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MANN LIBRARY AT CORNELL UNIVERSITY DATE DUE Lae aa pec |27 989 LPO | DEMCO 38-297 ‘ornell University Libra’ the relation of phyllotaxis to mechan ie ere tlie sale as aM Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. 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 yen ok QR lo44 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 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 PLATE XV. ‘ “IayotTRIp "Wd OZ ‘pertds-z eoryqgauurdsy MeRy ‘/NU2UM9 DILOISDH—'9LG “Oli “(1+ [) Teotqowurds ‘ssoroe ‘Wo 9% querd sory “MEET ‘w2p0fisngqo m2.19)8nH— "VLG ‘OT PLATE XVI. ‘ssoloe ‘MIO ZT peottjemuuAse ato ‘sSulppees ¢ “MRET ‘vnl2790 DIL9j8D))—"98G ‘OL us Tspeureip “mo Og “patids-Z ‘Teowgourssy Mey ‘npwavsm/ mwapsnp—"ngg oly i XVIT. PLATI ‘qooys a[t}te} Jo xedy *LOqUIAAO NT ssnyofiusoyn snsadhg—'96g “OTL ‘squerd pards-g ‘seynujds vind —"V69 “OTA 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. é “9t/OT ed 44 ayy jo oomedsetoguy = "T ‘svsoped snovsdig7—"q09 “91 r “9T/01/9 adfy ayy Jo outoo aqesnliq ‘orrund snurg—09 “LL (b+é) 82.6989 [fis snonsdigT—"QT9 “D1 © aqjasor Surppeag = "T Suyeunasey (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