of equal action in such a system, the cell-area would be most
simply mapped out by log. spiral lines, as in fig. 17.
But beyond the progressive increase in the size of the successive
members, no trace of the spiral remains in the construction; the
apex is committed to the formation of cell-members by a dichotomy
from the tetrahedral-cell, formed theoretically by three curved
segment-walls intersecting at right angles in an endodernal cell of
the parent axis; while no sooner are the segments cut off than
other forces come into play ;—each cell by its own individual growth
would tend to round off and become a sphere, but is prevented from
doing so by being in close contact with adjacent members; each
younger segment, again, is capable of becoming turgid at the expense
of an older one, and thus the apical cell retains all its walls convex
outwards, and each segment bulges out so that it is broadest in the
middle ; further, the orthogonal intersections of the segment-walls,
fairly obvious in segments 1, 2, 3 (fig. 18), forming angles of 90°,
90°, and 180°, are rapidly pulled into the symmetrical position, 120°,
120°, and 120°, as in segments 4, 5, 6; the orthogonal segments
thus early become irregularly hexagonal; while in the case of such
members—and the transverse section shows only twelve segments
or four complete coils—it becomes impossible to tell by observation
whether the symmetry has not become perfectly circular (fig. 18).
In the similar case of the stem of Hguisetwm, this secondary as-
sumption of circular symmetry is indicated by the formation of a
whorl of leaves from each cycle of three segments. Continued
formation of cell-membranes takes place orthogonally within the
primary segments, without: reference to the original spiral, and
40 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
thus five-sided cells are produced at the ends of the segments, which
on further orthogonal division render the system quite irregular
(fig. 18; 8, 9).
Spiral-vortex construction is thus almost entirely masked by
secondary phenomena due to—
(1) The shape of the initial cell.
i
Fig. 17.—Hypothetical log. spiral construction for 3 series of segments at an apex ;
construction for 3-sided cells by circles intersecting at 120° and 90°.
(2) The effect of the individual growth of the lateral members.
(3) The effects of tensions produced by fused lateral contact.
The effect of dividing up the initial cell into several initials
segmenting contemporaneously would be to render the symmetry
circular, and the formation of such a system of cell-division pro-
ceeding from a single initial in a tissue-mass, in which circular
GEOMETRICAL REPRESENTATION OF GROWTH. 41
symmetry is almost immediately attained, may be regarded as a
transition from the loose aggregate of lateral members in more
simple plant-forms to the symmetrical condition demanded by a
radially symmetrical bulky axis.
Now, whatever holds for lateral members of one cell only should
also hold for lateral members of cell-aggregates, and the principles
of symmetrical and asymmetrical growth in the production of cells
in Zone I. should apply equally well to the formation of emergences
Fig. 18.—Root-apex of Aspidium Filix-Mas.
based on these cells, which move over the paraboloid apex in the
same manner, but subsequently grow into more massive lateral
members, so that the same transition from a primitive spiral-vortex
construction to a symmetrical circular one should be traced in the
progressive specialization of phyllotaxis.
In the same manner, lateral members would be formed: (1)
along paths of equal action ; (2) in a mechanical system, independ-
ently of any aim on the part of the plant, with a view to their
subsequent functions; which (3) might require secondary alteration
42 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
to suit special requirements; while (4) secondary packing would
ensue as a result of unequal growth pressures, and (5) the same
tendency towards perfection of symmetrical construction of a radial
system would result in the substitution of secondary circular-vortex
construction for the primitive spiral system.
Although much of these deductions may appear at first sight
fanciful, it is evident that if such generalizations can be successfully
applied to the special case of leaf-distribution, the original con-
ception becomes much strengthened. Nor are the hypotheses here
put forward more imaginative than that of the paraboloid apex of
Sachs which remains incapable of proof, or his construction for the
apical cell of Pteris which does not satisfy the evidence of his
own drawings.
The proposition is then that the genetic spiral is a logarithmic
spiral, homologous with the line of current flow in a spiral-vortex ;
and that in such a system the action of orthogonal forces will be
mapped out by other orthogonally intersecting log. spirals,—the
“ narastichies.”
In comparing such a leaf-producing spiral-vortex with that of
the cell-producing vortex of Pteris, the differences will be due to
the absence of the disturbing secondary phenomena. Thus, there
is clearly no apical construction to be impressed on the series of
members, and the members again are wholly free from one another ;
they are so far at liberty to assume as far as possible the form of
spheres, but when formed in close contact they will exert lateral
pressures on each other which, when they cease to be orthogonally
distributed, will induce slipping. The theoretical spiral-vortex of
phyllotaxis will thus have one disturbing factor only, namely the
pressures due to any unequal individual growth of the component
members.
GROWTH.
So far, the mass of protoplasm constituting the growing apex has
been considered as a constant mass of fluid presenting radiating and
circular or spiral-vortex phenomena, as it appears, that is to say, at
any given moment of observation,
GROWTH. 43
It is, however, undergoing a constant increase by an expansion
throughout the entire mass; and restricting the diagram to the
plane circular expression of a transverse section, it is clear that such
uniform expansion must be represented by a circular meshwork of
similar figures, in which any given zone of particles in unit time
increases to the next outer zone of the same number of particles of
similar character. Such a construction may be represented by a
circular network of “squares” formed by the intersection of an
indefinite number of concentric circles by a constant number (7) of
radii. If “circles” be inscribed in the “square” areas the con-
struction becomes more obvious (fig. 19), since any given “circle”
must expand to the one next it on the same radius, which subtends
the same angle, and the whole system expands uniformly in all
directions. Thus, if two concentric circles are taken infinitely near
together, the space between them can be divided into infinitesimal
figures, which, even if magnified to finite size, would differ
infinitesimally from squares. For small distances, when n is very
large they may be regarded therefore as practically squares.
It follows from the construction that the concentric circles are
in geometrical progression, while the areas of the similar figures,
“ squares” or inscribed “ circles,” are also in geometrical progression
along the radial paths.
The law of uniform growth is therefore expressed by a geometrical
progression and not an arithmetical, and the fact that the parabola
of the apex-section cannot be considered compounded of a trans-
verse growth-velocity is so far evident.
Sachs constructed his diagrams on the basis of arithmetical pro-
gression, and, regarded from a geometrical standpoint, it is evident
that such a construction is correct for mature plant organs. Thus,
on comparing the structure of a plane circular plant such as Coleo-
’ 154chaete (fig. 87) with a theoretical construction, the cell walls are
marked out by radii which intersect concentric circles orthogonally,
and these latter increase by equal increments from the centre
outwards. But a little consideration shows that such construction
is not the result of uniform growth, but is the expression of the fact
that individual cells attain a certain constant bulk and then stop
growing. The plant is thus not continuing to grow throughout its
44 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
whole area, but only at the periphery, and the construction by
circles in arithmetical progression is therefore the expression of
peripheral growth, since if all the cells continued to grow equally,
they would form a series in geometrical progression and no new
radial walls would be laid down.
The several cases of symmetrical and asymmetrical construction
in an apex presenting uniform growth may now be considered in
order, commencing with the symmetrical forms, since these present
the simplest diagrams.
Restricting the diagram to a plane expression, it is clear that a
circular-vortex will be represented by concentric and radial series
of similar figures; a spiral-vortex by similar figures arranged along
intersecting logarithmic spirals.
If orthogonal figures (“squares”) are used in the circular con-
struction, they will also be represented by “square” areas bounded
by log. spirals in the spiral-vortex.
The curves (“circles”) inscribed in these areas, which approach
true circles as the “squares” approach true squares, may be repre-
sented by inscribed circles, the difference being within the error of
drawing when the angle subtended is small.
APPLICATION OF SPIRAL-VORTEX CONSTRUCTION. 45
IV. Application of Spiral-vortex Construction,
1. ARRANGEMENT A,
UNDER this heading may be comprised the simplest construction in
which the lateral members are formed in a free circular vortex, and
present the appearance known as “superposed whorls.” If the
members are formed freely on the paraboloid surface of the cone of
growth and exert no pressure whatever on each other, they will, if
destined to produce radially symmetrical organs, tend to assume
the shape of spheres, and may be thus represented by concentric
series of circles.
Such a construction is, however, rarely met with; it occurs in
flowers (ef. Primula), and is in such cases commonly regarded as
of secondary origin. Expressed in terms of single cells, it occurs
frequently in the endocortex of roots (ef Zea Mais), in which the
absence of any considerable pressure from the peripheral layers
enables the cells to retain the original orthogonal system in which
they were developed.
The mechanical construction of the system, again, as expressed
geometrically, indicates that if the mutual pressures of the component
members are equal for every member, no disturbance of the system
can take place. The circles will tend to become squares, this being
again well seen in Zea root; but no hexagonal packing can be
initiated unless some additional force is brought into requisition ;
such, for example, as may be seen in the Zea root, where unequal
growth and the pressure of the outer layers induce hexagonal pack-
ing in the exocortex when the endocortex may still remain’
unaffected.
46 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
ss.
ieee
Fig. 20.—Systems of uniform growth: Scheme B ; asymmetrical.
APPLICATION OF VORTEX CONSTRUCTION. 47
2. ARRANGEMENT B.
As previously indicated, the arrangement of superposed whorls in
a circular-vortex system can only be regarded as a special case of the
more general asymmetrical form of a spiral vortex. In this latter
system, therefore, the primordia take the arrangement of “ swper-
posed cycles.”
The circles become log. spirals, and the radii also portions of log.
spirals intersecting the former orthogonally.
As the spiral approaches a circle, the difference between the two
constructions will be scarcely observable to the eye, if a portion
only of the coil is seen; while even when the deflection of the
main spiral is considerable, as in fig. 20, where the width of a
member is gained in one quarter of a complete revolution, the
deflection of the “orthostichy” lines is almost imperceptible to the
eye when the construction lines are omitted. This figure thus
illustrates very forcibly the standpoint that lines which appear per-
fectly straight to the eye may still be very definitely spiral, and
the orthostichies of Schimper and Braun may have no real basis.
On the other hand, a glance at the construction of such a typical
phyllotaxis system as that of a capitulum of Helianthus is sufficient
to show that this type of spiral construction does not obtain in the
plant. Nor will any amount of mutual pressure in the primordia
produce any change in the system beyond squeezing the spheres
into cubes, since they are by construction orthogonally arranged ;
although it is conceivable that additional external pressures might
produce secondary hexagonal packing.
Since, then, such a spiral construction will not meet the require-
ments of normal phyllotaxis, an alternative method of orthogonal
arrangement may be considered.
3. CONCENTRATION SYSTEMS.
That the primordia at the apex of a growing stem were “con-
densed.” into a confined space, in order that they might be more
48 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
securely protected * in the bud, has long been a favourite biological
explanation of bud-structure; to the common disregard of the fact
that actively dividing cells are capable of sustaining enormous
pressures from the surrounding tissues; and that as shown in the
apex of Pteris, younger cells can grow and remain turgid at the
expense of all older ones. The case of pressure against a sclerosed
framework may be considered separately, but as far as parenchy-
matous structures are concerned, there is no reason to suppose
that primordia are not strong enough to resist all the pressures
that can be brought to bear on them in the bud, and the greatest
pressures are of their own making. As already indicated, so long
as they are formed in orthogonal series, all such mutual pressures
will only tend to alter their shape but not their arrangement.
From this standpoint of a special packed system, Airy formulated
a scheme of phyllotaxis, in which all systems were to be derived
from a type presenting a constant “ideal” divergence angle by
longitudinal compression. The same idea has been put forward
by Schwendener,t and his first figure illustrates the action of a
vertical condensing force on a spiral series of the Schimper type,
the natural effect of the latter being to change an orthogonal
system into a hexagonal one. Without going into further detail
as to Schwendener’s standpoint, or considering how such a vertical
condensing force could be obtained at a plant-apex, the problem
may be attacked in a different manner.
If a set of equal spheres be arranged in orthogonal series, all
forces of contact will act at right angles to the curved surfaces,
in this case circles, and will be represented by the sides of the
exscribed square areas. The whole system is in equilibrium.
But since the forces acting along the sides of a square are also
represented by the resultant forces along the diagonals, it follows
that the same contact pressures will give rise to lines of equal
pressure in a secondary orthogonal system. In other words, two
methods of arrangement are interchangeable (fig. 21) and equally
in equilibrium without any disturbance of the original forces.
The diagonal arrangement is equally in equilibrium as is the
* Of. Airy, Proceedings of the Royal Society, vol. xxii., 1874, pp. 297-307.
+ Mechanische Theorie der Blattstellungen, Leipzig, 1878, Taf. 1, figs. 1-4.
APPLICATION OF VORTEX CONSTRUCTION. 49
vertical one, and the plant is at liberty to choose either construction
along two sets of orthogonally acting lines of pressure in the same
system.
4. ARRANGEMENT C.
Such an arrangement, the familiar case of “alternating whorls,”
is, in fact, the one found in the vast majority of symmetrical plant
constructions. It presents all the advantages of a “condensed”
Fig. 21.—‘‘ Concentration ” system.
system, in that the same number of elements is so arranged that
it is shorter longitudinally and broader transversely. The geo-
metrical construction of fig. 21 shows that in such a simple case
the shortening of the system will represent a vertical gain of
6:3,/2::2: J2::100: 70-7, or a gain of up to 30 per cent. as
expressed in the number of internodes which can be laid down
on a given apex.
Without insisting on any such biological interpretation, as that
D
50 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
the plant actually selects such an arrangement in order to secure
this gain of concentration, the fact remains that the system is the
one found in the vast majority of whorled types, which are again
regarded as secondary derivatives from an ancestral asymmetrical
condition.
In selecting this alternative whorled construction (fig. 22), note
that—
(1) The system remains orthogonal, and is thus equally de-
pendent on lines of equal pressure.
(2) No additional pressure is required beyond those already in
action in arrangement A.
(3) However great the growth pressures of the equally developing
component primordia, no hexagonal packing will be produced, but
the circles will approach square rhombs, unless, as before, growth is
unequal, or takes place inside a closed system.
Again, since the whole bulk remains constant, the arrangement
is not due to any condensation, implying packing, but may be
perhaps better expressed by the term concentration.
When expressed in the form of a plane circular diagram (fig. 22),
the diagonals become orthogonally intersecting log. spirals in that
they cross the radii at a constant angle, and a suitable number of
these paired spirals will in turn map out the system by their
orthogonal intersections, which will give figures which are in the
limit squares with inscribed circles.
5. ARRANGEMENT D.
The general case of spiral phyllotaxis is now reached by taking the
asymmetrical expression of the construction C. The whole system
remains orthogonal, but is expressed by a spiral vortex-construction
in which the genetic spiral remains as in B the grand log. spiral
current line. The radial “ orthostichies’’ will intersect this spiral
orthogonally. The two symmetrical log. spirals which mark out
the paths along which the members are actually laid down in a
concentration system become an asymmetrical pair and form the
Contact-Parastichtes of the system. When the genetic spirals
APPLICATION OF VORTEX CONSTRUCTION.
Fig. 23.—Scheme D: asymmetrical.
51
52 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
cannot be traced, and the orthostichies are equally incapable of
observation, the whole system is plotted out by taking the observed
number of parastichies and expressing them as mutually intersecting
log. spirals.
- A method is thus attained which gives perfect mathematical
expression to such a system as that deduced from observation of the
Pine cone or Sunflower capitula, so perfect that any deviations
from it in the actual plant must be due to the influence of some
extraneous force not yet considered.
The following construction which expresses a (34+455) Sunflower
head may be taken as a type :—
Since the whole construction hinges on the diagrammatic
representation by intersecting log. spirals, a simple method is
required for drawing these curves with a degree of accuracy which
will at least cover the error of observation; and, as the graphic
constructions will be found to afford sufficient geometrical evidence
of the truth of the method, it will not be necessary to include any
strict mathematical proof. A simple way of obtaining very accurate
results is as follows:—Describe a large circle and divide it into
a conveniently large number of parts (50-100); draw the same
number of radii through these points, and then, proceeding from the
circumference inwards, draw, with the same centre, a series of
concentric circles, making with the radii a meshwork of squares,
as near as can be judged by the eye. In such a circular network
of squares, arranged in radial series in geometrical progression,
all lines which are drawn through the points of intersection in
any constant manner are logarithmic spirals, and when drawn in
reciprocal fashion intersect at al] points orthogonally, the simplest
case being that in which symmetrical diagonals are drawn across
the meshes, which gives, in fact, the preceding case for the structure
of alternating whorls (fig. 24).
An wnequal pair of curves may be selected by taking a diagonal
across two squares in one direction on one side, and across two in
a converse way on the other; by continuing these, two asym-
metrical log. spirals will be obtained having by construction the
ratio 1:2. By filling in such curves all round the figure, it may
be proved experimentally that by using the full number of short
APPLICATION OF VORTEX CONSTRUCTION. 53
curves (1), and half the long curves (2), an asymmetrical network
of similar “squares” will be obtained. To produce these results
the number of points into which the original circle was divided
must be divisible by two,
AOS
0 Pee
Me
Fig. 24.—Geometrical construction of log. spiral curves :
ratios (1 : 1), (1: 2), and (5 : 8).
In the same way a pair of curves in the ratio of 34:55 may be
obtained by dividing the circle into a number of parts, of which
34 and 55 are factors, and drawing two continuous curves across
the meshes in the ratio 34:55 on one side, and 55:34 on the
other. As these numbers are unwieldy, a simple method will
54 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
give equally correct results; the ratio 34:55 may be taken as
33:55 within the error of drawing in small squares, or 3:5, The
curves (3:5), taken in a circle divided into 90 parts, will give
results well within the limit of drawing the square meshwork
correctly. So close are the ratios of the stages of the continuous
1
fraction 1+1 that within the error of drawing any one will
1+ete.,
give satisfactory results; the error being considerable only in the
centre of the system, where the difficulty of measuring the squares
is also greatest. The true curve for 34:55, and in fact that for
the ideal angle of the continued fraction, lies between (3:5) and
(5: 8), and may be closely approximated from the ratios 3:5 in a
circle with 90 radii, or 5:8 in one of 80.
Such a pair of curves is, then, well within the error of drawing,
accurate for a (34+55) system, and may be used to map out a
spiral orthogonal construction; for practical purposes a pair of
curves may be cut out in card, fixed to the paper by a pin through
the centre of the circle, and used as a rule. By taking a circle of
radius equal to that of the curve pattern, and dividing it into 55
and also into 34 equal parts, so that one point may be common to
the two sets, and using the curves as a rule to mark 55 short
curves and 34 long ones, the whole circle will be plotted out into
a spiral meshwork of squares in orthogonal series, corresponding
to the parastichies of the Sunflower capitulum taken as a type, and
the plan may now be used as a check on the actual phyllotaxis
(fig. 25).
It is obvious that either the points of intersection may be re-
garded as the centres of construction of the lateral members, or
the square areas themselves as the actual members, if packing
is so close that no interspaces are left; and the appearance
of circular flower- primordia may be indicated by describing
‘circles in the approximately square areas. Regarding the point
on the circumference common to both sets as No. 1, the
whole system may be numbered up by Braun’s method, the
meshes along the short spirals differing by 55, those along the
long by 34,
APPLICATION OF VORTEX CONSTRUCTION. 55
In such a numbered diagram, the parastichies (34 + 55) are
observed to be by construction a concentrated system comple-
mentary to two other curves, the members of which differ hy 89
and 21 respectively.
Fig. 25.—Log. spiral construction for capitulum of Helianthus, (34+ 55)
vapitulum taken as a type: genetic spiral winds right.
That the orthogonal system adopted by the plant is the one
which makes most nearly for optimum concentration is shown by
the comparison of other orthogonal systems which would pass
through the same points of intersection. These systems may be
deduced from the functions of the summation series of numbers, or
taken from the diagram.
56 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Thus taking the parastichies noted, 34 + 55
and 21 + 89,
the same points may also (with others) be mapped out by
13 + 144,
or 8 + 233,
5 + 377,
3 + 610,
2 + 987,
and ultimately 1 + 1597, the least con-
centrated system.
Thus, on reducing the whole system to one grand spiral only,
which is, in fact, the genetic spiral, passing through 1, 2, 3, etc.,
the nearest approach so far to a true orthostichy line is the log.
spiral cutting the genetic spiral orthogonally, which passes through
1 and 1598, a point too near the centre to mark clearly, though
378, 611, 988, are indicated.
The error of attempting to define such a system by the Schimper-
Braun systém of fractions, in which it might pass as a 24 type, is
at once apparent: the 89 curve bears no obvious relation to the
genetic spiral, though it bisects the angle between the 34 and 55
curves. The system of intersections may be defined in terms of
any pair of these orthogonally intersecting spirals but not in terms
of one of each of two different sets. The only true expression for
the Sunflower head lies in taking the nearest numerical ratio,
which is that in which the squares, or the circular primordia
contained in them, are in actual lateral contact. There is in the
whole system figured no member radially superposed to number
one, and the construction is thus in agreement with Hofmeister’s
law; while other important deductions are—
(1) The genetic spiral follows the direction of the short curves (55).
(2) One coil of the “ genetic spiral” approximates a circle, almost
within the error of drawing the system.
(3) The false “ orthostichy” line (89) is very nearly a radius, or
within the expression of hypothetical “ torsion” in a specimen.
(4) The nearest approach to an orthostichy line (1597) is as
near a radius as the genetic spiral is to a circle.*
* For the true orthostichy line, ¢f. Mathematical notes.
APPLICATION OF VORTEX CONSTRUCTION. 57
(5) The concentration system adopted is the best possible under
the circumstances ; and
(6) As indicated by the proximity of the numbers, 34, 55, the
nearest approach to the symmetrical condition, when such a
summation series of constants has to be followed.
In the same manner, making use of the same curve, and dividing
the circle into the appropriate number of parts, the other systems
for the capitula of Helianthus may be plotted out, within the error
of drawing, and numbered up with identical results following the
geometrical construction; that there are no members radially
superposed to form a true “ orthostichy” line, so that each one of
the constructions naturally fulfils all the demands of Hofmeister’s
law. It will be noticed, however, that with a constant direction
for the curves, the “ genetic spiral” does not run the same way, so
that while in fig. 25 (Ps.=34+455) the genetic spiral was right-
handed, in similar constructions for (55+ 89) and (21434) it will
work out left-handed. Also, the lowest number always gives the
number of the longer curves, since in constructions in which the
longer curves are formed with the higher numeral the meshes,
although orthogonal, are not “ squares.”
Taking these Sunflower heads alone as matured structures, it
appears, then, to be evident that the axes bearing them have
impressed on them, at an early stage of their development in the
first zone of growth, a certain fixed ratio of curves which follow the
lines of equal action in the semi-fluid protoplasmic mass, the trans-
verse components of which may be represented by the construction
of a similar number of orthogonally intersecting logarithmic spirals.
The numbers of the curves employed and their ratio appear to
be an inherent property of the protoplasm of the plant apex, and
may vary from shoot to shoot, but within the same capitulum the
phenomena remain constant, except in so far as they may be
disturbed by secondary changes due to unequal rates of growth of
the members composing the system.
The numbers of the parastichies are the only constants which
define the system ; since, although the same points of intersection
may be plotted out by other related systems of spirals, it can readily
be proved by constructing the diagrams that the curves indicated
58 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
by the parastichies are the only ones which give these intersections
alone, 1.¢., give the minimum of intersections.
It remains to be considered why such numbers and ratios are
found, and to what extent the plant is restricted to such a series.
Fig. 26.—Log. spiral construction for system (8+13); with curve setting of
preceding the genetic spiral winds left.
Again, by using the same curve, and drawing eight long and
thirteen short spirals, a set of parastichies may be marked out which
will give equally correct results for the Pine cone (P. austriaca,
P. Pinea); and, on numbering the areas, they may be checked by the
actual specimen: all observed appearances are accurately imitated
in the diagram ; the axes of the rhombs of rhombs forming spirals
APPLICATION OF VORTEX CONSTRUCTION. 59
instead of radii as demanded by the Schimper-Braun theory, while
it is clear that any attempt to correct the Schimper construction
by demanding hypothetical displacement or torsion in the cone is
entirely unnecessary. The balance of evidence must fall on the
side which explains most facts with the least amount of straining.
Fig. 27.—The same, with inscribed circles as representatives of the primordia,
By inscribing “circles” in the “square” areas (fig. 27) the
diagram may be arranged to fit the case of spherical flower-
primordia, as in the inflorescence of Scabiosa atropurpwrea, in
which the bracts are omitted; while it is equally correct for the
androecium (8+13) of Helleborus miger, or for the bractless
60 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
spadix (8+13) of Anthurium crassinervium, so widely is this type
of construction distributed among flowering plants.
The diagrams are less satisfactory in the lower ratios, owing to
the fact that but few members can be represented on a circle, this
being correlated with the production of relatively large lateral
members over a greater vertical extent of a narrow axis; but they
are sufficiently clear to show (1) the approximate alternation of
successive cycles, (2) the fact that the first member of each successive
cycle falls on a spiral line and that there are no radial orthostichies
present. On the other hand, when parastichies are drawn through
the points of intersection of radii and circles as demanded by the
Schimper-Braun construction, these curves, which irritated Sachs,*
are seen to be neither log. spirals nor mutually orthogonal, and the
essential points of their construction are lost.t
The series of common phyllotaxis expressions can therefore only
be represented in terms of the intersecting contact-parastichies, in
the form: Ps.= (1+1), (1+2), (2+3), (8+5), (6+8), (8413), ete.,
in which the first number (the lowest of the pair) gives the long’
* Sachs, loc. cit., pp. 497-498 : “ Among the errors of this (Spiral) theory is the
one that the spiral arrangement of all organs on a common axis must necessarily
follow from its so-called parastichies.” ‘Even ordinary wall-papers show such
parastichies, and in the same way the arrangement of scales on the bodies of
fishes, of the hairs on the skin of mammals, and of the tiles on a roof, exhibit
such parastichies clearly enough.”
+Van Tieghem, Tratté de Botanique, vol. i. p. 63. A construction of a 2
system with the genetic spiral represented asa Spiral of Archimedes gives points
along 5 radii vectores which are the orthostichy lines of Schimper. Curves
drawn through the points differing by 2 and 3 respectively are again by con-
struction Archimedean spirals in the ratio 2:3. Such a simple spiral construc-
tion was evidently present in the minds of Bonnet and Calandrini in proposing
the original quincuncial system, and the fact that they observed that leaves did
not obey such a construction accurately was thus glossed over as a secondary bio-
logical phenomenon. Similarly all the divergence fractions of Schimper and
Braun clearly imply constructions by Spirals of Archimedes, and these spiral
systems are thus based on the fact that orthostichies are often fairly accurate to
the eye.
The Archimedean spirals, it is important to note, are based on hypothecated
orthostichies, and not the orthostichies on postulated Spirals of Archimedes,
Since, then, these spirals are usually associated with torsion phenomena and the
formation of screws, various torsion-hypotheses become superimposed on the
original unproven premises.
APPLICATION OF VORTEX CONSTRUCTION. 61
curves, and the ratio of the numbers indicates the degree of
asymmetry of the system.
Any attempt to express a fractional ratio leads to misconception
of the phenomena. Such a construction, however, can only be
applied to a system in which it is still possible to observe the
Fig, 28.—Log. spiral construction (3+ 5).
actual curves of contact, either on the mature plant or in the bud;
the Schimper system may still remain a plausible account of the
mature leafy axis, which has passed through a possibly uneven
zone of elongation, since it certainly supplies a demand; and, so
long as it is remembered that it is only an approximation, no great
harm will be done, and it will continue to be as useful as in the past.
62 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
6. HELICES AND SPIRALS OF ARCHIMEDES.
Mature cylindrical axes exhibiting spiral phyllotaxis in which
the lateral members are closely set and of uniform character,
present remarkably beautiful appearances of helices with parallel
screw-threads winding in converse directions (cf. figs. Stangeria,
(29), Cereus (30), Huphorbia (31), Araucaria (32).).
Such spirals with equidistant coils continued upwards on a cone,
would on the unrolled surface constitute portions of Archimedean
spirals as pointed out by the Bravais, and the projection on a trans-
verse plane would similarly give intersecting Archimedean spirals.
The fact that similar helices are produced by torsion action
apparently forms the basis of all torsion theories of phyllotaxis,
whether in the obvious form of Airy’s hypothesis or in the veiled dis-
placement system of Schwendener. As in other instances, however,
the same effect may be produced by widely different causes, but the
fact that the curves exhibited in phyllotaxis in the horizontal
plan may be spirals of Archimedes leading on to helices on the
cylindrical stem has been very generally accepted, and repre-
sented in diagrams in which concentric circles are taken in
arithmetical progression.
So far, in fact, as such curves can be judged by the eye the
approximation is very close, and not only so, but the curve drawn
on a specimen (cf. figs. 2, 3, 4) is clearly more like such a construc-
tion than the theoretical log. spiral system previously postulated.
Further consideration, however, shows wide differences; thus it is
clear to begin with, that the phyllotaxis helices observed on a shoot
are not torsion spirals in any sense, but are merely the result of a
uniform development in both lateral member and internode
whereby a certain constant volume is reached and then further
growth is checked. The helices are thus not produced by the uni-
form growth of all the lateral members which are initiated at
different times, and would, if the rate of growth were constant in
all, remain always unequal, but they are the result of a progressive
cessation of growth,—that is to say, the helices are of secondary
origin, and any spiral series of members, whatever the primary
PLATE IX.
Fa, 80. —Cereus ehilensis (7-+ 9).
(445
Staminate inforescence
Fie. 29. —Stangeria paradoaa,
PLATE X.
(F4 §) vsopnpunpling matoydug— "Le Ola
HELICES AND SPIRALS OF ARCHIMEDES. 63
spiral curve may be, must necessarily pass into a parallel screw
thread type when all the members become and remain equal.
If the helices are secondary productions, it is very possible that
the Archimedean spirals which would represent them on a plane
system are equally secondary. The fact that a stem may go on
producing leaves to infinity, without producing a terminal member,
and that the leaves develop as similar primordia, is alone sufficient
to suggest that the genetic spiral is a log. spiral, rather than a
spiral of Archimedes which winds directly to the centre of the
system and allows for no further development.
From the equation to the spiral of Archimedes (r=a0), by
taking a as different values of the 2, 3, 5, 8 series, while r and 6
are constant, it is easy to construct a series of spirals to correspond
to these ratios (fig. 33).
In such a series the intersections of successive members of the
series, drawn in the opposite direction, are seen to be, in accordance
with the closeness of the ratios 3: 5:8 : 13, ete., practically identical
within the limit of construction error.
A tracing from such a pair may therefore be used to map a
system corresponding to the data observed in the given plant, either
as a symmetrical or asymmetrical construction.*
In such a diagram it is at once observed that the intersections are
not orthogonal, and therefore afford no clue to the distribution of
pressures; while the rhombs are relatively much flatter at the
circumference, but become very steep towards the centre: so steep
do they become as all the spirals fall into the centre, that not only
cannot they be adequately represented in the diagram, but it is at
once obvious that it is impossible that such rhombs can in any way
indicate the structure of the actual primordia arising on a growing
apex, which are either isodiametric or elongated tangentially.
“ A familiar example of the former is seen in the chasing on a watch-case,
and will serve to illustrate the weak points of the system.
These curves also present a beautiful example of a subjective effect produced
by an indirect method of construction.
Engraved as wavy circles which have radii differing by a constant increment,
the sloping curves fall into series as spirals of Archimedes ; the number of waves
being constant in each circle, the construction is symmetrical and the spirals
thus appear equal in number in either direction.
64 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
For expressing the facts of actual development, Archimedean
spirals are therefore absolutely useless. But while this is so, the
progressive flattening of the rhombs at the periphery of the system
bears a remarkable similarity to the phenomena observed in the
progressive dorsiventrality of foliage leaves, and it is to this fact,
Fig. 33.—Geometrical construction for spirals of Archimedes, asymmetrical
and symmetrical pairs, a= 2, 3, 5, 8, 18, 21, 34.
combined with the production of members which attain a fairly con-
stant bulk, that the close approximation to an appearance of Archi-
medean spirals is due as the members attain their adult form.
In fact these spirals appeal to the eye, in the macroscopic
appearance of such a plant as Sempervivwm, or the cone of Pinus,
because the members observed cover a fairly uniform area; and
the parastichies approach Archimedean spirals in a transverse view
HELICES AND SPIRALS OF ARCHIMEDES. 65
for the same reason as the curves become helices on a cylindrical
stem. As soon as the section of the actually growing primordia is
observed, the resemblance to spirals of Archimedes vanishes.
Whether the log. spiral system is so far satisfactory or not,
it is thus quite evident that helices and spirals of Archimedes will
not satisfy the requirements of ontogenetic observation ; the ultimate
approximation to such curves is a secondary phenomenon; and
with the rejection of helices and Archimedean spirals, as implying
anything more than a subjective effect, must disappear all prejudices
in favour of the application of hypothetical torsion-agenctes.
66 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
V. Ideal Angles.
THE “IDEAL ANGLE.”
As previously indicated, the mechanical problem set the plant
in building up a system of lateral members is primarily dependent
on the fact that the phylogenetic tendencies limit the apex to the
construction of one member at a time; but, with this restriction,
radial symmetry is required in the structure as it progresses. The
corresponding metaphor would thus be the one of building a
cylindrical chimney, placing one brick at a time, and yet keeping the
top always level. To meet such a difficulty, it is clear that growth
must oscillate from side to side, and that Hofmeister’s law is a very
good expression of the phenomena observed.
From his fractional series of divergences, 4, 4, etc., Schimper de-
duced the “ideal angle,” and the brothers Bravais suggested that
this angle, 137° 30’ 27”-936, an angle irrational to the circum-
ference, might be regarded as the sole angle of normal phyllotaxis,
and the same line of argument has been followed up by C. de
Candolle. With the formation of other fractional series, other
“ideal angles” were added, and the importance of the first one
proposed became much impaired, while the possibility of there
being several “ideal angles” appeared very like a contradiction in
terms. All these angles followed from summation series express-
ing values of continued fractions of the type 1_
e+1
1+1
141, etc.
where z might be any whole or fractional number.
It has been noticed that a remarkable interpolation of the theory
IDEAL ANGLES. 67
of leaf-distribution, and the tendency to a biological interpretation
of phenomena, is responsible for the hypothesis that a nearer and
nearer approach to the “ideal angle” of each series implied a
better distribution of leaves in relation to their external environ-
ment, by preventing overlapping. The suggestion that biological
aim on the part of the plant may to a great extent control
the protoplasmic mechanism of phyllotaxis cannot be wholly
neglected; and the formation of a “concentration-system” has
already been placed in such a light, although it was not necessarily
accepted as proved. But it cannot be too strongly insisted, that
in any spiral, that is to say, any asymmetrical series, whatever
unequal ratios the parastichies may have, every system is equally
an ideal one so far as leaf-distribution is concerned, in that no two
leaves are ever vertically superposed within the limit of practical
observation and construction, a fact which follows from mathe-
matical deduction and geometrical construction by log. spirals.*
Every asymmetrical system equally obeys Hofmeister’s law, the
logical consequence of which is, again, that no superposition ever
takes place. The whole theory of an ascending series reaching to a
perfect type of leaf-distribution thus falls to the ground; and not
only so, but the symmetrical condition, which has been put forward
as possibly the true aim of the plant, implies an actual formation of
vertically superposed series of members, and therefore, according to
the original hypothesis of Bonnet, an immediate departure from
the maximum exposure. Nor is there any reason to doubt that
biological causes may induce such a result, when the maximum ex-
posure ceases to be the optimum; the remarkable production of a
decussate phyllotaxis in the assimilating shoots alone of types
which show other xerophytic adaptations being the most obvious
example.t
Wiesner,t who approached the subject from this very standpoint
of leaf-distribution, was led to very remarkable results.
He pointed out that the series in which w had a minimum value
* Of. note on Mathematical Orthostichies.
+ Cf Clematis, Labiatae, Euphorbia Lathyris, Jasminwm nudiflorum,
Crassula perfoliata.
t Flora, 1875,=Nos. 8/and 9.
68 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
(2) gave the most equal distribution of leaves on an axis, and made
use of the minimum number of members, Although Wiesner’s
views were thus brought into line with Schimper’s series, 3, 4, 2,
etc., and pointed to the ideal angle 137° 30’ 28”, it is clear that his
generalization implies much more than the simple statement that
this angle is better than any of the limiting angles of other series.
With a given number of leaves, this angle gives the optimum
approach to a symmetrical construction; that is to say, reverting
to the metaphor of building one brick at a time, this angle gives the
optimum method of oscillation across the system while laying down
the stated number of units, so that radial symmetry is most nearly
attained. Radial symmetry is, in fact, the grand aim, and not the
biological requisitions of leaf-distribution, which would be equally
well served by any other series, and, when unsatisfactory, may be
readily compensated by secondary zones of elongation either in the
main axis or in the lateral members themselves.
To suit the theory of Schimper, Wiesner made # = 2; but the
same results obviously follow when z=1, since the ratios 4, 2, 3, 2,
etc., are all complementary of those of the previous set, and the
limiting angle (=) of 360° = 222° 29’ 32” is the inverse angle
2
of 137° 30’ 28”.*
The objection previously taken to the Schimper-Braun theory
series of fractions was that they were used either to express angular
measurements which could not be measured, or orthostichies which
could not be proved to be vertical. It has now been seen that the
so-called orthostichies are, in all cases of asymmetrical phyllotaxis,
themselves log. spiral curves, and the divergence angles between
them are therefore contained by curved lines. In theory, the
angular measurements still hold,t but they not only become im-
i
* The curious fact that the ratio e :
is also that by means of which
5-1
v/ Z ) formed the subject of
botanical speculation on the part of Kepler in 1611. Ludwig, “ Weiteres tiber
Fibonaccicurven,” Bot. Centralb., 68, p. 7. ‘
+ Of. note on Oscillation Angles.
Euclid constructed the pentagon (sin 18° =
IDEAL ANGLES. 69
possible of observation, being contained by spiral curves, but also
of representation on a diagram when the curve equations are not
given. Similarly, the “‘orthostichies” cannot be represented on the
diagram until the form of the log. spiral is known. It has further
been shown that each of the determining ratios of the Schimper
series comprises two log. spirals which have, as a rule, no simple
relation to each other, so that neither can be drawn while each
is imperfectly defined.
The system can only be accurately planned by the parastichy
ratios, which, on the other hand, are much more readily observed
than an ambiguous orthostichy; while, in addition, the fact that
the curves used form a mutually intersecting orthogonal pair admits
of a simple method of geometrical construction.
The method of presentation by means of angles of divergence
and “orthostichies” must therefore be placed wholly on one side,
and it is, at the same time, clear that all observations on phyllo-
taxis constants, in which this method has alone been used to
determine them, are open to considerable error.
The parastichy ratios will therefore be alone used to define any
given system, and the normal system thus becomes:—
Ps. =(1+1), (1+ 2), (243), (8+5), (6+8), (8+13), (13+ 21), ete.
By tabulating these as simple ratios, the idea of angular
divergence is eliminated and a further fact is brought into
prominence :—
Leber s1
1: 2:11: OS
2y 3:41: 1960 L.5~
8: 5::1:16¢¢
5: 8::1:1:625
8212 2: 1e16Jb
13:21::1:1619~
and the limiting ratio
Bs f5-1 eee 1618
The simplest summation series thus implies a practical constancy
of parastichy ratios in its higher terms, while the axis and the
lateral primordia may be variable quantities.
Expressing this practically, in terms of the spiral-vortex
Ny
70 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
metaphor,—in any Angiosperm apex, whatever its bulk and the
relative size of the primordia placed on it, the same system of
curved lines of equal action is normally utilized, and the con-
struction may thus be planned, within an extremely small range
of error, with the same log. spiral curves, for all but the lowest
terms of the series; further, any other parastichy ratio which
approaches 1:1:62 may be built up with the same curve, but
By) es -L/
j
§5:%9— R
Fig. 34.—Table showing direction of contact parastichies, and the genetic 6 c
spa fo successive systems of the Fibonacci series. {3
special curves must be drawn for a closer hpproximation to a
symmetrical pair. ; I
The theoretical interest of the table lies in the fact, that if it is
the approximation of a certain ratio which is the essential point
in the scheme, the appearance of other parastichy numbers be-
comes conceivable so long as their ratio approximates 1: 1°618.
That such is actually the case may be checked in the case of a
certain proportion of the inflorescence of Dipsacus and Helianthus.
The generalization of Wiesner, therefore, when applied more
legitimately to morphological relations without reference to
\
\
\ ;
IDEAL ANGLES. 71
biological conditions, acquires a greatly enhanced value, in that
the ratios naturally adopted by the plant for its parastichies are
those which, being the successive terms of the simplest continuous
fraction 1
T+1
1+1
1, etc, give the optimum approach to symmetry in
an asymmetrical system.
Just as it has been previously shown that the plant in normal
asymmetrical phyllotaxis makes use of—
(1) The optimum concentration system.
(2) Those ratios of a set series which more nearly approximate
the symmetrical position of equality ;
so algo,
(3) It utilizes that summation series of ratios which allows the
optimum approach to radially symmetrical construction.
All these three factors appear then inherent in the protoplasm,
and wholly independent of extraneous forces. From the fact that all
of them are illustrated in Helianthus, for which a normal structure
was postulated, they may be regarded as constituting the funda-
mental principles of normal phyllotaxis; while cases in which
any one of them happens to be omitted or impaired may be re-
garded as secondary and induced by subsequent specialization or
degeneration.
OTHER SERIES.
While the series (1:2), (2:3), (3:5), ete, thus becomes the
normal system for asymmetrical phyllotaxis, the fact that other
continued series have been proposed, and are generally accepted,
remains to be considered. At the same time, it must be pointed
out that their recognition in virtue merely of the method of
“ orthostichies” is wholly unreliable, and it is only in those cases
in which the fractions have been determined by the method of
parastichies that the ratios can be regarded as correct. Many
such cases were recognised by the latter method by the brothers
Bravais, though more recently the orthostichy method has been
considered the most important (Schwendener, Weisse). Further,
since such cases, though widely distributed, are relatively less
72 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
common, and are met with in anomalous specimens of plants other-
wise exhibiting the normal series, and especially in plants growing
under unfavourable conditions of environment, it is extremely
doubtful whether the determination of any particular set of
parastichy ratios affords any adequate reason for constructing a
series to contain it.
Such series expressed in the Schimper form are—
a4, 4 % fo ve ee 99° 30° 6”
Bat dy By ate oe kl OP
ad, 4) Baws oe ow oe ba oP age
Go 2 8 Ts ek Ol Cm AIP
In the form of parastichy ratios—
a 1, 3, 4, 7, 11, 18, 29, 47.
1 9, 14, 23, 37, 60.
1, 5, 6, 11, 17, 28, 45, 73.
d. 2, 5, 7, 12, 19, 31, 50, 81.
Expressed in the former system, by turns of the genetic spiral and
indefinite orthostichies, the fact is elicited that the first three tend
to a smaller limiting angle than the normal series, while the last
(d) approaches a larger one; but when expressed as parastichies, it
becomes clear that from the standpoint of an approach to symmetry
they form very uneven systems. In the higher terms the ratios
are practically identical with those of the normal series (1 : 1:6) :—
(@) (0)
3: 4::1:1:3 4: 5::1:1:25
4: 7::1:1°75 5: 9::1:18
Gell ssl oT 9:14::1:15
11:18::1:1:63 14:23::1:1-642
18:29::1:161 23 :37::1:1°608
29:47::1:1°62
(¢) (d)
526 221212 2: 5::1:2°5
6:11::1:1:83 Be Pred ed
11:17::1:1°54 7:12::1:1-714
17:28::1:1:647 12:19::1:1:583
28:45 ::1:1-607 19: 31::1:1-632
IDEAL ANGLES. 73
and this is borne out by the observation that these higher numbers
are found, intermingled with the normal series, in such inflorescences
as those of Helianthus, and especially of Dipsacus fullonum.
Others are met with in the leafless Cactaceae, bractless spadices of
Aroids, and xerophytic types, species of Lycopodiwm, Sedum,
Euphorbia, ete.
Only in some of the lower terms is any marked advance toward
symmetrical curve-construction exhibited; eg. (3:4), (4:5), and
(5:6). Cases in which such types occur are again no commoner
than even closer approximations to equality, for which the necessity
of constructing continuous fractional expressions is still less apparent.
Thus :—
Parastichies (3+4) | Ratiol : 13 Echinocactus Willcamsit (with 2/3).
( Stangerta paradoxa, g cone; ¢
? cone also 4/5/9.
(25) a cay : Echinocactus Wr eitiamsit
| Pothos spadix (Bravais).
(5-+6) 1-2 | Cereus candicans.
” LEchinopsis tubtflora (varies 6/6).
( Acorus gramineus, spadix (with
7/7).
| Raphi Ruffia, fruit scales (with
F 7
(6+7) » 116 |5 Batlnopss multiplec (with 6/6,
| 6/8, 6/9).
Echinopsis tubiflora (with 5/7,
\ 6/6, 6/8).
(7+8) ‘5 1148 oe a Eyriestt (with 7/7, 8/8,
7/6, 6/9).
(8+9) 9 1125 | Richardia africana. and 9
on spadix.
(9+10) | 5, vi
(1On1T) | a Tt
(11 +12) 4 1:09 Cyathea Dregei (1 specimen, C.
de Candolle).
(12+13) 4s 1:083 | Hchinops dahuricus (small capit-
ulum).
(13+14) a 1:079 | Echinocactus sp. secondary effect
by addition of new ridges.
(14415) 6 1071 | Acorus Calamus (with 15/15).
(15+16)] ,, 106
Also 13/15, 13/16, with 16/16, in Echinops capitula, 22/23 in ? flowers of
a spadix of Pothos.
74 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
The exceptional occurrence of such parastichy ratios has the
greatest interest when taken into account with the possible
existence of intermediate conditions between the case of normal
asymmetry and true symmetry, since the fact that such may be
found might throw light on the causes which tend to induce sym-
metry; but, clearly, nothing is to be gained by formulating
hypothetical fractional series to contain them.
Symmetrical construction is quite definite and stands by itself;
the normal asymmetrical series, again, selects the optimum
ratios for the construction of an asymmetrical system to the best
advantage within certain restrictions: other ratios may requisition
a more symmetrical set of curves, but at the expense of an inferior
working angle (¢/: fig. 63, (6+'7).
Examination of the higher members of such series which necessarily
approach the ratios 1: (1°62) shows that they may be regarded as
composite systems in which two or more ratios of the normal series
are compounded: eg.
a9: 47= | "5h : Fes 17 28={ 74 : { 6
23: a7= {75h ee 19: s1={ 7} ee ete.
It is true that such manipulation of ratios is open to the objec-
tion that it is “playing with figures”; but from such relations it
follows that these systems might be expected to occur in plants
which also exhibit bijugate constructions; and, in fact, most of
them have been described in Dipsacus, the typical multijugate in-
florescence, while they occur as exceptions in Helianthus (fig. 54),
which again presents occasional bijugate capitula.*
The idea that there might be only one normal phyllotaxis series,
from which all others were derived by slight deviations, occurred
first to the brothers Bravais; and, when one recognises the possi-
bility of the addition of new parastichy lines, one at a time, or the
elision of one, as in the inflorescence of Dipsacus (figs. 38a, 6), just as
Cacti may add or lose ridges according to circumstances of nutrition,
it is clear that many of these so-called phyllotaxis constants must be
of local and secondary origin. Further discussion of such forms may
be left until multijugate types have been more fully considered.
* Of. Weisse, Pringsheim’s Jahrbiicher, xxx. p. 474,
ASYMMETRY. 75
VI. Asymmetry.
SYMMETRY AND ASYMMETRY.
THE whole subject of phyllotaxis is thus restricted to a question
of the symmetrical or asymmetrical growth of the plant-apex during
the process of originating the impulses which give rise to new
centres of lateral growth.
That the framers of the original spiral theory were undoubtedly
correct in demanding that symmetry is in all cases secondary, and
asymmetry the primitive condition, is shown by comparative mor-
phology ; and transition, if not actually ontogenetic, is clearly so
phylogenetically. All Phanerogams, Cryptogams and the bulk of
the Algze, with whorled series of lateral members, either commence
with an asymmetrical condition or show traces of it in subsequent
development; thus, among Angiosperms, in some types the vegeta-
tive system becomes symmetrical while the spiral condition is
retained in the reproductive members (Calycanthus); in others the
latter are wholly whorled, while the former retain the primitive
asymmetrical condition (Aguilegia) : even when both become wholly
whorled, the presence of a pentamerous flower, or a type derived
from such a structure (Labiatae), which is a form which does not
mechanically pack in the sense that 2-3-4 and 6—merous types do,
these being referable to rhomboidal and hexagonal systems, is
sufficient evidence of a primitive quincuncial construction.
Among lower forms, the Dasycladaceae alone present types in
which symmetrical construction possibly obtains from the earliest
stages (eomeris), and even in these the coenocytic structure is itself
admittedly secondary.
76 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Again, the impulses in the mass of growing protoplasm, constitut-
ing the first zone of growth, have no necessary connection with
the presence of cell-walls or any subsidiary current-movements
within the component cells of the apex, where these are
present.*
Thus, in Helianthus or Hippuris, each impulse, when first observ-
able, involves a whole group of cells; in Hquwisetwm several impulses
forming about one-third of a complete whorl of members can be
definitely localized as belonging to the derivatives of a single seg-
ment of the initial cell; while in the coenocytic Dasycladaceae,
the impulses are clearly independent of the non-existent cell walls,
but the phyllotaxis is none the less perfect in its symmetrical
relations, and bears comparison as a symmetrical concentrated type
with members exhibiting cellular structure. Thus Meomeris dumetosa
presents alternating whorls of thirty members, Equisetum Telmateia
30-40, Hippuris 6-12. Without insisting on any special dynami-
cal interpretation of the complicated phenomena exhibited by such
a living mass of protoplasm, it appears evident that the transverse
components of the forces involved in the production of symmetrical
phyllotaxis may be expressed by a diagram illustrating the uniform
motion of a free circular vortex, the construction lines of which are
* Modern researches (Véchting, Weisse, Schumann) have tended mainly to
the observation of the actual facts of ontogeny, on the lines laid down by
Hofmeister. But such methods have one weak point, they can only result in
the statements of the facts observed without giving any reason for such pheno-
mena, Thus in the case of the rise of a semi-fluid protoplasmic protuberance on
a similar semi-fluid mass, it is clear that the causes which led to the initiation of
such a formation are practically over so far as that protuberance is concerned,
as soon as it becomes visible, and other forces may come into play quite different
from the primary cause. Phyllotaxis is therefore concerned primarily with the
forces which produce new-growth centres at or below the surface of the proto-
plasmic mass of the growing point, and suggestions as to their modes of operation
can only be deduced from physical standpoints. The large broad apex (8-10
mim.) of a full-grown specimen of Aspidiwm Filix-Mas shows, in the early part
of the year, the primordia of the leaves of the next succeeding year already
commencing as slight elevations spaced out without any contact relations to one
another in the spiral series (5+8). The protuberances are visible to the naked
eye, without the section-cutting required for smaller buds, but no amount of
observation of the facts of development will explain the reason why these eleva-
tions appear in their appointed places. (Cf (8+5) system of fig. 35, Plate VIIL.)
ASYMMETRY. 77
circles and radii. When asymmetrical, as in the more general
theorem, the structure is illustrated by the phenomena of a spiral
vortex, and all the lines of construction are orthogonally inter-
secting logarithmic spirals.
In all cases the symmetrical must be regarded as due to second-
ary specialization of the asymmetrical case, just as the circle may
be regarded mathematically as a limiting case of an infinite log.
spiral curve.
In both systems, “concentrated” and “non-concentrated” con-
structions may be possible: consideration of the schemes (B) and
(D) shows immediately that all spiral types are more or less con-
centrated constructions, and that (A) and (C) are only the limiting
cases in either direction. The same fact is illustrated by the
numerical relations of the curves contained in the capitulum of
Helianthus taken as a type (fig. 15), in which (84+55) give the
optimum concentration (14+1597), the minimum. The optimum
concentration, produced in an asymmetrical system by the approxi-
mation of the number of intersecting curves in either direction to
equality, being thus a secondary effect of an approach to symmetry,
actual equality gives the perfect symmetrical condition of scheme
(C). The fact that this is the normal symmetrical case found in
plants, while it is also regarded phylogenetically as a secondary
specialization, is satisfactory evidence that the “ concentration” system
is after all not due to any hypothetical biological demands of bud-
construction, but the natural outcome of its evolution from a spiral
series in which the claims of symmetry are expressed by an approach
to equality in the number of parastichies as indicating orthogonal
lines of equal action.
The biological demand for a concentration construction is thus as
completely eliminated from the study of phyllotaxis as Bonnet’s
original biological demand for equal transpiration space has already
been seen to be unnecessary.
The terms “symmetrical ” and “asymmetrical ” are further prefer-
able to the older corresponding terms “whorled” and “spiral,” in
that it will appear, as Sachs suggested, all spiral appearances are
subjective, and not the representation of any spiral aim on the part
of the plant; while the term “whorled” can only paradoxically
78 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
include the symmetrical formation of one member only at a
node.
Four types may thus be distinguished—
(1) (Normal phyllotaxis) Asymmetrical concentrated.
(2) (Specialized phyllotaxis) Symmetrical concentrated.
(3) Asymmetrical least-concentrated.
(4) Symmetrical non-concentrated.
While the consideration of primary phyllotaxis will be completed by
reference to
(5) Muliyugate Types.
(6) Anomalous Types, including ratios other than those of the
normal series.
(To be continued.)
On the Relation of Phyllotaxis to Mechanical
Laws.
By
ARTHUR H. CHURCH, M.A., D.8c.,
Lecturer in Natural Science, Jesus College, Oxford.
PART II.
ASYMMETRICAL AND SYMMETRICAL PHYLLOTAXIS.
In the previously published chapters,* the theory was elaborated
that the arrangement of lateral members on a shoot-apex was
possibly the expression of the symmetrical or asymmetrical distri-
bution of growth-energy in the growing apex, and in a system
for which uniform growth was postulated, the appearances were
to be mapped in terms of the phenomena of vortex construction,
and represented graphically by the same geometrical construction
as the lines of equal pressure and flow in circular or spiral vortices
respectively.t+
That such conditions of uniform growth do not obtain to any
great extent in a plant-apex is sufficiently obvious, since the apex
is never absolutely plane, nor again do the curves seen in a
* On the Relation of Phyllotaxis to Mechanical Laws, Part I. Construction by
Orthogonal Trajectories ; A. H. Church, 1901, pp. 1-78. Cf. Note on Phyllotaxis,
Annals of Botany, vol. xv. p. 481, 1901.
+ The relation of logarithmic spirals to phenomena of growth is very neatly
expressed in a mathematical form, in that in two dimensions the logarithmic
spiral is the only curve in which one part differs from another in size only but
not in shape. This property naturally follows from the definition of an equi-
angular spiral, but it brings out very vividly the essential character of such
a curve as a line of growth,
F
80 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
section cross at right angles throughout the entire area. On the
other hand, the approximation to orthogonal intersection increases
as the centre of the system is approached, and on every curved
surface a portion may be regarded as a plane, if it be taken
sufficiently small. Although the conditions of uniform growth do
not obtain therefore in the whole system, it is a legitimate hypo-
thesis that they become more and more so at the initial growth-
centre, and that for practical purposes the laws of uniform growth
may for the present be assumed approximately constant on the
portion of the apex which is flat.
At the same time it is clear that the consideration of uniform
growth must precede that of varying and diminishing rates of
growth; and in the general discussion of primary phyllotaxis
phenomena, uniform growth may thus be assumed to obtain at
some point however small, at the apex of a plant, in the First
Zone of Growth in which new growth-centres are being initiated
independently of cell-structure, before the primordia they produce
become visible on the surface of the protoplasmic mass.
It was further pointed out that the necessity for a new method
of construction arose from the fact that, granted that such lateral
members as the leaves on a shoot arose as similar protuberances,
and under conditions of uniform growth would always remain so,
it was not mathematically possible to place a spiral series of such
bodies in contact on a plane surface in strict terms of the diver-
gence formule of Schimper and Braun, which were again originally
postulated for cylindrical systems.
Very close approximations to the true curve may be found in
shoots which show little longitudinal extension, and may be
plotted from sections of the plant in the rosette condition, For
example, in a section of the broad apex of a perennating rosette
of Verbascwm Thapsus (fig. 36), the curve drawn empirically
through the centres of the leaf-members is very similar to the
true curve, differing only in the fact that it is a little shorter
~adially.* ;
* In making such preparations a general method has been adopted which
appears to give sufficiently satisfactory results. Hand-cut sections of spirit-
material are cleared in potash and Hau de Javelle, and thus restored very
ASYMMETRICAL AND SYMMETRICAL PHYLLOTAXIS. 81
Fig. 36.—Verbascum Thapsus, L. Transverse section of the apex of a perennating
shoot: system (3+5). Thecurve drawn through empirical central points of the leaf
sections approximates the true curve of uniform growth very closely (cam. lucid,
82 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS,
The special cases of phyllotaxis may now be considered under
the headings :—
(1) Asymmetry of the normal Fibonacci series.
(2) Symmetrical construction, in which the Fibonacci ratios
are lost.
(3) The special case of “ Least concentrated” asymmetry.
(4) Non-concentrated symmetry.
While separate sections will be devoted to—
(5) Multijugate systems.
(6) Anomalous systems.
Subsequent sections will include the consideration of secondary
disturbances in the primary system, the relations of dorsiventral
primordia, deductions from the mathematical investigation of the
log. spiral systems, and the relation of all these factors to the
interpretation of floral structures in the form of floral diagrams.
nearly to the volume occupied when fresh. By making a careful camera
lucida drawing on a large scale, about 12 ins. by 8, the construction lines can
be followed with a considerable amount of accuracy. The diagrams are
subsequently blocked in with a brush, as the contour is thus more accurately
kept than by a pen. The errors of the diagrams are due to—(1) contraction
of the spirit-material ; (2) error of the lens and camera lucida; (3) error of
cutting the section strictly transverse. The last mentioned, being judged only
by eye and hand, is clearly the greatest source of error, and can only be
eliminated by comparison of several sections. On such figures, distances can
be measured in millimetres, and angles to half a degree, with a reasonable
amount of accuracy. All the figures used as illustrations are much reduced
in reproduction.
NORMAL FIBONACCI PHYLLOTAXIS. 83
I. Normal Fibonacci Phyllotaxis.
THE type corresponds to the case of cycles in spiral series derived
by Schimper and Braun from the fractional series of divergences
by the assumption of slight hypothetical “Prosenthesis,’ and by
Schwendener from the same fractional series by equally hypo-
thetical “contact-pressures.”
It can only be strictly defined by the number of intersecting
parastichies, the ratios of which mark successive values of the
stages of the continuous fraction 1
1+1
1+1
T, ete., and is figured dia-
grammatically by the corresponding number of log. spirals drawn
with the appropriate curve tracing of the series 1, 1, 2, 3, 5, 8, 13,
21, 34, 55, 89, 144, ete.
From the fact that this is the system found in the Sunflower,
which was regarded as par excellence a normal plant, it may be
regarded as the normal type for all Phanerogams, without
necessarily implying that it is also the phylogenetically primitive
one.
Since the construction diagrams are correct well within the
error of drawing, and far within that of any actual observation on
the plant, geometrical plans may be utilised for the further investi-
gation of the properties of such a system.
Since also the spiral construction (Scheme D*) was derived
geometrically from the symmetrical case (Scheme C), and that for
all mathematical deductions from the latter case, homologous pro-
* Part L., figs. 22, 23, p. 51.
84 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
positions will hold for the asymmetrical condition, and since, again,
uniform growth in a plane circular system may be indicated by a
series of “circles” enclosed in a “square” meshwork, in which
successive “squares” and “circles” are in geometrical progression, it
follows that growth is equally uniform throughout the spiral
system, and successive members along any spiral path are also in
geometrical progression. These relations, following from the
system of construction, are indicated for a (3+5) system in
fig, 28.
Although circles have been inscribed in the orthogonal areas, it
is clear that the proper figure which only becomes a circle as the
orthogonal area becomes a square, is of an “ovoid” form,* while a
point which may be found for each area by drawing twice the
number of curves in either direction, and therefore represents the
intersection of the intermediate spirals, will at the same limit
become the centre of the circle. It may be now termed the “ Centre
of Construction” of the ovoid. These centres of construction fall on
circles the radii of which are again in geometrical progression, and
growth in each lateral member is uniform with that of the circle
which represents the main axis; and in such an expanding system,
circles drawn through the centres of construction of the lateral
members will indicate the relative size of the axis when the
member was laid down. Since growth is uniform throughout the
system, this ratio is a constant and may be used to define the
system; and for the same reason, the same diagram which ex-
presses the relative size of the developing primordia will also
represent the relative position of the areas in which the first
impulses originated as possibly mathematical points corresponding
to the centres of construction. The ovoid figures approach circles
so closely when the angle they subtend is small, that the error is
almost beyond the error of drawing; below 60° it is practically
unnoticeable, and the inscribed curves may be thus represented by
actual circles. At the same time it must be remembered that,
though these figures are in a spiral system ovoid curves in their
relation to the parent: axis, they represent growth-centres with a
partially individualised activity, and may therefore by their own
* Cf. Mathematical Notes, for the equation and construction of the true curve.
NORMAL FIBONACCI PHYLLOTAXIS. 85
inherent growth movements tend to become circular in section, as
for example they would if they were merely semi-fluid masses
contained in a uniformly elastic membrane.
A circle inscribed within the orthogonal area with a definite centre
of ws own, which is not the true centre of construction, may be thus
at es : af e
Fig. 28.—Log, spiral construction (3 +5). Lae
taken as representing the actual primordiwm within the error of
drawing or observation.*
Since also the true curve is an ovoid, and the ratio of distances
between the new growth centres must be measured along the
* That is to say, since comparison can only be made between similar figures,
Cf. Mathematical Notes.
86 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
spiral paths which pass to infinity, a method may be adopted which
gives a mean ratio with approximate accuracy, dependent on the
initial error of drawing the curve-systems; and although destitute
of any strict mathematical value, a simple method of comparing
the ratios of the lateral members to their parent axis is obtained, if
this secondary circle and growth-centre be regarded as representing
the lateral member, and affording a mean value of distances which
in the limit approach equality. Thus in the (3+5) system (fig. 28),
measurements may be taken in millimetres from a carefully con-
structed figure within the error of drawing the true log. spiral. In
the case figured, the radius of the circle representing member 1
was 22°5 mm., while the distance between the centre of this circle
and the centre of the whole system was exactly 45 mm., and since
all the orthogonal areas are similar, the ratio is a constant for every
member and equals 1 : 2.*
Taking this ratio as a sufficiently correct approximation, @
(3+5) system is thus the natural consequence of a spiral-vortex
construction when the diameter of the primordium of the lateral
member is one-half that of the axis at the point on which %# ts
apparently inserted.
Again, by using the metaphor of an actual spiral-vortex move-
ment, it is clear that the impulses which originated in the fluid
mass of protoplasm may be regarded as at first of the nature of
mathematical points from each of which a new vortex motion was
initiated, expanding in all directions until it came into contact with
adjacent vortices; further expansion in each would then result in
the heaping up of a mass of protoplasm which becomes the obvious
external sign of the origin of a new member.t
The diameter of the primordium is thus only the expression of
the distance between the new impulses, and it is the ratio of such
distances to the diameter of the main axis which is the fundamental
constant which determines subsequent phyllotaxis phenomena. As
* Of. Mathematical Notes. More accurately the ratio may be taken as
1 : 1:95, the amount of error in the geometrical construction being J,th.
+ Hofmeister pointed out that the position of new members might be
indicated by changes in the tensions of the superficial cells before any elevation
of a primordium was observable.
NORMAL FIBONACCI PHYLLOTAXIS. 87
there is no evidence that there is any real vortex movement in the
protoplasm, the hypothesis that the lateral primordia must be
necessarily produced in close lateral contact cannot be maintained
on these grounds, and it does not follow that the ratio of the
diameters of axis and primordium will adequately represent the
distance-ratio of the new impulses.* Thus in the broad flat apex
of Aspidiwm Filix-Mas, the new primordia arise obviously at points
spaced at a considerable distance from each other, and yet fall
along the well-defined paths of a (5+8) curve system. (3-+5), fig.
35+; of. also Senwpervivum, fig. 82; Nymphaea, fig. 94.)
Although in such cases the elevation of the protuberances may
be imperceptible at their periphery, it is probable that each
primordium is strictly localized from the first at or below the
surface, even if this is not obvious to the eye, and in the great
majority of cases the actual close lateral contact of the primordia
is undoubted. The ratio of the diameter of the axis to that of
the primordium arising on it may therefore be conventionally used
as a constant and may be termed the Bulk-ratio.
* It will be noticed that this view of bulk-ratio is an entirely artificial one,
and can only be useful so far as it is regarded as a convention which may make
discussion easier, Widely differing results are given when comparison is made
between the area of the true ovoid curve and that of the circle drawn through
the centre of construction, which is the true ceutre of insertion. Two standpoints
are involved : one, that of the completed system in which the bulk reached by
the lateral members may acquire some relation to that of the axis on which they
are inserted ; the other, that of the similarity of growth-extension from all
growth-centres. In the former case, each lateral member would he regarded as
possessed of a certain relative size to begin with, and the same view would be
adopted so long as the growth may be considered uniform, since one part cannot
grow faster than another. In the latter, granted an increased rate in the lateral
members, each lateral growth-centre would continue to expand uniformly until
contact was made with adjacent members ; ultimate extension would thus be only
limited by the distance between adjacent impulses ; and this for a (1+1) system,
for example, is relatively enormous.
The importance of such lateral contact has been emphasised by Schwendener,
and it obtains in the vast majority of constructions ; but it is also clear that ina
system for which the rate of growth is not uniform, the growth of a lateral
primordium might be so affected that contact may either never be established or
only be attained at a subsequent stage. (Aspidiwm.)
+ Part I. Plate VIII.
88 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Examination of similar diagrams, constructed with the same
approximately correct curve, shows that these ratios follow the ©
same ascending series.
Thus actual measurements gave :—
mm. mm.
(3+ 5) . 22:5: 45 or very approximately 1: 2
(5+ 8) ‘ . 18 : 546 7 ' : 1: 3
(8+13) : . 12:5 : 625 ‘ : : 1: 5
(13+21) ‘ . 85: 685 : : : 1: 8
(21434) : . 55: 725 : : : 1:13
(34+55) : . 88: 755 ; : 1:21
As the error of measurement increases with the rise of the series,
it may be assumed that (55+ 89) corresponds to 1 : 34
(89+ 144) is 1:55
(144+ 233) 1:89
With the lower ratios (2 + 3), (1 + 2), (1 + 1), the error of
construction becomes obvious, since circles cannot be inscribed
in the “squares” to represent the ovoid curves with sufficient
accuracy.*
, Lateral primordia are thus to be regarded as bodies presenting
a definite bulk; and since the last given ratios are the highest
known in the plant, it follows that, in dealing with bulky prim-
ordia, in contradistinction to the mathematical points and series
tending to approach an “ideal angle” of the Schimper-Braun
theory, such a limitation of the series while the bulk of the
lateral member is still relatively considerable, must admit of the
possibility of a certain degree of structural error in the systems.
This becomes more obvious when the bulk-ratio is expressed in
terms of the angle subtended by a primordium.
Since the ratio of the radii of the primordium and parent axis is
* A comparison of these values on a carefully constructed diagram three
feet in diameter, in which the error of drawing became apparent, showed that
these general results hold within a very trifling error; and that this error is
not much more than the difference between these results and calculated mathe-
matical ratios. (Cf. Note IV.)
NORMAL FIBONACCI PHYLLOTAXIS, 89
the sine of half the angle subtended, the previous series be-
comes :—
System.* Bulk-ratio. Angle-subtended.
(8+5 1:2 60° or roughly 60°
(5+8 1:3 38° 57’ 39°
(8+13) 1:5 93° 5’ 23°
(13 +21) 1:8 14° 20'
(21 +34) 1:13 8° 48’
(34455) 1:21 5° 26!
(55+89) 1:34 3° 24’
(89 +144) 1:55 | 2° 5!
There is reason to believe, however, that these highest systems
are never formed directly at the apex of a young axis, but are
gradually built up along similar curve systems; and the occurrence
of such a system as (8+13) asa direct formation is the limit of
the capacity of the plant. When such a system is regarded as
composed of primordia each subtending an are of 23°, the possi-
bility of a range of error amounting to even more than one degree
is at once apparent. It will not be necessary so far to regard the
plant as working to any divergence angle of very exact degree,
and the approximation in building the system to such an angle as
187°°5 may be very rough. This becomes still more obvious when
the ease with which the primordia make very considerable adjust-
ments is taken into consideration.
* Of. Mathematical Notes, 1V.
90 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
II. Constant Phyllotaxis.
Wir such an assumption that growth is more or less uniform so
long as the apex remains in the condition of Zone I., and absolutely
so at some central point at which the new impulses are originated,
it is possible to build up any system of phyllotaxis, the results of
uniform growth being expressed by taking circles whose radii are in
the requisite geometrical progression. For example :—Taking a case
in which the lateral members are assumed to have a bulk-ratio with
the main axis of 1 : 2,a (3+5) system may be produced by adding
one new member at a time, and allowing each to grow along the
same series of circles in geometrical progression.
Fig. 37, 1, shows the insertion of one such primordium. Now, if
the second member be laid down on the side exactly opposite the
first, the system assumes at once a symmetrical condition, and the
symmetrical construction of one member at a node thus induced,
giving rise to the phenomenon of exactly alternating leaves in two
rows, is general among Monocotyledonous types (ris, Canna,
Gramineae.) If the axis is growing asymmetrically, however, the
primordium is formed at an approximate angle of 137° on a log.
spiral system, the ratio of the curves of which approaches 1 : 1°62 ;
since this angle and ratio gives, as concluded from Wiesner’s obser-
vations, the optimum oscillation effect in constructing the nearest
approach to a radial system, one member at atime. The second
member thus falls (fig. 37, 2) on one side of the diameter passing
through the first one.
As it must fall either right or left of this line, it would appear
probable that the chances are equal in either case, and that in
CONSTANT PHYLLOTAXIS. 91
accordance with the laws of chance, the side on which it falls, which
Fig. 37.—Scheme for constructing a (8+5) system by uniform rate of expansion in
axis and primordia: the asymmetrical addition of one new member at a time
produces a subjective appearance of spirals, or may be regarded as the result
of a growth oscillation across the apex. // 4y °/
clearly indicates the course of the ontogenetic spiral, Right or Left,
should result in the fact that, given a large number of specimens,
92 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
right- and left-hand spirals should be approximately equally dis-
tributed among them.*
* Bonnet (loc. cit., p. 179) gives 48 Right and 30 Left as the result of the
examination of 73 Chicory plants, or 59:41 per cent.
Observations on Pine cones do not yield very conclusive results, thus :—
T. 100 cones from one tree of Pinus austriaca (B.G.O. 1900) were counted ;
the curves on the base of the dry expanded cone are (8+13), the
direction of one of the shorter (13) curves is readily checked:
comparison of the table (fig. 34) shows that in such a system the
genetic-spiral winds in the same direction as the long curves. As the
cone is viewed from below, the spiral of the diagram will be the reverse,
or the same as that of the short curves observed. Such a set of cones
gave 59 Right and 41 Left.
II. 100 cones from one tree of Pinus pumilio (B.G.O. 1900) showing similar
construction gave 53 Right and 47 Left.
III. 100 cones from one tree of Pinus laricio (E. G. Broome, 1900), the “8”
curves being marked gave 68 Left and 32 Right.
The inequality of these numbers may therefore be the expression of the fact
that homodromy is more usual than heterodromy, and further observations are
needed.
These have been made for the same trees by Mr E. G. Broome for 1901 with
closely similar results :—1000 cones from the same tree of P. austriaca gave in
batches of 100 each—
RL.
54:46 @ (1)
56 : 44 oo. (3)
51 : 49 : (3)
49: 51 . 4 ——
Z : . f Average 53°6 : 46-4
54: 46 (7)
56: 44 ‘ ‘ (8)
61:49 2, (9)
54:46 . (10)
Although very nearly equal, there was thus a distinct tendency in this tree
to form a right-hand spiral.
The case of the specimen of P. laricto was even more remarkable.
Four hundred cones were counted ; the first hundred gave a result identical
with that of the previous year, . 68 : 32
75 : 25
Successive batches of 100 each, . { 79:21
63. : 37
Or an average of 71 Left : 39 Right, a result in which the element of chance
appears quite out of the question, so far as the members of one tree are concerned.
CONSTANT PHYLLOTAXIS. 93
In the same way, the third member falls into the largest gap
between members 1 and 2, but asymmetrically, being on the side
of the gap farthest from the last formed member. Hofmeister’s
law, which is clearly a simple way of expressing what is observed
in such a series of diagrams, may be thus enlarged by adding the
point that the new member falls asymmetrically into the next
largest gap.
The fourth member establishes lateral contact with No. 1, and
thus initiates the first contact-parastichy—the long curve of the
system, while the sixth member similarly commences the first
short curve parastichy, the system being (3+5); the parastichies
thus appear as subjective spirals joining the centres of members
in orthogonal contact (fig. 37, 9).
The subjective spiral joining the centre of successive members
becomes the ontogenetic spiral (fig. 37, 7); but the growth move-
ment is equally clearly defined as an oscillation across the apex of
the stem at an approximately constant angle (fig. 37, 8).
The ninth member falling-on a new log. spiral line passing
through No. 1, gives the so-called “ orthostichy ” line of Schimper.
Of these spirals, then, the genetic spiral is of interest solely that
it marks the ontogenetic path of construction; the contact-
parastichy spirals are, however, structural and map out the system,
in that they may be regarded as representing lines of equal and
asymmetric distribution of growth-energy in the protoplasm of the
apex. These are not necessarily the cause of the appearances, but
rather both phenomena are equally expressions of the fact that growth
is asymmetrical, and therefore represented by the geometrical con-
struction of a spiral-vortex.
UNIFORM AND VARYING-GROWTH PHENOMENA.
It would so far appear that the inherent property of protoplasm,
which determines the phyllotaxis of lateral members, thus reduces
to the fact that in every shoot the bulk-ratio of the lateral
primordia, or rather the ratio of the distance between two initial
points to the diameter of the axis, may be a constant, and definite
for each shoot, though less definite as a specific constant. Of such
94 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
a fact the secondary parastichy curves are a necessary consequence ;
and the ratios that these exhibit present a remarkable analogy
to the well-known phenomena observed in root-apices, in which the
number of the protoxylem points which here subsequently deter-
mine the arrangement of the lateral members is also an inherent
property of the protoplasm, determined for each root-apex, constant
within a certain range for the species, but varying in actual branches
with the amount of nutrition: for example, the constant for the
typically tetrarch root of Ranunculus repens sinks to 3 and 2
in a weak lateral root, or rises to 5 or 6 in a strong one. In the
same manner, the foliage leaves of a shoot of a given plant may
be laid down with a low number of curves, eg. (2+3), but in
strong axes the ratio may rise to (3+5), and in reduced axes may
fall to (1+2). The commonly observed fact that the main axes
of many plants present a “2” phyllotaxis, while secondary lateral
ones reduce to “2,” as first noticed by Bonnet, is thus explained
by the assumption that the protoplasm of any asymmetrically grow-
ing shoot possesses a definite curve-system, usually of a low ratio,
and ranging for the great majority of plants between (2+3)
and (8+13).
In certain cases, however, as in the capitula of Helianthus and
other Composites, the phyllotaxis ratios reach very high numbers,
and these exhibit the same phenomena in correlation with the
nutrition of the axis in which they obtain. Thus, the medium
capitulum of Helianthus taken as a type, presents (34+55) par-
astichy curves, but in weaker plants these reduce to (214+ 34) and
(13421), while in exceptionally well-nourished plants the con-
struction curves are (55+ 89) and (894144). The same progressive
reduction in the curve system is also noticed in the capitula
terminating lateral branches of the tirst and second degree.
Either, therefore, these high ratios are determined by the apex
at its first formation, or a transition must take place from one
series to another; the former alternative might be homologised
with the production of a polyarch root in many plants, and from
this point of view, the shoot of Helianthus might be regarded as
containing a large number of potential curves, of which, if con-
ditions were unfavourable, only a few would be utilised; and,
PLATE XI.
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CONSTANT PHYLLOTAXIS. 95
when mature plants are alone considered, the fact that the total
number of foliage leaves and sterile involucral scales borne on a
stem is with few exceptions less than one complete cycle of floral
members,* as indicated by the parastichies of the disk, shows how
difficult such an idealistic conception may be to disprove.
On the other hand, seedlings of Helianthus,+ well supplied with
reserve food materials, evidently lay down their primary curve
system under very similar conditions, and a section of the terminal
bud of a young plant shows a low ratio of parastichy contact-curves,
while definite contact curves of higher ratios, but yet lower than
those of the disk, are seen on cutting across the involucral scales of
the capitulum.
It remains, therefore, to consider the preceding curve-systems
from the standpoint of—
(1) Transition from one series to another.
(2) The phenomenon already indicated as possibly representing
a “rising phyllotaxis ” characteristic of the involucre of Composites
(Helianthus, fig. 14; Cynara, fig. 9).
To take an example of the former, the difference between a
(34455) construction and that of a (21+34) might be expressed
by retaining the 34 long curves and reducing the short curves to
21—that is, by dropping out 34 of the latter parastichy lines of equal
growth. If this be attempted on a diagram, the parastichy curves
and the ontogenetic spiral still follow the same direction, but the
orthogonal areas are obviously no longer “ squares,” and would not
therefore contain the homologues of circles—that is to say, in the
plant the circular primordia would be distorted to broad ellipses.
The (21+ 34) system can thus only be represented by 21 long curves
and 34 short. By dropping out 11 long and 21 short from the
* Of. Weisse, Tables of Helianthus, Prings. Jahrb., xxx. p. 474.
t Seedlings of Helianthus annuus produce, beyond the cotyledons, usually
3-4 pairs of decussating foliage leaves. It is not clear why such a symmetrical
construction should be found in the Sunflower and subsequently converted into
a spiral system ; in allied species, however, the decussate system is continued
throughout the whole of the assimilating region, but ceases at the branching of
the inflorescence region (cf. H. strumosus and H. rigidus, garden varieties).
When the secondary change takes place the curves (2+2) become typically
(2+8) or (8+5), rarely (8+4) or (2+4).
G
96 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
(34455) system, uneven orthogonal figures will, it is true, be
produced, and it might be conceivable that the plant could adjust
these to circular primordia as growth proceeded, so long as the
direction of the curves remains the same, but it will be found that
the genetic spiral is reversed.*
“Ls
1:2 —R \
2:3 -b :
3:5 —
5:6 -L
8213 —
R
B:a—-b
21: 34 —
34:65 -L
55 1 Bq —
Fig. 34.—Table showing direction of contact parastichies, and the genetic
spiral for successive systems of the Fibonacci series.
There is clearly no evidence whatever that such reversal occurs
in normal growth, since the genetic spiral (Cynara) may be checked
* This remarkable property of the curved systems, by which the spiral selected
as the ontogenetic spiral is reversed in successive ratio-systems, is tabulated in
fig. 34 (Part I. p. 70). Comparison of the structural diagrams for (5+8) and
(8+ 18), for example, with the same direction of parastichy curves, shows that the
displacement of the first member of successive cycles in the latter case follows the
direction of the genetic spiral, but in the former is in the reverse direction.
These appearances may be readily checked on the dry cones of Pinus austriaca
(8+18) and Pinus laricto (54-8).
Such reversal will again lend additional complexity to phenomena of homo-
dromy and heterodromy in lateral shoots.
CONSTANT PHYLLOTAXIS. 97
leaf by leaf* If then in the same construction the genetic spiral
remains constant, the direction of the parastichy curves must be
reversed; although it is possible that the plant-apex might stop and
reverse to complementary lines of equal distribution, such an
arrangement would undoubtedly appear in the form of a sharp line
of demarcation between the cycles, and would be recognised, if the
system were continued to any distance, as a reversal of parastichies.t
There does not appear to be, then, any practicable method of
passing from one cycle to the next without distortion being appar-
ent. Thus, in the inflorescence of Dipsacus fullonum, a new paras-
tichy curve may be initiated in the middle of the capitulum
(fig. 38a), or curves may be dropped out one at a time (fig. 380).
The distortion is very obvious, and appears, in the former case,
to be initiated by the development of two primordia in the place
of one when the error of construction, due to the greater increase
of the axis bearing a constant lateral member, becomes sufficiently
large. An identical phenomenon, in which a new ridge is
added, or dropped out, probably in correlation with conditions of
nutrition, is afforded by the stems of Cacti, the distortion produced
being again considerable t (fig. 39a, 6).
From the standpoint of uniform growth, no transition from one
phyllotaxis series to another is possible without distortion. Any
such changes, therefore, when they occur, must be secondary, and
the compensatory allowances must render the distortion more or
less obvious.
Thus, given a certain ratio between the diameter of the primordiwm
and that of the axis producing it, the growing apex works out the
system, until, by constant repetition, certain members fall into series
which give the subjective appearance of log. spiral parastichies.
Any secondary change in the system must result in the disturbance
of these parastichy curves, and either appear as a “ distortion” of the
series, or completely break wp the system.
In other words, so long, and so long only, as the ratio between
axis and lateral primordium remains constant, and a given axis
* Of. Bellis (fig. 47).
+ Cf. Saxtfraga wmbrosa (fig. 52).
{ Of. Vochting, Prings. Jahrb., xxvi. 438.
98 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
produces a uniform system of members, it becomes possible to give
a mathematically accurate account of the more immediate causes
of the phenomena observed, in that the members, as they are
formed, fall into a definite spiral system, which by constant
repetition produces the subjective effect of “ genetic-spiral,” “ para-
stichy ” spirals, and so-called “ orthostichy ” spirals.
The formal description of phyllotaxis thus becomes restricted
within very narrow limitations. As soon as the relative bulk of
axis and primordium varies to any considerable extent, changes
must ensue in the system, and the phyllotaxis formula becomes
altered, and is only again capable of mathematical expression when
the ratio once more attains constancy. At the same time it is
possible to deduce from these relations the fact that the change,
when it does occur in such a mechanically produced system, must
normally be a gradual one, while any abrupt transition from
one system to another, involving a very considerable alteration in
the bulk-ratio, is undoubtedly to be regarded as the expression of
an active interference in the working mechanism by the inherent
form-determining properties of the protoplasm of the organism.
And not only so, but such a violent disturbance of the system
must be regarded as the expression of a break in the ontogenetic
recapitulation of a phylogenetic change which was originally a
gradual progression. Abrupt disturbances in the bulk-ratio, at
any given point on a plant axis, thus imply a break in the con-
tinuity of a mechanical system of member production, which may
be taken as the sign of extreme biological specialisation ; and thus
the production of an Aroid spadix (Acorus, Richardia), or the
arrangement of the essential organs in certain flowers (Clematis,
Papaver, Paeonia, Cereus), in which the relative volume of the
lateral member with regard to that of the axis is abruptly lowered
to a very considerable degree, indicates a highly specialised line of
descent ; and such rapid transition-phases of phyllotaxis cannot be
accepted as expressions of the mechanical Jaws controlling normal
phyllotaxis change, but must be considered later, when the rules
governing the changes in simpler cases are understood.
CONSTANT PHYLLOTAXIS. 99
PHYLLOTAXIS OF ARAUCARIA EXCELSA.
Araucaria excelsa, R. Br—A good example of this relation of the
“ bulk-ratio” to phyllotaxis phenomena is afforded by such a plant-
form as Araucaria excelsa, in which, owing to the remarkably small
development of the foliage-leaves, and the absence of special
growth-modifications in them, the symmetrical development of
the plant is allowed free scope.
The main axis of a young plant grows erect and produces leaf-
members in well-defined (8+13) system, the primordia being
pressed into uniform rhomboidal areas, and the transverse section
shows a very close approximation to the theoretical vortex-con-
struction.
The secondary axes are spaced symmetrically round the main
stem and hence follow the Fibonacci series, 5 or 8 being the most
usual, When growth is vigorous these axes similarly present the
(8+13) system but develop horizontally (figs. 40, 41).
Tertiary axes are produced in two rows only, along the flanks of
the secondary axes, which may be so far regarded as presenting a
certain dorsiventrality in the space-form of the branches. Section
of the apex shows that the spiral construction is utilised and not
disturbed, the lateral branches of this order being arranged along
two lateral “ orthostichy ” lines (Phyllody spirals) towards the upper
surface, the spiral character of which may be observed on the
plant. It will be observed that these lateral shoots arise in the
axils of two successive members of each full cycle (eg. 12, 13, and
33, 34, in fig. 40).
These tertiary axes produce leaves in the system (5+8) (fig. 41),
and as a general rule develop no further. But when a leading bud
of a secondary axis is damaged, the end branch may assume its
place and similarly produce lateral shoots in two rows. An example
of such a bud whose construction showed the (5+8) system gave
rise to a large number of laterals, all of which were constant and
homodromous at (3 +5) (fig. 41).
Apices of lateral branches of the first degree may also be found
on the same plant exhibiting the anomalous ratio (7 +11) (fig. 41).
100 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Fig. 40.—Araucaria excelsa, R. Br. Transverse section of the growing point of a
lateral branch, system (8+13), camera lucida drawing. The origin of branches
of the next degree follows definite lines ; the first leaves of these axes are repre-
sented in outline only.
CONSTANT PHYLLOTAXIS. 101
Similar sections of successively smaller axes ; systems
(7+11), a branch of the first degree ; (5+8) a branch of the second degree ;
(3+5) special lateral axis of the third degree. ;
Fig. 41,—Araucaria excelsa.
102 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
On drawing sections of such a series of buds under the same lens,
with camera lucida, it becomes clear that the leaf-members in all
are practically identical, and the leaf-primordium a fairly constant
quantity, while the axis is variable.
From the previous table of the curve-systems—
(8+13) corresponds to a bulk-ratio of 5: 1,
(5+ 8) ” wu ” 3 : 1,
(38+ 5) ” ” ” 2 a f,
respectively ; while by drawing a special figure for the system
(7+11) it may be shown that the bulk-ratio in such case is very
approximately 4:1 (or 4:2 : 1).
From this standpoint, therefore, Araucaria excelsa builds its
phyllotaxis system according to the relative size of the axis
concerned, the ratio being expressed with remarkable accuracy to
the nearest whole number, according as the diameter of the axis is
5-, 4-, 3-, or 2- times as large as that of the primordium to be
placed on it.
The occurrence of the ratio (7+11) is of especial interest in
that it is associated with the normal series, not so much as an
exceptional member of another hypothetical series, but definitely
as the ratio which gives the missing intermediate bulk-ratio of 4:1.
Again, comparison of these diagrams shows a striking analogy
between the number of primordia of fairly constant bulk placed on
axes of different sizes and the segmentation of the tissues of such
growing points into constituent cells which also have an approxi-
mately equal volume in all. In fact, just as a growing point which
is well nourished contains a large number of cell units, and an im-
poverished one only a relatively small number, so one is bound to
conclude, the number of primordia arranged on an apex is an
expression of the strength of the growing point.
Whether dealing with cell-segments or lateral primordia, a plant-
apex, to use a common metaphor, must cut its coat according to its
cloth, and so long as the lateral members remain approximately
constant in volume, so the number must vary. In correlation with
the relative bulk-ratio of axis and member, a certain number of
curves are therefore selected.
CONSTANT PHYLLOTAXIS. 103
Thus a lateral axis which might normally lay down (8+ 13) curves
along which the lateral members would be built owing to a bulk-
ratio of (5: 1), would if impoverished reduce the number of curves
in correlation with the lowering of the bulk-ratio.
Thus (8+13) would reduce simply by dropping out one short
curve to (8+12); but the difference in the bulk-ratio thus implied
would be very small.* On the other hand, by dropping out one
of the long curves, 7.¢., 8 to 7, it would be necessary to take for the
short curves the nearest whole number which gives the same ratio
and working-angle; this would be 11, since 8: 13:: 7 : 11:375,
The log. spiral construction diagram shows that by so doing the
bulk-ratio is lowered to (4:2): 1.
In the same way, if the long curves were dropped to 6, the
nearest number in the required ratio is 10, since 8:13::6: 9°75;
and such a change would be correlated with a fall of the bulk-ratio
to (3°8):1. The next stage of reduction would give as in the
normal case (5+ 8) with the bulk ratio 3:1.
The systems’ (7+11) and (6+10) therefore represent approxi-
mations to a bulk-ratio (4:1) intermediate between (5+8) and
(8+13), and may be expected to occur together with these con-
structions: thus (7-+11) was noted among plants of Monanthes
polyphylia as equally common with (5+8). In the same way (6+
10) occurs commonly in Pinus (Pinea, pumilio), Podocarpus japonica,
etc., also as an enlargement of a (5+ 8) system rather than as an im-
* The case of such a construction as (8+12) would be of special interest, in
that, while it presents no difficulty from the point of view of the loss of one curve
from an (8+18) system, the practical result is the immediate formation of a
tetrajugate system, since (8+12)=4 (243), and such a system would work out as
four concurrent genetic spirals producing four members simultaneously instead of
one ata time. The occurrence of such a construction among normal specimens
thus either involves a fundamental change iu the building-mechanism, or else it
implies that the genetic spiral is now comparatively unimportant and only
secondary to the curves of the parastichies.
Among a batch of several hundred cones of Pinus austriaca, one such cone was
found in which a short curve was lost below the widest diameter of the closed
cone ; farther up, as in the general case, other curves including long ones were
also lost, but the effect of the first loss was, in the case of the dry expanded cone,
to render the parastichy system (8+ 12) as seen on the base, over a considerable
area.
104 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
=== Z
Fig. 42.—Podocarpus japonica, Siebold, Sections of leading apices (548) varying
(6+10), and weaker axes varying (+5) and (3 +4).
CONSTANT PHYLLOTAXIS. 105
poverished (8+ 13); but as (6+10) would constitute a bijugate type,
it will be considered under the special section. Taken in connec-
tion with the approach in these systems to the bulk ratio (4: 1)
which is not provided for in the normal Fibonacci series, special
interest attaches to an exceptionally fine closed cone of Pinus Pinea
in which the parastichies, though somewhat irregular, were for a
considerable distance undoubtedly (7 +-10) (Broome, 1900).
Although thus starting from a standpoint of a dulk-ratio
constant, it now appears that the number of parastichy cwrves becomes
of increased practical importance, in that, while the bulk-ratio may
be expressed by fractional quantities, the actual ratio of the curves as
representing paths of distribution of growth must be eapressed by
whole numbers. The very smallest corrections in any system must
thus be made in the bulk-ratio, the parastichy curves remaining
constant until some very considerable alteration becomes necessary.
That is to say, so long as the plant is condemned by phylogeny
to build asymmetrically, one member at a time, and so produce a
spiral series, the optimum attempt at symmetrical growth, in-
volving symmetrical nutrition, would be given by the limiting
ratio of the Fibonacci series a In such an ideal system
the ratio of the parastichies which map out the orthogonal paths of
distribution of growth should therefore be ( ,/5-1: 2); but as a frac-
tional number of curves is impossible in practice, the nearest approach
to this ratio, as expressed in whole numbers, is selected in correlation
unth the size of the lateral member required. The number of the
curves is therefore more important in practice than a perfect
oscillation angle of 137° 30’ 27"-936. On the other hand,* the
approach to the “Ideal Angle” is wonderfully close even in low
ratios, being within one minute for a (5+8) system.
Constant phyllotaxis may thus be considered from two entirely
different points of view; either a single growth-oscillation producing
members with a definite bulk-ratio is the determining cause, and
thus involves the fact that so long as one spiral is in operation the
numbers of the parastichy curves are only divisible by unity ; or else
the parastichy ratios are primary, and being normally successive
* Gf. Mathematical Notes on Log. Spiral Constructions,
106 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Fibonacci ratios which have 1 only as a common factor, it becomes
possible to draw a single genetic-spiral throughout tbe entire
series.
It will be noticed that the formation of lateral appendages one
at a time, which was put forward as being possibly a phylogenetic
reason for asymmetrical growth, is not wholly satisfactory ; but if
it will not hold, it leaves no clear justification for regarding the
tendency of plants, in the more general case, to grow asymmetrically
rather than symmetrically, as anything beyond the mere expression
of the mathematical fact that symmetrical construction is only to
be regarded as a special case of the proposition of growth. This
phylogenetic conception was based on the observation that in the
great bulk of lower cellular forms, the segmentation of the plant-
body into component cells is relatively on so large a scale that it
apparently controls the spave-form of the entire organism, although
it is still possible to regard it as the mere mechanism by which the
space-form is divided into units. The range of bulk-variation in
the working cells of green plants is remarkably restricted in
comparison with the range of bulk-variation in: the adult organisms ;
and just as in building a small house the size of the bricks may
become an important factor, while in a large one it would be
negligible, so in the construction of a small plant, histological
details are more striking than in immense plant forms. In these
simpler constructions, the mechanism of which is apparently
controlled by a single apical cell, the practical details of karyo-
kinesis and cell-formation require that new members initiated as
single cells should be formed one at a time from the initial cell,
and it is quite possible to regard this mode of production of a
space-form comprised of serially produced members as becoming
fixed, and then being retained even after the relative increase in
the bulk of the organism and its growth activities would admit of
the formation of massive members in which the cell segmentation
would be of subsidiary importance. By starting from this first
standpoint of a controlling genetic growth movement, which comes
into line with the theories of Schimper and Braun, the strength or
weakness of the hypothesis should become manifest. It is true
that the genetic spiral apparently never reverses, although a
CONSTANT PHYLLOTAXIS. 107
reversal of parastichies may be observed (cf. Saxifraga wmbrosa,
Cyperus) ; but such phenomena as the transitory variation of an
Fig. 43. —Sedum elegans, Le}. Section of shoot-apex (6+10),
(8 +13) Pine-cone to (8+ 12), implying the formation of four genetic-
spirals, or the bijugate variation (6+10) of a (5+8) system
108 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
(Podocarpus, etc.), become obscured rather than explained; while
the assumption of true circular symmetry from a spiral construction
is an entirely anomalous proceeding.
On the other hand, the second standpoint, that the parastichies
are the controlling feature, certainly gives broader views of the
whole range of phyllotaxis phenomena, in that it—
(1) Allows for addition or loss of curves to such an extent that it
is immaterial whether the system works out as one or many
“genetic spirals.”
(2) It suggests a simple transition to true symmetry by equalisa-
tion of the number of curves in either direction.
(3) It also presents a clear view of all transitional systems, and
especially those included under the term “expansion systems,”
which will now be considered.
Putting on one side, therefore, any discussion as to what is, after all,
to be regarded as the prime cause of the asymmetrical growth which
thus expresses itself in terms of subjective spirals, it is so far clear
that as a matter of practical observation of the phenomena, as well
as the discussion and tabulation of the results, this method of re
garding the parastichy ratio as being the more immediate controlling
factor not only affords more accurate information from a theoretical
standpoint, but is even simpler practically than the established
method of genetic spiral and so-called “ orthostichies’ as a means
of determining, tabulating, and reproducing the constructions.
RISING PHYLLOTAXIS. 109
IIT. Rising Phyllotaxis.
As previously indicated, any definite alteration of the system of a
constant phyllotaxis must result in a distortion of the parastichy
curves. The case of Dipsacus in which, with a constant lateral
member and varying axis, new members were added or lost to
compensate this change in the bulk-ratio, to the very slight extent
of losing or adding one or two curved rows, shows that the dis-
tortion effect may be considerable (figs. 38a, -6); and it is clear
that the same law of adding one member at a time to the system,
which controls the asymmetry of the whole construction, must also
result in the addition of such compensatory rows one at a time, if
the whole system is to change from one curve-ratio to another. If
now these changes are initiated on a rapidly expanding apex, such
as that presented by the developing capitulum of Helianthus, there
can evidently be no attainment of symmetry resembling that which
previously obtained in the apex, until every member of the cycle has
similarly divided, all round the axis.
A “Zone of Transition” must thus be intercalated between the
two systems, and will form the outward sign of the passage of one
system of asymmetry to another with the same approximate curve
construction. Previous considerations have further shown that, if
the curves remain constant in direction, and the ontogenetic spiral
is also constant, change can only take place between alternate
members of the ratio series; thus (5+8) rises with a minimum
effort and least distortion to (13+21); (¢ table in fig. 34).
Confirmation of such a view is very obvious in Helianthus, in
which the ratio of the contact-curves of the inner sterile involucral
scales of the capitulum, bears constantly this relation to the para-
110 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
stichy ratio of the disk-florets. (Cf fig. 54a, showing the anomalous
ratio (11+18), the zone of transition being marked by the line of
ray-florets, at which the parastichy curves are uniformly increased
to form those of the disk-florets (fig. 54).
With these data it is now possible to construct the diagram for
any capitulum of Helianthus, and to exhibit the relations between
the disk-florets and the sterile involucral scales, as well as the
position and number of the ray-florets, for any particular phyllo-
taxis ratio. The compensatory allowances for the inevitable dis-
tortion cannot be completely shown, since these are necessarily
corrected as the new members are formed, one at a time, and the
curve-construction must therefore exhibit the asymmetry of either
the former or latter condition. Since secondary changes are very
marked in the former system, which produces dorsiventral members
only, the curve-system may be constructed from the ratios pre-
sented by the parastichies of the disk.
The diagram (fig. 44) is thus drawn for a small capitulum in
which the disk-florets have the phyllotaxis (21+ 34), and the inner
sterile involucral scales (8 + 13).
By selecting the proper (8+13) curves from the (21+34)
system, it will be seen that the amount of malformation in the
diagram is not large.
Since a complete cycle of (8+13) contains 21 members, the
transition to (21+ 34) will be most economically effected, as already
indicated, by an approach to quadrant division in each area; while,
to effect the transition, it is necessary to add 13 long curves and 21
short ones. If then one long one and one short are added in each
of the first 13 members of the cycle, the last 8 will only require to
add one short one each to complete the requisite number.
This is done in the diagram (fig. 44), and the whole construction
is closely comparable with the segmentation of a layer of cell-
tissue into new lines of cells, as seen in the familiar examples of
the cell layer of Melobesia or Coleochaete. The diagrams here work
centripetally instead of centrifugally, but the method of segmenta-
tion is identical. Each area is thus seen to be subdivided by two
new lines, constituting the new paths in either direction, and these
are directly homologous with the characteristic T-shaped wall of
RISING PHYLLOTAXIS. 111
Algal segmentation; the whole thus affording a remarkable con-
firmation of the original hypothesis, that the formation of massive
primordia follows the same laws which control the production of
lateral members consisting of single cells, The members which,
Fig. 44.—Expansion system : log. spiral scheme for the introduction of new paths
which determine the rise of phyllotaxis ratios in the capitulum of Helianthus
annuus from (8 +18) to (21+34). A smallcapitulum is taken as a type; the
genetic spiral winds left ; the small amount of unavoidable error in construction
is admitted in the (8 +13) system.
in the diagram drawn by this method, exhibit the greatest deviation
from “square” areas, are the transitional members which subse-
quently produce ray-florets. With the view of making the
H
112 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
construction clearer, these have been indicated by triangular areas
corresponding to their appearance in the capitulum section.
The close agreement of an actual specimen with the preceding
Fig. 45.—Helianthus annuus. Section of a young capitulum, 10 mm. in diameter:
cam. lucid. drawing of half the disk.
generalisation can be checked on a drawing made from a similar
capitulum in the bud stage (fig. 45), and also in the anomalous
capitulum (fig. 54).
Fig. 45, drawn under the camera-lucida from a young bud
RISING PHYLLOTAXIS. 113
10 mm. in diameter, shows a very close approximation to such a
theoretical construction. By taking a curve-tracing (fig. 46) from
this figure, the agreement becomes much more obvious, and the
8 —
a 0
42 AS
* me:
26 .
5
%»
1g
3
10
Wd
Be
2
‘s
28
4 Yd
33 7
20
12
Fig. 46.—Curve-tracing of the preceding ; the ray-florets are blocked in and the
areas numbered in agreement with the theoretical construction of fig. 43.
areas are observed to agree number by number with the theoretical
system. The figure affords a point of additional interest in that
one ray-floret is in excess, and clearly pushed into the circle of
rays out of its place. Comparison of the numbered tracing shows
114 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
that this belongs to member 0, and represents a case of the
fertility of the last involucral member. The fact that the number
of sterile ray-florets is not absolutely constant is thus illustrated,
while it may be noted that such irregularities will also affect
the observed number of involucral scales as a constant quantity.
Deviations from theoretical construction are seen to be due
merely to:—(1) The assumption of dorsiventral symmetry by the
sterile involucral scales, which further exhibit sliding growth over
one another in the tangential direction to such an extent that
a ray-floret is sometimes isolated from its subtending bract, while
the members themselves are greatly flattened radially. (2) Owing
to the sharp line of demarcation between these dorsiventral
members and the fertile bracts, which with their enclosed radially-
symmetrical florets retain their original relative volume, the edge
of the disk becomes a fairly rigid circular boundary against which
the decadent ray-florets are subsequently pressed. The ultimate
assumption by these latter of a form adjusted to the room allotted
them in the bud is thus not due to the fact that they are subtended
by members of transition ; while since the same influences may act
to a greater or less extent in other peripheral florets, the number
of ray-florets is not absolutely constant, although the approximation
to the theoretical number is very close.*
Transition to a higher scale of phyllotaxis, on an axis which is
rapidly expanding without correlated expansion in the lateral
members, thus takes place in alternate stages of the ratio series,
Thus in the specimen of Cynara Scolymus figured, the large
involucral scales (figs. 9, 52) showed on the sides of the capitulum
the obvious parastichies (8+13), and these are further seen to be
contact parastichies in section of the head. After production of
79 of these the phyllotaxis rose to (21+34) and 102 smaller
sterile scales were laid down, while at the edge of the disk the
phyllotaxis rises to (55+89), with which curve-construction the
disk-florets are produced without subtending bracts (fig. 52).
The total number of sterile scales (181) being relatively large,
and greater than the number of florets in one complete cycle (144),
* Of. Weisse, Variation Curves, p. 480, tables ; Ludwig, Ueber Variations-
kurven, Bot. Centralb., lxiv. p. 102.
RISING PHYLLOTAXIS. 115
shows that the involucral region is here much more developed than
in Helianthus, while two complete transitions are included in the
construction.
The relation of two such transitional systems to one another
therefore remains to be considered. Does the transition take place
rapidly and with a minimum number of members, or is it irregular
and spread over a large number? In Cynara the number of
members observed is very large, and in absence of further data
either method may be possible; the remarkable accuracy with
which the transition has been planned for Helianthus suggests that
the latter plant may be taken as a type.
Before passing on to the case of the double transition, the
mechanism of the normal expansion of Helianthus capitulum re-
quires to be carefully considered. As already noted, such a
transitional system presents the appearance of a cell-segmentation,
or even a process by means of which primordia are as it were
separated out along specially formed lines of cleavage. The whole
construction is influenced rather by the preceding parastichy lines
than by a definite genetic spiral, and the system grows throughout
along its pre-established paths of asymmetry. This agrees with the
facts observed in the actual ontogeny, so many new members being
formed simultaneously that the genetic spiral is hopelessly lost
sight of, and can only be traced by numbering the members.
It would appear now that the genetic spiral as a line of building
has lost its significance ; the question of phyllotaxis becomes one of
continuing an expansive development along similar curved paths,
and the plant continues to work out a definite pattern quite
mechanically.
It is clear that while such expansion may be pursued indefi-
nitely, the same ratio system will always be constantly maintained,
and constructions in the very highest terms of the Fibonacci series
thus become conceivable, not by accuracy of the building-mechanism
however, which in such case would come down to a question of
working at an angle correct to minutes and seconds, but by simply
following up a system of construction which maintains at all stages
an approximately identical degree of symmetry.
The method of employing this expanding mechanism may now
116 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
be considered, the process being rendered obvious by the fact that
the ray-florets in the case of the Sunflower occupy the transitional
areas between two cycles.
Comparison of the figure in which 21 ray-florets are indicated as
black patches shows that these follow a remarkable sequence which,
counting from No.1 in the direction of the genetic spiral, may be
represented by the figures—
2-1°2:1°2 | 2-1°2-1°2| 2-1-2
The fact is thus brought out that the essence of the Fibonacci
series consists in the manner in which it may be regarded as com-
posed of the expression 241424142 treated as a recurring
quantity. Thus 3 of these members add up to 5, 5 to 8, 8 to 13,
any 13 to 21, and any 21 to 34, etc. Any ratio of the series may
undergo subdivision in this sense to produce the next higher mem-
ber. From this it follows that the law of arranging members of a
higher cycle on a preceding lower one consists in the method of
dividing them in sequence in the order indicated. And conversely,
whenever increase of members takes place in such a manner, it is
at once clear that a transitional sequence of the Fibonacci series is '
implied (ef. Cactaceae).
To subdivide a phyllotaxis system so as to retain the Fibonacci
ratio, it is therefore only necessary to start from No.1, in the
direction of the genetic spiral, and put in new paths in the
sequence (2'1:2-°1°2), or graphically—
bo. . MIVIVVIVIV
When two such transitions are involved, the sequence becomes
(3°2°3°2°3), or—
it | VIV IV VIV
Vivtv
By noting this property, the (8+ 13) curves are selected from the
(21+ 34) set, the sequence being carried out along the direction of
the spiral concerned; thus in the practical construction of diagrams,
it is necessary to start from 1 and proceed:from the concave side of
, ete.
, ete.
RISING PHYLLOTAXIS. 117
the curve, marking out sections of (3 -2°3-2-3), etc., all round the
system (fig. 44).
The same relation will be found to hold for other ratio-systems.
Formula IL. thus gives a special case of the law of subdivision of
paths of growth, corresponding to the T-shaped segmentation of
Algal membranes; while the graphic method of representation
furnishes a key to the construction of a normal transition system.
Deviations from it will imply irregularity in expanding phyllotaxis ;
while a guide is again provided for checking the anomalous addition
of new ridges as a consequence of the formation of new parastichies
in many Cactaceae.
The areas on such a construction diagram are readily numbered,
the members of the original (8+13) system being marked out by
their differences of 8 and 13. Although the diagram was con-
structed empirically to begin with, the correspondence of such a
theoretical construction with the phenomena actually observed in a
capitulum (fig. 45) is so striking that the accuracy of the method is
beyond doubt, and its mechanism may be further analysed. Thus,
the system was originally (8+13); each of the first 13 new
members adds a new long curve and the system is thus gradually
changed to (21+13).
It is important to note that when forking of the paths takes place,
it is the external or peripheral portion of the subdivided segment
which must always be regarded as the “member adding the new
curve.” ,
Thus 1-13 each addsa new long curve in 13 segments of the
original (8+13) system.
The system is now practically (21+13), which, it must be noted,
is quite a different construction from (13+ 21), although the possi-
bility of an adjustment on the part of the plant is not to be
neglected, since the transition takes place in a growing system.
No. 14 again adds the first short curve; 21 of the same type are
required, and thus 33 adds the last one of the set, and at 34 the
new system is completed, and the Fibonacci ratio is again perfected
and may be continued indefinitely. But if expansion is to again
commence, it is clear that it cannot begin before the 34th member.
In a normal system, therefore, cach member adds one new path to the
118 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
system ; the difference in the number of curves between two transition
stages gives the number of members involved in the change, thus :—
(3+5) passes into ( 8+13) in 13 members,
(5+8) , (13421)in21
(8413) _,, (214+34)in 34 ,, ete, ete.
The law of normal expansion is so simple, and works in Helian-
thus capitulum with such remarkable accuracy, that there can be
little doubt that it represents in some way a mechanical distribution
of growth-energy which is a common property of all plants grow-
ing under conditions in which these mechanical relations are allowed
free scope.
As soon as the Fibonacci ratio is disturbed, the system gradually
and uniformly proceeds, one member at a time, to put it right again.
The addition of all the long curves before the short curves are put
in is perhaps the most curious feature. It may be also noted that
though the initiation of the change makes for symmetry, by raising
the lower number of the ratio first, the ratio is equalised, then again
rendered unequal, and again equalised at one point in putting in the
short paths; but the system does not remain stationary at these
points of symmetry, it passes on and only rests in the condition of
equilibrium of the completed ratio. Although the acquisition of a
Fibonacci ratio may be regarded as the optimum attempt at sym-
metry in an asymmetrical system, the plant does not so far show
any preference for a symmetrical relation attained during a period
of transition. Nor is it clear that such a point of symmetry, al-
though isolated in the construction by considering one new curve
at a time, is at all comparable with a true symmetrical construction.
The whole system is growing together in a correlated method, and
the metaphor of crystallisation is perhaps the only one which fits
the phenomena observed.
Now that the number of members involved in making any
given normal expansion is known, it remains to see to what extent
one expansion can rapidly succeed another. Thusin the Helianthus
capitulum taken as a type, it appeared that a new system could
only commerce at the 34th member or beyond it. The data given
for Cynara show that the number of members of each system was
RISING PHYLLOTAXIS. 119
considerably in excess of the theoretical minimum, and the conclu-
sion is warranted that the expansions may take place at intervals
at irregular distances from one another, although the actual tran-
sition may be limited to the minimum number of members; the
new ratio being continued for an indefinite number of members
before a new change takes place.
Thus the Daisy (fig. 47) shows a double transition, from (2+3)
foliage leaves, (5+8) involucre, to (13+21) florets. The ends of
the system are not obtainable, but it is obvious that 13 involucral
members of the (5+8) system are retained when the complete
transition would have been completed in 8.
Very remarkable relations occur in Helianthus in the number of
members constituting the involucre, and the tendency of Helianthus
to approximate a continuous expanding system is very marked.
Thus the involucre of a (55+89) capitulum should contain 34
sterile scales ; the range (Weisse) is 26—42, but it must be borne
in mind that the exact localisation of the involucral region is a
matter of difficulty, since the reason which determines which scales
shall be the first to produce florets is not known, and the range is
thus as equally open to minor variations as the number of the ray-
florets themselves. The variation curves tabulated by Weisse show
that similar ratios hold for capitula exhibiting the other parastichy
systems on the disk, and the number of sterile involucral scales is
thus one grade lower along the ratio series 3, 5, 8, etc., than the
theoretical number of ray-florets. The involucre of Helianthus
thus apparently includes the whole of the members involved in
one complete transition, while a.second transition takes place
immediately after, of which the ray-florets represent those transi-
tional members which have to negotiate the bulk change. Since
the head taken as a type in fig. 45 is only of a low order (21+34),
and the involucre therefore (8+13), it would follow that some of
the foliage leaves were laid down with a (3+5).
But much larger heads are found, so that it becomes probable
that more than two such transitions may take place in such a rapidly
expanding axis.
' It has already been shown, for example, that Cynara shows two6
transitions in the involucre, and another at the edge of the disk:
120 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Fig. 47.—Transverse section of the whole central portion of a perennating plant of
Bellis perennis (January), showing expansion systems (2+3), (5+8), and
(134+21): cam. lucid. drawing.
RISING PHYLLOTAXIS. 121
the disk being (55 +89), inner involucre (21+ 34), outer involucre
(8+13), and the upper foliage leaves (3 +5).
Comparison of the Daisy (fig. 47) again shows foliage leaves
(2+8), involucre of 13 members (5+8), and florets (13+21), the
rays being not yet determinable.
A large (89+ 144) Sunflower would therefore be built up by the
regular expansion (2+3), (5+8), (13+21) foliage leaves, (34+55)
involucre, and (89+ 144) florets of the disk, thus giving five com-
plete transitions along the axis. The difficulty of giving a definite
phyllotaxis constant for the leafy stem is thus rendered obvious.
It now becomes possible to give a connected account of the phyllotaxis
phenomena of Helianthus, based on this remarkable limitation of the
involucre to the minimum number of members which represents the
transitional period between two cycles, and the following scheme is borne
out by the data tabulated in the Variation Curves of Weisse (loc. cit.,
p. 478). The weakest Sunflower axis resembles the Daisy in presenting
two changes only : 1.¢., beyond the 3-4 pairs of decussating (2 + 2) leaves of
the seedling, the system assumes normal asymmetry by laying down (2+3)
curves.. After producing a total average of 20-22 foliage leaves (average
of 20 plants=22), the system expands to (5+8); since 8 new curves
are to be added, 8 members represent the minimum number of leaves
before another transition can be initiated. Such an involucre would
therefore normally contain 8 leaves and the second transition would give
(18+21) florets system, of which the first 13 would form the ray-florets.
If, for example, after forming 8 members of the transition to (5+8) the
vegetative condition was still vigorous, a further rise to (18+21) would
be effected in 21 leaves, the involucre would now be 21, the total number
of leaves about 22+8=30, and the parastichies of the disk (34+55)
(Weisse, p. 478). Again, with the same proviso another change in the
vegetative region to (84+55) would be effected in 55 leaves, and if these
represent the involucre the next rise (89+144) would give the largest
‘capitulum, Such a plant should show (22+8+21)=51 foliage leaves
and 55 involucral scales, total 106. It is of interest to note that one
such plant counted by Weisse gave 46 of the former and 62 of the latter,
total 108 ; when allowance is made for the ill-defined character of the
involucral region, the correspondence is remarkably close.
That Heltanthus presents a progressively rising phyllotaxis involving the
minimum number of members with a remarkable degree of accuracy
appears therefore fairly clear. Similarly (+5) as a stronger system
would give other terms of the capitulum series, and expanding systems
derived from variation derivatives (2+4), (8+4), would lead on to
capitula of the series 16, 26, 42, 68, and 29, 47, 76 respectively. These
variations are however comparatively rare, the former representing the
122 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
bijugate capitula described, and the latter including all the anomalous
constructions yet observed. On the other hand, while it is probable that
such generalisations represent the general tendency of the phyllotaxis
phenomena in Helianthus with a considerable amount of accuracy, it is
not necessary to assume that they will give absolutely accurate results
in the case of individuals, Examination of the apices of a large
number of young plants is sufficient to show the possibility of a con-
siderable amount of variation, the causes of which are still unknown.
The agreement of the preceding data with the observations of
Weisse shows that even if continuous expansion does not take place,
a margin of about twenty leaves suffices to cover all the members
formed in any Sunflower plant which do not belong to such a con-
struction, including the first leaves of the seedling and odd members
filling the gaps between transitional systems.
These generalisations require therefore to be tested on actual
apices in which the transition is being effected. The difficulty of
the method will lie in the fact that only one section is obtainable
from any given plant, and there is thus no telling what the system
observed would have passed on to if the plant had been left alone
until the capitulum was reached.
Owing however to the remarkable uniformity of growth in the
bud of Helianthus, it is possible to cut sections which practically
include the whole of the phyllotaxis system so far as it has gone.
The first and most remarkable aberration observed in the Sun-
flower is that the apex commences in the plumule a definitely
symmetrical construction, and changes to normal symmetry at a
subsequent and varying date.
Thus seedlings which have not as yet produced their first leaves
beyond the green cotyledons, show on sectioning two or three pairs
of decussating leaves constituting the terminal bud of the plumule
and contained in the cotyledon tube (fig. 48a, three pairs).
Asymmetry may commence early ; at the 5th leaf while still en-
closed in the tube (fig. 480), or it may be delayed until the first
leaves are well grown.
Seedlings in which two pairs of leaves are well developed on the
elongating axis should afford suitable material for the onset of
asymmetry. In such buds most usually the (243) system is
directly assumed (fig. 48d, for 10 leaves), more rarely symmetry
RISING PHYLLOTAXIS. 123
a3!
Ie
Fig. 48.—Apices of seedlings of Helianthus annuus, cam. lucid. drawings. A and B,
cotyledons only showing ; A, symmetry for three pairs of leaves ; B, commence-
ment of asymmetry at the 5th leaf; C, older seedling, symmetry maintained for
5 pairs ; D, asymmetry (2+3) above a symmetrical pair ; E, asymmetry (3 +5)
following a decussate series and already changing at 8 to a higher series.
124 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
may be continued (fig. 48¢, five pairs), and in other cases the (3+5)
system may be produced directly beyond the decussating members
(fig. 48). This last example is of interest in that the (3+5)
system immediately commences a further rise, and examination of
the method shows that the leaf numbered 8 is the first member
definitely concerned, in that it does not fall accurately between 3
and 5, in the manner that 7 fell between 4 and 2, but beyond its
normal station, so that 13 afterwards falls behind it, in the manner
tabulated in fig. 44 as the type expansion.
Another example (fig. 49) shows a similar transition only taking
place after a formation of 13 members of the asymmetrical system
(Nos. 14 and 19 falling in the gap between 9 and 11), The fact
that older leaves in Helianthus are spaced out owing to a great
development of hairs, militates against accurate observation of the
contact lines in sections of large area, but by remembering that
each leaf fills its own rhomb, and that the development of hairs
thus compensates the diminution in the rate of growth of the leaf
itself, the contact-lines of the hairs of adjacent leaves may be taken
as a very approximate representation of the theoretical members.
Since the sudden change from a decussate (2+2) to an
asymmetrical (3+ 5), implying the sudden intercalation of four new
curves, is much less easy to understand than the change to (243)
which only adds one, a capitulum of (21434), similar to the one
for which a diagram has already been constructed, may be selected,
in that it should present a double expansion of the type (3+5),
(8+13), (21434).
Such a capitulum in the bud condition, the whole 6 mm. in
diameter, is sectioned in fig. 50; every member being accounted
for beyond the uppermost pair of decussating leaves.
As a (21434) capitulum should present 21 ray-florets, 13
involucral scales, and an average of 25 foliage leaves, including the
primary decussating pairs (Weisse), the bud in question is evidently
well within normal range in that it shows a total of 27 vegetative
members between the uppermost pair and the first ray-floret. By
adding 8 for an average number of four pairs, the total number of
leaves should have been 35, which is again sufficiently close to the
theoretical number 254+13=38; the specimen is thus a normal
RISING PHYLLOTAXIS. 125
plant so far as it goes, and well within the range of the Variation
Curves. Again, in the system (3+5), (8+13), (21434), the
\
Fig, 49.—Seedling of Helianthus annwus, (2+8) system above decussating leaves :
transition to a higher series after asymmetrical production of 13 members.
members along the long curves differ by 3, 8, or 21 respectively,
and since in dorsiventral constructions the long curves are more
126 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
obvious than the shorter ones, these paths may be studied as a
whole and mapped out by curved lines passing through the members
differing by these values, and the method of the forking of the long
curves of the system made more obvious (fig. 50). The method
of numbering may be checked by noting the proper differences in
adjacent members, and also by taking an approximate divergence
angle of 137°.
Bearing in mind the convention previously laid down in the
type capitulum of fig. 44, that the member which originates a new
curve is always the external member of the two paths produced by
the forking, the system may be further analysed.
Starting from the uppermost pair of decussate leaves 1 and 2,
No. 3 commences the asymmetrical condition and adds a new
curve ; three long curves are now established and the original (2 + 2)
system completes the transition to (3+5) in the members 3, 4
and. 5.
Normal expansion commences immediately in 6, since 6 is the
external member of the fork eH
Thus 6 puts in the first new long curve, and similar relations
hold for 7, 8, 9, and 10. Five long curves are now added, and
the system is at this moment (8+5). As in the normal case, the
short curves immediately commence to be laid down, commencing
at 11, which is the internal member of the previous fork, but the
11 (difference of 5)
19 ( 55 13).
Similar relations are established for the members 11-18 inclusive,
and 19 adds no curve, being the inside member of the last forking,
and falls in the gap left for it. No change takes place till 30 is
reached; that is to say, 20 to 29 inclusive, or 10 leaves are formed
in a uniform (8+13) system. At 30, however, a long.curve is
again added, since 30 is the outside member of the fork
99 ee (difference of 8)
43 ( 3 21):
as the first member of the ultimate transition, it also correctly
subtends a ray-floret, Similar relations hold for 30-42 inclusive,
external member of the branch 6
Helhanthus annuuc.(21+34) 6mm bud. Cam Lucid Oy 3_AH Church Jeit igor
Vie, 60.—Helianthus anuvus. 6 mm. bud of a (21+84) capitulum: cam, lucid. drawing of the whole
plant, in a transverse section, beyond the uppermost pair of decussate leaves. Two complete
expansion systems (3+5),(8+13), (21 +34) are included: the more obvious paths of the longer
construction curves are dotted, and the introduction of new paths rendered clearer.
RISING PHYLLOTAXIS. 127
or the 13 members required to add the long curves of the system
now momentarily (21+13). At 43 (the internal member of the
22 fork), a short curve is added and 21 new short ones are put in
between 43 and 63, the system being completed at 64. This
latter change may be better checked on a construction diagram
similar to that of fig. 44, in which No. 1 is replaced by 30. The
correspondence is again exact, aud the actual construction of the
diagram and the addition of the new curves one at a time, as
they are observed on the specimen, is required to fully comprehend
the symmetrical relations of the construction. The ray-florets, it
will be noted, should extend from 30-50 inclusive; a minor
variation is of interest in that in this specimen an extra ray-floret,
since 22 are actually present, is formed from a disk-member, while
in the previous capitulum of fig. 44 an extra one was produced by
an involucral member. Thus the floret in the axil of 53 forms
a ray, and the member flattens out to an involucral scale, while
51, 52, and 54 are normal disk-florets.
The identity of the expansion mechanism, which is thus twice
repeated in one section, with the construction previously postulated
is very striking, and nothing is more remarkable than this rhythmic
interchange in the addition of the curves in the two directions.
It will be noted that in the case of the production of (3+5)
from (2+2) the transition was very rapid, and not conformable
to the rule for asymmetrical expansion unless the decussate con-
dition be regarded as a variation of (1+ 2), in which case expansion
should be completed in 5 members. In such case the next ex-
pansion commences immediately at 6, while a wide gap separated
(8+18) transition from the (21+ 34), 10 members being laid down
with constant phyllotaxis. The previous estimates for the phyllo-
taxis phenomena of Helianthus are thus subject to an error of a
variable number of members between the systems, and the involucre
does not necessarily include a definite transitional period. The
limitation of the involucre of Helianthus to a minimum number
of leaves has nothing to do with the expansion mechanism, but
represents that number of members which makes a complete cycle
of contact around the axis before the last transition commences
at the region of ray-florets. Helianthus may then be taken asa
I
128 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
type of plant in which very considerable expansive changes take
place, and the optimum succession of the transitions may be very
closely approximated.
That such perfection of transition is not the general rule is seen
in Cynara, in which a change was only effected after a considerable
number of members of the different system had been laid down.
The case of the Daisy is also of interest, in that 13 involucral
members are retained in a capitulum which would in Helianthus
average 8; this involucre of 13 being practically constant for the
Daisy (Ludwig).*
Again, the essential feature of such transitions lies in the fact
that, given a ratio of the Fibonacci series, the change is rapid, and
when completed gives another member of the Fibonacci system.
From this point of view, the ratios of the Fibonacci series may be
regarded as stations of stable equilibrium, in that they give the
-optimum working angle and set of curves plotting the system, and
any alteration of such a system produces a state of instability
which as rapidly as possible resumes the Fibonacci relationship.
The changes may take place with the minimum number of
members intervening (Helianthus), or they may be separated by a
larger or even variable number (Bellis, Cynara); but the change
when it does take place is in these types rapidly negotiated and
the “stable equilibrium” of a Fibonacci ratio regained.
So long, therefore, as it is regarded as a mere convention, which
describes phenomena without explaining them, it is convenient to
regard a normal plant as possessed of what may be termed a
Fibonacci sense, by which any alteration in the phyllotaxis system,
whether due to alterations in the bulk-ratio or not, is controlled
and corrected to a system of the normal optimum series,
In other plants, there may be no evidence of any such control-
ling power: examples being afforded by the stems of tree-ferns and
* The relation between these constructions may be obtained from the contact
relations of dorsiventral members. It consists in the fact that while 8 members
of a (5+8) system form a minimum single investment to an axis, 13 will form
a double one. The limitation of the calyx of a pentamerous flower to 5 members
of a (8-+5) system is thus curiously repeated in the case of the similarly pro-
tective involucre of the Sunflower capitulum.
RISING PHYLLOTAXIS. 129
Cactaceae. As an extreme case a specimen of Cephalocereus senilis
may be cited, in which new curves were added singly, and without
rule, at intervals of about 700 leaves.
While, however, each normal expanding system presents the
appearance of growing out of its predecessor so that the curve
ratio remains and will thus remain practically constant, the point
of view of bulk-ratio becomes lost. An expanding type may
represent, as in Helianthus, a fairly constant bulk-ratio affected by
an expanding axis, and this as a special case may be separated from
types in which the bulk-ratio is the only variable quantity. That
such may occur is shown by the rising phyllotaxis of such inflor-
escences as those of Dipsacus, in which very small florets occur
almost immediately after large “decussate” foliage leaves on an
axis which does not continuously dilate; and in the same way the
members of a flower may be laid down with a varying bulk-ratio
on either a constant or a variable axis. In such case, if the change
of ratio is sufficiently large, it is evident that given a constant
genetic-spiral, the direction of the parastichies may remain un-
affected as in the preceding example of normal expansion.
The possibility is, however, not eliminated that the change in
bulk-ratio may not be correlated with the previous system, and
that a- definite break in the phyllotaxis will thus be produced.
With the same genetic spiral, that is to say, the bulk-ratio may
be so independently affected that the parastichies will reverse and
the system show more obvious distortion.
Such systems may be included under the term Discontinuous
PHYLLOTAXIS, and will be characterised by a reversal of the contact-
parastichies.
Two examples of such phenomena may be considered.
I. Cyperus alternifolius. The strap-shaped foliage leaves are pro-
duced in a (142) system giving three spiral series (spires)
(fig. 596), which owing to the approximate equality in
radial depth of the developing members become approxi-
mate spirals of Archimedes, without necessarily implying
any torsion phenomena; but beyond these biologically
specialised members, small unmodified scale leaves subtend
the spikelets of the terminal cluster (fig. 51), These
180 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Fig. 51.—Cyperus alternifolius, R. Br. Section of shoot-apex (November), three-
spired system =(1 +2), enclosing terminal system of scale-leaves (2+3). Left-
hand spiral apparently broken at No, 12.
RISING PHYLLOTAXIS. 131
present a phyllotaxis (2+3) which is evidently correlated
with the assumption of a lower bulk-ratio. The transition
is abrupt, and takes place at the member numbered 12,
which is thrown forward to such an extent, that the
genetic spiral at first appears to have been reversed; the
new curve established at this point allows No. 14 to fill
in the gap behind 12. The parastichies are reversed and
a very definite distortion of the system is apparent at the
junction of the two systems.
II. Saxifraga wmbrosa (London Pride). Young shoots produce
leaves in a rosette of the type (243), older ones may
exhibit (83+5). Sections of a bud taken in January show
the new year’s growth already laid down within the
rosette of the previous season and already terminated by
a developing inflorescence. In such cases a break occurs
at the junction of the two seasons’ growth (fig. 52).
This is exaggerated, owing to the fact that the last formed
leaves of the previous season were poorly developed and
never attained adult form (11 and 12). The same break
in the genetic spiral, and reversal of the parastichy curves,
is observable : the genetic spiral continues in a right-hand
. direction, but the long curves are now differing by 3,
the short by 5; the dislocation of the junction is shown
by the curious relations of the members numbered
(along the curves) 13 and 15; and this, taken into con-
nection with the rudimentary condition of 11 and 12,
militates against any view that the (2+3) appearance of
the older members is merely an effect of sliding growth
due to the formation of a sheathing leaf-base.
These two examples suffice to show—
(L) A new system may be originated at a new period of growth
producing similar members on a presumably increased axis.
(2) That on a constant axis the bulk-ratio may be lowered so
as to admit of a system which has no direct relation in the
distribution of its curves to the previous construction.
The latter case is of great importance in dealing with the
insertion of floral members in floral structures, while the former
132 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
is of special interest in connection with renewed growths. Both
phenomena again may be involved in the general rise of phyllo-
taxis observed in young plants which develop mature axes with a
comparatively high ratio of curves.
Thus it is clear that the youngest plants of Aspidiwm Filix-Mas,*
Fig. 52.—Sazxifraga wmbrosa, L. Section of perennating foliage-shoot (January) :
(24-3) system of previous year enclosing (3+5) system of the new season’s
growth : the latter separated by a dotted line.
or first-year seedlings of Nymphaea, do not possess the (5+8)
system of the adult (cf fig. 94).
The transition may take place at the commencement of a new
season’s growth as in Saxifraga wmbrosa, or new curves may be
added from time to time, as in the inflorescence of Dipsacus, to
* De Bary, Comparative Anatomy, p. 285,
RISING PHYLLOTAXIS. 133
keep up with the expansion of the axis. The preceding examples
of Cyperus and Saxifraga seem to show that as in the general case
of normal expanding phyllotaxis, the stations of stable equilibrium,
which are constituted by ratios in the Fibonacci series, are at-
tained as quickly as possible at any given transition.
FALLING PHYLLOTAXIS.
It now remains to consider to what extent the converse of these
generalisations will hold for reduction of the phyllotaxis systems.
That the same general principles obtain is shown by the inflor-
escence of Dzpsacus, in which curves are dropped out with facility
equal to their interpolation, and here as in the stems of Cacti no
definite law holds, but one is put in or taken out as required to
adjust the bulk-ratio.
Comparison of the capitula of Compositae which exhibit the
perfection of expanding construction, shows that equal accuracy
does not obtain in the reducing stages, and that these are rather
of the type already included under the term “ discontinuous.” So
far as can be determined, the reduction takes place more or less
irregularly, but on the whole the ratios of the Fibonacci series
continue to mark stations of equilibrium, and these when reached
may remain constant for a considerable period, though the arrange-
ment is not marked equally well in all cases. The alteration in
the bulk-ratio is not sufficient to present the optimum transition
which would maintain the set of the contact-parastichies unchanged ;
but as this rises, reversal of the parastichies takes place, and a
consequent distortion of the system is therefore noticed at a greater
or less distance from the periphery of the system.
The reason for this is obvious: the log. spiral construction
which goes on for an infinite extent, however suitable for the
production of vegetative leaves, will not answer the purpose of the
capitulum, all the florets of which are required to produce seed
within a relatively short period.
In the Helianthus capitulum it is true that a uniform succession
of flowers in successively younger and smaller stages is maintained
134 RELATION OF PHYLLOTAXIS IO MECHANICAL LAWS.
to a considerable degree, but it caunot go on for ever. The system
has a limit, and the system of progressive growth comes into conflict
with the demand for the production of a mass of flowers approxi-
mately simultaneously.
To limit the series, the system must therefore be broken at
certain intervals, and a lower series of curves, implying that the
bulk-ratio has been raised, will be the sign of such modification.
Cynara Scolymus may be taken as a type (fig. 53a, 6). The
large capitulum commences formation of disk-florets with the ratio
(55+ 89), and this is continued as shown by the shorter curves for
8-10 members along these paths—that is to say, for an aggregate of
about 9 x 89=800 florets. Since the meshes of the log. spiral net-
work have all this time been getting smaller, though a theoretical
construction diagram shows that the difference is almost imper-
ceptible, a point is now reached at which adjustments in the lateral
primordia can no longer be strained, and an alteration of the system
correlated with a marked modification of the bulk-ratio is required.
While the 55 long curves are thus carried on, the 89 short ones
break off and present an approximate reduction to 34 still longer
spirals—that is to say, the system is now (34+55) and the set of
the parastichies reversed. The transition is, however, not clearly
marked, and it cannot be traced with sufficient accuracy to show
whether it is rapid or irregular, owing to the fact that the actual
leaves are absent. The point to note is that a Fibonacci ratio is
soon regained. The “34” curves are carried almost to the centre
of the disk, but at about six florets along the “55” curves these in
turn break away, suggesting a reduction to 21, and implying there-
fore a second reversal to (21+34). The central part of the disk
becomes still more obscure.
The phenomena of falling phyllotaxis in Cynara somewhat re-
sembles the rising phyllotaxis of the same capitulum, but with
diminished accuracy. Expansion followed normal lines, but re-
duction gives discontinuous systems.
In the same way Helianthus presents reduction reversals, but
they are less marked. For example, the head (34+ 55), taken as a
type (fig. 15), shows the short curves remaining unbroken for 11-12
members. The number of florets in which small adjustments were
PLATE NIII.
Pitas Es
Portion of the same capitu-
ghtly, showing falling systems toward
lum magnified sli
Fie. 53b.—Cynara Scolymis,
the centre.
Fie. 58¢.—Cynara Seolymus, L. Capitulum (55+ 89)
shaved down to the ovaries of the disk-florets.
PLATE NIV,
*pe}jop aie sutaysAs
Suey eq} jo syzed oxy : ATWYSs peyruseur wnyn4
-1dvo oures ay} Jo UOTJIOg = *sunwwD SNYQDNIE——"QEG “OT
“(LP + 63) 8]210Y-YSIp 0} Futstt (T+ TT) atonpoaut
‘cnngyideos snojemouy ‘sniwun snyjunyayy—"vFG “OT
RISING PHYLLOTAXIS. : 135
made before the system was altered and distortion ensued was
thus 6-700. The large capitulum (89+144) (fig. 13), similarly
shows unbroken short curves for 13-14 members, giving a total of
nearly 1900 florets before the reduction set in. An anomalous
head (29 +47) (fig. 54), is only constant for about seven members
along the shorter curves, or for a total of 320, the reduction taking
place about half way in from the edge of the disk. In rare cases
alteration may commence right on the edge and the parastichies
then become too irregular to count. Although the general plan of
reduction is clear, it does not appear to be sufficiently accurate to
warrant the construction of theoretical diagrams. It is possible,
however, that the actual change is still rapidly effected, and the
mechanism of transition should again be denoted by a reduction in
terms of the (2, 1, 2, 1, 2), etc, expression which marked the
transit from one Fibonacci ratio to another.
ASYMMETRICAL CONSTRUCTIONS IN FLORAL DiaGRAms.
Referring back now to the general scheme for the orientation
of the cycles of the Schimper-Braun series (fig. 1), it becomes
increasingly obvious that such constructions and their inter-
pretations have no necessary connection with spiral systems, but
are merely the expression of the relationship of successive terms
of the Fibonacci series; and as already noted, Schimper and Braun
added nothing to the Spiral Theory of Bonnet, but intercalated the
Fibonacci ratios, which thus constitute an entirely independent
generalisation. The fact that the tabulated orientations do agree
with phenomena observed in the plant is really the expression of
the rise of the Fibonacci ratios in the sequence 2, 1, 2, 1, 2, etc.,
as presenting the most symmetrical approximation to equal
division of the new paths of growth.
It is further apparent that it is impossible to construct any
accurate presentation of a spiral system in terms of circles; and as
soon as circles are adopted, a source of error is introduced which
leads one on unconsciously to further fallacies. It is impossible
to interpret an asymmetrical system other than by spiral con-
1386 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
struction, and the character of the spiral must therefore be known ;
yet the circular plan has been generally adopted in the construction
of floral-diagrams. Take, for example, the “2” spiral, or quincwn-
cial type, which characterises the calyx or perianth of the great
majority of Dicotyledonous types: in the floral literature of the
period which marks the evolution of the floral-diagram at the
hands of the Wydler-Eichler school, from a mere transverse section
of the flower-bud, one constantly comes across the tendency of
older observers to deduce an abstract ideal type of construction
which represented the general average of a mass of observations.
Thus the remarkable prevalence of whorled arrangements in
the floral members of the majority of floral types, which also holds
to such an enormous extent in the ontogeny of the essential organs
even when the calyx remains spiral, led to the assumption of a
series of concentric circles as the basis of the floral-diagram ; since
again such circles were easy to draw, and spiral construction was
not particularly obvious except in the case of a few of the Ranun-
culaceae and allied families, for which the application of theoretical
diagrams was recognised as being extremely difficult. ‘The intro-
duction of the doctrines of evolution led to the result that these
ideal generalised types were frequently interpreted as actual
primitive forms, without any further phylogenetic evidence being
required ; circles were thus retained as being sufficiently accurate,
and the “2?” spiral thus became conventionally represented in
terms of two circles, two members being placed on the outer and
three in the inner, constituting the Dicyclic calyx.*
* Such a theory as that of the Dicyclic calyx affords a good example of the
manner in which an abstract morphological generalisation, obtained from an
average of a large number of observations, becomes mistaken for a phylogenetic
one and ends by obscuring the very phenomena it was intended to elucidate.
It is clear that the number of members selected by the plant to serve as a pro-
tective investment, or an attractive cycle, depends primarily on the relative
tangential extension of these members: that 4=(2+2) are most frequently
selected in a dimerous flower, or 6=(3+8) in the case of trimery, merely
indicates that the members are relatively narrow, and subtend, when adult, an
angle which is less than 180° or 120° as the case may be. Where the width is
relatively considerable (cf. 2 sepals, Papaver, 3 sepals, Tradescuntia), a single
cycle suffices and is therefore employed ; and the necessity for a dicyclic calyx
thus falls to the ground. This subject will be further considered under the
RISING PHYLLOTAXIS. 137
Few systematists probably ever troubled about the question as to
whether the helix of Bonnet and Schimper became an Archimedean
spiral in the ground-plan; they accepted the construction and
expressed the “orthostichies” of the genetic spiral as radii of the
circle, assuming therefore that they were also radii vectores of the
spiral.*
So long as this is regarded as a simple method of constructing a
spiral system, and the convention is granted, no particular harm is
done, and for conventional floral-diagrams the same construction
may be retained so long as it is clearly understood that such a
diagram does not present the facts of the given floral-structure
accurately, but only a symmetrical version of them. Thus in a
floral-diagram of the Buttercup, it is possible to place 8 oblique
rows of stamens on a certain number of circles, or emphasise 13
rows on a smaller number, but the true (8+13) construction,
giving 8 oblique rows one way and 13 the other, requires a spiral
curve. On the other hand, if the circular plan be adopted and the
convention forgotten, error creeps in and may become magnified in
the course of further deductions. Thus, having placed three members
of a “2” spiral on one circle, and two on another, the next false
step was readily made in assuming that the spiral either actually
consisted of two such whorls in the plant, or might be interpreted as
heading “ Varying growth in lateral members”; it thus becomes referable to
the laws which control the tangential extension of foliar members, and
discussion must therefore be postponed until the angles normally subtended by
free and packed primordia have been tabulated for the different systems. (Cf
Mathematical Notes.)
* It is of interest here to compare the views of Schleiden (Grundriss der
Botantk, Eng. trans., p. 264), who in his masterly analysis of the principles of
phyllotaxis as they were discussed in his day, stood alone among German
botanists in his support of the theories of the Bravais, in that they were
logically based on mathematical laws, and deduced from the properties of a
“mathematical spiral” ; the fact that an almost indefinite number of mathe-
matical spirals may be proposed appears to have been completely forgotten.
The spiral in question was a helix wound on a cylinder, which has a parallel
screw-thread, but also makes equal angles with vertical lines drawn on the
cylinder ; it may thus be continued up a cone as a curve, which would on a
projection give either a spiral of Archimedes or a logarithmic spiral, according
as the former or latter property was allowed to determine the curve.
188 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
homologous to two whorls of three and two respectively; and this
spiral series (2+3) thus represented an intermediate condition
between a whorled (2+ 2) and a whorled (3+ 3).
These were, in fact, the views of Eichler,* and constituted his
theory as to the origin of a pentamerous flower, the effect of which
is noticeable throughout the whole of his classical systematic work.
That the (24+3) spiral system is intermediate between the
(242) and the (3+3) systems is obvious. The first is an asymmet-
rical construction, the two latter symmetrical systems; but it does
not follow that the “intermediate condition” is therefore a second-
ary derivative, while the (24+ 2) and (3+3) are primary formations.
Eichler’s interpretation naturally implies this peculiar standpoint,
and trimery and dimery are thus regarded as more primitive than
pentamery, notwithstanding the enormous preponderance of five
‘orthostichies” in vegetative shoots, a fact first noticed by Bonnet.
However, this view being granted, it follows that Monocotyledons
and trimerous and dimerous apetalous Dicotyledons must be the
* Reference to Eichler’s Introduction (Bliithendiagramme I., II.) shows that
these views apparently arose from a misinterpretation of the facts represented
by the three types of flower in Berberis vulgaris, Thusa transverse section of the
winter-bud (September) of a shoot which will flower in the next season shows
foliage leaves in the contact relations of a (2+3) system while the inflorescence
bracts are apparently (3+5). A longitudinal section of a later stage (November)
shows the flowers developing, and brings out the remarkable fact that the
terminal flower is the only one which has free space to grow. That this
terminal flower should under these circumstances continue the system of the
axis it terminates, and therefore present floral-members in five slightly spiral
“vertical rows,” as in Delphiniwm Ajacis (androecium), is perfectly natural,
and this normally occurs, On the other hand, the lateral flowers, whether
owing to specific tendencies, or the stimulus of close packing, tend to vary, and
so give the natural symmetrical variants (2+2) and (8+3) of the (2+3)
system. One would not be surprised to find the normal type in a lateral
flower, or the (+3) in a terminal, if these variations are determined solely by
the plant, and such again are the facts observed. The fact that the plant
selects a certain number of members to act as perianth segments and sporophylls
in the different cases, is wholly secondary to the mechanical construction which
originally produced them ; the rule the plant adopts is to take as many as will
completely fill a circle round the axis for each kind of member it requires, this
being 4, 5, 6 for the three types respectively, as shown by the proper geometrical
construction in terms of the numbers of curves.
RISING PHYLLOTAXIS. 139
primitive floral types; the fact that the majority of these again are
whorled, while in a great many the vegetative shoots remain in the
spiral condition, apparently presenting no difficulty. Simul-
taneously with the introduction of these ideas, Gymnosperms were
becoming finally separated from Dicotyledons, and the further
fallacy was interpolated, that since Angiosperms must have passed
through a Gymnospermic condition, that condition must necessarily
be the same as that observed in recent Gymnosperms which
happen to be all diclinous anemophilous tree-types; and thus in
spite of the teaching of the heterosporous Vascular Cryptogams,
dicliny as a primitive condition became added to dimery. The
general lines of the Decandollean classification, followed in this
country by Bentham and Hooker, were consequently more or less
inverted ; and it is so far possible that the systematic work of the
Eichler-Engler school rests on a very fallacious basis. It is to be
hoped that at no very distant date the pendulum will again swing
back to the Decandollean standpoint, since it is increasingly
evident that no hypothesis as to the phylogeny of Angiosperm
families can ever be acceptable to morphologists which is not based
on the standpoint that floral axes and members primarily obey the
same mechanical .laws of construction as obtain in asymmetrical
vegetative shoots, and therefore originally followed the same
simple Fibonacci ratio systems: that all primitive floral types
were therefore necessarily asymmetrical in construction, and
produced a considerable number of members in an indefinite
system ; the often abused term “indefinite” being used in the sense
pointed out by Schleiden as its only logical meaning, 2.¢., a con-
struction in which lateral members are set apart for their special
functions in an indefinite number along the same genetic spiral.
The fact that the present position of Systematic Botany may be
wholly erroneous, and that its errors may be traced back, in part
at least, to the neglect of the correct presentation of spiral con-
struction, may prove to be historically interesting.
Without going into further detail at present as to the con-
struction of floral-shoots, it may be pointed out that evidence will
be subsequently adduced to show that the vast majority of floral-
structures may be reduced to derivatives of two common asym-
140 RELATION OF PHYLLOTAXIS TO MEUHANICAL LAWS.
metrical constructions only,—the (243) and the (3+5), from both
of which pentamerous forms may be obtained.
I. The rules for constant phyllotaxis may be observed; ey., in
Delphinium Ajacis, (34+5) throughout; Calycanthus, (5+8)
throughout.
II. Normal expansion types are general among the primitive
forms; eg., Ranunculaceae and allied families exhibiting very
commonly (2+3) (5+8), Nigella, Magnolia; (3+5) (8413)
Helleborus, Aconitum Napellus.
III. The symmetrical variations of the form,—
(2+3) becomes (2+2) or (3+3),
(3+5) becomes (8+3) or (5+5),-
follow normal lines of production, by the equalisation of the
parastichy curves; the full
type thus commences by
adding normally to the long
curves, but remains station-
ary on the first attainment of
symmetry, while true dimery
and trimery are seen to be
secondary, as dicliny also un-
doubtedly is as well.
IV. The case of falling
phyllotaxis is rendered espe-
cially interesting, in that such
a reduced series naturally
closes the production of
lateral members at the end
of the floral axis and will be
noticed usually in the mem-
bers of the gynoecium (Ran-
Fig. 1.—Symmetrical version of falling unculaceae).
phyllotaxis in terms of circular con. The general phenomena of
struction instead of spiral. such a falling phyllotaxis hav-
ing been described as theoretically representing a fall of the Fibonacci
ratios along their normal sequence 2, 1, 2, 1, 2, etc., the scheme
RISING PHYLLOTAXIS. 141
already presented in fig. 1, as including such a descending series of
terms arranged:on a circular plan, becomes a valuable convention
for the satisfactory completion of the floral diagram of such a type,
in that, when the circles are cleaned out, it cannot be distinguished
by the eye from the appearances presented by the plant; while at
the same time it retains in its construction the theoretical suggestion
of what true reduction should have been if it had taken place
equally all round the axis.
142 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
IV. The Symmetrical Concentrated Type.
Tuts, the most highly perfected condition of phyllotaxis, is met
with more especially in the mechanisms of flowers, and also in
specialised assimilating shoots (Hippuris, Labiatae, Oleaceae,
Equisetineae, Dasycladeae, Characeae) presenting the case of “ alternat-
ing whorls.” In the theory of Schiraper and Braun it can only be
derived from a spiral system, by complicated processes of “ Pro-
senthesis”; while, on the other hand, very curious hypotheses that
spiral arrangement is secondary and derived ontogenetically from
primitive whorled conditions have been put forward by Henslow
and Airy.*
It is sufficiently clear, however, that these latter views ignore the
normal facts of spiral development as expressed by Hofmeister ;
and previous considerations of the normal asymmetrical concen-
trated type indicate that the “concentration” of the system is one
of the surest marks of its origin from the corresponding asymmetri-
cal case, by the assumption of true circular symmetry as a special
case of log. spiral construction. The system is thus simply defined
by the number of members in the alternating series, and accurately
planned in a diagram by symmetrical pairs of mutually orthogonally-
intersecting log. spiral parastichies; «¢, when the number of
members is very large, the construction is checked by counting an
equal number of contact-parastichies in either direction (fig. 55).
It may be derived from the preceding type either phylogenetically
or ontogenetically, by the growing zone becoming at any stage
* Trans, Linn. Soc., ser. ii, vol, i, 1875. Proc. Roy. Soc. vol. xxii., 1874,
297-307.
THE SYMMETRICAL CONCENTRATED TYPE. 143
definitely symmetrical, and thus producing a series of members
simultaneously instead of one by one.
The specialisation of the construction is thus simply indicated :
the cumbrous method of building a radial system by the addition of
individual members, which initiates the mechanism of the pre-
.
Fig. 55.—Symmetrical construction (5+5), representing the condition of true
pentamery as observed in the flowers of many Dicotyledons.
ceding type, is swept away ab initio, and with it vanishes all the
preceding geometrical considerations of spiral-vortices, ideal angles,
and restricted attempts at symmetry.
How small the actual change may be in a given apex, is indi-
cated by the close approximation of the genetic spiral to a circle,
K
144 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
and the approximately equal bulk of the members of one cycle in
the higher ratios of spiral phyllotaxis; as also, in actual ontogeny,
by the imperceptible differences of time between their serial for-
mation when growth is at all rapid.
The determination of the number of members composing a whorl
remains wholly a property of the protoplasm, though the marked
constancy in the retention of the number 5 in the flowers of
Dicotyledons, corresponding to an advance on the asymmetrical
system (3+5), indicates the influence of phylogeny on subsequent
specialisation ; on the other hand, the readiness with which whorled
pentamerous flowers vary to 6- and 4-merous, and trimerous to 4- and
2-merous, shows an increased plasticity in the system, and that
with the assumption of a true symmetrical construction, the causes
which led to the adoption of the series of Fibonacci must also have been
elommnated.
The same elimination of this series of numbers is observable in
the progressive rise of members in a whorl in correlation with an
expanding axis bearing more constant members (Meomeris), and
again in the progressive reduction of members along successive
degrees of ramification (Zquisetum).
Special interest attaches to the lower members of the series, «e.,
those in which one or two pairs of symmetrical curves plot out the
construction. The former gives the symmetrical formation of alter-
nating members in two rows at angles of 180°, generally known as
the distichous condition.
The latter, produced by two pairs of curves, on axes at 180°, is
that usually known as the Decussate system, giving four vertical
orthostichies. This appears to be very constantly correlated with
xerophytic specialisations in the assimilating system, and is often
quite independent of any dimery in the floral members (Clematis,
Calycanthus, Labiatac). It is remarkable in that it presents the
first doubled system ; and while it could not be produced from two
concurrent genetic spirals, it is possible that, given such a doubled
curve-system, secondary reversion to the asymmetrical condition
might be initiated from either pair, and a doubled spiral system be>
the result.
That the decussate system may be also produced as a variation
THE SYMMETRICAL CONCENTRATED TYPE. 145
of whorled trimery is further shown by the case of reduced
Monocotyledonous flowers; eg., individual flowers of Jris,
Lilium.
The case of three symmetrical pairs of curves at angles of 120°
which gives the typical trimerous Monocotyledonous flower, here
represents the full symmetrical case of the system (2+3), as is
shown by the partial retention of the spiral in ontogeny (Liliwm
candidum, etc.); but it may also occur as a variation of a decussate
type, as in the assimilating shoots of Fuchsia gracilis, Fraxinus,
Impatiens, and again as an extreme reduction of a pentamerous
flower passing through the tetramerous phase and thus independent
of the ratio series (Oenothera biennis).
Similarly the case of whorls of four members may have a threefold
origin, to be separated carefully in the consideration of floral
phylogeny: firstly, as an extreme variation of the decussate system
(foliage shoots of Fuchsia gracilis); secondly, an advance variation
of trimery, flowers of Crocus, Iris, Leucojwm, Lilium (more constant
in Paris); and lastly, a reduction variation from pentamery, the
most general case of tetramery, as found in the flowers of Oenothera,
Alchemilla, Cruciferae; and less frequently, Ruta, Jasminum,
HLuonymus, Ampelopsis, Viburnum, ete., etc. In the same way true
hexamery may be produced as a variant of pentamery, as in
flowers of Ruta, Jasminum, Ampelopsis, Viburnum, Heraclewm,
etc., supplying increasing evidence that with perfect symmetry
in construction the value of the series of Fibonacci is com-
pletely lost, although the phylogenetic relics persist to a very
considerable degree; due, no doubt, in many cases to the fact
that symmetry is only attained in the specialised floral mechanism,
while the parent shoot still retains its unmodified asymmetrical
and mechanical construction, so long as there is no direct ad-
vantage to be gained by substituting either radial or dorsiventral
symmetry.
As in the case of asymmetrical constructions, it is easy by
making geometrical drawings to obtain an idea of the bulk-ratio
for any given symmetrical system with a degree of accuracy quite
sufficient for any practical purposes, the ovoid curves inscribed
in the log. spiral meshes being taken as circles.
146 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
The following table expresses these results :—
Angle
Whorls of Bulk-ratio. ee pease subtended by
pa rhomb (“square”).
2 T2261. 115° 180°
3 lis : 1) ( a) 120°
4 2 a) 60° 90°
5 24:1 48° 72°
6 28:1 41° 60°
7 33:1 35° 51°3
8 37:1 31° 45°
9 41:1 28° 40°
10 46:1 25° 36°
Inspection of the bulk-ratio column, which may be assumed to
be fairly accurate when the angle subtended is 60 degrees or less,
is sufficient to show that the rise from pentamery to hexamery,
for example, would represent a comparatively small variation as
expressed in the formation of a larger and better nourished axis
which tended to produce members of a constant type.
The diagrams also illustrate the fact that whorled tetramery has
almost identically the same bulk-ratio as the (3+5) asymmetrical
system from which a spiral pentamerous flower is phylogenetically
derived ; while whorled hexamery almost equally approximates the
bulk-ratio 3 : 1 of the asymmetrical (5+8) system.
It is easy to adduce facts which fall into line with such generalisa-
tions, although they do not necessarily add any proof of the theory ;
for example, the latter case is of interest in connection with the
readiness with which terminal flowers of Campanula media vary
to symmetrical hexamery when the vegetative main shoot presents
the (5+ 8) asymmetry.
As an example of the perfect irregularity of the symmetrical
expanding construction, and its absolute independence of the
Fibonacci series, the vegetative shoots of Lquisetum Telmateia
afford conspicuous illustration.
For example: a weak foliage shoot of 32 nodes, the continuation
of a rhizome bearing leaves in whorls of 10-11, showed a rapid
THE SYMMETRICAL CONCENTRATED TYPE. 147
rise at first, culminating in a maximum at the 13th node, with a
gradual fall towards the slender apex; the whole shoot being of a
spindle shape in the bud and the leaf members approximately
constant in volume.
The leaves at successive nodes were as follows :—
11, 13, 14, 14,17, 20, 20, 22, 24, 27, 28, 29, 30; 29, 30, 26, 26;
26, 23, 23, 21, 19, 16, 14, 12, 9, 8, 6, 6, 4, 3.
The number thus ultimately falls to 3, which possibly represents
the ancestral number derived from the three segments of the apical
cell, as in the similarly constructed apex of the leafy gametophyte
axis of many mosses; although it is difficult to prove, even in
Equisetum, that since the protuberances which indicate the prim-
ordia appear to involve these segments, they are necessarily
dependent on the histological segmentation.
Another strong shoot (May 1901) including 40 internodes gave
similar results: springing from a rhizome of uniform construction
with 13 members in a whorl, the shoot reached the level of the
soil in 5 internodes, 13, 13, 16, 18, 22 respectively ; the maximum
was reached in 12 internodes, the additional ones being 24, 27, 28,
30, 33, 34, 36 respectively. As in the previous example, this
maximum condition was succeeded by a region in which variation
took place, the numbers for the next 5 nodes being 34, 36, 32, 34,
35. A steady descent then set in and was continued for the
remaining 23 nodes :—36, 34, 32, 30, 30, 30, 29, 27, 26, 24, 24, 22,
20, 17, 14, 12, 9, 6, 5, 5, 4, 4, 4.
Equisetum Telmateia thus affords an excellent example of the
possible independence of each nodal-formation of a symmetrical
system ; the bulk-ratio is independently arranged for each cycle
of members, and although it may remain constant when only a few
large primordia are inserted, the curious oscillation period between
the rising and falling series shows that in the case of relatively
very small primordia the mechanism is imperfect and only
approximates the number at each node. Similar phenomena occur
in the Dasycladaceae (Neomeris) ; in Hquisetum they become the more
striking, in that very regular constructions are often postulated for
the stelar system, which is only secondary to that of the leaves,
and thus less accurate than is generally supposed.
148 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Applying these generalisations to the simpler members of the
series, it would appear probable, then, that given a primordium
subtending a considerable angle, the chances of variation between
successive whorls of members would be correspondingly decreased.
The difference between three members in a whorl and four, for
Fig. 56. Descending symmetrical phyllotaxis: geometrical representation of the apex
of a shoot of Egwisetum Telmateia, showing irregular parastichies.
example,- being considerable, it would be expected that. individual
or specific variations to tetramery, or from pentamery to hexamery,
should, when they occur, affect all the whorls of the shoot, and this
in fact is the general case. The occurrence of such a condition as
that observed in a starved plant of Cucurbita Pepo, which is normally
THE SYMMETRICAL CONCENTRATED TYPE. 149
very constantly pentamerous, in which 4 sepals were followed by 3
petals and the anther lobes of 2 stamens, would form so marked an
exception that it would be readily recognised as a deformity. It
will further be noted that, expressed in terms of the parastichy
curves, a symmetrical whorl of 4 members will be contained by (4
+4) curves, and a whorl of 5, (545), ete. It thus follows that any
change in a symmetrical system, in which symmetry is retained
from node to node, implies the addition or loss of at least two curves
simultaneously, one in either direction, since the addition or loss of
an uneven number would at once throw the construction into an
asymmetrical form.
That the increase or reduction of the members of a whorled system
may often be due to variations in nutrition, so that the bulk-ratio
may be involved in a manner similar to that described for asym-
metrical types, is clearly suggested by the enormous range of varia-
tions observable in some flowers, especially Papaver somniferum ;
under varying conditions of cultivation the number of carpels which
may reach 15 in a strong plant readily falls toa minimum 4 in
progressively starved plants, while the aggregate number of stamens
which present an irregularly symmetrical system may be simul-
taneously reduced from over 500 to 8.
Better examples are met with in the progressive reduction along
successive axes of the same plant, homologous again with the reduc-
tion along the members of the Fibonacci ratios in successive
ramifications previously noted (Helianthus, etc.), but differing again
in the complete absence of these values, and thus affording a much
more gradual decline: eg., Ruta commonly produces terminal 6-5-
merous flowers, while 4-merous are practically constant in the
ultimate scorpioid cymes. Most striking is the case of Sempervivum :
three plants of S. italicwm, growing in the same pot, gave the
following numbers for the sepals and petals of successive floral axes
(the construction is not absolutely constant throughout individual
flowers) :—
T. T Te Tue Tie ve Tn.
(12 13 12 11 ll 10
I 14 13 12 12 12 11
12 13 11 11 1
150 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
T. Ti. Tu. Tin. Tiv. Tv. Tv.
14 12 12 12 11 11
IZ 12 12 11 11 10
M5 é 15 “ 11 10 10
12 12 12 12 11
12 12 11 11
F 12 13 12 12
III. . , 14 12 12 7
(12 13 11 13
the reduction being thus fairly progressive from 15 to 10 along the
ultimate ramifications.
The case of Hgwisetum further illustrates the mechanism of
addition and loss of members. No rules are here applicable; the
number added may be quite irregular, and in the case of falling
symmetrical phyllotaxis, the amount of adjustment required in the
mechanism must be very considerable. So marked is the rising and
falling sequence in the vegetative shoot of #. Telmateia, and so
relatively short is the region over which a constant phyllotaxis
would be possible, that it may be said that this plant never possesses
anything better than an irregularly symmetrical construction; the
obvious part of the whorled appearance being produced by the
adjustment of the secondary zones of growth which constitute the
internodes. The apical portion of such a plant may be taken as a
type of “irregular symmetry,” which is again a distinct phenomenon
from normal asymmetry, but, as will be seen, incapable of distinction
as a primary phyllotaxis construction from irregular asymmetry.
Taking the latter example of #. Telmateia,* and translating it into
* Tt will be noticed that this affords what may be termed an architectural
conception of the Lqguisetwm shoot, based on the view that all the leaf members
are of equal value, and that Hquisetum is only a modern highly xerophytic
edition of a plant which once presented normal vegetative leaves ; on the other
hand, it does not accord with the accepted versions of the construction of the
apex of such a shoot, usually found in text-books, the older researches on this
plant having been conducted from the standpoint of the dominant influence of
the apical cell (Hofmeister, Reess, Cramer). Once this cell is deposed from
authority, it will be seen that it is extremely difficult to prove whether the
annular ridge really belongs to a cycle of three segments which have been “a little
displaced” (Reess), or may not equally well be regarded as the result of an
independent symmetrical annular impulse which must nearly approximate these
superficial cells. The same annular ridge again represents such an early gamo-
THE SYMMETRICAL CONCENTRATED TYPE. 151
a diagram in the transverse plane,in the form of a floral-diagram,
by placing the observed number of apparently perfectly similar
members on a series of concentric circles to represent the whorls, a
scheme will be plotted out which is therefore identically that of the
telescoped axis (fig. 56). In such an irregular system the paras-
tichies present a hopeless medley, straight in places, curved in others,
but still roughly equal in number when counted in either direction,
at a given level, so far as they can be counted. When the con-
struction circles are rubbed out,no interpretation of such a condensed
system is possible to the eye; in Hquisetwm, the symmetrical
condition is rendered obvious by precocious gamophylly and second-
ary elongation of the system ; but in the absence of such a second
zone of growth, it is evident that it could only be included under
the loose term “indefinite,” and that no such system can be verified:
nor can the construction be described. When such constructions
occur in flowers, as very noticeably for example in Clematis, it is
possible to regard it as a degenerate symmetrical one, when as in
this case the whole of the other phyllotaxis relations of the plant are
symmetrical ; but it is clear that all reduction systems must closely
resemble one another in their capacity for becoming undeterminable.
In the presence of a primary phyllotaxis system, therefore, whether
in the case of asymmetry or symmetry, it is only possible to give an
phylly of the lateral members that it is evident that leaf-production is no longer °
normal, while the interpretation of “ primary and secondary teeth” (Reess) is
also doubtful. Thus Hofmeister, when he first investigated the apex, called the
whole sheath one leaf which produced more teeth as it became older (which is
certainly one way of interpreting the termination of the shoot as expressed in
fig. 56), and that 4 was the primary number (Higher Cryptogamia, 1862, p. 270).
Reess futher admitted the possibility of the formation of a number of primary
teeth (7-8), of which 3 was not a factor, so that the sextant segments must have
been unequally affected. It is remarkable that these views should have been
accepted without any drawings or accurate evidence in favour of them ; the
fact that the annular ridge is formed quite independently of the apical segmenta-
tion being sufficiently clear to the unprejudiced eye. (Cf. Cramer’s drawings,
Pflanzenphys. Unters. Nigeli und Cramer, iii. plate xxxiii. figs. 19, 20; xxxiv.
1-8 ; also in text-books.) The same fact has been pointed out by Schwendener
(Botanishche Mittheilungen, vol. i. p. 153), who examined the critical case of E.
scirpoides, with the cycle of three members. There is thus no doubt that the
symmetrical formation of the impulses which produce the lateral members is
wholly independent of the asymmetrical segmentation into cell-units,
152 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
account of the phenomena when the construction remains constant for
a period sufficient to give by recapitulation the appearance of definite
contact-parastichies ; an unequal number of intersecting curves means
asymmetrical construction, an equal number implies true symmetry.
The slightest deviation from absolute symmetry produces an
apparent spiral effect, just as the failure of a circle to come round
on itself in the smallest degree would produce a spiral curve,
and the subjective effect, as judged by the eye and interpreted in
terminology, is quite disproportionate to the cause,
Thus the parastichies of wall-papers and tiles on a roof, quoted
by Sachs, are as clearly the expression of a symmetrical con-
struction as the vertical and horizontal lines of the pattern.
Equally good examples are often seen in the arrangement of
imbricating ovules in an ovary (Aselepias) or scale-emergences on
fruits, etc. (Raphia, Acorn-cup); so long as the construction is
regular, the secondary “ parastichies” present an equal number in
either direction; but the slightest deviation from strict regularity
at once renders these curves unequal or irregular, and a spiral
system is the result. Thus in the Sago-Palm fruit (Raphia, fig.
72), the emergences are relatively very large, and when regularly
formed they fall into series giving symmetrical curves (6+6),
(7+7); but any trifling irregularity in formation spoils these
rows, and thus (6+7) is equally common: the secondary spiral
appearance thus produced does not imply that the scales constitute
a phyllotaxis system, or that the members are leaves, although
regarded merely as adult structures the resemblance is very
striking; the suggestion that this similitude in lateral appendages
of different value morphologically may be the outcome of a
common Jaw of growth is very obvious.
The phyllotaxis phenomena of whorls and spirals observed on
the plant are thus merely the outward expression of the distinc-
tion between symmetrical and asymmetrical construction. In the
primary system, seen in Zone I., when the original lateral contacts
are maintained, the most obvious sign of the mode of growth is
the equality or inequality of the diagonal construction lines
(parastichies), these being more readily checked by the eye than
the complementary lines of construction, which may be circles or
‘
THE SYMMETRICAL CONCENTRATED TYPE. 153
spirals hard to differentiate. The mathematical fact that the
number of members represented by the integer which is a common
factor of this parastichy ratio are of identical value, becomes
expressed in the number of members left at a node when the
internodes are subjected to secondary elongation. If the paras-
tichy ratios are equal, the system pulls out as rings of members
of the same number, and a similar number of subjective spirals
may be drawn diagonally from node to node; if they are unequal,
but divisible by a common factor, for example 3, then 3 members
are left at each node and 3 spirals may be so drawn in one direction;
but if divisible by unity only, a single member is left isolated at
a node, and the one subjective spiral which may be drawn through
the whole system becomes dignified by the name of ‘genetic-
spiral,” in that it attains an enhanced ontogenetic value according
as the rate of production of the system in time becomes decreased.*
* Since the postulated change in the mechanism of symmetry involves the
addition or loss of construction curves at least two at a time, it becomes of
interest to see to what extent deviations from such a symmetrical change may
be found. Thus the addition or loss of one curve only would produce imme-
diate asymmetry which would be expressed by a transition from whorls to
spirals. Such a spiral series would again be of the maximum-concentrated
type, since the contact-parastichies would only differ by 1, and would possess
as a complementary system the least-concentrated type, in which one spiral
passes through all the members as a contact-line, and winds around the stem
(cf. fig. 36). The extent to which such a genetic spiral becomes obvious to the
eye may differ according to circumstances. Cf. Lycopodiwm Selago (5+ 6) and
Cactaceae (6+ 7), in which the construction is not seen on the cylindrical axis,
but is readily observed in section of the apex, or on the apex as in Cacti. On
the other hand, it has already been pointed out that the symmetrical develop-
ment of the foliar members in Lqwisetum is marked by congenital gamophylly ;
a transition to the asymmetrical condition would therefore be expected to show
similar gamophylly along the course of the ontogenetic path, and the lateral
members thus form a spiral fan winding round the axis. Such variations have
been frequently described as monstrosities (Milde, Reinsch), and spiral portions
may thus be intercalated in a whorled system.
Of. Reinsch, Equisetum Telmateia ; Flora, 1858, Taf. it fig. 3, p. 69 (a
spiral for 203 members intercalated between whorls of 30 and 28); Flora, 1860,
p. 737, Taf. vii. fig. 9. A similar reversion to asymmetry is described for
Hippuris and Casuarina (Reinsch) ; while it is of interest to compare the
spiral ridge thus formed in Zquzsetum along the genetic-spiral of such systems
with the ridges of Cacti which often follow the paths orthogonal to the genetic
spiral (phyllody spirals, fig. 63). ;
154 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
V. Asymmetrical Least-Concentrated Type.
In its simplest form, as expressed in terms of single cells, this is
the condition which obtains in the derivatives of the three-sided
apical cell of Ferns, Equiseta, and Muscineae, where the three series
of segments form superposed series; a line joining their centres of
construction becomes the ontogenetic log. spiral, while the three lines
passing radially through the centres of construction of the super-
posed segments also form three log. spirals, so that no two members are
mathematically superposed, within any limit of construction. The
system is thus defined by the number of these “vertical” spiral
rows.
In the case of the cell-segments of Pteris root-apex, these log.
spirals were not obvious, owing to the fact that only a few members
are shown in one transverse section, although, owing to their
rectangular construction closely approximating 1: 5, more members
were seen than can be plotted out in a normal orthogonal curve
system. The fact that the arrangement naturally follows from the
presence of a three-sided cell, in which each segment produces a
foliar outgrowth, while the presence of the three-sided apical cell
may itself be a sign of a primitive method of concentrating the
terminal ramifications of a filamentous Algal type, lends consider-
able weight to the view that this method may be phylogenetically
one of the oldest constructions, so far as it occurs in Mosses.*
* Cases in which the relative size of the cell constituents of the plant-body is so
great that the arrangement of the lateral members is apparently within the control
of single cells, may be conveniently left for the present, and the discussion of
phyllotaxis confined to those cases in which the space form of the organism is
ASYMMETRICAL LEAST-CONCENTRATED TYPE. 155
In Filicineae and Equiseta, however, considerable departures have
been made from the type so far as the origin of the lateral foliar
membersisconcerned. Thus, while in Zguisetum circular symmetry
is apparent almost immediately behind the apex, and the number
of members in a whorl of leaves is by no means necessarily a number
of which three is a factor, nor bears any relation to the series of
Fibonacci; on the other hand, in Ferns, a specialised concentrated
system may be in full operation, and thus Aspidiwm Filix-Mas, with
a three-sided apical cell, produces foliar members and a correlated
stelar meshwork in the system (5+8), (3+5), or (2+3) (fig. 35).*
independent of its histological composition. The limitation, for the present, of
the term “leaf” to such a massive protuberance avoids the difficulty of dis-
tinguishing between leaves, branches, or mere hairs in Algal forms. Special
interest attaches to the three-sided cell of the Muscineae, since this directly cuts
off the segments which become lateral members, and in that in the majority of
forms, the arrangement of leaves becomes discussed in terms of the Fibonacci
series. Thus Fontinalis antipyretica, with a tetrahedral cell like that of Equisetum,
gives keeled leaves in three well-marked spires, which straighten out on elongated
shoots, but on short thick ones compare with the spires of Pandanus (Goebel,
Leitgeb.). In other cases (Polytrichum formosum type) the apical cell divides by
oblique septa in a constant manner, giving three series of oblique segments, the
three spires being so much exaggerated that “ orthostichies” may be expressed,
in high ratios, of both Fibonacci and anomalous series (Hofmeister, Miiller,
Lorentz, Goebel). Torsion is admittedly absent (Goebel), and the leaf-traces in
the stem follow the same coiled three-spired series. It thus becomes a question
as to whether this oblique segmentation is really the cause or a consequence of
the formation of new growth-centres in a definite manner within the substance
of the apical cell, and that the whole mechanism of asymmetrical growth, which
in more massive plants produces a Fibonacci system of cell-aggregates, is not here
enchained by the necessities of cell-segmentation, so that the new lateral growth-
centres are never sufficiently free to assume the homologue of a spherical form,
correlated with a centric distribution of growth-energy ; and the exigences of
histological division may thus effectually mask the true asymmetrical construc-
tion. It may be noted that the oblique leaf segments ultimately produce very
fairly symmetrical leaf-forms, and that the space-form of the adult shoot com-
pares very favourably with that of ordinary leafy stems. (Cf. Goebel in Schenk’s
Handbuch, vol. ii. p. 373; Miiller, Prings. Jahrb., vol. v. p. 247; Engler and
Prantl, Nat. Pflanz. Fam., Musci, p. 178).
* De Bary, Comparative Anatomy, p. 285.
Hofmeister attempted to derive the ;°; phyllotaxis of Aspidium Filia-mas from
the segmentation of the three-sided apical cell, although the segment walls were
clearly parallel to the sides. Hofmeister’s view that the genetic spiral was
necessarily homodromous with the cell-spiral and each segment gave a leaf, is
156 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Tt has been previously pointed out that the concentrated and
non-concentrated symmetrical conditions are only the limiting cases
of spiral constructions which vary in the degree of concentration,
all being concentrated to a certain extent in relation to the case of
superposed whorls; the most concentrated asymmetrical system
being that in which the number of intersecting parastichies most
nearly approximates equality ; the least concentrated, that in which
they differ most widely.
It is thus clear that the least concentrated types must have one
of the members of the ratio unity, and the lowest members of the
normal phyllotaxis series (1+1),(1+2) may be therefore isolated
as representatives of such systems. In this construction other
contact parastichies are necessarily wholly absent (¢f Scheme B,
fig. 20); the one long curve becomes the ontogenetic spiral, and the log.
spiral shorter curves become vertical spiral rows which may be con-
ventently described as “ spires.”
Thus the two-spired type occurs in Gasteria (figs. 570, 58a),
and the three-spired type in Cyperus, Pandanus, Apicra spiralis
(fig. 59a, 6). Such two-spired plants occur in species of Gasteria
mingled, on the one hand, with specimens exhibiting normal ratio-
series (3+5) or (243), Gasteria ensifolta, G. candicans; and, on the
other hand, with the special case of symmetrical (1+ 1) construction,
G. obtusifolia (fig. 57a).
So closely are these connected that seedlings vary in the same
batch (fig. 580). As the succulent dorsiventral leaves spread out,
the two spires become very pronounced; but any assumption of
torsion in one plant more than another, or, in fact, in any such
obviously put out of court by the fact that the phyllotaxis spiral is often antt-
dromous, and normal Fibonacci phyllotaxis phenomena may be found associated
with a two-sided apical cell. (Cf. Schwendener, Botanische Mittheilungen, vol. i.
p. 156.)
Nor was there ever any evidence in support of the older view beyond the
standpoint of the dominance of so special a mode of cell-construction. On the
other hand, comparison of Equisetum and Aspidiwm show that whatever the
“orowth-centre” may be, or whatever its nature, it is not localised in the nucleus
of the apical cell, but must be either a finite mass larger in these cases than a
single cell, or else represents a general function of the whole protoplasmic sub-
stance of the apex comparable with the somewhat allied conception of Polarity.
ASYMMETRICAL LEAST-CONCENTRATED TYPE. 157
succulent forms, in order to space out these leaves to better
advantage with regard to light, is clearly out of the question, when
the xerophytic structure indicates that such exposure is not desired
and is as purposely avoided by assumption of the symmetry as in
the parallel decussate type.
Since (1+1) gives a normal symmetrical construction with one member
only at a node, it is difficult to bring these two-spired types into line
with the normal asymmetrical series. The deflection of the members
is so slight that it appears possible to regard the case as one in which
the (1+1) generating curves become slightly unequal, and thus produce
asymmetry of the form 1 : (1+ a), where a is very small. From this
point of view the two-spired Gasteria becomes of greatest interest, in that
it appears to present an example of secondary symmetry which is with
difficulty maintained from node to node, t.¢., the curve does not keep
true.
The three-spired type, familiar in the leafy shoots of Pandanus
and Cyperus, is apparently similarly derived from a (1+2) system.
A transverse section of the foliage-bud of Cyperus alternifolius
shows the three spires very clearly (fig. 51), while the course of
the genetic-spiral is as clearly marked as in the case of the seg-
ments of the apical cell of the Fern-root (fig. 51, left-hand
spiral through 1-9). The spires become again obvious when the
axillary reproductive axes are developed in ascending series in
November—December (fig. 596).
The leaves of Cyperus are highly specialised from a biological standpoint.
The first formed members on a shoot are wholly sheathing, so that their
phyllotaxis cannot be determined in the full-grown buds; the foliage
leaves elongate tangentially and fold in a peculiar manner without
increasing in radial depth to any extent after their first formation. As
a consequence the curves soon become approximate Archimedean spirals
so far as they can be judged by the eye ; but, as previously pointed out,
it does not necessarily follow that such spirals of Archimedes imply
torsion. The formation of special folded strap-shaped members is a
secondary biological phenomenon which almost effectually masks the
orthogonal system so far as it is visible at the apex.
Thus, it is impossible to say from the direction of the spirals whether the
three spirals seen are the complementary “spires” of a (1+2) system
or the three shorter curves of a (2+8), since a left-hand genetic spiral
would work out these same curves in either case. The interpretation
taken, that the (1+2) system is adopted, is based on the fact that two
158 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
members make contact round the axis, and the five “spires” of a (2+3)
system cannot be traced. The special type of folding may be regarded
as the biological exaggeration of the “bean-like” form of the ovoid
curve in a (1+2) system. (Cf. Mathematical Notes.)
A similar spired appearance will also be secondarily produced
in all types in which the numbers of the contact-parastichy ratio
differ by unity: thus—
(2+8) exhibit the complementary systems (1+ 5) Cereus hybrids, seedlings.
3 +4) ” ” ” ql + 7) Sedum reflecum.
(4+5) ” ” » (1+ 9) Cereus pasacana.
Lycopodium Selago.
(+6) » ” » (+11) Echinopsis
Zuccarinianus.
Lycopodiwm Selago.
(6+7) 3 $5 » (1+183) LEchinopsis multiplex.
(7+8) Rs ‘ » (1415) LEchinopsis Eyriesiz.
so that, while the parastichies may be readily counted, the one
curve of the secondary system becomes the ontogenetic spiral,
while the log. spiral “ orthostichies” orthogonal to this curve form
the respective number of “spires” (fig. 63, (6+7)).
Such types are best seen in the Cactaceae, where the latter
curves are frequently emphasised by a biological production of
ridges along their course; the primary parastichies are then
counted by taking members in succession along adjacent ridges;
the secondary curve which gives the genetic spiral, along alternate
ridges, forming an obvious spiral winding around the apex of the
plant.
Examples of the seven-spired type occur in vegetative shoots of
Sedum reflecwm and Euphorbia biglandulosa, in which a (3+4)
system occurs as a specific variation; in the former, the repro-
ductive shoots assume the symmetrical constructions (8+8) and
(6+6), while in the latter a normal Cyathiwm is produced.
The seven-spired effect produced in the (3+4) system of Sedwm
reflecwm (fig. 76), and Euphorbia biglandulosa (fig. 77), as also the
five-spired system of (243) Huphorbia myrsinites, is directly com-
parable with the three-spired screw of Pandanus, the special
formation of the last case being intensified by the condensation
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ASYMMETRICAL LEAST-CONCENTRATED TYPE. 159
of the axis and the imbrication of the stout folded leaves. A
similar condensation in the case of E. biglandulosa would produce
equally good screw twists (fig. 776), while an identical exaggerated
spiral is seen in the winter-shoots of S. reflexwm (fig. 76d).
Apicra spiralis, most commonly a five-spired form, (2+83),
varies to the three-spired form (1+2), and is then very similar
to Cyperus and Pandanus; owing to the greater succulence of
the leaves, however, the system appears less telescoped, and the
spiral twist less striking (fig. 59a). Section of the apex shows
the same phenomena as those figured for the three-spired Cyperus.
Pandanus is also identical.
Other good examples of such constructions are afforded by
the shoots of Lycopodiwm Selago (4+5), (5+6) (fig. 78), where
they occur in conjunction with true whorled systems (3+3),
(4+4), (5+5) (figs. 79, 80).
The production of “spired” types is of special interest in that
in several cases the spires are extremely well-marked (Pandanus),
and from their approximation to helices have been made the
chosen examples of torsion theories. “As previously noted, the
appearance of Archimedean spirals, or true helices in the case of
cylindrical axes, will only be produced when the members attain
accurately equal bulk; «2, when they definitely cease further
growth on reaching a certain specific volume. In such cases the
“orthostichy” curves become straight lines; but so long as any
growth is taking place, however little it may be, the similar mem-
bers will retain the gradated series in which they were formed,
although the difference between adjacent members, consequent on
the retarded rate of growth, may be so small as to be inappreciable
to the eye. .
Ultimate appearances are complicated by the fact that cessation
of growth may take place in two ways: either, as in typical and
theoretically uniform leaf production over a considerable length
of axis, members grow to a certain size and then stop; or, growth
may diminish and cease uniformly throughout the whole system,
with the result that the completed system retains to a very con-
siderable degree the graduated sequence of its ontogeny; this
being well seen in seasonal cessation of growth, as in the production
L
160 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
of buds and similar arrested systems, of which the Pine-cone affords
a good illustration.
In the first-mentioned case the resultant curves will be spirals
of Archimedes or helices; in the latter, modified logarithmic
spirals, which may be conveniently described as “retardation
spirals.”
In the former case, again, the visible result will be the straighten-
ing out of the * orthostichy lines,” and the spires of a spired system
may thus become ultimately quite straight, such an effect being
well marked in typical Cacti whose seedlings have obvious spiral
ridges. The consideration of such growth-forms as these also
illustrates the fact that the final effect is due not only to the
assumption of equal volume in the members themselves, but also
to the attainment of equal length in the secondary zones of elonga-
tion which constitute the internodes. So long, therefore, as the
internodes are growing, the same appearance as that presented by
a gradated series of members will be maintained, even when these
members are practically equal. The spires of Euphorbia biglandu-
losa (fig. 77), and Sedum reflecum (fig. 76), thus continue to be
well marked after the leaves have reached the adult condition
owing to continued growth in the main axis. In the limit, the
rows become much straighter, but usually only after the fall of
the leaves. The “orthostichy” lines thus appear to become
straighter and straighter, as growth slows down in successive
members and internodes; but they will always be spiral lines so
long as growth continues throughout the whole system.
Spiral “ orthostichy lines” and “spires” are thus usually more
obvious in buds and bud-sections than in adult structures, as
originally noted by Bonnet; while if the whole system stops
growth simultaneously these spiral orthostichy lines or spires
become fixed, and the resulting structure has the appearance of a
permanent bud.
In such a construction the secondary phenomenon of dorsiven-
trality produces very striking results. Thus the fact that a leaf
increases tangentially to a greater degree than in radial extent may
be regarded as due to a diminished radial rate of growth. With-
out going into further detail at present with regard to such a
ASYMMETRICAL LEAST-CONCENTRATED TYPE. 161
standpoint, it may be noted that the effect of progressive dorsi-
ventrality in a growing system will be to exaggerate the curvatures
of all the spiral paths. Thus the attainment of a degree of dorsi-
ventrality sufficient to make a member about twice as broad as
thick, as in the leaves of Abies, ete., will result in the fact that the
“ orthostichy”” lines or “spires” become as curved as the shorter
paths of the normal curve tracing, while these latter become as
markedly curved as the normal longer paths. With a still greater
degree of dorsiventrality the spires become still further pronounced,
so long, that is to say, as the system is either still growing, or else
has stopped altogether.
The difficulty in the case of Cyperus and Pandanus is, however,
not to prove that the curvature of the so-called “orthostichies,”
which is sufficiently clear in a section of the apex, may be due to
torsion,* since in theoretical construction they should be curved
and not straight; the question is why, with so great an assumption
of dorsiventrality, these lines are not much more curved? This
may be possibly very largely due to the special mode of folding the
strap-shaped leaves into one another; as they grow they slip over
each other in such a way that they must form three rows in the
bud, and the assumption of a divergence angle of 120°-126°
(Schwendener) may be thus quite secondary. For example, in
Cyperus (fig. 51), the last leaves being rudimentary do not fold, and
in a section cut apparently quite transversely the divergence angle
between 6 and 7 was 134°; beyond these members the angles
vary owing to change of system, while other irregularities are
observable in the last folded members. There is no real necessity
to postulate torsion, nor is there any ready method of proving it.
So great is the alteration in such systems owing to the effects
of rapid retardation in the rate of growth behind the apex, that
the log. spiral construction, founded on theoretical uniform growth,
completely fails to represent the results attained in the plant.
One fact alone remains clear: in a construction in which growth
is rapidly slowing down, and the members acquiring approximately
equal radial depth, but still elongating tangentially, the appearance
* For torsion theory ¢f. Schwendener, Botanische Mitthetlungen, vol. i. p. 163,
162 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
of Archimedean spirals will be subjectively produced, and the
orthostichy lines thus appear as if they ought to be straight. But
until such radial equality is produced the curves cannot be spirals
of Archimedes, and the “orthostichies” cannot be straight, what-
ever else the nature of the spiral may be. The assumption that
the orthostichies should be primarily straight thus entirely falls
to the ground, and torsion theories based on such hypotheses are
unnecessary.
SYMMETRICAL NON-CONCENTRATED TYPE. 163
VI. Symmetrical Non-concentrated Type.
LikE the preceding, a comparatively rare formation, this forms
the system known as superposed whorls.
Similarly, also, it is more general as expressed in terms of cells,
than of lateral members of more massive character, being, in fact,
the conceivably theoretical case for the primary arrangement
of isodiametric cells in the growing points of all Phanerogams,
and well seen in the unmodified tissues of many roots (¢f. Zea).
The remarkable absence of concentration systems in cell-tissues,
while these form the characteristic feature of the arrangement
of massive primordia, affords confirmation of the hypothesis that
concentration is always derived secondarily through a spiral
construction. The presence of superposed whorls in the vegetative
shoot is doubtful, but in floral mechanisms it is more general,
and in a large number of cases generally accepted as being of
secondary origin. From the standpoint of the theory of Schimper
and Braun, superposition of the members of successive whorls
naturally followed from their constructions for superposed spiral
cycles, and any deviation from such superposition had to be
accounted for by prosenthesis. The present standpoint, that
alternation is the normal and primitive condition, thus renders
many phylogenetic generalisations improbable. The fact that
in higher plants, whorled types appear to be always reached
vid a concentrated asymmetrical construction, suggests, therefore,
that true superposition is always secondary. The logical con-
sequences of such a view have an important bearing on the
structure of floral organs. It becomes necessary to distinguish
164 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
between superposition which is mathematically accurate and that
which is only apparent to the eye.
The determination of phyllotaxis systems in flower-shoots
in which the construction is not continued for a sufficient
number of members to judge whether the apparent orthostichies
are truly vertical or really spiral, may present a difficulty.
Thus, spiral flowers may be constructed in the systems (1+ 2),
(2+3), (3+5), giving respectively cycles of 3,5, or 8 apparently
superposed members, on the lines of the three-spired Cyperus,
five-spired Apicra, or an eight-ridged Cactus or Euphorbia melo-
Sormis.
If the number of members is few, and their relative bulk
very nearly equal, superposition may be sufficiently accurate
to the eye, or may actually become so by secondary growth
changes, as possibly in the flowers of Beta and Amaranthus with
superposed perianth and androecium.
Thus the five-spired terminal flower of Berberis vulgaris presents
cycles sufficiently superposed to the eye in the expanded flower,
but in development the spires are better marked, so that the first-
formed sepals would not be said to be at all superposed to the
petals: the construction being, in fact, as markedly (3+5) as
in the case of Delphinium Ajacis. Similarly in Nigella damascena,
in which the androecium is constructed in a (5+8) system, the
eight shorter curves, which are well marked in the expanded
flower, have been interpreted as “oblique orthostichies.”
On the other hand, mathematical superposition can only
be produced in a symmetrical construction in which circles
and straight lines really are present as the orthogonal construction
paths of the system, and in such cases the superposition takes place
between members of alternate whorls, the construction being that
of a concentrated system.
Again, when some of the floral members show true alternation
and others do not (Ruta, Primula), some secondary change must
be implied; the presence of (5+5) formation in part shows that
at one time the essential organs of the flower must have attained
this symmetrical construction throughout, and a definite stand-
point is thus opened up for the consideration of obdiplostemony,
SYMMETRICAL NON-CONCENTRATED TYPE. 165
for example. In all cases, the correct solution can only be deduced
from the observation of the contact-parastichies in actual develop-
ment; but it becomes increasingly evident that the extension
to the flower of the hypothesis that lateral members are primarily
produced in a mechanical system and subsequently adapted to their
special functions affords a satisfactory since well-defined basis
on which to establish theories of the morphology and phylogeny
of floral-structures.
166 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
VII. Multijugate Types.
WuEN the type of normal asymmetrical phyllotaxis is thus com-
pletely isolated as consisting of systems mapped out by log. spiral
curves in the ratio series of Braun and Fibonacci, 2, 3, 5, 8, etc.;
and the type of normal symmetrical phyllotaxis is equally clearly
delimited as.a secondary construction, physiologically indépendent
of the ratio-series, though connected with it phylogenetically, the
greatest interest attaches to all other phyllotaxis phenomena, which
though less common, may throw light on the causes which tend to
induce symmetry, before postulating, as a last resource, some
hypothetical inherent tendency in the protoplasm itself.
These types may be included under two series: firstly, the multi-
jugate systems of Bravais; and secondly, systems in which the
parastichy ratios belong to series other than that of Braun and
Fibonacci, eg., the 3, 4, 7, 11... .,4, 5, 9,14...., or still
higher series.
The term multijugate was applied by the brothers Bravais to types
of phyllotaxis in which the numbers expressing the parastichy
ratios are divisible by a common factor; so that 2 (13+21)=(26
+42), a bijugate system; while 3 (13+21)=(89+63) would be a
trijugate one.
Expressed in angular measure, there is clearly no difference
between such divergences and the expression 33, and in the spiral
theory of Schimper there was in fact-no room for such types,
except as anomalous expressions of transitional whorled stages or
“twisted whorls” of 2, 3, etc., in which successive whorls wére
neither superposed nor exactly alternating.* The simple method
* Cf. Wydler, Flora, 1851, p. 125.
PLATE XVIII.
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r
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(b+é)
82.6989 [fis snonsdigT—"QT9 “D1
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aqjasor Surppeag = "T
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(p+Z) ssodoe yna yuerd
y Cmnuogns snomsdxg—"PL9 “D1
MULTIJUGATE TYPES. 167
of regarding them as derived from two or more concurrent genetic
spirals did not suit the spiral theory, which demanded one spiral
line of growth.
Such forms of phyllotaxis are, however, not so rare as supporters
of the Schimper-Braun hypothesis incline to suppose; they may
occur in all types in which anomalous series are met with, and are
most widely distributed.
They were first fully described by Bravais (loc. cit., p. 96), although
examples had previously been noted by Schimper and Braun, and
also by De Candolle,* instances being observed in the inflorescences
of Dipsacus, Scabiosa, Arnica, Zinnia, Spilanthus, Piper, Veronica,
Verbena; flowers of Cactus, Calycanthus; cones of Pinus maritima,
and foliage shoots of P. palustris.
The possibility of an approach to bijugate capitula in Composites
is further shown by secondary maxima on the variation curves of
Ludwig; while Weisse, out of a batch of 140 plants of Helianthus,
obtained one bijugate example (16+ 26).
“The fact that they may occur in the plant which has already
been found to exhibit normal phyllotaxis phenomena most com-
pletely, lends additional interest to these constructions. Thus, out
of a batch of capitula, collected at haphazard by E. G. Broome, two
were bijugate, (26442) and (42+ 68) respectively ; the others were
quite normal;+ while out of the total crop of 130 cones on a
plant of Pinus pumilo (B. G. O., 1900), one cone only was
(6/10/16), fig. 60a, the rest being normal (5/8/13).
The bractless spadices of Aroids have already been noted as pre-
senting anomalous types of phyllotaxis, and among six inflorescences
growing on the same plant of Anthurium Crassinervium, the paras-
tichies of three were (8+13), a fourth one was irregular, the other
two multijugate of the types (6/12/18) and (6/9/15) respectively.
*De Candolle, Org. Veg., vol. i. p. 326, 1827. “Leaves opposite in spiral
pairs” in Globulea obvallata, and also according to Roeper in Ajuga genevensis,
+A case of extreme reduction in Helianthus annuus is of interest :—A seed
germinated in a crevice of a stone wall, four feet from the ground (B. G. O,,
1901) and developed a small starved plant : the impoverished terminal capitulum
produced 10 ray-florets and 28 disk-florets. The contact curves of these were
only (6+10) as taken from a section-drawing. The capitulum was thus bijugate,
although the 2-3 foliage leaves beyond the primary decussating pairs were not,
168 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Dipsacus fullonum, having been very fully investigated by Bravais,
may be taken as a type of the bijugate condition. Seedlings flower
in the second summer, and the plants usually die after fruiting;
the seedlings of the first year form a tuft of leaves with a very
definite spiral arrangement. In other species, D. sylvestris,
D. laciniatus, a well-marked radical-rosette is produced, in the latter
the aggregate of leaves being flattened out on the soil to form a
rosette two feet in diameter, in which apparently no two leaves are
superposed, and to all appearances the spiral construction is that of
the normal series (fig. 610). If the plant be cut across (fig. 61a), the
contact parastichies are seen to be well marked in the bud and
constantly (2+4); the first year’s plant being thus bijugate as a
seedling without any apparent reason.
In the second year a tall leafy shoot is sent up which bears leaves
of the specialised “bucket” type, most marked in D. laciniatus ;
this shoot is at first sight symmetrical with “decussate” phyllotaxis,
and beyond the vegetative leaves the apex produces a complex
terminal capitulum which in well-nourished plants is practically
constantly bijugate of the type (26+ 42).
Thus Bravais found this type in 272 out of 350 capitula, or over
77 per cent. In the progressively smaller lateral heads, other
systems appear, often trijugate, but sometimes of the normal series,
and equally often anomalous systems or quite undeterminable types
occur. Bravais tabulates 4 per cent. normal series, 45 trijugate, 7
per cent. undeterminable, and 6 per cent. anomalous systems. The
facts, then, show that Dipsacus presents an example of a plant with
a specialised leafy axis, springing from an asymmetrical system, and
exhibiting, when the vegetative period is over, another asymmetrical
system which, like the first, is normally bijugate.
The construction of a Dipsacus plant, then, is very remarkable if
these facts are true,—that it commences with a (244) rosette,
becomes symmetrical (decussate) in the leafy shoot, and then
produces a bijugate inflorescence (26+42); since this would imply
that a double transition from asymmetry to symmetry takes place in
the life of the plant in passing firstly from leafy rosette to tall leafy
shoot, and secondly, from inflorescence to flower. The assump-
tion of symmetry in the floral members is so general that it
MULTIJUGATE TYPES. 169
affords no difficulty. The “decussate” axis requires further
investigation.
Examination of the rosette of a seedling (fig 62a, Divsacus sylves-
tris) shows that the (2+4) system is well defined, and results in
the formation of alternating pairs of leaves in four spiral rows.
By taking lines drawn through the centre of the median vascular
bundles of each leaf on a drawing carefully made under the camera
lucida, the angle at which the planes of successive pairs of leaves
intersect may be measured with sufficient accuracy. That perfect
accuracy is not attainable is shown by the fact that such lines do
not intersect over the centre of the growing point; such disturb-
ances being the effect of unequal growth, further evidence of which
is seen in the drawing of Scabiosa plumosa, in which the spirals are
not equally curved (fig 620).
The angle measured in such a diagram averages 75° (73°-77°);
by constructing a log. spiral theoretical system of (2+4) by means
of log. spiral curves (1: 2), a similar system may be plotted out, and
lines drawn similarly through the “centre of construction ” of the
“square” areas; on sucha figure the theoretical divergence angle
thus measured was found to approximate 73° (more correctly 72°).
Observation of a Dipsacus plant which is commencing to send up
an erect axis shows that the terminal bud maintains the same
hijugate construction unchanged, and that the same system is
continued in all the foliage leaves until the terminal inflorescence
is produced. The leaves are therefore not decussate at all, alternat-
ing pairs crossing at about 72°, and not at 90°. True symmetry is
thus never attained, the apparent decussation being due to a
bijugate (2+4) formation in which, owing to the fact that a
bijugate construction implies two concurrent ontogenetic spirals,
two members are produced at each node at points diametrically
opposed.
The expansion of the system in the inflorescence is apparently
not so accurate as the system deduced for Helianthus. Thus (2+4)
should normally expand to (6+10), (16426), (42+68), but
Dipsacus fullonwm gives terminal heads of the system (26+42)
as the type, and D. pilosus is even more constantly (10+16)
(fig. 600). In noting this irregularity it may be pointed out that
Fig. 62a.—Transverse section of perennating seedling of Dipsacus sylvestris ; cam.
lucid. drawing showing angle of oscillation of system (2+4).
Fig. 62b.—Transverse section of perennating shoot of Scabiosa plumosa ; cam, lucid.
MULTIJUGATE TYPES. 171
while normal (16+ 26) and (42468) have been already noticed in
Helianthus, one capitulum was recorded as presenting the type
(26 + 42),*
Per ease cies,
:
Fig. 68.—Geometrical construction for a system (6+7) with complementry system
(1413): Lchinopsis multiplex, giving a 13-spired shoot with genetic spiral
winding on apex.
As these anomalous capitula are rare in Helianthus, but the rule
in Dipsacus, the section of the expanding series should prove of
* Such numbers being derived from observation of the external characters of
the mature capitula, do not necessarily give the construction system, since if the
facets observed subtend a smaller angle than the original primordia at their
insertions, a higher series of curves will be apparent as the contact lines judged
by the eye.
172 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
special interest. Owing to the subsequent tendency of Dipsacus
to insert or lose curves on the main capitulum axis in order to
compensate local growth variations, each capitulum requires to be
taken on its own merits. The one selected suffices to indicate the
normal procedure as well as the possibility and extent of local and
individual variations.
Dipsacus fullonum (Anomalous Expansion System). A ter-
minal head of a remarkably fine plant was taken at the end of
March, when the inflorescence was just becoming visible among the
leaves of the terminal bud. The plant had been growing fully
exposed during a mild winter, and should have flowered in the
preceding summer ; a series of hard frosts (22° F.) had also set in
just as the head commenced to develop. Very little protection is
afforded by the surrounding foliage leaves, and if external environ-
ment has any effect in producing anomalies, anomalous construction
should be expected, and as a matter of fact it was very marked.
A section of such a capitulum (6 mm. in height), taken towards
the lower part, includes the whole of the involucre, and may readily
be drawn with considerable accuracy (fig. 64). Comparison of
the involucral members shows two large median members (1 and
1’), and on the sides of the drawing 3 and 3’, and 5 and 5’, fairly
clearly indicated, and diverging at something like the proper angle ;
but careful measurement shows that the angle between 1 and 3 is
only 60°, and that between 3 and 5, 70°. That a transition is in
progress is obvious from the regular segmentation by T-shaped
walls, which might be easily mistaken for a tissue-drawing. This,
again, is much clearer than in Helianthus, owing to the fact that the
true primary members are here alone present and fill their rhombs,
while the florets they subtend are only just commencing and have
not as yet commenced to squeeze their bracts into the interstices
between them.
The only modification of the theoretical orthogonal construction
is found in the dorsiventrality of the members, which includes a
slight normal slipping across the paths of the shorter curves. The
leaf-pairs 1, 3, and 5 having been determined, it is easy, by approxi-
mating the divergence angle, to locate 7, 9, and 11. The three
members 1, 11, 7 being in contact, it is clear that the phyllotaxis
MULTIJUGATE TYPES. 173
of the capitulum commences as (6+10), the normal expansion
derivative from the (2+4) of the rosette and leafy axis.
Fig. 64.—Section taken near the base of a young inflorescence of Dipsacus fullonwm,
6 mm. long; cam. lucid. drawing; curve system up to point of section
(23 +24).
174 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
On counting the curves of the diagram, the remarkable fact is
brought out that these are (23+ 24) in the central portion of the
system, with the possibility of further division in some of them.
The anterior part of the}figure shows clearly, however, the normal
Fig. 65.—Theoretical curve construction for inflorescence of Dipsacus fullonwm,
(16 +26), as expansion derivative of (6+10) continued from the (2+4) of the
vegetative shoot. A curve has been adopted which imitates the progressive
dorsiventrality of the members.
appearance of a transition system (cf. Helianthus), which should
therefore have been (16+ 26).
It will therefore be an advantage if the (16+26) system is
constructed, and used as a means of comparison with this anomalous
MULTIJUGATE TYPES. 175
system, so that the point at which error crept in may be located.
A construction on lines similar to those used for Helianthus may
be arranged ; a still closer approximation to the observed phenomena
Fig. 66.—Dipsacus fullonum. Section of developing capitulum showing normal
expansion (6+10) to (16+26) in agreement with the theoretical construction
(genetic spiral reversed). On one side ofa line drawn through 5 and 5’, the 3 long
curves expand normally to 8 ; on the other side (shaded) 5 short curves become
3° 2° 3° 2*3=18, the two halves of the bijugate system being inverted images of
each other.
M
176 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
being obtained by using a transitional curve-tracing which expresses
progressive dorsiventrality. The system would be theoretically
mapped out by taking 16 long and 26 short curves in the ratio
8:13; the ordinary (3+5) curve being approximately accurate.
From the (16+26) curves, the (6+10) set are readily selected
by taking paths in the system from No. 1 in the manner described
for Helianthus; complexity coming in with the presence of two
points of origin 1 and 1’ (fig. 65),
The construction may now be compared with the section;
segments 1, 3, 5 and 1’, 3’, 5’ are clearly determined by these
primary curves alone, and the transition commences with the pair
7 and 7’. Thus 7 and 7’ each add a long curve, and 9 and 9 follow
the same rule, with the result that at this moment the system is
(10+10); as in Helianthus, however, such transitional symmetry is
ignored and the new curves go on being added. A difference, how-
ever, is now noticed, evidently due to the clashing of two Fibonacci
series: 11 is bounded by two new curves, that is to say, adds one long
and one short. The system is now (12+12); similarly 13 adds two
curves, and so does 15, the system thus maintaining symmetry at
(144+14) and (16416). Beyond this point, 17 adds one short
curve only, and is followed by 19, 21, 23, and 25; the system again
becoming asymmetrical and ending as (16426). The transition
from (6+10) to (16426) is thus effected on the diagram at the
26th member; but the first six did not enter this expanding series,
but represent the members of transition from the previous (2+4)
foliage shoot.
The number of transitional members is thus apparently lower
than in the normal series, ¢f. Helianthus, but agreement is shown in
the fact that the outer 16 members which establish contact around
the axis, constitute a species of protective involucre to the base of
the inflorescence, and it is remarkable that their relative bulk is
very approximately indicated by the area of the rhombs correspond-
ing to them, a curious confirmation of the uniform character of
growth in unspecialised members.
There can be little doubt but that this construction represents
the actual distribution of growth in originating the inflorescence
of Dipsacus, and any deviations from it must be regarded as
MULTIJUGATE TYPES. 177
anomalies. It is now possible to consider the actual specimen
(fig. 64) in relation to the theoretical scheme; this particular
head (23+ 24) represents a range of variation not included in
the observations of the Bravais, and the (164+26+442) type is
fairly constant for strong terminal heads.
It will be noted that on tracing the ramifications of the long
curves in the manner adopted for Helianthus, the areas leading from
1, 5 and 7, as also 1’, 5’ and 7’, correspond member for member,
but not these leading from 3 and 9.
This is further seen to be due to the fact that 3 overlaps 7,
instead of falling clear of it; so that 3 is possibly the member
which has gone wrong, and thé fact that the divergence angle be-
tween 1 and 3 was only 60° would be confirmed by the subsequent
error of the system. New paths are being opened up from 9 as
compensation at this point, but it appears that the construction has
been thrown out by this displacement of one particular pair of
leaves. To what extent such an effect might be ascribed to the
action of the frosts at the time the capitulum was commencing is of
course not evident from the consideration of one specimen alone.*
* It is clear, on the other hand, that too much importance must not be attached
to the low divergence angle between 1 and 3, when it is borne in mind that these
members are also contained in an expansion system derived from the (2+4) of
the vegetative shoot.
That the new (16+26) system commences at 7, 7’, suggests that the (6+ 10)
system was only completed at 5, 5’, these members adding the last two short
curves of the system. Allowing a new curve for each member, on the lines of
Helianthus, this would imply that 1, 1’ added short curves, and four long
curves were put in with the upper two pairs of foliage leaves: thus on a
capitulum (fig. 66) which agreed with the postulated construction of fig. 65, the
divergence between 1 and 3 was 593°, which agrees with the preceding within
the limit of the error of observation.
It will also be noted that subsequent growth is not uniform: the members
tend to come away from full contact, and a small amount of sliding growth
accompanying the dorsiventrality must be allowed for. It is possible that the
expansion from (2+ 4) to (6+10), as in the succeeding phase, is more rapid than
the Helianthus type, since lateral capitula of Dipsacus show the inflorescence
commencing immediately beyond the two vegetative prophylls. The case of the
(234-24) capitulum, granting a loss in the shorter curves, suggests that another
expansion had commenced and added extra long curves beyond the type
(16 +26).
178 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Further discussion of these effects and the anomalies of a large
series of such capitula would be beyond the range of the present
paper, which only seeks to trace out the general lines of phyllotaxis
as indicated by the homologies of cell-segmentation. Two points
are specially striking in the expansion series of Dipsacus: first, the
extent of the stations of symmetry in the expanding system, which
subsequently give way to a renewal of the original ratio; and’
secondly, the beautiful approximation of the normal part of the
diagram to the segmenting blocks of protoplasm characteristic of
the tissues of many Algal forms (Melobesia, Ralfsia, Coleochaete).
In such a working mechanism, again, as in Helianthus, the
genetic spiral is completely lost sight of and forgotten, although
the two concurrent lines may be traced in numbering up the
members; even the oscillation-theory is weakened, and the con-
clusion that the system grows and segments along new paths of
distribution dependent on the pre-existing system, with the mathe-
matical accuracy of the “crystallisation” of the Micellar Theory,
is almost unavoidable.
Dipsacus thus presents an example of a plant in which the
(2+3) system of the Fibonacci series is replaced by (2+4). This
phenomenon, rare in Helianthus, here becomes the rule, and the
whole construction of the main axis is bijugate. The reason for
this is still wanting, but it is clear that what in Helianthus re-
presents only an individual variation, is in Dipsacus a specific and
even family character (¢f. Scabiosa, Cephalaria *).
As will be described later, similar specific variations occur in
anomalous series, a8 for example, the (3+-5) of Sedum acre, in con-
trast to the (3+4) of S. refleaum (ef. figs. 76a, 0).
That the true expansion type 16/26/42, given by the Bravais for
the great majority of the capitula of Dipsacus fullonum, does
actually obey the theoretical construction of fig. 65, is shown by a
similar section of a developing capitulum in fig. 66; the agreement
is perfect, and the addition of new curves is seen to follow the
* Variation to a true (2+2) system was also found in Cephalaria tartarica;
while the variation in one plant of Dipsacus sylvestris to (8+-6), giving “ twisted-
whorls” of 3, is of special interest in connection with the readiness with which
(2 +2) is in some plants replaced by (3+ 3).
MULTIJUGATE TYPES. 179
3° 2° 3° 2:3 law in the case of both the longer and shorter paths ;
the two halves of the capitulum, on either side of any line drawn
through one pair of members, are images the one of the other; while
in this particular case, the whole diagram is taken as the reverse of
fig. 65, as the two systems are useful for reference.
Identical constructions, tending to anomalous formations, occur in bijugate
species of Silphiwm among the Compositae; thus S. perfoliatum and
S. connatum are wholly bijugate in their foliage shoots, and present the
same pairs of “bucket” leaves as are characteristic of Dipsacus, while
S. lacinvatum obeys the normal Fibonacci ratios.
Silphium perfoliatum, L., normally produces terminal capitula which are
bijugate of the same Dipsacus type (16+26) with variations. All sub-
sequent lateral capitula of the inflorescence system, which goes on rami-
fying to the third degree in the type of a symmetrical dichasium (the
ultimate ramifications being reduced as one prophyll alone remains
fertile), are of the (13+21) type, and attain this phyllotaxis by pro-
gressive expansion from beyond the insertion of the fertile prophylls.
The distinction between the bijugate and the normal capitula is obvious
on looking at the involucre from behind ; the normal capitulum pre-
senting a 3-5-8-star pattern, while the bijugate heads have four outer
members arranged in a cross (fig. 670).
These terminal capitula commence the bijugate character normally in the 2,
4, 6, 10, etc., series, but the construction subsequently becomes irregular :
heads of S. perfoliatum, taken after the flowering-period (fig. 67a), show
a remarkable similarity to the Dipsacus pattern of fig. 64; and a similar
uniformity of growth in the leaf-members results in the fact that the
transitional members become successively smaller in opposite pairs. The
central portion is not clear, but the fact that irregularity may commence
at an early stage is shown by the feeble development of 15, 15’, while
17, 17’ are still well-marked. Sections of such capitula, taken in the
bud-stage, do not show the construction so clearly as in the case of
Dipsacus ; irregularity in the curve system is very marked, an average
of 22-25 being observed among six capitula, and thus follows lines similar
to those already described (fig. 64); the same addition of anomalous
longer paths may be observed, and a similar loss of short curves; the
capitula do not, however, present so typical an appearance, owing (1) to
the fact that dorsiventrality of the outer members is excessive, the four
external members meeting round the axis, so that new contacts are
established beyond those of the theoretical construction ; (2) the capitu-
lum is almost plane, and the number of members inserted on it limited,
hence the system commences to be destroyed almost as soon as the curve
paths reach their maximum. Many capitula are thus rendered incapable
of being counted.
The essential point to note is that the system commences regular expansion
180 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
as in Dipsacus, but in all the cases observed produced ‘ultimately an
anomalous and perfectly indefinite construction, the only point in
common being the much closer approximation of the parastichy ratios
to equality, so far as they could be estimated. On the other hand, the
capitula in subsequent ramifications, right down to the smallest formed
buds, appeared to be constantly (13+21). There is so far, then, a distinct
tendency for the first-formed and best-nourished capitula, to not only
carry on the bijugate construction of the foliage-shoot, but to become
further anomalous. It thus becomes of interest to compare allied species
in order to see how far these irregularities are of local and individual or
specific importance.
S. connatum, L., closely resembles S. perfoliatum, possesses the same “ bucket-
type” of paired leaves, as also the same general habit and size, but flowers
about a month later. Of the terminal capitula, (B.G.O. 1901) some of the
first-produced showed the bijugate 2, 4, 6, 10 pattern in the involucre,
but the majority were of the normal (13+ 21) type, as in the rest of the
inflorescence. As the plants were growing side by side with S. perfoliatum,
it is possible that a different period of flower-development may have had
a local influence.
8. laciniatum, L., with normal asymmetrical phyllotaxis, has a more reduced
inflorescence, while the size of the individual capitula is correspondingly
increased. The terminal head of a strong shoot gave 34 short curves, but
the longer were too irregular to count, although the approximation to
equality was evidently very close ; a lateral head gave (28+ 34), suggest-
ing a slight rise beyond a (21+34) system ; while the last formed heads
presented (21+34) exactly. -It is thus clear that in Stlphium, especially
in leading capitula, the capacity for the addition of excess curves is very
well marked, and the stations of the Fibonacci ratios are not observed
under conditions of special nutrition with the accuracy of the normal
plant for which Helianthus was regarded as a type. In other words,
adopting the previous convention, the Fibonacci sense is less well-de-
veloped in Silphiwm, and anomalous constructions due to the interpolation
of excess curves are readily produced ; but the tendency is again always
towards a nearer approximation to symmetry, as exhibited by an approach
to equality in the parastichy ratios. From these facts it is thus possible
to argue that the irregularities in the particular capitulum of Dipsacus
(fig. 64) were not due to the stimulus of external environment in the
form of low temperature variations, but are to be correlated with the
extra vigour in the main axis, due to the fact that the flowering period
had been delayed. Once more, also, it may be pointed out how hopeless
it is to express any of these irregular constructions in terms of “ genetic-
spirals,” while they are readily discussed from the standpoint of paras-
tichy ratios.
Similar relations between terminal bijugate inflorescences, which under
excess nutrition tend to become anomalous, and lateral capitula of the
normal Fibonacci character, are general among other members of the
MULTIJUGATE TYPES. 181
Dipsaceae ; species of Cephalaria affording good illustrations. C. tartarica,
Schrad., typically presents terminal capitula of the (10+16) type,
and all the laterals (8+13) (fig. 68a, 6); the appearance of these
is sufficiently obvious in the bud-condition (fig. 69a), and the fact that
the bijugate expansion commences normally is shown by fig. 690 ;
specially fine terminal heads again show subsequent variations and
irregularities, Similarly C. radiata gave (12+19) with slight
irregularities for the terminal capitulum of a strong plant, (10+16)
for all weaker ones, while all lateral (T’, T”) were (8+183).
C. leucantha, also terminals (JT) (10+16), laterals (T’, T”) (8+13), and
Scabiosa atropurpurea, terminals (10+16), or (6+10) in fruit, and
laterals (8+-13), becoming (5+8) in fruit.
In these plants, however, bijugate construction is only apparent in the
terminal capitulum which closes a bijugate (2+4) vegetative shoot ;
this type of asymmetry being lost in the lateral branches in which
normal Fibonacci relations are restored beyond the prophylls. On the
other hand, bijugate capitula occur in Dipsacus terminating branches
of the first, second, and even third degree as well. The tabulation
of the parastichy ratios observed in typical specimens of the commoner
species will give the clearest idea of the distribution of multijugate,
normal, and anomalous or irregular systems.
(1.) Dipsacus sylvestris, an average plant, 5 feet high, the terminal capitulum
95 mm. by 45 mm. in diameter, over the spines, showed an irregular
construction about (30+ 38) at the broadest diameter ;* six other lateral
capitula gave in order :—
1: T.. ,
27+36 (irreg.)
30+ 33 (irreg.)
30+381 (irreg.)
26 +42
26 + 42
23+ 28
24+36
Irregularity thus occurred in the leading capitula, and also in the last-formed
basal ones ; two lateral capitula were exactly right, and all would appear
to be derivates of the full 16, 26, 42 system.
(2.) A much finer plant, which had been growing in the open, 6 feet high,
* Note that in counting irregular systems, the eye is readily misled in
following the wavy curves, and an approximation to equality in the ratios is thus
often a consequence of confusion of two sets of parastichies. No data for such
systems which are not taken from sections can be considered absolutely reliable.
182 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
with terminal capitulum 100 mm. by 50 mm. in diameter, and ten
lateral branches fertile, and thirty-five lateral capitula well developed,
.
gave :—
T. T’. F
26-+42
26 +38
26442
26442. . 16426
26442 . . 16426
26 +42
264+42 . . 16426
27444
16+26
25 +42 “te 4-8
26+40
+41. . 16426
26-+42
2+42 . . 16426
26 +42 . 16426
26 +42
16+26
(26442 . .4 91431
16+26
ee . . 16426
26442 . . 20431
26-+42
26442 . 16 +26
26+42 . 14421
The plant was thus remarkably constant throughout to the bijugate con-
struction ; in a few capitula, as counted at the broadest diameter, slight
irregularities occurred, but only four tertiary heads suggested a reversion
to normal (21+34). As previously noted, the difference between the
prominent sets of curves is due to the fact that the florets which are here
counted do not necessarily present the same contact-relations as the
primary leaf-members which subtend them.
(3.) Dupsacus fullonum, a strong plant grown in the open, 5 feet high ; the
terminal capitulum perfectly cylindrical and 100 mm. long by 40 mm. in
diameter, exhibited the type 16, 26, 42, unchanged throughout almost the
entire length of the capitulum: 14 lateral capitula gave :—
XX.
PLATE
“paraquintt so[eos
Jedonpoaut ayy f wofoq Woy e]NyIdeo [wIaze] PUL [BUTUT
1a] Jo uosiiwdwog ‘wnzor7ofuad wnrydjis— 919 “OI
‘uotsuvdxe o4vSnliq
Io} pataqwunMu sapeos [BONJOAUI dy} { SuLoMoY loqje
(peutmrey) wnpngyiden YT ‘wnpvpofsad wnrydpeygy— "P19 “LT
er SST.
PLAT!
‘(ST -+ §) unpngideo peteqey
NIDDM VILBVYTIG—"Y8Q A
]BULIOT,
(OL+0T) wungides
pRUpog ‘narunpwng nitvpoydag—"Eg “1g
{ XNIT.
0)
PLATI
“MOTaq Woy WHUyIdeo peUTUIa] aYTL— “969 “OL
‘eqesnliq Sutoq LoUL1o} 91[4 ‘epnqideo [Blaqv] OM) pu TeUTUL
a10} Jo soMadsaLOPUT SUN M/LepLNy MRD NITI;I— "BQ ‘Oy
MULTIJUGATE TYPES. 183
T- T’. TT”.
8 ae
f (20+33) ' maultenated
16/26/42 ’
21 and irreg.
[ 16/26/42 . } 14+18 irreg.
ja } 13420
| {13421
[ decree ) undevl,
é
(ieree 4 JE
16+26
1e-¢26 | undevl.
This plant thus showed a distinct tendency to revert to normal (13+21) in the
ultimate branches ; the irregularities are otherwise very slight.
(4.) Dopsacus laciniatus, an average plant grown in the open, 8 feet high ; the
terminal capitulum, 100 mm. by 50 mm. over the spines, was irregular, the
number of curves counted round the broadest diameter being (26438) ;
23 lateral capitula were borne on branches of the first, second, and third
degree :—
me T’. i fae am aaa
{ 16426 j oo
( 23437 . . 2 (20481) Ges
13421
16 +26
264384 ea ee
| :
18+31 (irreg.) :
[25+40 cee (imeg) ee
26442. 16 +26
25 +40 . 17424
21434
22+31
16 +26 16+26
21434 17+26
Here also the constancy to the (16+26) type is remarkable ; while individual
irregularities are small, there is in several cases, apparently without rule, a
marked reversion to capitula of the 13/21/34 series, and as in D. sylvestris
the leading heads are more usually anomalous.
(5.) Dipsacus pilosus, a strong plant 7 feet high; the terminal capitulum
spherical and 50 mm. in diameter over the spines showed around its
184 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
greatest diameter a perfect (16+26) system. D. pilosus presents a more
primitive type than the preceding species, in that the stem is branched
freely to the third and fourth degree, and lateral branches continue the
structure of the main axis, also retaining the bijugate construction. The
capitula are smaller and contain relatively fewer flowers, the ultimate
heads, in fact, often producing so few that the parastichies are too ill-
defined to be counted.
The plant produces a multitude of small capitula instead of specialising a few
large ones in the terminal region, and the type of construction is re-
markably constant. Thus the plant selected, producing branches to the
fourth degree from ten nodes, gave a total of 176 capitula sufficiently
well developed to be counted : the Jast small heads remain undeveloped
as the plant exhausts itself at the end of the summer.
Of the 175 lateral capitula, 112 were accurately (10+16) around the middle ;
30 were (6+10), the difference between these constructions being subject
to secondary error in counting adult structures, 8 were only one or two
-curves out in either direction, and 25 were of the (8+18) type ; thus, in
all, 80 per cent. were bijugate capitula, and about 15 per cent. reverted
to the normal Fibonacci ratio.
The general phenomena of all muléiugate systems can be readily
studied from their structural diagrams, and though in many cases
the systems are not necessarily constant for any considerable period,
it is only by expressing the construction geometrically that the sig-
nificance of a common factor to the ratio is made obvious. Thus
a (10+16) system, characteristic of the inflorescence of Dipsacus
ptlosus and Cephalaria tartarica (fig. 68a), may be represented by
drawing the 10 and 16 log. spirals in the requisite ratio 5:85; and
since the mathematical fact that these curves plot the system is the
only definite statement that can be made with regard to it, it
follows that the system must be numbered by Braun’s method, by
taking members as differing by 10 and 16 along their respective
paths: on so doing (fig. 70) it will be found that no interpretation
in terms of “genetic-spirals” is possible save that which admits the
presence of two equal and concurrent paths orientated at points
diametrically opposed. Taking one of these as No. 1, the
members are represented by odd numerals only, there are two
Nos. 3, for example, but no No. 2, and by taking a divergence angle
of 137° from 1, it will be found that each system has its own path
1, 3, 5, etc., and 1’, 3’, 5’, etc., and these “ genetic-spirals ” work out in
a direction the reverse of that of a normal (5+8) system.
MULTIJUGATE TYPES. 185
Other systems may be similarly constructed, and the essential
point of the mathematical proposition rendered clear, that in
multijugate systems there is no longer a single genetic spiral.
Such systems may now be viewed from the standpoint of a
transition to symmetrical construction, in that while the Fibonacci
Fig. 70.—Geometrical construction for bijugate system (10 +16).
ratios are to a certain extent retained, the construction is modified,
with the result that two members are simultaneously produced, and
the system, to continue a previous metaphor, is now built two
bricks at a time, the members of each series being formed at the
proper divergence angle. While again there is no nearer approach
186 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
to equality of the ratios, there is a distinct sign of symmetrical
construction, in that any change involving a rise or fall in the
system must, in order fo retain the bijugate construction, take
place by adding or losing two curves simultaneously, since if a
single path be gained or lost, the ratio may become divisible by
unity only and thus work out as a single genetic spiral.
From the point of view that a decussate system represents a
doubled construction, (2+2)=2 (1+1), the possibility of the
secondary reversion to the doubled spiral construction implied in
bijugate systems is very apparent. The examples met with in the
inflorescence of Verbena and the flower of Calycanthus might be
thus explained; but it must be pointed out that the rule does not
hold for Helianthus annuus, which, though decussate at first, reverts
to normal Fibonacci ratios with almost perfect constancy ; nor, again,
does it apply to H. rigidus and H. strwmosus, which are decussate
almost to the terminal capitulum. The fact that the more obvious
parastichies of garden Verbenas may vary from (5+8) to (6+10),
and the floral members of Calycanthus in the same manner, is quite
independent of the decussate phyllotaxis of the vegetative shoot,
and comparable with similar variation in Sedwm elegans, Podocarpus
japonica, etc.
Again, the distinction between a truly decussate (2+2) system
and the bijugate variant (2+ 4) is often indistinguishable to the eye,
so far as the general appearance of the adult shoot is concerned.
That very considerable displacements may take place in the former
symmetrical construction is shown, for example, by taking sections
of a decussate bud of Epilobiwm angustifolium (perennating shoot):
on cutting a section a little above the actual apex (fig. 71, 1), very
considerable changes may be seen to follow irregular growth and
twisting of the older leaves. Such distortion is very general in
decussate leafy shoots, and requires to be carefully separated from
bijugate construction. Thus in the typically decussate family of
the Labiatae, this external deformation of the symmetrical con-
struction is very common, and the original case of Ajuga genevensis
evidently comes under this head: in rosette-forming mem-
bers of this and other families, or in their seedlings and peren-
nating foliage shoots, the apparent reversion to an asymmetrical
MULTIJUGATE TYPES. 187
system is often very marked. (Dianthus, Phlomis (fig. 73),
Urtica, etc.)
These irregularities in petiole formation, etc., might evidently
occur to an equal extent in asymmetrical systems, but they would
Fig. 71.—Epilobium angustifolium, L.—I., section some distance above the apex of a
perennating shoot. II., section exactly at the apex, symmetrical (2+ 2) system.
not be so readily noticed, owing to the difficulty of judging the
error of such constructions by the eye alone.*
* If two equal and similar leaf-primordia meet around an axis and tend to
pack, the chances are that, if the ends are well developed and rounded, one will
slip under the other on one side and over the other on the opposite side. The
two developing members thus become pushed askew with regard to their true
position and that of adjacent members, and an irregular effect is produced. To
test true symmetry (2+2) as opposed to bijugate (2+4) construction, it is
necessary to cut the primordia at the apex before they commence to overlap
(fig. 71, 2). Again, such secondary confusion will be greater in a symmetrical
construction where the primordia of the same whorl should exactly meet, since
in the case of asymmetry the paths for slipping are provided in the spiral con-
struction. Hence a symmetrical system tends to give greater secondary irregu-
larity than an asymmetrical one, and it is thus rather the exception than the rule
for a decussate plant to show four strict orthostichies. The externally visible
188 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
The multijugate systems, so far considered, have been either the
(2+4) system regarded as a variant of the (2+3), or the cases,
6/10, 16/26/42, which represent the normal expansion along the
lines already indicated for Helianthus.
More elaborate systems, divisible by 3, 4, etc., occur chiefly among
the Cactaceae and similar growth-forms, as variations of anomalous
systems which become divisible by common factors, and these will
be noticed under the special heading.
Among bijugate types, two cases call for special mention; the
(6+10) of foliage shoots, and the expansion type (10+16) which
does not represent the normal sequence, but apparently indicates
a stoppage at an intermediate stage in the normal Fibonacci expan-
sions.
The (6+10) appears to be initiated directly on vegetative shoots,
in which it may be regarded as a variant, possibly often local, of
the normal ratio (5+ 8).
Thus Pinus pumilio cone, normal (5+8), varied to (6+10) (fig.
60a).
Sedum elegans shoots, vary (5+8) and (6410) (fig. 43).
Pinus Pinea seedlings vary (5+ 8) and (6+ 10).
Podocarpus japonica leading shoots vary (5+8) and
(6+10) (fig. 42).
In dealing with Araucaria, it has already been shown that from
the standpoint of bulk-ratio, (6+10) represents an intermediate
stage approximating 4:1 (or 38:1), and is therefore equally
possible as an alternative construction with (7+11), which
approximates 4:1. The conclusion that (6+10) may therefore
represent an enlargement of a (548) system, in which the bulk
of the axis is increased without the lateral primordia taking their
relative share in the increased nutrition, is unavoidable, and the
manner in which (6+10) is found associated with (548) in the
examples given strongly supports it; on the other hand it may be
regarded with equal probability as the expression of an inherent
result depends on the extent to which the members more than fill their full are or
fail to do so. In the latter case four straight rows of narrow leaves are observed
(Euphorbia Lathyris, 4-ridged Cacti and succulent Huphorbiae); in the former
with broad or sheathing leaves the rows may be perfectly irregular (Dianthus).
MULTIJUGATE TYPES. 189
variation capacity on the part of the plant, and entirely indepen-
dent of circumstances of nutrition: experimental evidence may
throw light on the point.
The (10+16) type was found to be constant to a remarkable
extent for lateral capitula of Dipsacus pilosus; such capitula are
easily cut in early stages, and owing to the relative length of the
spiny bracts, the whole of the system may be obtained in one
section. As in other examples, growth is extremely uniform, and
although the members lose their lateral contact except at their
bases, they maintain their relative positions with great accuracy.
A section of such a capitulum, taken near the base, shows
unmistakably, however, that the contact edges of the rhomboid
members lie along the paths (16+ 26) (fig. 24, 2), and that the
appearance of (10+16) is therefore secondary, and due to the
fact that in the adult head the curves are counted from the con-
tact lines of florets rather than of the bracts. When these florets,
which tend to be more constant in volume on capitula of different
sizes, subtend a greater angle than the original member in whose
axil they arose, it is clear that new contact lines will be empha-
sised and the system apparently altered. A similar result occurs
in the elongated fruit-heads of Scabiosa atropurpurea, these in the
flowering condition show most usually terminal heads (10+16),
and laterals (8 +13), as contact-lines for the florets which diminish
in size towards the centre; in the fruit-head, owing to the greater
development of the involucels, the more obvious curves reduce
to (6+10) and (5+8), while the fact that the fruit must be all
equal in bulk is correlated with an elongation of the axis and the
tendency for the conversion of the curves into intersecting
helices on a cylindrical surface.
A section of a similar capitulum of D. wilosus, taken at the
insertion of the terminal members, is of further interest in that
the fall of the bijugate system is shown to be absolutely regular,
and the last two sterile members are diametrically opposed. The
system, that is to say, remains bijugate to the end; this may be
more strikingly demonstrated by numbering the members back-
wards ; the contact paths will be seen’ to change from differences
by two to six, and by four to ten, as perfectly as in the number-
190 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
ing of the expansion system of the involucral region (fig.
74, 1).
The terminal capitulum of Cephalaria tartarica may be taken
as typical of a definite (10+16) system. The vegetative shoot is
normally (244), as shown by the paired leaves, and the terminal
head presents the six-parted pattern (fig. 75) characteristic of
Silphium, etc., and is much flatter than that of Dipsacus. A
section of such a capitulum may be taken in the bud condition,
8 mm. in diameter, just at the level of the insertion of the
last formed bracts, to include every leaf on the head, owing to
the close imbrication of the well-developed peripheral members.
The subtending bracts are markedly dorsiventral, and the slight
amount of sliding growth has operated normally, with the result
that the longer paths become more pronounced; and where the
florets are cut, the curves, as in Helianthus, approach the ortho-
gonal construction more obviously owing to the similar character
of the more or less circular florets. Such a section (fig. '75) affords
a beautiful example of rising and falling phyllotaxis, and this
particular capitulum shows a descending system with the accuracy
of the diagram of Dipsacus pilosus, the terminal members being
two sterile scales orientated in the same plane as the first invol-
ucral pair.
On such a diagram every leaf may be numbered by taking an
approximate oscillation-angle of 137° from 1 and 1’, whichever
end of the system be taken as a starting-point; the figure is thus
numbered from the outer involucral scales 1 and 1’ to 137 and
137’, the capitulum thus including 136 members. Owing to the
marked dorsiventrality of the members and slight sliding-growth
across some of the curve-paths, it is not possible to accurately
follow the interposition of new paths, according to the convention
adopted in the previous cases of Helianthus and Dipsacus, The
system, however, commences as (2+4), and 3 and 3’ do not make
complete contact, but open up room for 5 and 5’: thus according to
the convention, each 5 may be said to add a new curve to the
system. That the maximum attained is really (10416), as shown
in the section, appears to be fully warranted by the comparison
of other sections, although it is true that only the tips of the
PLATE XXIII.
“dorioysip Arepuodes Y4LA
(3+) qooys Suyeuusriag “T ‘vsoogntf s2woyyg—'SL ‘DIA
“(9 +9) My WO sapeos Jo
quoweSterre peoreurut hg
ABN
‘
ofnar vrydoy—'SL "OA
MULTIJOGATE TYPES. 191
Fig. 74.—Dipsacus pilosus.—l., section of young capitulum at level of terminal
members, numbered backwards as an expansion system. II., a similar capitulum
cut near the base, showing contact parastichies (16 +26).
N
192 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
majority of the members are cut, and the original construction
does not necessarily follow from such a section. Taking the rise
from (2+4) to (10+16) as the expression of the addition of 20
Fig. 75.—Cephalaria tartarica, Schrad. Section of terminal capitulum 6/10/16
type, numbered throughout.
new curves, the system should be complete at about twenty-six
members, and this is possibly the case, but the proof is not definite :
MULTIJUGATE TYPES. 193
a fall apparently commences at about 101, and the curves are
evidently dropped out with the regularity postulated for “dis-
continuous phyllotaxis” in the Fibonacci ratio.
Such a diagram presents, in fact, an elegant epitome of the
phenomena which any theory of phyllotaxis is called upon
to interpret, and if possible explain. It includes a bijugate con-
struction, rising from a known constant system of (2+4) to an
equally definite (10+16) system, as shown by the contact lines
of the rhomboid members, and then falling equally symmetrically
towards the close of the construction to two leaves placed opposite
each other in the median line, just like the initial pair of the
series.
Treated as the product of a spiral ontogenetic line of de-
velopment, or an oscillating growth movement across the apex,
laying down new growth-centres at an approximately equal
divergence angle, it is clear that two such genetic paths must
be in operation, producing members in diametrically opposed
pairs, and that the adjustment of members with a progressively
lowered bulk-ratio must also involve slight changes in the
oscillation-angle, since the angle which builds a (2+4) system is
not the same as that which builds a (10+16); how these angular
changes may be controlled by the plant is at present quite
inexplicable.
Treated, on the other hand, as a system in which new growth-
centres are formed at the points of intersection of indefinitely
continued asymmetrical construction curves, among which new
paths may be opened up or subsequently closed according to a
simple law for the spacing out of the added members around the
axis, as already hypothecated for Dipsacus, the number of “ genetic
spirals” which work out the system in point of time, as also the
exact oscillation-angle, becomes immaterial, and the subject admits
of clearer expression and is easier to handle. Such a standpoint is
here put forward solely on account of these reasons ; it is sufficiently
obvious that it does not follow that the simplest method of dealing
with facts necessarily involves any account of their actual evolu-
tion or causation. To suggest that the plant knows what it is doing
in laying down a stated number of curved paths is of course as futile
194 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
as was the original demand for a spiral line of growth as an expres-
sion of the plant aim.
Inherent asymmetrical growth entails the phenomena of a spiral
system, and the number of the curved paths is determined by the
mathematical claims of radial symmetry in construction, limited by
the relative size of the new members. Individual or accidental
variations on such a theme will produce more or less definite modi-
fications; and such, if markedly beneficial, may no doubt become
stereotyped as specific constants. There is so far no reason there-
fore why (2+4) as a variant of a (2+3) system should not be
almost as common as the symmetrical (2+ 2); it does not give the
symmetry which protects lower leaves from vertical light, but it
does give two opposite members which become localised at a node,
and this in Dipsacus and Silphium (sp.) appears to be a definite
biological advantage, although it is not apparent in Scabiosa and
Cephalaria. Once given the (2+4) system, the expansion deriva-
tives follow rules as perfect as those deduced for Helianthus and
Cynara, while the descending system is again the most perfect yet
described.
The phenomena of multijugate systems thus indicate even more
clearly than in the case of expansion systems and falling phyllotaxis
of the normal series, the weakness of the “ genetic-spiral ” hypothesis
as interpreting changes and variations either local or specific in
asymmetrical construction.
How the asymmetrical system is actually originated in a shoot-
apex is not yet apparent, but the conventional standpoint of bulk-
ratio, in which a member is formed of a certain relative size at an
approximately accurate divergence angle, so far summarises the
facts. But once a working system is produced and the members
of a full cycle laid down, it becomes increasingly clear that the
subsequent history of the system is controlled much more by these
existing curves than by any “spiral line of growth.” New paths
are added regularly according to the Fibonacci law, or quite
irregularly, with the result that the numbers indicated by the
contact-parastichies alone express the system, and if these happen
to vary so as to be divisible by a common integral factor, multi-
jugate systems result.
MULTIJUGATE TYPES. 195
The method of regarding such systems as controlled by two or
more genetic spirals neither presents any further explanation of
the phenomena, nor is more generally useful in practice, than it
would be if every parastichy line were called a genetic spiral, since
all équally go on winding indefinitely. It is interesting to com-
pare such a standpoint with the original conception of “ Multiple
Spirals” put forward by Bonnet and Calandrini.
196 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
VIII. Anomalous Series.
Unper this heading may be included all ratios not divisible by
a common factor which are not included in the Fibonacci series.
The formation of imitation summation series has been previously
described, as for example :—
3, 4, 7, 11, 18, 29, 47;
4, 5, 9, 14, 23;
5, 6, 11, 17, 28, ete.
And it has been pointed out that such series differ from the
Fibonacci series in that the ratios of successive terms are neither
approximately constant, nor do they always approach 1: 1°62,
although this ratio is approached as the series proceed.
It has further been shown that the number of parastichy curves
is usually low, and it follows that among low numbers almost any
ratio must be capable of expression in one series or the other.
For example, in such a series as—
6: 6
6: 7\ on system would be symmetrical,
6: 8; two bijugate, one trijugate, and
6: 9 | one anomalous ;
6:10
and the close relation of such forms as variation types, is seen
among Cactaceae. (Cf. special section.)
But it does not follow that all the ratios of such hypothetical
series actually exist in plant structures.
For example, (3 +4) is found not uncommonly (Sedum, Euphorbia,
Cereus), and (7+11) also occurs (Araucaria), but (4+7) is very
ANOMALOUS SERIES. . 197
rare. Similarly (4+5), (5+6) may be found in Lycopodium and
Cacti as constants, but not the rest of the series; although their
occurrence as transitional stages is not impossible (Cacti), the
general rule which may be formulated at this stage of the con-
sideration of anomalous series being, that any anomalous system
represents an equal or a nearer approach to equality in the ratios
than those of the normal series, and that their occurrence may be
taken as a sign of a nearer approximation to symmetry.
The following cases may be considered separately :—
1. High ratios approximating equality and associated with
symmetry.
. High ratios produced as expansion systems.
Low ratios as specific or individual variations.
. Production of anomalous systems by irregular introduction or
loss of curves.
5, Acquisition of symmetry.
i oO bo
I. High numbers the ratio of which is considerably nearer equality
than the normal 1: 1:62.
That these represent variations on all but perfect attainment of
the symmetrical condition is shown by the fact that they occur side
by side with true whorled specimens.
For example :—Acorus Calamus commonly presents parastichies of the form
(15+15), but almost equally (14+15) may be counted. Lchinops
dahurtcus, often described as whorled in its inflorescence, shows paras-
tichies very clearly on the almost spherical receptacle after the fall of
flowers and fruit in autumn : five primary heads gave (16+16), (16+13),
(16+ 16), (15+12), and (15+13), while smaller lateral inflorescences only
(12+ 13) and (13413).
It is difficult to avoid the conclusion that these numbers represent slight
deviations from a symmetrical construction based on an asymmetrical
system (10+16) or (8+13).
The scales on the fruits of Raphia Ruffia, again, vary between (6+6), (6+7),
(7+7) and (6-+8) on the same inflorescence, fig. 72, (6 +6).
It is of interest to note that in these cases the question of normal
phyllotaxis is entirely put on one side. In the two first, the prim-
ary members are, so far as is visible, ontogenetically absent, and the
secondary radial floral axes cannot be expected to necessarily follow
identical laws; while in the last, the lateral members are emer-
198 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
gencies more or less symmetrically placed on a foliar structure
which only resembles a shoot in that the aggregated carpels con-
stitute a mass exhibiting radial symmetry.
IL. Higher members of the series 3, 4, 7, etc. and 4, 5, 9, ete.
Examples of such constructions occur in Dipsacus and Helianthus,
side by side with bijugate representatives, and clearly represent in
the latter the expansion-series of seedling variations.
As pointed out by Bravais (Joc. cit., p. 100), great care is required
in the case of Dipsacus in which single curves are readily dropped
or added in the middle of the inflorescence (figs. 38a, 6), and the
ratios derived from the number of parastichies will often vary in
-the different portions of the head. Many examples are given by
Bravais; thus a capitulum presenting (23+ 37) would be a member
of the 1,4,5 .... series, but the omission of one curve in either
direction, by reducing the system to (22+36), would cause it to be
included under a bijugate construction of the 1,3,4... . series.
In the case of Helianthus the curved systems acquire a greater
degree of constancy, and the ratios, with rare exceptions, are per-
fectly definite. Thus Weisse obtained one bijugate and six anoma-
lous capitula of the types (18 +29) and (47+76) among 140 plants,
Although Weisse’s pot-plants were obviously very poorly nourished,
the percentage of anomalous capitula was no greater than in plants
grown in the open, so that it does not appear that such anomalies
are directly induced by bad environment. As previously noted, two
bijugate capitula were found among a batch of 15 from one garden,
while another batch of 15 plants, grown under the most unfavour-
able conditions (B. G. O., 1900), included three anomalous heads (29
+47) and (47+76), as well as one which could not be counted at
all.* From the point of view that variations are initiated in the
seedling, these results would not be surprising, and they would seem
to imply that the expansion series proceeded normally in spite of
bad environment. That these constructions are not merely due to
* Pot plants were placed in an open bed late in June, and remained without
water throughout the whole of a dry hot season. They grew about 3 feet high
and produced capitula which only began to expand early in November, when
they were all cut down by a hard frost, The remaining twelve were half
(34455), the others (55 +89).
PLATE XXIV.
*‘poyluseur
A[AyBrys ‘ATesaeasueay yuo yooys ous ayT[—"99s “Old
= (p+) ooys ederpoy
“poutds- J
T ‘wnxayas wnpsgy—'vg) “OI
i XXV.
PLATI
“lO PUD Pamela Joors owes a L—"912 “OTT
“paards- 4
= ooys (+E) ‘ysaq ‘wsoynpungbrg niqsoydng—'v Ly “Olt
PRR
ANOMALOUS SERIES. 199
an anomalous mode of forming the transitional 21 series in the
capitulum itself is clear from the form (29+ 47), fig. 54, in which
the contact parastichies of the involucre are seen to be (11+18),
and the rise of phyllotaxis so far follows the normal course. The
(29+47) capitulum is again of special interest in that it does not 9
represent the normal sequence of expansion from the (3+4), which -
includes all the other anomalous heads.
III. Low ratios of the anomalous series.
Such constructions occur more commonly in plants which ex-
hibit marked xerophytic specialisations, and are associated with
normal spiral systems in closely allied species, but less generally
with the whorled condition in the assimilating shoots. There is
little reason for regarding them as markedly beneficial to the plant,
although it is clear that the nearer the ratios approach equality
the less exposure there will be in the long run to intense light, if
the axis is condensed, although possibly no two leaves are mathe-
matically superposed ; the assumption that they represent variations
in the production of down-grade assimilating shoots appears more
probable. They should thus be especially characteristic of the
leafless Cactaceae and Euphorbiae, and such is in fact found to be
the case. (Cf. special section.)
Thus the very beautifully seven-spired Euphorbia biglandulosa closely re-
sembles in habit and glaucous foliage H. myrsinites, which possesses
normal (2+8) structure, and both form normal Cyathium inflorescence
shoots. E. myrsinites varies from (2+3) in weak axes to (3+4) in the
strongest: it is thus difficult to avoid the conclusion that (3+4) repre-
sents a weakened form of (3+5) (fig. 77).
Similar variations occur in succulent Saxifrages and Crassulaceae.
Sedum acre with normal (3+5) foliage shoot passes into a whorled (5+5)
flower, symmetry being attained as usual beyond the calyx members.
Sedum reflecum with a (3+4) or seven-spired shoot, produces terminal 8-merous
flowers, while the lateral scorpioid cymes contain 6-merous flowers, the
variation of the assimilating shoot being thus passed on to the reproduc-
tive shoot (figs 76a, 6).
Sedum elegans, as previously noted, varies from (5+8) to (6+10).
Monanthes polyphylla forms rosettes of (8 +18) or (7+11), the flowers being
symmetrical and 6-—8-merous: 7-merous flowers are common also
on (7+11) shoots.
Such phenomena present a close parallel to the case of multijugate
200 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
types, and are evidently due to a change in the bulk-ratio of the
seedling, which may be rare in “normal plants,” but becomes common
Fig. 78.—Lycopodium Selayo, L. Shoot-apex (5+ 6).
in plants showing marked xerophytic adaptations, and even a specific
constant in certain forms. In such variations the “ Fibonacci sense ”
202 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
may be again said to be lost, and the system may be explained as in
Araucaria (7 +11), and the bijugate (6+ 10) system, from the stand-
point of a change in the bulk-ratio; but the question is only
removed one degree farther on, seeing that the reason is now
required as to why in such plants the bulk-ratio becomes
modified.
One of the most beautiful examples of such variation is afforded
by Lycopodium Selago. The leafy apices are easy to cut, the leaf
members are all uniform and very little modified, and branching of
the main axis takes place by dichotomy of the apex, and not by the
reduced axillary shoots.
Parastichy systems are exhibited in the forms—
(5-46), (445), (3+3),
(5+5), (4+4), (2+2), (the last being found in the axillary
shoots), and transitional stages may be observed.
Thus out of 20 apices, 7 were (5+5),
5 , (5+6),
5 , (4+5),
2 4 (4+4),
1 was (348).
Comparison of a series of such apices, drawn under the same power,
shows at once that the round leaf-primordia are constant through-
out, but the diameter of the apex varies, and becomes gradually
smaller in correlation with the lowering of the bulk-ratio (figs. 78,
79, 80).
The special point of interest, however, is the close approximation
to symmetry, and the large proportion of symmetrical cases found.
Thus 10 out of 20 apices were symmetrical, while the small lateral
bulbils appear to be constantly (2-+-2).
In such cases, where, as theoretical diagrams indicate, the primor-
dia subtend an angle of between 50° and 60°, small changes in the
bulk-ratio cannot explain the whole of the phenomena. As already
shown, the bulk-ratio for (4+4) is practically identical with that of
(3+5), and the bulk-ratio in such constructions cannot therefore
be regarded as the sole determining factor; but behind these pheno-
mena there appears a controlling power which is aiming at a
a
ANOMALOUS SERIES. 208
still greater approximation to adult symmetry than that afforded by
the Fibonacci series,
Similarly a still closer approach to symmetry may be indicated
by the assumption of such ratios as (6+7), (7+8), (8+9), (9+10),
etc., and these are to be observed more especially among the
Cactaceae,in which any biological effect implied in decreasing the
leaf surface exposed to light is nil. (Cf. special section.)
IV. Once it is granted that a new row of members, implying the
opening up of a new spiral path, may be initiated at any point on
any expanding axis, or again dropped out on a decreasing one,
without necessarily implying the corresponding change all round
the system, it is obvious that a vast number of anomalous systems
may be secondarily produced, as in the case of Dipsucus taken by
the Bravais. Among the variety of ratios thus obtained, some, as
soon as they happen to be divisible by a common factor, would be
classed as multijugate; so that it now becomes clear that the
multyugate condition is only a special case of an anomalous con-
struction, and often no doubt produced by the same causes.
While, however, the multijugate primary condition has been re-
garded as a break in the direction of symmetry consequent on the
loss of the Fibonacci series, it does not follow that such will always
explain anomalous secondary systems. The very fact that new curves
may be added singly, without compensation, throughout the rest of
the system, shows that the sense of symmetry has deteriorated.
In dealing with any given case, therefore, it becomes of interest
to see what alteration is made at any given change of system.
Does the change, that is to say, make for symmetry, or the
reverse ?
In other words, is a long curve added or a short? Similarly in
reduction, the loss of a short curve makes for symmetry, as expressed
by equality in the ratios; the loss of a long curve renders the con-
struction more asymmetrical.
Remarkable examples are afforded among the Cactaceae, in
which any alteration in the phyllotaxis system is rendered obvious
by a corresponding addition or loss of a vertical ridge. The change
will often be observed to make for asymmetry; the following ex-
amples suffice :—
204 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
1. Melocactus communis: semi-globular form, showing 21 ridges,
formed by a system (9+12); a new short curve added raised the
system to (9+13)= 22 ridges.
2. Cereus chilensis: specimen forming a cylindrical shaft 6 feet
high, ridges at level of ground 14=(7+7). The axis was thus
symmetrical and remained unchanged for a height of 5 feet, including
about 1200 members. A new ridge was then put in, and the system
raised to (7+8), and this remained constant for about 75 members.
A second new ridge was then put in (fig. 390), raising the total
to 16, and this system was continued to the growing point.
It becomes, therefore, a point of interest to note whether the sym-
metrical condition of the greater portion of the shaft was regained,
or whether the change was quite aimless.
The latter proved to be the case, the new parastichy system
being (7+ 9).
On the other hand, a variation which makes for symmetry is
shown in Lycopodium Selago (fig. 79). Twin branches, one of
which, as is frequently the case, develops more rapidly than the
other, showed at their apices the systems (4+5) and (4+4), the
former being about 1 mm. taller than the latter. The asymmetrical
shoot thus shows 9 spiral series of leaves; the symmetrical one 8
theoretically vertical orthostichies. As a matter of fact, small
growth movements connected with the assumption of dorsiventrality
and unequal development render the lines drawn through the
centres of construction slightly distorted (fig. 79, 2, 3).
On examination of these lines in the (4+5) system, it will be
seen that a break is commencing at the member numbered 12.
Thus 21 falls too much on one side of 12, so that 26 is still more
on one side of 17, and does not make contact with 22, its pre-
decessor along the “4” line. The visible system is thus preparing
for the intercalation of a new long path, which will raise the
curves to (5+5). In contrast, again, to the case mentioned of the
symmetrical Cereus chilensis,a shoot of ZL. Selago, with the sym-
metrical construction (3+3), was observed to change directly to
(4+4) (fig. 80), so that the symmetry was purposely retained.
V. Finally, just as accidental variation may give a bijugate
system, or anomalous systems with very nearly equal ratios, so, as
ANOMALOUS SERIES. 205
soon as equality is reached, the symmetrical construction follows as a
mathematical consequence. How small the change may be is shown,
for example, by comparing the structural diagram for a (6+) with
a (6+6). The result, however, is very striking in that an accu-
rately simultaneous formation of a whole cycle of members is
substituted for a serial formation; but it serves to bring out the
fact that the actual appearance of the members, in time, has possibly
little to do with the mechanism which produces them within the
protoplasmic mass of the apex. It is important to note that the
Fig. 80.—Lycopodium Selago, (3 +3) and (3+38), changing to (4+ 4).
simultaneous formation is a mathematical fact dependent on the
manner in which the construction is directly changed from a pre-
sentation in terms of a spiral-vortex to that of a circular one,
In many Cactaceae, such an assumption of symmetry appears to
be entirely accidental (¢f special section), and asymmetry may be
again produced. In the case of specialised decussate assimilating
shoots, the fact that reversion to asymmetry may take place in the
sporophylls (Calycanthus) has been held to support the view that.
the decussate condition is of biological utility. An example, again
taken from Lycopodiwm Selago (fig. 80), shows that symmetry is
206 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
retained, and new paths are added symmetrically (as in Hqutsetwm)
by the bifurcation of old ones, although the change of bulk-ratio
which requires to be negotiated in adding two new curves is very
considerable.
To sum up, anomalous ratios are rare in normal plants, but are
especially characteristic of specialised inflorescences and xerophytic
assimilating shoots of such plants as Sedum, Huphorbia, Lycopodium,
Cactaceae.
So long as they are primary constructions, they imply a reduction
of the Fibonacci sense; but with the loss of the Fibonacci ratios,
there is correlated a general independent attempt at symmetry
as expressed by equality of the ratios, with the general result
that these are less than (1:1°62); while in extreme cases the
approximation is so close that the anomalous system may often
be regarded as the exception.
They represent modifications of the formal phenomena of
phyllotaxis, and oecur as local, individual, or even specific variations,
Taken in connection with multijugate systems, they may be re-
garded as a second case of a break towards adult symmetry, as
opposed to a symmetrical building mechanism. The loss of the
Fibonacci series is more complete, and the capacity for independent
approximation to actual equality in the ratio is correspondingly
increased.
Special interest attaches to the case of Lycopodiwm Selago in that
here the “Fibonacci sense” appears to be entirely lost, and the.
approximation to a construction which involves a nearer aproach
to adult symmetry is so close that strictly symmetrical examples
are as general as the asymmetrical approximations. Viewed from
the standpoint of a plan of building, it is clear that the hypothesis
of an oscillation-angle can no longer explain the mechanism (¢/, fig.
78), since the system is built on a distinct spiral path; and on the
other hand, the view that the “ genetic-spiral” is the determining
factor, while it gives an interpretation of the asymmetrical cases,
only exaggerates the gap which has been held to exist between
asymmetrical and symmetrical constructions. That such con-
structions may be really separated by only very trivial distinctions
appears to be shown by the occurrence of cases like that of the twin
ANOMALOUS SERIES. 207
shoots of a dichotomy, (4+ 4) and (4+5) (¢f fig. 79, 2, 3); and that
this is not a rare or exceptional occurrence is shown by the fact that
identical appearances may be found among the shoots of Cactaceae
(ef. special section, Echinopsis). The conclusion appears fully war-
ranted, that these apices have impressed on them a set of curves,
adjusted to the relative size of the lateral member required, which
give an approximately symmetrical construction; any accidental
variation in the ratio which involves inequality necessarily pro-
duces an effect of spirals, while equality in the number of inter-
secting curves implies the subsequent appearance of whorls.*
Thus in dealing with anomalous constructions, the interpretation
of the facts observed in terms of a genetic spiral is only possible
when the system remains constant, and even in comparatively
simple cases the enumeration of the parastichy ratios may prove to
sytematists a simpler method of describing the facts observed.t
In all cases, in fact, except among the very simplest constructions,
the “ genetic-spiral”” hypothesis becomes somewhat of an incubus ;
it is quite useless, but still one does not like to throw it over com-
pletely. It is true that all complicated constructions are more
simply regarded as systems of intersecting curves, and that once such
a system is in working order it appears to act along the curved paths
of the parastichies, adding or losing these curves as required; but
in the simplest cases on which the spiral construction for asym-
* Lycopodium Selago presents a point of great interest in that the terminal
growth-centre, which clearly is not expressed in terms of an apical cell on the
broad flat apex, definitely bifurcates and two independent growth-centres result,
each of which initiates its own curve system, with little regard to the other or
to the parent centre. These relations have been investigated by Cramer (Pflan-
zenpys. Unters., Nigeli und Cramer, iii. p. 10), and not only may the shoots
of the dichotomy give dissimilar systems, either symmetrical or asymmetrical,
but in cases of both being asymmetrical the genetic spiral may work out either
homodromous or antidromous, and thus in one case antidromous to the parent
axis. The suggestion is obvious, therefore, that all such new growth-centres
produce their systems quite independently and adjust their own bulk-ratio and
symmetrical relations. The new systems may with difficulty be expressed as
bifurcations of older paths, so far as these reach round each half; but in terms
of genetic spirals they become still more involved, in that, as already seen, true
symmetry is readily attained. (Cf. Cramer, Plate XXIX. figs. 9-13.)
+ Cf. Schumann, Monographia Cactacearwm.
0
208 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
metrical growth was founded, as in the case of the three-sided apical-
cell of the Fern, the genetic spiral is present and apparently actually
represents the asymmetrical formation of new growth-centres, one
at atime. To what extent this can be regarded as holding for the
more complicated production of the growth-centres of more massive
primordia must necessarily be obscure, until more is known as to
what is really implied by the convention “ growth-centre,” and how
far such a centre has any material existence, or possesses a finite
_ character.
It is meanwhile interesting to note that the genetic spiral asa
single determining path was the creation of Schimper, and that the
older writers, including Bonnet, were content with the expressions
“ Multiple Spirals,” “ Parallel Spires,” for even slightly complicated
constructions. The deduction of a single genetic spiral is, in fact,
the result of the assumption of a spiral of Archimedes as the funda-
mental growth spiral. The utilisation of such a spiral, passing
through equidistant points on the radii vectores, is clearly the
simplest mode of expressing such a construction; and Sachs is so
far correct in stating that the orthostichy system of Schimper and
Braun is preferable to the parastichy system of the Bravais: if a
given set of points can be defined in terms of two sets of spirals, but
also in terms of one spiral and definitely straight lines, the latter is
certainly preferable. But with the elimination of spirals of Archi-
medes straight lines vanish (for practical purposes), and the points of
intersection of log. spirals can only be defined in terms of two of the
orthogonally intersecting curves; the genetic spiral thus becomes
useless theoretically, since its complementary orthogonal path is not
obvious, while the parastichy ratios are simple and readily observed
and tabulated. The genetic spiral thus tends to vanish as the log.
spiral theory replaces that of Schimper and Braun, but at the same
time the “ orthostichy ” curves are often so nearly straight that the
Schimper-Braun formule will remain very useful in a large number
of cases for descriptive purposes; nor can there be any objection to
such a proceeding so long as the convention is recognised.
The error of the older phyllotaxis systems which postulate spirals
of Archimedes is, however, more deeply seated than appears at first
sight ; it now becomes evident that its introduction into Botany
ANOMALOUS SERIES. 209
was due to an entire misapprehension of the phenomena of proto-
plasmic growth, as was only natural when protoplasm was still un-
known (1754-1835). By regarding growth as the addition of
layers of equal thickness in equal times, as in the conventional
representation of the addition of annual rings to a tree, expressed
in terms of concentric circles with equal increments on the radii,
a conception of arithmetical progression was introduced, which
naturally resulted in the adoption of the spiral of Archimedes.
A clearer recognition of the interstitial growth of a mass of proto-
plasm throughout its whole substance, by becoming expressed as a
series of concentric circles in geometrical progression which may
contain a network of similar figures, leads equally naturally to the
assumption of a log. spiral as the actual curve of asymmetrical
growth.
Finally, it must be pointed out that the whole of the observations
and deductions hitherto given for phyllotaxis constructions, in-
cluding systems expanding and falling according to the Fibonacci
law, are the expression of the geometrical properties of intersecting
spiral curves, without necessarily adding any further information
with regard to the character of the spirals; and almost any pair of
unequal curves will give approximate results. The appearance of
log. spirals will be produced subjectively by arranging any collection
of similar figures in spiral series; and it is thus necessary to keep
in mind Sachs’ original observation that the subjective appearance
does not necessarily tell anything of the mode of formation of a
given construction. The log. spiral theory demands orthogonal
intersection, and this has so far not been proved, although it might
be legitimately hypothecated from the analogy of the orthogonal-
intersection theory of cell-formation proposed by Sachs; since it is.
sufficiently clear that if the segmentation of the plant-body in
terms of celly and cell-layers can be expressed by orthogonal
trajectories, there must be some law behind these phenomena
which controls the distribution of growth-energy, and this may
prove to be in some way comparable to that which governs more
strictly physical phenomena.*
* “Sections through growing, and especially through young parts of the plant,
always show arrangements of the cells which are quite definite, and in the
210 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
The point therefore remains,—How far is this appearance of
orthogonally intersecting log. spirals possibly a secondary effect pro-
duced by building a system of approximately similar protuberances ?
This problem may be attacked by assuming the orthogonal log.
spiral construction as expressing a distribution of growth energy
and seeing whither it will lead—that is to say, by deducing the
proper curves for the transverse component of the members, build-
ing the corresponding mathematical systems of what such phyllo-
taxis should be, and then comparing these constructions, and any
deductions which may be made from them, with the familiar
phenomena observed in a transverse section of a shoot-apex.
If the appearances agree, or can be made to agree within an in-
telligible range, when other secondary factors are allowed for, the
orthogonal system may be regarded as proved for phyllotaxis, as
one special case of a theory of growth distribution; and while
proving this, the same deductions would further involve a con-
firmation of the original views of Sachs, which still remain some-
what hypothetical, in that they are based on appearances judged by
the eye; and it at once becomes evident that this conception of
the distribution of growth-energy in orthogonally-intersecting
planes must be of the utmost importance in determining the
primary space-form of the whole of the plant-body.
In thus dealing with phyllotaxis phenomena which present the
appearance in transverse section of a system of intersecting curves,
two points of view may be established. One, that of the builder,
in which the addition of new elements in time is made the leading
feature; the other, that of the architect, to whom the actual order
of construction may be- immaterial. Is, that is to say, the space-
form of a plant determined by the visible structure of the growing
point—or is it an invisible property of the shoot, and the same
growth form may be worked out in terms of different units? The
highest. degree characteristic ; the directions of the cell divisions are by no
means accidental, and an observer sufficiently acquainted with geometrical and
mechanical science at once recognises in the structures presented by the totality
of cell-walls within an organ, cut in the proper manner, that we have here to
do with a conformity to law, the true meaning of which, however, is difficult to
decipher” (Sachs, Physiology, Engl. trans., p. 432).
ANOMALOUS SERIES. 211
presence of complicated growth forms in such plants as Fungi,
Florideae, Siphoneae, and Lichens suggests what may be termed the
architectural view, which Sachs has so greatly strengthened by his
recognition of the fact that the apical-cell of Vascular Cryptogams,
so far from being “the ruler of the whole growth in the growing-
point,” represents merely “a break in the constructive system.”
The more general standpoint has undoubtedly been that of watch-
ing the building processes, and this usually finds expression in the
discussion of the fate of cell-segments.*
It is this possibility of drawing a distinction between the con-
sideration of a given phyllotaxis system, as the product of one or
more genetic spirals, or as a complex of intersecting contact-
parastichies, which is so far the most valuable feature of the log.
spiral theory ; in that, by regarding the same construction from
two different standpoints, prejudice in favour of either one of them
may be avoided.
* Sachs, Physiology, p. 483: “It was formerly supposed to be possible to
characterise the true morphological or phylogenetic nature of an organ by the
way in which cell-division took place, and hundreds of treatises and laboriously
drawn plates were devoted to the purpose.”
(To be continued.)
Page 6, line 31, for become
12,
ERRATA. (Parr 1)
13,
10,
12,
17,
26,
15,
read becomes.
interesting ,, intersecting.
produced ,, produce.
parobolas ,, parabolas.
endodernal ,, endodermal.
1:15 » 1:2
(The omission of 1 : 2 affects all the ratios in
the column.)
inflorescence read inflorescences.
On the Relation of Phyllotaxis to Mechanical
Laws.
By
ARTHUR H. CHURCH, M.A., DSc.,
Lecturer in Natural Science, Jesus College, Oxford.
PART III.
SECONDARY GROWTH PHENOMENA.
I. Notation.
In the preceding general survey of the phenomena of Phyllotaxis
it has been observed that the arrangement of the lateral members
(appendages) of the plant body of higher plant-forms exhibits
remarkable phenomena of Rhythm, and the arrangement, that
is to say, thus works out as a definite pattern. The exceptions
to this generalisation are so few that these may be safely regarded
as cases in which the rules have been complicated by further
specialisation, or possibly by degeneration in the construction
mechanism, and in the vast majority of cases the rhythmic
character of the phenomena is their most distinguishing feature.
In so far as the phenomena are rhythmic, the observed facts
admit of mathematical expression ; but at the outset it becomes
extremely important to distinguish what exactly are the data
afforded by the plant itself, and what conceptions may have been
gratuitously introduced into the study of the subject. In the
historical development of botanical science it was unavoidable
that the first generalisations of plant-morphology should have
been founded on the contemplation of adult plant structures, on
shoots, for example, which possessed nodes and internodes: a
P
216 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
curiously academic view of a plant thus survives very generally
in text-books which bears little reference to the facts of ontogeny
and the manner in which a leafy shoot is actually constructed.
The fact that all internodes are secondary and subsidiary growths,
and that the elongation of a typical shoot is a secondary and
extremely complicated phenomenon, is often forgotten or unex-
pressed. The fact that the arrangement of leaves on such shoots
produces the subjective effect of circles or winding spirals is also
entirely secondary, the primary construction system only being
observed at the apex of the shoot, or on shoots which exhibit no
secondary elongation whatever.
Leaving on one side, therefore, all academic prejudices in favour
of whorls and a single genetic-spiral traced on an elongated leafy
axis by drawing a subjective line through successive members,
the actual data of the rhythm exhibited by the plant in building
its leafy shoot system reduce merely to the enumeration of a
certain number of curves which intersect in either direction. No
further data can be obtained from the living organism than such
observation of these intersecting curves, the contact-parastichies.
These: are therefore simple numerical expressions involving two
whole numbers only; and not only so, but every additional
factor read into the subject comprises, to use Sachs’ expression,
“gratuitously introduced mathematics.”
There can be, however, no objection to the introduction of the
mathematical properties of the numbers, since the numbers are
given; and the fact that mathematics may be introduced follows
directly from the presence of continued rhythmic phenomena.
But error creeps in as soon as the bare numerical data afforded
by the plant are combined to constitute a mathematical expression
or formula. The facts of observation supply an intersecting
system of equally distributed spiral curves, the number of which
must be integral and can usually be readily checked. The only
additional mathematical data that can be introduced, therefore,
are the mathematical properties of such intersecting curves. But
in expressing the relation of the numbers of these intersecting
curves, care must be taken to render the resultant expression
mathematically harmless, To this end, the notation has been
NOTATION. 217
adopted of connecting the two numbers by the sign +, which may
be taken as meaning simply and, or more pictorially as a cross.
The data are simply that so many curves cross so many, nothing
is added as to the angle of intersection, and such a formula
includes the simple facts of observation.
Any attempt to indicate a ratio introduces a source of error ;
the formula (5:8) would mathematically imply a construction by
log. spirals in that ratio; and, although it has been suggested that
such is actually the case at the growth-centre, the expression has
been avoided until the proof appears more satisfactory. Still
greater is the error of the old notation which states that 5 and 8
parastichies imply an 38; genetic spiral with orthostichies as
straight lines. Such a mathematical statement is alone possible
for the spirals of Archimedes and helices originally postulated by
Bonnet and Calandrini for adult constructions.
It has been pointed out that in no growing system is any helical
construction possible, and that the retention of the old fractional
notation constitutes a hopeless state of confusion which still
vitiates much of the literature of the subject ; since it is clear that
no theory which implies unstated the mathematics of helical
construction, and which therefore deals with members of equal
bulk or points equally spaced, can ever afford any insight into
the construction of a growing system of gradated primordia. In
no instance is the unfortunate error of this gratuitous interpola-
tion of helical mathematics more conspicuous than in dealing
with the phenomena of contact-pressures ; the two things cannot
coexist in the plant. Contact-pressures must be growth-pressures,
and equal volume is only attained in the lateral members at the
close of their growth period, when growth-pressures cease with the
growth of the members: when they are mutually pressing one
another they are not equal in size, and the Archimedean notation
becomes so misleading that deductions involving this standpoint
are often quite unintelligible.*
While, again, the genetic-spiral hypothesis only includes a certain
number of phyllotaxis constructions, all such rhythmic patterns
may be considered from the standpoint of the simplest method of
* Of. Schwendener, Berichte Deutsch, Bot, Gesell., 1902, p. 264,
218 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
reading the pattern—that is, as a complex of intersecting spiral
curves. The mathematical properties of such intersecting spirals
are readily deduced mathematically, and still more obviously by
simple geometrical constructions, of which several examples have
been previously given (figs. 25, 26, 28, 55, 63, 70). From these it
becomes clear that, in dealing with such intersecting curves, three
cases are mathematically possible, and all occur widely distributed
in the plant-kingdom.*
First, if the two integers which express the spiral curves in
either direction are divisible by unity only, one spiral of the same
class can be drawn through the entire series of intersections. A
numerical value can be given to all the points of intersection by
counting along the spirals in either direction numerals differing
by the number of the same spirals in the set. The fact that such
a numerical value can be given is a mathematical consequence of
the peculiar curve construction ; and in this case, since one spiral
passes through the entire series of points, the numerals utilised
are successive numbers (Braun’s method).
Secondly, if the two integers are divisible by a common factor
(x), » spirals of the same class can be each drawn through = of
the points of intersection (fig. 70); the same method of numbering
up does not utilise successive numerals, but gives 7 sets.
Lastly, as a special case of the preceding, equality of the
integers results in the same number of spirals passing each
through its own share of the points; but each set of points lies
on a common and readily observed circular path.
These three sets of mathematical phenomena are properties not of
plants but of intersecting spiral curves. They follow in the plant
because the rhythmic expression of phyllotaxis takes this
particular form of distribution. Why it should take this form,
is of course the next fundamental question. But so far it will
be seen that the first case constitutes the condition of normal
spiral phyllotaxis, extremely general because the Fibonacci ratios
* For the general proof of these statements in mathematical form I am
indebted to Mr H. Hilton ; for log. spirals or spirals of Archimedes it can be
shown geometrically on the diagrams,
NOTATION. 219
commonly utilised agree with the rule. The second case is that
of the multijugate system, while the most special third case is the
familiar one of whorled arrangement in which successive whorls
alternate.
All these phenomena, again, are more simply and correctly de-
scribed in terms of the curve systems. It has been noted that
only in the first case is there a single spiral which can be isolated
as an ontogenetic spiral; and the fact that such a spiral can be
isolated, and is consequently seen when the whole system under-
goes a very general, though entirely secondary, elongation, is a
geometrical accident of the construction, however useful such
secondary elongation may be in the plant economy. The recog-
nition of this spiral on adult plant-forms by Bonnet is thus
necessarily responsible for the peculiar inverted manner of re-
garding phyllotaxis phenomena, and although the inverted mode
of expression is common to many branches of plant-morphology,
there is no justification for the continuation of such lines of
thought at the present day.
220 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
II. Rhythm.
In previous chapters a general theory of growth was enunciated,
according to which the production of new members might be
capable of mathematical expression and of geometrical representa-
tion in a diagrammatic form. That growth is distributed at the
apex of a shoot in such a manner that its transverse component may
be expressed by a plane circular construction around a central
point (the growth-centre) is sufficiently clear, in that the circular
section of the vast majority of plant axes is evidently the outcome
of such a regular and symmetrical distribution from the “ growing-
point”: so much so, in fact, that any stem which is not circular
in section is generally recognised as the result of secondary
inequalities in the rate of transverse growth. On the other hand,
it is clear that such a generalisation is based on an unexpressed
physical conception of radial growth; and although it is thus
possible to imagine a stem which will be mathematically circular
in section, it does not necessarily follow that such a stem ever
occurs in nature; nor would it be expected, owing to the recognised
frequency of secondary irregularities in every growth-system. The
fact that no stem is mathematically circular in section does not
affect this well-established generalisation; but it is necessary to
point out that such ideas involve a physical conception which, as
in other cases, must ever be the basis of any system of morphology.
Exceptional cases, apart from the production of angular and
ridged stems, and the band-like forms produced by uneven
secondary growth in thickness, may be included under three
types :—
RHYTHM. 221
I. The cladode form, in which the shoot becomes secondarily
flattened in one plane, by a more rapid growth in that
direction than in any other (¢f. Opuntia).
II. The fasciated stem, usually though not necessarily classed
as a monstrosity.
III. The so-called dorsiventral shoot, in which centric growth is
replaced by an eccentric distribution which involves the
phyllotaxis system.
In the first of these cases (Opuntia), section of the apex (fig. 81)
shows that the original phyllotaxis pattern is normal, and only
becomes distorted at a subsequent stage.
In the “fasciated” system, the centric distribution around a
point (the single growth-centre) is changed for an attempt at
similar distribution around a number of such centres (¢f. monstrous
flowers of Buttercups with two or three distinct gynoecial cones,
and double Daffodils) or around a longer or shorter series of such
points constituting a line, with the result that great disturbances
ensue, owing to the impossibility of normal uniform growth ex-
pansion in such a system; the resultant paths of which would,
along the flanks of the crested apex, be represented by parallel
straight lines replacing the intersecting curves, which would still
appear at the ends of the system. These appearances are well
shown in the case of a fasciated shoot of Oenothera (fig. 82); the
whole of the curved crested apex, over an inch in length, could
not be cut in one transverse plane, but a small portion suffices to
show the marked irregularities produced both in shape and series
of the foliage-leaves as their growth expansion brings them into
new and anomalous contacts.
The case of the change from centric distribution to eccentric,
the so-called dorsiventral shoot, may be left for subsequent dis-
cussion; it is only necessary so far to point out that the con-
struction lines of its phyllotaxis system should continue to be
represented by orthogonal trajectory curves, just as those of the
eccentric starch-grain apparently follow the same laws as those
presented by centric forms.
In such a circular growth diagram, again, the result of a uniform
rate of growth in the whole system may be expressed by a
222 RELATION OF PHYLLOTAXIS TO MECHANICAL“LAWS.
Fig. 81.—Opuntia leucotricha, P. DC. Apex of spring-shoot, system (8+13),
rendered bilateral by secondary cladode formation (sections of the spines
dotted): cam. lucid.
RHYTHM. 223
Fig. 82.—Qcenothera sp. Apex of fasciated shoot (perennating rosette) ; portion of
section 7th the whole length and 4 mm. long: cam. lucid. drawing showing
irregular expansion curves ; axillary fower-buds tetramerous and trimerous,
224 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
circular meshwork of quasi-squares, in which all the similar
meshes are produced in equal times—it being evident, as pre-
viously pointed out, that the consideration of an ideal condition
of uniform growth should precede any attempt at a closer
approximation to the facts of growth actually presented by
living organisms, While, again, this geometrical presentation
of uniform growth is so far simple in its radially symmetrical
relations, a geometrical device admits of homologous cases of
asymmetrical growth also being plotted, thus giving, as already
described, a figure identical with the geometrical representa-
tion of lines of equal pressure and paths of flow in a spiral
vortex.
In such a system the introduction of lateral growth-centres
may be next considered. That such a secondary growth-centre
should repeat the construction of the primary centre appears
fully warranted as a sound hypothesis. The phenomenon of a
lateral growth-centre is thus to be similarly planned by a circular
meshwork of quasi-squares, and the figure illustrates similar
relations expressed in terms of equal time-units. It may thus
be taken that a lateral growth-centre may be similarly represented
either by a true circle, or possibly by the homologue of a circle:
the two cases may be subsequently distinguished.
In the arrangement of such lateral members, again, one of two
conditions must obtain: either the system is wholly irregular,
or it is regular and systematic. The former case is apparently
presented in certain specialised inflorescences (Ficus) and floral
axes (Clematis), androecium (Paeonia, Cereus), but not in
positions in which it can present any claim to be regarded as
representative of a phylogenetically primitive arrangement; and
when the construction is thus irregular, little can be said about
it beyond the fact that the impulses apparently obey no law
which can be formulated, other than the statement that they
appear to be very approximately equidistant.
On the other hand, in the vast majority of plants, especially
in unmodified vegetative shoots, as previously pointed out, the
regularity of phyllotaxis formations is their most remarkable
and distinctive feature; and this clearly implies at least an equal
RHYTHM. 225
regularity in the initiation of the impulses which produce new
centres of lateral growth.
Thus it follows from observation of plant systems that such
lateral growth-centres producing a sequence of similar foliage
members are always similar figures at similar ages, and that these
are so arranged that they make similar contacts with adjacent
members. In other words, if the lateral growth-centres are repre-
sented as circles, they must be arranged in some manner after
such schemes as shown in figs. 19, 20, 22, 23, so long as the
simple case of uniform growth is postulated. That is to say, in
that the lateral members are similar figures they will fall along
lines plotted by equiangular spirals, intermediate between the
limiting cases of the straight line and the circle; and in that
they may be represented by “circles” in lateral contact which
would be contained in the quasi-squares, the contact lines of such
series must necessarily be orthogonally intersecting equiangular
spirals. The log. spiral theory of phyllotaxis is thus the necessary
outcome of :—
I. The theory of the geometrical representation of a uniform
growth-movement.
II. The hypothesis that a lateral growth-centre is essentially of
the same nature as the symmetrical growth-centre origin-
ally postulated.*
*It has been stated above that the lateral growth-centres would be ex-
pressed as true circles or as the homologues of circles inscribed in the meshes
of the square meshwork. That the latter is probably the case in the formation
of leaf-members is very clear from the fact that circles cannot be placed in
the accurate contact relations required ; this being especially noticed in low
systems which in the plant are apparently as regular in formation as higher
ones.
A simple and fundamental conception of a leaf as opposed to a branch is
thus brought out, which constitutes, in fact, a true mathematical distinction
between an axis and an appendage. A leaf is a primary appendage belonging
to a system controlled by a central growth-centre, a subsidiary development
of it, differing from it in its increased rate of growth, and is thus represented
by the quasi-circle homologue, the controlling growth-centre remaining at
the apex of the shoot. A branch or lateral axis, on the other hand, is repre-
sented by a true circle, that is to say, as a new growth-centre wholly uncon-
trolled by the growth-centre of the parent shoot, and maintaining its own
226 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Although uniform growth may be postulated for the main
shoot, or at any rate in some part, however small, of the First
Zone of Growth in which the new impulses are being initiated,
it is clear that if growth proceeds at the same rate from the
lateral growth-centres as well, these will never make any relative
progress nor produce any visible result, although they may have
been mapped out in the construction system. Nor is a rate of
growth in the lateral members at first slower than that of the
parent axis conceivable, since the insertion of the lateral members
constitutes the surface of the axis itself. In botanical phrase-
ology, therefore, so long as the rate of growth in the primordium
and axis is equal, the lateral growth-centres remain “dormant.”
No visible effect, then, can be produced by growth from a lateral
growth-centre unless its rate of growth be greater than that of
the system as a whole. In such case the expansive development
of each lateral centre will be continued until contact is established
with .adjacent members. Thus, in the simplest conception of a
growing system of stem and leaves, uniform growth may be
postulated for the main shoot, and uniform growth, but at an
increased rate, for all the lateral members, the result being that
the growth of the lateral member becomes visible as a disturbance
of the original equable system, and protuberances are formed
which come sooner or later into close lateral contact.
Observation of the plant shows that such methods of arrange-
ment actually prevail, and the regularity of the construction,
especially as indicated by the contact-lines, is its most fundamental
and important feature. Nor again is it possible that any such
regularity can ever be a secondary effect ; comparison of systems
in which primordia are less regularly formed, and exert unequally
distributed contact-pressures on one another, as in the case of
the growth of fasciated shoots, and in the apparently centrifugally
growth-centre at its apex as a perfectly independent system. This view
further suggests that the imperfectly individualised growth-centre which
gives rise to a leaf outgrowth remains at the point of its insertion, and the
apparent presence of an apical cell in certain leaves would thus appear to have
nothing to do with their space-form, but is, as in the case of the shoot itself,
only a part of the mechanism by means uf which the architectural form is
worked out.
RHYTHM. 227
developed androecium derived from a circular zone of growth
(Paeonia, Cereus), shows that such secondary influences will only
increase the primary irregularity.
Since, as hypothecated, the geometrical construction of a circular
meshwork of quasi-squares indicates a time-diagram, that is to
Say, one expressed in terms of rate of growth, and the above
constructions follow the lines of such a diagram or its asymmetrical
homologues, it is clear that the system must be first interpreted
in terms of time, and that the regularity of the system is the
expression of a remarkably beautiful periodicity or rhythm in
member production.
That regular phyllotaxis phenomena are really the expression of
such accurate periodicity in member production will be readily
granted; but such a statement does not take one very far, since it
is only another way of expressing an obvious fact. The point is,
—to what is this periodicity due, and will it afford any further
insight into the phenomena? Thus, once such periodicity is
granted, it is clear that the phenomena of “rising” and “falling”
phyllotaxis may be very elegantly expressed from this standpoint,
in that a rising phyllotaxis and high ratios would imply an in-
creased activity of production of new growth-centres on a given
area, correlated with an increased vigour in the axis ; while falling
phyllotaxis and low ratios become a sign of enfeebled growth—
that is to say, growth-centres are only produced at greater in-
tervals of time, with the result that they each influence a wider
tract, and thus give rise to members of relatively greater bulk,
so that the system presents the subjective appearance of a smaller
number of intersecting curves. But, on the other hand, it affords
little further insight into the causes affecting other phenomena of
symmetry, bijugate systems, etc. Thus, in dealing with symmetri-
cal as opposed to asymmetrical systems, periodicity can go no
further than the expression of the simple fact that in the former
case several members are simultaneously produced at equal in-
tervals of time, while in the latter case only single members are
produced at equal intervals.
There must, in fact, be some still more hidden meaning in the
construction, from which the periodicity as expressed in a time-
228 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
diagram, and in actual ontogeny, follows as naturally as does the
geometrical construction by logarithmic spirals from the addition of
similar members.
The perfect regularity of the system shows that it is not the
ultimate shape or the lateral contact of the members which is the
essential controlling feature; the form may vary with subsequent
growth changes, and the similarity of the contact-relations is again
only the expression of regular periodicity of formation. Whatever
subsequent changes take place, the primary curves drawn through
the centres of construction of the lateral members, in the great
majority of cases, retain their numerical relations, the only differ-
ence being in the form of the curves themselves. Still more
remarkable is the fact that in many cases even the secondary
lateral axes subtended by these primary members (Helianthus),
or emergences based on them (Pine-cone), maintain the original
curve system with such constancy that phyllotaxis theories are
discussed from the standpoint of these structures, which have only
a secondary relation to the true lateral members.*
The essential point to note is that in order to produce such a
degree of regularity the actual centres of lateral growth must
have been initiated at definitely established points; that is to say,
an infinite number of causes might produce secondary irregulari-
ties once a system were laid down: the fact that any system can
be traced in the adult condition implies that the initial impulses
must have been not only equally regularly placed, but presumably
far more so.
Thus, in the postulated construction of circular growth-centres
plotted along orthogonally intersecting log. spirals, the numbers of
which are taken from observation of the plant, it follows that
these initial centres must also have been laid down at the inter-
section of orthogonally intersecting log. spirals of the same ratio.
The main question at issue, therefore, is to determine why these
points should be found at the intersection of certain orthogonal
* Of. Jost, Bot. Zeitung, 1902, p. 21, “Die Theorie der Verschiebung
seitlicher organe durch ihren gegenseitigen Druck” ; Leisering, Flora, 1902,
p. 378, “Die Verschiebungen an Helianthuskopfen in Verlaufe ihrer Entwicke-
lung vom Aufbliihen bis zur Reife.”
RHYTHM. 229
trajectory paths, and what may such paths and intersections
possibly mean from a physical standpoint—that is to say, to
what extent may the diagrams be also taken as the expression of
a field of distribution of growth-energy, comparable, for example,
to manifestations of distribution of the physical energy of the
electro-magnetic field ?
To what extent one may be justified in thus passing from a
kinematic to a kinetic standpoint is, of course, very questionable ;
and similarly little can be said beyond mere speculation until
more is known as to what is actually meant by the expression
growth-energy, or the energy of life, and how far it is comparable,
for example, with “electrical” energy. One point may, however,
be conceded: that in the case of living matter, the actual mechani-
cal energy accompanying life obeys physical laws just as surely as its
material substance obeys chemtical laws.
The data afforded by the plant are these :—
I. A growing, expanding system, containing, therefore, moving
particles ; in which
II. Growth-energy is being introduced from a central “grow-
ing-point”; and
III. A construction which, as expressed in the transverse com-
ponent of the formation of lateral members, has been
put forward as implying primarily the geometrical pro-
perties of orthogonal trajectories.
How far, then, can analogues be found for such a system in the
domain of physics; and how far is it possible to press such an
analogy, as indicating some fundamental law of protoplasmic
growth ?
Further, in the discussion of symmetrical and asymmetrical
phyllotaxis (cf. Part II.), it became increasingly evident that, while
the hypothesis of a single controlling ontogenetic spiral gives no
satisfactory clue to the general phenomena of all varieties of
phyllotaxis, all such systems might be readily interpreted and
discussed in terms of series of intersecting curves—the contact-
parastichies. These curves should, therefore, have some meaning
attached to them. If, as the log. spiral theory suggests, these
curves imply lines of equal distribution of growth-energy, it may
230 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
be possible to give an explanation in physical terms; but, on the
other hand, it is clear that, if the intersections are never ortho-
gonal, the data given by the plant are so obscure that the
phenomena of phyllotaxis only become the more hopeless of
explanation. So strongly is this standpoint suggested, that it
appears well worth while to assume the consequences of ortho-
gonal intersection and base all hypotheses on them; since, if
the view is a mistaken one, the error must become apparent
sooner or later; while, so long as no such error appears, it
may be assumed that the hypothesis of energy-distribution is a
workable one.
The use of the word action in previous chapters (cf Part I.
p. 36), as a generalised expression, has been since avoided, as in
its strict mathematical sense the term undoubtedly places the
subject in too complex a light to be at present available for
botanical purposes: its introduction was mainly based on the
necessity for indicating that the systems presented phenomena
of movement, without reference to any obviously unattainable
data as to the actual velocity of any units which might be
regarded as component particles. Since the actual velocities of the
particles of a growing plant-apex are extremely small,a closer analogy
may perhaps be found, so far as the present purpose is concerned,
in a two-dimensional electrostatic magnetic field whose properties
may be considered as depending on each quasi-square portion of
space, enclosed by lines of force and equipotential lines, possessing
the same amount of potential energy. Just, in fact, as in the
above case the same amount of potential energy may be con-
sidered to be situated in each quasi-square, so in the plant-apex
the same amount of growth-energy, 7¢. that required for the
production of a single leaf-primordium, is localised in a single quasi-
square of the phyllotaxis diagram.* Or, on the other hand, if
growth-energy be considered as more analogous to kinetic energy
* A botanist would probably be more inclined to state the converse pro-
position: the fact that an equal amount of energy is presumably directed
into each lateral primordium, granted a constant relation between axis and
primordium, would involve such a geometrical construction. Either way of
looking at it is sufficient for present purposes.
RHYTHM. 231
than potential energy, a similar distribution of energy will be
found in the. two-dimensional motion of an incompressible
fluid.*
But it must always be borne in mind that such hypotheses of
equal energy-distribution only deal with the hypothetical region
included under the conventional expression “ growth-centre.”
Away from this region, which represents a more or less gratuitous
conception, and which, being beyond the range of actual observa-
tion, must also always remain hypothetical, retardation of growth
ensues, and tends to produce rapid deformation of the log, spiral
systems. Similarly, in the case of eccentric growth, deformation
immediately sets in at different rates on different sides. Hence
any theory of energy-distribution involving equal amounts of
energy on every square must still remain hypothetical, though
the quasi-square system, whether deformed by retarded or unequal
growth - rates, will continue to indicate equivalent growth-areas ;
and such areas mapped by the intersecting curves, whatever the
* Again, even the homology of vortex construction is open to objection,
since, although it was expressly stated (Part I. p. 36) that the terminology of
spiral and circular vortices was introduced as a metaphor to make clear what
was implied by a certain type of geometrical construction, the idea of a
spiral vortex appears to carry with it an impression of spiral movement.
It cannot be too strongly insisted that no spiral growth-movement either exists
an the plant or is implied by the log. spiral theory.
The theory may be a spiral one, the phyllotaxis may be justly termed
spiral, since the pattern seen may be expressed as spirals, but the growth-move-
ment is absolutely radial. (Of. Weisse, Prings. Jahrb., 1904, p. 419.)
It is in this sense that the suggestion of Sachs is so valuable and correct,
that “all the spirals are subjective” ; and as a purely psychical phenomenon
it is interesting to note how the spiral pattern of a moving mass insensibly
leads many observers on to the interpretation of a spiral motion (cf. Goéthe)
just as phyllotaxis has been for a similar reason inundated with torsion
theories.
It is, in fact, one of the best points of the log. spiral theory here put forward
that not only is the growth-movement regarded as radial, but it can be shown
mathematically that even in a centric spiral system such lateral primordia are
bilaterally symmetrical about the radius along which they travel away from
the growing-point. (Cf. Mathematical Notes, Form of the Ovoid Curve.)
Further, in order to avoid the repetition of a “spiral” standpoint, the
expression asymmetrical is definitely adopted as a better mathematical mode
of expression.
Q
232 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
subsequent mathematical form of these may be, will exhibit the
results of equal growth in equal times.
Two points may be here conceded: there must be, as already
stated, some mechanical law implying a fundamental property of
force and matter underlying these phenomena of rhythm; and
it will again be hardly possible to discuss such speculations without
trespassing on the terminology of some branch of physical science,
the fundamental laws of which are really equally obscure. Thus,
choice has been suggested between the terminology of the electro-
static field, vortex-motion, or even the crystallisation * which
constituted the basis of Nigeli’s micellar theory. There is no
suggestion that phyllotaxis has anything to do with any of these
physical phenomena; but certain features capable of geometrical
presentation by orthogonal trajectories, common to these physical
phenomena, appear also to result from the determining causes of
phyllotaxis. The essential point at present is,—granted the
geometrical theory can be established for phyllotaxis, what
inferences can be drawn from it from a physical standpoint, any
or none? When physicists are in a position to state that the
conceptions by means of which they are led to the mathematical
laws of phenomena are necessarily absolutely correct, it may be
possible to further discuss what ultimate bearing the similar
orthogonal construction may have in the case of living protoplasm.
Till then it is at any rate remarkable that such similarity should
be found, and few will doubt that, as Sachs pointed out for cell-
structure, some law evidently controls the whole series of
phenomena, which must again be a fundamental property of
living matter. If the introduction of a mathematical conception
of growth and growth-centres can lead to any better method of
dealing with the facts, there will be no harm in trying to apply
* The general facts of crystallisation are even more remarkable in that they
refer to inanimate matter. Thus it may be possible to deduce mathematically
the number of crystalline forms, but the prime cause which determines why
crystallisation should ever take place, or why some forms should be commoner
than others, or why a given substance should select a special form, is as remote
as any indication of the prime cause of phyllotaxis. The number of arrange-
ments possible in phyllotaxis is relatively small, and the observation and
tabulation of their occurrence comparatively simple.
RHYTHM. 233
it, so long as “growth-movement” and “ growth-energy ” are re-
cognised as being in some way comparable, though not necessarily
identical, with more strictly physical phenomena. While, again,
the application of the strictly mathematical conception of a
uniform distribution of growth-energy around an initial growth-
centre must remain necessarily in the condition of a working
hypothesis, since it can only apply to a region which is itself
somewhat hypothetical, in which the rate of growth is conceivably
uniform, there can be no doubt that such an hypothesis must con-
tinue to form the basis of all considerations of the geometrical repre-
sentation of the growth-phenomena presented by the plant-body ;
and before passing on to the discussion of the numerous conditions
which may be superimposed on such an elementary phyllotaxis
system, it may perhaps be as well to sum up the points which so
far appear definitely established.
Thus, in Part I. (Construction by Orthogonal Trajectories), it
appeared increasingly evident that the general method of accumula-
ting phyllotaxis data by the observation of orthostichies was
hopeless, not only from the standpoint of actual observation, but
a consideration of the mathematical propositions of Schimper and
Braun showed that helical constructions had become applied to
something they were never intended for, ce. to the developing
systems at the growing-points, in which, since the spirals are
obviously neither helices nor spirals of Archimedes, the postulated
helical mathematics no longer held, and the systems of orthostichies
as vertical lines vanish for theoretical reasons, as also for practical
purposes. The study of orthostichies thus became eliminated
from phyllotaxis, while the value of parastichies and the genetic-
spiral remained unassailed.
In Part II. (Asymmetry and Symmetry), on the other hand, a
general consideration of the phenomena of the phyllotaxis systems
most commonly exhibited in the plant-kingdom clearly brought
out the fact already noted in the preceding chapter, that mathe-
matical systems of intersecting curves presented different
phenomena, with the result that the genetie-spiral only held for
one out of three possible cases, and this again only so long as
the system remained constant. Since the genetic-spiral conven-
234 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
tion only applies to one special case, and does not hold for the
whorled and multijugate systems, it is clear that the so-called
genetic-spiral also vanishes from theoretical considerations ; since
now it is seen to be merely a property of a special arrangement
of intersecting spirals, it can tell no more as to the meaning of the
phenomena then was previously known. The creation of Schimper,
it retains a certain interest as a relic of the past, but can now only
be regarded as a convention which is often useful in practice owing
to the fact that it admits of a method of attributing a numerical
value to the members which, so long as growth is distributed
equally around the growth-centre (ie. centric), is actually a tume-
sequence, and expresses the order of ontogeny as checked by
observation. It must, however, be remembered that this
sequence, obtained by resolving a certain number of inter-
secting curves along a single path, will necessarily cease to
be a time-sequence if once the growth-system becomes eccentric
(cf. Eccentricity).
With orthostichies and the genetic-spiral both eliminated from
the subject, the parastichies alone remain, not only as the data
to be accumulated by observation of the plant, but as the ex-
pression of the working mechanism of the construction. Using
again what must be perfectly metaphorical language, since
borrowed from strictly physical conceptions, the log. spiral theory
suggests that new centres of lateral growth are originated at the
points of intersection of curves, which may be regarded as indi-
cating a type of segmentation of the protoplasmic mass, wholly
independent of cell-formation, along paths of distribution of
equal growth-potential which may be so far homologised with
“Lines of Force.” To bring these curves into homology with
equipotential lines it is required to prove that the intersec-
tions are primarily orthogonal. The method adopted consists
in assuming the fact, and continuing subsequent mathematical
deductions with a view to render the error of the theory
apparent. So long as no marked discrepancy appears, the
theory may be regarded as a fair approach to the description
of the conditions actually prevailing in the field of a “ growth-
centre,”
RHYTHM. 235
The log. spiral theory again clearly differs fundamentally from
all conceptions of “induction” in that the initiation of the new
centres which work out the pattern remains wholly within the
control of the construction centre at the apex of the main shoot,
the living protoplasm of which would thus appear to possess a
certain power of numerical choice. In other words, the paths of
the construction forces are centrifugal, and not, as the induction
theory would suggest, centripetal.
236 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
III. Contact-Pressures.
THE existence and great importance of contact-pressures has been
emphasised by Schwendener, as also the fact that the contact-lines
follow those of what has previously been described as a “ concen-
tration-system.” It has also been seen that such growth-pressures
may be referred to an increased rate of expansion in the lateral
primordium as compared with that of the parent axis. This
increased rate of growth implies that growth initiated from a new
growth-centre extends radially and equally in all directions until
contact is made with adjacent centres of growth distribution; and
in the great majority of cases it would appear that the visible rise
of a primordium has some relation to the formation of contact
surfaces ; although in other cases (cf. Aspidiwm, fig. 35) there can
be no doubt that the primordia rise from a central region before
any lateral contacts are effected.
It is clear that the existence of such undoubted cases of the
complete absence of any lateral contact whatever, combined with
the production of perfectly normal Fibonacci phyllotaxis, com-
pletely puts out of court all theories of phyllotaxis which demand
the close lateral contact of primordia as being of fundamental
importance in determining the initiation of new growth-centres,
which has been such a favourite standpoint from the time of
Hofmeister to that of Schwendener and his pupils. The construc-
tion of such an apex as that of Aspidiwm (fig. 35) is alone sufficient
to disprove any contact theory, whether it be taken in the original
form of mechanical contact-action, or in the diluted and still more
hypothetical form of contact-stimulus. The essential point, how-
CONTACT-PRESSURES. 237
ever, is the determination of the ultimate value of contact-pressures
when these do obtain.
From the general hypothesis of a uniform rate of growth in
centric systems, it follows that all contact-pressures may be
resolved into components acting along the orthogonally inter-
secting construction lines of the system; and so long as growth is
uniform, no displacement can ensue, the only result being a change
of form; the lateral members being, in fact, squeezed into the
shape of quasi-squares. That contact-pressures may exist between
growing primordia is undoubted, and that contacts are made in a
“ concentration-system ”: these are facts of observation. But it
does not follow that they are in any way pre-eminently important
in producing any displacements whatever in the developing
system.
All theories of the effect of contact-pressures imply that the
primordia just formed by the growing apex exert an influence,
whether of the nature of a direct mechanical pressure or an
“induction ” (Weisse), on the centre which gave them birth. That
such secondary centres of admittedly limited growth should thus
impress their individuality on the parent centre of unlimited
growth activities and control its subsequent operations appears at
first sight somewhat preposterous; but this view has appealed
to many botanists, and however much such a standpoint may be
regarded with suspicion, since it represents an ideal post hoe ergo
propter hoc type of argument, the essential point is to see how
such a conception may have been treated from a physical or
mechanical standpoint, and further, what may be deduced from it.
It is clear to begin with that the amount of a contact-pressure
cannot be estimated by the eye alone, and yet observations of
effects which may or may not be due to such pressures constitute
the only means of tracing such a theory. How shallow such
interpretations may be is well seen, for example, in a criticism
of Winkler by Weisse * in which a three-angled apex is said to be
clearly due to the pressure exerted on it by three leaves which
have been just produced from it, and are naturally moving away
from it with the continued expansion of the growing-point. Nor
* Prings. Jahrb., 1903, p. 413.
238 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
can there ever be much use in such observations unless the amount
of pressure can be put into the mathematical theory. The so-
called mechanical theory is thus not mechanical in any sense; it
is based on pressures which cannot be measured, or even proved
to exist, and may therefore be wholly imaginary, and such theories
are as useless as any other standpoint to which the stigma of
“Nature Philosophy ” may be attached.*
In considering the special standpoint taken up by Schwendener,
and the importance attributed by him to displacements, it must be
remembered that Schwendener formulated the Dachstuhl theory
to explain the well-known observation that the general facts of
phyllotaxis phenomena as seen in growing shoots did not agree
with the postulated accurate angular divergences of the Bonnet-
Schimper helical system: and also that the most important piece
of obvious evidence of such alteration was afforded by the very
general displacement of the angles of primordia which become
angular under mutual pressure. This latter feature may be con-
sidered separately; at present it is only essential to point out
that displacement of angles does not necessarily imply displace-
ment of the whole member, and that, the Schimper-Braun Archi-
medean formule having been shown to be fundamentally incorrect
for developing systems,—the error of the construction being
rendered clear by the log. spiral theory,—the correction of such
constructions by hypothetical secondary displacement becomes
wholly unnecessary.
Schwendener’s theory, put forward in 1875, has long held the
field, since from the complexity of its assumptions its application
to the plant was not easy to understand and still more difficult
to disprove. The conception of what has been termed “ bulk-
ratio” was introduced as a factor in determining phenomena of
spiral phyllotaxis ; but as previously shown, however valuable
such a convention may be, it affords no clue whatever to the still
more fundamental phenomena of asymmetry and the true sym-
metry of whorled construction (¢f. Part IT.).
Schwendener also assumed as facts of observation certain dis-
placements of the lateral members, and close lateral contact
* Of. Weisse, Pringsheim’s Jahrb., vol. xxxix. p. 419.
CONTACT-PRESSURES. 239
between the developing primordia: the fact that the Schimper-
Braun formula did not hold for developing systems was common
knowledge, but his method of connection of these factors into
causal relation was extremely vague, and it may be noted that it
never appealed to the critical acumen of Sachs (cf. also Pfeffer,
Physiology, Eng. trans., vol. ii. p. 144). It may also be pointed
out that, whatever importance be attributed to Schwendener’s
conception of the alteration of primary systems by hypothetical
pressures, whether intrinsic, of the members themselves, or extrinsic,
of some compressing agency, they have after all little to do with
the fundamental facts of phyllotaxis, which is only concerned
with the production of the primary system itself, all secondary
alterations being subsidiary phenomena. Schwendener, in fact,
still requires to prove :—
I. The existence of any force producing displacement ;
II. The fact that true displacements really are produced ;
III. That such displacements are the result of the postulated
force,
whether, again, the force be regarded as a mechanical agency or
a still vaguer phenomenon of stimulation. The second of these
points has been attacked by Schumann and Jost,* their object
being to establish the fact, always sufficiently obvious to the
unprejudiced mind, that such extensive displacements do not take
place, and that the initial curve-system, as it first becomes visible
at the plant-apex, persists in the adult condition unless rendered
ambiguous by secondary elongation of the shoot.
The standpoint here taken up is not so much that Schwendener’s
theory is impossible,—it is founded on certain definite premises
from which mathematical results ensue,—but that it is entirely
gratuitous and unnecessary, since the phenomena it was intended
to explain, 7.2. the secondary alterations of the Schimper-Braun
constructions, are non-existent; while the premises themselves
more than include all the data from which the log. spiral theory
is mathematically derived—the very data, in fact, for which in
previous pages stricter evidence has been demanded.
* L. Jost, Bot. Zeit., 1902, p. 21; B. Leisering, Flora, 1902, p. 378; Prings.
Jahrb., 1902, p. 421.
240 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
As Schwendener’s standpoint is somewhat involved and subject
to modification, the following example of his original method may
serve to illustrate the difference between the two conceptions. A
well-known figure from Schwendener’s first plate has been copied
by Weisse into Goebel’s Organography, and appears in the English
translation (p. 75). A phyllotaxis system is supposed to be repre-
sented by (1) a set of spheres—a legitimate hypothesis, but still
purely a hypothesis, since there is no evidence to show that the
transverse section of a primordium is ever mathematically circular.
(2) The spheres are taken as being all the same size: a condition
which is never reached in the plant until growth has uniformly
ceased, and the pressures with it. (3) They are arranged according
to a helical divergence system of Schimper and Braun, which is
all right once equal spheres have been postulated. (4) It is
assumed that such an arrangement will give orthogonal loose
packing; and finally, (5) an outside vertical force, an entirely
hypothetical conception so far as the plant is concerned, is applied,
with the natural result that the system may be ultimately thrown
into close hexagonal packing. It is difficult to see what exact
bearing such a conception, involving so many doubtful assumptions,
can have on the arrangement of the gradated primordia arising
on a radially symmetrical plant-apex; but, by taking the vertical
force as a tension instead of a compressing force, it becomes clear
that such a construction might approximately represent the
changes produced in an adult system by passing it through the
second zone of elongation, and which have been previously regarded
as wholly outside the province of phyllotaxis, except in so far as
it may concern the descriptive writer. In discussing Schwendener’s
standpoint, the first thing which requires to be clearly defined is
the exact significance of what is to be included under the term
phyllotaxis ; is it to include all secondary changes in the system
which may appeal to the eye, or has it to do solely with the
actual forces which produce the primary system within the proto-
plasmic mass of the apex, without any reference to the details
of cell-construction? Thus all phenomena of packing must be
secondary: primordia must have been made and have reached a
certain bulk before they can be packed. The agencies which
CONTACT-PRESSURES. 241
determine the initiation of new growth-centres are perfectly
distinct from those which come into operation once they are
formed and have produced members of a definite visible
bulk,
The weakness of Schwendener’s argument is sufficiently clear—
the mere assumption of a cylindrical surface which may be
unrolled at once puts all developmental phenomena out of court:
the apex of a plant can never be regarded as a cylinder, although
on the other hand it may never be quite flat; the unrolled
cylinder representing, in fact, the longitudinal component of the
growth-system which is solely due to a retardation in the rate of
growth in a system which would remain always plane so long as
uniform growth persisted. Similarly, the primordia can never be
represented during development as equal spheres, nor possibly as
truly circular in section. The assumption of circular figures,
which will also be similar, and the orthogonal arrangement is
alone all that is required to mathematically deduce the log. spiral
theory; since, when transferred toa plane projection of a growing-
point, no other spirals except log. spirals drawn in the manner
previously postulated will continue to give either similar figures or
orthogonal intersections,*
As already pointed out, the attempt to eliminate inconvenient
spiral curves by unrolling the helix of Bonnet on to a plane is
the point at which the initial error crept in. The helix represents
the secondary stage of phyllotaxis, in which the members have
attained constant volume by a progressive cessation of growth.
A growing system is necessarily a log. spiral system or a derivative
of one, and the helix drawn on a cylinder is mathematically related
to both the spiral of Archimedes and the log. spiral of a plane
projection, and may, therefore, be derived from either. Two of
the five hypotheses of Schwendener, therefore, when applied to
the transverse component of a phyllotaxis system, are sufficient to
give the log. spiral theory, which agrees so closely with observed
facts that no external agency, whether of contact-pressures, con-
tact-stimulation, or anything else, is required to make the system
* IT am indebted to Mr H. Hilton for the mathematical proof of these
statements.
242 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
more in accord with a mathematical conception of what the
relations of lateral members to one another should be.
As previously noted, the orthogonal quality of the system, and
the possibility of representing primordia as either circles or the
homologues of circles, the two points assumed by Schwendener,
before postulating the disturbing agency, are just the two factors
for which a more rigid proof has been sought. It will thus be
seen that Schwendener’s Dachstuhl theory can only apply to the
displacement of members after they have been formed, and such
apparent displacements are, no doubt, very general; they may be
due to secondary bilateralityy of the members, inequalities in the
rates of growth of different parts of the members, as well as to
different growth-relations between primordia and the axis. But
all these features are secondary, and require to be carefully
separated from the mode of initiation of the new impulses which
produce the growth-centres of new members before these become
visible.
Such secondary relations of displacement, so far as they may
be due to contact-pressures, may be briefly considered from the
standpoints of :—
I. The pressure of older members on younger ones as they are
formed.
II. The reciprocal pressures of growing primordia against all
with which they come into contact.
III. The effect of a rigid boundary on a growing system.
So many entirely diverse phenomena have been included under
the heading Phyllotaxis, that some consideration of these secondary
relations of phyllotaxis systems is required in order to clear the
ground before the primary and essential features can be treated
without prejudice. To repeat the present standpoint,—Phyllotaxis
has to deal with the processes which determine the rhythmic
origin and regular arrangement of the primary lateral members
(appendages) of a plant-shoot in the first Zone of Growth; the
arrangement of primary members and secondary derivatives
(lateral axes) on the adult stem being merely the relic of such a
formation, which may have no obvious relation to the primary
system. The fact that secondary axillary shoots, or formations of
CONTACT-PRESSURES. 243
the type of the Pine-cone scales, really do give a system apparently
identical with the true phyllotaxis relations of the primary
members, affords a curious witness of the deep-seated faith of
observers in the laws of uniform growth; and thus the Composite
capitulum, the Aroid spadix, and the Pine-cone have always been
favourite examples of theories with which they have after all only
a secondary connection (Schwendener, Jost, Leisering).
I. Tue PRESSURE OF OLDER MEMBERS ON PROGRESSIVELY
YOUNGER ONES.
Hofmeister first put forward the view that the presence of older
members must affect the position of new ones; and that new
members in the vast majority of cases arise ontogenetically in
close contact with older ones is sufficiently obvious; it remained,
however, for Schwendener to make such close contact the basis of
a definite mechanical theory. But the value of contact-pressure
theories is greatly discounted if examples can be adduced in which
the primordia do not arise in contact at all, and yet present the
normal appearances of spiral phyllotaxis (Schumann). Thus, in
many large shoots with broad flat apices such is apparently the case
(Nymphaea, Sempervivum, fig. 83); the latter may be taken as a
type of these constructions. The youngest visible primordia are
low elevations which show no boundary-line along the shallow
depressions between them ; but so long as the primordia show any
inclination to rhomboidal shape, a certain amount of contact must
be admitted, and contraction in the spirit-material allowed for.
In a broad apex such contraction may be greater in the longitudinal
direction, and in this and other cases frequently has the effect of
pulling down the growing-point into a slight depression. The only
evidence that can be accepted of complete absence of lateral
contact will be the retention by the primordia of their original
approximately circular outline. Such primordia occur noticeably
in the apices of species of Opwntia, where the leaves, though
rudimentary, are better developed than in most Cactaceae (0.
cylindrica, O. leucotricha); but most remarkably and easiest of
observation in such Ferns as the common Aspidiwm Filix-Mas,
244 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Fig. 83.—Sempervivum tectorum, L. Perennating rosette of adult plant cut above
the apex, and the members numbered, showing secondary system (5+8); the
smaller figure being the actual apex, showing system (3 +5).
CONTACT-PRESSURES. 245
Perennating rhizomes taken from January to March show, within the coiled
leaves of the current year, the primordia of the next season just com-
mencing. On removing the chaffy scales, these appear as papillae, some
quite visible to the naked eye, and also quite isolated from one another :
Hofmeister’s empirical generalisation, that each new member falls
asymmetrically in the widest gap between two older ones, is as patent
as in any small bud that requires to be sectioned. The system is made
clearer by removing the entire apex, about 4 mm. thick, and rendering
it transparent in Eau de Javelle (fig. 35).
Although destitute of lateral contact-pressures, the primordia
arise each in normal position for (8+ 5) or (5+8) systems, and the
lines drawn through empirical centres of construction form spirals,
which intersect in the central region very approximately at right
angles, so far as can be judged by the eye: the fact that this is the
true structural condition being checked by examination of the
stellar meshwork in the adult part of the shoot.
Again, consideration of the cell-structure of the apex of Aspidium
root (fig. 18) shows clearly the general law of pressure as affecting
younger members. Any younger cell can always grow successfully
against all the pressures of older ones of the same character, and
the apical cell grows against the pressure of the entire mass, and
retains its walls always convex outwards. Similarly, any younger
member can always compress an older one, and is therefore not
essentially affected by it. In transverse sections of free leaf-
producing buds, the primordia are again always convex outwards,
and the preceding members become flattened: an example is
afforded by Araucaria (fig. 41); new lateral buds flatten their
subtending leaves, which would otherwise have remained rhom-
boidal in section. In more typical plants the older leaves them-
selves tend to assume a flattened appearance owing to their
progressive bilaterality, so that the effect seen may not be due to
one cause alone. In fact, when it is borne in mind that in a
typical foliage-bud the axis which includes the leaf-insertions is
growing and expanding simultaneously with the young primordia
arising from it, it is clear that the subject of bud-pressures requires
very careful handling, since when the whole system is growing
uniformly there may be absolutely no pressures in the bud at all,
the conventional expression “ packing in the bud” being largely
246 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
due to a subjective impression which has no real basis. The very
existence of bud-pressures requires to be proved in any given case.
The influence of the pressure of older members of the same system
on younger ones may therefore be completely disregarded ; nor
have visible primordia any directive influence on other primordia
as they become visible: the relations of adjacent members being
established, before the protuberances appear, in the actual substance
of the protoplasmic mass.
II. ReEcIPpROcAL PRESSURES IN OLDER MEMBERS.
Such pressures, as already seen, can only be due to an increased
rate of growth in the primordia as compared with that of the axis,
or to different rates of growth in different directions, or in different
parts of the members. If, as by hypothesis, the members are
primarily arranged in orthogonal series (loose-packing), all mutual
pressure may be resolved into components along the orthogonal
paths of the system : these, if equally distributed, can have no effect
on the packing of the members, but if at all marked the shape will
be altered, and the “circles” will become “ squares.” Only when
the pressures are unequally distributed will any sliding effect be
noticed, culminating possibly in close-packing of the hexagonal
type. So long as growth is uniform, and the mathematical con-
struction holds, the disturbing effect will be nil ; change of shape
may take place, but no change of position.
In a great many leafy shoots this obtains to a considerable
degree, and the leaf-primordia assume a rhomboidal form, as seen
in transverse section, approaching that of a “square” of the log,
spiral meshwork; and this holds so long as growth proceeds uni-
formly throughout the whole shoot. The fundamental section-
form of all leaves developed in closely packed systems is therefore
that of a quasi-square with more or less rounded angles, the median
line of orientation passing along one diagonal. Beautiful examples
of such undifferentiated members persist especially among some
Coniferae, in which bilaterality is small or wanting, and the leaf
elongates to a “needle” type (Cedrus atlanticus, Araucaria excelsa ;
cf. also Mamillaria and the Pine-cone).
CONTACT-PRESSURES. 247
On the other hand, evidence that growth ceases to be uniform is
seen in the majority of leafy shoots. The quasi-square rhombs
become flattened, the system is no longer orthogonal, a peculiar
“ sliding-growth” usually takes place, and the spirals tend to pass
into spirals of Archimedes as the members attain equal volume and
are spaced at equal intervals. Such cessation of uniform growth
is produced by a lowering of the rate of growth in the lateral
members; and such reduction, if equal in all directions, will tend
to loosen the members from their close contact, and the bud
“opens out.” As the rate of growth is thus lowered in the
primordia, contact-pressures necessarily vanish (cf. Opuntia).
A special case is, however, general among leafy shoots: the rate of
growth diminishes more rapidly in the radial direction than in the
tangential, while in the latter the rate of growth may be apparently
relatively increased. The same effect would be produced if the
radial growth of the axis be diminished at a greater rate than the
tangential growth of the leaf, owing to an apparent contraction of
the whole system. These phenomena constitute the special case
of the bilaterality of the so-called dorsiventral leaf, and may be
considered separately. It is so far clear, however, that the effect
of an increased tangential growth, real or apparent, must induce
sliding of the members over one another ; but it does not follow
that an internal thrust on the part of the members themselves
can ever convert the system into any approximation to the
hexagonal packing of the “pile of shot” type.*
* That is to say, if an orthogonal system of vertical and horizontal rows of
bodies, free to roll over each other, be acted on by an external horizontal force,
the horizontal rows are retained, but the vertical ones are displaced so as to
intersect the horizontal at 60°. In the circular system of a transverse section,
the vertical rows are represented by radii, and the horizontal by the circular
paths : in the corresponding asymmetrical case, the vertical rows may therefore
be represented by the shorter curves, the horizontal by the longer ones. The
general result of any lateral thrust on the part of the members themselves
will be that the shorter contact-curves become broken ; and this again is the
phenomenon usually observed as soon as the primordia become markedly
bilateral ; while the longer curves are retained unaffected, and are thus rendered
increasingly conspicuous. On the other hand, a vertical compressing force
(Schwendener), acting along the shorter curves therefore, would have
produced similar flattening appearances, but would have tended to maintain
R
248 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
No evidence exists for the formulation of any theory of phyllo-
taxis to explain the origin of a normal asymmetrical system which
involves the conception of the application of some external pressure.
That the application of an external pressure on an empirically
constructed system will produce results somewhat analogous to
those seen in the plant, can never be an acceptable argument.
While the action of an internal pressure set up by the members
themselves may, it is true, subsequently alter the appearance
of the system, but it can have no relation to the mode of its
formation.
III. THe INFLUENCE oF A RIGID BOUNDARY.
A boundary more or less resistant may be formed by older
members of a character dissimilar to that of the uniform system
previously considered. It has been previously pointed out that
the youngest cells of a plant tissue will grow against practically
any pressure that may be brought to bear on them in the living
plant, and the same should hold good for the youngest members.
This constitutes, in fact, the conception of youth. But such
vitality is not necessarily long continued ; this power of resistance
usually rapidly diminishes. There is thus always a point at which
cells or primordia begin to yield to surrounding pressures, and both
cells and primordia as they grow older begin to assume the form
adapted for least resistance to surrounding more rigid bodies, and
fill the space available to them.
Such diminution of vitality is the more rapid in members
which attain no great specialisation; or, more correctly, the
the vertical or shorter paths and have broken the approximately horizontal
ones. Further, it must be noted that a cylindrical system of spheres arranged
orthogonally would not pack by any pressure into a perfect hexagonally
arranged one, in the sense of the accurate packing of the “pile of shot.” The
original contact-lines would necessarily be broken somewhere, and the
resultant contact-curves would not present the regular arrangement which,
on the other hand, as normally obtains in the adult plant as it does m the
developing system. Nor, again, was there ever any reason to suppose that the
whole leaf-primordia would slide over each other to such an extent when their
bases constitute the surface of the axis,
CONTACT-PRESSURES. 249
converse should be stated—it is the diminution of vitality which
renders them degenerate. This tendency to yield to outside
pressure becomes, in fact, a measure of the decadence of growth-
vitality and constitutes the phenomenon of “packing.” Packing
thus takes place in the case of both cells and lateral primordia
as they attain their adult condition; and the phenomena ob-
served in the packing of cells composing ordinary parenchymatous
tissue may be taken as a type of what is to be expected in the
analogous case of lateral members. All growing-points lay down
cells conceivably endowed at first with equal growth-energy, and
arranged in layers the main periclinal and anticlinal construction
lines of which, as Sachs pointed out, are probably orthogonal
trajectory curves. As the rapid maturation of the specialised
peripheral layers involves a reduction in their capacity for main-
taining the rate of growth of the cells composing the inner
tissues, and these latter tend to round off in order to produce
the necessary intercellular spaces, the system falls into the
irregular arrangement, approximating hexagonal packing, familiar
in tranverse sections of a typical stem or root. Further pressures,
especially well seen in the case of thickened members exhibit-
ing sliding-growth, produce cell-forms often very approximately
hexagonal in section.
Results somewhat similar should therefore be obtained in the
case of lateral primordia which develop within a closed space
and show feeble growth specialisations. To further satisfy the
conditions, an apex is required in which the primordia are pro-
duced in a system with a fairly high ratio of curves, and a simple
example is afforded by the winter-bud of Cedrus.
In Cedrus Libant, as in many other Conifers, the advanced xerophytic
specialisation of the perennating foliage-buds takes the form of the
protection of the young primordia of the foliage-leaves by means of a
ring-growth of the stem which constitutes a well-marked cup, identical,
in fact, with the circular zone of growth which in floral shoots
represents the first stage in the development of perigynous and
epigynous floral structures. Such a ring-growth may be conveniently
termed the crater type of apex, as opposed to the normal production
of a cone apex. In Cedrus Libani the crater is well marked, and,
following the mechanical law of growth for such a lateral structure,
950 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
the more it develops, the more it tends to close over the apex, and
hence exerts all the greater pressure on the enclosed primordia, the
normal curve-systems for which should be (5+8) in small shoots.
Longitudinal sections of such a bud (December) show the well-marked
crater, the external surface and rim clothed with protective scale-
Fig. 84.—Cedrus Libani, Transverse section of winter bud, system (5 + 8),
showing effect of pressure of crateriform axis,
leaves, the base of the inner surface producing the young foliage-leaves
of the next season, and a conical growing-point rising from the base
(fig. 85).
A transverse section just above the apical cone, passing through the crater
wall, will show sections of leaf-primordia about three cycles deep
(fig. 84). The nature of the packing is obvious, irregular hexagonal
figures being produced, as in the packing of parenchymatous tissue ;
PLATE XXVI.
sxade WIOJTIe} e419 Surmoys ‘pnq-o8e1[o]
Sarjeuuaiod Jo uoroos [eurpnysuoy 2vvqrT snupag—'eg “Fry *(¢+¢) ymod-Surmois ‘sozy-wapag wnipidsp —'Ge ‘Buy
CONTACT-PRESSURES. 251
and at first sight the (5+8) system has been quite destroyed ; but
coinparison of lines drawn through the central bundles of the leaves
shows that it may still be traced with sufficient accuracy to admit of
numbering the members,
The Cedrus bud thus represents a case of simple radial com-
pression of the older leaves of a (5+8) system against a circular
boundary, the result being merely to produce irregular and often
hexagonal figures; that is to say, owing to the effect of radial
pressure the plastic masses of the older leaves are squeezed into
irregular shapes, but they do not roll over one another to any
extent. The phenomenon is rather one of adjustment of growth
than of actual displacement, the centres of construction indi-
cated by the vascular bundles remaining very fairly in their
places. In dealing with such packing of leaf-primordia no
analogy can be drawn corresponding to that of the packing of
spheres into the hexagonal arrangement of the “pile of shot.”
The action of a radial compressing force, here provided by the
overarching of the crater wall on the developing system, does not
tend to produce displacements in any way comparable to those
of the original Dachstuhl theory; the only result of such
additional radial compression being the production of irregular
figures resembling those seen on cutting a piece of ordinary
parenchymatous tissue: the system tends to become irregular,
but it is quite clear that no radial (ae. vertical) compressing
force acting on such a circular asymmetrical system would ever
so change it that the system would after displacement retain
a regular construction. The fact that in the general case
phyllotaxis systems normally retain regular parastichy curves is
therefore the proof that no extra pressures beyond those of the
growing primordia are normally in operation.
Similarly, pressure against the relatively greatly developed
cotyledons to a certain extent affects the shape of the first small
needle-leaves of the seedlings of Conifers (Pinus, Cedrus). That
such seedlings possess, so far as can be seen, an irregular phyllo-
taxis system may be due to more than one cause: that the
actual curve constructions are at first anomalous and even some-
times symmetrical may be traced from sections which show the
252 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
curve-system of the vascular bundles supplying them, as they
pass down the axis. Further, there can be no doubt that any
production of an “expansion system” will give the appearance of
irregularity, when the number of members developed before
the change takes place is not sufficient to give the appearance of
definite parastichies.
Thus Pinus sylvestris * commences very commonly with an
approximation to a (2+3) system, although the older shoots show
(5+8); and P. Pinea after initial irregularity settles down to
(5+8). In such specialised seedlings, as in the case of species of
Helianthus, for some reason, the phyllotaxis is irregular at first,
though this is by no means the general case for all plants. But
the effect of pressure against the cotyledons so long as the
plumule is enclosed between their bases, and these again by the
endosperm, is well seen in the case of the first leaves which are
initiated while the seedling is wholly within the endosperm and
testa. Sections of such seedlings show very marked irregular
packing shapes produced by pressure, much as in the bud of
Cedrus Libani. Such pressures add, therefore, to the complexity
of the determination of the systems as they appear at any given
time ; but they clearly have nothing to do with the origination
of the first impulses which determined the formation of the
leaf-members in the substance of the broad embryo apex (fig. 86).
Comparison of the broad apex of a seedling in which the radicle has alone
protruded (fig. 86, I.), with that of older seedlings, suggests most
strongly that primordia are already being formed within it, but have
not yet arisen above the surface. The space between the bases of the
cotyledons is usually somewhat elliptical, and the primordia at the
ends of the ellipse are distinctly more advanced than the others, so that
here, as in other cases in which true centric growth does not obtain, the
actual ontogenetic order of appearance gives no clue to the order of
formation. There is as yet no regular system observable, since the
number of leaves already in sight is insufficient to show contact-
parastichies: the arrangement thus appears somewhat irregular, and,
owing to the conical shape of the apex, is not readily observed in
transverse section. Slightly older seedlings, in which the cotyledons
* Cf. Schwendener, Bot. Mittheilungen, i. p. 89, Taf. v. The number of
leaves seen in section being too small to give any reliable pattern.
254 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
have been wholly withdrawn from the testa and are beginning to
expand, present a very remarkable appearance (fig. 86, II., LII., IV.).
Parastichies are still wanting, as indicating any definite system, and the
primordia assume irregular forms under pressure ; so that the resem-
blance to a section of ordinary packed parenchyma is very close.
Growth continues to be irregular in the individual primordia (IIT.),
but as the plants become older it appears more regular and parastichies
begin to appear. How far these appearances are partly due to ir-
regularities in the phyllotaxis system itself is thus obscured ; but the
irregularity in the phyllotaxis is associated with irregularity in the
shape of the members. That the phyllotaxis system is itself irregular
is rendered probable by the comparison of other types (Cedrus Atlanti-
cus), but this would not necessarily lead to irregular shapes in the
members, As the contact-parastichies become increasingly obvious,
they give very anomalous results: for example, (IV.) is apparently a
system (6+7) with irregular packing among the first leaves; but
when the cotyledons fully expand, and the plumule becomes visible
between them, the presence of a definite system of the normal series
becomes clear for the first time. The central portion of the bud is now
unmistakably (5+8) (V.), although in the example figured this
appears to have been only rendered normal by the opening up of a new
curve by the member numbered 1.
Seedlings vary from (5 +8) to (6+10), (5+8) being the usual type. Section
of a plant in which the primary
shoot had reached the length of
6 inches, shows a normal (5+8)
system with remarkable perfection,
the members retaining to a very
considerable degree the form of the
quasi-square of the theoretical con-
struction, owing to the very small
extent to which progressive bilater-
ality has been carried. Such a bud,
however, grown in a warm house,
retains the primary construction
to a much greater degree than the
foliage-buds produced on older plants
in the open air and exposed to desic-
cation : these primary shoots lacking
the protection of the bud-system
Fig. 87.—Coleochaete scutata, Young plant, subsequently developed in the adult
showing radial and circular v7 a condition of the plant.
A very instructive case is afforded by the arrangement of the
florets on bractless Composite capitula, and may be well observed
CONTACT-PRESSURES. 255
in large heads of Cynara Scolymus (fig. 530). Here the primary
members are entirely wanting, but their axillary shoots neverthe-
less present perfect curve-systems of constant (55+ 89) and falling
a
Fig. 88.—Pinus Pinea. Transverse section of the apex of the young seedling,
6 inches high: system (5+ 8).
systems (fig. 53b). That the growth-centres of the primary
subtending leaves are actually existent, though they may
not be visible, appears undoubted: their rate of growth has not
become sufficiently great to raise them above the surface of
256 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
the inflorescence receptacle, and they remain so far dormant
growth-centres.*
But the discussion of the mutual relations of the florets of
Cynara does not enter into the question of primary phyllotaxis at
all, except in so far that it is a derivative arrangement, which
follows the primary non-developed system so uniformly and
closely that it may be taken as a perfect guide to the original con-
struction. These florets, though not primary phyllotaxis elements
at all, are not in any close lateral contact, but are packed round
with hairs so that each develops independently and maintains its
own normal orientation, without any angular alterations produced
by mutual pressures. The curves are perfect Fibonacci systems,
and two features of special interest may be noticed :—
I. The rotation of the peripheral florets (fig. 536) owing to
secondary pressure against the smooth, firm involucre
edge; no slipping is involved nor displacement, only a
readjustment.
II. The elongated oval shape of the ovaries, in the slightly
spiral “median plane,” due, again, not to any mutual
pressure, but to the inherent structural tendency of
two “median carpels” to build an oval rather than a
circular organ.
Comparing Helianthus now with Cynara, it will be seen that the
curve-systems are identically accurate in both types, but Helian-
thus differs (1) in having subtending bracts present and visible,
although pressed out of their original positions by their axillary
florets; (2) the ovaries of the florets are definitely rhomboidal by
mutual pressure; (3) those of the decadent ray-florets change
their shape but not their position, and thus become packed into
triangular facets. The angular ovaries are of special interest:
theoretically they should have presented the same oval shape as
in Cynara, since they are constituted by the same two “median
carpels.” But in consequence of growth-pressure, each oval has
* Cf. the interesting case of the missing subtending bract of the flower of
Nymphaea ; the axillary flower thus appears to fall in the normal phyllotaxis
system, as if the flower replaced a leaf. But the young flower-bud does not
fill the quasi-square left empty, and is packed into it with woolly hairs,
CONTACT-PRESSURES. 257
to fit into a square mesh of the phyllotaxis system together with
the crushed subtending bract. The result is that the ovals with
their long axis in the “ median plane” of the flower are so adjusted
that they come to lie obliquely across the quasi-squares, but with-
out otherwise interfering with the curve construction. The radial
extension of the ovaries, that is to say, breaks the long curves,
giving them a serrated or stepped appearance, but the short curves
remain unaffected.
This implies, however, no displacement whatever of the orthogonal
construction system: the centres of construction remain unaffected,
there is a change of shape, but not of position, so that the pheno-
menon is again not one of displacement but rather of readjustment.
Section of a young capitulum, for example (fig. 89), at the level
of the style canals, gives a series of points which can be taken
accurately for each flower; the curves drawn empirically through
these points show the theoretical square meshwork with as great
a degree of accuracy as could be expected from a plant. The dis-
placement of the florets is thus apparent and not real, and the
effect of any radial elongation of members arranged in a spiral
phyllotaxis series will be to step the long curves, while the short
curves remain unaftected.
That some alteration in the phenomena of the curve-systems
should therefore be observed in Helianthus in passing from the
flowering condition with florets circular in sevtion to the fruiting
condition with radially elongated achenes, is sufficiently obvious.*
But such alterations have no reference whatever to the causes
which produced the primary system of subtending bracts; nor
does such a phenomenon enter into the question of the primary
importance of contact-pressures. The bicarpellary ovaries do not
assume the flattened form in consequence of mutual pressure ;
their flattened form is as much an inherent growth function as
that of the two-carpelled fruits of the Umbelliferae: it is easy to
see by cutting a capitulum in two, or by noting the shape of fruits
adjacent to ovaries which have proved sterile, that the flattening
of the fruit is quite independent of any pressures, although mutual
pressures may make the angles more pronounced. The changes
* Bot. Zeit., 1902, pp. 226, 230.
258 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
observed in the curve-systems are optical effects produced by
elongating the members of the system in a particular way, and the
Fig. 89.—Helianthus annuus. Portion of a young 6-mm. capitulum, system
(84+55). Section at the level of the style canals of the developing flowers,
showing construction quasi-squares, and the oblique setting of the rhomboid
ovaries thus ‘‘stepping” the ‘‘34” curves, Quasi-squares of the comple-
mentary (21+89) system dotted.
CONTACT-PRESSURES. 259
amount of mutual pressure is determined solely by the relative
rate of growth of the receptacle and the fruits; and if the growth
of the former be only sufficiently active, contact-pressures would
be entirely eliminated.*
Fig. 90.— Euphorbia Wulfenti, Hoppe, Terminal system of a strong shoot (8 +13),
showing progressive dorsiventral bilaterality of the members.
* A good example is afforded by the growth of the capitulum receptacle
in Rudbeckia and Scabiosa: e.g. in Scabiosa atropurpurea the florets form a
gradated series, but the fruits are required to be all the same size. The axis
260 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
The change of angle observed in the intersection of the para-
stichy curves of such inflorescences is thus due to the relative
growth of the rapidly enlarging receptacle and its fruits. These
changes are therefore tertiary effects in the phyllotaxis appear-
ances, and are merely the expression of the mathematical
properties of intersecting spiral curves, directly comparable to
similar curve changes observed in shoots passing through a
secondary zone of growth. Not only will any small amount of
slipping of the angles of such ovaries, in the readjustment of
radially elongated growth forms, tend to bring a third set of curves
into view,* but the same appearance necessarily follows as soon as
any growth change lowers the angle of the intersection of the
contact-parastichies to 60° or raises it to 120°. These again are
geometrical phenomena due to different rates of growth in a
system which was previously considered to be adult, and have no
bearing on the formation of the initial curve-systems observed
in the first stage of development of the capitulum.
An identical phenomenon of growth adjustment, or “ packing,”
occurs in the typical Aroid spadix, and is well seen in the case of
the dimerous flowers of Anthurium. These inflorescences, like
Cynara, possess their bract growth-centres clearly existent and
functional, although, save for exceptional cases such as <Acorus,
they are only traced through the intermediary of their axillary
flower-shoots ; and these in many species (cf. fig. 93) present a
normal curve-system of (8+13). But the dimerous flowers of
Anthurium, (2+ 2) in construction, and orientated in the “ median
plane,” present two outer perianth segments as the first-developed
members, and hence in a growing system these will be larger
than the others, so that the flower is always slightly elongated
in the “ median-plane,” that is to say, approximately along the
long axis of the spadix, and presenting a radial extension of the
elongates in compensation sufficiently to loosen all contact-pressures, but the
curve-systems remain undisturbed ; although a new set of spirals may appeal
to the eye, owing to the large relative size of the involucels. Hence a lateral
(8+13) capitulum becomes (5+ 8) in the fruiting condition.
* When the long curves become thus stepped, and so tend to be less con-
spicuous, a (384+ 55) system is readily confused with a (55+89) in the latter
fruiting condition. Cf. Jost, Bot. Zeit., 1902; Leisering, Prings. Jahrb., 1902.
CONTACT-PRESSURES. 261
floret. It would therefore be expected from theoretical construc-
tion that the long curves should prove to be stepped, owing to
apparent displacement, as the lozenge-shaped flowers readjust to
fit into the square meshes of the system.*
Comparison of fig. 93 shows that such is the case; the 8-curves
are slightly stepped, but the 13-curves run clean-edged. At the
apex of the spadix the phyllotaxis system falls normally to (548),
and here again the 8-curves are the smooth ones, while the
5-curves are stepped.
In the case of ¢rimerous flowers, similar secondary relations
between the florets produce phyllotaxis effects, which again have
little bearing on the primary construction system, in that the
trimerous flowers produce a secondary appearance of hexagonal
facets which therefore present three lines of contact; this being
again the general case for trimerous Monocotyledonous types
(Acorus, Muscart), In dealing with examples of hexagonal facetting
it must, however, be clearly borne in mind that hexagonal facetting
has no necessary connection with hexagonal packing. Hexagons
which appear very fairly regular may still be orthogonally
arranged. By taking a normal orthogonal (8+13) system
(fig. 95), and cutting off corners at the point of contact,
i.e by pressing the members against each other instead of
sliding them, an orthogonal series of hexagons may be obtained,
which show a third line of contact according to the mode of
operation, 5/8/13 or 8/13/21/; the latter being shown in the
figure, which is designed more particularly to illustrate the relation-
ship of the prismatic cone scales of P. Pinea.
* Qf. Schwendener for Displacement Theory, in Goebel’s Urganography,
Eng. trans. p. 80. This does not account for the whole of the phenomena:
the adult florets are almost square lozenges (¢f. fig. 93), and the greater part of
the slip is now due to the radially symmetrical orientation of a decussate (2+2)
system, individual florets being constructed in median and transverse planes.
The effect noticed being thus connected with the subject of the orientation of
lateral axes, can with difficulty be included under the heading of Phyllotaxis ;
but it is sufficiently clear that though helical curves may intersect to give
square facets, it is only when the number of curves in either direction is
equal that the diagonals of the facets will be median and transverse planes
and no readjustment will be necessary (¢f. fig. 110, chap. vi.).
262 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
All these hexagonal structures are, however, secondary pro-
ductions in the phyllotaxis system; another example of primary
hexagonal leaf-facetting is afforded by Huphorbia mamillaris
(fig. 92).
This plant is of further interest in that, as in analogous ridged Cactoid
forms, the phyllotaxis is anomalous: eg. the specimen figured was
7-ridged =(3+4) system at the level of the soil, but after the pro-
duction of about 300 members, two new curves are added quite
irregularly (fig. 925), and the system becomes (4+5) or 9-ridged, this
formation being continued to the apex for over 500 members. The
new curves are interpolated without rule, and the lateral branches
commence as (2+3), which again soon rises to (3+4).
The case of the Pinus cone (fig. 7) is of interest, again, in that
the rhomboid scales are not leaves but secondary structures obeying
the same laws of uniform growth as their reduced parent members ;
but these cone-scales are not elongated radially like the Helianthus
ovaries, but tangentially. The converse phenomenon will there-
fore be observed in the phyllotaxis pattern; that is to say, the
long curves will now remain clear, while the short curves will
exhibit stepping. The readjustment of a tangentially elongated
member in the square meshwork of the growth-system thus tends
to change the shape but not the position of the member. Such
tangential extension, again, becomes the normal condition in all
“dorsiventral” foliage-leaves, and will be considered again from
another standpoint. The essential point at present is to note that
such readjustments, and sliding-growth effects, do not imply any
displacements of the growth-centres. The primary phyllotaxis
relations are unaffected, and any secondary appearances which
may be involved in the pattern follow geometrically from the
properties of intersecting spiral curves. All contact-pressures
must be growth-pressures, and must be studied, therefore, from
the standpoint of growing systems.
Schwendener’s Dachstuhl theory, as a working hypothesis, thus
disappears as completely as the original one of Schimper and
Braun it was designed to correct. Apart from the mathematical
conception of helices inherited from these botanists, it was founded
on two perfectly definite facts of general observation :—
CONTACT-PRESSURES. 263
(1) The contact-relations of the young primordia.
(2) The fact that an actual displacement of the corners of
members growing in close contact does take place.
Fig. 91,—Euphorbia Wulfentt (Hoppe). Terminal system of weaker shoots,
(5 +8) and (3 +5).
From the literature of the subject (Schumann, Schwendener,
Jost), it is clear that it is impossible to prove the first point of
contact-relation to the satisfaction of all, since many cases occur in
which the question of lateral contacts would be described differently
iS)
264 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
by different observers. The fact remains that in many cases the
primordia not only do not appear to touch, but are certainly
not in any relation of mutual pressure (Aspidivm). While this
standpoint of origin in contact creates a prejudice in favour of
regarding the primordia as of a bulky nature from their first
inception, the growth-theory here proposed, by postulating an
initial growth-centre of the nature of a mathematical point, leads
directly to the view that contact-relations are secondary, and that
theoretically growth must proceed around each centre until it
reaches the field of adjacent growth-centres, although the actual
boundary may be beyond observation. What is really established
by the observations of Schwendener is the fact that these contact-
relations are characteristically of a special type, to which the
conventional terminology of a “concentrated system” has been
applied. It is clear, however, that the growth-centres of the
primordia are always quite distinct from each other, and that the
contact-relations must be ultimately produced if growth only
continues long enough, and in the great majority of leafy shoots
contacts are soon established. But such contacts will not
necessarily imply any displacements, and the phenomena
of displacement of the edges of the members according to
a definite plan is perhaps one of the most remarkable features
of ordinary spirally constructed leafy shoots; and there can be
no doubt that the existence of such slipping or sliding-growth
effect, which apparently implies a forcible displacement of the
members, was the fundamental fact which led Schwendener to
postulate a forcible displacement under mutual pressure. The
actual significance of the reyular displacement of leaf edges in
asymmetrical systems, which it must also be noted does not take
place in symmetrical constructions, may be left for the present,
since it only becomes apparent from the mathematical considera-
tion of the subject. It is only necessary to point out that the
necessity for such displacement naturally follows from the general
conception of a phyllotaxis system as built up of primary and
secondary growth-centres, and the fact that the required dis-
placement effect. does occur is one of the strongest proofs of the
practicability of the log. spiral theory (¢/. Mathematical Notes).
PLATE XXVII.
*s1098f [BUOBLXAT[ JO SAUIT 9} JO HaLys0z
ayy Aq polwoIpte ‘saamMo Mou Jo TWoIyoNpo1UL snoyeumoge
‘SLD LULU DUQ.LOYENT — "966 “BUA
Surmoys “Jooys- suarpouvag
“WU 0%
maysds
‘
“107 eWIRIP
1 Sya0R] [BUOHeXeY UO saAva, SaLMoys ‘(G+ Pp)
yooys jo xed y ‘s1apprumU matoydngy — DEE ‘BIT
PLATE XXVIII
Q t
Pee 2c 9G@OIAID
0d ¢
Ge 6 O GO |
Spadix 12 mm, in
g, 93.—Anthurium crasstnervium,
diameter
5
Fig,
Fig, 53b.—Cynara Scolymus.—Portion of capitulum.
, system (8 +13), reduced at apex to (5+8).
CONTACT-PRESSURES. 265
Fig. 94.—Nymphea alba, perennating bud of year-old seedling, 3 mm. in diameter,
system (2+3). Transverse section for measurement of the divergence angles
on lines drawn through the xylem of the smal] bundles.
266 RELATION OF PHYLLOTAXIS TO MECHANICAL LA
ECCENTRIC GROWTH. 267
IV. Eccentric Growth.
THE special type of symmetry observed in a plant body which
presents a distinct upper and under surface, differing in appear-
ance and physiological function, has been very generally described
by the term dorsiventral (Sachs). This term, borrowed from the
usage of Animal Morphology, in which this type of symmetry is
the rule rather than, as in plants, the exception, naturally implies
the possession of a dorswm and a venter, or at least a hypothetical
plane dividing the body into two regions which will ultimately
become these surfaces. Even in Zoology the convention is
admittedly based on the body structure of higher animals, and
from these is carried down to less differentiated types; as implying
a certain bilaterality of structure. In its application to Botany,
therefore, in the absence of anything which corresponds to either
dorsum or venter, which appear to be regions ultimately correlated
with the evolution of a locomotile body seeking solid food, the
term must remain a purely metaphorical expression; and, as in
the case of all borrowed terms, care must be taken lest the
suggested analogy be ultimately accepted as an actual fact. The
term dorsiventral apparently presented itself to Sachs in prefer-
ence to the simpler term bilateral, in that it contained the sugges-
tion that the two surfaces must be dissimilar; and the wider term
bilateral would, according to him, include the case, for example, of
an erect shoot bearing members in two rows, or with what has
been previously termed a symmetrical (1+1) construction. For
the same reason a diarch root would be bilateral, though few
would nowadays object to a diarch root being still classed as only
a special case of radially symmetrical root construction. In
268 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Animal Morphology, again, the conception of an axis and its
appendages, though admittedly a convention, is based on the
simple fact of observation that in higher animals the body is
constructed of one main axis and its limbs, or in Arthropoda, for
example, of one main axis and segmental appendages. The fact
that the main axis exhibits dorsiventral symmetry leads to the
application of the same theoretical conception to the symmetry
of the appendages. But the term dorsiventral is an axial designa-
tion; the idea of an appendage being dorsiventral in itself is
wholly outside the range of the convention. Thus, while the
transference of the term dorsiventral as a metaphor to a special
case of shoot symmetry may prove helpful, its common application
to the case of a bifacial foliage leaf, which is typically an appen-
dage borne on a radially symmetrical axis, remains open to grave
objection. Similarly, the extension of the term to special floral
systems (inflorescences), and even to zygomorphic flowers (Goebel),*
may be of very questionable advantage ; since the application of a
metaphor to constructions in which it is not otherwise the custom
to discuss the dorsal and ventral surfaces, adds little to descriptive
power; while there is no obvious criterion as to which surface
is to be regarded as the dorsal and which the ventral, and further
conventions require to be introduced. Sachs first pointed out the
fact that the so-called dorsiventrality of shoots as a type of
morphological construction was correlated with a difference in
physiological reaction to stimuli: the change of symmetry being
the outward and visible sign, or even the cause (Physiology, pp. 489
and 493), of a fundamental change in protoplasmic reaction, and
that dorsiventral shoots were plagiotropic. The special case of
isobtlateral leaf-members here again presented an objection to the
use of the wider term bilateral; while difacial, on the other
hand, scarcely conveys the impression of a dorsiventral shoot
* According to Goebel, the term dorsiventral implies an upper and an under
side, while the conception of zygomorphy is based on a subordinate feature, the
existence of a right and left side! Both phenomena are obviously equally the
consequence of eccentricity, the primary feature being the displacement of the
growth-centre, from which all secondary phenomena of “irregularity” or
asymmetry naturally follow.
ECCENTRIC GROWTH. 269
which may still be practically circular in section. The term
dorsiventral thus became applied by Sachs as indicating a definitely
physiological conception of plant symmetry, which might be ex-
pressed by a morphological bilaterality, but the bilaterality must
not be “ double,” that is to say, a dorsiventral organ must have
two unlike surfaces.
It is clear that not only have several special cases been included
under the same terms, but the existing terms are not very strictly
defined. Thus the case of a symmetrical (1+1) leafy shoot pre-
sents little difficulty ; there is no need to regard it as bilateral
any more than a (5+5) shoot would gain by being called ten-
sided. The cases of the cladode and the fasciated stem are again
only special cases of radial symmetry, while the latter has much
in common with the often-quoted type of Marchantia and Algal
forms, as Fucus. Two cases remain, the bifacial foliage leaf, and
the “dorsiventral” shoot bearing leaves on the so-called dorsal
surface: the former presenting a phenomenon which is the
property of an appendage without reference to the symmetry of
the axis bearing it, the latter a special case of axis construction.
In fact, it now becomes apparent that the so-called dorsiventrality
of these structures may be due to entirely different causes, and
that two distinct phenomena have been included under the same
metaphorical expression. The bilaterality of an appendage is a
mathematical property of the primordium, and may be expressed
as either a dbifacial or isobilateral flattening ; while the so-called
dorsiventrality of a leafy shoot system is merely the expression
of its structural eccentricity. The term dorsiventral may be there-
fore conveniently eliminated from botanical phraseology altogether ;
the physiological conception of Sachs being clearly defined by his
later expression plagiotropism. At any rate, before committing
oneself too far to the pursuit of academic abstractions as to the
symmetrical relations of living and growing organisms, it will be
well to consider the mathematical consequences of certain simple
types of growth.
Returning to the original proposition of growth, it is obvious
that mathematically centric growth is only a special and ideal
case, which is possibly extremely rare in nature, If perfect radial
270 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
growth at a uniform radial distribution from a growth-centre
implies the uniform action of various surrounding stimuli, which
in the vast majority of cases are unequally distributed, the
wonder remains that centric growth should be on the whole so
closely approximated that it appeals to the senses sufficiently to
justify this simple mathematical conception as a general starting-
point. Thus, although the majority of tree trunks and branches
may be fairly circular in section, few would show an even
approximately central pith, while the eccentricity of large starch
grains becomes the type. That the growth-centre of a shoot
exposed to varying environment should become eccentric is
therefore not to be wondered at; but the degree of eccentricity
can only be judged by the after effects and by the eye, so that a
slight alteration of the system would not necessarily be noticed,
and it becomes very difficult to draw any sharp line between what
may be taken as sufficiently centric and constructions which are
obviously eccentric.
Such cases of eccentricity and their relations as expressed in
the phyllotaxis systems may, however, be as readily followed by
geometrical constructions as the allied cases of the centric and
eccentric tree trunk, or the centric and eccentric starch grain,
the construction lines of which are generally accepted as being
represented by orthogonal trajectories (Sachs), though it is true
that no absolute proof has yet been given. Further, it becomes
possible, by geometrical constructions similar to those already put
forward as explanatory of the relations of symmetric (whorled)
and asymmetric (spiral) centric phyllotaxis systems, to deduce the
properties of the same system when the whole growth-system
becomes eccentric: the whole series of phenomena representing,
in fact, definite mathematical cases of growth construction which
would naturally be expected to occur in organisms exhibiting
growth under different aspects.
Just as, in dealing with the growth and phyllotaxis of the main
shoot axis, the chances would appear to lie mathematically in
favour of asymmetry rather than symmetry as the fundamental
case, so that perfectly symmetrical construction, as exhibited in
the (2+ 2) or decussate system, or the (5+5) symmetry of flowers,
ECCENTRIC GROWTH. 271
is probably to be regarded as a special case of construction evolved
by secondary specialisation for a special set purpose,—the exist-
ence of a capacity for becoming symmetrical being the essential
character which may be increased by natural selection,—so again,
in the case of centric and eccentric construction, a tendency
toward eccentricity would be expected, in that it presents the most
general mathematical construction; while the chances that nutri-
tion or circumstances of environment would not be absolutely
constant on all sides of the organism are enormously greater than
those of uniformity.
The essential question is rather, Is there such a thing as
absolutely centric growth at all in nature? Hypotheses of
centric growth have been read into the plant as presenting simple
mathematical cases ; but should not the eccentric be rather taken
as the starting-point, since, by assuming a simple case which is
not the primitive one, the subject may be approached from a
misleading point of view? The marvel is, in fact, not that some
shoot systems should show marked eccentricity of growth, but
that any should ever approximate the centric, wnless the inherent
growth faculty of the organism in working out its specific form
according to simple laws of growth 1s, as a rule, far stronger than its
tendency to respond to external stimult. However, the fact remains
that the difference between centric and eccentric growth pheno-
mena is only loosely judged by the eye, which is not a mathe-
matical guide, and for practical purposes growth is often so
approximately centric that there is no harm in calling it so, so
long as it is remembered that it is not possible to draw a sharp
distinction between the two cases by actual observation.
So far as eccentricity of growth may be induced by inequality
in the action of external stimuli, such as light and gravity, quite
apart from the equally possible conception of inherent irregularity
of construction, it becomes clear that every lateral flower shoot,
borne on an erect axis, would have a natural tendency to become
eccentric, just as every lax shoot which is not strong enough to
support its own weight. A tendency to more or less obvious
zygomorphy should, for example, be the rule in any short lateral
floral axis, and not the exception ; and here again with no reference
272 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
whatever to special external agencies such as insect visitors, but
simply as a structural feature. Natural selection may increase this
tendency, but it was always present in the growing-point of the
shoot. By stating the converse for “terminal flowers” on centric
shoots, one begins to arrive at the truth underlying such gener-
alisations as that of De Candolle that no zygomorphic flower
could be terminal, notwithstanding the fact that every flower
terminates its own axis.*
It is also a point of interest that the more general occurrence
of eccentric growth in flower shoots, in that it involves the sub-
jective appearance of a certain bilateral symmetry, is responsible
for the introduction of conventional conceptions of “median
planes,” etc, which, though transferable to symmetrical phyllo-
taxis systems, have absolutely no existence in the more generalised
case of asymmetrical construction.
Because the genetic-spiral of Schimper would not explain the
phyllotaxis of a dorsiventral shoot, Sachs was willing to throw
over the entire spiral theory, although he was fully aware of the
fact that cases of markedly “dorsiventral” phyllotaxis are rela-
tively few, occurring usually in growth constructions specialised
in other directions; and the elucidation of such “ dorsiventral”
constructions remains a test case for any suggested hypothesis
which claims to interpret the facts of shoot construction and leaf
arrangement. The present standpoint is sufficiently clear; there
can be little doubt that the habitual use of the term “dorsiven-
tral” introduces a fallacious standpoint; one incautiously argues
that the distinction of a dorsal and ventral surface as in an animal
must be a very remarkable specialisation, and so undoubtedly its
consequences appear in the adult shoot, though the separation of
the two surfaces is not apparent at the growing-point where the
construction commences. Once the term “dorsiventrality” is
eliminated, or only retained as a harmless metaphor, and it begins
to be obvious that all that the phenomena include is an unequal
distribution of growth in different directions, the subject becomes
much clearer from the point of view of strict morphology; while
the physiological conception of plagiotropism, and whether this is
* Cf. Goebel, Organography of Plants, Eng. trans, p. 133.
ECCENTRIC GROWTH. 273
the cause or the effect of the unequal growth, may also be placed
on one side.
The log. spiral hypothesis, based on the laws of uniform growth,
which so readily established the connection between spiral
(asymmetric) and whorled (symmetrical) centric leaf arrangement,
is equally readily applicable to the case of eccentric growth-systems.
The eccentricity involves the whole growth-system of axis and
appendages (leaves), and thé mathematical properties of the quasi-
square systems remain unaffected, the only alteration produced
being a co-ordinated change in the form of the whole shoot-system ;
just, for example, as in the case of an unequally developed Pine-
cone, every scale on the cone is affected, and takes its share in the
structural eccentricity.
Just as uniform centric growth is a definite mathematical
conception, the geometrical properties of which may be readily
investigated by drawing suitable log. spiral constructions on a
groundwork of a circular meshwork of quasi-squares, and the
geometrical properties of such constructions may be deduced
before making any further observation of the plant; so it is well
to put together the general facts of the homologous cases of
eccentric growth, in order to see what phenomena will be char-
acteristically expected, before any attempt is made to bring plant
constructions into line with such a hypothesis.
The difficulties in the way of getting a satisfactory geometrical
construction for an eccentric growing system, in which, that is to
say, the eccentricity is progressive, and becomes more marked as
growth proceeds, although it may not be visible to the eye to
begin with, are naturally considerably greater than in the first-
studied simple case of centric distribution (fig. 24); since all the
periclinal curves in the most general case would cease to be circles,
and become complicated ovoid curves very much of the type
observed in typical starch grains, while the diagonal construction
lines cease to be log. spirals, although all the lines may still
be regarded as derivatives of these curves. A useful figure
which appears to combine all the essential facts of eccentricity,
together with a simple geometrical method of construction, in
that it is wholly constructed in terms of orthogonally intersect-
974 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
ing circles, may be adapted from a special case of electrical
distribution.*
Such a diagram is then an eccentric homologue of fig. 24, and
any phyllotaxis system for uniformly progressive eccentric dis-
tribution may be taken from it. It must be noted, however, that
although a diagram in terms of circles has been utilised because
it is easier to construct, there is no suggestion that the circle
represents the true shape of the periclinal curves. This diagram
is a special case, and is only taken so far as it goes, in that it will
give the correct appearances within the error of estimation by the
eye, and is, at the same time, a mathematically correct orthogonal
* For the construction of this figure I am indebted to Mr E. H. Hayes.
It represents one half of two systems of coaxial circles which intersect ortho-
gonally, which would represent in electricity the lines of magnetic force around
two equal and parallel currents travelling in opposite directions. The data
for drawing it are as follows :—
Let XX’ and YY’ be rectangular ordinates intersecting at the origin O.
From O along the axis OX take a point C, 5 inches from O. The centres
of the intersecting circles fall along CX and OY, OY’ at the following dis-
tances, C =the centre of construction, on OC produced describe circles with
centres . . at distance from C . . . with radius.
3°399 inches . 6°749 inches.
1°565 ” 4255 »
‘778 ” 2°896 ”
401, 2041 =,
210 4 ; 1464,
‘Wl 4s 1059,
0588, . 769,
0313 ,, ‘560,
0167 ,, ‘409,
0089 i; 298,
0047 ,, 218,
The intersecting circles pass through C with centres on YY’ at distances
from O of
+ 1625 niches.
3633,
6882,
15388,
also the circle through C with centre O, and the straight line OC. With
these data, intermediate meshes may be filled in empirically within the
accuracy of drawing the figure.
ECCENTRIC GROWTH. 275
system of quasi-squares. Since, again, an eccentric system in
which growth is unequally distributed on different sides conveys
to the eye the subjective appearance of a displacement of the growth-
centre towards one side, while the general approximation to a
circular outline may not be affected, it may be convenient to make
use of this phraseology as a simple way of describing the construc-
tion, although it has no causal significance. For example, by
selecting the (6+5) system of construction curves a figure will be
obtained (fig. 96) which is an eccentric homologue of fig, 55, and
represents definitely, therefore, the general construction mechanism
of a zygomorphic pentamerous flower. In this special case the
growth-centre has apparently been displaced towards the upper
surface (posterior side of the floral diagram); the converse con-
struction is seen on turning the figure the other way round.
The diagram will now be seen to illustrate several points of
interest which will be useful in the interpretation of floral
construction :—
I. The system has lost its radial symmetry and become definitely
bilateral, or, as it has been termed, “dorsiventral” ; that is to say,
as soon as the growth-centre is displaced, a line may be drawn
dividing the construction into two halves as a simple geometrical
consequence of the eccentricity.
II. Notwithstanding this the construction remains definitely
(5+5); that is to say, all previous deductions based on the
corresponding centric type continue to hold. The system is
growing, and the growth-centres in each whorl of five are still
initiated simultaneously: the fact that they may afterwards grow
at different rates is wholly secondary.
III. But while the strict alternation, the contact-relations, and
the simultaneous initiation of five new centres remain unaffected,
the appearance of the system at any given moment will always
present the subjective effect that the largest members must have
started first! Each cycle of five has in fact been described as a
“successive whorl”; while a construction of the type of fig. 96
would be termed “ascending development,’ and its inverted
homologue a case of “descending development.” It is at once
clear that a “successive whorl” is a contradiction in terms, and
276 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
that accepted interpretations of the facts of ontogeny are very
conventional.
Fig. 96.—Eccentric growth ; system (5+ 5).
IV. The essential point in such a construction is that the
eccentricity is progressive, and is therefore increasingly obvious at
the periphery of the figure, while towards the centre it may be
ECCENTRIC GROWTH. 277
indistinguishable from centric growth. A developing shoot-
system, that is to say, may be so approximately centric in the
region of the growth-centre that it may appear perfectly normal
to the eye, and yet a very obvious amount of inequality may
appear in the older members.
Fig. 55.—Centric symmetrical construction (5 +5).
V. The lateral members themselves share in the distortion, in
that the ovoid-curves which fit into the quasi-squares are no longer
symmetrical in themselves unless they fall in the new plane of
bilaterality. That is to say, the members proceeding from the
sides of the system will themselves be asymmetrical ; or, in tech-
278 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
nical phraseology, anisophylly is a necessary accompaniment of
dorsiventrality.”
The subject of anisophylly (Wiesner, Goebel *) is, however, a
very wide one, and no more is intended at this stage than the
indication of the fact that all such cases of asymmetry require
to be studied from a definitely structural basis;* and that
anisophylly is a necessary consequence, both in the form of
unequal leaf-members (Wiesner) and unequal-sided leaves (Goebel),
just as in the corresponding case of the formation of the lateral
pinnules of a compound leaf. And further, the study of eccentric
growth phenomena affords a guide to the type of anisophylly to be
expected, that is to say, to the fundamental and primary form
of anisophylly, any variations from which must be regarded as
secondary specialisations. For example, if the limiting log. spiral
curves of a (1+1) mesh (fig. 24) be taken as possibly representing
a general fundamental symmetrical growth-form for a leaf-lamina,
of which asymmetrical homologues would be produced by unequal
rates of growth of the general form of the asymmetrical curves
used in plotting an asymmetrical construction, the alteration of
such a symmetrical form consequent on its development in an
eccentric growth-system should be similarly expressed by the
curves bounding the quasi-squares of the eccentric system. Such
a suggestion, based on the standpoint that the fundamental growth-
form of the typical leaf-lamina (the Urblatt of Goéthe) is to be
derived from a unilateral retarded growth distribution initiated
from a basal growth-centre on the surface of the axis, is purely
tentative; but the homologues of the log. spiral curves of the
centric (5+5) system, seen in fig. 96, serve to indicate with
sufficient approximation the general fact that the lateral leaf-
members of an eccentric growth-system must themselves be
asymmetrical, and only those which develop in the plane of
eccentricity truly equilateral. The anisophylly, again, is of the
general type that the largest side of the leaf is on the side of the
* Of. Goebel, Organography of Plants, Eng. trans., p. 99: “ Anisophylly
occurs exclusively on plagiotropous shoots and is a character of adaptation which
has an evident relation to the direction of the shoot and especially its position
with regard to light.”
ECCENTRIC GROWTH. 279
axis showing the greatest growth; and when, as is so commonly
the case, the displacement of the growth-centre is toward the
upper side of the shoot, the largest side of the leaf, as also the
largest leaves, will be toward the lower surface.
Similar generalisations apply to the structure of leaf-pinnules,
in which the largest lobe is necessarily on the side nearest the
base. This was, in fact, the conclusion reached by De Candolle
from empirical observation, and may be taken to indicate that the
great majority of cases obey these structural conditions. The
reverse effect does, however, occur in a few cases, notably in
the example of Goldfussia, a decussate type,* and to these the
teleological explanation may be more safely applied: the only
point to which attention is directed at present being that such
apparent exceptions do not disprove the general mathematical
basis of the construction mechanism. Teleological interpretations
which seem to satisfy some cases but not others are never wholly
satisfactory; but that secondary specialisations may be super-
imposed on the primary construction would be naturally expected.t
* Cf. Goebel, Organography, p. 112.
+ As an example of a comparable phenomenon in which the primary
condition of eccentricity is apparently reversed by a secondary specialisation
which takes the form of a later development of eccentricity in a diametrically
opposite direction, the case of the development of the typical Papilionaceous
flower may be taken, and the same holds for many cases of specialised
zygomorphic flower-shoots. Thus, in Cytisus Laburnwm, longitudinal sections
of the perennating buds in January show that all the floral members are laid
down in position on a markedly eccentric receptacle ; the anterior side being
twice as large as the posterior, the anterior members also exaggerated ; so that
the conventional interpretation of ascending development is obvious. The
same eccentricity persists on until the anthers are fully formed and the ovules
produced inside the carpel (March) ; but as the mechanism receives its final
adjustment in the colouring buds, the growth of the posterior side of the flower
is considerably increased, i.e. the eccentricity is reversed, and the posterior
petal becomes the largest, and the posterior side of the receptacle is thrown up
as a considerable elevation, giving the axis that semi-crateriform condition which
has induced systematists to describe the type as “ perigynous.”
Similarly, in Viola odorata, longitudinal sections of young buds cut in March
show all stages in floral development, and may be accurately plotted by cam.
lucida, The same phase of “ascending development” indicates a structural
eccentricity in the system, and is also expressed in the unequal development
of the floral receptacle, the anterior side being again considerably larger than
T
280 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
The general application of this assumption of anisophylly in
the phyllotaxis system is plotted in fig. 97 for the simple decussate
sO
Fig. 97.—Eccentrie growth, system for orientation and approximate anisophylly
of the members of (1+1), (2+2), and (3+3) systems,
the posterior. Ata slightly older stage, when the flower stalk is 2 mm. long,
the construction appears equalised, and the receptacle takes on a uniformly
crateriform outline. By the time, however, that the ovules arise in the ovary
and the bud attains a length of 1:5 mm., the eccentricity is definitely reversed
and the posterior side is distinctly larger than the anterior, and this again
becomes increasingly exaggerated in the older bud and flower.
ECCENTRIC GROWTH. 281
type of Selaginedla, in which the general form relations of the
lateral members are indicated with a degree of accuracy quite
sufficient to illustrate the main principles. It may be noted also
that such a type, in which the leaf-members are not specialised to
any extent, but remain in the primary “ leaf-base” type of member,
illustrate such simple mathematical construction relations much
more obviously than is to be expected in more specialised petiolate
forms in which the primary leaf-primordium may have undergone
secondary segmentation. From the construction of the diagram it
also follows that the greater the degree of eccentricity in the
construction the more marked will be the anisophylly; so that in
Selaginelia, for example, the difference in the size of the two pairs
of leaves becomes a measure of the amount of secondary divergence
from the primitive radially symmetrical type.
That a shoot-system should become obviously eccentric in its
growth and exhibit phenomena variously included under the terms
“ dorsiventrality,” “zygomorphy,” and “anisophylly,” in the sense
defined by both Wiesner and Goebel, is, however, after all not the
most remarkable feature of such shoot construction; or again that
such eccentricity should be possibly induced by external environ-
ment, whether light action, as in the case of certain foliage-shoots,
or as an adaptation to insect-visits, as in the case of many flower-
shoots. The eccentric tendency may be inherent in any shoot, and
especially so in lateral ones, and if at all advantageous may become
exaggerated by natural selection with the production of its more
or less marked after effects; but the most remarkable feature is
not the existence of the eccentricity itself, so much as the manner
in which the eccentric construction when markedly developed
becomes established with a constant orientation in the phyllotaxis
system. In other words, the displacement of the growth-centre
in such cases is not accidental, but follows a definite direction
which must often be accurate to a remarkable degree. These
phenomena include both the orientation of the foliage of the
“ dorsiventral ” shoot and the “ plane of zygomorphy “ of the eccentric
flower-shoot. In the case of foliage-shoots, the production of an
amount of structural eccentricity sufficient to appeal to the eye in
the anisophylly of the leaves in the case of normal asymmetrical
282 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
phyllotaxis is very rare (cf. Abies sp.); and the fact that such a
degree of “dorsiventrality” may be legitimately regarded as a
secondary specialisation is suggested by the curious fact that in
the great majority of such phyllotaxis systems the construction is
symmetrical of the distichous (1+1) type, the decussate (2+ 2), and
very rarely (3+3) (cf Salvinia, Catalpa syringaefolia) ;* thus
giving, in the first case, two rows of leaves on the upper side of
the shoot, and in the second case four lateral rows of the Selaginella
type. But on comparing the orientation of such lateral shoots
with the normal examples of distichous and decussate orientation,
it will be noted that while in the former case the leaves may lie
either in the transverse or median planes, and in the latter are
cruciately orientated in these same planes, the assumption of
eccentric development is accompanied by a displacement of the
growth-centre in such a manner that the leaf-members are always
left on the flanks of the now bilateral construction, and the growth-
centre is displaced towards the upper side in such a way that
distichy is replaced by two rows of leaves now apparently on the
upper surface, and decussation by an accurate displacement at 45°,
giving a diagonal orientation; while the (3+3) system displaces
at 30°, with the result that three anisophyllous rows are left on
either flank. In fact, as opposed to the use of the term “ dorsi-
ventral,” the more immediate purpose of the operation would
really appear to be the ereution of a right and left side, so that one
would be as fully justified in calling a “dorsiventral” shoot
zygomorphic, as a“ zygomorphic ” flower dorsiventral (cf. Goebel).
Such displacements have been variously described in terms of
angular changes and “displacements,” according as the general
effect is judged by the eye;t but there can be little doubt that
the same general construction principles continue to obtain, and
that the scheme of fig. 97 conveys a very good summary of the
facts of the case, the orthogonal intersection lines of the respective
systems remaining unaffected. For example, it is clear that the
greater the eccentric specialisation of a distichous shoot, the more
nearly would the two rows of leaves appear as a single median line
* Goebel, Organography, p. 108.
+ Of. Ibid., p. 112.
ECCENTRIC GROWTH. 283
on the upper side of the shoot, and their anisophylly might tend
to disappear.*
While, again, a marked degree of eccentricity is rarely met with
in the case of asymmetrically constructed foliage-shoots, such a
phenomenon is frequent in the case of spirally constructed flowers ;
and in such types the very general attempt to convert the floral
diagram into a symmetrical circular expression has not only led
to confusing results, but in many instances has served to conceal
the essential asymmetry of the floral structure. Thus, in a typical
Angiosperm flower the assumption of symmetry in the sporophyll
region is so remarkable and so definite, that the circular plan of
the floral diagram becomes the common convention, to which a
quincuncial calyx or perianth as an outer investment appears
almost as an accident of the construction; and when the asym-
metrical region is thus limited to the members of a single contact-
* The case of Salvinia is of special interest: the embryo, as is well known,
commences a normal asymmetrical development which is continued for three
leaves constituting a single cycle. At the fourth node, symmetrical alternating
whorls of three commence ; that is to say, a (2+3) primary system adds after
one complete cycle around the axis a new path of distribution, the system thus
immediately becoming symmetrical of the type (83+3): less frequently (2+2)
symmetry is first attained at the third node (Pringsheim). Normal eccentricity
is, however, superimposed on the construction, thus giving three rows of leaves
on either flank. It is interesting to compare this architectural scheme for the
position and even approximate primary shape of such leaf-members and the
apparent order of development of these successive whorls with the ontogenetic
or building account given by Pringsheim (Gesammelte Abhandlungen, vol. ii.
p. 354), although so little of any scheme can be definitely checked at the
actual algal-like apex. Thus, according to Pringsheim, the water-leaf arises
first, and is followed by the foliage-leaf farthest removed from it, the other
leaf on the same side as the submerged leaf last. The whorls then alternate
in the same fashion. This agrees with the geometrical construction (fig. 97),
but Pringsheim’s theoretical schemes are based on a preconception of the
importance of quadrant division in the segmenting cells behind the apex.
Once it is admitted that the apical cell is not the ruler of the space-form, but
an accident due to the special type of segmentation into relatively bulky cells,
and that lateral members are not localised accurately to special segments,
beyond the fact that regularity in production of cells and members must
necessarily involve a certain coincidence between the two, the value of these
segmental schemes is much diminished, and the drawings on which they are
based do not afford any convincing evidence of their theoretical importance.
284 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
cycle the deviation from perfect radial symmetry may escape
notice.
That many floral types, on the other hand, are asymmetrical
throughout the whole sporophyll region, including the corolla, is
undoubted ; although, again, owing to the general tendency of the
plant to promote efficiency in the floral mechanism by the reduc-
tion of the number of its parts, the number of members produced
is often insufficient to give any spiral pattern to the eye. In such
cases the Hichlerian convention of assuming circles as much as
possible has been adopted: thus, while Aconitum cannot be regarded
as anything else but asymmetrical throughout and _ slightly
eccentric (perianth (3+5), sporophylls (8+13), growth-centre
displaced anteriorly in the plane of sepal 2), many diverse views
have been proposed with regard to the interpretation of the flower
of Tropacolum majus.
As in previous instances, a geometrical construction diagram may
be readily prepared which will illustrate the phenomena to be
expected in the development of an eccentrically growing asym-
metrical construction. Thus, fig. 98 is drawn for an eccentric
(3+5) system, and fig. 99 for a (5+8), in both of which, as in
these zygomorphic flowers, the growth-centre has been displaced
anteriorly owing to an increased rate of growth of the posterior
side of the flower, and the plane of eccentricity follows that of the
second sepal, which is not necessarily the median plane of the
diagram, but sufficiently near it for practical purposes.
Consideration of fig. 99 at once shows important features; the
curve construction, as before, is wholly built on orthogonally
intersecting curves, and it follows from the mathematical pro-
perties of the numbers of the curves employed that the quasi-
squares may be serially numbered. But the mathematical order
of such enumeration is no longer that of increasing size; that is
to say, wis not the order of actually visible ontogeny, although it
still remains the theoretical order of the initiation of the growth-
centres. In such a flower, therefore, some members would appear
to arise as protuberances out of their normal spiral sequence, owing
to the fact that those on the posterior side are growing at an
increased rate, and those on the anterior side at a diminished rate.
ECCENTRIC GROWTH. 285
It may further be shown by diagrams that by varying the amount
Fig. 98.—Eccentric growth, system (3+5) orientated for the plane of No. 2.
Growth-centre anterior. Cf. Tropeolum majus,
of eccentricity the details of such ontogeny would also vary, so
that no common rule could be given for the serial ontogeny of
286 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
floral types of varying degrees of eccentricity, while other com-
plications occur, owing to the fact that eccentricity may not
Fig. 99.—Eccentric growth-system (5+ 8), orientated for plane of No. 2, Growth-
centre anterior ; a portion of the construction-network indicated on one side,
always follow the same plane of sepal 2. Every flower requires
to be considered on its own merits, according to the time at
which eccentric growth commences and the degree of eccentricity
ECCENTRIC GROWTH. 287
attained. Thus, in fig. 99, eccentric in the plane of sepal 2,
the increased rate of development of the posterior side of the
flower involves the ontogenetic origin of members numbered
according to their theoretical value to appear to the eye as if they
had been produced in the sequence 2, 5,7, 4, 10, 1, 3, 8, 12, ete.
If, again, in the same figure, numbers 1-5 represent a normal
quincuncial calyx, and 6-10 five spirally arranged succeeding
petals, approximately alternating with them, 11-18 would be
eight stamens and 19-21 three carpels, all in normal spiral series.
Of these stamens, however, the apparently oldest would be 12, 13,
15 on the posterior side, 11, 18, and finally 14, 16, 17.
It now becomes of interest to compare the interpretations put
forward to explain the construction of the flower of Z'ropacolum
(Freyhold, Buchenau, Eichler, Van Tieghem, Rohrbach, Cela-
kovsky). The simple view which regards the androecium as
consisting of two whorls of five, and the flower as cyclic, but with
two median stamens suppressed (Van Tieghem, Eichler), affords
no suggestion whatever as to the peculiar irregular ontogeny of
the eight stamens. Nor can any attempt at manipulation of a
“8 divergence,” which would follow, according to Schimper
and Braun, from the presence of eight members, account for the
anomalous and yet fairly constant order of development, and
more particularly for the postero-lateral position of the first to
appear. Thus the sequence for a left-hand flower is:
8|4 *
2 1
6 7
3] 5
According to the excellent account and figures of Rohrbach,+
the position of 4, 5, 6 may vary somewhat. The calyx is ad-
mittedly spiral, but the axis is not apparently eccentric at this
stage; the corolla is also spiral, as shown by Rohrbach; the second
petal being distinctly larger than the others at first, shows that
eccentricity now setsin. The fact that the stamens arise singly
in an irregular order, and not strictly “ ascending” or “ descending,”
implies that whorled symmetry is out of the question and that
* Kichler, Bluthendiagramme, li. p. 298. + Bot. Zeit., 1869, p. 848.
288 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
spiral construction must be involved. Since the members are
few and the calyx apparently normal (3+5), there appears little
reason to doubt that 7ropacolum is a spiral (3+5) type throughout
all its parts, retaining 5 sepals, 5 petals, 8 stamens, and usually
3 carpels (less frequently 4-5) in normal spiral sequence, which
is, however, affected in so far as the numerical order of apparent
development is concerned by a pronounced degree of eccentricity.
Thus, according to the diagrams, figs. 98, 99 show that the
amount of eccentricity of the developing flower lies between these
two figures ; the former shows eccentricity proceeding at a greater
rate, while in the latter it has affected a greater range of members.
The agreement of fig. 98 with the data of Kichler and Rohrbach is
not only very close, but it conveys in itself the reason why the
data may slightly vary (Rohrbach, Bot. Zeit., 1869, p. 848, figs.
1, 516, 17); while observation of the error of the geometrical
construction shows the difficulty of accurately gauging the relative
size of the primordia by the eye, the error of the geometrical
method being again less than that of observation of the protuber-
ances on the actual shoot apex. Thus in fig. 98 the sequence of
apparent origin would be considered to be 12, 11, 13, 14, 15, 16,
17, 18; or
8 | 5
3 1
6 vi
2/4
while in fig. 99 it would be 12, 13, 15, 11, 14, 18, 16,17; or
6/3
2 1
7 8
4/15
It is clear that if the eccentricity of the former diagram had proceeded
further 13 would have become larger than 11, that is, 3 would have
replaced 2 in the first scheme ; while if the eccentricity of the latter
had not been carried so far 15 would not have appeared larger than
11 and 18 would not have been larger than 16 and 17, these being
the changes required to bring either scheme into agreement with
that of Rohrbach.
ECCENTRIC GROWTH. 289
There can be little doubt that the explanation of the 7'ropaeolum
flower is remarkably simple, once the eftect of an unequal rate of
growth on one side of the whole shoot and its appendages is
understood, and that such construction diagrams not only include
between them the facts of observation, but point out the degree
to which variation may be expected, according to the amount of
eccentricity obtaining at the moment of observation.
The flower of Z'ropacolum majus is thus in all probability a
(3+5) asymmetrical type throughout; its eccentricity being still
further complicated by: (1) the delayed development of the
corolla, so frequent in petaloid types, these members being greatly
retarded at first, so that the relation of the corolla cycle to the
spiral sequence is not apparent; (2) the stamens also present a
degree of growth retardation which causes them to lose at an
early date the normal contact-relations of a (3+5) system, and
so loosen out until the contact-relations of the next type (5+8)
are approximated (fig. 99), in which eight members are required to
fill a contact-cycle. A ninth stamen, if produced (No. 19). would
be median anterior, and the orientation of the carpels slightly
oblique.
As opposed to the Eichlerian type of diagram, a convention
based on the visible structure of the bud, these schemes become
structural diagrams for the primary distribution of growth-energy
in the initiation of the floral members. An attempt has been
made in fig. 99 to approximate the amount of eccentricity re-
quired to agree with known data, if the system had retained
uniform growth in all its parts; the fact that the flower illustrated
is “ left-hand” resulting from the direction in which the funda-
mental construction curves cross one another (fig. 34).
Given these fundamental phyllotaxis phenomena as the basis
of the construction of the flowering axis, it now becomes possible
to isolate all superimposed variations and alterations in the
relative rates of growth which collectively determine the formation
of a floral mechanism from a mere collection of uniform lateral
appendages.
290 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
V. The Bilaterality of Appendages,
SINCE it follows from the mathematical construction of a circular
system of lateral members, making close contact and exhibiting
uniform growth expansion, that no absolutely radially symmetrical
member can be primarily produced from the growth-centre and
remain a part of the system, a comprehensive definition of a foliar
member is thus obtained, in that all leaf-structures must be by
construction inherently bilateral from their first inception; and
although the true curve of the transverse section of a member
may differ from a circle only within the error of drawing, these
primordia are always eccentric with regard to the point which
has been termed the “centre of construction.”
The biological observation that “stems” bear “leaves” in
acropetal series is thus correlated with the mathematical fact
that such inherently bilateral primordia can alone be primarily
produced by a growing apex so as to satisfy the observed
phenomena. Radially symmetrical “ branches,” on the other hand,
are to be regarded as secondary productions, and arise at a
greater distance behind the growing-point; as, for example, in
the apparent axil of a previously formed primary primordium,
these areas being the only spots left vacant in a normal growing
system.
Every leaf, or lateral member of the first degree, whether borne
on the axis of gametophyte or sporophyte, whether an assimilatory
appendage or a sporophyll, is thus structurally bilateral from the
first ; and however much certain types may subsequently become
“centric” in form, they remain nevertheless eccentric internally,
with regard to the centre of construction, so long as any growth
THE BILATERALITY OF APPENDAGES. 291
persists; although, again, the amount of eccentricity may be very
slight, and, when the bulk-ratio is small, quite inappreciable to
the eye. In the case of sporophylls, more especially stamens,
the approximation to a circular section in the developing prim-
ordium may be very close;* but in the majority of foliar members
the structural bilaterality becomes increasingly exaggerated in
the form of the so-called dorsiventrality of the leaf, as these
members become specialised as assimilating laminae, exposing
the maximum surface to gaseous interchange.
In fact, the primordia seen at the apex of a typical leafy shoot
are usually obviously bilateral from their first appearance, and
thus apparently flattened in a tangential direction; but as
previously indicated, their first appearance tells little of their
first inception. And, just as it has been shown that a uniform
cessation of growth at a certain stage in all the lateral members
may, as in the case of Coleochaete (fig. 87), suggest the appearance
of a uniform yrowth-increment comparable to the effect of a
uniform growth-movement expressed as a uniform velocity; or
avain by the subsequent attainment of a constant bulk in the
lateral members may create a subjective impression of torsion-
spirals, intersecting as helices on a cylindrical axis where no torsion
exists; so this flattening of the members will necessarily produce
appearances to a certain extent suggestive of the action of a
strong compressing force. There is no need whatever to assume
that the first production of bilateral symmetry in a leaf-primordium
is caused by the stimulus of any mechanical pressure in the bud;
it depends primarily on the actual mechanical construction of
the growing zone, The primordia which subsequently grow
according to their inherent dispositions are able at first to resist
all pressures of adjacent primordia; and so long as these are
equally distributed along orthogonal paths of construction, the
* An interesting example is afforded by the development of the sporophylls
of Clematis (C. integrifolia, Fackmannt, etc.), and a comparison with the
formation of the primary branches of the terminal umbel of such a form as
Heraclewm gigantewm. The former presents a system of leaf-members, the
latter a system of branches whose subtending bracts are suppressed ; but to
the eye the appearances presented by the two cases, and the shape of the
primordia and their contact-relations, are identical so far as can be judged.
292 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
greatest mutual pressures can only press them into close rhom-
boidal contact and convert their section into the form of quasi-
squares.
The appearance presented by a typical foliage-bud, however, is
very different from any such theoretical construction; and it is
clear that the assumption of the considerable amount of flattening
included under the conventional use of the term “ dorsiventrality,”
which is much greater than that of the original primordium, must
entail correlated alterations in the rates of growth. The secondary
flattening of the member is most simply regarded as the effect of
a diminution in the rate of radial growth of the whole system
(fig. 100) ; and as soon as the members diminish in radial growth
at a greater rate than the axis does, the bud loosens its contacts
and begins to open out. Such diminution of radial growth may
also produce the effect of a tangential extension where this does
not really exist, or again it might be associated with such an
increased tangential rate of growth. The several cases may thus
be considered from the standpoint of differences in the rates of
growth-expansion in two directions, the radial and the tangential,
these being represented in any given system by the diagonals of
the rhomboid meshes, which in the case of spiral systems are
both spiral lines.
In a typical bud, again, this “ flattening” is also always associated
in spiral systems with a phenomenon of “sliding-growth,” which is
one of the most remarkable properties of a leafy shoot, in that
the method adopted is perfectly definite. The leaf-members
exhibit a certain amount of slipping at their edges, and the
arrangement is carried out with the greatest precision, so long as
the construction is asymmetrical and spiral. It must be noted,
however, that the corresponding phenomena in the case of whorled
symmetrical constructions is either wanting (cf, valvate preflora-
tion), wholly irregular, or very rarely according to a definite
scheme (¢f. convolute prefloration). In fact, it appears possible
even at this point to make the generalisation that a certain
primary sliding-growth must be a mathematical necessity of
assymmetrical construction in phyllotaxis systems.*
* Of. Mathematical Noles.
THE BILATERALITY OF APPENDAGES. 293
In dealing with such a phenomenon it is necessary, to begin
with, to distinguish between facts of observation and any inter-
Fig. 100.—Geometrical construction, including progressive ‘‘dorsiventrality” and
sliding-growth effect for system (8+13). Drawn with construction curve
(Type II.) for uniformly retarded growth. : Jf
pretations which may have been ascribed to them. Now the
facts observable are very definite: the rhomboidal primordia
apparently thrust laterally along their tangential diagonal, and as
THE BILATERALITY OF APPENDAGES. 295
they thus appear to become tangentially elongated a certain
amount of slipping takes place at the angles. Such sliding-growth
Fig. 43.—Sedum elegans{(6 + 10).
is most simply described as comprising a stepping of the shorter
construction curves, while the longer ones remain unaffected and
: U
296 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
are thus rendered increasingly obvious; so much so, that when
the sliding effect is considerable they can alone be readily checked.
From this standpoint it is easy to introduce the phenomena of
normal sliding-growth into the construction diagram, as in fig. 110
(5+8), or again as in the (8+13) system of fig. 100, in which the
Fig. 102.—Sedwm acre, L. Perennating bud, system (3 +5).
scheme consists of a normal phyllotaxis system platted: however,
with a curve of the second type,—of uniformly retarded growth,
—which conveys the information that a progressive retardation in
the radial rate of growth of the system as a whole produces the
effect of a progressive “dorsiventrality ” in the leaf-members.
But while the progressive increase of dorsiventrality in the
THE BILATERALITY OF APPENDAGES. 297
members is thus associated with a certain amount of sliding-
growth, it does not follow that the two phenomena are in any way
dependent one on the other; the progressive flattening may be
clearly ascribed to a reduction in the rate of radial growth, which
is common to symmetrical as well as asymmetrical constructions,
and must obtain in the transverse component of all dome-shaped
apices; the sliding effect may be entirely isolated from such a
retardation effect, just as in the diagram the reduction in the rate
of radial growth-expansion can be imitated by using a curve
expressing this factor, but the sliding effect has to be put in
subsequently.
It becomes apparent that the consideration of the phenomena
thus included under the general term “sliding-growth,” in con-
nection with lateral appendages, requires very careful handling,
in that it might evidently be the result of several distinct growth-
factors; and as in other cases, the first-suggested interpretations
may not be the right ones. In fact, the true interpretation of such
lateral slipping is of great importance, since it constitutes the
most important evidence in connection with Schwendener’s dis-
placement theory: the Dachstuhl theory assumed that flattening
implied a pressure, and that such readjustment of the angles of
the primordia indicated a displacement of the whole member, and
thus affected the divergence-angle and thereby altered the postu-
lated Schimper-Braun construction. The diagrams (figs. 100, 110)
sufficiently indicate that no amount of such sliding-growth really
affects the primary construction system which is taken from the
initial points, ze. the centres of construction, while, as has been
already repeatedly stated, mere flattening is only the expression
of reduced radial growth.
In considering the meaning of these phenomena it may be
pointed out, to begin with, that the displacement of the points of
contact of four boundary planes does not necessarily involve any
displacement of the original centres of construction ; this, in fact,
is the usual result of the effect. of lateral contact-pressures, com-
parable with the special case of fitting previously indicated as
giving to an otherwise orthogonal construction the appearance of
hexagonal facetting (fig. 95, Pine-cone), or the typical adjustment
298 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
of the radially elongated ovaries of Helianthus, Even in the case
of four cell-walls meeting at right-angles, such slipping normally
Fig. 103.—Sedwm pruinatum, Brot. Leafy shoot, system (5+ 8).
obtains, and is always allowed for in a tissue-drawing,—the four
cell-walls at 90° tending to pull into two sets of three: this
THE BILATERALITY OF APPENDAGES. 299
appearance has been noted by Sachs as a secondary phenomenon
presenting no real objection to the primary construction of tissue
masses by orthogonal trajectory curves (cf. Sachs, Physiology,
p. 433). Thus a slight displacement in a primary orthogonal (5 +8)
system has little effect in altering the construction as a whole,
and the diagram closely represents the fact observed in a section
of the adult stem-apex of Nymphaea (fig. 110).
Secondly, the existence of similar slipping in the form of a growth
adjustment has already been shown to occur in connection with
the arrangement of lateral axes which have only a subsidiary
connection with the primary system of foliar appendages. Thus,
in the stock examples of the Helianthus capitulum, the Pine-cone,
and the Aroid spadix, it was evident that any secondary ex-
tension of a member which was not a foliar appendage, either
radially or tangentially, must produce a similar sliding effect ; the
“stepping ” affecting different curves according to the geometrical
necessities of the construction. That is to say, any variation in
the bulk of a lateral appendage at a point beyond its insertion will
produce alterations in the system if section takes the members at
this point; and it must be remembered that the transverse section
of a typical bud with dome-shaped apex cuts the peripheral
members at a higher level of their course than in the case of the
youngest primordia. Further, a tendency of a leaf-primordium to
become wider tangentially in some part of its course, above its
insertion, may also be taken as typical for the great majority of
foliage-buds. Any such increased tangential growth of a prim-
ordium at a point above its base, while these insertion-areas still
constitute the surface of the axis, must necessarily involve a
readjustment slipping of the same type as that found in the scales
of the Pine-cone; that is to say, the short curves will become
stepped. Variations in the bulk of the appendage at different
parts of its length will thus produce sliding effects in both
asymmetrical and symmetrical systems: in the former the sliding
will follow the tangential diagonals in an orderly manner, since
the paths of such sliding are left obvious in the constrnction; but
in the case of symmetry the overlapping will be quite irregular,
since normally the primordia should accurately meet at their
300 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
edges (cf. valvate prefloration), and there is no reason why the
slip should take place in one way more than the other.
Fig. 104.—Sedum reflecwm, L. Perennating shoot, system (3-+-4)=7-spired :
the vascular bundles follow the spiral of dorsiventrality, which is here also the
ontogenetic spiral,
ale
THE BILATERALITY OF APPENDAGES. 301
But even these effects are only of secondary importance, since
it is clear that any change in the volwme of the lateral appendage
beyond the area of its insertion can have no reference whatever to
the primary construction of the system, although it may tend to
produce all the normal appearances of sliding-growth. Much less,
therefore, can similar effects, produced by changes in the volume
of axillary shoots, etc., as in the case of the Pine-cone, and
aggregated inflorescences (Helianthus), have any bearing whatever
on the subject: such changes can only be included as tertiary
factors; and one wonders more and more at the curious standpoint
which has sought to find in these tertiary effects a basis for
the Dachstuhl theory. On the other hand, the possibility still
remains that in the theoretical and mathematical construction of
asymmetrical phyllotaxis systems there may still exist a primary
cause for these displacement movements ; and as a matter of fact,
the necessity for such readjustment of the free portion of the
appendage follows directly from the mathematical consideration of
the log. spiral theory.*
The discussion of this readjustment, which implies a slight
rotation of the primordia about their centres of construction, and
is also only applicable to cases of asymmetrical construction, may
therefore be postponed until the equation to the quasi-circle
primordium has been deduced: at present it will be sufficient to
consider the geometrical consequences of adding the compensation
for such “ sliding-growth ” to the theoretical diagrams.
The spiral construction diagrams, as expressed by intersecting
curves giving rhomboid meshes, present a good working idea
of a typical foliage-bud in which the members have, in consequence
of mutual growth-pressures, filled all the room available to each
primordium, and thus assume an obliquely rhomboidal section
(ef. Pinus, fig. 88; Araucaria, fig. 41). A section of such a bud
at the level of the growing-point includes the insertion-areas of
the members close to the apex; but away from this the members
will be cut at some distance above their bases, owing to the
character of the dome-shaped apex. The plane diagram, therefore,
represents insertion-areas only, and these alone are now under
* Of. Mathematical Notes.
802 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
consideratibn; that is to say, these figures include the study of
the relationships of adjacent leaf-bases.
Observation shows that, as the rhombs are progressively ex-
tended along their transverse diagonal, the shorter construction
curves are “stepped”; and since these diagonal paths were also
originally log. spirals, the curve of a value one stage lower in the
summation series than the numbers expressing the ratio of the
curves composing the system (ic. their difference) may be con-
ventionally termed the “Spiral of Dorsiventrality.” From this
standpoint a bifacial leaf is only flattened in a strictly horizontal
plane when it is produced in a symmetrical phyllotaxis system ;
in which case the paths of lateral extension are concentric circles:
in the more general case of asymmetry, structural dorsiventrality
becomes exaggerated along a spiral path, which has therefore no
direct relation to external environment, as, for example, the action
of vertical light, although it is the nearest approach possible to
a horizontal line in each rhomb. In other words, the architectural
scheme of each shoot is controlled by the growth-centre of the
axis, which is the fundamental growth-centre of the whole shoot-
system, and here, as in the case of eccentricity, the influence of
external environment, if this is the determining agent, must act
on the primary centre at the end of the shoot, and all subsequent
architectural details are worked out according to strict geometrical
principles. These construction diagrams further show that the
result of the sliding-growth effect is here to place this tangential
diagonal more and more in a horizontal line: a teleologist might
at this point even make the suggestion that the object of the
sliding effect was of the nature of a biological “ adaptation” which
would render the surface of the leaf-lamina more strictly
horizontal; but such an explanation is wholly gratuitous. The
distinction which is here drawn between the geometrical plan of
leaf-base insertions and the geometrical properties of the free
portions of the primordia, as expressed in the lamina portion of
the leaf, will be further discussed from the standpoint of the
mathematical equations of the theoretical curves (cf. Mathematical
Notes, VIII.).
A few interesting details are also more clearly exhibited by
THE BILATERALITY OF APPENDAGES. 303
such theoretical diagrams; thus, as each leaf-rhomb is extended
laterally, over an older primordium and under a younger member,
the shorter curves assume their characteristic serrated appearance,
and the long curves remain with an unaltered clean contact edge.
The recognition of the phyllotaxis ratio of such a growing shoot
then becomes increasingly obvious; the smooth edges of the lines
of contact of one set of curves directly indicating that they must
be the longer curves of the construction.
Again, the lower the parastichy ratio in the Fibonacci or any
other series, the more marked is the “spiral of dorsiventrality ”
as an obliquely horizontal path; while with higher ratios this so
nearly approaches a circle that the flattening and position of the
leaf-insertions along a spiral path are not readily observed; nor
will it be noticed when dorsiventrality becomes excessive, or again
is correlated with phenomena of sliding growth.
Thus, in the case of the sporophylls of Stangeria (fig. 29), the
rhombs are obviously extended along an obliquely horizontal
spiral, and the same fact may be noticed in Pine-cones; while
in the lowest cases of asymmetrical phyllotaxis, the oblique
insertion of the members becomes very marked (cf. Gasteria, fig.
58a). Also, comparison of the diagram (fig. 110) shows how
little the amount of sliding may be that is sufficient to bring the
transverse diagonal of each rhomb very approximately parallel
with the circle, «ec. truly horizontal.
It is also obvious that the “spiral of dorsiventrality ” does not
bear any necessary relation to the “ontogenetic spiral” in a given
shoot: it is the spiral which is orthogonal to the curve previously
termed Schimper’s orthostichy line, and since this latter, which
bisects the angle of the intersection of the construction spirals,
winds in the direction of the longer curves, the “ spiral of dorsiven-
trality” always follows the direction of the shorter spirals, and
bears the same relation to the genetic-spiral as do these (cf. Table,
fig. 34). With a right-hand genetic-spiral, that is to say, the
foliage-leaves on a (2+3) and a (3+5) shoot are obliquely inserted
in converse directions, and their bases remain obliquely inserted
even after the shoot has passed through a zone of elongation
which renders Schimper’s orthostichy line sufficiently straight
304 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
to the eye, this being especially well seen in the case of
leaf-scars.
Similarly, the exaggeration of the primary bilaterality of a
primordium may follow the path of the so-called orthostichy
spiral, which intersects the spiral of dorsiventrality orthogonally.
Fig. 105.—Acacia cultriformis. Young shoot of phyllodes, system (5+ 8).
The insertion of phyllodes and “isobilateral” members in packed
asymmetrical phyllotaxis constructions does not necessarily follow
an exactly vertical plane corresponding to any direct stimulus
of vertical light, but follows the path of the spiral diagonal, which
is very nearly vertical. The spiral path becomes more approxi-
mately radial (vertical) as the phyllotaxis ratios approach equality,
but in Fibonacci systems the spiral nature of the path is obvious
THE BILATERALITY OF APPENDAGES. 305
in the diagram: thus in young growing shoots of Phyllodinous
Acacias (fig. 105) the development of the phyllode lamina along
slightly spiral vanes (spires) is readily recognisable; and the
same generalisation holds for the secondary protuberances which
constitute the ridges of the Cactaceae (spiral of phyllody).
On the other hand, with the assumption of the special case
of true symmetrical construction, these geometrical relationships
vanish, in that the complementary diagonals of the quasi-squares
become circles and radii respectively; so that in a whorled type
the leaf-laminae lie in a strictly horizontal plane from the first,
and a whorled phyllode is also wholly orientated in a vertical
plane.
Finally, it may again be noted that all these generalisations,
being applicable to the rhomboidal section of a leaf-primordium
presented in a phyllotaxis system in which the leaves tend to
take the form of quasi-squares under mutual pressure, do not
directly concern the relationships of the free primordia of the
primary system. The nature and symmetrical properties of the
primary primordia, which in section present the form of an ovoid
curve, which in the theoretical construction is to be regarded
as a quasi-circle, require to be considered separately, when the
mathematical properties of such constructions have been more
fully described (cf. Mathematical Notes); the special point,
of interest being that, while in asymmetrical constructions the
rhomboid sections are also asymmetrical and obliquely placed,
the fundamental curve of the primordium is mathematically
orientated from the first along a radius of the whole system
passing through its centre of construction, and about which
radius the member is truly bilaterally symmetrical.
A clear distinction is thus required to be drawn between the
behaviour of the leaf-base, as seen at the insertion-area (or leaf-
scar), which is the surface of the axis, and the properties of the
free portion of the lateral appendage.
As general examples of these various phenomena, comparison
may be made of the sections of the terminal buds of Pinus (fig.
88) and Araucaria (fig. 41), in which the amount of sliding-
growth is relatively small, The transition is shown very perfectly
306 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
in Podocarpus (fig. 42) and Huphorbia Wulfenii (fig. 90), in
which a very considerable degree of “dorsiventrality,” accompanied
r
Fig. 106.—Huphorbia biglandulosa, Spring shoot, system (3 + 4).
by a well-marked sliding readjustment, is attained at a short
distance from the apex. Species of Sedum with so-called centric
leaves present a point of interest in that, while the primordia are
THE BILATERALITY OF APPENDAGES. 307
at first approximately isodiametric, “dorsiventrality ” at first pro-
ceeds normally, and the leaves only attain their adult pseudo-
circular outline at a later date (ef Sedum pruinatum (5+8),
fig. 103; S. elegans (6410), fig. 43; S. reflecum (8+4), fig.
140). The “spiral of phyllody” is clearly indicated in spring
shoots of Acacia cultriformis (fig. 105), though rapid elongation
in the main axis tends to prevent the full effect from being
observed in a section passing exactly transverse to the growing-
point: the diagram illustrates a section cut slightly obliquely on
one side.
It will also be noted that, just as “dorsiventrality” becomes
normally associated with a “stepping” of the shorter curves, so
phyllody must similarly be connected with a stepping of the long
curves, the phenomenon being identical with that previously
described for the disk-florets of Helianthus (fig. 89).
GEOMETRICAL REPRESENTATION OF BILATERALITY.
Owing to the extreme development of bilaterality in typical
foliage-members, the number of leaves seen in a transverse section
of a bud is usually so greatly increased that the primary log.
spiral construction curves are clearly wholly inadequate for the
expression of the modified construction (fig. 3). At the same
time, as the members enlarge with a diminishing rate of radial
growth, the curves cease to be log. spirals, and when growth ceases,
and all the members have attained an approximately uniform bulk,
the construction lines necessarily pass into curves which cannot
be readily distinguished by the eye from spirals of Archimedes
with equidistant coils, and in the theoretical case would be such
curves (cf. fig. 4).
A construction by definite spirals of Archimedes in the suitable
ratio thus presents to the eye a system much more in accord with
what is actually seen over the greater part of the area of a section
of a foliage-bud than that of the primary hypothetical log. spirals,
owing to the fact that the rhombs plotted by the parastichy curves
may have very nearly equal radial depth. The appearance of
progressive “dorsiventrality,” as the rhombs become relatively
308 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
flatter and flatter at the periphery of the system, is also very
nearly approximated; while it is clear that such Archimedean
spirals would be replaced by helices on a cylindrical axis, and
thus constitute the ultimate spirals observed on the adult plant,
in which growth may be considered to have wholly ceased. A
curve-tracing for the expression of these secondary growth-
phenomena may thus be constructed by taking the central portion
as normally log. spiral, while at the periphery the curves grade
into the corresponding spirals of Archimedes. With such an
empirically constructed curve-pattern a system of dorsiventral
primordia may be plotted, which, when due allowance is made
for the phenomena of sliding-growth adjustments along the spiral
of “dorsiventrality,” presents a most accurate imitation of the
phenomena observed in the macroscopic view of a plant which
presents only these modifications of its construction system in
the adult condition (cf. Sempervivum spinulosum, fig. 4).
On the other hand, it is equally clear that, so long as any
growth persists, the curves will never really become spirals of
Archimedes, although the approximation may be very close to
the eye, and the previous construction will not correctly interpret
the phenomena observed in the section of a growing apical system,
as seen, for example, in a transverse section of the apex of
Euphorbia Wulfenii (fig. 90) or Podocarpus japonica (fig. 42)
which comprises young growing members only. Since “dorsi-
ventrality” may be regarded as the expression of a decrease
in the radial growth of the primordia, the log. spiral construction
curve may be modified by giving it a radial retardation; and for
present purposes it may be sufficient to assume that such retarda-
tion may be uniform. A curve of this form (Type II.) may
therefore be used to plot a system which is still growing, but
at a progressively slower rate (fig. 100), and by adding the
sliding adjustment which steps the shorter curves, a very close
approximation is afforded to the Huphorbia section of fig 90—
at any rate, one so close that the amount of error is not appreciable
to the eye, the actual rate of retardation not being known.*
* Thus, fig. 100 represents a simple geometrical construction in which
uniform growth at the hypothetical growth-centre undergoes subsequently
THE BILATERALITY OF APPENDAGES. 309
Extreme “ dorsiventrality.”—As the lateral extension of members
becomes excessive, and their radial depth as seen in transverse
section more and more approximately uniform, it is clear that a
construction by spirals of Archimedes, which give on intersection
rhombs which are extremely flattened out in a tangential direction,
will closely simulate all observed phenomena, so far as the eye can
judge, although they may never be absolutely correct for growing
systems. Thus, by constructing such diagrams ((2+ 3), fig. 107,
(3+5), fig. 109) using a pair of Archimedean spirals (fig. 33) con-
tinued to the second and third intersection respectively, the
structure of “dorsiventral” leaves of the extreme form found in
foliage-buds is very closely ¢mitated, and by adding the theoretical
slipping across the paths of the shorter curves, it will be seen that
all the phenomena observed are fairly accurately planned. Good
results are thus obtained for floral diagrams of adult flowers,
although for buds better ones would result from a retardation
curve. At the same time it must be noted that Schimper-Braun
constructions are being utilised, in which the structural error now
becomes too small to be noticed.
Nothing is more remarkable in dealing with the sections of a
large number of buds, than the extent to which growth is normally
so correlated in the whole shoot-system, and the amount of
lateral sliding remains so fairly constant in the section that the
a uniform retardation, with a consequent alteration of the curve-system. It
must be remembered that uniform growth remains a purely mathematical
conception, and the log. spirals drawn to express it may be distinguished as
curves of the First Type. Similarly, the assumption of a uniform retardation
is equally a mathematical conception, which, though it may represent a nearer
approximation to the truth, evidently does not yet contain the whole truth,
since, whatever such a retarded growth may be theoretically, a general know-
ledge of growth-processes in living organisms suggests at once that such
protoplasmic growth would show daily and even hourly variations on the
curve the more accurately it could be plotted. These considerations, however,
will not prejudice the attempt to reach a solution of the phenomena by
mathematical conceptions, so far as it may be possible: they serve to indicate
that the conditions become more and more complicated, and the simpler
hypotheses require to be taken first. Assuming a uniform retardation as a
secondary conception, the curves which are used to plot; such a construction
may be distinguished as curves of the Second Type.
310 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
original contact-relations may remain largely unaffected, and the
recognition of the phyllotaxis constant for any given bud may be
rendered easy, however much the members may be apparently
extended tangentially. As a general rule, the original contact-
phenomena remain unaffected near the points of insertion, and the
clean-edged long curves and “stepped” shorter curves are readily
distinguished. As noted previously, however, and as in the case of
hexagonal facetiting, such sliding effects always bring a third set of
contact-curves into view ; so that, when excessive, some confusion
may be produced in the primary system (ef. Sedum acre, fig. 102).
Again, as soon as the amount of “dorsiventrality” and the
accompanying sliding-growth becomes considerable, the original
spiral “ orthostichies”’ become extremely vague, owing to the
difficulty of judging the centres of construction, to which the
vascular system does not always afford a sufficient guide; and
although theoretically the curvature of this spiral increases with
progressive “dorsiventrality,” the superposition of the extended
members is so close, to the eye, that any deviation from the
superposition demanded by the Schimper-Braun hypothesis is
inappreciable. It is thus evident that the Schimper-Braun
formulae for estimating and describing adult phyllotaxis continue
to hold with a considerable amount of accuracy for shoots with
markedly dorsiventral members in which the rate of growth is
considerably lessened, which constitute, in fact, the normal type
of foliage-shoots; but the appearances regarded by Bonnet and
Schimper and Braun as primary are now seen to owe their
existence to a series of secondary growth-phenomena.
Contact-cycles—The empirical constructions given in figs. 107
and 109, for systems plotted by Archimedean spirals of the second
and third intersection, further suffice to bring into prominence
a valuable indication of the relation of the individual members of
one cycle of a phyllotaxis system, from the standpoint of their
overlapping to form continuous investments of the axis. These
relations necessarily hold whatever may be the nature of the
spirals used to plot the system; but by using a form of curve
which exaggerates the tangential lines of contact, in the manner
seen in section of a foliage or flower bud, the relations become
THE BILATERALITY OF APPENDAGES. 311
much more striking than in the original log. spiral constructions
(fig. 28 (3-+5)).
Thus the (2+3) system constructed by Archimedean spirals of
the second intersection agrees sufficiently well with the phenomena
Fig. 107.—Geometrical construction (2+ 8) for contact-cycles, in terms of
Archimedean spirals of the second intersection.
observed in the apex of Sempervivum calcaratum (2+ 8) (fig. 108),
while the seedling of Mymphaea alba is also clearly (2+3) (fig.
94). Three members form a cycle in contact round the axis, one
being half covered, the other two meeting at their pointed angles
x
812 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
in the primary construction, but slightly overlapping owing to the
secondary sliding-growth effect. The presence of such contact-
relations therefore implies a modification of a (2+3) log. spiral
system.
Similarly, for the (3-+5) system (figs. 28, 109), five members
overlap by secondary sliding, although in the original condition
two would be half covered and the other three would just touch :
Fig. 108.—Sempervivum calcaratum, Hort. Apex of one-year-old shoot, (2+8).
eight members again form an investment everywhere two members
deep. In the same manner, by using higher ratios, any spiral
constructions will point to the generalisation that the number of
the shorter curves is given by the number of members sufficing to
form a single investment of the axis, in the case of an unmodified
system, or overlapping slightly to form a single closed cycle in the
more usual case of flattened members.
It will be at once noticed that such a (3 + 5) system, for example,
THE BILATERALITY OF APPENDAGES. 313
presents a striking picture of the familiar quincwncial calya of the
great majority of pentamerous Dicotyledonous flower-types ; and
given an axis producing such a system of lateral members, the
reason for retaining jive with such constancy to constitute a pro-
tective investment becomes increasingly obvious.* These relations
are necessary consequences of the utilisation of lines of equal
distribution in spiral series, and are of especial interest in connec-
tion with the Fibonacci system, in that they give the clue to the
number of members requisite to give the best and most equal
arrangement, whether in the form of a single cycle or many, so
that in dealing with the numerical relations of the parts of asym-
metrically constructed flowers it becomes possible to deduce. a
normal or average type of construction.
The (3+5) system, in an adult condition, and represented by
spirals of Archimedes, from the standpoint of the Schimper-Braun
formulae, becomes a 3 type, and not a 2 as it is usually reckoned ;
the error being introduced with the assumption that a specialisa-
tion of five members implies a cycle of 2, although, as previously
noted (p. 15), there was no possible criterion for such an assump-
tion. That is to say, in dealing with a spirally constructed
pentamerous flower, the (3+5) system brings the first petal to
the front in the gap between sepals 1 and 3, while by regarding
* The apparent mimicry of pentamerous flower mechanism observed in the
inflorescence of many Composites, in which a calyx-like involucre of five
segments is succeeded by a corolla-like series of ray-florets, 5 or 8, and a
series of disk-members resembling the spirally arranged sporophylls of a
Ranunculaceous type, is thus solely due to the working out of corresponding
phyllotaxis rules in the two cases ; one full cycle of protective members being
succeeded by one full cycle of decorative ones and one or more cycles of
reproductive members. If the phyllotaxis system is low, (2+3) or (3+5), the
retention of 5 or 8 members in one full cycle of contact is as normal as the re-
tention of full cycles of higher: terms of the series in Sunflower capitula. There
is no proof that this so-called mimicry is intentional, or even biologically
advantageous ; it is the necessary outcome of a similar low phyllotaxis system
combined with an attempt to reduce the members of each kind to a minimum ;
of. the 5-star flower-like capitula of Chrysogonum virginianum, and Tagetes
signata ; asalso Dahlia coccinea—5 involucral segments, 8 rays, and (8+13)
disk-florets—with Aconitum napellus—5 sepals, 8 petals, and (8+13) sporo-
phylls.
314 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
the quincuncial calyx as a 2 series this proceeding required the
assistance of “ prosenthesis.” Similarly, the fundamental type of a
trimerous Monocotyledonous type is a (2+3) system, giving a
cycle of three members in lateral contact ; so that, when expressed
Fig. 109.—Geometrical construction for contact-cycles, (3+5) system, in terms of
Archimedean spirals of the third intersection.
in the adult condition by spirals of Archimedes, it would give a 2
divergence formula rather than the 4 of systematists.
The extent, however, to which reliance can be placed on such
contact-phenomena in bud-sections remains to be further con-
sidered from the standpoint of varying growth-phenomena observed
in primordia which do not necessarily retain their primary
contact-relations.
THE BILATERALITY OF APPENDAGES. 3815
Fig. 110.—Geometrical construction (5+ 8) by log. spirals, with added correction for
sliding-growth effect. Cf. Nymphaea alba, old perennating stock.
3816 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
VI. Varying Growth in Lateral Members,
So far, it may be noted, all secondary changes, whether those
included under the previous headings of secondary pressure
effects or phenomena of bilaterality, have for their result the
obliteration of the theoretical log. spiral construction figure, and
tend to obscure the primary system. To such an extent, and
so rapidly, does this deformation usually take place, that the
primary theoretical construction for the distribution of growth
becomes increasingly difficult of accurate observation; and it is
clear that the special case of bilaterality is but one of a large series
of phenomena in which varying rates of growth produce secondary
displacement effects. While, in fact, these special cases of
“dorsiventrality” and “phyllody” include the disturbances set
up in the system consequent on varying rates of growth in
different planes in the primordia themselves, other variations
are also possible, and may be comprised more especially under
two main sections :—
(1) Varying rates of growth between the axis and the primordia.
(2) Varying rates of growth in the primordia at different
portions of their length.
That is to say, although the log. spiral construction was founded
on a physical and mathematical conception—the assumption of
a uniform growth-expansion in the protoplasm of the shoot-apex—
it does not necessarily follow that such a uniform rate of growth
is actually present to any great extent in the growing-point of
any given plant, any more than that all radially constructed stems
should prove to be mathematically circular in section. The case
of unrform growth, however, requires to be considered first, just
VARYING GROWTH IN LATERAL MEMBERS. 317
as Newton’s first law of motion comes before the second: granted
the conditions for uniform growth are known, any variations from
these must imply the existence of secondary agencies; and thus,
if primordia are once started in a certain direction, and continue
to expand at a uniform rate of growth, very nearly uniform also
with that of the main axis, the transverse section of the bud at
any time would be marked out along orthogonally intersecting
log. spirals; and conversely, if the construction observed does not
consist of true log. spirals, or the intersections are not orthogonal,
it must follow that some secondary change in the rate or direc-
tion of growth must have taken place subsequent to the time
at which the primordia were laid down, or perhaps became visible.
As already noted, a deep-seated faith in this very fact, that a
primordium which has been set growing in any given direction
will, in the absence of any secondary disturbing force, continue
to do so, and so retain its relative station with perfect accuracy,
has led to the utilisation of such structures as the Pine-cone, the
Aroid spadix, and bractless capitula of Composites as the most
typical examples of phyllotaxis.
While a log. spiral construction thus possibly represents the
arrangement of certain “lines of force” which are intimately
connected with the primary cause of phyllotaxis, the most usual
disturbing factor will be the capacity for variation in the rate of
growth in the members themselves; while alterations in the
direction of growth will be less marked, but may follow pressure
changes as the members become adult, and the action of external
conditions of environment, as in the case of the light-induced
eccentricity of certain spirally constructed leafy shoots (Abies),
or still better, a Pine-cone which grows unequally as a whole
as it bends down after pollination.
It is so far evident that the original phyllotaxis construction
along the log. spiral paths conceivably laid down in the protoplasm
at the actual growing-point, will be exactly maintained so long,
and so long only, as growth remains uniform in the system, and
the members retain the same general shape as that of the apex
on which they are borne, or tend by mutual pressure to take
the form of quasi-squares as seen in section. Such an ideal
818 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
construction is not easy to find, owing to the fact that members
conventionally recognised as Jeaves always subserve special
functions, if developed to any extent, for which they become
secondarily adapted; but the possibility of the conception of
such a theoretical and mathematical primary type of leaf-member
becomes increasingly clear, although now in a sense somewhat
different from that of the Ideal Leaf of Goéthe. A remarkably
close approximation to such a hypothetical structure is found in the
protuberances associated with the primary leaf-points of a typical
Mamillaria. These conoidal masses, which physiologically replace
the primary leaves, are secondary outgrowths which certainly re-
present a leaf “idea” in the plant, worked out in a primitive
mechanical form: their special point of interest is that they
convey a very clear conception of the theoretical system, in that
they, unlike the primary leaf-appendages, have usually no special
tendency to become obviously bilateral. Thus Mamillaria, having
lost its primary “dorsiventral” assimilating members, in repeat-
ing its lateral outgrowth scheme a second time, expresses itself in
’ terms of the simplest possible type of appendage.
Again, if growth in the whole system were uniform in the
transverse plane, all primordia, once they were formed, would
persist with their bases always in the same plane, and all would
appear in transverse section; but owing to special phenomena of
retardation in the main axis, they fall back along it, and the tips
of the peripheral members gradually become lost in transverse
section of the apex: similar phenomena will also represent the
general result of an intercalated “second zone of elongation.”
It is almost unnecessary to state the case in which growth is
uniform between axis and primordium (dormant centres), although
by assuming the proposition of uniform growth it becomes possible
to check the aberration from such a theoretical construction ; and
the rate of growth in the lateral members in the transverse
direction cannot be much greater than that of the axis at their
bases, since their insertions constitute the surface of the axis,
though at a higher level the growth may be unequally distributed
(bilaterality). The longitudinal rate may, however, be considerably
greater, and in such case the lateral members close over the apex
VARYING GROWTH IN LATERAL MEMBERS. 319
and constitute a “bud,” so that the formation of a terminal
bud-cluster of members and its subsequent expansion thus becomes
a question of the correlated sequence of these growth-variations.
While, then, the general effect of elongation in the main axis
will be to diminish the number of members seen in transverse
section, this may be compensated to a certain extent by extensive
growth in length on the part of the lateral members themselves.
Simultaneously, the approach of adult members to a uniform bulk
results in the secondary production of an effect of spirals of
Archimedes, as already noted.
Hence, the best phyllotaxis patterns will persist in shoots whose
leaves undergo little special modification, and remain either in a
primitive condition as protuberances of rhomboidal section, or are
elongated to needle-like members of similar rhomboidal section.*
Good examples are therefore afforded by Araucaria excelsa (fig.
41), Cryptomeria japonica, Pinus Pinea (primary leaves) (fig.
88), Cedrus atlanticus (primary leaves).
Araucaria eacelsa (figs. 41, 42).—The leaves retain the rhom-
boidal section of the “square” areas with considerable accuracy,
the orthogonal intersections are fairly well marked, and the
solitary vascular bundles of each member are formed very obvi-
ously at the “centre of construction” (most clearly observed in
fig. 42 (7-+11)).
Similar phenomena are even better marked in the seedling of
Pinus Pinea (fig. 88); the orthogonal intersections are in some
cases closely approximated toward the centre of the system, and
so clearly is the original construction retained that the disturbing
factors are readily isolated. These are seen to consist of: (1) the
diminution of tangential growth of members as they reach the
periphery of the plane of section, so that each member of a spiral
path subtends a sinaller angle than its successor, except in the
* A source of error is also introduced owing to the fact that the primordia,
however perfect in themselves, are inserted on a curved base, and do not,
therefore, extend in a vertical direction. A longitudinal section is therefore
necessary in order to see which primordia are sufficiently accurately placed
to be taken as typical. This error tends to be reduced as the apex increases
in diameter ; hence the value of Composite capitula (Helianthus) as a type.
320 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
case of the first pressure of the primordia to fill their quasi-
square areas. Thus, in the specimen figured, the first primordia
along a spiral path subtended angles of 47°5°, 52°, 53° (the theor-
etical angle for a quasi-square of a (5+8) system being 51°5°);
but nearer the periphery the series fall off to 47°, 40°, 40°, 38°,
35°, respectively. (2) Diminution of radial growth implying
progressive “dorsiventrality ” is relatively slight, and the sliding
effect small, the preceding data suggesting 2° only; and this is
due to the fact that the leaves tend to round off at the angles
instead of sliding over each other by sharp-edged laminae. (3)
Growth of the axis pulls the older members down out of the plane
of section; this growth being much more rapid in the seedling of
Pinus Pinea than in a lateral shoot of Araucaria, the compensating
effect of a longer needle in the former is not noticed.
Further modification of the primary phyllotaxis pattern ensues
with the advance of “ dorsiventrality” in the members; in cases
in which the leaf-member is markedly dorsiventral, but also
elongated, thus constituting a strap-shaped member, the optimum
effect will be observed in bud-section (ef. Podocarpus japonica, fig.
42). In the more general case, however, the “ dorsiventrality ” of
the leaf-lamina is not regular throughout the whole extent of the
member, but it may present the following cases :—
(1) Very great at the base (type of the sheathing leaf-base).
(2) Small in the petiole, or absent.
(3) Considerably exaggerated in the assimilating lamina.
The distinction between lamina and midrib may be small or
wanting (Sempervivum, fig. 83) ; or, again, very marked (Campanula,
fig. 101; Verbascwm, fig. 36); and in the limit, the former may be
complicated by compound segmentation. In such case, the seg-
ments of each leaf are restricted to its own rhomboid area, and
each leaf is packed or crumpled independently within its own
rhomboidal domain (Ranunculus repens, Nigella, Rheum). Similar
phenomena are observable on a smaller scale in the case of the
simple leaves of Verbaseum and Campanula (fig. 101); in the
figure of the latter instances of anomalous sliding effects have also
been indicated. When the sliding of dorsiventral leaves is ex-
cessive, it is clear that new contact-relations will be established in
VARYING GROWTH IN LATERAL MEMBERS. 321
this part of the bud-section, just as the rounding off of the
members from their original contact may also open up another
set of contact-parastichies.
Again, since the transverse section must pass through the
insertion of the youngest members, the development of the
sheathing basal portion as a dorsiventral lamina will often form
the most conspicuous feature. Nymphaea alba (fig. 94) may be
taken as a type: the construction system for the seedling (2+ 3) is
identical with that of Sempervivwm calcaratum (fig. 108), and in the
older members is closely imitated by the theoretical construction
(ef. fig. 110). The rhombs retain their original position, rounding
off somewhat in the petiole, and the dorsiventral lamina sprouts
on the younger members and slides among the older ones, cutting
off the original lateral contact-relations. Such a figure is, again,
practically identical with the bud-section of Ranunculus repens
(perennating axis), in which the older leaves become compound,
but each remains packed within its own area; also with that of
Lsoétes lacustris, in which the sheathing effect is somewhat greater
and the contact much closer; and finally, by complete fusion of the
sheathing portion of each leaf around the axis, the construction
becomes that of the apex of Rheum undulatum.
To see the general effect of the diminution in the rate of
tangential growth of lateral members, unmodified by other agencies,
a shoot will be required which either bears uniform members in
large numbers, or in virtue of a slow rate of elongation in the
axis will admit of a large number of members being cut in one
transverse section. An example of such a type is found in the
rosette of Sempervivum,; the leaves are uniformly dorsiventral,
with no distinction of midrib and lamina, and are retained in an
unmodified bud-type of growth. The seedling of Pinus should
also afford suitable material, as also species of Sedum. The
general result will be that, if the rate of growth diminishes tan-
gentially, the apparent phyllotaxis system will be raised; while if
the tangential rate be increased, the apparent construction system
will present a lower ratio system than the one actually laid down
at the apex.
Sempervivum tectorum (figs. 2, 3)—The succulent leaves expand
322 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
upwards to form a lamina in which “dorsiventrality . is not so
excessive as in a more typical foliage-leaf. The growing-point is
sunk to the level of the bases of all the leaves, and in spirit-
material may be contracted below the insertion of the great
majority. Sections of the Lud may therefore be compared from
different levels (fig. 83, I., IT.). A section across the whole bud
portion of a full-grown plant shows for the most part five clear
curves, pointing therefore to a phyllotaxis system (5+8). The
members may be thus readily numbered up along these curves
and the system checked to the centre by tracing a hypothetical
ontogenetic spiral with a divergence angle of 137°: the parastichies
are, however, seen to be imperfect at the centre, and the overlap-
ping of members in sets of three points to a phyllotaxis of (2+3)
(fig. 83, members 28-36). On the other hand, a section which
just grazes the top of the axis shows that the actual system in
which the members are laid down is (3+5). The apex is broad
and flat, the members arise apparently without close contact, and
their boundaries are difficult to recognise along the shallow groove
which separates them; it is, in fact, only by taking a section that
their shape can be defined. There is no doubt, however, as to the
construction of the curve-system being (3+5). The explanation
of these phenomena appears therefore quite simple; the lamina of
the younger members increases rapidly at first, and sliding-growth
is sufficient to bring three members into lateral contact; a section
taken through the upper part of the younger members in this
condition presents the appearance of a (2+3) system. On the
other hand, the bulk of the section passes through the outer
leaves lower in their course, at a point at which tangential
extension is at a minimum; the members thus apparently draw
away from each other laterally, and the “8” curves are thus
opened up, and the system assumes the form of a (5+8) type with
the “5” curves smooth-edged.
A similar effect is seen in species of Sedum, in which the
members tend to round off and form so-called “centric” leaf-
forms. Thus S. pruinatum (fig. 103), developing as a (5+8)
system, presents the secondary appearance towards the periphery
of the section of an (8+13), the “8” curves being now the smooth-
VARYING GROWTH IN LATERAL MEMBERS. 323
edged ones instead of the “5” series, and the angle subtended by
the members diminishes from 37° in No. 66 to 33° in the member
numbered 1.
A still clearer example of this effect of the diminution of the
angle subtended by older members is afforded by the previously
cited seedling of Pinus Pinea, on which the angles were carefully
measured. The fall ranged from 53°, the maximum angle sub-
tended by young sliding dorsiventral members, to 35° and even 30°
at the extreme periphery: from the measurement of theoretical
construction diagrams, the angle subtended by a member of a
(5+8) system is 51'5°, that by a member of an (8+13) system
32°. When the small amount of sliding-growth is regarded as in
this example compensated by a rounding off of the angles of the
leaves, the completeness of the transition is remarkable, and the
corresponding apparent alteration in the system is obvious, the
“8” long smooth-edged curves being the most prominent feature
of the section.*
A similar simple case of great interest is afforded by the
comparison of the appearances observed on a closed (wet) and
open (dry) cone of Pinus. Thus, in P. austriaca the scales on
the closed cone present facets averaging 12 mm. in diameter,
while the cone itself is about 30 mm. in diameter at the widest
part; the angle subtended by a scale varies between 45° and 50°,
and the apparent phyllotaxis system is therefore (5+8), as seen
in the contact-parastichies. When the cone is fully expanded
(fig. 5), the diameter of the structure is increased to 60 mm. or
* Note that the angles subtended by rhombs of the theoretical log. spiral
construction, as also any divergence angles measured from the centre of the
system, will continue to hold good for the plane projection of the transverse
section, whatever subsequent changes may take place in the rate of radial
expansion. While, that is to say, all allowance for the radial retardation of the
actual specimen is omitted from the theoretical quasi-square construction, all
angular measurements continue to hold for members in the same transverse
plane, and thus the calculated divergence angles of the different systems
(Mathematical Note V.) are the true divergence angles of plant phyllotaxis,
however much the radial rate of growth may be affected, since a reduction in
the tangential rate, by producing a dome-shaped apex, pulls the members
involved down out of the transverse plane.
324 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
more, the scale areas remaining the same; the angle subtended
by each scale thus falls to 25° or 30°, with the necessary result
that the parastichy system now observed is that of (8+13).
Similarly, in Pinus Pinea the closed cone 75 mm. in diameter
presents scale-facets 23-24 mm. across, subtending an angle of
36° to 38°. When correction is made for overlapping, it is clear
that the system observed will be (8+13); but when the cone
expands to a diameter of 110 mm., the angle subtended by a
scale falls to 20° or 25°, which approaches the 19°8° of the
(138+21) system.
The apparent contact-curves of the Pine-cone therefore present
in themselves no reliable evidence whatever of the actual system
with which the carpels were laid down at the apex, beyond the
fact that ratios of the Fibonacci series were utilised; the fact
that they are the ones really employed is only to be proved by
an examination of the curves of the developing shoot-apex ; and
similarly, it is impossible to say from the mere examination of
the free tips of any cluster of leaf-members what the true
phyllotaxis system might have been (ef. Sempervivum spinulosum,
fig. 4, and cones of Araucaria), No satisfactory generalisations
concerning phyllotaxis can be made for any given asymmetrically
constructed plant until the curve-system, or the contact-relations
of the members at their insertion on the actual growing-point,
has been determined. The system is not necessarily obvious on
the adult shoot, and the appearances seen in the case of adult
structures need bear no direct relation to the true construction
system ; the number of parastichies only helps in that it affords
a guide to the ratio-series concerned (cf fruiting heads of
Helianthus, Scabiosa, and Dipsacus, in which the assumption of a
uniform type of fruit may cause the construction as judged in
terms of contact-parastichies to vary one stage in the ratio-series).
The primary curve-system of the growing-point thus tends to
be more or less destroyed by the action of the following factors,
all of which may be present to a greater or less extent in the
production of a typical foliage leaf-bud :—
(1) Diminution in the tangential rate of growth, resulting in
the lowering of the angle subtended by the member.
VARYING GROWTH IN LATERAL MEMBERS. 325
(2) Diminution in the radial direction, giving rise to phenomena
of “dorsiventral” bilaterality, including sliding-growth
effects.
(3) Secondary elongation of the main axis in passing through
the “Second Zone of Growth.”
(4) Intercalary growth of petiole formation.
(5) Cessation of growth-activity, leading to the production of
members of a constant bulk in the adult condition.
(6) All irregular and local secondary pressure relations in
members approaching maturity, producing distortion of
spirals in asymmetrical systems, or the true orthostichies
in symmetrical constructions (¢/. fig. 79).
(7) Special differentiation of individual members or portions
of them.
So greatly is the discussion of theoretical phyllotaxis limited
in its general application to descriptive purposes, that these
generalisations may be almost taken as suggesting that the
original use of the term phyllotaxis as applied to all arrangements
and effects, as seen by the eye on young or adult shoots, may
after all be retained with great advantage for such phenomena;
and the word be still used to express the general relations of
members as presented to the eye, and judged either by the loose
and approximate method of Schimper and Braun, or preferably
by the observation of the more obvious parastichies at any given
point; while the true primary system, which is the first visible
sign of the hidden forces which initiate new growth-centres in
the actual substance of the protoplasm, and which can only be
satisfactorily determined by examination of a transverse section
at the level of the growing apex, may possibly be preferably
restricted to such a term as mechanotaais, in that it indicates
in the most concise form the actual growth-mechanism, while the
parastichies of such a section may further represent paths of
equal distribution of growth-energy, existing in a system in
which vital energy follows general laws of orthogonal distribution,
comparable with those which obtain in the case of manifestations
of forms of physical energy, the discussion of which is: brought
within the range of mathematical conceptions in the mathematical
3826 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
treatment of mechanics. The otherwise expressive and suggestive
term growth-vortec may be placed on one side, in that the ex-
pression “ vortex ” has only been used as a geometrical metaphor ;
though, on the other hand, it is this vortical condition of growth
which is the essential character of a shoot-system, and the term
would serve to emphasise the standpoint that a leaf is thus to
be regarded as a specialised growth-phase of the parent axis,
with which, so long as its primary growth is maintained, it
remains ‘co-ordinated, as also with its adjacent members, and
within certain limitations under the control of the parent
growth-centre.
Mathematical Notes on Log. Spiral Systems and
their Application to Phyllotaxis Phenomena.
By
E. H. HAYES, M.A.,
Fellow of New College, Oxford ;
AND
A. H. CHURCH.
ANY application of mathematical methods to such a subject as
that of Phyllotaxis must necessarily be limited by the hypotheses
taken as the basis of any conception of the relationship of the
phenomena observed, and clearly no further information can be
deduced than follows from the original premises.
Thus, as already seen (p. 6), the mathematical conception of
a helix winding on a cylinder, which was assumed by Bonnet to
be a satisfactory interpretation of the facts observed on adult
shoots,—although it did not hold for younger ones,—forms the
basis of the Schimper-Braun formulae; and the assumption of a
spiral with parallel screw-thread led to the adoption of spirals of
Archimedes when the phenomena were required to be represented
as a plane circular system.
Similarly, the introduction of the Fibonacci series of ratios by
Schimper and Braun naturally brought with it all the mathematical
properties of this curious series of numbers, and other observed
ratios were readily fitted into similar summation series. But in
deducing all the mathematical properties of such combinations,
which necessarily follow from the presence of the numbers
themselves, it does not follow that the plant, in possibly selecting
327 Y
328 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
such ratios for one reason only, is in any way aiming at all
the mathematical consequences of its choice, although they must
all necessarily follow; and it thus becomes increasingly difficult
to draw the line between the tabulation and interpretation of
actual observations and the pursuit of abstract mathematical
functions, which rapidly degenerates, as Sachs pointed out, into
mere “ playing with figures.”
So, again, the introduction of eguiangular spirals as indicating
curves of growth necessarily brings with it all the mathematical
properties of these curves. One admittedly makes no further
advance toward the interpretation of the causes of phyllotaxis by
the mere introduction of equiangular, ontogenetic, and parastichy
spirals,
But if the primary mathematical conception is based on a
legitimate foundation, such as that of uniform growth-expansion
appears to be, the properties of log. spiral systems become
increasingly important as indicating symmetrical or asymmetrical
cases of perfect growth, although such spirals may never be
measured or even really exist in actual phyllotaxis phenomena,
since the modification of the primary construction spirals may be
made the subject of subsidiary hypotheses.
While, therefore, the purely mathematical investigation of log.
spiral constructions can add nothing to the explanation of the
phenomena, it becomes of interest to tabulate the properties
of intersecting systems of these curves, in that functions may
be deduced mathematically which are not readily apparent in
geometrical constructions, just as geometrical constructions, on
the other hand, may confirm or make more obvious a mathematical
generalisation.
It. remains, therefore, to consider what the properties and
appearance of such abstract ideal phyllotaxis systems would be,
‘the relation of their parastichies and orthostichies, as also the
form of the curves which would represent the homologues of
circles inscribed in the orthogonal meshes, and the angles sub-
tended by these in the different systems: the whole set of
phenomena thus affording a view of an ideal uniformly expanding
system of lateral appendages on a growing axis, which may then
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 329
be used as a Standard of Reference for the comparison of the
phenomena actually observed on any given shoot.
It cannot, perhaps, be too strongly insisted that the log. spiral
theory is of value solely in so far that it affords such a standard of
reference. The same mathematical conception which assumes
the possibility of an abstract uniform protoplasmic growth, also
takes cognisance of the fact that protoplasmic growth is never
uniform, although the approximation may be very close in certain
special cases ; and just how close this approximation may be, the
mathematical investigation of the log. spiral systems should help
to disclose.
While, again, the mathematical study of these curves may be
fairly regarded as outside the province of the botanist, it is clear
that the empirical results obtained in previous pages (Part II.)
by means of geometrical constructions, more especially in dealing
with the convention of budk-ratio, will have little value unless they
can be checked by mathematical methods.
Nore I.—General Equation to the Ovoid Curve in a Log. Spiral
Quasi-Square Mesh—the Quasi-Circle.
Taking the asymmetrical construction as more primitive and
mathematically a more general case of construction, of which
whorled symmetry is only a special case, the system can be dis-
cussed mathematically in the following terms :—
In an m:n network of logarithmic spirals, the equation of one
set of spirals may be written,
n log r=n log c+mO+(2k—1)z,
and of the other,
m log r=m log e—nO+(21—1)7;
where &, J, are any integers, positive or negative.
Values of & differing by m refer to consecutive turns of the same
spiral, and similarly values of / differing by n.
The “centres of construction” of the network are given by the
intermediate spirals,
n log r=n log c+ m0 + 2k’r,
and m log r=m log e—nO+ QU ar.
330 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
To find the curve inscribed in a mesh of the network analogous
to a circle inscribed in a square, use the orthogonal transformation
log r=, O=y: also put log c=a.
The equations of the two sets become,
ne — my = na+(2k—1)7,
many =ma+(2l—1)z,
so that in the transformed system the meshes are equal squares of
Sm+n®
Consider the mesh whose sides are given by k=0, k=1;
i=0, /=1.
Its “centre of construction” is at the intersection of the spirals
given by k’=0, l’=0, we. is at the point r=c, 0=0.
The sides of the corresponding square in the transformed system
are,
ne—my = nok,
ML+ NY =Ma-kT,
and its centre is at the point z=a, y=0.
The equation of the circle touching the sides is
Pd
mE
Transforming back, the equation of the required curve is found
to be :—
(@—-aPty=
p\ 2 2
log“) +@=—7 . .. (L)
Cc m+n
The logarithm is the natural logarithm, and @ is measured in
circular measure: when the logarithm is the tabular log and @ is
measured in degrees the equation may be written :—
‘ 1
log r=log c+1 36438, / 7 00003086462, (IT.)
The point corresponding to the centre of the circle is =c, 0 =0,
ae. is the “ centre of construction.”
Since all the meshes are similar, differing only in size, the above
equation will apply to any mesh, if ¢ is the distance of the centre
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 331
of construction from the origin, and @ is measured from the line
joining these points. The curve touches the sides of the mesh
where they are met by the two intermediate spirals which deter-
mine the “centre of construction.”
From the character of the above equation it follows that the
curve is symmetrical with respect to the line joining the origin to the
centre of construction.
This mathematical deduction is of the greatest botanical interest,
in that it brings out the remarkable fact that every lateral
primordium is primarily dilaterally symmetrical with regard to a
radius drawn through the centre of construction and the centre
of the main axis ; and thus, whether in a whorled or spiral phyllo-
taxis system, us primary structural peculiarities will be identacal.
Thus, no change whatever is involved in the properties or shape
of the lateral members themselves, when the phyllotaxis system
passes from an asymmetrical construction to a symmetrical one;
that is to say, change of symmetry in the radial axis system does
not directly affect the symmetry of the appendage, and whatever
the curve-ratio of the construction, the leaf-members would be
equally isophyllous, although eccentric growth of the whole shoot,
by affecting the shape of the ovoid curve itself, involves antsophylly.
On the other hand, the contact-pressures of adjacent growing
primordia, which cause them to approach the shape of the quasi-
square meshes, results in making the primordia secondarily
asymmetrical to a certain extent when the curve-system is
asymmetrical.
The general result of this mathematical investigation is to
establish the fact that certain essential properties are common to
all leaf-primordia expressed as quasi-circles ; and these may now
be expressed in botanical phraseology.
I. All such appendages are bilaterally symmetrical about a
median line, the radius drawn through their own centre of
construction and the growth-centre of the axis itself. The
appearance of radial flattening to which they are subject in the
main growing system thus exaggerates this symmetry in two
orthogonal directions—one a radius of the system, the other a
circular path of the same central system.
332 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
II. The appendages further possess that peculiar attribute
called by Sachs their dorsiventrality—a term which, as previously
shown, requires to be taken in a purely metaphorical sense, and
which only holds its own in that it is as useful as any other
expression for indicating the fact that the members possess two
unlike surfaces, and that these are upper and lower surfaces, a
point not implied by the term bifacial. The form of the quasi-
circle shows that the peripheral portion of the appendage, as
seen in section, is larger than the interior part; so that, in
carrying these members over the slope of a dome-shaped apex,
the exaggerated side becomes the lower surface of the leaf: this
again being the mathematical consequence of the fact that the
growth-centre is transferred to a point nearer the inner side of
the curve.
III. The term isophylly indicates still more concisely that
property of the members in which the bilaterality of the
appendage is expressed in the form of two equal sides about an
axis of the member in the tangential plane of the system, and is
usually applied to the shape of the lamina surface rather than to
its section: this again is equally a mathematical property of a
growing primordium possessing such a curve-section.
The mathematical investigation thus shows that all primordia,
whatever value be given to m and n, present these properties as
fundamental and unavoidable features of construction. Every
appendage is mathematically bilateral, dorsiventral, and isophyllous,
wih regard to the shape of the curve and the position of its centre
of construction. These are mathematical necessities of the type of
growth-system adopted by the shoot as a centric growth-centre
producing a rhythmic series of subsidiary centric growth-centres.
A growing system might evidently have one such centre or more
than one. One is the simplest case, and as a matter of observa-
tion is the general rule; on the other hand, the case of multiple
growth-centres is included under the botanical title of fasciation
phenumena. Here at last is a definite foundation on which to
build the morphology of the shoot; and it now becomes possible
to draw a distinction between the necessary properties and the
accidentia, or phenomena of subsequent adaptation. Why the
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 333
shoots of higher plants should have evolved such a growth-
construction as a means of increasing the body surface remains
unsolved; but so long as a shoot possesses a persistent embryonic
growing apex, some such construction would appear necessary :
in fact, if the growth-centre is to remain a point, dichotomy would
be the only simple alternative. Why, again, certain curves should
be selected, their number, ratio, inequality or approximate or
actual equality, still constitute further problems apparently
hopeless of any immediate solution, although teleological sug-
gestions may be put forward to explain the frequency of simpler
forms of symmetry, eg. (2+2) and (1+1). But granted the
initiation of such construction systems by the plant itself as the
elaborated response to some general co-operation of external
agencies, these fundamental characters are mathematical conse-
quences, however much or little the subsidiary action of special
influences may tend to subsequently mould the growth-forms
thus initiated by the shoot-apex. The influence of external
environment, of which so much is expected in these days by
enthusiastic materialists, must have something to act upon: the
use of the favourite expression adaptation implies the pre-existence
of a certain something which can be modified; and just what this
something is, and how far it goes, is thus defined in mathematical
terms by such a generalisation as that of the growing system of
growth-centres.
Leaf-appendages in centric growth-systems are therefore
bilateral, dorsiventral, and isophyllous, not from direct relation to
the action of any such agencies as gravity or vertical light, but
from the mechanical laws controlling the distribution of the
material substance of which they are composed. Further, a
plant shoot builds such primary appendages of one kind only;
whether they are all to be classed under the general term /eaf-
members, or whether this term is to be restricted to the specialised
assimilating organs, is a matter of little consequence. The abstract
Urblatt of Goéthe is now exchanged for an actual concrete and
mathematically defined appendage, the quasi-circle primordiwm,
and the futility of any discussion as to the priority of foliage-leaf
or sporophyll becomes obvious.
334 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Nore Il.—WMathematical Orthostichies in Log. Spiral Systems.
To obtain a collinear intersection of m and n spirals, 14 a
second point of intersection in the same straight line with the
origin, it is clear that in the Fibonacci series, for example, it is
only when m:n:: 5—1: 2 that such an intersection will take
place at infinity.
But with m and n finite integers, radially superposed inter-
sections will take place; and taking the case of m spirals crossing
n, the nearest point collinear with the origin and any given point
on the same side of it will be the (m?-+-n?)th term: that is to say,
in the general case in which m and ” have no common factor.
Thus, in the system (3+5) a true orthostichy will exist between
any member taken as 0 and the (9+25)=34th. In the case of
Sempervivum tectorum, for example (figs. 1, 2, 83), the contact-
parastichies at the apex being (3+5) (fig. 83), the line drawn
through No. 1 and No. 35 (fig. 2) should be mathematically a
radius, and a true orthostichy line to produce which the onto-
genetic spiral would wind 13 times (ie, i):
It will be noticed that such points are beyond the range of the
construction diagrams, which only include a portion of one
revolution of the pair of generating log. spirals; and also beyond
observation on the plant, owing to the fact that minute differences
in the growth of older leaves would suffice to produce slight
displacements which would destroy the effect of these mathemati-
cally straight lines. For practical purposes these true orthostichies
pass beyond the limit of consideration, but the fact that such are
possible is still of botanical interest; while the curious relation of
such an empirical formula of the Schimper-Braun series to the
actual construction in the case of Sempervivum is noteworthy.
The phyllotaxis may here be thus accurately written in the
Schimper-Braun terms os but such a formula can only be deduced
from the consideration of the properties of a (3+5) system, and
not from any inspection of the external characters of the leaf-
arrangement on the plant itself, in which superposition effects
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 335
are increased by the gradual transition to apparent spirals of
Archimedes as the members attain a uniform bulk; nor could any
\
approximation to the eye of a = orthostichy line be ever taken
as an indication of a (3+5) apical construction.
Similarly, other orthostichies may be tabulated :—
(m?+n7) for (1+ 2)system= 5
(2+ 3) , = 18
(34+ 5), = 34
(5+ 8) , = 89
( 8413) , = 233
(844+55) , =4181 etc.
The first case (1+2) is of interest, in that it should be the
phyllotaxis of Pandanus and Cyperus, which it obviously is not
(of. figs. 51,595). This has been suggested as due, as in the case of
Sempervivum, to the change in the spirals consequent on the rapid
attainment of approximately equal radial depth (fig. 51). The
last case being that of the capitulum of Helianthus taken as a
type (fig. 15), in which the system was only carried for between
6-700 members before it broke down; so that even if this type
of formula were retained in the descriptive account of phyllotaxis,
it becomes quite useless in all high series.
Note II].—The Form of the “ Ovoid” Curve.
From the equation of Note I., the curve for any given system
may be plotted out. Five such curves, those for the
asymmetrical (3+5),
asymmetrical (2+ 3),
symmetrical (2+ 2),
asymmetrical (1+ 2),
symmetrical (1+ 1),
are represented in fig. 111.
It will be noticed that the form of the (3+5) quasi-circle
scarcely differs to the eye from a circle, and this approximation
is shown by the dotted line. The curve is, however, slightly
336 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
flattened in the plane passing through the origin, and is thus
broader than long; that is to say, the primordium of such a
system is already by construction slightly “ dorsiventrally ”
Fig. 111.--Set of five quasi-circles of the systems (3+5), (2+8), (2+2), (1+2),
(1+1), arranged for convenience of illustration in diminishing series, 1, 2, 3,
4, 5 respectively, along the plane of median bilaterality XY.
Cy, Ca, Cs, Cy, Cs, the centres of construction, and O;, 0, O;, the origins of
respective curves,
A circle AB, with centre C, has been drawn in contact with the (3+5) curve for
purposes of comparison.
bilateral in the median plane. Thus the extreme breadth of the
curve as plotted was 11-796 inches, the extreme length 11:304
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 337
inches ; the curve thus approaches a circle in which the amount
of error is not greater than jth of the radius.
In all higher systems the approximation to the circle will be
successively closer, so that, as previously pointed out, the use of
circles in the quasi-squares of any system above (3+5) is beyond
any error of constructing a small diagram. The relation of the
curve to its “centre of construction” homologous with the centre
of the circle has also been previously indicated, since it is readily
noted on the geometrical diagram by taking the point of inter-
section of the intermediate spirals.
The (2+3) curve, again, shows a marked distortion, and the
flattening on the side towards the origin is excessive, the general
outline obviously differing from a circle. This is still further ex-
aggerated in the (2+ 2) curve, in which a distinct dimple begins to
appear at this point (fig. 111, IIT.), and the “ centre of construction ”
shows still greater displacement. The (1+ 2) curve becomes dis-
tinctly kidney-shaped, with the centre of construction very close to
the depression (fig. 111, IV., C'”); and the limiting case is met with in
the (1+ 1) curve (fig. 112, A, the centre of construction being at C’).
It thus follows that lateral primordia may be represented in
theoretical construction diagrams as circles, within any error of
drawing, in any system from (3+5) upwards. In lower systems
the bilaterality of the ovoid is very marked, so much so, in fact,
that the occurrence of such a form at the apex of a plant-shoot
would not readily strike the observer as in any sense due to
the production of a primordium, the section of which would be
homologous with a circle, and within its sphere of growth
possessed of the same physical properties.*
-* The same curves, or similar figures for any given ratio system whatever,
may be easily drawn within the error of drawing by a simple geometrical
method. For example, to draw the curve for the ratio 3 : 4, make a curve-
tracing for this ratio from the circular network of squares (p. 53), and with the
curve-tracing mark out a quasi-square mesh of the system. Divide this into
12 equal parts in either direction by describing 11 intermediate spirals in each
direction, and into these smaller squares transfer square by square the circle
inscribed in a true square similarly divided into 144 meshes, the points where
the circle cuts the meshes being judged by the eye with sufficient accuracy.
Since the curves of the higher ratios so nearly approach a circle, the lower
ones figured are really the only ones which possess a special interest.
338 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Note [V.—Bulk-Ratio.
The bulk-ratio, defined as the ratio between the sectional
diameter of the lateral primordium and that of the axis at the
point at which the member was apparently inserted, was taken
as represented with approximate accuracy by the ratio of the
radius of a circle empirically described in a quasi-square mesh to
that of the circle drawn through the centre of the former
one (p. 88), and from such empirical constructions the angle
subtended by a primordium of any given system was approxi-
mately measured, well within any degree of observation error
on the plant.
Since there can evidently be no ready comparison of bulk-
ratio between the circular section of an axis and the ovoid
section of a lateral member, this method will prove sufficiently
accurate, But it is possible to approach the subject from a
different standpoint. Thus, the angle subtended at the origin
by each ovoid primordium may be calculated mathematically,
and this angle may be compared with the empirical geometrical
construction.
Data tor true curve. Data tor empirical circle.
Systems of m Bulk-ratio
and m spirals, |Angle subtended Sine of half approximated |Correspond- Correspond-
—_ 36 . angle from geo- | ing sine of | ing angle ;
NV m2+n2 metrical con- | half angle. | of. p. 89.
struction.
(1+2) 161° ar
(2+3) 99° 52’ see 28 aia 283
(3+5) 6L° 44’ 513 lL: 2 500 60°
(5 +8) 38° 10’ 327 1: 3 333 38° 57’
(8 +13) 23° 35’ "204 1: 6 ‘200 23° 6’
(13 +21) 14° 35’ "1269 1: 8 "125 14° 20’
(21+34) 9° 0785 1:13 ‘0769 8° 48”
rab 5° 34’ “0486 1:21 0476 5° 26"
55 + 89) 3° 26’ “0300 1:34 0294 3° 24’
(89+ 144) 2° 7 01855 1:55 01818 2° 8’
(144 +233) 1° 19’ 01147 1:89 01124 1°17’
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 339
The angle subtended by a primordium belonging to any system
of orthogonally intersecting log. spirals is given by the following
formula :—
Qar 360°
im? +n? oe m+n?
The bulk-ratio for circular primordia was represented by the
sine of half the angle subtended at the origin; the same expression
may be regarded as representing the approximate bulk-ratio in
the case of the ovoid curves which so nearly approach circles.
From the above table it will be seen that the difference between
the angles subtended by the ovoid curves and those subtended by
circles having the respectively simple bulk-ratios obtained approxi-
mately by geometrical construction is a very small one. Such an
error is quite within any limit of construction error in small dia-
grams, and is far within the error of checking systems in the case of
the plant, in which, as previously noted, the construction adjustments
that must be made in the bulk-ratio before a new spiral path is
introduced must be necessarily often very considerable; so that
for practical purposes the integral bulk-ratios of the respective
systems may be taken as sufficiently accurate statements of the
phenomena observed. The empirical geometrical method of
estimating the bulk-ratio of any given system is therefore suffi-
ciently reliable, if the convention can be of any assistance, and
does not involve any special mathematical knowledge.
Note V.—The Oscillation Angle.
Taking the construction of a constant asymmetrical spiral
phyllotaxis system as the result of adding members at a constant
divergence angle, or as a phenomenon of growth oscillation,—a
convention which only holds, however, as has been previously
made clear, for integral ratios only divisible by unity as a common
factor,—the measurement of the true angular divergence of the
members of the systems constituting the Fibonacci series becomes
of special interest from the standpoint of comparison with the
helical divergences of the Schimper-Braun-Bravais convention.
3840 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
The angle is given by the formula :—
2mn—M 6 69°
m+n?
Thus for (1+ 2) the divergence angle = of 360° = 216° or 144°
(2+ 3) ” ” fs ” = 138° 27’ 42”
(3+ 5), » 31, = 187° 38’ 50"
(5+ 8), 5 83, =187° 31’ 41”
(8+13) 3 5 344, =137° 30° 38”
while the limiting angle
i= 1 = 137° 30’ 98”
The “ideal angle” of the Fibonacci series remains the same as
in the Schimper-Braun series; but the angles for lower members
of the series are not only very different from the conventional
series, but they are definitely very much more like the angles
obtained in measurements of plant specimens.
The standpoint of the Bravais, that there might be quite reason-
ably only one angular divergence for normal Fibonacci phyllotaxis,
and that one the “ideal angle” of Schimper, is thus seen to
be well within the experimental facts. Since, it may be again
pointed out, these angles hold for growing systems, they do not
hold for systems which show progressive cessation of growth; but
so long as growth proceeds uniformly in the system, that is to say,
the nearer the apex of a plant approximates uniform growth, so
will these angles be the true angles of phyllotaxis, and will be
found well within any error of observation on the plant. The
(1+2) system alone differs from the ideal angle by about 7°, and
as already noted, a (1+2) system which can be regarded as
approximately exhibiting uniform growth is not readily obtained,
owing to the effect of growth-retardation and secondary cessation.
It will be noticed that already at the (2+3) system the “ideal
angle” may be attained within an error of about one degree
(of. fig. 94) in a primordium which subtends 100°; while an error
of one per cent. is practically beyond consideration in dealing with
the plant. A slightly higher ratio (5+8) gives the “ideal angle”
within a theoretical error of one minute, While the possible
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 341
physical accuracy of construction which is represented by an
(8+13) system, probably the highest ratio ever directly initiated
at the apex of a shoot, suggests that in the extremely minute
growth-centre in the first zone of growth, beyond any visible
primordia, the mechanism at the hypothetic growth-centre might
become a question of even molecular aggregation, and thus may
be again fairly comparable to phenomena of crystallisation.
For practical purposes the angle 1374° may thus be assumed
approximately constant for all Fibonacci systems beyond (2+3).
For this system the value 138°5° obtained from the geometrical
construction is sufficiently accurate to suggest that similar con-
structions will be equally satisfactory in the case of anomalous
ratios. For example, the system (7+11) of Araucaria excelsa as
represented on a geometrical diagram gave 99°6° for the oscillation
angle, while the calculated divergence was 99°53°.
So long, therefore, as a log. spiral construction is postulated, the
botanist may investigate the subject without any need of special
mathematical knowledge; the simple geometric diagrams taken in
the preceding pages being far within any error of observation on
the plant, and having the additional advantage of presenting a
difficult subject in a simple and concrete form.
Note VI.—The Fibonacci Series.
The most remarkable feature in connection with plant phyllo-
taxis, whatever view be taken of its origin or final cause, is after
all the predominance of the numbers of the Fibonacci series. That
the series is not by any means indispensable is shown by the wide
range of variation into anomalous systems, and the complete
elimination of the series in the case of symmetrical constructions.
The following two points may be here brought forward to throw
light, if possible, on this peculiarity of plant construction :—
I. The numbers of the construction curves must be integers and
low numbers, or else the lateral appendages will be relatively very
small; and asa matter of fact in all seedlings the lateral append-
ages are relatively large as compared with the main axis. These
are facts derived from observation of the plant, and from the
conception of bulk-ratio,
3842 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Next, as a matter of observation also, the ratio of the con-
struction curves must not show any great inequality; on the
contrary, a very general approximation of equality in the numbers
of the curves appears to be the general rule. The rule appears to
be that one number must not be more than double the other: this
being again the generalisation of Schimper and Braun, which places
the ratio 1:2 as the limit. The highest range of this type of
ratio has been recorded as (3 : 6) for a trijugate plant of Dipsacus.*
Hence the choice of higher plants is really restricted in the great
majority of cases to such combinations as—
1
es 2, ;
2:2,2:3,2:4
3, 3:4, 3:5, 3:6, ete,
these being the only low combinations possible. Taking these
nine ratios, it will be observed that three are cases of true
symmetry, three are Fibonacci pairs, while the (1:1) may also
be regarded as in the Fibonacci series; the (3: 4) is the
commonest anomalous ratio, and the (2 : 4) the common “ bijugate ”
one. Taking only these simple expressions, then, the balance
of construction is in favour of the Fibonacci series, which when
once laid down lead on naturally to higher expansion derivatives
of the system, which follow with mathematical precision as con-
sequences of the properties of systems of intersecting spiral curves.
A predominance of Fibonacci ratios, so far as asymmetrical phyllo-
taxis is alone concerned, would thus be expected to obtain; and
this quite apart from any possible biological utility of the series or
of a spiral distribution or building mechanism, prejudices in favour
* There is a suggestion that other ratios occurred in lower types: a wider
range of ratio, e.g. 1 : 3, occurs in Mosses, as also in the apical cell of the Fern ;
ratios of 1: 4 also in Florideae. These and a few isolated cases (cf.
Cheirostrobus, Scott) require to be taken separately: the general standpoint
obviously being that all mathematical possibilities should be equally expected
to occur, and the fact that certain types obtain in present vegetation rather
than others may indicate the gradual effect of natural selection on the con-
struction mechanism, the general trend appearing to be, as already indicated,
towards either symmetry or ratios of the Fibonacci series,
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 343
of which standpoints have been so frequently built up by a use of
the post hoc ergo propter hoc line of argument.
IL It will be seen to follow from the remarkable property of
the Fibonacci ratios—that the ratios of any successive pair are
almost constant, and that 3:5:8:13: 21, etc., with a considerable
degree of accuracy, so far as integers are alone concerned, as again
is the case in the curve-systems of phyllotaxis—that in the case of
all expansion systems derived from an initial pair with a view to
lessen the relative size of the lateral appendage, these numbers
alone give a minimum loss of regularity at every step in the change ;
while with any other series, such as 3:4:7:11, etc., the transition
would involve a large step and a small step alternately. In other
words, any aim on the part of the plant at uniformity of con-
struction in a system which is liable to change by the addition or
loss of paths, as in cases of very active growth (cf. Helianthus) in
which new curves are continually being added to reduce the
relative size of the lateral member as the growing-point gains in
bulk, can only be satisfied in one of two ways. Either the plant
acquires true symmetry and maintains it by adding curves in
either direction simultaneously (cf Hguwisetum), or that asym-
metrical system must be adopted in which the expansion
transitions can be effected with the least loss of regular construction.
The system which fulfils these demands is the Fibonacci series ;
and from merely numerical reasons there appears to be a balance
in favour of the chance of the initiation of curves in these ratios
to begin with. So that, granted the asymmetrical condition of
phyllotaxis is the primitive one, the general occurrence of curve-
ratios in the Fibonacci series would be mathematically expected
to occur. The choice of the plant for optimum phyllotaxis relations,
in fact, lies between true symmetry and the Fibonacct type of
asymmetry; hence when true symmetry obtains the special
numbers of the latter sequence are no longer to be noticed as more
usual than others, and all other systems become rightly classed as
anomalous, in that they deviate from the two optimum conditions.
One thus becomes mathematically justified in regarding anomalous
variations, including the peculiar bijugate constructions, as ex-
pressive of a state of degeneration in the mechanism of shoot
Zz
344 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
construction; and when this condition becomes the rule, it is
possibly the result of unfavourable conditions of environment, and
may thus be correlated, as in the case of Cacti, etc., with other
xerophytic peculiarities.
Nore VIIl.—Continued Fractions,
Since the time of Schimper and Braun much importance has
been attached to the formulation and presentation of the ratios
commonly found in plants in the form of a summation-series
presenting certain mathematical properties; the ratios being
successive values of the stages of a continued fraction, the limiting
value of which became expressed as an “ideal angle.” Hence
mathematical statements became read into the subject with which
Botany has nothing whatever to do. The formation of these
summation-series from observation of the plant kingdom, which
represents the great botanical discovery of Schimper on which all
his contributions to the theory of phyllotaxis were based, is a
mathematical consequence of the phenomena of intersecting spiral
curves radiating round a central point. The preceding geometrical
diagrams have rendered this sufficiently obvious (cf. figs. 25, 26).
Thus, if a certain number of curves cross another set, the same
points of intersection will be mapped out by two other sets re-
presenting the sum and difference of the first set (cf. p. 56); oy,
if m curves cross n, (m+n) and (m—n) curves will also pass
through the same points and form diagonals of the original meshes :
four terms of a summation-series are thus involved, and other
terms may be obtained in the manner already described. Given
the intersecting curves, the mathematical manipulation and
description of continued fractions becomes a feature with which
Botany has nothing to do, nor is it at all helpful in any direction.
Such expressions may attract the mathematician, but they repel
the botanist, and it is hoped that the method of constructing
_ geometrical diagrams of the types indicated, on which the rela-
tions of the numbers can be more directly traced, will tend to
eliminate these expressions from botanical literature, together
with the curious prosenthesis formulae of older writers and
the Dachstuhl angles of a later school.
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 345
Nore VIII.—Shding-Growth.
From Note I. it is clear that each primordium is primarily
bilaterally symmetrical about two axes, represented respectively
by aradius of the system and a circle passing through its centre
of construction; and so long as the primordia are free from
adjacent members each will retain this form, except in so far as it
becomes affected by secondary alterations in the rate of growth in
these respective directions ; and this holds for the case of symmetry
and also for the more general case of asymmetrical construction ;
that is to say, in an asymmetrical construction in which no
contacts are made, no sliding-growth effect takes place, and the
leaf-members would be horizontally extended and isophyllous
on the adult cylindrical shoot. But as soon as the members
make lateral contact, the mathematical conditions undergo a
change, as previously noticed, the members become represented
by oblique rhombs, obliquely placed and anisophyllous; the
packed leaf of an asymmetrical system thus becomes secondarily
an asymmetrical structure, while in a symmetrical system,
on the other hand, it still retains its original symmetry in
relation to the radial and circular paths of the system: in other
words, the free portion of a leaf in a spiral system is free to obey
its structural properties as a quasi-circle, but so long as it is packed
and makes close contact with adjacent members its growth-form
becomes distorted into the form of a quasi-square rhomb. In a
spirally constructed bud, therefore, with leaves growing more or
less in contact below, but free from one another above, as the
expression of a conoid growth-form which was initiated from a
point and extended until it reached the product of an adjacent
centre, the change from the lower distorted region to the upper
symmetrical part, when this takes place throughout the whole
system, will convey the impression that a slight twist has taken
place in the members, as the tangential diagonal of the oblique
rhomb changes to a true circular path. It is this appearance of a
tendency to a readjustment on the part of the free portion of
the appendage which gives the primary tendency to slip in the
bud, and the phenomenon provisionally included under the term
7%
346 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
“ sliding-growth” is essentially nothing more than the appearance
of these circular paths in the spiral asymmetrical diagram. That
this appearance of sliding-growth effect tends to change the
“spiral of dorsiventrality” into a circular path has already
been shown, but this is not a “biological adaptation”; it is not
the direct result of any external conditions of environment ; it is
simply the expression in the free primordium of those fundamental
properties of a quasi-circle which become masked so long as the
primordium tends under pressure to take the properties of a
quasi-square.
The primary sliding effect is thus defined as the result of the
free portion of all appendages produced in contact-systems
attempting as they become free to return to their original
position of symmetry along radial and circular paths. The occur-
rence of oblique spiral symmetry in a contact-system is a
phenomenon of distortion, and all such effects are increased by any
secondary growth-relations of the appendage, either as it becomes
larger above its insertion, or as it tends to tangential extension, as
in the differentiation of the leaf-lamina from the midrib.
It is of interest, therefore, to compare these deductions from the
mathematical equation with the facts observed in the plant when
plotted into a large drawing under the camera Incida; it must,
however, be remembered that in the section at the level of the
growing-point transitional stages will be found, but if the members
make contact from the first, the amount of “sliding-growth ” is
fairly constant (cf. figs. 101, 106), and may be put into the’
theoretical diagram with a log. spiral, as in fig. 100; the amount
of sliding-growth which may be regarded as normal for a given
asymmetrical construction being the amount which will make
the tangential diagonal of any appendage a circular path.
MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 347
Fig. 112.—Distichous (1+1) phyllotaxis system. Geometrical construction for
uniform growth, with derivative curves showing radial retardation; A, B, C,
free members ; D, E, ‘‘ packed” members.
3848 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
General Conclusions.
I. THE log. spiral theory, as already indicated,* is put forward
solely as a mathematical conception, admittedly gratuitously
introduced into plant-morphology, as a fundamental hypothesis
founded on a mathematical theory of centric growth (Part L.,
p. 16), and is intended to replace the helical theory of Bonnet, «''' ''
which, being deduced from an ideal adult construction, was only
required to imitate appearances of leaf-distribution on a full-
grown shoot. In the preceding pages this new view of the
growth of a plant-shoot and its appendages from one “growth-
centre” has been elaborated from a simple standpoint of uniform
growth which does not necessarily ever obtain in any living
body.
II. The mathematical data corresponding to such a standpoint
have been deduced and tabulated, in accordance with the simple
numerical data afforded by plant phyllotaxis systems; and thus it
has been shown that the divergence angles of such uniformly
growing asymmetrical systems can be deduced mathematically,
while by simple geometrical constructions very reliable results
may be obtained by one quite ignorant of mathematics. Also,
granting the reasonable hypothesis that the lateral members of a
plant are formed in one growth-system controlled by the shoot-
apex, the curves of the transverse components of such members
for different systems have been deduced and figured.
III. Given these data, it now remains to take them to the
plant and note how far confirmation of the theory of orthogonal
construction can be obtained, as implying an orthogonal distribu-
* New Phytologist, 1902, p. 49; Annals of Botany, vol. xviii., p. 227.
GENERAL CONCLUSIONS. 349
tion of growth-energy comparable with that of the electro-magnetic
field.
The curves plotted by Mr E. H. Hayes should thus represent the
fundamental shape of the leaf-sections, and the general equation
is definitely put forward as a mathematical definition of a “leaf”
outgrowth.* Such curves, if seen in sections, would not at first
be regarded as equivalent to circles, while closely packed members
approach under similar conditions the form of quasi-squares ;
the conducting tissue (vascular bundles) being orientated around
a point described as the centre of construction. With the help
of these curves and data it now becomes possible to pass on to
the next phase of growth and study the phenomena of varying
rates of growth, and more especially the retarded growth-systems
and unilateral modes of distribution which clearly characterise the
growing-points of shoots and the formation of leaf-laminae as
studied in planes other than the transverse, which has so far alone
been considered.
* Annals of Botany, vol. xviii., p. 227.
.
THE publication of the preceding section concludes the essential
part of a memoir commenced in November 1900 and submitted,
unsuccessfully, to the Linnwan Society in May 1901. Since that
time sections including suggestions on floral construction have
been added in order to place the subject on a wider basis, and
many additional figures have been included in the text, but
publication has been delayed for lack of funds. The first two
parts were published by the author in 1901-2, and the assistance
of a grant from the Royal Society (August 1904) has admitted of
the completion of the present volume.
Grateful acknowledgment is here made of the suggestions and
assistance of many mathematical friends who have been perhaps
more interested in the subject than botanists, and especially to
Mr H. Hilton of Magdalen College and to Mr E. H. Hayes of
New College, without whose interesting discussion of the curves
the subject would have remained in a rudimentary phase.
A. H. C.
NOTES anp ERRATA. (Part II.)
Page 92. Cones from the same two trees of Pinus austriaca and P. laricio
have been counted by Mr E. G. Broome for 1902 with similar one-sided
results,
Thus P. Laricto (1902) :—
lst 100 . . 34R.and66L.
Qnd yy . . 82 4 68,,
ard, =. ts d8isstCts«C,
4th ,, 29 , T1,,
bth, . . 41 4 59,,
6th, . . 31 , 69,
or an average of
69:16 L. and 30°33 R:
a result practically identical with that of the two preceding years for the same
tree: the average for 1100 cones during the three years being
69°82 L. and 30°18 R.
Pinus austriaca (B.G.O, 1902).—A poor crop only admitted of 500 cones
being counted :—
Ist 100. . 54R.and 46 L.
and y a ee
3rd_,, 54g, «= 46,
4th ,, 58, 42,,
sth ,, a ee oe
the average being 54 R. and 46 L. The result again is sufficiently identical
with that of 1901. It may be pointed out that two of these batches gave very
widely different results, in one batch a preponderance of left-hand cones being
found, a fact which serves to show that 100 is too small a number to give a
satisfactory conclusion. The average result for 1600 cones during three years
for this particular tree is thus
54°06 R. and 45°94 L.
351
352 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS.
Page 116. Fibonacci Ratios.—The relation of the sequence 2°1‘2°1° 2, etc., is
not quite so definite as stated, the summation being correct up to 34; since 34
members of the series do not add up to 55 but 54; similarly, 55 members add
up to 88. To keep the ratio correct in an expansion system an extra curve
must be put in: this does not affect the value of the convention, since the series
must be arranged around a circle, and the sequence must be broken somewhere.
Since a strict adherence to the 2°1°2°1°2 sequence would in these cases
result in the formation of multijugate systems, this property of the numbers
involved renders the accuracy with which the expansion ratios succeed one
another still more remarkable,
Page 153, line 25, for 36, read 63.
Note on Dichotomous Systems in Helianthus annuus,
Dichotomy of the shoot-apex of the type described in Lycopodium Selago
(fig. 79, II., ITI.), and found characteristically in the Lycopodiaceae and allied
forms, is possibly to be regarded as the most primitive type of ramification of
the main axis of aerial plants. At any rate the causes which have directed the
evolution of the axillary branching of the strict type met with in higher
plants still remain far to seek, though there can be little doubt as to the
biological utility of the method so widely adopted. Dichotomy of such a
strict type is less frequent among higher plants, and its occurrence would as
a matter of fact be usually classed as a monstrosity. It is clear, however, that
such dichotomy, however anomalous it may be considered, represents the first
step in the production of the still more complicated growth-systems included
under the heading of Fasciation, and that the division of a growth-centre into
two equal centres is the simplest case of irregularity.
Helianthus annuus, which has so frequently been taken as the most typical
representative of phyllotaxis phenomena, owing to the marvellous accuracy of
its inflorescence construction scheme, has already been shown to present in
addition all the typical phenomena of symmetrical and asymmetrical construc-
tions, the perfection of Fibonacci relationships, and also the peculiar relation-
ships of bijugate construction : it again becomes a plant of special interest
from the frequency of the occurrence of strict. dichotomy, which in garden
specimens may affect the whole of the lateral branch system of the plant.
Similar phenomena, but in a less perfect manner, may be noticed in such
allied forms as H. rigidus and H. strwmosus, in their garden varieties. In
such cases the dichotomy may occur (I.) in the foliage region, giving long-
stalked pairs of capitula ; (II.) close behind the involucral region, giving twin-
heads; and (III.) within the involucre, resulting in the phenomenon of
“two-eyed” capitula with a more or less perfect ray series between the
two disks.
Observation of such systems shows that the irregularities recorded for
Lycopodium Selago (p. 207) also hold good for Helianthus ; there being thus
no necessary connection between the distribution of the primordia of the
secondary centres, either between themselves or between these and that of the: ° °
NOTES AND ERRATA. 353
parent shoot before this segmentation of the growth-centre took place, That
is to say, the “genetic-spiral” may work out as homodromous or hetero-
dromous, and the two capitula may be true “twins,” ¢.¢. images one of the
other, or they may not ; and the latter is possibly the commoner case.
When the systems are homodromous, in the long-stalked form (Case I.), the
close agreement of the position of the foliage-leaves is readily checked ; the
only variations being clearly due to the secondary unequal elongation of the
different internodes. Where heterodromy occurs, the point of bifurcation will
be associated with a pair of equal leaves close together. In the case of the
capitula themselves, homodromy or heterodromy is readily checked by noticing
the course of the long and short curves of the disk, though this is often
rendered impossible owing to the addition of structural irregularities.
The fact that phenomena of dichotomy, identical with those obtaining
normally in Lycopodium, should occur as anomalous constructions in such a
plant as Helianthus, would thus appear to suggest, not so much the extreme
antiquity of the dichotomous method, as that this represents an alternative
system of ramification which is worked out equally thoroughly in accordance
with certain definite mechanical laws; and different plants have at different
periods selected that method which in the long run proved most satisfactory
to them. Thus Helianthus, like other Phanerogams, exhibits normally a
system of axillary branches, but still retains the power to arrange an alter-
native method of construction, just as L. Selago is typically dichotomous in its
assimilative region, but presents in addition a copious formation of true
axillary shoots which are subsequently utilised as gemmae for a secondary
biological purpose,
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