Sphericity

“ This is very clearly inferred from the fact already established, viz., that a material point acts all around itself in accordance with the Newtonian law; for this fact compels the conception of a material element as a virtual sphere, of which the matter is the central point, while its virtual sphericity must be traced to the special character of the form. Now, although the centre of a sphere borrows all its centric reality from the sphericity of which it is the intrinsic term, yet the sphericity itself cannot be confined within its own centre ”

“ If the sphere be the one surface of complete symmetry and therefore of independent equi­lib­rium, it follows that in every cell which is otherwise conformed there must be some definite force to cause its departure from sphericity; and if this cause be the very simple and obvious one of the resistance offered by a solidified envelope, such as an egg-shell or firm cell-wall, we must still seek for the deforming force which was in action to bring about the given shape, prior to the assumption of rigidity. ”

“ But if the subjects of geometry are removed from matter, are pure forms, and are separated from [86] sensible objects: they will be all of them, without doubt, void of parts, incorporeal, and destitute of magnitude. For extension, tumor, and interval, approach to forms, on account of the material receptacle in which they are involved, and which receives things destitute of parts, distributed into parts; things void of dimension, extended into dimension; and immoveable natures accompanied with motion. How then, if this is the case, shall we cut a right line, triangle, and circle? ”

“ We have a division of Concrete Science which generalizes those concomitants of this re-dis­tri­bu­tion that hold good among all orders of concrete objects—a division which explains why, along with a {92} predominating integration of matter and dissipation of motion, there goes a change from an indefinite, incoherent homogeneity, to a definite, coherent heterogeneity ”

“ We get it when we pass from a short vertical line to a long horizontal one, from a small square to a large circle, as well as when we pass between those figures whose shapes are congruous. But when the shapes are congruous our consciousness of the relation is a good deal more distinct, and it is most distinct of all when, in the exercise of our analytic attention, we notice, first, a part, and then the whole, of a single line or shape. ”

“ The separation of form and function is an abstraction, as is that of matter and motion; we cannot suppose the connection of particular functions with particular forms,—particular organization,—to be accidental, any more than we can suppose the particular properties of particular inorganic composites and elements to be accidental or these particular properties to be without result in the organic matter into which the particular composites and elements are taken up. ”

“ The symmetrical relations of space in crystals correspond to the simple relations of number in synthesis; and thus, because there is rule, there is regularity, and regularity assumes the form of beauty. ”

“ Now here we have four different species of curve—circle, ellipse, parabola, and hyperbola—each having its peculiar properties and its separate equation, and the first and last of which are quite opposite in nature, connected together as members of one series, all producible by a single process of insensible modification.
But the blindness of those who think it absurd to suppose that complex organic forms may have arisen by successive modifications out of simple ones, becomes astonishing when we remember that complex organic forms are daily being thus produced. ”

“ It is in form that we must seek the differences between species, and not in the combinations of matter, which are almost the same in all. [53] The differences are to be sought at the level of the second and third degrees of composition.
The existence of differences of form introduces a new problem, the problem of diversity. There are only a few possible combinations of the principal organs, but as you get down to less important parts the possible scope of variation is greatly increased, and most of the possible variations do exist. ”

“ Every existent thing, that is, every individual object, is a compound of matter and form. We may compare them in this respect to the material and the shape of a thing, though we must be careful not to think that form is merely shape. Geometry considers shapes as if they existed by themselves. ”

“ The classification of these different kinds of symmetry has been most attended to in Crystallography, in which science it is the highest and most general principle by which the classes of forms are governed. ”

“ In the study of nature, for example, we distinguish matters, forces, genera and the like, and stereotype each in its isolation. Thought is here acting in its analytic capacity, where its canon is identity, a simple reference of each attribute to itself. It is under the guidance of the same identity that the process in knowledge is effected from one scientific truth to another. Thus, for example, in mathematics magnitude is the feature which, to the neglect of any other, determines our advance. Hence in geometry we compare one figure with another, so as to bring out their identity. ”

“ Morphology considers not only forms, but it also considers kinds, forces, causes, and concepts. Dynamics considers not only forces, but it also considers kind, forms, causes, and concepts. Evolution considers not only causes, but it considers kinds, forms, forces, and concepts; and ideation considers not only concepts, but it also considers kinds, forms, forces, and causes, and the difference between these five concomitants is the point of view when every one of the essentials and its derivatives is considered abstractly. ”

“ I trust that some one with the right intellectual equipment will endeavor to follow this guide; and I am sanguine enough to think that a comprehensive study of the geometrical phenomena of differentiation will suggest to a penetrative mind that critical experiment which may one day reveal the meaning of spontaneous division, the mystery through which lies the road, perhaps the most hopeful, to a knowledge of the nature of life. ”

“ The use of mathematics gave a new aspect to crystallography, and the development of the optical relationships opened a magnificent field of wonderful phenomena which can be described as a garden gay with flowers of light, charming in themselves and interesting in their relations to the forces which guide and govern the regular structure of matter. ”

“ We shall infer that the particular structures which have been formed by the activities carried on will not grow more heterogeneous and more definite; and that if other orders of activities, implying other sets of forces, commence, answering structures of another kind will begin to make their appearance, to grow more heterogeneous and definite, and to replace the first. And it will be manifest that while the transition is going on—while the first structures are dissolving and the second evolving—there must be a mixture of structures causing apparent confusion of traits. ”

“ Morphology then is not only a study of material things and of the forms of material things, but has its dynamical aspect, under which we deal with the interpretation, in terms of force, of the operations of Energy. And here it is well worth while to remark that, in dealing with the facts of embryology or the phenomena of inheritance, the common language of the books seems to deal too much with the material elements concerned, as the causes of development, of variation or of hereditary transmission. ”

“ It will of course be understood that by the term Symmetry I here intend, not that more indefinite attribute of form which belongs to the domain of the fine arts, as when we speak of the ‘symmetry’ of an edifice 68 or of a sculptured figure, but a certain definite relation or property, no less rigorous and precise than other relations of number and position, which is thus one of the sure guides of the scientific faculty, and one of the bases of our exact science. ”

“ All things which possess extension, or, in other words, which fill space, are subject to geometrical laws. Possessing extension, they possess figure; possessing figure, they must possess some figure in particular, and have all the properties which geometry assigns to that figure. If one body be a sphere and another a cylinder, of equal height and diameter, the one will be exactly two-thirds of the other, let the nature and quality of the material be what it will. Again, each body, and each point of a body, must occupy some place or position among other bodies ”

“ What, then, is the significance of all these facts, and why should we gather them from the highways of morphology but for the lesson which they teach, that all forms of animals, plants, rocks, and stars are traced to the substrate of extension in the particle? Extension traced through all its complicated relations of space, form, metamorphosis and organism is found to be the ultimate substrate of them all. ”

“ The sphere of change is universally found to be limited in such a way that any a changes only into α, α1, α2, . . ., any b only into β, β1, β2; and, in general, every real being changes only into such a ‘closed series’ of forms as, taken collectively, are deducible from the original nature of the being; while no being can ever pass out of the series of its own forms over into a series of forms belonging to another being. ”

“ Even the laws of geometry are chiefly of practical importance to us as being a portion of the premisses from which the order of the succession of phenomena may be inferred. Inasmuch as the motion of bodies, the action of forces, and the propagation of influences of all sorts, take place in certain lines and over definite spaces, the properties of those lines and spaces are an important part of the laws to which those phenomena are themselves subject. ”

“ If now it be asked how, if triangles, squares, square roots, genera, and the like, are but improvised human 'artefacts,' their properties and relations can be so promptly known to be 'eternal,' the humanistic answer is easy. If triangles and genera are of our own production we can keep them invariant. ”

“ WE have seen in the History of the Sciences, that the principle which I have there termed1 the Principle of Developed and Metamorphosed Symmetry, has been extensively applied in botany and physiology, and has given rise to a province of science termed Morphology. In order to understand clearly this principle, it is necessary to obtain a clear idea of the Symmetry of which we thus speak. But this Idea of Symmetry is applicable in the inorganic, as well as in the organic kingdoms of nature; it is presented to our eyes in the forms of minerals, as well as of flowers and animals ”

“ In solid geometry, however, the reasoning is no more difficult than in plane geometry, but the conceptions are far more complex, and, if the teacher insisted that the pupil should not take a single step until his conceptions were perfectly clear, all the difficulties would disappear. Of this I am fully persuaded, for I have had to encounter the same difficulties over and over again in teaching crystallography. In beginning the study of geometry, of course the power of conception should be helped in every possible way. ”

“ To get rid of this difficulty, and at the same time to save the credit of the supposed system of necessary truth, it is customary to say that the points, lines, circles, and squares which are the subject of geometry, exist in our conceptions merely, and are part of our minds; which minds, by working on their own materials, construct an à priori science, the evidence of which is purely mental, and has nothing whatever to do with outward experience. ”

“ This small specimen of the geometry of visibles, is intended to lead the reader to a clear and distinct conception, of the figure and extension which is presented to the mind by vision; and to demonstrate the truth of what we have affirmed above, namely, that those figures and that extension which are the immediate objects of sight, are not the figures and the extension about which common geometry is employed; that the geometrician, while he looks at his diagram, and demonstrates a proposition, hath a figure presented to his eye, which is only a sign and representative of a tangible figure ”

“ What interpretation is to be put on these truths of classification? We find that organic forms admit of an arrangement everywhere indicating the fact, that along with certain attributes, certain other attributes, which are not directly connected with them, always exist. How are we to account for this fact? And how are we to account for the fact that the attributes possessed in common by the largest assemblages of forms, are the most vitally-important attributes? ”

“ When we revert to experience for an understanding of congruence, we find it exemplified in the rigid bodies of nature, whose geometrical dispositions yield, more or less precisely, the results of pure Euclidean geometry. If we idealise congruence, as thus defined, and express it mathematically, we may say that perfectly rigid bodies are those whose measurements would yield Euclidean results with absolute precision. ”

“ All properties which physical bodies in contradistinction to geometric bodies possess can be traced back to a fundamental concept, which, in conjunction with the concepts explained in the former chapter, serves to characterize and distinguish the physical structure. ”