roar “viel 5 teem saat een tea ee: hee prreey Ma ran ae ier tela Ma ete ma ALBERT R. MANN LIBRARY AT CORNELL UNIVERSITY DATE DUE DEC [27 1989 |U, ornell University Libra the relation of phyllotaxis to mechan Pha ra as ae ce ae 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 yon ok QR lo4-4 CSG 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. Euphorbia 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. ‘ ¢ 3 : 75-78 PART II. ASYMMETRICAL AND SYMMETRICAL PHYLLOTAXIS. J. Normal Fibonacci Phyllotaxis: Conception of Bulk- Ratio 5 ; : 5 , . 83-89 TI. Constant Phyllotaxis: Araucaria, Podocarpus . 90-108 Vv vl III. IV. VI. VI. VIII. IT. IIT. IV. CONTENTS. Rising Phyllotaxis : Helianthus Capitulum ; Fibonacci Expansion ; Helianthus Seedling; Cyperus ; Falling Phyllotaxis; Cynara; Asymmetrical Floral Diagrams The Symmetrical Concentrated Type: Hqwisetwm . Asymmetrical Least Concentrated Type: Cyperus, Gasterta Symmetrical Non-concentrated Type . Multijugate Type: Bravais; Dipsacus; Expansion System ; Stlphium, Cephalaria Anomalous Series: Sedum vreflexcum,; 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 : . 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 . 829-333 II. Mathematical Orthostichies in Log. Spiral Systems . 334-335 III. The Form of the ‘‘ Ovoid” Curve : : . 835-337 IV. Bulk-Ratio . . : ‘ 3 . 338-339 V. The Oscillation Angle . ‘ : ; 339-341 VI. The Fibonacci Series. 3 ‘ . 341-344 VII. Continued Fractions . : : ‘ 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. —tH— 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 faHing 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 og 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 Churcth.—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 @ 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 szmzlar 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”. Ludlk-ratio of axis to primordium=O04., AB.=1: 5 within a small error, or=Sin 402 =.204 for the true curve. S 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 : . ¢ 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 Church.—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 ' aaene- 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. Zon 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 dijugate 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 dae a Lown fun--nn. 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 es 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), (848), &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 1 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 on ey a 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 oh 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 similar 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 guasv- 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 Phyllotaxts. section of m spirals crossing , in the manner required, is given in such’ a form as, lo T= lo c+I: 64 8 ——_———, — .0000 0864 62, e S =_ 3 3 = 1 n* 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 m+n 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 ot 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 itis 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 -otiginal 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 eccen¢ric 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. wf 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 uxilateral 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. oS 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. Linneus 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 lusage 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 multiwugate 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 :— 3, 4% 8 Ys dp 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, ?= 144° 21 = 137° 27 16”:36 2= 135° 4 = 187° 31’ 4112 is = 138° 27’ 41-54 be = 137° 30’ 0” f= 137° 8’ 34-28 | “Ideal angle” =137° 30’ 27-936 44= 187° 38’ 49"-41 It will be noticed that the differences become extremely minute in the higher fractions, and that beyond 3; 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 iiber 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. A 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 34;, 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 ,°s, 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 beié& 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 55;, 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 * Kichler, 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 3, 4, #, 7p etc., leading on to an ideal angle of 99° 30’ 6”, and 4,14, 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. Sci. 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 abswrdum 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 pie th Mow caren 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. Ifa 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 7;, 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 5; 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, Luonymus, 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. oar pea a subtended by y 7 thomb (“square”). 2 TQ: 1. 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 Equisetum 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.