By change of single cyphers in this, the simplest of all continuous fractions, the expressions were also obtained for all measures of position that deviate from the usual main series. The common occurrence of so-called leaf-whorls seemed at once to be opposed to the principle of special growth and to the doctrine of position founded upon it, especially in the cases in which it was supposed that all the leaves of a whorl arise simultaneously. But the founders of the doctrine, relying on their geometrical constructions, declared that every theory is incorrect, which sets out from the whorl as a simultaneous formation. But the way in which the different leaf-whorls of a stem are arranged among themselves, and are connected with continuous spiral positions, required new geometrical constructions; it was necessary to assume a supplementary relation (prosenthesis), which the measure of the phyllotaxis adopts in the transition from the last leaf of one cycle to the first of the next. Artificial as this construction appears, it has the advantage of saving the spiral principle, and the prosenthetic relation itself admits of being again expressed in highly simple fractions,—a great advantage for the formal consideration of the relative positions of the parts of the flower, and their relation to the preceding positions of the leaves. The great skill shown by the founders of the doctrine in the morphological consideration of the whole plant-form appears equally in the establishment of the rules, according to which the relations of position of the leaves of a side-shoot connect with those of the mother-axis, and which made it possible to represent the nature of inflorescences especially with extreme clearness by means of geometrical figures. An expressive and elegant terminology not only made the whole theory attractive, but fitted it in a high degree to supply a suitable, plain, and precise phraseology for describing the most varied forms of plants. That the theory possesses such advantages as these may be gathered from the fact, that since 1835 the morphological examination and comparison not only of flowers and inflorescences, but also of vegetative shoots and their ramification, has reached great formal completeness. A thorough acquaintance with the principle of this doctrine has made it possible to explain to reader or hearer the most intricate forms of plants so clearly, that they may be said to reveal the law of their formation themselves, and to grow before the eye of the observer, while at the same time the most recondite relations of the organs of the same or of different plants were brought out distinctly and in elegant phraseology. When this mode of description was combined with De Candolle’s views on abortion, degeneration, and adherence, and at the same time took into consideration the chief physiological forms of leaf-structures, according as these were developed as scales, foliage-leaves, bracts, floral envelopes, staminal and carpellary leaves, it was possible to give such an artistic account of every form of plant, as made it visible to sense in its entirety, and at the same time brought out the morphological law of its construction. Whoever reads the writings of Alexander Braun and Wydler, and especially of Thilo Irmisch (after 1873), who knew how to combine his descriptions in a variety of ways with remarks on the biological relations of plants, cannot fail to admire the extraordinary skill displayed by these men in describing plants. Compared with the dry diagnoses of the systematists, their descriptions attain to the dignity of an art, and present the commonest forms to the reader in a new and attractive light. But the theory had a further advantage; it seemed not only to present the form of the plant in its matured state, but to treat it genetically; and in fact it did possess an element of historical development, inasmuch as it made the genetic succession of the leaves and of their axillary shoots, which is at the same time the succession from the base to the summit, the foundation of all consideration of the plant-form. But it is also true that in this lay one of the weak sides of the theory; as long as it was a question only of continuous spirals, the succession of matured leaves does also represent the succession of their formation in time; but this was not actually proved in the case of leaf-whorls, and here, to save the theory, genetic relations had to be presupposed for which no further proof was forthcoming, while fresh researches have repeatedly shown that a strict application of Schimper’s theory is found frequently to contradict the facts of development as directly observed[47]. Moreover, regard was had only to those measurements of divergence on the continuous genetic spiral which were taken on the matured stem, while there was always the possibility that the divergences might have been different at the first, and been afterwards modified, as Nägeli subsequently suggested[48]. And again, the theory had a dangerous adversary to encounter in the frequent occurrence of leaves that are strictly alternate or crossed in pairs, and to conceive of this as a spiral arrangement must at once appear to be an arbitrary proceeding both from the mathematical point of view and from that of historical development; the assumption of a return of the genetic spiral from leaf to leaf, as for instance in the Grasses, like the prosenthesis in the change of divergence, afforded, it is true, a construction which was geometrically correct, but which could hardly be made to agree with the history of development and the mechanical forces concerned. Again, it was a great and essential defect in the theory, that in assuming the spiral arrangement it entirely neglected the relations of symmetry of the plant-form, which are in many cases clearly expressed, and their connection with the outer world, on which Hugo von Mohl had already published some excellent remarks in 1836,—a defect, which unhappily is not yet sufficiently appreciated. A due consideration of these objections, and of the cases in which the history of development is opposed to the constructions of the theory, must have led to the conviction that the idea of a spiral tendency in the growth of plants is at least not borne out in all cases, and more profound reflexion would show, that a scientific principle, really explaining the phenomena, is no more to be found in the assumption of such a general tendency, than in a like assumption with regard to the heavenly bodies, that they have a tendency to elliptic movement because they commonly move in ellipses. Hence Hofmeister, the latest investigator of the doctrine of phyllotaxis on the basis of the history of development, comes to the conclusion that the notion of a screw-shaped or spiral course of evolution of lateral members of plants is not merely an unsuitable hypothesis, but an error. Its unreserved abandonment is, he considers, the first condition for attaining an insight into the proximate causes of the varieties of relative position in the vegetable kingdom. But this judgment, correct as it is, was pronounced thirty years after the appearance of Schimper’s theory; history, which speaks from another point of view, and not only enquires into the correctness of a theory but has to appraise its historical importance, speaks in a less unfavourable manner. The chief point here is not whether the theory was right, but how far it contributed to the advance of the science. It was distinctly fruitful in results, for it brought the important question of the relative positions of organs for the first time into the front rank in the study of morphology; we may even say that a large part of the results of the study of the history of development were first brought into the true light by the consistent application of the theory, or in the effort to disprove it. With all its fundamental errors, Schimper’s theory remains one of the most interesting phenomena in the history of morphology, because it was carried out with thorough logical consistency. We should as little wish to omit it from our literature, as modern astronomy would wish to see the old theory of epicycles disappear from its history. Both theories served to connect together the facts that were known in their time.

The fundamental error of the theory lies much deeper than appears at first sight. Here too we have the idealistic conception of nature, which refuses to know anything of the causal nexus, because it takes organic forms for the ever-recurring copies of eternal ideas, and in accordance with this platonic sphere of thought confounds the abstractions of the mind with the objective existence of things. This confusion shows itself in Schimper’s doctrine, inasmuch as he takes the geometrical constructions, which he transfers to his plants and which, though they may be highly suitable from his point of view, are nevertheless purely arbitrary, for actual characters of the plants themselves, in other words, takes the subjective connection of the leaves by a spiral line for a tendency inherent in the nature of the plant. Schimper in making his constructions overlooked the fact that, because a circle can be described by turning a radius round one of its extremities, it does not follow that circular surfaces in nature must really have been formed in this way; in other words, he did not see that the geometrical consideration of arrangements in space, useful as it may otherwise be, gives no account of the causes to which they are due. But this was not properly an oversight in Schimper’s case, for he would have scarcely admitted efficient causes in the true scientific sense into his explanations of the form of plants. How far Schimper was from regarding plants as something coming into being in time and according to natural laws, how profoundly he despised the principles of modern natural science is shown in his judgment of Darwin’s theory of descent and of the modern atomic theory, the coarseness of which is the more surprising, because Schimper was a man of refined and even poetic feeling. ‘Darwin’s doctrine of breeding,’ he says, ‘is, as I discovered at once and could not help perceiving more and more after repeated and careful perusal, the most shortsighted possible, most stupidly mean and brutal, much more paltry even than that of the tesselated atoms with which a modern buffoon and hired forger has tried to entertain us.’ Here is the old platonic view of nature flying at modern science; the sternest ‘opposites’ that culture has ever produced.

The theory of Schimper, which should rather be called the theory of Schimper and Braun, considering the active part which Braun took from the first in framing and applying it, was capable of further development only in the mathematical and formal direction, as was shown especially in Naumann’s essay, ‘Ueber den Quincunx als Grundgesetz der Blattstellung vieler Pflanzen’ (1845). The defects above described, but not the merits of the theory were shared by the doctrine of phyllotaxis laid down about ten years later by the brothers Louis and Auguste Bravais. Their theory makes use of mathematical formulae to even a greater extent than that of Schimper without paying any attention to genetic conditions, and yet it is less consistent with itself, for it assumes two thoroughly different kinds of phyllotaxis, the positions in which are arranged in a straight and in a curved line; for the latter without any apparent reason a purely ideal original divergence is assumed which stands in irrational relation to the circumference of the stem, and from it all other divergences should be derivable; and this ultimately degenerates into mere playing with figures which in this form afford no deeper insight into the causes of the relations of position. As regards serviceableness in the methodic description of plants the theory of the brothers Bravais is much inferior to that of Schimper[49].

The genetic morphology founded about the year 1840 had to make the best terms it could with the doctrine of phyllotaxis, which was constructed on a totally different principle; the two went their way on the whole side by side without disturbance from one another till the year 1868, when Hofmeister in his general morphology attacked the principle of Schimper’s theory, and endeavoured to substitute a genetic and mechanical explanation of the relative positions for the purely formal account of them; this attempt however, which from the nature of the case has not yet led to a finished theory but nevertheless contains the germ of a further development of this important doctrine, does not come within the scope of this history.

The doctrine of phyllotaxis of Schimper and Braun, as it appeared after 1830, had clearly presented only one side of the theory of metamorphosis; what other elements there were in it capable of being turned to speculative account were further cultivated by Alexander Braun between the years 1840 and 1860. In this period fresh points of view were asserting themselves in botanical research; the founding of the doctrine of cells, the study of the more delicate anatomy of plants and of the history of development, and increased methodical knowledge of the Cryptogams were enlarging the repertory of botanical facts, while the physico-mechanical method of investigation was being more and more adopted. Braun, who took an active part by his own researches in this revolution in morphological botany, remained true nevertheless to idealistic views; and in his frequent and comprehensive discussions of the general results of the new investigations in accordance with these views he has shown how far the idealistic platonising contemplation of nature is in a condition to do justice to the results of exact inductive enquiry. The opposition between his point of view and that of the most eminent representatives of the inductive method became more and more pronounced as years went on, and must be treated here as a historical fact. But if the new tendency in botany pursued especially by von Mohl, Schleiden, Nägeli, Unger, and Hofmeister may be called inductive in the absence of a better term, and be contrasted with the idealistic tendency represented by Braun and his school, it must not be supposed that the latter did not equally contribute in matters of detail to the enriching of the science by the method of induction; on the contrary, Braun himself was the author of a series of important works conceived in this spirit. When the new method is here called inductive, it should be understood that the word is used in a higher than the usual sense, and some explanation of this point will not be superfluous in this place. Idealistic views of nature of all times, whether they present themselves as Platonism, Aristotelian logic, Scholasticism or modern Idealism, have all of them this in common, that they regard the highest knowledge attainable by man as something already won and established; the highest axioms, the most comprehensive truths are supposed to be already known, and the task of inductive enquiry is essentially that of verifying them; the results of observation serve to elucidate already received views, to illustrate already known truths; inductive enquiry has only to establish individual facts. But in the sense in which inductive enquiry was understood by Bacon, Locke, Hume, Kant, and Lange, its task is one that goes essentially farther than this; it must not be content with establishing individual facts, but it must employ them in the critical examination of the most general notions that have come down to us, and do its best to deduce new and comprehensive theories from them, even where these may be entirely opposed to traditional views. But it is part of the very nature of this method of investigation, that its general results are subject to constant modification and improvement; each more general truth has only a temporary value, and endures as long as no new facts militate against it. The distinction therefore between idealism and the inductive method in the domain of natural science comes to this, that the former fits new facts into a scheme of old conceptions, the latter deduces new conceptions from new facts; the one is in its nature dogmatic and intolerant, the other eminently critical; the one is conservative, the other always pressing forwards; the one inclines to philosophic contemplation, the other to vigorous and productive investigation. To this must be added one point of great importance; the idealistic view of nature, rejecting causality, explains nature from notions of design, and is teleological; ethical and even theological elements are thus introduced into natural science.

It is in this form that the distinction between the idealistic view represented by Braun and the modern inductive morphology presents itself to us. If it were the task of this history only to record the discovery of new facts, it would be superfluous to allude to these differences here; but then it would also be impossible to estimate rightly that portion of Braun’s long scientific labours which is at once the most original and the most interesting from the historical point of view, and which is to be found not so much in his many descriptive and monographic works, as in his philosophic efforts in the domain of morphology; these moreover deserve our consideration, because they carry out Goethe’s half-explained conceptions to their remotest consequences, and express in purer form the idealism which lies at the foundation of the older nature-philosophy. No botanist since Cesalpino has so thoroughly endeavoured to leaven the entire results of inductive investigation with the principles of an idealistic philosophy, and to explain them in its light.

Braun’s philosophical views not only accompany his knowledge of facts, but everywhere permeate and colour it; in his writings, contributions, and monographs on the most various subjects, facts are regarded from the point of view of his philosophy. He has given a general view of his philosophical principles and illustrated them by a vast variety of facts in his famous book, ‘Betrachtungen über die Erscheinung der Verjüngung in der Natur, insbesondere in der Lebens-und Bildungsgeschichte der Pflanze’ (1849-50). He himself directs attention to the opposition between his own stand-point and the modern induction in the tenth page of the preface, where he replies to the obvious objection, that his ideas may be regarded as antiquated, in the words, ‘A more living contemplation of nature, such as is here attempted, which seeks in natural bodies not merely the operation of dead forces, but the expression of a living fact, does not lead, as is supposed, to airy structures of fancy, for it does not pretend to gain a knowledge of life in nature in any other way than as it is revealed in phenomena,’ etc. This thought is still more distinctly uttered in page 13 of the text; ‘As external nature without mankind presents to us only the spectacle of a labyrinth without a guide, so too scientific contemplation, which denies the inner spiritual principle in nature and the intimate connection of nature with the informing spirit[50], leads to a chaos of substances and forces, which are unknown because divorced from spirit, or, to speak more precisely, to a chaos of nothing but unknown causes, which work together in an inexplicable manner.’ In a note to this passage he points expressly to ‘the comfortless character of such an unreal mode of viewing nature, which must necessarily endeavour to root out everything in the conceptions and language of science which appears from its own point of view to be anthropopathic,’ and he requires a tender, ethical element as essential to botanical investigation. The chief object of the volume is to prove that everything in organic life may be resolved into rejuvenescence, of which idea no definition is actually given, though the whole contents of the book are a search after a definition. We may regard the idea of rejuvenescence, as presented by Braun, as an extension of the idea of metamorphosis, in which extended form it is adapted to take in even the results of the cell-theory, of the history of development, and of the modern knowledge of the Cryptogams from the idealistic point of view. One peculiarity of his mode of expounding his views is observed here, as on other occasions, namely, that he gives no precise and arbitrary definition to a word, for instance, like rejuvenescence in the present place, and in a later work to the word individual, but looks behind the word for a profound or even mysterious meaning, which is to be perceived and brought to light by contemplation of the phenomena. In page 5 he says, ‘Thus we see youth and age appear alternately in one and the same history of development; we see youth burst through age, and by growth or transformation step into the middle of the development. This is the phenomenon of rejuvenescence, which is repeated in endless multiplicity in every province of life, but nowhere appears more clearly expressed or more accessible to investigation than in the vegetable kingdom. Without rejuvenescence there is no history of development.’—‘If then we ask for the causes of the phenomena of rejuvenescence (page 7), we shall indeed allow that nature, into which special life enters in its various manifestations, excites, awakes, and works by the influences which the years and even the days bring with them; but the true and inner cause can only be found in the desire after perfection which belongs to every being in its kind, and urges it to bring the outer world, which is strange to it, more and more into complete subjection to itself, and to fashion itself in it as independently as its specific nature admits.’ Further on he says (page 17), ‘The impulse or tendency to development in each creature is likewise no direction of activity impressed from without, but one given from within and working as an inner determination and force from the depth of the inner nature.’ A passage also from page 111 of his treatise on polyembryony, published in 1860, may be quoted here; ‘Though the organism, in the process of realising itself, is subject to physical conditions, yet the proper causes of its morphological and biological characteristics do not lie in these conditions; its laws belong to a higher stage of development of its being, to a sphere in which the faculty of self-determination is distinctly manifested. If this is so, the laws of an organic being appear as tasks imposed, the fulfilling of which is not absolutely necessary but only in relation to the attainment of a definite end, as precepts, to which strict obedience may possibly not be paid.’ To return once more to the idea of rejuvenescence, we find at page 18 the words, ‘As regards the idea of rejuvenescence, from the foregoing considerations we draw the conclusion, that the surrender of growths already accomplished and the going back to new beginnings, the commencement of rejuvenescence, indicate only the outer side of the proceeding, while the essential part of it is an inner gathering up of forces, a new creating, as it were, out of the individual principle of life, a fresh reflecting upon the specific task or the gaining renewed hold upon the type which is to be presented in the outer organism. By this means rejuvenescence maintains its fixed relation to development, which can and ought to present in gradually attained perfection that only which lies in the nature of the creature, and is most intimately its own.’ And at the conclusion of the work (page 347) he says, ‘The way in which the inner spiritual nature of life is specially manifested in the phenomenon of rejuvenescence may be defined as reminiscence in the true sense of the word, as the power of grasping anew in the phenomenon the inner destination of life as contrasted with its daily alienation and decay, and applying it with renewed strength towards that which is without,’ etc.

This conception of rejuvenescence is, then, applied to all the phenomena of life in plants; not only the metamorphosis of leaves, the formation of shoots and their ramification, and the different modes of cell-formation, but even palaeontological facts are manifestations of rejuvenescence, which in the sequel puts off the form of an abstract idea, and becomes personified into an active personality, as is seen in page 8 in the expression, ‘activity of rejuvenescence.’

The relation of Braun’s views to the question of the constancy of species may to some extent appear doubtful; some utterances of his may be interpreted to admit a transmutation of species accomplished in the course of ages, while others are opposed to this, and it is the latter which appear to be consistent with the idealistic position. We read, for instance, at page 9, ‘The appearance, as though the like was always repeating itself in nature, is suggested when we glance back from our station in time upon the succession of former epochs. Here we find the real first beginnings of species and genera, and even of orders and classes in the vegetable and animal kingdoms; we see at the same time that more or less thorough transformations are connected with the appearance of the higher grades in the organic kingdom, so that genera and species of the old world disappear, and new ones step into their place. All this change expresses not the mere accident of convulsions, which, while they destroy, at the same time prepare new ground for the prosperity of organic nature, but rather definite laws whose action pervades all the individual detail of the development of organic life.’ On the other hand we find at the conclusion of the treatise on polyembryony, written a short time before the appearance of Darwin’s memorable work, a sentence which makes the assumption of a transmutation of species appear very doubtful; it says (page 257), ‘If we are justified in assuming a general organic connection in the history of development in plant-forms, can we imagine that the type of the Mosses and of the Ferns has come from the Algae, or vice versa, that the Alga-form owes its origin to the Mosses and Ferns?’