CHAPTER XVII
ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON OF RELATED FORMS[641]
In the foregoing chapters of this book we have attempted to study the inter-relations of growth and form, and the part which certain of the physical forces play in this complex interaction; and, as part of the same enquiry, we have tried in comparatively simple cases to use mathematical methods and mathematical terminology in order to describe and define the forms of organisms. We have learned in so doing that our own study of organic form, which we call by Goethe’s name of Morphology, is but a portion of that wider Science of Form which deals with the forms assumed by matter under all aspects and conditions, and, in a still wider sense, with forms which are theoretically imaginable.
The study of form may be descriptive merely, or it may become analytical. We begin by describing the shape of an object in the simple words of common speech: we end by defining it in the precise language of mathematics; and the one method tends to follow the other in strict scientific order and historical continuity. Thus, for instance, the form of the earth, of a raindrop or a rainbow, the shape of the hanging chain, or the path of a stone thrown up into the air, may all be described, however inadequately, in common words; but when we have learned to comprehend and to define the sphere, the catenary, or the parabola, we have made a wonderful and perhaps a manifold advance. The mathematical definition of a “form” has a quality of precision which was quite lacking in our earlier stage of mere description; it is expressed in few words, or in still briefer symbols, and these {720} words or symbols are so pregnant with meaning that thought itself is economised; we are brought by means of it in touch with Galileo’s aphorism (as old as Plato, as old as Pythagoras, as old perhaps as the wisdom of the Egyptians), that “the Book of Nature is written in characters of Geometry.”
Next, we soon reach through mathematical analysis to mathematical synthesis; we discover homologies or identities which were not obvious before, and which our descriptions obscured rather than revealed: as for instance, when we learn that, however we hold our chain, or however we fire our bullet, the contour of the one or the path of the other is always mathematically homologous. Lastly, and this is the greatest gain of all, we pass quickly and easily from the mathematical conception of form in its statical aspect to form in its dynamical relations: we pass from the conception of form to an understanding of the forces which gave rise to it; and in the representation of form and in the comparison of kindred forms, we see in the one case a diagram of forces in equilibrium, and in the other case we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other. Here, since a change of material form is only effected by the movement of matter, we have once again the support of the schoolman’s and the philosopher’s axiom, “_Ignorato motu, ignoratur Natura_.”
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In the morphology of living things the use of mathematical methods and symbols has made slow progress; and there are various reasons for this failure to employ a method whose advantages are so obvious in the investigation of other physical forms. To begin with, there would seem to be a psychological reason lying in the fact that the student of living things is by nature and training an observer of concrete objects and phenomena, and the habit of mind which he possesses and cultivates is alien to that of the theoretical mathematician. But this is by no means the only reason; for in the kindred subject of mineralogy, for instance, crystals were still treated in the days of Linnaeus as wholly within the province of the naturalist, and were described by him after the simple methods in use for animals and plants: but as soon as Haüy showed the application of mathematics to {721} the description and classification of crystals, his methods were immediately adopted and a new science came into being.
A large part of the neglect and suspicion of mathematical methods in organic morphology is due (as we have partly seen in our opening chapter) to an ingrained and deep-seated belief that even when we seem to discern a regular mathematical figure in an organism, the sphere, the hexagon, or the spiral which we so recognise merely resembles, but is never entirely explained by, its mathematical analogue; in short, that the details in which the figure differs from its mathematical prototype are more important and more interesting than the features in which it agrees, and even that the peculiar aesthetic pleasure with which we regard a living thing is somehow bound up with the departure from mathematical regularity which it manifests as a peculiar attribute of life. This view seems to me to involve a misapprehension. There is no such essential difference between these phenomena of organic form and those which are manifested in portions of inanimate matter[642]. No chain hangs in a perfect catenary and no raindrop is a perfect sphere: and this for the simple reason that forces and resistances other than the main one are inevitably at work. The same is true of organic form, but it is for the mathematician to unravel the conflicting forces which are at work together. And this process of investigation may lead us on step by step to new phenomena, as it has done in physics, where sometimes a knowledge of form leads us to the interpretation of forces, and at other times a knowledge of the forces at work guides us towards a better insight into form. I would illustrate this by the case of the earth itself. After the fundamental advance had been made which taught us that the world was round, Newton showed that the forces at work upon it must lead to its being imperfectly spherical, and in the course of time its oblate spheroidal shape was actually verified. But now, in turn, it has been shown that its form is still more complicated, and the next step will be to seek for the forces that have deformed the oblate spheroid. {722}
The organic forms which we can define, more or less precisely, in mathematical terms, and afterwards proceed to explain and to account for in terms of force, are of many kinds, as we have seen; but nevertheless they are few in number compared with Nature’s all but infinite variety. The reason for this is not far to seek. The living organism represents, or occupies, a field of force which is never simple, and which as a rule is of immense complexity. And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of _ideal_ simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies far beyond our powers of analysis. Nor must we forget that the biologist is much more exacting in his requirements, as regards form, than the physicist; for the latter is usually content with either an ideal or a general description of form, while the student of living things must needs be specific. The physicist or mathematician can give us perfectly satisfying expressions for the form of a wave, or even of a heap of sand; but we never ask him to define the form of any particular wave of the sea, nor the actual form of any mountain-peak or hill[643]. {723}
For various reasons, then, there are a vast multitude of organic forms which we are unable to account for, or to define, in mathematical terms; and this is not seldom the case even in forms which are apparently of great simplicity and regularity. The curved outline of a leaf, for instance, is such a case; its ovate, lanceolate, or cordate shape is apparently very simple, but the difficulty of finding for it a mathematical expression is very great indeed. To define the complicated outline of a fish, for instance, or of a vertebrate skull, we never even seek for a mathematical formula.
But in a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the _deformation_ of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalysed and undefined. This process of comparison, of recognising in one form a definite permutation or _deformation_ of another, apart altogether from a precise and adequate understanding of the original “type” or standard of comparison, lies within the immediate province of mathematics, and finds its solution in the elementary use of a certain method of the mathematician. This method is the Method of Co-ordinates, on which is based the Theory of Transformations.
I imagine that when Descartes conceived the method of co-ordinates, as a generalisation from the proportional diagrams of the artist and the architect, and long before the immense possibilities of this analysis could be foreseen, he had in mind a very simple purpose; it was perhaps no more than to find a way of translating the _form_ of a curve into _numbers_ and into _words_. This is precisely what we do, by the method of co-ordinates, every time we study a statistical curve; and conversely, we translate numbers into form whenever we “plot a curve” to illustrate a table of mortality, a rate of growth, or the daily variation of temperature or barometric pressure. In precisely the same way it is possible to inscribe in a net of rectangular co-ordinates the outline, for instance, of a fish, and so to translate {724} it into a table of numbers, from which again we may at pleasure reconstruct the curve.
But it is the next step in the employment of co-ordinates which is of special interest and use to the morphologist; and this step consists in the alteration, or “transformation,” of our system of co-ordinates and in the study of the corresponding transformation of the curve or figure inscribed in the co-ordinate network.
Let us inscribe in a system of Cartesian co-ordinates the outline of an organism, however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull. We may now treat this complicated figure, in general terms, as a function of _x_, _y_. If we submit our rectangular system to “deformation,” on simple and recognised lines, altering, for instance, the direction of the axes, the ratio of _x_/_y_, or substituting for _x_ and _y_ some more complicated expressions, then we shall obtain a new system of co-ordinates, whose deformation from the original type the inscribed figure will precisely follow. In other words, we obtain a new figure, which represents the old figure _under strain_, and is a function of the new co-ordinates in precisely the same way as the old figure was of the original co-ordinates _x_ and _y_.
The problem is closely akin to that of the cartographer who transfers identical data to one projection or another; and whose object is to secure (if it be possible) a complete correspondence, _in each small unit of area_, between the one representation and the other. The morphologist will not seek to draw his organic forms in a new and artificial projection; but, in the converse aspect of the problem, he will inquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shown to be a transformed representation of the other. This once demonstrated, it will be a comparatively easy task (in all probability) to postulate the direction and magnitude of the force capable of effecting the required transformation. Again, if such a simple alteration of the system of forces can be proved adequate to meet the case, we may find ourselves able to dispense with many widely current and more complicated hypotheses of biological causation. For it is a maxim in physics that an effect ought not to be ascribed to {725} the joint operation of many causes if few are adequate to the production of it: _Frustra fit per plura, quod fieri potest per pauciora._
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It is evident that by the combined action of appropriate forces any material form can be transformed into any other: just as out of a “shapeless” mass of clay the potter or the sculptor models his artistic product; or just as we attribute to Nature herself the power to effect the gradual and successive transformation of the simplest into the most complex organism. In like manner it is possible, at least theoretically, to cause the outline of any closed curve to appear as a projection of any other whatsoever. But we need not let these theoretical considerations deter us from our method of comparison of _related_ forms. We shall strictly limit ourselves to cases where the transformation necessary to effect a comparison shall be of a simple kind, and where the transformed, as well as the original, co-ordinates shall constitute an harmonious and more or less symmetrical system. We should fall into deserved and inevitable confusion if, whether by the mathematical or any other method, we attempted to compare organisms separated far apart in Nature and in zoological classification. We are limited, not by the nature of our method, but by the whole nature of the case, to the comparison of organisms such as are manifestly related to one another and belong to the same zoological class.
Our inquiry lies, in short, just within the limits which Aristotle himself laid down when, in defining a “genus,” he showed that (apart from those superficial characters, such as colour, which he called “accidents”) the essential differences between one “species” and another are merely differences of proportion, of relative magnitude, or (as he phrased it) of “excess and defect.” “Save only for a difference in the way of excess or defect, the parts are identical in the case of such animals as are of one and the same genus; and by ‘genus’ I mean, for instance, Bird or Fish.” And again: “Within the limits of the same genus, as a general rule, most of the parts exhibit differences ... in the way of multitude or fewness, magnitude or parvitude, in short, in the way of excess or defect. For ‘the more’ and ‘the less’ may be represented as {726} ‘excess’ and ‘defect[644].’ ” It is precisely this difference of relative magnitudes, this Aristotelian “excess and defect” in the case of form, which our co-ordinate method is especially adapted to analyse, and to reveal and demonstrate as the main cause of what (again in the Aristotelian sense) we term “specific” differences.
The applicability of our method to particular cases will depend upon, or be further limited by, certain practical considerations or qualifications. Of these the chief, and indeed the essential, condition is, that the form of the entire structure under investigation should be found to vary in a more or less uniform manner, after the fashion of an approximately homogeneous and isotropic body. But an imperfect isotropy, provided always that some “principle of continuity” run through its variations, will not seriously interfere with our method; it will only cause our transformed co-ordinates to be somewhat less regular and harmonious than are those, for instance, by which the physicist depicts the motions of a perfect fluid or a theoretic field of force in a uniform medium.
Again, it is essential that our structure vary in its entirety, or at least that “independent variants” should be relatively few. That independent variations occur, that localised centres of diminished or exaggerated growth will now and then be found, is not only probable but manifest; and they may even be so pronounced as to appear to constitute new formations altogether. Such independent variants as these Aristotle himself clearly recognised: “It happens further that some have parts that others have not; for instance, some [birds] have spurs and others not, some have crests, or combs, and others not; but, as a general rule, most parts and those that go to make up the bulk of the body are either identical with one another, or differ from one another in the way of contrast and of excess and defect. For ‘the more’ and ‘the less’ may be represented as ‘excess’ or ‘defect.’ ”
If, in the evolution of a fish, for instance, it be the case that its several and constituent parts—head, body, and tail, or this fin and that fin—represent so many independent variants, then our co-ordinate system will at once become too complex to be intelligible; we shall be making not one comparison but several {727} separate comparisons, and our general method will be found inapplicable. Now precisely this independent variability of parts and organs—here, there, and everywhere within the organism—would appear to be implicit in our ordinary accepted notions regarding variation; and, unless I am greatly mistaken, it is precisely on such a conception of the easy, frequent, and normal independent variability of parts that our conception of the process of natural selection is fundamentally based. For the morphologist, when comparing one organism with another, describes the differences between them point by point, and “character” by “character[645].” If he is from time to time constrained to admit the existence of “correlation” between characters (as a hundred years ago Cuvier first showed the way), yet all the while he recognises this fact of correlation somewhat vaguely, as a phenomenon due to causes which, except in rare instances, he can hardly hope to trace; and he falls readily into the habit of thinking and talking of evolution as though it had proceeded on the lines of his own descriptions, point by point, and character by character[646].
But if, on the other hand, diverse and dissimilar fishes can be referred as a whole to identical functions of very different co-ordinate systems, this fact will of itself constitute a proof that variation has proceeded on definite and orderly lines, that a comprehensive “law of growth” has pervaded the whole structure in its integrity, and that some more or less simple and recognisable system of forces has been at work. It will not only show how real and deep-seated is the phenomenon of “correlation,” in regard to form, but it will also demonstrate the fact that a correlation which had seemed too complex for analysis or {728} comprehension is, in many cases, capable of very simple graphic expression. This, after many trials, I believe to be in general the case, bearing always in mind that the occurrence of independent or localised variations must often be considered.
We are dealing in this chapter with the forms of related organisms, in order to shew that the differences between them are as a general rule simple and symmetrical, and just such as might have been brought about by a slight and simple change in the system of forces to which the living and growing organism was exposed. Mathematically speaking, the phenomenon is identical with one met with by the geologist, when he finds a bed of fossils squeezed flat or otherwise symmetrically deformed by the pressures to which they, and the strata which contain them, have been subjected. In the first step towards fossilisation, when the body of a fish or shellfish is silted over and buried, we may take it that the wet sand or mud exercises, approximately, a hydrostatic pressure—that is to say a pressure which is uniform in all directions, and by which the form of the buried object will not be appreciably changed. As the strata consolidate and accumulate, the fossil organisms which they contain will tend to be flattened by the vast superincumbent load, just as the stratum which contains them will also be compressed and will have its molecular arrangement more or less modified[647]. But the deformation due to direct vertical pressure in a horizontal stratum is not nearly so striking as are the deformations produced by the oblique or shearing stresses to which inclined and folded strata have been exposed, and by which their various “dislocations” have been brought about. And especially in mountain regions, where these dislocations are especially numerous and complicated, the contained fossils are apt to be so curiously and yet so symmetrically deformed (usually by a simple shear) that they may easily be interpreted as so many distinct and separate “species[648].” A great number of described species, and here and there a new genus (as the genus Ellipsolithes for an obliquely deformed Goniatite or Nautilus) are said to rest on no other foundation[649].
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If we begin by drawing a net of rectangular equidistant co-ordinates (about the axes _x_ and _y_), we may alter or _deform_ this {729} network in various ways, several of which are very simple indeed. Thus (1) we may alter the dimensions of our system, extending it along one or other axis, and so converting each little square into a corresponding and directly proportionate oblong (Fig. 353). It follows that any figure which we may have inscribed in the original net, and which we transfer to the new, will thereby be _deformed_ in strict proportion to the deformation of the entire configuration, being still defined by corresponding points in the network and being throughout in conformity with the original figure. For instance, a circle inscribed in the original “Cartesian” net will now, after extension in the _y_-direction, be found elongated {730} into an ellipse. In elementary mathematical language, for the original _x_ and _y_ we have substituted _x__{1} and _c_ _y__{1}, and the equation to our original circle, _x_^2 + _y_^2 = _a_^2, becomes that of the ellipse, _x__{1}^2 + _c_^2 _y__{1}^2 = _a_^2.
If I draw the cannon-bone of an ox (Fig. 354, A), for instance, within a system of rectangular co-ordinates, and then transfer the same drawing, point for point, to a system in which for the _x_ of the original diagram we substitute _x′_ = 2_x_/3, we obtain a drawing (B) which is a very close approximation to the cannon-bone of the sheep. In other words, the main (and perhaps the only) difference between the two bones is simply that that of the sheep is elongated, along the vertical axis, as compared with that of the ox in the relation of 3/2. And similarly, the long slender cannon-bone of the giraffe (C) is referable to the same identical type, subject to a reduction of breadth, or increase of length, corresponding to _x″_ = _x_/3.
(2) The second type is that where extension is not equal or uniform at all distances from the origin: but grows greater or less, as, for instance, when we stretch a _tapering_ elastic band. In such cases, as I have represented it in Fig. 355, the ordinate increases logarithmically, and for _y_ we substitute ε^{_y_}. It is obvious that this logarithmic extension may involve both abscissae and ordinates, _x_ becoming ε^{_x_}, while _y_ becomes ε^{_y_}. The circle in our original figure is now deformed into some such shape as that of Fig. 356. This method of deformation is a common one, and will often be of use to us in our comparison of organic forms.
(3) Our third type is the “simple shear,” where the rectangular co-ordinates become “oblique,” their axes being inclined to one another at a certain angle ω. Our original rectangle now becomes such a figure as that of Fig. 357. The system may now be described in terms of the oblique axes _X_, _Y_; or may be directly referred to new rectangular co-ordinates ξ, η by the simple transposition _x_ = ξ − η cot ω, _y_ = η cosec ω.
(4) Yet another important class of deformations may be represented by the use of radial co-ordinates, in which one set of lines are represented as radiating from a point or “focus,” while the other set are transformed into circular arcs cutting the radii orthogonally. These radial co-ordinates are especially applicable {731} to cases where there exists (either within or without the figure) some part which is supposed to suffer no deformation; a simple illustration is afforded by the diagrams which illustrate the flexure of a beam (Fig. 358). In biology these co-ordinates will be especially applicable in cases where the growing structure includes a “node,” or point where growth is absent or at a minimum; and about which node the rate of growth may be assumed to increase symmetrically. Precisely such a case is furnished us in a leaf of an ordinary dicotyledon. The leaf of a {732} typical monocotyledon—such as a grass or a hyacinth, for instance—grows continuously from its base, and exhibits no node or “point of arrest.” Its sides taper off gradually from its broad base to its slender tip, according to some law of decrement specific to the plant; and any alteration in the relative velocities of longitudinal and transverse growth will merely make the leaf a little broader or narrower, and will effect no other conspicuous alteration in its contour. But if there once come into existence a node, or “locus of no growth,” about which we may assume the growth—which in the hyacinth leaf was longitudinal and transverse—to take place radially and transversely to the radii, then we shall
at once see, in the first place, that the sloping and slightly curved sides of the hyacinth leaf suffer a transformation into what we consider a more typical and “leaf-like” shape, the sides of the figure broadening out to a zone of maximum breadth and then drawing inwards to the pointed apex. If we now alter the ratio between the radial and tangential velocities of growth—in other words, if we increase the angles between corresponding radii—we pass successively through the various configurations which the botanist describes as the lanceolate, the ovate, and finally the cordate leaf. These successive changes may to some extent, and in appropriate cases, be traced as the individual leaf grows {733} to maturity; but as a much more general rule, the balance of forces, the ratio between radial and tangential velocities of growth, remains so nicely and constantly balanced that the leaf increases in size without conspicuous modification of form. It is rather what we may call a long-period variation, a tendency for the relative velocities to alter from one generation to another, whose result is brought into view by this method of illustration.
There are various corollaries to this method of describing the form of a leaf which may be here alluded to, for we shall not return again to the subject of radial co-ordinates. For instance, the so-called unsymmetrical leaf[650] of a begonia, in which one side of the leaf may be merely ovate while the other has a cordate outline, is seen to be really a case of _unequal_, and not truly asymmetrical, growth on either side of the midrib. There is nothing more mysterious in its conformation than, for instance, in that of a forked twig in which one limb of the fork has grown longer than the other. The case of the begonia leaf is of sufficient interest to deserve illustration, and in Fig. 360 I have outlined a leaf of the large _Begonia daedalea_. On the smaller left-hand side of the leaf I have taken at random three points, _a_, _b_, _c_, and have measured the angles, _AOa_, etc., which the radii from the hilus of the leaf to these points make with the median axis. On the other side of the leaf I have marked the points _a′_, _b′_, _c′_, such that the radii drawn to this margin of the leaf are equal to the former, _Oa′_ to _Oa_, etc. Now if the two sides of the leaf are {734} mathematically similar to one another, it is obvious that the respective angles should be in continued proportion, i.e. as _AOa_ is to _AOa′_, so should _AOb_ be to _AOb′_. This proves to be very nearly the case. For I have measured the three angles on one side, and one on the other, and have then compared, as follows, the calculated with the observed values of the other two:
_AOa_ _AOb_ _AOc_ _AOa′_ _AOb′_ _AOc′_ Observed values 12° 28.5° 88° — — 157° Calculated values — — — 21.5° 51.1° — Observed values — — — 20 52 —
The agreement is very close, and what discrepancy there is may be amply accounted for, firstly, by the slight irregularity of the sinuous margin of the leaf; and secondly, by the fact that the true axis or midrib of the leaf is not straight but slightly curved, and therefore that it is curvilinear and not rectilinear triangles which we ought to have measured. When we understand these few points regarding the peripheral curvature of the leaf, it is easy to see that its principal veins approximate closely to a beautiful system of isogonal co-ordinates. It is also obvious that we can easily pass, by a process of shearing, from those cases where the principal veins start from the base of the leaf to those, as in most dicotyledons, where they arise successively from the midrib.
It may sometimes happen that the node, or “point of arrest,” is at the upper instead of the lower end of the leaf-blade; and occasionally there may be a node at both ends. In the former case, as we have it in the daisy, the form of the leaf will be, as it were, inverted, the broad, more or less heart-shaped, outline appearing at the upper end, while below the leaf tapers gradually downwards to an ill-defined base. In the latter case, as in _Dionaea_, we obtain a leaf equally expanded, and similarly ovate or cordate, at both ends. We may notice, lastly, that the shape of a solid fruit, such as an apple or a cherry, is a solid of revolution, developed from similar curves and to be explained on the same principle. In the cherry we have a “point of arrest” at the base of the berry, where it joins its peduncle, and about this point the fruit (in imaginary section) swells out into a cordate outline; while in the {735} apple we have two such well-marked points of arrest, above and below, and about both of them the same conformation tends to arise. The bean and the human kidney owe their “reniform” shape to precisely the same phenomenon, namely, to the existence of a node or “hilus,” about which the forces of growth are radially and symmetrically arranged.
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Most of the transformations which we have hitherto considered (other than that of the simple shear) are particular cases of a general transformation, obtainable by the method of conjugate functions and equivalent to the projection of the original figure on a new plane. Appropriate transformations, on these general lines, provide for the cases of a coaxial system where the Cartesian co-ordinates are replaced by coaxial circles, or a confocal system in which they are replaced by confocal ellipses and hyperbolas.
Yet another curious and important transformation, belonging to the same class, is that by which a system of straight lines becomes transformed into a conformal system of logarithmic spirals: the straight line _Y_ − _AX_ = _c_ corresponding to the logarithmic spiral θ − _A_ log _r_ = _c_ (Fig. 361). This beautiful and simple transformation lets us at once convert, for instance, the straight conical shell of the Pteropod or the _Orthoceras_ into the logarithmic spiral of the Nautiloid; it involves a mathematical symbolism which is but a slight extension of that which we have employed in our elementary treatment of the logarithmic spiral.
These various systems of coordinates, which we have now briefly considered, are sometimes called “isothermal co-ordinates,” from the fact that, when employed in this particular branch of physics, they perfectly represent the phenomena of the conduction of heat, the contour lines of equal temperature appearing, under appropriate conditions, as the orthogonal lines of the co-ordinate system. And it follows that {736} the “law of growth” which our biological analysis by means of orthogonal co-ordinate systems presupposes, or at least foreshadows, is one according to which the organism grows or develops along _stream lines_, which may be defined by a suitable mathematical transformation.
When the system becomes no longer orthogonal, as in many of the following illustrations—for instance, that of _Orthagoriscus_ (Fig. 382),—then the transformation is no longer within the reach of comparatively simple mathematical analysis. Such departure from the typical symmetry of a “stream-line” system is, in the first instance, sufficiently accounted for by the simple fact that the developing organism is very far from being homogeneous and isotropic, or, in other words, does not behave like a perfect fluid. But though under such circumstances our co-ordinate systems may be no longer capable of strict mathematical analysis, they will still indicate _graphically_ the relation of the new co-ordinate system to the old, and conversely will furnish us with some guidance as to the “law of growth,” or play of forces, by which the transformation has been effected.
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Before we pass from this brief discussion of transformations in general, let us glance at one or two cases in which the forces applied are more or less intelligible, but the resulting transformations are, from the mathematical point of view, exceedingly complicated.
The “marbled papers” of the bookbinder are a beautiful illustration of visible “stream lines.” On a dishful of a sort of semi-liquid gum the workman dusts a few simple lines or patches of colouring matter; and then, by passing a comb through the liquid, he draws the colour-bands into the streaks, waves, and spirals which constitute the marbled pattern, and which he then transfers to sheets of paper laid down upon the gum. By some such system of shears, by the effect of unequal traction or unequal growth in various directions and superposed on an originally simple pattern, we may account for the not dissimilar marbled patterns which we recognise, for instance, on a large serpent’s skin. But it must be remarked, in the case of the marbled paper, that though the method of application of the forces is simple, yet in the aggregate the system of forces set up by the many {737} teeth of the comb is exceedingly complex, and its complexity is revealed in the complicated “diagram of forces” which constitutes the pattern.
To take another and still more instructive illustration. To turn one circle (or sphere) into two circles would be, from the point of view of the mathematician, an extraordinarily difficult transformation; but, physically speaking, its achievement may be extremely simple. The little round gourd grows naturally, by its symmetrical forces of expansive growth, into a big, round, or somewhat oval pumpkin or melon. But the Moorish husbandman ties a rag round its middle, and the same forces of growth, unaltered save for the presence of this trammel, now expand the globular structure into two superposed and connected globes. And again, by varying the position of the encircling band, or by applying several such ligatures instead of one, a great variety of artificial forms of “gourd” may be, and actually are, produced. It is clear, I think, that we may account for many ordinary biological processes of development or transformation of form by the existence of trammels or lines of constraint, which limit and determine the action of the expansive forces of growth that would otherwise be uniform and symmetrical. This case has a close parallel in the operations of the glassblower, to which we have already, more than once, referred in passing[651]. The glassblower starts his operations with a _tube_, which he first closes at one end so as to form a hollow vesicle, within which his blast of air exercises a uniform pressure on all sides; but the spherical conformation which this uniform expansive force would naturally tend to produce is modified into all kinds of forms by the trammels or resistances set up as the workman lets one part or another of his bubble be unequally heated or cooled. It was Oliver Wendell Holmes who first shewed this curious parallel between the operations of the glassblower and those of Nature, when she starts, as she so often does, with a simple tube[652]. The alimentary canal, {738} the arterial system including the heart, the central nervous system of the vertebrate, including the brain itself, all begin as simple tubular structures. And with them Nature does just what the glassblower does, and, we might even say, no more than he. For she can expand the tube here and narrow it there; thicken its walls or thin them; blow off a lateral offshoot or caecal diverticulum; bend the tube, or twist and coil it; and infold or crimp its walls as, so to speak, she pleases. Such a form as that of the human stomach is easily explained when it is regarded from this point of view; it is simply an ill-blown bubble, a bubble that has been rendered lopsided by a trammel or restraint along one side, such as to prevent its symmetrical expansion—such a trammel as is produced if the glassblower lets one side of his bubble get cold, and such as is actually present in the stomach itself in the form of a muscular band.
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We may now proceed to consider and illustrate a few permutations or transformations of organic form, out of the vast multitude which are equally open to this method of inquiry.
We have already compared in a preliminary fashion the metacarpal or cannon-bone of the ox, the sheep, and the giraffe (Fig. 354); and we have seen that the essential difference in form between these three bones is a matter of relative length and breadth, such that, if we reduce the figures to an identical standard of length (or identical values of _y_), the breadth (or value of _x_) will be approximately two-thirds that of the ox in the case of the sheep and one-third that of the ox in the case of the giraffe. We may easily, for the sake of closer comparison, determine these ratios more accurately, for instance, if it be our purpose to compare the different racial varieties within the limits of a single species. And in such cases, by the way, as when we compare with one another various breeds or races of cattle or of horses, the ratios {739} of length and breadth in this particular bone are extremely significant[653].
If, instead of limiting ourselves to the cannon-bone, we inscribe the entire foot of our several Ungulates in a co-ordinate system, the same ratios of _x_ that served us for the cannon-bones still give us a first approximation to the required comparison; but even in the case of such closely allied forms as the ox and the sheep there is evidently something wanting in the comparison. The reason is that the relative elongation of the several parts, or individual bones, has not proceeded equally or proportionately in all cases; in other words, that the equations for _x_ will not suffice without some simultaneous modification of the values of _y_ (Fig. 362). In such a case it may be found possible to satisfy the varying values of _y_ by some logarithmic or other formula; but, even if that be possible, it will probably be somewhat difficult of discovery or verification in such a case as the present, owing to the fact that we have too few well-marked points of correspondence between the one object and the other, and that especially along the shaft of such long bones as the cannon-bone of the ox, the deer, the llama, or the giraffe there is a complete lack of easily recognisable corresponding points. In such a case a brief tabular statement of apparently corresponding values of _y_, or of those obviously corresponding values which coincide with the boundaries of the several bones of the foot, will, as in the following example, enable us to dispense with a fresh equation.
_a_ _b_ _c_ _d_ _y_ (Ox) 0 18 27 42 100 _y′_ (Sheep) 0 10 19 36 100 _y″_ (Giraffe) 0 5 10 24 100
This summary of values of _y′_, coupled with the equations for the {740} value of _x_, will enable us, from any drawing of the ox’s foot, to construct a figure of that of the sheep or of the giraffe with remarkable accuracy.
That underlying the varying amounts of extension to which the parts or segments of the limb have been subject there is a law, or principle of continuity, may be discerned from such a diagram as the above (Fig. 363), where the values of _y_ in the case of the ox are plotted as a straight line, and the corresponding values for the sheep (extracted from the above table) are seen to form a more or less regular and even curve. This simple graphic result implies the existence of a comparatively simple equation between _y_ and _y′_.
An elementary application of the principle of co-ordinates to the study of proportion, as we have here used it to illustrate the varying proportions of a bone, was in common use in the sixteenth and seventeenth centuries by artists in their study of the human form. The method is probably much more ancient, and may even be classical[654]; it is fully described and put in practice by Albert Dürer in his _Geometry_, and especially in his _Treatise on Proportion_[655]. In this latter work, the manner in which the {741} human figure, features, and facial expression are all transformed and modified by slight variations in the relative magnitude of the parts is admirably and copiously illustrated (Fig. 364).
In a tapir’s foot there is a striking difference, and yet at the same time there is an obvious underlying resemblance, between the middle toe and either of its unsymmetrical lateral neighbours. Let us take the median terminal phalanx and inscribe its outline in a net of rectangular equidistant co-ordinates (Fig. 365, _a_). Let us then make a similar network about axes which are no longer at right angles, but inclined to one another at an angle of about 50° (_b_). If into this new network we fill in, point for point, an outline precisely corresponding to our original drawing of the middle toe, we shall find that we have already represented the main features of the adjacent lateral one. We shall, however, perceive that our new diagram looks a little too bulky on one side, the inner side, of the lateral toe. If now we substitute for our equidistant ordinates, ordinates which get gradually closer and closer together as we pass towards the median side of the toe, then we shall obtain a diagram which differs in no essential respect from an actual outline copy of the lateral toe (_c_). In short, the difference between the outline of the middle toe of the tapir and the next lateral toe may be almost completely expressed by saying that if the one be represented by rectangular equidistant co-ordinates, the other will be represented by oblique co-ordinates, whose axes make an angle of 50°, and in which the abscissal interspaces decrease in a certain logarithmic ratio. We treated our original complex curve or projection of the tapir’s toe as a function of the form _F_ (_x_, _y_) = 0. The figure of the tapir’s lateral {742} toe is a precisely identical function of the form _F_ (_e_^{_x_}, _y__{1}) = 0, where _x__{1}, _y__{1} are oblique co-ordinate axes inclined to one another at an angle of 50°.
Dürer was acquainted with these oblique co-ordinates also, and I have copied two illustrative figures from his book[656].
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In Fig. 367 I have sketched the common Copepod _Oithona nana_, {743} and have inscribed it in a rectangular net, with abscissae three-fifths the length of the ordinates. Side by side (Fig. 368) is drawn a very different Copepod, of the genus _Sapphirina_; and about it is drawn a network such that each co-ordinate passes (as nearly as possible) through points corresponding to those of the former figure. It will be seen that two differences are apparent. (1) The values of _y_ in Fig. 368 are large in the upper part of the figure, and diminish rapidly towards its base. (2) The values of _x_ are very large in the neighbourhood of the origin, but diminish rapidly as we pass towards either side, away from the median vertical axis; and it is probable that they do so according to a definite, but somewhat complicated, ratio. If, instead of seeking for an actual equation, we simply tabulate our values of _x_ and _y_ in the second figure as compared with the first (just as we did in comparing the feet of the Ungulates), we get the dimensions of a net in which, by simply projecting the figure of _Oithona_, we obtain that of _Sapphirina_ without further trouble, e.g.:
_x_ (_Oithona_) 0 3 6 9 12 15 — _x′_ (_Sapphirina_) 0 8 10 12 13 14 —
_y_ (_Oithona_) 0 5 10 15 20 25 30 _y′_ (_Sapphirina_) 0 2 7 3 23 32 40
In this manner, with a single model or type to copy from, we may record in very brief space the data requisite for the production of approximate outlines of a great number of forms. For instance the difference, at first sight immense, between the attenuated body of a _Caprella_ and the thick-set body of a _Cyamus_ is obviously little, and is probably nothing, more than a difference of relative magnitudes, capable of tabulation by numbers and of complete expression by means of rectilinear co-ordinates.
The Crustacea afford innumerable instances of more complex deformations. Thus we may compare various higher Crustacea with one another, even in the case of such dissimilar forms as a lobster and a crab. It is obvious that the whole body of the former is elongated as compared with the latter, and that the crab is relatively broad in the region of the carapace, while it tapers off rapidly towards its attenuated and abbreviated tail. In a general way, the elongated rectangular system of co-ordinates {744} in which we may inscribe the outline of the lobster becomes a shortened triangle in the case of the crab. In a little more detail we may compare the outline of the carapace in various crabs one with another: and the comparison will be found easy and significant, even, in many cases, down to minute details, such as the number and situation of the marginal spines, though these are in other cases subject to independent variability.
If we choose, to begin with, such a crab as _Geryon_ (Fig. 369, 1), and inscribe it in our equidistant rectangular co-ordinates, we shall see that we pass easily to forms more elongated in a transverse {745} direction, such as _Matuta_ or _Lupa_ (5), and conversely, by transverse compression, to such a form as _Corystes_ (2). In certain other cases the carapace conforms to a triangular diagram, more or less curvilinear, as in Fig. 4, which represents the genus _Paralomis_. Here we can easily see that the posterior border is transversely elongated as compared with that of _Geryon_, while at the same time the anterior part is longitudinally extended as compared with the posterior. A system of slightly curved and converging ordinates, with orthogonal and logarithmically interspaced abscissal lines, as shown in the figure, appears to satisfy the conditions.
In an interesting series of cases, such as the genus _Chorinus_, or _Scyramathia_, and in the spider-crabs generally, we appear to have just the converse of this. While the carapace of these crabs presents a somewhat triangular form, which seems at first sight more or less similar to those just described, we soon see that the actual posterior border is now narrow instead of broad, the broadest part of the carapace corresponding precisely, not to that which is broadest in _Paralomis_, but to that which was broadest in _Geryon_; while the most striking difference from the latter lies in an antero-posterior lengthening of the forepart of the carapace, culminating in a great elongation of the frontal region, with its two spines or “horns.” The curved ordinates here converge posteriorly and diverge widely in front (Figs. 3 and 6), while the decremental interspacing of the abscissae is very marked indeed.
We put our method to a severer test when we attempt to sketch an entire and complicated animal than when we simply compare corresponding parts such as the carapaces of various Malacostraca, or related bones as in the case of the tapir’s toes. Nevertheless, up to a certain point, the method stands the test very well. In other words, one particular mode and direction of variation is often (or even usually) so prominent and so paramount throughout the entire organism, that one comprehensive system of co-ordinates suffices to give a fair picture of the actual phenomenon. To take another illustration from the Crustacea, I have drawn roughly in Fig. 370, 1 a little amphipod of the family Phoxocephalidae (_Harpinia_ sp.). Deforming the co-ordinates of the figure into the {746} curved orthogonal system in Fig. 2, we at once obtain a very fair representation of an allied genus, belonging to a different family of amphipods, namely _Stegocephalus_. As we proceed further from our type our co-ordinates will require greater deformation, and the resultant figure will usually be somewhat less accurate. In Fig. 3 I show a network, to which, if we transfer our diagram of _Harpinia_ or of _Stegocephalus_, we shall obtain a tolerable representation of the aberrant genus _Hyperia_, with its narrow abdomen, its reduced pleural lappets, its great eyes, and its inflated head.
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The hydroid zoophytes constitute a “polymorphic” group, within which a vast number of species have already been distinguished; and the labours of the systematic naturalist are constantly adding to the number. The specific distinctions are for the most part based, not upon characters directly presented {747} by the living animal, but upon the form, size and arrangement of the little cups, or “calycles,” secreted and inhabited by the little individual polypes which compose the compound organism. The variations, which are apparently infinite, of these conformations are easily seen to be a question of relative magnitudes, and are capable of complete expression, sometimes by very simple, sometimes by somewhat more complex, co-ordinate networks.
For instance, the varying shapes of the simple wineglass-shaped cups of the Campanularidae are at once sufficiently represented and compared by means of simple Cartesian co-ordinates (Fig. 371). In the two allied families of Plumulariidae and Aglaopheniidae the calycles are set unilaterally upon a jointed stem, and small cup-like structures (holding rudimentary polypes) are associated with the large calycles in definite number and position. These small calyculi are variable in number, but in the great majority of cases they accompany the large calycle in groups of three—two standing by its upper border, and one, which is especially variable in form and magnitude, lying at its base. The stem is liable to flexure and, in a high degree, to extension or compression; and these variations extend, often on an exaggerated scale, to the related calycles. As a result we find that we can draw various systems of curved or sinuous co-ordinates, which express, all but completely, the configuration of the various {748} hydroids which we inscribe therein (Fig. 372). The comparative smoothness or denticulation of the margin of the calycle, and the number of its denticles, constitutes an independent variation, and requires separate description; we have already seen (p. 236) that this denticulation is in all probability due to a particular physical cause.
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Among the fishes we discover a great variety of deformations, some of them of a very simple kind, while others are more striking and more unexpected. A comparatively simple case, involving a simple shear, is illustrated by Figs. 373 and 374. Fig. 373 represents, within Cartesian co-ordinates, a certain little oceanic fish known as _Argyropelecus Olfersi_. Fig. 474 represents precisely the same outline, transferred to a system of oblique co-ordinates whose {749} axes are inclined at an angle of 70°; but this is now (as far as can be seen on the scale of the drawing) a very good figure of an allied fish, assigned to a different genus, under the name of _Sternoptyx diaphana_. The deformation illustrated by this case of _Argyropelecus_ is precisely analogous to the simplest and commonest kind of deformation to which fossils are subject (as we have seen on p. 553) as the result of shearing-stresses in the solid rock.
Fig. 375 is an outline diagram of a typical Scaroid fish. Let us deform its rectilinear co-ordinates into a system of (approximately) coaxial circles, as in Fig. 376, and then filling into the new system, space by space and point by point, our former diagram of _Scarus_, we obtain a very good outline of an allied fish, belonging to a neighbouring family, of the genus _Pomacanthus_. This case is all the more interesting, because upon the body of our _Pomacanthus_ there are striking colour bands, which correspond in direction very closely to the lines of our new curved ordinates. In like manner, the still more bizarre outlines of other fishes of the same family of Chaetodonts will be found to correspond to very slight modifications of similar co-ordinates; in other words, to small variations in the values of the constants of the coaxial curves.
In Figs. 377–380 I have represented another series of Acanthopterygian fishes, not very distantly related to the foregoing. If we start this series with the figure of _Polyprion_, in Fig. 377, we see that the outlines of _Pseudopriacanthus_ (Fig. 378) and of _Sebastes_ or _Scorpaena_ (Fig. 379) are easily derived by substituting a system of triangular, or radial, co-ordinates for the rectangular ones in {750} which we had inscribed _Polyprion_. The very curious fish _Antigonia capros_, an oceanic relative of our own “boar-fish,” conforms closely to the peculiar deformation represented in Fig. 380.
Fig. 381 is a common, typical _Diodon_ or porcupine-fish, and in Fig. 382 I have deformed its vertical co-ordinates into a system of concentric circles, and its horizontal co-ordinates into a system of curves which, approximately and provisionally, are made to resemble a system of hyperbolas[657]. The old outline, transferred {751} in its integrity to the new network, appears as a manifest representation of the closely allied, but very different looking, sunfish, _Orthagoriscus mola_. This is a particularly instructive case of deformation or transformation. It is true that, in a mathematical sense, it is not a perfectly satisfactory or perfectly regular deformation, for the system is no longer isogonal; but nevertheless, it is symmetrical to the eye, and obviously approaches to an isogonal system under certain conditions of friction or constraint. And as such it accounts, by one single integral transformation, for all the apparently separate and distinct external differences between the two fishes. It leaves the parts near to the origin of the system, the whole region of the head, the opercular orifice and the pectoral fin, practically unchanged {752} in form, size and position; and it shews a greater and greater apparent modification of size and form as we pass from the origin towards the periphery of the system.
In a word, it is sufficient to account for the new and striking contour in all its essential details, of rounded body, exaggerated dorsal and ventral fins, and truncated tail. In like manner, and using precisely the same co-ordinate networks, it appears to me possible to shew the relations, almost bone for bone, of the skeletons of the two fishes; in other words, to reconstruct the skeleton of the one from our knowledge of the skeleton of the other, under the guidance of the same correspondence as is indicated in their external configuration.
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The family of the crocodiles has had a special interest for the evolutionist ever since Huxley pointed out that, in a degree only second to the horse and its ancestors, it furnishes us with a close and almost unbroken series of transitional forms, running down in continuous succession from one geological formation to another. I should be inclined to transpose this general statement into other terms, and to say that the Crocodilia constitute a case in which, with unusually little complication from the presence of independent variants, the trend of one particular mode of transformation is visibly manifested. If we exclude meanwhile from our comparison a few of the oldest of the crocodiles, such as _Belodon_, which differ more fundamentally from the rest, we shall find a long series of genera in which we can refer not only the changing contours of the skull, but even the shape and size of the many constituent bones and their intervening spaces or “vacuities,” to one and the same simple system of transformed co-ordinates. The manner in which the skulls of various Crocodilians differ from one another may be sufficiently illustrated by three or four examples.
Let us take one of the typical modern crocodiles as our standard of form, e.g. _C. porosus_, and inscribe it, as in Fig. 383, _a_, in the usual Cartesian co-ordinates. By deforming the rectangular network into a triangular system, with the apex of the triangle a little way in front of the snout, as in _b_, we pass to such a form as _C. americanus_. By an exaggeration of the same process we at once get an approximation to the form of one of the sharp-snouted, {753} or longirostrine, crocodiles, such as the genus _Tomistoma_; and, in the species figured, the oblique position of the orbits, the arched contour of the occipital border, and certain other characters suggest a certain amount of curvature, such as I have represented in the diagram (Fig. 383, _b_), on the part of the horizontal co-ordinates. In the still more elongated skull of such a form as the Indian Gavial, the whole skull has undergone a great longitudinal extension, or, in other words, the ratio of _x_/_y_ is greatly diminished; and this extension is not uniform, but is at a maximum in the region of the nasal and maxillary bones. This especially elongated region is at the same time narrowed in an exceptional degree, and its excessive narrowing is represented by a curvature, convex towards the median axis, on the part of the vertical ordinates. Let us take as a last illustration one of the Mesozoic crocodiles, the little _Notosuchus_, from the Cretaceous formation. This little crocodile is very different from our type in the proportions of its skull. The region of the snout, in front of and including the frontal bones, is greatly shortened; from constituting fully two-thirds of the whole length of the skull in _Crocodilus_, it now constitutes less than half, or, say, three-sevenths of the whole; and the whole skull, and especially its posterior part, is curiously compact, broad, and squat. The orbit is unusually large. If in the diagram of this skull we select a number of points obviously corresponding {754} to points where our rectangular co-ordinates intersect particular bones or other recognisable features in our typical crocodile, we shall easily discover that the lines joining these points in _Notosuchus_ fall into such a co-ordinate network as that which is represented in Fig. 383, _c_. To all intents and purposes, then, this not very complex system, representing one harmonious “deformation,” accounts for _all_ the differences between the two figures, and is sufficient to enable one at any time to reconstruct a detailed drawing, bone for bone, of the skull of _Notosuchus_ from the model furnished by the common crocodile.
The many diverse forms of Dinosaurian reptiles, all of which manifest a strong family likeness underlying much superficial diversity, furnish us with plentiful material for comparison by the method of transformations. As an instance, I have figured the pelvic bones of _Stegosaurus_ and of _Camptosaurus_ (Fig. 384, _a_, _b_) to show that, when the former is taken as our Cartesian type, a slight curvature and an approximately logarithmic extension of the _x_-axis brings us easily to the configuration of the other. In the original specimen of _Camptosaurus_ described by Marsh[658], the anterior portion of the iliac bone is missing; and in Marsh’s restoration this part of the bone is drawn as though it came somewhat abruptly to a sharp point. In my figure I {755} have completed this missing part of the bone in harmony with the general co-ordinate network which is suggested by our comparison of the two entire pelves; and I venture to think that the result is more natural in appearance, and more likely to be correct than was Marsh’s conjectural restoration. It would seem, in fact, that there is an obvious field for the employment of the method of co-ordinates in this task of reproducing missing portions of a structure to the proper scale and in harmony with related types. To this subject we shall presently return.
In Fig. 385, _a_, _b_, I have drawn the shoulder-girdle of _Cryptocleidus_, a Plesiosaurian reptile, half-grown in the one case and full-grown in the other. The change of form during growth in this region of the body is very considerable, and its nature is well brought out by the two co-ordinate systems. In Fig. 386 I have drawn the shoulder-girdle of an Ichthyosaur, referring it to _Cryptocleidus_ as a standard of comparison. The interclavicle, which is present in _Ichthyosaurus_, is minute and hidden in _Cryptocleidus_; but the numerous other differences between the two {756} forms, chief among which is the great elongation in _Ichthyosaurus_ of the two clavicles, are all seen by our diagrams to be part and parcel of one general and systematic deformation.
Before we leave the group of reptiles we may glance at the very strangely modified skull of _Pteranodon_, one of the extinct flying reptiles, or Pterosauria. In this very curious skull the region of the jaws, or beak, is greatly elongated and pointed; the occipital bone is drawn out into an enormous backwardly-directed crest; the posterior part of the lower jaw is similarly produced backwards; the orbit is small; and the quadrate bone is strongly inclined downwards and forwards. The whole skull has a configuration which stands, apparently, in the strongest possible contrast to that of a more normal Ornithosaurian such as _Dimorphodon_. But if we inscribe the latter in Cartesian coordinates (Fig. 387, _a_), and refer our _Pteranodon_ to a system of oblique co-ordinates (_b_), in which the two co-ordinate systems of parallel lines become each a pencil of diverging rays, we make manifest a correspondence which extends uniformly throughout all parts of these very different-looking skulls.
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We have dealt so far, and for the most part we shall continue to deal, with our co-ordinate method as a means of comparing one known structure with another. But it is obvious, as I have said, {757} that it may also be employed for drawing hypothetical structures, on the assumption that they have varied from a known form in some definite way. And this process may be especially useful, and will be most obviously legitimate, when we apply it to the particular case of representing intermediate stages between two forms which are actually known to exist, in other words, of reconstructing the transitional stages through which the course of evolution must have successively travelled if it has brought about the change from some ancestral type to its presumed descendant. Some little time ago I sent to my friend, Mr Gerhard Heilmann of Copenhagen, a few of my own rough co-ordinate diagrams, including some in which the pelves of certain ancient and primitive birds were compared one with another. Mr Heilmann, who is both a skilled draughtsman and an able morphologist, returned me a set of diagrams which are a vast improvement on my own, {758} and which are reproduced in Figs. 388–393. Here we have, as extreme cases, the pelvis of _Archaeopteryx_, the most ancient of known birds, and that of _Apatornis_, one of the fossil “toothed”
birds from the North American Cretaceous formations—a bird shewing some resemblance to the modern terns. The pelvis of _Archaeopteryx_ is taken as our type, and referred accordingly to {759} Cartesian co-ordinates (Fig. 388); while the corresponding coordinates of the very different pelvis of _Apatornis_ are represented in Fig. 389. In Fig. 390 the outlines of these two co-ordinate systems are superposed upon one another, and those of three intermediate and equidistant co-ordinate systems are interpolated between them. From each of these latter systems, so determined by direct interpolation, a complete co-ordinate diagram is drawn, and the corresponding outline of a pelvis is found from each of these systems of co-ordinates, as in Figs. 391, 392. Finally, in Fig. 393 the complete series is represented, beginning with the known pelvis of _Archaeopteryx_, and leading up by our three intermediate hypothetical types to the known pelvis of _Apatornis_.
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Among mammalian skulls I will take two illustrations only, one drawn from a comparison of the human skull with that of the higher apes, and another from the group of Perissodactyle {760} Ungulates, the group which includes the rhinoceros, the tapir, and the horse.
Let us begin by choosing as our type the skull of _Hyrachyus agrarius_, Cope, from the Middle Eocene of North America, as figured by Osborn in his Monograph of the Extinct Rhinoceroses[659] (Fig. 394).
The many other forms of primitive rhinoceros described in the monograph differ from _Hyrachyus_ in various details—in the characters of the teeth, sometimes in the number of the toes, and so forth; and they also differ very considerably in the general {761} appearance of the skull. But these differences in the conformation of the skull, conspicuous as they are at first sight, will be found easy to bring under the conception of a simple and homogeneous transformation, such as would result from the application of some not very complicated stress. For instance, the corresponding co-ordinates of _Aceratherium tridactylum_, as shown in Fig. 395, indicate that the essential difference between this skull and the former one may be summed up by saying that the long axis of the skull of _Aceratherium_ has undergone a slight double curvature, while the upper parts of the skull have at the same time been {762} subject to a vertical expansion, or to growth in somewhat greater proportion than the lower parts. Precisely the same changes, on a somewhat greater scale, give us the skull of an existing rhinoceros.
Among the species of _Aceratherium_, the posterior, or occipital, view of the skull presents specific differences which are perhaps more conspicuous than those furnished by the side view; and these differences are very strikingly brought out by the series of conformal transformations which I have represented in Fig. 396. In this case it will perhaps be noticed that the correspondence is not always quite accurate in small details. It could easily have been made much more accurate by giving a slightly sinuous curvature to certain of the co-ordinates. But as they stand, the correspondence indicated is very close, and the simplicity of the figures illustrates all the better the general character of the transformation.
By similar and not more violent changes we pass easily to such allied forms as the Titanotheres (Fig. 397); and the well-known series of species of _Titanotherium_, by which Professor Osborn has {763} illustrated the evolution of this genus, constitutes a simple and suitable case for the application of our method.
But our method enables us to pass over greater gaps than these, and to discern the general, and to a very large extent even the detailed, resemblances between the skull of the rhinoceros and those of the tapir or the horse. From the Cartesian co-ordinates in which we have begun by inscribing the skull of a primitive rhinoceros, we pass to the tapir’s skull (Fig. 398), firstly, by converting the rectangular into a triangular network, by which we represent the depression of the anterior and the progressively increasing elevation of the posterior part of the skull; and secondly, by giving to the vertical ordinates a curvature such as to bring about a certain longitudinal compression, or condensation, in the forepart of the skull, especially in the nasal and orbital regions.
The conformation of the horse’s skull departs from that of our primitive Perissodactyle (that is to say our early type of rhinoceros, _Hyrachyus_) in a direction that is nearly the opposite of that taken by _Titanotherium_ and by the recent species of rhinoceros. For we perceive, by Fig. 399, that the horizontal co-ordinates, which in these latter cases became transformed into curves with the concavity upwards, are curved, in the case of the horse, in the opposite direction. And the vertical ordinates, which are also curved, somewhat in the same fashion as in the tapir, are very nearly equidistant, instead of being, as in that animal, crowded together anteriorly. Ordinates and abscissae form an oblique {764} system, as is shown in the figure. In this case I have attempted to produce the network beyond the region which is actually required to include the diagram of the horse’s skull, in order to show better the form of the general transformation, with a part only of which we have actually to deal.
It is at first sight not a little surprising to find that we can pass, by a cognate and even simpler transformation, from our Perissodactyle skulls to that of the rabbit; but the fact that we can easily do so is a simple illustration of the undoubted affinity which exists between the Rodentia, especially the family of the Leporidae, and the more primitive Ungulates. For my part, I would go further; for I think there is strong reason to believe that the Perissodactyles are more closely related to the Leporidae than the former are to the other Ungulates, or than the Leporidae are to the rest of the Rodentia. Be that as it may, it is obvious from Fig. 400 that the rabbit’s skull conforms to a system of {765} co-ordinates corresponding to the Cartesian co-ordinates in which we have inscribed the skull of _Hyrachyus_, with the difference, firstly, that the horizontal ordinates of the latter are transformed into equidistant curved lines, approximately arcs of circles, with their concavity directed downwards; and secondly, that the vertical ordinates are transformed into a pencil of rays approximately orthogonal to the circular arcs. In short, the configuration of the rabbit’s skull is derived from that of our primitive rhinoceros by the unexpectedly simple process of submitting the latter to a
strong and uniform flexure in the downward direction (cf. Fig. 358, p. 731). In the case of the rabbit the configuration of the individual bones does not conform quite so well to the general transformation as it does when we are comparing the several Perissodactyles one with another; and the chief departures from conformity will be found in the size of the orbit and in the outline of the immediately surrounding bones. The simple fact is that the relatively enormous eye of the rabbit constitutes an independent variation, which cannot be brought into the general and fundamental transformation, but must be dealt with {768} separately. The enlargement of the eye, like the modification in form and number of the teeth, is a separate phenomenon, which supplements but in no way contradicts our general comparison of the skulls taken in their entirety.
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Before we leave the Perissodactyla and their allies, let us look a little more closely into the case of the horse and its immediate relations or ancestors, doing so with the help of a set of diagrams which I again owe to Mr Gerard Heilmann[660]. Here we start afresh, with the skull (Fig. 402, _A_) of _Hyracotherium_ (or _Eohippus_), inscribed in a simple Cartesian network. At the other end of the series (_H_) is a skull of Equus, in its own corresponding network; and the intermediate stages (_B_–_G_) are all drawn by direct and simple interpolation, as in Mr Heilmann’s former series of drawings of _Archaeopteryx_ and _Apatornis_. In this present case, the relative magnitudes are shewn, as well as the forms, of the several skulls. Alongside of these reconstructed diagrams, are set figures of certain extinct “horses” (Equidae or Palaeotheriidae), and in two cases, viz. _Mesohippus_ and _Protohippus_ (_M_, _P_), it will be seen that the actual fossil skull coincides in the most perfect fashion with one of the hypothetical forms or stages which our method shews to be implicitly involved in the transition from _Hyracotherium_ to _Equus_. In a third case, that of _Parahippus_ (_Pa_), the correspondence (as Mr Heilmann points out) is by no means exact. The outline of this skull comes nearest to that of the hypothetical transition stage _D_, but the “fit” is now a bad one; for the skull of _Parahippus_ is evidently a longer, straighter and narrower skull, and differs in other minor characters besides. In short, though some writers have placed _Parahippus_ in the direct line of descent between _Equus_ and _Eohippus_, we see at once that there is no place for it there, and that it must, accordingly, represent a somewhat divergent branch or offshoot of the Equidae[661]. It may be noticed, especially in the case of _Protohippus_ {769} (_P_), that the configuration of the angle of the jaw does not tally quite so accurately with that of our hypothetical diagrams as do other parts of the skull. As a matter of fact, this region is somewhat variable, in different species of a genus, and even in different individuals of the same species; in the small figure (_Pp_) of _Protohippus placidus_ the correspondence is more exact.
In considering this series of figures we cannot but be struck, not only with the regularity of the succession of “transformations,” but also with the slight and inconsiderable differences which separate the known and recorded stages, and even the two extremes of the whole series. These differences are no greater (save in regard to actual magnitude) than those between one human skull and another, at least if we take into account the older or remoter races; and they are again no greater, but if anything less, than the range of variation, racial and individual, in certain other human bones, for instance the scapula[662].
The variability of this latter bone is great, but it is neither {770} surprising nor peculiar; for it is linked with all the considerations of mechanical efficiency and functional modification which we dealt with in our last chapter. The scapula occupies, as it were, a focus in a very important field of force; and the lines of force converging on it will be very greatly modified by the varying development of the muscles over a large area of the body and of the uses to which they are habitually put.
Let us now inscribe in our Cartesian co-ordinates the outline of a human skull (Fig. 404), for the purpose of comparing it with the skulls of some of the higher apes. We know beforehand that the main differences between the human and the simian types depend upon the enlargement or expansion of the brain and braincase in man, and the relative diminution or enfeeblement of his jaws. Together with these changes, the “facial angle” increases from an oblique angle to nearly a right angle in man, {771} and the configuration of every constituent bone of the face and skull undergoes an alteration. We do not know to begin with, and we are not shewn by the ordinary methods of comparison, how far these various changes form part of one harmonious and congruent transformation, or whether we are to look, for instance, upon the changes undergone by the frontal, the occipital, the maxillary, and the mandibular regions as a congeries of separate modifications or independent variants. But as soon as we have marked out a number of points in the gorilla’s or chimpanzee’s skull, corresponding with those which our co-ordinate network intersected in the human skull, we find that these corresponding points may be at once linked up by smoothly curved lines of intersection, which form a new system of co-ordinates and constitute a simple “projection” of our human skull. The network represented in Fig. 405 constitutes such a projection of the human skull on what we may call, figuratively speaking, the “plane” of the chimpanzee; and the full diagram in Fig. 406 demonstrates the correspondence. In Fig. 407 I have shewn the similar deformation in the case of a baboon, and it is obvious that the transformation is of precisely the same order, and differs only in an increased intensity or degree of deformation.
In both dimensions, as we pass from above downwards and from behind forwards, the corresponding areas of the network are seen to increase in a gradual and approximately logarithmic order in the lower as compared with the higher type of skull; and, in short, it becomes at once manifest that the modifications of jaws, braincase, and the regions between are all portions of one continuous and integral process. It is of course easy to draw the {772} inverse diagrams, by which the Cartesian co-ordinates of the ape are transformed into curvilinear and non-equidistant co-ordinates in man.
From this comparison of the gorilla’s or chimpanzee’s with the human skull we realise that an inherent weakness underlies the anthropologist’s method of comparing skulls by reference to a small number of axes. The most important of these are the “facial” and “basicranial” axes, which include between them the “facial angle.” But it is, in the first place, evident that these axes are merely the principal axes of a system of co-ordinates, and that their restricted and isolated use neglects all that can be learned from the filling in of the rest of the co-ordinate network. And, in the second place, the “facial axis,” for instance, as ordinarily used in the anthropological comparison of one human skull with another, or of the human skull with the gorilla’s, is in all cases treated as a straight line; but our investigation has shewn that rectilinear axes only meet the case in the simplest and most closely related transformations; and that, for instance, in the anthropoid skull no rectilinear axis is homologous with a rectilinear axis in a man’s skull, but what is a straight line in the one has become a certain definite curve in the other.
Mr Heilmann tells me that he has tried, but without success, to obtain a transitional series between the human skull and some prehuman, anthropoid type, which series (as in the case of the Equidae) should be found to contain other known types in direct linear sequence. It appears impossible, however, to obtain such a series, or to pass by successive and continuous gradations through such forms as Mesopithecus, Pithecanthropus, _Homo neanderthalensis_, and the lower or higher races of modern man. The failure is not the fault of our method. It merely indicates that no one straight line of descent, or of consecutive transformation, exists; but on the contrary, that among human and anthropoid types, recent and extinct, we have to do with a complex problem of divergent, rather than of continuous, variation. And in like manner, easy as it is to correlate the baboon’s and chimpanzee’s skulls severally with that of man, and easy as it is to see that the chimpanzee’s skull is much nearer to the human type than is the baboon’s, it is also not difficult to perceive that the series is not, {773} strictly speaking, continuous, and that neither of our two apes lies _precisely_ on the same direct line or sequence of deformation by which we may hypothetically connect the other with man.
As a final illustration I have drawn the outline of a dog’s skull (Fig. 408), and inscribed it in a network comparable with the Cartesian network of the human skull in Fig. 404. Here we attempt to bridge over a wider gulf than we have crossed in any of our former comparisons. But, nevertheless, it is obvious that our method still holds good, in spite of the fact that there are various specific differences, such as the open or closed orbit, etc., which have to be separately described and accounted for. We see that the chief essential differences in plan between the dog’s skull and the man’s lie in the fact that, relatively speaking, the former tapers away in front, a triangular taking the place of a rectangular conformation; secondly, that, coincident with the tapering off, there is a progressive elongation, or pulling out, of the whole forepart of the skull; and lastly, as a minor difference, that the straight vertical ordinates of the human skull become curved, with their convexity directed forwards, in the dog. While the net result is that in the dog, just as in the chimpanzee, the brain-pan is smaller and the jaws are larger than in man, it is now conspicuously evident that the co-ordinate network of the ape is by no means intermediate between those which fit the other two. The mode of deformation is on different lines; and, while it may be correct to say that the chimpanzee and the baboon are more brute-like, it would be by no means accurate to assert that they are more dog-like, than man. {774}
In this brief account of co-ordinate transformations and of their morphological utility I have dealt with plane co-ordinates only, and have made no mention of the less elementary subject of co-ordinates in three-dimensional space. In theory there is no difficulty whatsoever in such an extension of our method; it is just as easy to refer the form of our fish or of our skull to the rectangular co-ordinates _x_, _y_, _z_, or to the polar co-ordinates ξ, η, ζ, as it is to refer their plane projections to the two axes to which our investigation has been confined. And that it would be advantageous to do so goes without saying; for it is the shape of the solid object, not that of the mere drawing of the object, that we want to understand; and already we have found some of our easy problems in solid geometry leading us (as in the case of the form of the bivalve and even of the univalve shell) quickly in the direction of co-ordinate analysis and the theory of conformal transformations. But this extended theme I have not attempted to pursue, and it must be left to other times, and to other hands. Nevertheless, let us glance for a moment at the sort of simple cases, the simplest possible cases, with which such an investigation might begin; and we have found our plane co-ordinate systems so easily and effectively applicable to certain fishes that we may seek among them for our first and tentative introduction to the three-dimensional field.
It is obvious enough that the same method of description and analysis which we have applied to one plane, we may apply to another: drawing by observation, and by a process of trial and error, our various cross-sections and the co-ordinate systems which seem best to correspond. But the new and important problem which now emerges is to _correlate_ the deformation or transformation which we discover in one plane with that which we have observed in another: and at length, perhaps, after grasping the general principles of such correlation, to forecast approximately what is likely to take place in the other two planes of reference when we are acquainted with one, that is to say, to determine the values along one axis in terms of the other two.
Let us imagine a common “round” fish, and a common “flat” fish, such as a haddock and a plaice. These two fishes are not as nicely adapted for comparison by means of plane co-ordinates as {775} some which we have studied, owing to the presence of essentially unimportant, but yet conspicuous differences in the position of the eyes, or in the number of the fins,—that is to say in the manner in which the continuous dorsal fin of the plaice appears in the haddock to be cut or scolloped into a number of separate fins. But speaking broadly, and apart from such minor differences as these, it is manifest that the chief factor in the case (so far as we at present see) is simply the broadening out of the plaice’s body, as compared with the haddock’s, in the dorso-ventral direction, that is to say, along the _y_ axis; in other words, the ratio _x_/_y_ is much less, (and indeed little more than half as great), in the haddock than in the plaice. But we also recognise at once that while the plaice (as compared with the haddock) is expanded in one direction, it is also flattened, or thinned out, in the other: _y_ increases, but _z_ diminishes, relatively to _x_. And furthermore, we soon see that this is a common or even a general phenomenon. The high, expanded body in our Antigonia or in our sun-fish is at the same time flattened or _compressed_ from side to side, in comparison with the related fishes which we have chosen as standards of reference or comparison; and conversely, such a fish as the skate, while it is expanded from side to side in comparison with a shark or dogfish, is at the same time flattened or _depressed_ in its vertical section. We proceed then, to enquire whether there be any simple relation of _magnitude_ discernible between these twin factors of expansion and compression; and the very fact that the two dimensions tend to vary _inversely_ already assures us that, in the general process of deformation, the _volume_ is less affected than are the _linear dimensions_. Some years ago, when I was studying the length-weight co-efficient in fishes (of which we have already spoken in Chap. III, p. 98), that is to say the coefficient _k_ in the formula _W_ = _k_ _L_^3, or _k_ = _W_/_L_^3, I was not a little surprised to find that _k_ was all but identical in two such different looking fishes as our haddock and our plaice: thus indicating that these two fishes, little as they resemble one another externally (though they belong to two closely related families), have approximately the same _volume_ when they are equal in _length_; or, in other words, that the extent to which the plaice’s body has become expanded or broadened is _just about {776} compensated for_ by the extent to which it has also got flattened or thinned. In short, if we could permit ourselves to conceive of a haddock being directly transformed into a plaice, a very large part of the change would be simply accounted for by supposing the former fish to be “rolled out,” as a baker rolls a piece of dough. This is, as it were, an extreme case of the _balancement des organes_, or “compensation of parts.”
Simple Cartesian co-ordinates will not suffice very well to compare the haddock with the plaice, for the deformation undergone by the former in comparison with the latter is more on the lines of that by which we have compared our Antigonia with our Polyprion; that is to say, the expansion is greater towards the middle of the fish’s length, and dwindles away towards either end. But again simplifying our illustration to the utmost, and being content with a rough comparison, we may assert that, when haddock and plaice are brought to the same standard of length, we can inscribe them both (approximately) in rectangular co-ordinate networks, such that _Y_ in the plaice is about twice as great as _y_ in the haddock. But if the volumes of the two fishes be equal, this is as much as to say that _xyz_ in the one case (or rather the summation of all these values) is equal to _XYZ_ in the other; and therefore (since _X_ = _x_, and _Y_ = 2_y_), it follows that _Z_ = _z_/2. When we have drawn our vertical transverse section of the haddock (or projected that fish in the _yz_ plane), we have reason accordingly to anticipate that we can draw a similar projection (or section) of the plaice by simply doubling the _y_’s and halving the _z_’s: and, very approximately, this turns out to be the case. The plaice is (in round numbers) just about twice as broad and also just about half as thick as the haddock; and therefore the ratio of breadth to thickness (or _y_ to _z_) is just about four times as great in the one case as in the other.
It is true that this simple, or simplified, illustration carries us but a very little way, and only half prepares us for much greater complications. For instance, we have no right or reason to presume that the equality of weights, or volumes, is a common, much less a general rule. And again, in all cases of more complex deformation, such as that by which we have compared Diodon with the sunfish, we must be prepared for very much more {777} recondite methods of comparison and analysis, leading doubtless to very much more complicated results. In this last case, of Diodon and the sunfish, we have seen that the vertical _expansion_ of the latter as compared with the former fish, increases rapidly as we go backwards towards the tail; but we can by no means say that the lateral _compression_ increases in like proportion. If anything, it would seem that the said expansion and compression tend to vary inversely; for the Diodon is very thick in front and greatly thinned away behind, while the flattened sunfish is more nearly of the same thickness all the way along. Interesting as the whole subject is we must meanwhile leave it alone; recognising, however, that if the difficulties of description and representation could be overcome, it is by means of such co-ordinates in space that we should at last obtain an adequate and satisfying picture of the processes of deformation and of the directions of growth[663].
{778}
EPILOGUE.
In the beginning of this book I said that its scope and treatment were of so prefatory a kind that of other preface it had no need; and now, for the same reason, with no formal and elaborate conclusion do I bring it to a close. The fact that I set little store by certain postulates (often deemed to be fundamental) of our present-day biology the reader will have discovered and I have not endeavoured to conceal. But it is not for the sake of polemical argument that I have written, and the doctrines which I do not subscribe to I have only spoken of by the way. My task is finished if I have been able to shew that a certain mathematical aspect of morphology, to which as yet the morphologist gives little heed, is interwoven with his problems, complementary to his descriptive task, and helpful, nay essential, to his proper study and comprehension of Form. _Hic artem remumque repono._
And while I have sought to shew the naturalist how a few mathematical concepts and dynamical principles may help and guide him, I have tried to shew the mathematician a field for his labour,—a field which few have entered and no man has explored. Here may be found homely problems, such as often tax the highest skill of the mathematician, and reward his ingenuity all the more for their trivial associations and outward semblance of simplicity.
That I am no skilled mathematician I have had little need to confess, but something of the use and beauty of mathematics I think I am able to understand. I know that in the study of material things, number, order and position are the threefold clue to exact knowledge; that these three, in the mathematician’s hands, furnish the “first outlines for a sketch of the Universe”; that by square and circle we are helped, like Emile Verhaeren’s carpenter, to conceive “Les lois indubitables et fécondes Qui sont la règle et la clarté du monde.”
For the harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural {779} Philosophy are embodied in the concept of mathematical beauty. A greater than Verhaeren had this in mind when he told of “the golden compasses, prepared In God’s eternal store.” A greater than Milton had magnified the theme and glorified Him “who sitteth upon the circle of the earth,” saying: He measureth the waters in the hollow of his hand, he meteth out the heavens with his span, he comprehendeth the dust of the earth in a measure.
Moreover the perfection of mathematical beauty is such (as Maclaurin learned of the bee), that whatsoever is most beautiful and regular is also found to be most useful and excellent.
The living and the dead, things animate and inanimate, we dwellers in the world and this world wherein we dwell,—πάντα γα μὰν τὰ γιγνωσκόμενα,—are bound alike by physical and mathematical law. “Conterminous with space and coeval with time is the kingdom of Mathematics; within this range her dominion is supreme; otherwise than according to her order nothing can exist, and nothing takes place in contradiction to her laws.” So said, some forty years ago, a certain mathematician; and Philolaus the Pythagorean had said much the same.
But with no less love and insight has the science of Form and Number been appraised in our own day and generation by a very great Naturalist indeed:—by that old man eloquent, that wise student and pupil of the ant and the bee, who died but yesterday, and who in his all but saecular life tasted of the firstfruits of immortality; who curiously conjoined the wisdom of antiquity with the learning of to-day; whose Provençal verse seems set to Dorian music; in whose plainest words is a sound as of bees’ industrious murmur; and who, being of the same blood and marrow with Plato and Pythagoras, saw in Number “la clef de la voûte,” and found in it “le comment et le pourquoi des choses.”
NOTES:
[1] These sayings of Kant and of Du Bois, and others like to them, have been the text of many discourses: see, for instance, Stallo’s _Concepts_, p. 21, 1882; Höber, _Biol. Centralbl._ XIX, p. 284, 1890, etc. Cf. also Jellett, _Rep. Brit. Ass._ 1874, p. 1.
[2] “Quum enim mundi universi fabrica sit perfectissima, atque a Creatore sapientissimo absoluta, nihil omnino in mundo contingit in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum quin omnes mundi effectus ex causis finalibus, ope methodi maximorum et minimorum, aeque feliciter determinari queant atque ex ipsis causis efficientibus.” _Methodus inveniendi_, etc. 1744 (_cit._ Mach, _Science of Mechanics_, 1902, p. 455).
[3] Cf. Opp. (ed. Erdmann), p. 106, “Bien loin d’exclure les causes finales..., c’est de là qu’il faut tout déduire en Physique.”
[4] Cf. p. 162. “La force vitale dirige des phénomènes qu’elle ne produit pas: les agents physiques produisent des phénomènes qu’ils ne dirigent pas.”
[5] It is now and then conceded with reluctance. Thus Enriques, a learned and philosophic naturalist, writing “della economia di sostanza nelle osse cave” (_Arch. f. Entw. Mech._ XX, 1906), says “una certa impronta di teleologismo quà e là è rimasta, mio malgrado, in questo scritto.”
[6] Cf. Cleland, On Terminal Forms of Life, _J. Anat. and Phys._ XVIII, 1884.
[7] Conklin, Embryology of Crepidula, _Journ. of Morphol._ XIII, p. 203, 1897; Lillie, F. R., Adaptation in Cleavage, _Woods Holl Biol. Lectures_, pp. 43–67, 1899.
[8] I am inclined to trace back Driesch’s teaching of Entelechy to no less a person than Melanchthon. When Bacon (_de Augm._ IV, 3) states with disapproval that the soul “has been regarded rather as a function than as a substance,” R. L. Ellis points out that he is referring to Melanchthon’s exposition of the Aristotelian doctrine. For Melanchthon, whose view of the peripatetic philosophy had long great influence in the Protestant Universities, affirmed that, according to the true view of Aristotle’s opinion, the soul is not a substance, but an ἑντελέχεια, or _function_. He defined it as δύναμις _quaedam ciens actiones_—a description all but identical with that of Claude Bernard’s “_force vitale_.”
[9] Ray Lankester, _Encycl. Brit._ (9th ed.), art. “Zoology,” p. 806, 1888.
[10] Alfred Russel Wallace, especially in his later years, relied upon a direct but somewhat crude teleology. Cf. his _World of Life, a Manifestation of Creative Power, Directive Mind and Ultimate Purpose_, 1910.
[11] Janet, _Les Causes Finales_, 1876, p. 350.
[12] The phrase is Leibniz’s, in his _Théodicée_.
[13] Cf. (_int. al._) Bosanquet, The Meaning of Teleology, _Proc. Brit. Acad._ 1905–6, pp. 235–245. Cf. also Leibniz (_Discours de Métaphysique; Lettres inédites, ed._ de Careil, 1857, p. 354; _cit._ Janet, p. 643), “L’un et l’autre est bon, l’un et l’autre peut être utile ... et les auteurs qui suivent ces routes différentes ne devraient point se maltraiter: _et seq._”
[14] The reader will understand that I speak, not of the “severe and diligent inquiry” of variation or of “fortuity,” but merely of the easy assumption that these phenomena are a sufficient basis on which to rest, with the all-powerful help of natural selection, a theory of definite and progressive evolution.
[15] _Revue Philosophique._ XXXIII, 1892.
[16] This general principle was clearly grasped by Dr George Rainey (a learned physician of St Bartholomew’s) many years ago, and expressed in such words as the following: “......it is illogical to suppose that in the case of vital organisms a distinct force exists to produce results perfectly within the reach of physical agencies, especially as in many instances no end could be attained were that the case, but that of opposing one force by another capable of effecting exactly the same purpose.” (On Artificial Calculi, _Q.J.M.S._ (_Trans. Microsc. Soc._), VI, p. 49, 1858.) Cf. also Helmholtz, _infra cit._, p. 9.
[17] Whereby he incurred the reproach of Socrates, in the _Phaedo_.
[18] In a famous lecture (Conservation of Forces applied to Organic Nature, _Proc. Roy. Instit._, April 12, 1861), Helmholtz laid it down, as “the fundamental principle of physiology,” that “There may be other agents acting in the living body than those agents which act in the inorganic world; but those forces, as far as they cause chemical and mechanical influence in the body, must be _quite of the same character_ as inorganic forces: in this at least, that their effects must be ruled by necessity, and must always be the same when acting in the same conditions; and so there cannot exist any arbitrary choice in the direction of their actions.” It would follow from this, that, like the other “physical” forces, they must be subject to mathematical analysis and deduction. Cf. also Dr T. Young’s Croonian Lecture On the Heart and Arteries, _Phil. Trans._ 1809, p. 1; _Coll. Works_, I, 511.
[19] _Ektropismus, oder die physikalische Theorie des Lebens_, Leipzig, 1910.
[20] Wilde Lecture, _Nature_, March 12, 1908; _ibid._ Sept. 6, 1900, p. 485; _Aether and Matter_, p. 288. Cf. also Lord Kelvin, _Fortnightly Review_, 1892, p. 313.
[21] Joly, The Abundance of Life, _Proc. Roy. Dublin Soc._ VII, 1890; and in _Scientific Essays_, etc. 1915, p. 60 _et seq._
[22] Papillon, _Histoire de la philosophie moderne_, I, p. 300.
[23] With the special and important properties of _colloidal_ matter we are, for the time being, not concerned.
[24] Cf. Hans Przibram, _Anwendung elementarer Mathematik auf Biologische Probleme_ (in Roux’s _Vorträge_, Heft III), Leipzig, 1908, p. 10.
[25] The subject is treated from an engineering point of view by Prof. James Thomson, Comparisons of Similar Structures as to Elasticity, Strength, and Stability, _Trans. Inst. Engineers, Scotland_, 1876 (_Collected Papers_, 1912, pp. 361–372), and by Prof. A. Barr, _ibid._ 1899; see also Rayleigh, _Nature_, April 22, 1915.
[26] Cf. Spencer, The Form of the Earth, etc., _Phil. Mag._ XXX, pp. 194–6, 1847; also _Principles of Biology_, pt. II, ch. I, 1864 (p. 123, etc.).
[27] George Louis Lesage (1724–1803), well known as the author of one of the few attempts to explain gravitation. (Cf. Leray, _Constitution de la Matière_, 1869; Kelvin, _Proc. R. S. E._ VII, p. 577, 1872, etc.; Clerk Maxwell, _Phil. Trans._ vol. 157, p. 50, 1867; art. “Atom,” _Encycl. Brit._ 1875, p. 46.)
[28] Cf. Pierre Prévost, _Notices de la vie et des écrits de Lesage_, 1805; quoted by Janet, _Causes Finales_, app. III.
[29] Discorsi e Dimostrazioni matematiche, intorno à due nuove scienze, attenenti alla Mecanica, ed ai Movimenti Locali: appresso gli Elzevirii, MDCXXXVIII. _Opere_, ed. Favaro, VIII, p. 169 seq. Transl. by Henry Crew and A. de Salvio, 1914, p. 130, etc. See _Nature_, June 17, 1915.
[30] So Werner remarked that Michael Angelo and Bramanti could not have built of gypsum at Paris on the scale they built of travertin in Rome.
[31] Sir G. Greenhill, Determination of the greatest height to which a Tree of given proportions can grow, _Cambr. Phil. Soc. Pr._ IV, p. 65, 1881, and Chree, _ibid._ VII, 1892. Cf. Poynting and Thomson’s _Properties of Matter_, 1907, p 99.
[32] In like manner the wheat-straw bends over under the weight of the loaded ear, and the tip of the cat’s tail bends over when held upright,—not because they “possess flexibility,” but because they outstrip the dimensions within which stable equilibrium is possible in a vertical position. The kitten’s tail, on the other hand, stands up spiky and straight.
[33] _Modern Painters._
[34] The stem of the giant bamboo may attain a height of 60 metres, while not more than about 40 cm. in diameter near its base, which dimensions are not very far short of the theoretical limits (A. J. Ewart, _Phil. Trans._ vol. 198, p. 71, 1906).
[35] _Trans. Zool. Soc._ IV, 1850, p. 27.
[36] It would seem to be a common if not a general rule that marine organisms, zoophytes, molluscs, etc., tend to be larger than the corresponding and closely related forms living in fresh water. While the phenomenon may have various causes, it has been attributed (among others) to the simple fact that the forces of growth are less antagonised by gravity in the denser medium (cf. Houssay, _La Forme et la Vie_, 1900, p. 815). The effect of gravity on outward _form_ is illustrated, for instance, by the contrast between the uniformly upward branching of a sea-weed and the drooping curves of a shrub or tree.
[37] The analogy is not a very strict one. We are not taking account, for instance, of a proportionate increase in thickness of the boiler-plates.
[38] Let _L_ be the length, _S_ the (wetted) surface, _T_ the tonnage, _D_ the displacement (or volume) of a ship; and let it cross the Atlantic at a speed _V_. Then, in comparing two ships, similarly constructed but of different magnitudes, we know that _L_ = _V_^2, _S_ = _L_^2 = _V_^4, _D_ = _T_ = _L_^3 = _V_^6; also _R_ (resistance) = _S_ ⋅ _V_^2 = _V_^6; _H_ (horse-power) = _R_ ⋅ _V_ = _V_^7; and the coal (_C_) necessary for the voyage = _H_/_V_ = _V_^6. That is to say, in ordinary engineering language, to increase the speed across the Atlantic by 1 per cent. the ship’s length must be increased 2 per cent., her tonnage or displacement 6 per cent., her coal-consumpt also 6 per cent., her horse-power, and therefore her boiler-capacity, 7 per cent. Her bunkers, accordingly, keep pace with the enlargement of the ship, but her boilers tend to increase out of proportion to the space available.
[39] This is the result arrived at by Helmholtz, Ueber ein Theorem geometrisch ähnliche Bewegungen flüssiger Körper betreffend, nebst Anwendung auf das Problem Luftballons zu lenken, _Monatsber. Akad. Berlin_, 1873, pp. 501–14. It was criticised and challenged (somewhat rashly) by K. Müllenhof, Die Grösse der Flugflächen, etc., _Pflüger’s Archiv_, XXXV, p. 407, XXXVI, p. 548, 1885.
[40] Cf. also Chabrier, Vol des Insectes, _Mém. Mus. Hist. Nat. Paris_, VI–VIII, 1820–22.
[41] _Aerial Flight_, vol. II (_Aerodonetics_), 1908, p. 150.
[42] By Lanchester, _op. cit._ p. 131.
[43] Cf. _L’empire de l’air; ornithologie appliquée à l’aviation_. 1881.
[44] _De Motu Animalium_, I, prop. cciv, ed. 1685, p. 243.
[45] Harlé, On Atmospheric Pressure in past Geological Ages, _Bull. Geol. Soc. Fr._ XI, pp. 118–121; or _Cosmos_, p. 30, July 8, 1911.
[46] _Introduction to Entomology_, 1826, II, p. 190. K. and S., like many less learned authors, are fond of popular illustrations of the “wonders of Nature,” to the neglect of dynamical principles. They suggest, for instance, that if the white ant were as big as a man, its tunnels would be “magnificent cylinders of more than three hundred feet in diameter”; and that if a certain noisy Brazilian insect were as big as a man, its voice would be heard all the world over: “so that Stentor becomes a mute when compared with these insects!” It is an easy consequence of anthropomorphism, and hence a common characteristic of fairy-tales, to neglect the principle of dynamical, while dwelling on the aspect of geometrical, similarity.
[47] I.e. the available energy of muscle, in ft.-lbs. per lb. of muscle, is the same for all animals: a postulate which requires considerable qualification when we are comparing very different _kinds_ of muscle, such as the insect’s and the mammal’s.
[48] Prop. clxxvii. Animalia minora et minus ponderosa majores saltus efficiunt respectu sui corporis, si caetera fuerint paria.
[49] See also (_int. al._), John Bernoulli, _de Motu Musculorum_, Basil., 1694; Chabry, Mécanisme du Saut, _J. de l’Anat. et de la Physiol._ XIX, 1883; Sur la longueur des membres des animaux sauteurs, _ibid._ XXI, p. 356, 1885; Le Hello, De l’action des organes locomoteurs, etc., _ibid._ XXIX, p. 65–93, 1893, etc.
[50] Recherches sur la force absolue des muscles des Invertébrés, _Bull. Acad. E. de Belgique_ (3), VI, VII, 1883–84; see also _ibid._ (2), XX, 1865, XXII, 1866; _Ann. Mag. N. H._ XVII, p. 139, 1866, XIX, p. 95, 1867. The subject was also well treated by Straus-Dürckheim, in his _Considérations générales sur l’anatomie comparée des animaux articulés_, 1828.
[51] The fact that the limb tends to swing in pendulum-time was first observed by the brothers Weber (_Mechanik der menschl. Gehwerkzeuge_, Göttingen, 1836). Some later writers have criticised the statement (e.g. Fischer, Die Kinematik des Beinschwingens etc., _Abh. math. phys. Kl. k. Sächs. Ges._ XXV–XXVIII, 1899–1903), but for all that, with proper qualifications, it remains substantially true.
[52] Quoted in Mr John Bishop’s interesting article in Todd’s _Cyclopaedia_, III, p. 443.
[53] There is probably also another factor involved here: for in bending, and therefore shortening, the leg we bring its centre of gravity nearer to the pivot, that is to say, to the joint, and so the muscle tends to move it the more quickly.
[54] _Proc. Psychical Soc._ XII, pp. 338–355, 1897.
[55] For various calculations of the increase of surface due to histological and anatomical subdivision, see E. Babak, Ueber die Oberflächenentwickelung bei Organismen, _Biol. Centralbl._ XXX, pp. 225–239, 257–267, 1910. In connection with the physical theory of surface-energy, Wolfgang Ostwald has introduced the conception of _specific surface_, that is to say the ratio of surface to volume, or _S_/_V_. In a cube, _V_ = _l_^3, and _S_ = 6_l_^2; therefore _S_/_V_ = 6/_l_. Therefore if the side _l_ measure 6 cm., the ratio _S_/_V_ = 1, and such a cube may be taken as our standard, or unit of specific surface. A human blood-corpuscle has, accordingly, a specific surface of somewhere about 14,000 or 15,000. It is found in physical chemistry that surface energy becomes an important factor when the specific surface reaches a value of 10,000 or thereby.
[56] Though the entire egg is not increasing in mass, this is not to say that its living protoplasm is not increasing all the while at the expense of the reserve material.
[57] Cf. Tait, _Proc. R.S.E._ V, 1866, and VI, 1868.
[58] _Physiolog. Notizen_ (9), p. 425, 1895. Cf. Strasbürger, Ueber die Wirkungssphäre der Kerne und die Zellgrösse, _Histolog. Beitr._ (5), pp. 95–129, 1893; J. J. Gerassimow, Ueber die Grösse des Zellkernes, _Beih. Bot. Centralbl._ XVIII, 1905; also G. Levi and T. Terni, Le variazioni dell’ indice plasmatico-nucleare durante l’intercinesi, _Arch. Ital. di Anat._ X, p. 545, 1911.
[59] _Arch. f. Entw. Mech._ IV, 1898, pp. 75, 247.
[60] Conklin, E. G., Cell-size and nuclear-size, _J. Exp. Zool._ XII. pp. 1–98, 1912.
[61] Thus the fibres of the crystalline lens are of the same size in large and small dogs; Rabl, _Z. f. w. Z._ LXVII, 1899. Cf. (_int. al._) Pearson, On the Size of the Blood-corpuscles in Rana, _Biometrika_, VI, p. 403, 1909. Dr Thomas Young caught sight of the phenomenon, early in last century: “The solid particles of the blood do not by any means vary in magnitude in the same ratio with the bulk of the animal,” _Natural Philosophy_, ed. 1845, p. 466; and Leeuwenhoek and Stephen Hales were aware of it a hundred years before. But in this case, though the blood-corpuscles show no relation of magnitude to the size of the animal, they do seem to have some relation to its activity. At least the corpuscles in the sluggish Amphibia are much the largest known to us, while the smallest are found among the deer and other agile and speedy mammals. (Cf. Gulliver, _P.Z.S._ 1875, p. 474, etc.) This apparent correlation may have its bearing on modern views of the surface-condensation or adsorption of oxygen in the blood-corpuscles, a process which would be greatly facilitated and intensified by the increase of surface due to their minuteness.
[62] Cf. P. Enriques, La forma come funzione della grandezza: Ricerche sui gangli nervosi degli Invertebrati, _Arch. f. Entw. Mech._ XXV, p. 655, 1907–8.
[63] While the difference in cell-volume is vastly less than that between the volumes, and very much less also than that between the surfaces, of the respective animals, yet there _is_ a certain difference; and this it has been attempted to correlate with the need for each cell in the many-celled ganglion of the larger animal to possess a more complex “exchange-system” of branches, for intercommunication with its more numerous neighbours. Another explanation is based on the fact that, while such cells as continue to divide throughout life tend to uniformity of size in all mammals, those which do not do so, and in particular the ganglion cells, continue to grow, and their size becomes, therefore, a function of the duration of life. Cf. G. Levi, Studii sulla grandezza delle cellule, _Arch. Ital. di Anat. e di Embryolog._ V, p. 291, 1906.
[64] Boveri. _Zellen-studien, V. Ueber die Abhängigkeit der Kerngrösse und Zellenzahl der Seeigellarven von der Chromosomenzahl der Ausgangszellen._ Jena, 1905.
[65] Recent important researches suggest that such ultra-minute “filter-passers” are the true cause of certain acute maladies commonly ascribed to the presence of much larger organisms; cf. Hort, Lakin and Benians, The true infective Agent in Cerebrospinal Fever, etc., _J. Roy. Army Med. Corps_, Feb. 1910.
[66] _Zur Erkenntniss der Kolloide_, 1905, p. 122; where there will be found an interesting discussion of various molecular and other minute magnitudes.
[67] _Encyclopaedia Britannica_, 9th edit., vol. III, p. 42, 1875.
[68] Sur la limite de petitesse des organismes, _Bull. Soc. R. des Sc. méd. et nat. de Bruxelles_, Jan. 1903; _Rec. d’œuvres_ (_Physiol. générale_), p. 325.
[69] Cf. A. Fischer, _Vorlesungen über Bakterien_, 1897, p. 50.
[70] F. Hofmeister, quoted in Cohnheim’s _Chemie der Eiweisskörper_, 1900, p. 18.
[71] McKendrick arrived at a still lower estimate, of about 1250 proteid molecules in the minutest organisms. _Brit. Ass. Rep._ 1901, p. 808.
[72] Cf. Perrin, _Les Atomes_, 1914, p. 74.
[73] Cf. Tait, On Compression of Air in small Bubbles, _Proc. R. S. E._ V, 1865.
[74] _Phil. Mag._ XLVIII, 1899; _Collected Papers_, IV, p. 430.
[75] Carpenter, _The Microscope_, edit. 1862, p. 185.
[76] The modern literature on the Brownian Movement is very large, owing to the value which the phenomenon is shewn to have in determining the size of the atom. For a fuller, but still elementary account, see J. Cox, _Beyond the Atom_, 1913, pp. 118–128; and see, further, Perrin, _Les Atomes_, pp. 119–189.
[77] Cf. R. Gans, Wie fallen Stäbe und Scheiben in einer reibenden Flüssigkeit? _Münchener Bericht_, 1911, p. 191; K. Przibram, Ueber die Brown’sche Bewegung nicht kugelförmiger Teilchen, _Wiener Ber._ 1912, p. 2339.
[78] Ueber die ungeordnete Bewegung niederer Thiere, _Pflüger’s Archiv_, CLIII, p. 401, 1913.
[79] Sometimes we find one and the same diagram suffice, whether the intervals of time be great or small; and we then invoke “Wolff’s Law,” and assert that the life-history of the individual repeats, or recapitulates, the history of the race.
[80] Our subject is one of Bacon’s “Instances of the Course,” or studies wherein we “measure Nature by periods of Time.” In Bacon’s _Catalogue of Particular Histories_, one of the odd hundred histories or investigations which he foreshadowed is precisely that which we are engaged on, viz. a “History of the Growth and Increase of the Body, in the whole and in its parts.”
[81] Cf. Aristotle, _Phys._ vi, 5, 235 _a_ 11, ὲπεὶ γὰρ ἅπασα κίνησις ἐν χρόνῳ, κτλ. Bacon emphasised, in like manner, the fact that “all motion or natural action is performed in time: some more quickly, some more slowly, but all in periods determined and fixed in the nature of things. Even those actions which seem to be performed suddenly, and (as we say) in the twinkling of an eye, are found to admit of degree in respect of duration.” _Nov. Org._ XLVI.
[82] Cf. (e.g.) _Elem. Physiol._ ed. 1766, VIII, p. 114, “Ducimur autem ad evolutionem potissimum, quando a perfecto animale retrorsum progredimur, et incrementorum atque mutationum seriem relegimus. Ita inveniemus perfectum illud animal fuisse imperfectius, alterius figurae et fabricae, et denique rude et informe: et tamen idem semper animal sub iis diversis phasibus fuisse, quae absque ullo saltu perpetuos parvosque per gradus cohaereant.”
[83] _Beiträge zur Entwickelungsgeschichte des Hühnchens im Ei_, p. 40, 1817. Roux ascribes the same views also to Von Baer and to R. H. Lotze (_Allg. Physiologie_, p. 353, 1851).
[84] Roux, _Die Entwickelungsmechanik_, p. 99, 1905.
[85] _Op. cit._ p. 302, “Magnum hoc naturae instrumentum, etiam in corpore animato evolvendo potenter operatur; etc.”
[86] _Ibid._ p. 306. “Subtiliora ista, et aliquantum hypothesi mista, tamen magnum mihi videntur speciem veri habere.”
[87] Cf. His, On the Principles of Animal Morphology, _Proc. R. S. E._ XV, 1888, p. 294: “My own attempts to introduce some elementary mechanical or physiological conceptions into embryology have not generally been agreed to by morphologists. To one it seemed ridiculous to speak of the elasticity of the germinal layers; another thought that, by such considerations, we ‘put the cart before the horse’: and one more recent author states, that we have better things to do in embryology than to discuss tensions of germinal layers and similar questions, since all explanations must of necessity be of a phylogenetic nature. This opposition to the application of the fundamental principles of science to embryological questions would scarcely be intelligible had it not a dogmatic background. No other explanation of living forms is allowed than heredity, and any which is founded on another basis must be rejected ....... To think that heredity will build organic beings without mechanical means is a piece of unscientific mysticism.”
[88] Hertwig, O., _Zeit und Streitfragen der Biologie_, II. 1897.
[89] Cf. Roux, _Gesammelte Abhandlungen_, II, p. 31, 1895.
[90] _Treatise on Comparative Embryology_, I, p. 4, 1881.
[91] Cf. Fick, _Anal. Anzeiger_, XXV, p. 190, 1904.
[92] 1st ed. p. 444; 6th ed. p. 390. The student should not fail to consult the passage in question; for there is always a risk of misunderstanding or misinterpretation when one attempts to epitomise Darwin’s carefully condensed arguments.
[93] “In omni rerum naturalium historia utile est _mensuras definiri et numeros_,” Haller, _Elem. Physiol._ II, p. 258, 1760. Cf. Hales, _Vegetable Staticks_, Introduction.
[94] Brussels, 1871. Cf. the same author’s _Physique sociale_, 1835, and _Lettres sur la théorie des probabilités_, 1846. See also, for the general subject, Boyd, R., Tables of weights of the Human Body, etc. _Phil. Trans._ vol. CLI, 1861; Roberts, C., _Manual of Anthropometry_, 1878; Daffner, F., _Das Wachsthum des Menschen_ (2nd ed.), 1902, etc.
[95] Dr Johnson was not far wrong in saying that “life declines from thirty-five”; though the Autocrat of the Breakfast-table, like Cicero, declares that “the furnace is in full blast for ten years longer.”
[96] Joly, _The Abundance of Life_, 1915 (1890), p. 86.
[97] “_Lou pes, mèstre de tout_ [Le poids, maître de tout], _mèstre sènso vergougno, Que te tirasso en bas de sa brutalo pougno_,” J. H. Fabre, _Oubreto prouvençalo_, p. 61.
[98] The continuity of the phenomenon of growth, and the natural passage from the phase of increase to that of decrease or decay, are admirably discussed by Enriques, in “La morte,” _Riv. di Scienza_, 1907, and in “Wachsthum und seine analytische Darstellung,” _Biol. Centralbl._ June, 1909. Haller (_Elem_. VII, p. 68) recognised _decrementum_ as a phase of growth, not less important (theoretically) than _incrementum_: “_tristis, sed copiosa, haec est materies_.”
[99] Cf. (_int. al._), Friedenthal, H., Das Wachstum des Körpergewichtes ... in verschiedenen Lebensältern, _Zeit. f. allg. Physiol._ IX, pp. 487–514, 1909.
[100] As Haller observed it to do in the chick (_Elem._ VIII, p. 294): “Hoc iterum incrementum miro ordine ita distribuitur, ut in principio incubationis maximum est: inde perpetuo minuatur.”
[101] There is a famous passage in Lucretius (v. 883) where he compares the course of life, or rate of growth, in the horse and his boyish master: _Principio circum tribus actis impiger annis Floret equus, puer hautquaquam_, etc.
[102] Minot, C. S., Senescence and Rejuvenation, _Journ. of Physiol._ XII, pp. 97–153, 1891; The Problem of Age, Growth and Death, _Pop. Science Monthly_ (June–Dec.), 1907.
[103] Quoted in Vierordt’s _Anatomische ... Daten und Tabellen_, 1906. p. 13.
[104] _Unsere Körperform_, Leipzig, 1874.
[105] No such point of inflection appears in the curve of weight according to C. M. Jackson’s data (On the Prenatal Growth of the Human Body, etc., _Amer. Journ. of Anat._ IX, 1009, pp. 126, 156), nor in those quoted by him from Ahlfeld, Fehling and others. But it is plain that the very rapid increase of the monthly weights, approximately in the ratio of the cubes of the corresponding lengths, would tend to conceal any such breach of continuity, unless it happened to be very marked indeed. Moreover in the case of Jackson’s data (and probably also in the others) the actual age of the embryos was not determined, but was estimated from their lengths. The following is Jackson’s estimate of average weights at intervals of a lunar month:
Months 0 1 2 3 4 5 6 7 8 9 10 Wt in gms. ·0 ·04 3 36 120 330 600 1000 1500 2200 3200
[106] G. Kraus (after Wallich-Martius), _Ann. du Jardin bot. de Buitenzorg_, XII, 1, 1894, p. 210. Cf. W. Ostwald, _Zeitliche Eigenschaften_, etc. p. 56.
[107] Cf. Chodat, R., et Monnier, A., Sur la courbe de croissance des végétaux, _Bull. Herb. Boissier_ (2), V, pp. 615, 616, 1905.
[108] Cf. Fr. Boas, Growth of Toronto Children, _Rep. of U.S. Comm. of Education_, 1896–7, pp. 1541–1599, 1898; Boas and Clark Wissler, Statistics of Growth, _Education Rep._ 1904, pp. 25–132, 1906; H. P. Bowditch, _Rep. Mass. State Board of Health_, 1877; K. Pearson, On the Magnitude of certain coefficients of Correlation in Man, _Pr. R. S._ LXVI, 1900.
[109] _l.c._ p. 42, and other papers there quoted.
[110] See, for an admirable résumé of facts, Wolfgang Ostwald, _Ueber die Zeitliche Eigenschaften der Entwickelungsvorgänge_ (71 pp.), Leipzig, 1908 (Roux’s _Vorträge_, Heft V): to which work I am much indebted. A long list of observations on the growth-rate of various animals is also given by H. Przibram, _Exp. Zoologie_, 1913, pt. IV (_Vitalität_), pp. 85–87.
[111] Cf. St Loup, Vitesse de croissance chez les Souris, _Bull. Soc. Zool. Fr._ XVIII, 242, 1893; Robertson, _Arch. f. Entwickelungsmech._ XXV, p. 587, 1908; Donaldson. _Boas Memorial Volume_, New York, 1906.
[112] Luciani e Lo Monaco, _Arch. Ital. de Biologie_, XXVII, p. 340, 1897.
[113] Schaper, _Arch. f. Entwickelungsmech._ XIV, p. 356, 1902. Cf. Barfurth, Versuche über die Verwandlung der Froschlarven, _Arch. f. mikr. Anat._ XXIX, 1887.
[114] Joh. Schmidt, Contributions to the Life-history of the Eel, _Rapports du Conseil Intern. pour l’exploration de la Mer_, vol. V, pp. 137–274, Copenhague, 1906.
[115] That the metamorphoses of an insect are but phases in a process of growth, was firstly clearly recognised by Swammerdam, _Biblia Naturae_, 1737, pp. 6, 579 etc.
[116] From Bose, J. C., _Plant Response_, London, 1906, p. 417.
[117] This phenomenon, of _incrementum inequale_, as opposed to _incrementum in universum_, was most carefully studied by Haller: “Incrementum inequale multis modis fit, ut aliae partes corporis aliis celerius increscant. Diximus hepar minus fieri, majorem pulmonem, minimum thymum, etc.” (_Elem._ VIII (2), p. 34).
[118] See (_inter alia_) Fischel, A., Variabilität und Wachsthum des embryonalen Körpers, _Morphol. Jahrb._ XXIV, pp. 369–404, 1896. Oppel, _Vergleichung des Entwickelungsgrades der Organe zu verschiedenen Entwickelungszeiten bei Wirbelthieren_, Jena, 1891. Faucon, A., _Pesées et Mensurations fœtales à différents âges de la grossesse_. (Thèse.) Paris, 1897. Loisel, G., Croissance comparée en poids et en longueur des fœtus mâle et femelle dans l’espèce humaine, _C. R. Soc. de Biologie_, Paris, 1903. Jackson, C. M., Pre-natal growth of the human body and the relative growth of the various organs and parts, _Am. J. of Anat._ IX, 1909; Post-natal growth and variability of the body and of the various organs in the albino rat, _ibid._ XV, 1913.
[119] _l.c._ p. 1542.
[120] Variation and Correlation in Brain-weight, _Biometrika_, IV, pp. 13–104, 1905.
[121] _Die Säugethiere_, p. 117.
[122] _Amer. J. of Anatomy_, VIII, pp. 319–353, 1908. Donaldson (_Journ. Comp. Neur. and Psychol._ XVIII, pp. 345–392, 1908) also gives a logarithmic formula for brain-weight (_y_) as compared with body-weight (_x_), which in the case of the white rat is _y_ = ·554 − ·569 log(_x_ − 8·7), and the agreement is very close. But the formula is admittedly empirical and as Raymond Pearl says (_Amer. Nat._ 1909, p. 303), “no ulterior biological significance is to be attached to it.”
[123] _Biometrika_, IV, pp. 13–104, 1904.
[124] Donaldson, H. H., A Comparison of the White Rat with Man in respect to the Growth of the entire Body, _Boas Memorial Vol._, New York, 1906, pp. 5–26.
[125] Besides many papers quoted by Dubois on the growth and weight of the brain, and numerous papers in _Biometrika_, see also the following: Ziehen, Th., _Das Gehirn: Massverhältnisse_, in Bardeleben’s _Handb. der Anat. des Menschen_, IV, pp. 353–386, 1899. Spitzka, E. A., Brain-weight of Animals with special reference to the Weight of the Brain in the Macaque Monkey, _J. Comp. Neurol._ XIII, pp. 9–17, 1903. Warneke, P., Mitteilung neuer Gehirn und Körpergewichtsbestimmungen bei Säugern, nebst Zusammenstellung der gesammten bisher beobachteten absoluten und relativen Gehirngewichte bei den verschiedenen Species, _J. f. Psychol. u. Neurol._ XIII, pp. 355–403, 1909. Donaldson, H. H., On the regular seasonal Changes in the relative Weight of the Central Nervous System of the Leopard Frog, _Journ. of Morph._ XXII, pp. 663–694, 1911.
[126] Cf. Jenkinson, Growth, Variability and Correlation in Young Trout, _Biometrika_, VIII, pp. 444–455, 1912.
[127] Cf. chap. xvii, p. 739.
[128] “ ...I marked in the same manner as the Vine, young Honeysuckle shoots, etc....; and I found in them all a gradual scale of unequal extensions, those parts extending most which were tenderest,” _Vegetable Staticks_, Exp. cxxiii.
[129] From Sachs, _Textbook of Botany_, 1882, p. 820.
[130] Variation and Differentiation in Ceratophyllum, _Carnegie Inst. Publications_, No. 58, Washington, 1907.
[131] Cf. Lämmel, Ueber periodische Variationen in Organismen, _Biol. Centralbl._ XXII, pp. 368–376, 1903.
[132] Herein lies the easy answer to a contention frequently raised by Bergson, and to which he ascribes great importance, that “a mere variation of size is one thing, and a change of form is another.” Thus he considers “a change in the form of leaves” to constitute “a profound morphological difference.” _Creative Evolution_, p. 71.
[133] I do not say that the assumption that these two groups of earwigs were of different ages is altogether an easy one; for of course, even in an insect whose metamorphosis is so simple as the earwig’s, consisting only in the acquisition of wings or wing-cases, we usually take it for granted that growth proceeds no more after the final stage, or “adult form” is attained, and further that this adult form is attained at an approximately constant age, and constant magnitude. But even if we are not permitted to think that the earwig may have grown, or moulted, after once the elytra were produced, it seems to me far from impossible, and far from unlikely, that prior to the appearance of the elytra one more stage of growth, or one more moult took place in some cases than in others: for the number of moults is known to be variable in many species of Orthoptera. Unfortunately Bateson tells us nothing about the sizes or total lengths of his earwigs; but his figures suggest that it was bigger earwigs that had the longer tails; and that the rate of growth of the tails had had a certain definite ratio to that of the bodies, but not necessarily a simple ratio of equality.
[134] Jackson, C. M., _J. of Exp. Zool._ XIX, 1915, p. 99; cf. also Hans Aron, Unters. über die Beeinflüssung der Wachstum durch die Ernährung, _Berl. klin. Wochenbl._ LI, pp. 972–977, 1913, etc.
[135] The temperature limitations of life, and to some extent of growth, are summarised for a large number of species by Davenport, _Exper. Morphology_, cc. viii, xviii, and by Hans Przibram, _Exp. Zoologie_, IV, c. v.
[136] Réaumur: _L’art de faire éclore et élever en toute saison des oiseaux domestiques, foit par le moyen de la chaleur du fumier_, Paris, 1749.
[137] Cf. (_int. al._) de Vries, H., Matériaux pour la connaissance de l’influence de la température sur les plantes, _Arch. Néerl._ V, 385–401, 1870. Köppen, Wärme und Pflanzenwachstum, _Bull. Soc. Imp. Nat. Moscou._ XLIII, pp. 41–110, 1870.
[138] Blackman, F. F., _Ann. of Botany_, XIX, p. 281, 1905.
[139] For various instances of a “temperature coefficient” in physiological processes, see Kanitz, _Zeitschr. f. Elektrochemie_, 1907, p. 707; _Biol. Centralbl._ XXVII, p. 11, 1907; Hertzog, R. O., Temperatureinfluss auf die Entwicklungsgeschwindigkeit der Organismen, _Zeitschr. f. Elektrochemie_, XI, p 820, 1905; Krogh, Quantitative Relation between Temperature and Standard Metabolism, _Int. Zeitschr. f. physik.-chem. Biologie_, I, p. 491, 1914; Pütter, A., Ueber Temperaturkoefficienten, _Zeitschr. f. allgem. Physiol._ XVI, p. 574, 1914. Also Cohen, _Physical Chemistry for Physicians and Biologists_ (English edition), 1903; Pike, F. H., and Scott. E. L., The Regulation of the Physico-chemical Condition of the Organism, _American Naturalist_, Jan. 1915, and various papers quoted therein.
[140] Cf. Errera, L., _L’Optimum_, 1896 (_Rec. d’Oeuvres, Physiol. générale_, pp. 338–368, 1910); Sachs, _Physiologie d. Pflanzen_, 1882, p. 233; Pfeffer, _Pflanzenphysiologie_, ii, p. 78, 1904; and cf. Jost, Ueber die Reactionsgeschwindigkeit im Organismus, _Biol. Centralbl._ XXVI, pp. 225–244, 1906.
[141] After Köppen, _Bull. Soc. Nat. Moscou_, XLIII, pp. 41–110, 1871.
[142] _Botany_, p. 387.
[143] Leitch, I., Some Experiments on the Influence of Temperature on the Rate of Growth in _Pisum sativum, Ann. of Botany_, XXX, pp. 25–46, 1916. (Cf. especially Table III, p. 45.)
[144] Blackman, F. F., Presidential Address in Botany, _Brit. Ass._ Dublin, 1908.
[145] _Rec. de l’Inst. Bot. de Bruxelles_, VI, 1906.
[146] Hertwig, O., Einfluss der Temperatur auf die Entwicklung von _Rana fusca_ und _R. esculenta_, _Arch. f. mikrosk. Anat._ LI, p. 319, 1898. Cf. also Bialaszewicz, K., Beiträge z. Kenntniss d. Wachsthumsvorgänge bei Amphibienembryonen, _Bull. Acad. Sci. de Cracovie_, p. 783, 1908; Abstr. in _Arch. f. Entwicklungsmech._ XXVIII, p. 160, 1909.
[147] Der Grad der Beschleunigung tierischer Entwickelung durch erhöhte Temperatur, _A. f. Entw._ Mech. XX. p. 130, 1905. More recently, Bialaszewicz has determined the coefficient for the rate of segmentation in Rana as being 2·4 per 10° C.
[148] _Das Wachstum des Menschen_, p. 329, 1902.
[149] The _diurnal_ periodicity is beautifully shewn in the case of the Hop by Joh. Schmidt (_C. R. du Laboratoire de Carlsberg_, X, pp. 235–248, Copenhague, 1913).
[150] _Trans. Botan. Soc. Edinburgh_, XVIII, 1891, p. 456.
[151] I had not received, when this was written, Mr Douglass’s paper, On a method of estimating Rainfall by the Growth of Trees, _Bull. Amer. Geograph. Soc._ XLVI, pp. 321–335, 1914. Mr Douglass does not fail to notice the long period here described; but he lays more stress on the occurrence of shorter cycles (of 11, 21 and 33 years), well known to meteorologists. Mr Douglass is inclined (and I think rightly) to correlate the variations in growth directly with fluctuations in rainfall, that is to say with alternate periods of moisture and aridity; but he points out that the temperature curves (and also the sunspot curves) are markedly similar.
[152] It may well be that the effect is not due to light after all; but to increased absorption of heat by the soil, as a result of the long hours of exposure to the sun.
[153] On growth in relation to light, see Davenport, _Exp. Morphology_, II, ch. xvii. In some cases (as in the roots of Peas), exposure to light seems to have no effect on growth; in other cases, as in diatoms (according to Whipple’s experiments, quoted by Davenport, II, p. 423), the effect of light on growth or multiplication is well-marked, measurable, and apparently capable of expression by a logarithmic formula. The discrepancy would seem to arise from the fact that, while light-energy always tends to be absorbed by the chlorophyll of the plant, converted into chemical energy, and stored in the shape of starch or other reserve materials, the actual rate of growth depends on the rate at which these reserves are drawn on: and this is another matter, in which light-energy is no longer directly concerned.
[154] Cf. for instance, Nägeli’s classical account of the effect of change of habitat on Alpine and other plants: _Sitzungsber. Baier. Akad. Wiss._ 1865, pp. 228–284.
[155] Cf. Blackman, F. F., Presidential Address in Botany, _Brit. Ass._ Dublin, 1908. The fact was first enunciated by Baudrimont and St Ange, Recherches sur le développement du fœtus, _Mém. Acad. Sci._ XI, p. 469, 1851.
[156] Cf. Loeb, _Untersuchungen zur physiol. Morphologie der Thiere_, 1892; also Experiments on Cleavage, _J. of Morph._ VII, p. 253, 1892; Zusammenstellung der Ergebnisse einiger Arbeiten über die Dynamik des thierischen Wachsthum, _Arch. f. Entw. Mech._ XV, 1902–3, p. 669; Davenport, On the Rôle of Water in Growth, _Boston Soc. N. H._ 1897; Ida H. Hyde, _Am. J. of Physiol._ XII, 1905, p. 241, etc.
[157] _Pflüger’s Archiv_, LV, 1893.
[158] Beiträge zur Kenntniss der Wachstumsvorgänge bei Amphibienembryonen, _Bull. Acad. Sci. de Cracovie_, 1908, p. 783; cf. _Arch. f. Entw. Mech._ XXVIII, p. 160, 1909; XXXIV, p. 489, 1912.
[159] Fehling, H., _Arch. für Gynaekologie_, XI, 1877; cf. Morgan, _Experimental Zoology_, p. 240, 1907.
[160] Höber, R., Bedeutung der Theorie der Lösungen für Physiologie und Medizin, _Biol. Centralbl._ XIX, 1899; cf. pp. 272–274.
[161] Schmankewitsch has made other interesting observations on change of size and form, after some generations, in relation to change of density; e.g. in the flagellate infusorian _Anisonema acinus_, Bütschli (_Z. f. w. Z._ XXIX, p. 429, 1877).
[162] These “Fezzan-worms,” when first described, were supposed to be “insects’ eggs”; cf. Humboldt, _Personal Narrative_, VI, i, 8, note; Kirby and Spence, Letter X.
[163] Cf. _Introd. à l’étude de la médecine expérimentale_, 1885, p. 110.
[164] Cf. Abonyi, _Z. f. w. Z._ CXIV, p. 134, 1915. But Frédéricq has shewn that the amount of NaCl in the blood of Crustacea (_Carcinus moenas_) varies, and all but corresponds, with the density of the water in which the creature has been kept (_Arch. de Zool. Exp. et Gén._ (2), III, p. xxxv, 1885); and other results of Frédéricq’s, and various data given or quoted by Bottazzi (Osmotischer Druck und elektrische Leitungsfähigkeit der Flüssigkeiten der Organismen, in Asher-Spiro’s _Ergebn. d. Physiologie_, VII, pp. 160–402, 1908) suggest that the case of the brine-shrimps must be looked upon as an extreme or exceptional one.
[165] Cf. Schmankewitsch, _Z. f. w. Zool._ XXV, 1875, XXIX, 1877, etc.; transl. in appendix to Packard’s _Monogr. of N. American Phyllopoda_, 1883, pp. 466–514; Daday de Deés, _Ann. Sci. Nat._ (_Zool._), (9), XI, 1910; Samter und Heymons, _Abh. d. K. pr. Akad. Wiss._ 1902; Bateson, _Mat. for the Study of Variation_, 1894, pp. 96–101; Anikin, _Mitth. Kais. Univ. Tomsk_, XIV: _Zool. Centralbl._ VI, pp. 756–760, 1908; Abonyi, _Z. f. w. Z._ CXIV, pp. 96–168, 1915 (with copious bibliography), etc.
[166] According to the empirical canon of physiology, that (as Frédéricq expresses it) “L’être vivant est agencé de telle manière que chaque influence perturbatrice provoque d’elle-même la mise en activité de l’appareil compensateur qui doit neutraliser et réparer le dommage.”
[167] Such phenomena come precisely under the head of what Bacon called _Instances of Magic_: “By which I mean those wherein the material or efficient cause is scanty and small as compared with the work or effect produced; so that even when they are common, they seem like miracles, some at first sight, others even after attentive consideration. These magical effects are brought about in three ways ... [of which one is] by excitation or invitation in another body, as in the magnet which excites numberless needles without losing any of its virtue, _or in yeast and such-like_.” _Nov. Org._, cap. li.
[168] Monnier, A., Les matières minérales, et la loi d’accroissement des Végétaux, _Publ. de l’Inst. de Bot. de l’Univ. de Genève_ (7), III, 1905. Cf. Robertson, On the Normal Rate of Growth of an Individual, and its Biochemical Significance, _Arch. f. Entw. Mech._ XXV, pp. 581–614, XXVI, pp. 108–118, 1908; Wolfgang Ostwald, _Die zeitlichen Eigenschaften der Entwickelungsvorgänge_, 1908; Hatai, S., Interpretation of Growth-curves from a Dynamical Standpoint, _Anat. Record_, V, p. 373, 1911.
[169] _Biochem. Zeitschr._ II, 1906, p. 34.
[170] Even a crystal may be said, in a sense, to display “autocatalysis”: for the bigger its surface becomes, the more rapidly does the mass go on increasing.
[171] Cf. Loeb, The Stimulation of Growth, _Science_, May 14, 1915.
[172] _B. coli-communis_, according to Buchner, tends to double in 22 minutes; in 24 hours, therefore, a single individual would be multiplied by something like 10^{28}; _Sitzungsber. München. Ges. Morphol. u. Physiol._ III, pp. 65–71, 1888. Cf. Marshall Ward, Biology of _Bacillus ramosus_, etc. _Pr. R. S._ LVIII, 265–468, 1895. The comparatively large infusorian Stylonichia, according to Maupas, would multiply in a month by 10^{43}.
[173] Cf. Enriques, Wachsthum und seine analytisehe Darstellung, _Biol. Centralbl._ 1909, p. 337.
[174] Cf. (_int. al._) Mellor, _Chemical Statics and Dynamics_, 1904, p. 291.
[175] Cf. Robertson, _l.c._
[176] See, for a brief resumé of this subject, Morgan’s _Experimental Zoology_, chap. xvi.
[177] _Amer. J. of Physiol._, X, 1904.
[178] _C.R._ CXXI, CXXII, 1895–96.
[179] Cf. Loeb, _Science_, May 14, 1915.
[180] Cf. Baumann u. Roos, Vorkommen von Iod im Thierkörper, _Zeitschr. für Physiol. Chem._ XXI, XXII, 1895, 6.
[181] Le Néo-Vitalisme, _Rev. Scientifique_, Mars 1911, p. 22 (of reprint).
[182] _La vie et la mort_, p. 43, 1902.
[183] Cf. Dendy, _Evolutionary Biology_, 1912, p. 408; _Brit. Ass. Report_ (Portsmouth), 1911, p. 278.
[184] Lucret. v, 877. “Lucretius nowhere seems to recognise the possibility of improvement or change of species by ‘natural selection’; the animals remain as they were at the first, except that the weaker and more useless kinds have been crushed out. Hence he stands in marked contrast with modern evolutionists.” Kelsey’s note, _ad loc._
[185] Even after we have so narrowed the scope and sphere of natural selection, it is still hard to understand; for the causes of _extinction_ are often wellnigh as hard to comprehend as are those of the _origin_ of species. If we assert (as has been lightly done) that Smilodon perished owing to its gigantic tusks, that Teleosaurus was handicapped by its exaggerated snout, or Stegosaurus weighed down by its intolerable load of armour, we may be reminded of other kindred forms to show that similar conditions did not necessarily lead to extermination, or that rapid extinction ensued apart from any such visible or apparent disadvantages. Cf. Lucas, F. A., On Momentum in Variation, _Amer. Nat._ xli, p. 46, 1907.
[186] See Professor T. H. Morgan’s _Regeneration_ (316 pp.), 1901 for a full account and copious bibliography. The early experiments on regeneration, by Vallisneri, Réaumur, Bonnet, Trembley, Baster, and others, are epitomised by Haller, _Elem. Physiologiae_, VIII, p. 156 _seq._
[187] _Journ. Experim. Zool._ VII, p. 397, 1909.
[188] _Op. cit._ p. 406, Exp. IV.
[189] The experiments of Loeb on the growth of Tubularia in various saline solutions, referred to on p. 125, might as well or better have been referred to under the heading of regeneration, as they were performed on cut pieces of the zoophyte. (Cf. Morgan, _op. cit._ p. 35.)
[190] _Powers of the Creator_, I, p. 7, 1851. See also _Rare and Remarkable Animals_, II, pp. 17–19, 90, 1847.
[191] Lillie, F. R., The smallest Parts of Stentor capable of Regeneration, _Journ. of Morphology_, XII, p. 239, 1897.
[192] Boveri, Entwicklungsfähigkeit kernloser Seeigeleier, etc., _Arch. f. Entw. Mech._ II, 1895. See also Morgan, Studies of the partial larvae of Sphaerechinus, _ibid._ 1895; J. Loeb, On the Limits of Divisibility of Living Matter, _Biol. Lectures_, 1894, etc.
[193] Cf. Przibram, H., Scheerenumkehr bei dekapoden Crustaceen, _Arch. f. Entw. Mech._ XIX, 181–247, 1905; XXV, 266–344, 1907. Emmel, _ibid._ XXII, 542, 1906; Regeneration of lost parts in Lobster, _Rep. Comm. Inland Fisheries, Rhode Island_, XXXV, XXXVI, 1905–6; _Science_ (n.s.), XXVI, 83–87, 1907. Zeleny, Compensatory Regulation, _J. Exp. Zool._ II, 1–102, 347–369, 1905; etc.
[194] Lobsters are occasionally found with two symmetrical claws: which are then usually serrated, sometimes (but very rarely) both blunt-toothed. Cf. Calman, _P.Z.S._ 1906, pp. 633, 634, and _reff._
[195] Wilson, E. B., Reversal of Symmetry in _Alpheus heterochelis_, _Biol. Bull._ IV, p. 197, 1903.
[196] _J. Exp. Zool._ VII, p. 457, 1909.
[197] _Biologica_, III, p. 161, June. 1913.
[198] _Anatomical and Pathological Observations_, p. 3, 1845; _Anatomical Memoirs_, II, p. 392, 1868.
[199] Giard, A., L’œuf et les débuts de l’évolution, _Bull. Sci. du Nord de la Fr._ VIII, pp. 252–258, 1876.
[200] _Entwickelungsvorgänge der Eizelle_, 1876; _Investigations on Microscopic Foams and Protoplasm_, p. 1, 1894.
[201] _Journ. of Morphology_, I, p. 229, 1887.
[202] While it has been very common to look upon the phenomena of mitosis as sufficiently explained by the results _towards which_ they seem to lead, we may find here and there a strong protest against this mode of interpretation. The following is a case in point: “On a tenté d’établir dans la mitose dite primitive plusieurs catégories, plusieurs types de mitose. On a choisi le plus souvent comme base de ces systèmes des concepts abstraits et téléologiques: répartition plus ou moins exacte de la chromatine entre les deux noyaux-fils suivant qu’il y a ou non des chromosomes (_Dangeard_), distribution particulière et signification dualiste des substances nucléaires (substance kinétique et substance générative ou héréditaire, _Hartmann et ses élèves_), etc. Pour moi tous ces essais sont à rejeter catégoriquement à cause de leur caractère finaliste; de plus, ils sont construits sur des concepts non démontrés, et qui parfois représentent des généralisations absolument erronées.” A. Alexeieff, _Archiv für Protistenkunde_, XIX, p. 344, 1913.
[203] This is the old philosophic axiom writ large: _Ignorato motu, ignoratur natura_; which again is but an adaptation of Aristotle’s phrase, ἡ ἀρχὴ τῆς κινήσεως, as equivalent to the “Efficient Cause.” FitzGerald holds that “all explanation consists in a description of underlying motions”; _Scientific Writings_, 1902, p. 385.
[204] As when Nägeli concluded that the organism is, in a certain sense, “vorgebildet”; _Beitr. zur wiss. Botanik_, II, 1860. Cf. E. B. Wilson, _The Cell, etc._, p. 302.
[205] “La matière arrangée par une sagesse divine doit être essentiellement organisée partout ... il y a machine dans les parties de la machine Naturelle à l’infini.” _Sur le principe de la Vie_, p. 431 (Erdmann). This is the very converse of the doctrine of the Atomists, who could not conceive a condition “_ubi dimidiae partis pars semper habebit Dimidiam partem, nec res praefiniet ulla_.”
[206] Cf. an interesting passage from the _Elements_ (I, p. 445, Molesworth’s edit.), quoted by Owen, _Hunterian Lectures on the Invertebrates_, 2nd ed. pp. 40, 41, 1855.
[207] “Wir müssen deshalb den lebenden Zellen, abgesehen von der Molekularstructur der organischen Verbindungen welche sie enthält, noch eine andere und in anderer Weise complicirte Structur zuschreiben, und diese es ist welche wir mit dem Namen _Organisation_ bezeichnen,” Brücke, Die Elementarorganismen, _Wiener Sitzungsber._ XLIV, 1861, p. 386; quoted by Wilson, _The Cell_, etc. p. 289. Cf. also Hardy, _Journ. of Physiol._ XXIV, 1899, p. 159.
[208] Precisely as in the Lucretian _concursus_, _motus_, _ordo_, _positura_, _figurae_, whereby bodies _mutato ordine mutant naturam_.
[209] Otto Warburg, Beiträge zur Physiologie der Zelle, insbesondere über die Oxidationsgeschwindigkeit in Zellen; in Asher-Spiro’s _Ergebnisse der Physiologie_, XIV, pp. 253–337, 1914 (see p. 315). (Cf. Bayliss, _General Physiology_, 1915, p. 590).
[210] Hardy, W. B., On some Problems of Living Matter (Guthrie Lecture), _Tr. Physical Soc. London_, xxviii, p. 99–118, 1916.
[211] As a matter of fact both phrases occur, side by side, in Graham’s classical paper on “Liquid Diffusion applied to Analysis,” _Phil. Trans._ CLI, p. 184, 1861; _Chem. and Phys. Researches_ (ed. Angus Smith), 1876, p. 554.
[212] L. Rhumbler, Mechanische Erklärung der Aehnlichkeit zwischen Magnetischen Kraftliniensystemen und Zelltheilungsfiguren, _Arch. f. Entw. Mech._ XV, p. 482, 1903.
[213] Gallardo, A., Essai d’interpretation des figures caryocinétiques, _Anales del Museo de Buenos-Aires_ (2), II, 1896; La division de la cellule, phenomène bipolaire de caractère electro-colloidal, _Arch. f. Entw. Mech._ XXVIII, 1909, etc.
[214] _Arch. f. Entw. Mech._ III, IV, 1896–97.
[215] On various theories of the mechanism of mitosis, see (e.g.) Wilson, _The Cell in Development_, etc., pp. 100–114; Meves, _Zelltheilung_, in Merkel u. Bonnet’s _Ergebnisse der Anatomie_, etc., VII, VIII, 1897–8; Ida H. Hyde, _Amer. Journ. of Physiol._ XII, pp. 241–275, 1905; and especially Prenant, A., Theories et interprétations physiques de la mitose, _J. de l’Anat. et Physiol._ XLVI, pp. 511–578, 1910.
[216] Hartog, M., Une force nouvelle: le mitokinétisme, _C.R._ 11 Juli, 1910; Mitokinetism in the Mitotic Spindle and in the Polyasters, _Arch. f. Entw. Mech._ XXVII, pp. 141–145, 1909; cf. _ibid._ XL, pp. 33–64, 1914. Cf. also Hartog’s papers in _Proc. R. S._ (B), LXXVI, 1905; _Science Progress_ (n. s.), I, 1907; _Riv. di Scienza_, II, 1908; _C. R. Assoc. fr. pour l’Avancem. des Sc._ 1914, etc.
[217] The configurations, as obtained by the usual experimental methods, were of course known long before Faraday’s day, and constituted the “convergent and divergent magnetic curves” of eighteenth century mathematicians. As Leslie said, in 1821, they were “regarded with wonder by a certain class of dreaming philosophers, who did not hesitate to consider them as the actual traces of an invisible fluid, perpetually circulating between the poles of the magnet.” Faraday’s great advance was to interpret them as indications of _stress in a medium_,—of tension or attraction along the lines, and of repulsion transverse to the lines, of the diagram.
[218] Cf. also the curious phenomenon in a dividing egg described as “spinning” by Mrs G. F. Andrews, _J. of Morph._ XII, pp. 367–389, 1897.
[219] Whitman, _J. of Morph._ II, p. 40, 1889.
[220] “Souvent il n’y a qu’une séparation _physique_ entre le cytoplasme et le suc nucléaire, comme entre deux liquides immiscibles, etc.;” Alexeieff, Sur la mitose dite “primitive,” _Arch. f. Protistenk._ XXIX, p. 357, 1913.
[221] The appearance of “vacuolation” is a result of endosmosis or the diffusion of a less dense fluid into the denser plasma of the cell. _Caeteris paribus_, it is less apparent in marine organisms than in those of freshwater, and in many or most marine Ciliates and even Rhizopods a contractile vacuole has not been observed (Bütschli, in Bronn’s _Protozoa_, p. 1414); it is also absent, and probably for the same reason, in parasitic Protozoa, such as the Gregarines and the Entamoebae. Rossbach shewed that the contractile vacuole of ordinary freshwater Ciliates was very greatly diminished in a 5 per cent. solution of NaCl, and all but disappeared in a 1 per cent. solution of sugar (_Arb. z. z. Inst. Würzburg_, 1872, cf. Massart, _Arch. de Biol._ LX, p. 515, 1889). _Actinophrys sol_, when gradually acclimatised to sea-water, loses its vacuoles, and _vice versa_ (Gruber, _Biol. Centralbl._ IX, p. 22, 1889); and the same is true of Amoeba (Zuelzer, _Arch. f. Entw. Mech._ 1910, p. 632). The gradual enlargement of the contractile vacuole is precisely analogous to the change of size of a bubble until the gases on either side of the film are equally diffused, as described long ago by Draper (_Phil. Mag._ (n. s.), XI, p. 559, 1837). Rhumbler has shewn that contractile or pulsating vacuoles may be well imitated in chloroform-drops, suspended in water in which various substances are dissolved (_Arch. f. Entw. Mech._ VII, 1898, p. 103). The pressure within the contractile vacuole, always greater than without, diminishes with its size, being inversely proportional to its radius; and when it lies near the surface of the cell, as in a Heliozoon, it bursts as soon as it reaches a thinness which its viscosity or molecular cohesion no longer permits it to maintain.
[222] Cf. p. 660.
[223] The elongated or curved “macronucleus” of an Infusorian is to be looked upon as a single mass of chromatin, rather than as an aggregation of particles in a fluid drop, as in the case described. It has a shape of its own, in which ordinary surface-tension plays a very subordinate part.
[224] _Théorie physico-chimique de la Vie_, p. 73, 1910; _Mechanism of Life_, p. 56, 1911.
[225] Whence the name “mitosis” (Greek μίτος, a thread), applied first by Flemming to the whole phenomenon. Kollmann (_Biol. Centralbl._ II, p. 107, 1882) called it _divisio per fila_, or _divisio laqueis implicata_. Many of the earlier students, such as Van Beneden (Rech. sur la maturation de l’œuf, _Arch. de Biol._ IV, 1883), and Hermann (Zur Lehre v. d. Entstehung d. karyokinetischen Spindel, _Arch. f. mikrosk. Anat._ XXXVII, 1891) thought they recognised actual muscular threads, drawing the nuclear material asunder towards the respective foci or poles; and some such view was long maintained by other writers, Boveri, Heidenhain, Flemming, R. Hertwig, and many more. In fact, the existence of contractile threads, or the ascription to the spindle rather than to the poles or centrosomes of the active forces concerned in nuclear division, formed the main tenet of all those who declined to go beyond the “contractile properties of protoplasm” for an explanation of the phenomenon. (Cf. also J. W. Jenkinson, _Q. J. M. S._ XLVIII, p. 471, 1904.)
[226] Cf. Bütschli, O., Ueber die künstliche Nachahmung der karyokinetischen Figur, _Verh. Med. Nat. Ver. Heidelberg_, V, pp. 28–41 (1892), 1897.
[227] Arrhenius, in describing a typical colloid precipitate, does so in terms that are very closely applicable to the ordinary microscopic appearance of the protoplasm of the cell. The precipitate consists, he says, “en un réseau d’une substance solide contenant peu d’eau, dans les mailles duquel est inclus un fluide contenant un peu de colloide dans beaucoup d’eau ... Evidemment cette structure se forme à cause de la petite différence de poids spécifique des deux phases, et de la consistance gluante des particules séparées, qui s’attachent en forme de réseau.” _Rev. Scientifique_, Feb. 1911.
[228] F. Schwartz, in Cohn’s _Beitr. z. Biologie der Pflanzen_, V, p. 1, 1887.
[229] Fischer, _Anat. Anzeiger_, IX, p. 678, 1894, X, p. 769, 1895.
[230] See, in particular, W. B. Hardy, On the structure of Cell Protoplasm, _Journ. of Physiol._ XXIV, pp. 158–207, 1889; also Höber, _Physikalische Chemie der Zelle und der Gewebe_, 1902. Cf. (_int. al._) Flemming, _Zellsubstanz, Kern und Zelltheilung_ 1882, p. 51, etc.
[231] My description and diagrams (Figs 42–51) are based on those of Professor E. B. Wilson.
[232] If the word _permeability_ be deemed too directly suggestive of the phenomena of _magnetism_ we may replace it by the more general term of _specific inductive capacity_. This would cover the particular case, which is by no means an improbable one, of our phenomena being due to a “surface charge” borne by the nucleus itself and also by the chromosomes: this surface charge being in turn the result of a difference in inductive capacity between the body or particle and its surrounding medium. (Cf. footnote, p. 187.)
[233] On the effect of electrical influences in altering the surface-tensions of the colloid particles, see Bredig, _Anorganische Fermente_, pp. 15, 16, 1901.
[234] _The Cell_, etc. p. 66.
[235] Lillie, R. S., _Amer. J. of Physiol._ VIII, p. 282, 1903.
[236] We have not taken account in the above paragraphs of the obvious fact that the supposed symmetrical field of force is distorted by the presence in it of the more or less permeable bodies; nor is it necessary for us to do so, for to that distorted field the above argument continues to apply, word for word.
[237] M. Foster, _Lectures on the History of Physiology_, 1901, p. 62.
[238] _Op. cit._ pp. 110 and 91.
[239] Lamb, A. B., A new Explanation of the Mechanism of Mitosis, _Journ. Exp. Zool._ V, pp. 27–33, 1908.
[240] _Amer. J. of Physiol._ VIII, pp. 273–283, 1903 (_vide supra_, p. 181); cf. _ibid._ XV, pp. 46–84, 1905. Cf. also _Biological Bulletin_, IV, p. 175. 1903.
[241] In like manner Hardy has shewn that colloid particles migrate with the negative stream if the reaction of the surrounding fluid be alkaline, and _vice versa_. The whole subject is much wider than these brief allusions suggest, and is essentially part of Quincke’s theory of Electrical Diffusion or Endosmosis: according to which the particles and the fluid in which they float (or the fluid and the capillary walls through which it flows) each carry a charge, there being a discontinuity of potential at the surface of contact, and hence a field of force leading to powerful tangential or shearing stresses, communicating to the particles a velocity which varies with the density per unit area of the surface charge. See W. B. Hardy’s paper on Coagulation by Electricity, _Journ. of Physiol._ XXIV, p. 288–304, 1899, also Hardy and H. W. Harvey, Surface Electric Charges of Living Cells, _Proc. R. S._ LXXXIV (B), pp. 217–226, 1911, and papers quoted therein. Cf. also E. N. Harvey’s observations on the convection of unicellular organisms in an electric field (Studies on the Permeability of Cells, _Journ. of Exper. Zool._ X, pp. 508–556, 1911).
[242] On Differences in Electrical Potential in Developing Eggs, _Amer. Journ. of Physiol._ XII, pp. 241–275, 1905. This paper contains an excellent summary of various physical theories of the segmentation of the cell.
[243] Gray has recently demonstrated a temporary increase of electrical conductivity in sea-urchin eggs during the process of fertilisation (The Electrical Conductivity of fertilised and unfertilised Eggs, _Journ. Mar. Biol. Assoc._ X, pp. 50–59, 1913).
[244] Schewiakoff, Ueber die karyokinetische Kerntheilung der _Euglypha alveolata, Morph. Jahrb._ XIII, pp. 193–258, 1888 (see p. 216).
[245] Coe, W. R., Maturation and Fertilization of the Egg of Cerebratulus, _Zool. Jahrbücher_ (_Anat. Abth._), XII, pp. 425–476, 1899.
[246] Thus, for example, Farmer and Digby (On Dimensions of Chromosomes considered in relation to Phylogeny, _Phil. Trans._ (B), CCV, pp. 1–23, 1914) have been at pains to shew, in confutation of Meek (_ibid._ CCIII, pp. 1–74, 1912), that the width of the chromosomes cannot be correlated with the order of phylogeny.
[247] Cf. also _Arch. f. Entw. Mech._ X, p. 52, 1900.
[248] Cf. Loeb, _Am. J. of Physiol._ VI, p. 32, 1902; Erlanger, _Biol. Centralbl._ XVII, pp. 152, 339, 1897; Conklin, _Biol. Lectures_, _Woods Holl_, p. 69, etc. 1898–9.
[249] Robertson, T. B., Note on the Chemical Mechanics of Cell Division, _Arch. f. Entw. Mech._ XXVII, p. 29, 1909, XXXV, p. 692. 1913. Cf. R. S. Lillie, _J. Exp. Zool._ XXI, pp. 369–402, 1916.
[250] Cf. D’Arsonval, _Arch. de Physiol._ p. 460, 1889; Ida H. Hyde, _op. cit._ p. 242.
[251] Cf. Plateau’s remarks (_Statique des liquides_, II, p. 154) on the _tendency_ towards equilibrium, rather than actual equilibrium, in many of his systems of soap-films.
[252] But under artificial conditions, “polyspermy” may take place, e.g. under the action of dilute poisons, or of an abnormally high temperature, these being all, doubtless, conditions under which the surface-tension is diminished.
[253] Fol, H., _Recherches sur la fécondation_, 1879. Roux, W., Beiträge zur Entwickelungsmechanik des Embryo, _Arch. f. Mikr. Anat._ XIX, 1887. Whitman, C. O., Oökinesis, _Journ. of Morph._ I, 1887.
[254] Wilson. _The Cell_, p. 77.
[255] Eight and twelve are by much the commonest numbers, six and sixteen coming next in order. If we may judge by the list given by E. B. Wilson (_The Cell_, p. 206), over 80 % of the observed cases lie between 6 and 16, and nearly 60 % between 8 and 12.
[256] _Theory of Cells_, p. 191.
[257] _The Cell in Development_, etc. p. 59; cf. pp. 388, 413.
[258] E.g. Brücke, _Elementarorganismen_, p. 387: “Wir müssen in der Zelle einen kleinen Thierleib sehen, und dürfen die Analogien, welche zwischen ihr und den kleinsten Thierformen existiren, niemals aus den Augen lassen.”
[259] Whitman, C. O., The Inadequacy of the Cell-theory, _Journ. of Morphol._ VIII, pp. 639–658, 1893; Sedgwick, A., On the Inadequacy of the Cellular Theory of Development, _Q.J.M.S._ XXXVII, pp. 87–101, 1895, XXXVIII, pp. 331–337, 1896. Cf. Bourne, G. C., A Criticism of the Cell-theory; being an answer to Mr Sedgwick’s article, etc., _ibid._ XXXVIII, pp. 137–174, 1896.
[260] Cf. Hertwig, O., _Die Zelle und die Gewebe_, 1893, p. 1; “Die Zellen, in welche der Anatom die pflanzlichen und thierischen Organismen zerlegt, sind die Träger der Lebensfunktionen; sie sind, wie Virchow sich ausgedrückt hat, die ‘Lebenseinheiten.’ Von diesem Gesichtspunkt aus betrachtet, erscheint der Gesammtlebensprocess eines zusammengesetzten Organismus nichts Anderes zu sein als das höchst verwickelte Resultat der einzelnen Lebensprocesse seiner zahlreichen, verschieden functionirenden Zellen.”
[261] _Journ. of Morph._ VIII, p. 653, 1893.
[262] Neue Grundlegungen zur Kenntniss der Zelle, _Morph. Jahrb._ VIII, pp. 272, 313, 333, 1883.
[263] _Journ. of Morph._ II, p. 49, 1889.
[264] _Phil. Trans._ CLI, p. 183, 1861; _Researches_, ed. Angus Smith, 1877, p. 553.
[265] Cf. Kelvin, On the Molecular Tactics of a Crystal, _The Boyle Lecture_, Oxford, 1893, _Baltimore Lectures_, 1904, pp. 612–642. Here Kelvin was mainly following Bravais’s (and Frankenheim’s) theory of “space-lattices,” but he had been largely anticipated by the crystallographers. For an account of the development of the subject in modern crystallography, by Sohncke, von Fedorow, Schönfliess, Barlow and others, see Tutton’s _Crystallography_, chap. ix, pp. 118–134, 1911.
[266] In a homogeneous crystalline arrangement, _symmetry_ compels a locus of one property to be a plane or set of planes; the locus in this case being that of least surface potential energy.
[267] This is what Graham called the _water of gelatination_, on the analogy of _water of crystallisation_; _Chem. and Phys. Researches_, p. 597.
[268] Here, in a non-crystalline or random arrangement of particles, symmetry ensures that the potential energy shall be the same per unit area of all surfaces; and it follows from geometrical considerations that the total surface energy will be least if the surface be spherical.
[269] Lehmann, O., _Flüssige Krystalle, sowie Plasticität von Krystallen im allgemeinen_, etc., 264 pp. 39 pll., Leipsig, 1904. For a semi-popular, illustrated account, see Tutton’s _Crystals_ (Int. Sci. Series), 1911.
[270] As Graham said of an allied phenomenon (the so-called blood-crystals of Funke), it “illustrates the maxim that in nature there are no abrupt transitions, and that distinctions of class are never absolute.”
[271] Cf. Przibram, H., Kristall-analogien zur Entwickelungsmechanik der Organismen, _Arch. f. Entw. Mech._ XXII, p. 207, 1906 (with copious bibliography); Lehmann, Scheinbar lebende Kristalle und Myelinformen, _ibid._ XXVI, p. 483, 1908.
[272] The idea of a “surface-tension” in liquids was first enunciated by Segner, _De figuris superficierum fluidarum_, in _Comment. Soc. Roy. Göttingen_, 1751, p. 301. Hooke, in the _Micrographia_ (1665, Obs. VIII, etc.), had called attention to the globular or spherical form of the little morsels of steel struck off by a flint, and had shewn how to make a powder of such spherical grains, by heating fine filings to melting point. “This Phaenomenon” he said “proceeds from a propriety which belongs to all kinds of fluid Bodies more or less, and is caused by the Incongruity of the Ambient and included Fluid, which so acts and modulates each other, that they acquire, as neer as is possible, a _spherical_ or _globular_ form....”
[273] _Science of Mechanics_, 1902, p. 395; see also Mach’s article Ueber die physikalische Bedeutung der Gesetze der Symmetrie, _Lotos_, XXI, pp. 139–147, 1871.
[274] Similarly, Sir David Brewster and others made powerful lenses by simply dropping small drops of Canada balsam, castor oil, or other strongly refractive liquids, on to a glass plate: _On New Philosophical Instruments_ (Description of a new Fluid Microscope), Edinburgh, 1813, p. 413.
[275] Beiträge z. Physiologie d. Protoplasma, _Pflüger’s Archiv_, II, p. 307, 1869.
[276] _Poggend. Annalen_, XCIV, pp. 447–459, 1855. Cf. Strethill Wright, _Phil. Mag._ Feb. 1860.
[277] Haycraft and Carlier pointed out (_Proc. R.S.E._ XV, pp. 220–224, 1888) that the amoeboid movements of a white blood-corpuscle are only manifested when the corpuscle is in contact with some solid substance: while floating freely in the plasma or serum of the blood, these corpuscles are spherical, that is to say they are at rest and in equilibrium. The same fact has recently been recorded anew by Ledingham (On Phagocytosis from an adsorptive point of view, _Journ. of Hygiene_, XII, p. 324, 1912). On the emission of pseudopodia as brought about by changes in surface tension, see also (_int. al._) Jensen, Ueber den Geotropismus niederer Organismen, _Pflüger’s Archiv_, LIII, 1893. Jensen remarks that in Orbitolites, the pseudopodia issuing through the pores of the shell first float freely, then as they grow longer bend over till they touch the ground, whereupon they begin to display amoeboid and streaming motions. Verworn indicates (_Allg. Physiol._ 1895, p. 429), and Davenport says (_Experim. Morphology_, II, p. 376) that “this persistent clinging to the substratum is a ‘thigmotropic’ reaction, and one which belongs clearly to the category of ‘response.’ ” (Cf. Pütter, Thigmotaxis bei Protisten, _A. f. Physiol._ 1900, Suppl. p. 247.) But it is not clear to my mind that to account for this simple phenomenon we need invoke other factors than gravity and surface-action.
[278] Cf. Pauli, _Allgemeine physikalische Chemie d. Zellen u. Gewebe_, in Asher-Spiro’s _Ergebnisse der Physiologie_, 1912; Przibram, _Vitalität_, 1913, p. 6.
[279] The surface-tension theory of protoplasmic movement has been denied by many. Cf. (e.g.), Jennings, H. S., Contributions to the Study of the Behaviour of the Lower Organisms, _Carnegie Inst._ 1904, pp. 130–230; Dellinger, O. P., Locomotion of Amoebae, etc. _Journ. Exp. Zool._ III, pp. 337–357, 1906; also various papers by Max Heidenhain, in _Anatom. Hefte_ (Merkel und Bonnet), etc.
[280] These various movements of a liquid surface, and other still more striking movements such as those of a piece of camphor floating on water, were at one time ascribed by certain physicists to a peculiar force, _sui generis_, the _force épipolique_ of Dutrochet: until van der Mensbrugghe shewed that differences of surface tension were enough to account for this whole series of phenomena (Sur la tension superficielle des liquides considérée au point de vue de certains mouvements observés à leur surface, _Mém. Cour. Acad. de Belgique_, XXXIV, 1869; cf. Plateau, p. 283).
[281] Cf. _infra_, p. 306.
[282] Cf. p. 32.
[283] Or, more strictly speaking, unless its thickness be less than twice the range of the molecular forces.
[284] It follows that the tension, depending only on the surface-conditions, is independent of the thickness of the film.
[285] This simple but immensely important formula is due to Laplace (_Mécanique Céleste_, Bk. x. suppl. _Théorie de l’action capillaire_, 1806).
[286] Sur la surface de révolution dont la courbure moyenne est constante, _Journ. de M. Liouville_, VI, p. 309, 1841.
[287] See _Liquid Drops and Globules_, 1914, p. 11. Robert Boyle used turpentine in much the same way. For other methods see Plateau, _op. cit._ p. 154.
[288] Felix Plateau recommends the use of a weighted thread, or plumb-line, drawn up out of a jar of water or oil; _Phil. Mag._ XXXIV, p. 246, 1867.
[289] Cf. Boys, C. V., On Quartz Fibres, _Nature_, July 11, 1889; Warburton, C., The Spinning Apparatus of Geometric Spiders, _Q.J.M.S._ XXXI, pp. 29–39, 1890.
[290] J. Blackwall, _Spiders of Great Britain_ (Ray Society), 1859, p. 10; _Trans. Linn. Soc._ XVI, p. 477, 1833.
[291] The intermediate spherules appear, with great regularity and beauty, whenever a liquid jet breaks up into drops; see the instantaneous photographs in Poynting and Thomson’s _Properties of Matter_, pp. 151, 152, (ed. 1907).
[292] Kühne, _Untersuchungen über das Protoplasma_, 1864, p. 75, etc.
[293] _A Study of Splashes_, 1908, p. 38, etc.; Segmentation of a Liquid Annulus, _Proc. Roy. Soc._ XXX, pp. 49–60, 1880.
[294] Cf. _ibid._ pp. 17, 77. The same phenomenon is beautifully and continuously evident when a strong jet of water from a tap impinges on a curved surface and then shoots off it.
[295] See a _Study of Splashes_, p. 54.
[296] A case which we have not specially considered, but which may be found to deserve consideration in biology, is that of a cell or drop suspended in a liquid of _varying_ density, for instance in the upper layers of a fluid (e.g. sea-water) at whose surface condensation is going on, so as to produce a steady density-gradient. In this case the normally spherical drop will be flattened into an oval form, with its maximum surface-curvature lying at the level where the densities of the drop and the surrounding liquid are just equal. The sectional outline of the drop has been shewn to be not a true oval or ellipse, but a somewhat complicated quartic curve. (Rice, _Phil. Mag._ Jan. 1915.)
[297] Indeed any non-isotropic _stiffness_, even though _T_ remained uniform, would simulate, and be indistinguishable from, a condition of non-stiffness and non-isotropic _T_.
[298] A non-symmetry of _T_ and _T′_ might also be capable of explanation as a result of “liquid crystallisation.” This hypothesis is referred to, in connection with the blood-corpuscles, on p. 272.
[299] The case of the snow-crystals is a particularly interesting one; for their “distribution” is in some ways analogous to what we find, for instance, among our microscopic skeletons of Radiolarians. That is to say, we may one day meet with myriads of some one particular form or species only, and another day with myriads of another; while at another time and place we may find species intermingled in inexhaustible variety. (Cf. e.g. J. Glaisher, _Ill. London News_, Feb. 17, 1855; _Q.J.M.S._ III, pp. 179–185, 1855).
[300] Cf. Bergson, _Creative Evolution_, p. 107: “Certain Foraminifera have not varied since the Silurian epoch. Unmoved witnesses of the innumerable revolutions that have upheaved our planet, the Lingulae are today what they were at the remotest times of the palaeozoic era.”
[301] Ray Lankester, _A.M.N.H._ (4), XI, p. 321, 1873.
[302] Leidy, Parasites of the Termites, _J. Nat. Sci., Philadelphia_, VIII, pp. 425–447, 1874–81; cf. Saville Kent’s _Infusoria_, II, p. 551.
[303] _Op. cit._ p. 79.
[304] Brady, _Challenger Monograph_, pl. XX, p. 233.
[305] That the Foraminifera not only can but do hang from the surface of the water is confirmed by the following apt quotation which I owe to Mr E. Heron-Allen: “Quand on place, comme il a été dit, le dépôt provenant du lavage des fucus dans un flacon que l’on remplit de nouvelle eau, on voit au bout d’une heure environ les animaux [_Gromia dujardinii_] se mettre en mouvement et commencer à grimper. Six heures après ils tapissent l’extérieur du flacon, de sorte que les plus élevés sont à trente-six ou quarante-deux millimetres du fond; le lendemain beaucoup d’entre eux, _après avoir atteint le niveau du liquide, ont continué à ramper à sa surface, en se laissant pendre au-dessous_ comme certains mollusques gastéropodes.” (Dujardin, F., Observations nouvelles sur les prétendus céphalopodes microscopiques, _Ann. des Sci. Nat._ (2), III, p. 312, 1835.)
[306] Cf. Boas, _Spolia Atlantica_, 1886, pl. 6.
[307] This cellular pattern would seem to be related to the “cohesion figures” described by Tomlinson in various surface-films (_Phil. Mag._ 1861 to 1870); to the “tesselated structure” in liquids described by Professor James Thomson in 1882 (_Collected Papers_, p. 136); and to the _tourbillons cellulaires_ of Prof. H. Bénard (_Ann. de Chimie_ (7), XXIII, pp. 62–144, 1901, (8), XXIV, pp. 563–566, 1911), _Rev. génér. des Sci._ XI, p. 1268, 1900; cf. also E. H. Weber. (_Poggend. Ann._ XCIV, p. 452, 1855, etc.). The phenomenon is of great interest and various appearances have been referred to it, in biology, geology, metallurgy and even astronomy: for the flocculent clouds in the solar photosphere shew an analogous configuration. (See letters by Kerr Grant, Larmor, Wager and others, in _Nature_, April 16 to June 11, 1914.) In many instances, marked by strict symmetry or regularity, it is very possible that the interference of waves or ripples may play its part in the phenomenon. But in the majority of cases, it is fairly certain that localised centres of action, or of diminished tension, are present, such as might be provided by dust-particles in the case of Darling’s experiment (cf. _infra_, p. 590).
[308] Ueber physikalischen Eigenschaften dünner, fester Lamellen, _S.B. Berlin. Akad._ 1888, pp. 789, 790.
[309] Certain palaeontologists (e.g. Haeusler and Spandel) have maintained that in each family or genus the plain smooth-shelled forms are the primitive and ancient ones, and that the ribbed and otherwise ornamented shells make their appearance at later dates in the course of a definite evolution (cf. Rhumbler, _Foraminiferen der Plankton-Expedition_, 1911, i, p. 21). If this were true it would be of fundamental importance: but this book of mine would not deserve to be written.
[310] _A Study of Splashes_, p. 116.
[311] See _Silliman’s Journal_, II, p. 179, 1820; and cf. Plateau, _op. cit._ II, pp. 134, 461.
[312] The presence or absence of the contractile vacuole or vacuoles is one of the chief distinctions, in systematic zoology, between the Heliozoa and the Radiolaria. As we have seen on p. 165 (footnote), it is probably no more than a physical consequence of the different conditions of existence in fresh water and in salt.
[313] Cf. Doflein, _Lehrbuch der Protozoenkunde_, 1911, p. 422.
[314] Cf. Minchin, _Introduction to the Study of the Protozoa_, 1914 p. 293, Fig. 127.
[315] Cf. C. A. Kofoid and Olive Swezy, On Trichomonad Flagellates, etc. _Pr. Amer. Acad. of Arts and Sci._ LI, pp. 289–378, 1915.
[316] D. L. Mackinnon, Herpetomonads from the Alimentary Tract of certain Dungflies, _Parasitology_, III, p. 268, 1910.
[317] _Proc. Roy. Soc._ XII, pp. 251–257, 1862–3.
[318] Cf. (_int. al._) Lehmann, Ueber scheinbar lebende Kristalle und Myelinformen, _Arch. f. Entw. Mech._ XXVI, p. 483, 1908; _Ann. d. Physik_, XLIV, p. 969, 1914.
[319] Cf. B. Moore and H. C. Roaf, On the Osmotic Equilibrium of the Red Blood Corpuscle, _Biochem. Journal_, III, p. 55, 1908.
[320] For an attempt to explain the form of a blood-corpuscle by surface-tension alone, see Rice, _Phil. Mag._ Nov. 1914; but cf. Shorter, _ibid._ Jan. 1915.
[321] Koltzoff, N. K., Studien über die Gestalt der Zelle, _Arch. f. mikrosk. Anat._ LXVII, pp. 364–571, 1905; _Biol. Centralbl._ XXIII, pp. 680–696, 1903, XXVI, pp. 854–863, 1906; _Arch. f. Zellforschung_, II, pp. 1–65, 1908, VII, pp. 344–423, 1911; _Anat. Anzeiger_, XLI, pp. 183–206, 1912.
[322] Cf. _supra_, p. 129.
[323] As Bethe points out (Zellgestalt, Plateausche Flüssigkeitstigur und Neurofibrille, _Anat. Anz._ XL. p. 209, 1911), the spiral fibres of which Koltzoff speaks must lie _in the surface_, and not within the substance, of the cell whose conformation is affected by them.
[324] See for a further but still elementary account, Michaelis, _Dynamics of Surfaces_, 1914, p. 22 _seq._; Macallum, _Oberflächenspannung und Lebenserscheinungen_, in Asher-Spiro’s _Ergebnisse der Physiologie_, XI, pp. 598–658, 1911; see also W. W. Taylor’s _Chemistry of Colloids_, 1915, p. 221 _seq._, Wolfgang Ostwald, _Grundriss der Kolloidchemie_, 1909, and other text-books of physical chemistry; and Bayliss’s _Principles of General Physiology_, pp. 54–73, 1915.
[325] The first instance of what we now call an adsorptive phenomenon was observed in soap-bubbles. Leidenfrost, in 1756, was aware that the outer layer of the bubble was covered by an “oily” layer. A hundred years later Dupré shewed that in a soap-solution the soap tends to concentrate at the surface, so that the surface-tension of a very weak solution is very little different from that of a strong one (_Théorie mécanique de la chaleur_, 1869, p. 376; cf. Plateau, II, p. 100).
[326] This identical phenomenon was the basis of Quincke’s theory of amoeboid movement (Ueber periodische Ausbreitung von Flüssigkeitsoberflächen, etc., _SB. Berlin. Akad._ 1888, pp. 791–806; cf. _Pflüger’s Archiv_, 1879, p. 136).
[327] J. Willard Gibbs, Equilibrium of Heterogeneous Substances, _Tr. Conn. Acad._ III, pp. 380–400, 1876, also in _Collected Papers_, I, pp. 185–218, London, 1906; J. J. Thomson, _Applications of Dynamics to Physics and Chemistry_, 1888 (Surface tension of solutions), p. 190. See also (_int. al._) the various papers by C. M. Lewis, _Phil. Mag._ (6), XV, p. 499, 1908, XVII, p. 466, 1909, _Zeitschr. f. physik. Chemie_, LXX, p. 129, 1910; Milner, _Phil. Mag._ (6), XIII, p. 96, 1907, etc.
[328] G. F. FitzGerald, On the Theory of Muscular Contraction, _Brit. Ass. Rep._ 1878; also in _Scientific Writings_, ed. Larmor, 1902, pp. 34, 75. A. d’Arsonval, Relations entre l’électricité animale et la tension superficielle, _C. R._ CVI, p. 1740. 1888; cf. A. Imbert, Le mécanisme de la contraction musculaire, déduit de la considération des forces de tension superficielle, _Arch. de Phys._ (5), IX, pp. 289–301, 1897.
[329] Ueber die Natur der Bindung der Gase im Blut und in seinen Bestandtheilen, _Kolloid. Zeitschr._ II, pp. 264–272, 294–301, 1908; cf. Loewy, Dissociationsspannung des Oxyhaemoglobin im Blut, _Arch. f. Anat. und Physiol._ 1904, p. 231.
[330] We may trace the first steps in the study of this phenomenon to Melsens, who found that thin films of white of egg become firm and insoluble (Sur les modifications apportées à l’albumine ... par l’action purement mécanique, _C. R. Acad. Sci._ XXXIII, p. 247, 1851); and Harting made similar observations about the same time. Ramsden has investigated the same subject, and also the more general phenomenon of the formation of albuminoid and fatty membranes by adsorption: cf. Koagulierung der Eiweisskörper auf mechanischer Wege, _Arch. f. Anat. u. Phys._ (_Phys. Abth._) 1894, p. 517; Abscheidung fester Körper in Oberflächenschichten _Z. f. phys. Chem._ XLVII, p. 341, 1902; _Proc. R. S._ LXXII, p. 156, 1904. For a general review of the whole subject see H. Zangger, Ueber Membranen und Membranfunktionen, in Asher-Spiro’s _Ergebnisse der Physiologie_, VII, pp. 99–160, 1908.
[331] Cf. Taylor, _Chemistry of Colloids_, p. 252.
[332] Strasbürger, Ueber Cytoplasmastrukturen, etc. _Jahrb. f. wiss. Bot._ XXX, 1897; R. A. Harper, Kerntheilung und freie Zellbildung im Ascus, _ibid._; cf. Wilson, _The Cell in Development, etc._ pp. 53–55.
[333] Cf. A. Gurwitsch, _Morphologie und Biologie der Zelle_, 1904, pp. 169–185; Meves, Die Chondriosomen als Träger erblicher Anlagen, _Arch. f. mikrosk. Anat._ 1908, p. 72; J. O. W. Barratt, Changes in Chondriosomes, etc. _Q.J.M.S._ LVIII, pp. 553–566, 1913, etc.; A. Mathews, Changes in Structure of the Pancreas Cell, etc., _J. of Morph._ XV (Suppl.), pp. 171–222, 1899.
[334] The question whether chromosomes, chondriosomes or chromidia be the true vehicles or transmitters of “heredity” is not without its analogy to the older problem of whether the pineal gland or the pituitary body were the actual seat and domicile of the soul.
[335] Cf. C. C. Dobell, Chromidia and the Binuclearity Hypotheses; a review and a criticism, _Q.J.M.S._ LIII, 279–326, 1909; Prenant, A., Les Mitochondries et l’Ergastoplasme, _Journ. de l’Anat. et de la Physiol._ XLVI, pp. 217–285, 1910 (both with copious bibliography).
[336] Traube in particular has maintained that in differences of surface-tension we have the origin of the active force productive of osmotic currents, and that herein we find an explanation, or an approach to an explanation, of many phenomena which were formerly deemed peculiarly “vital” in their character. “Die Differenz der Oberflächenspannungen oder der Oberflächendruck eine Kraft darstellt, welche als treibende Kraft der Osmose, an die Stelle des nicht mit dem Oberflächendruck identischen osmotischen Druckes, zu setzen ist, etc.” (Oberflächendruck und seine Bedeutung im Organismus, _Pflüger’s Archiv_, CV, p. 559, 1904.) Cf. also Hardy (_Pr. Phys. Soc._ XXVIII, p. 116, 1916), “If the surface film of a colloid membrane separating two masses of fluid were to change in such a way as to lower the potential of the water in it, water would enter the region from both sides at once. But if the change of state were to be propagated as a wave of change, starting at one face and dying out at the other face, water would be carried along from one side of the membrane to the other. A succession of such waves would maintain a flow of fluid.”
[337] On the Distribution of Potassium in animal and vegetable Cells; _Journ. of Physiol._ XXXII, p. 95, 1905.
[338] The reader will recognise that there is a fundamental difference, and contrast, between such experiments as these of Professor Macallum’s and the ordinary staining processes of the histologist. The latter are (as a general rule) purely empirical, while the former endeavour to reveal the true microchemistry of the cell. “On peut dire que la microchimie n’est encore qu’à la période d’essai, et que l’avenir de l’histologie et spécialement de la cytologie est tout entier dans la microchimie” (Prenant, A., Méthodes et résultats de la Microchimie, _Journ. de l’Anat. et de la Physiol._ XLVI, pp. 343–404, 1910).
[339] Cf. Macallum, Presidential Address, Section I, _Brit. Ass. Rep._ (Sheffield), 1910, p. 744.
[340] In accordance with a simple _corollary_ to the Gibbs-Thomson law.
[341] It can easily be proved (by equating the increase of energy stored in an increased surface to the work done in increasing that surface), that the tension measured per unit breadth, _T__{_ab_}, is equal to the energy per unit area, _E__{_ab_}.
[342] The presence of this little liquid “bourrelet,” drawn from the material of which the partition-walls themselves are composed, is obviously tending to a reduction of the internal surface-area. And it may be that it is as well, or better, accounted for on this ground than on Plateau’s assumption that it represents a “surface of continuity.”
[343] A similar “bourrelet” is admirably seen at the line of junction between a floating bubble and the liquid on which it floats; in which case it constitutes a “masse annulaire,” whose mathematical properties and relation to the form of the _nearly_ hemispherical bubble, have been investigated by van der Mensbrugghe (cf. Plateau, _op. cit._, p. 386). The form of the superficial vacuoles in Actinophrys or Actinosphaerium involves an identical problem.
[344] In an actual calculation we must of course always take account of the tensions on _both sides_ of each film or membrane.
[345] Hofmeister, _Pringsheim’s Jahrb._ III, p. 272, 1863; _Hdb. d. physiol. Bot._ I, 1867, p. 129.
[346] Sachs, Ueber die Anordnung der Zellen in jüngsten Pflanzentheilen, _Verh. phys. med. Ges. Würzburg_, XI, pp. 219–242, 1877; Ueber Zellenanordnung und Wachsthum, _ibid._ XII, 1878; Ueber die durch Wachsthum bedingte Verschiebung kleinster Theilchen in trajectorischen Curven, _Monatsber. k. Akad. Wiss. Berlin_, 1880; _Physiology of Plants_, chap. xxvii, pp. 431–459, Oxford, 1887.
[347] Schwendener, Ueber den Bau und das Wachsthum des Flechtenthallus, _Naturf. Ges. Zürich_, Febr. 1860, pp. 272–296.
[348] Reinke, _Lehrbuch der Botanik_, 1880, p. 519.
[349] Cf. Leitgeb, _Unters. über die Lebermoose_, II, p. 4, Graz, 1881.
[350] Rauber, Neue Grundlegungen zur Kenntniss der Zelle, _Morph. Jahrb._ VIII, pp. 279, 334, 1882.
[351] _C. R. Acad. Sc._ XXXIII, p. 247, 1851; _Ann. de chimie et de phys._ (3), XXXIII, p. 170, 1851; _Bull. R. Acad. Belg._ XXIV, p. 531, 1857.
[352] Klebs, _Biolog. Centralbl._ VII, pp. 193–201, 1887.
[353] L. Errera, Sur une condition fondamentale d’équilibre des cellules vivantes, _C. R._, CIII, p. 822, 1886; _Bull. Soc. Belge de Microscopie_, XIII, Oct. 1886; _Recueil d’œuvres_ (_Physiologie générale_), 1910, pp. 201–205.
[354] L. Chabry, Embryologie des Ascidiens, _J. Anat. et Physiol._ XXIII, p. 266, 1887.
[355] Robert, Embryologie des Troques, _Arch. de Zool. exp. et gén._ (3), X, 1892.
[356] “Dass der Furchungsmodus etwas für das Zukünftige unwesentliches ist,” _Z. f. w. Z._ LV, 1893, p. 37. With this statement compare, or contrast, that of Conklin, quoted on p. 4; cf. also pp. 157, 348 (footnotes).
[357] de Wildeman, Etudes sur l’attache des cloisons cellulaires, _Mém. Couronn. de l’Acad. R. de Belgique_, LIII, 84 pp., 1893–4.
[358] It was so termed by Conklin in 1897, in his paper on Crepidula (_J. of Morph._ XIII, 1897). It is the _Querfurche_ of Rabl (_Morph. Jahrb._ V, 1879); the _Polarfurche_ of O. Hertwig (_Jen. Zeitschr._ XIV, 1880); the _Brechungslinie_ of Rauber (Neue Grundlage zur K. der Zelle, _M. Jb._ VIII, 1882). It is carefully discussed by Robert, Dév. des Troques, _Arch. de Zool. Exp. et Gén._ (3), X, 1892, p. 307 seq.
[359] Thus Wilson (_J. of Morph._ VIII, 1895) declared that in Amphioxus the polar furrow was occasionally absent, and Driesch took occasion to criticise and to throw doubt upon the statement (_Arch. f. Entw. Mech._ I, 1895, p. 418).
[360] Precisely the same remark was made long ago by Driesch: “Das so oft sehematisch gezeichnete Vierzellenstadium mit zwei sich in zwei Punkten scheidende Medianen kann man wohl getrost aus der Reihe des Existierenden streichen,” _Entw. mech. Studien, Z. f. w. Z._ LIII, p. 166, 1892. Cf. also his _Math. mechanische Bedeutung morphologischer Probleme der Biologie_, Jena, 59 pp. 1891.
[361] Compare, however, p. 299.
[362] _Ricreatione dell’ occhio e della mente, nell’ Osservatione delle Chiocciole_, Roma, 1681.
[363] Cf. some of J. H. Vincent’s photographs of ripples, in _Phil. Mag._ 1897–1899; or those of F. R. Watson, in _Phys. Review_, 1897, 1901, 1916. The appearance will depend on the rate of the wave, and in turn on the surface-tension; with a low tension one would probably see only a moving “jabble.” FitzGerald thought diatom-patterns might be due to electromagnetic vibrations (_Works_, p. 503, 1902).
[364] Cushman, J. A. and Henderson, W. P., _Amer. Nat._ XL, pp. 797–802, 1906.
[365] This does not merely neglect the _broken_ ones but _all_ whose centres lie between this circle and a hexagon inscribed in it.
[366] For more detailed calculations see a paper by “H.M.” [? H. Munro], in _Q. J. M. S._ VI, p. 83, 1858.
[367] Cf. Hartog, The Dual Force of the Dividing Cell, _Science Progress_ (n.s.), I, Oct. 1907, and other papers. Also Baltzer, _Ueber mehrpolige Mitosen bei Seeigeleiern_, Inaug. Diss. 1908.
[368] Observations sur les Abeilles, _Mém. Acad. Sc. Paris_, 1712, p. 299.
[369] As explained by Leslie Ellis, in his essay “On the Form of Bees’ Cells,” in _Mathematical and other Writings_, 1853, p. 353; cf. O. Terquem, _Nouv. Ann. Math._ 1856, p. 178.
[370] _Phil. Trans._ XLII, 1743, pp. 565–571.
[371] _Mém. de l’Acad. de Berlin_, 1781.
[372] Cf. Gregory, _Examples_, p. 106, Wood’s _Homes without Hands_, 1865, p. 428, Mach, _Science of Mechanics_, 1902, p. 453, etc., etc.
[373] _Origin of Species_, ch. VIII (6th ed., p. 221). The cells of various bees, humble-bees and social wasps have been described and mathematically investigated by K. Müllenhoff, _Pflüger’s Archiv_ XXXII, p. 589, 1883; but his many interesting results are too complex to epitomise. For figures of various nests and combs see (e.g.) von Büttel-Reepen, _Biol. Centralbl._ XXXIII, pp. 4, 89, 129, 183, 1903.
[374] Darwin had a somewhat similar idea, though he allowed more play to the bee’s instinct or conscious intention. Thus, when he noticed certain half-completed cell-walls to be concave on one side and convex on the other, but to become perfectly flat when restored for a short time to the hive, he says: “It was absolutely impossible, from the extreme thinness of the little plate, that they could have effected this by gnawing away the convex side; and I suspect that the bees in such cases stand on opposite sides and push and bend the ductile and warm wax (which as I have tried is easily done) into its proper intermediate plane, and thus flatten it.”
[375] Since writing the above, I see that Müllenhoff gives the same explanation, and declares that the waxen wall is actually a _Flüssigkeitshäutchen_, or liquid film.
[376] Bonnet criticised Buffon’s explanation, on the ground that his description was incomplete; for Buffon took no account of the Maraldi pyramids.
[377] Buffon, _Histoire Naturelle_, IV, p. 99. Among many other papers on the Bee’s cell, see Barclay, _Mem. Wernerian Soc._ II, p. 259 (1812), 1818; Sharpe, _Phil. Mag._ IV, 1828, pp. 19–21; L. Lalanne, _Ann. Sci. Nat._ (2) Zool. XIII, pp. 358–374, 1840; Haughton, _Ann. Mag. Nat. Hist._ (3), XI, pp. 415–429, 1863; A. R. Wallace, _ibid._ XII, p. 303, 1863; Jeffries Wyman. _Pr. Amer. Acad. of Arts and Sc._ VII, pp. 68–83, 1868; Chauncey Wright, _ibid._ IV, p. 432, 1860.
[378] Sir W. Thomson, On the Division of Space with Minimum Partitional Area, _Phil. Mag._ (5), XXIV, pp. 503–514, Dec. 1887; cf. _Baltimore Lectures_, 1904, p. 615.
[379] Also discovered independently by Sir David Brewster, _Trans. R.S.E._ XXIV, p. 505, 1867, XXV, p. 115, 1869.
[380] Von Fedorow had already described (in Russian) the same figure, under the name of cubo-octahedron, or hepta-parallelohedron, limited however to the case where all the faces are plane. This figure, together with the cube, the hexagonal prism, the rhombic dodecahedron and the “elongated dodecahedron,” constituted the five plane-faced, parallel-sided figures by which space is capable of being completely filled and symmetrically partitioned; the series so forming the foundation of Von Fedorow’s theory of crystalline structure. The elongated dodecahedron is, essentially, the figure of the bee’s cell.
[381] F. R. Lillie, Embryology of the Unionidae, _Journ. of Morphology_, X, p. 12, 1895.
[382] E. B. Wilson, The Cell-lineage of Nereis, _Journ. of Morphology_, VI, p. 452, 1892.
[383] It is highly probable, and we may reasonably assume, that the two little triangles do not actually meet at an apical _point_, but merge into one another by a twist, or minute surface of complex curvature, so as not to contravene the normal conditions of equilibrium.
[384] Professor Peddie has given me this interesting and important result, but the mathematical reasoning is too lengthy to be set forth here.
[385] Cf. Rhumbler, _Arch. f. Entw. Mech._ XIV, p. 401, 1902; Assheton, _ibid._ XXXI, pp. 46–78, 1910.
[386] M. Robert (_l. c._ p. 305) has compiled a long list of cases among the molluscs and the worms, where the initial segmentation of the egg proceeds by equal or unequal division. The two cases are about equally numerous. But like many other writers, he would ascribe this equality or inequality rather to a provision for the future than to a direct effect of immediate physical causation: “Il semble assez probable, comme on l’a dit souvent, que la plus grande taille d’un blastomère est liée à l’importance et au développement précoce des parties du corps qui doivent en naître: il y aurait là une sorte de reflet des stades postérieures du développement sur les premières phénomènes, ce que M. Ray Lankester appelle _precocious segregation_. Il faut avouer pourtant qu’on est parfois assez embarrassé pour assigner une cause à pareilles différences.”
[387] The principle is well illustrated in an experiment of Sir David Brewster’s (_Trans. R.S.E._ XXV, p. 111, 1869). A soap-film is drawn over the rim of a wine-glass, and then covered by a watch-glass. The film is inclined or shaken till it becomes attached to the glass covering, and it then immediately changes place, leaving its transverse position to take up that of a spherical segment extending from one side of the wine-glass to its cover, and so enclosing the same volume of air as formerly but with a great economy of surface, precisely as in the case of our spherical partition cutting off one corner of a cube.
[388] Cf. Wildeman, _Attache des Cloisons_, etc., pls. 1, 2.
[389] _Nova Acta K. Leop. Akad._ XI, 1, pl. IV.
[390] Cf. _Protoplasmamechanik_, p. 229: “Insofern liegen also die Verhältnisse hier wesentlich anders als bei der Zertheilung hohler Körperformen durch flüssige Lamellen. Wenn die Membran bei der Zelltheilung die von dem Prinzip der kleinsten Flächen geforderte Lage und Krümmung annimmt, so werden wir den Grund dafür in andrer Weise abzuleiten haben.”
[391] There is, I think, some ambiguity or disagreement among botanists as to the use of this latter term: the sense in which I am using it, viz. for any partition which meets the outer or peripheral wall at right angles (the strictly _radial_ partition being for the present excluded), is, however, clear.
[392] _Cit._ Plateau, _Statique des Liquides_, i, p. 358.
[393] Even in a Protozoon (_Euglena viridis_), when kept alive under artificial compression, Ryder found a process of cell-division to occur which he compares to the segmenting blastoderm of a fish’s egg, and which corresponds in its essential features with that here described. _Contrib. Zool. Lab. Univ. Pennsylvania_, I, pp. 37–50, 1893.
[394] This, like many similar figures, is manifestly drawn under the influence of Sachs’s theoretical views, or assumptions, regarding orthogonal trajectories, coaxial circles, confocal ellipses, etc.
[395] Such preconceptions as Rauber entertained were all in a direction likely to lead him away from such phenomena as he has faithfully depicted. Rauber had no idea whatsoever of the principles by which we are guided in this discussion, nor does he introduce at all the analogy of surface-tension, or any other purely physical concept. But he was deeply under the influence of Sachs’s rule of rectangular intersection; and he was accordingly disposed to look upon the configuration represented above in Fig. 168, 6, as the most typical or most primitive.
[396] Cf. Rauber, Neue Grundlage z. K. der Zelle, _Morph. Jahrb._ VIII, 1883, pp. 273, 274:
“Ich betone noch, dass unter meinen Figuren diejenige gar nicht enthalten ist, welche zum Typus der Batrachierfurchung gehörig am meisten bekannt ist .... Es haben so ausgezeichnete Beobachter sie als vorhanden beschrieben, dass es mir nicht einfallen kann, sie überhaupt nicht anzuerkennen.”
[397] Roux’s experiments were performed with drops of paraffin suspended in dilute alcohol, to which a little calcium acetate was added to form a soapy pellicle over the drops and prevent them from reuniting with one another.
[398] Cf. (e.g.) Clerk Maxwell, On Reciprocal Figures, etc., _Trans. R. S. E._ XXVI, p. 9, 1870.
[399] See Greville, K. R., Monograph of the Genus Asterolampra, _Q.J.M.S._ VIII, (Trans.), pp. 102–124, 1860; cf. IBID. (n.s.), II, pp. 41–55, 1862.
[400] The same is true of the insect’s wing; but in this case I do not hazard a conjectural explanation.
[401] _Ann. Mag. N. H._ (2), III, p. 126, 1849.
[402] _Phil. Trans._ CLVII, pp. 643–656, 1867.
[403] Sachs, _Pflanzenphysiologie_ (_Vorlesung_ XXIV), 1882; cf. Rauber, Neue Grundlage zur Kenntniss der Zelle, _Morphol. Jahrb._ VIII, p. 303 _seq._, 1883; E. B. Wilson, Cell-lineage of Nereis, _Journ. of Morphology_, VI, p. 448, 1892, etc.
[404] In the following account I follow closely on the lines laid down by Berthold; _Protoplasmamechanik_, cap. vii. Many botanical phenomena identical and similar to those here dealt with, are elaborately discussed by Sachs in his _Physiology of Plants_ (chap. xxvii, pp. 431–459, Oxford, 1887); and in his earlier papers, Ueber die Anordnung der Zellen in jüngsten Pflanzentheilen, and Ueber Zellenanordnung und Wachsthum (_Arb. d. botan. Inst. Würzburg_, 1878, 1879). But Sachs’s treatment differs entirely from that which I adopt and advocate here: his explanations being based on his “law” of rectangular succession, and involving complicated systems of confocal conics, with their orthogonally intersecting ellipses and hyperbolas.
[405] Cf. p. 369.
[406] There is much information regarding the chemical composition and mineralogical structure of shells and other organic products in H. C. Sorby’s Presidential Address to the Geological Society (_Proc. Geol. Soc._ 1879, pp. 56–93); but Sorby failed to recognise that association with “organic” matter, or with colloid matter whether living or dead, introduced a new series of purely physical phenomena.
[407] Vesque, _Ann. des Sc. Nat._ (_Bot._) (5), XIX, p. 310, 1874.
[408] Cf. Kölliker, _Icones Histiologicae_, 1864, pp. 119, etc.
[409] In an interesting paper by Irvine and Sims Woodhead on the “Secretion of Carbonate of Lime by Animals” (_Proc. R. S. E._ XVI, 1889, p. 351) it is asserted that “lime salts, of whatever form, are deposited _only_ in vitally inactive tissue.”
[410] The tube of Teredo shews no trace of organic matter, but consists of irregular prismatic crystals: the whole structure “being identical with that of small veins of calcite, such as are seen in thin sections of rocks” (Sorby, _Proc. Geol. Soc._ 1879, p. 58). This, then, would seem to be a somewhat exceptional case of a shell laid down completely outside of the animal’s external layer of organic or colloid substance.
[411] _C. R. Soc. Biol. Paris_ (9), I, pp. 17–20, 1889; _C. R. Ac. Sc._ CVIII, pp. 196–8, 1889.
[412] Cf. Heron-Allen, _Phil. Trans._ (B), vol. CCVI, p. 262, 1915.
[413] See Leduc, _Mechanism of Life_ (1911), ch. X, for copious references to other works on the artificial production of “organic” forms.
[414] Lectures on the Molecular Asymmetry of Natural Organic Compounds, _Chemical Soc. of Paris_, 1860, and also in Ostwald’s _Klassiker d. ex. Wiss._ No. 28, and in _Alembic Club Reprints_, No. 14, Edinburgh, 1897; cf. Richardson, G. M., _Foundations of Stereochemistry_, N. Y. 1901.
[415] Japp, Stereometry and Vitalism, _Brit. Ass. Rep._ (Bristol), p. 813, 1898; cf. also a voluminous discussion in _Nature_, 1898–9.
[416] They represent the general theorem of which particular cases are found, for instance, in the asymmetry of the ferments (or _enzymes_) which act upon asymmetrical bodies, the one fitting the other, according to Emil Fischer’s well-known phrase, as lock and key. Cf. his Bedeutung der Stereochemie für die Physiologie, _Z. f. physiol. Chemie_, V, p. 60, 1899, and various papers in the _Ber. d. d. chem. Ges._ from 1894.
[417] In accordance with Emil Fischer’s conception of “asymmetric synthesis,” it is now held to be more likely that the process is synthetic than analytic: more likely, that is to say, that the plant builds up from the first one asymmetric body to the exclusion of the other, than that it “selects” or “picks out” (as Japp supposed) the right-handed or the left-handed molecules from an original, optically inactive, mixture of the two; cf. A. McKenzie, Studies in Asymmetric Synthesis, _Journ. Chem. Soc._ (Trans.), LXXXV, p. 1249, 1904.
[418] See for a fuller discussion, Hans Przibram, _Vitalität_, 1913, Kap. iv, Stoffwechsel (Assimilation und Katalyse).
[419] Cf. Cotton, _Ann. de Chim. et de Phys._ (7), VIII, pp. 347–432 (cf. p. 373), 1896.
[420] Byk, A., Zur Frage der Spaltbarkeit von Razemverbindungen durch Zirkularpolarisiertes Licht, ein Beitrag zur primären Entstehung optisch-activer Substanzen, _Zeitsch. f. physikal. Chemie_, XLIX, p. 641, 1904. It must be admitted that further positive evidence on these lines is still awanting.
[421] Cf. (_int. al._) Emil Fischer, _Untersuchungen über Aminosäuren, Proteine_, etc. Berlin, 1906.
[422] Japp, _l. c._ p. 828.
[423] Rainey, G., On the Elementary Formation of the Skeletons of Animals, and other Hard Structures formed in connection with Living Tissue, _Brit. For. Med. Ch. Rev._ XX, pp. 451–476, 1857; published separately with additions, 8vo. London, 1858. For other papers by Rainey on kindred subjects see _Q. J. M. S._ VI (_Tr. Microsc. Soc._), pp. 41–50, 1858, VII, pp. 212–225, 1859, VIII, pp. 1–10, 1860, I (n. s.), pp. 23–32, 1861. Cf. also Ord, W. M., On Molecular Coalescence, and on the influence exercised by Colloids upon the Forms of Inorganic Matter, _Q. J. M. S._ XII, pp. 219–239, 1872; and also the early but still interesting observations of Mr Charles Hatchett, Chemical Experiments on Zoophytes; with some observations on the component parts of Membrane, _Phil. Trans._ 1800. pp. 327–402.
[424] Cf. Quincke, Ueber unsichtbare Flüssigkeitsschichten, _Ann. der Physik_, 1902.
[425] See for instance other excellent illustrations in Carpenter’s article “Shell,” in Todd’s _Cyclopædia_, vol. IV. pp. 550–571, 1847–49. According to Carpenter, the shells of the mollusca (and also of the crustacea) are “essentially composed of _cells_, consolidated by a deposit of carbonate of lime in their interior.” That is to say, Carpenter supposed that the spherulites, or calcospherites of Harting, were, to begin with, just so many living protoplasmic cells. Soon afterwards however, Huxley pointed out that the mode of formation, while at first sight “irresistibly suggesting a cellular structure, ... is in reality nothing of the kind,” but “is simply the result of the concretionary manner in which the calcareous matter is deposited”; _ibid._ art. “Tegumentary Organs,” vol. V, p. 487, 1859. Quekett (_Lectures on Histology_, vol. II, p. 393, 1854, and _Q. J. M. S._ XI, pp. 95–104, 1863) supported Carpenter; but Williamson (Histological Features in the Shells of the Crustacea, _Q. J. M. S._ VIII, pp. 35–47, 1860) amply confirmed Huxley’s view, which in the end Carpenter himself adopted (_The Microscope_, 1862, p. 604). A like controversy arose later in regard to corals. Mrs Gordon (M. M. Ogilvie) asserted that the coral was built up “of successive layers of calcified cells, which hang together at first by their cell-walls, and ultimately, as crystalline changes continue, form the individual laminae of the skeletal structures” (_Phil. Trans._ CLXXXVII, p. 102, 1896): whereas v. Koch had figured the coral as formed out of a mass of “Kalkconcremente” or “crystalline spheroids,” laid down outside the ectoderm, and precisely similar both in their early rounded and later polygonal stages (though von Koch was not aware of the fact) to the calcospherites of Harting (Entw. d. Kalkskelettes von Asteroides, _Mitth. Zool. St. Neapel_, III, pp. 284–290, pl. XX, 1882). Lastly Duerden shewed that external to, and apparently secreted by the ectoderm lies a homogeneous organic matrix or membrane, “in which the minute calcareous crystals forming the skeleton are laid down” (The Coral _Siderastraea radians_, etc., _Carnegie Inst. Washington_, 1904, p. 34). Cf. also M. M. Ogilvie-Gordon, _Q. J. M. S._ XLIX, p. 203, 1905, etc.
[426] Cf. Claparède, _Z. f. w. Z._ XIX, p. 604, 1869.
[427] Spicules extremely like those of the Alcyonaria occur also in a few sponges; cf. (e.g.), Vaughan Jennings, _Journ. Linn. Soc._ XXIII, p. 531, pl. 13, fig. 8, 1891.
[428] _Mem. Manchester Lit. and Phil. Soc._ LX, p. 11, 1916.
[429] Mummery, J. H., On Calcification in Enamel and Dentine, _Phil. Trans._ CCV (B), pp. 95–111, 1914.
[430] The artificial concretion represented in Fig. 202 is identical in appearance with the concretions found in the kidney of Nautilus, as figured by Willey (_Zoological Results_, p. lxxvi, Fig. 2, 1902).
[431] Cf. Taylor’s _Chemistry of Colloids_, p. 18, etc., 1915.
[432] This rule, undreamed of by Errera, supports and justifies the cardinal assumption (of which we have had so much to say in discussing the forms of cells and tissues) that the _incipient_ cell-wall behaves as, and indeed actually is, a liquid film (cf. p. 306).
[433] Cf. p. 254.
[434] Cf. Harting, _op. cit._, pp. 22, 50: “J’avais cru d’abord que ces couches concentriques étaient produites par l’alternance de la chaleur ou de la lumière, pendant le jour et la nuit. Mais l’expérience, expressément instituée pour examiner cette question, y a répondu négativement.”
[435] Liesegang, R. E., _Ueber die Schichtungen bei Diffusionen_, Leipzig, 1907, and other earlier papers.
[436] Cf. Taylor’s _Chemistry of Colloids_, pp. 146–148, 1915.
[437] Cf. S. C. Bradford, The Liesegang Phenomenon and Concretionary Structure in Rocks, _Nature_, XCVII, p. 80, 1916; cf. _Sci. Progress_, X, p. 369, 1916.
[438] Cf. Faraday, On Ice of Irregular Fusibility, _Phil. Trans._, 1858, p. 228; _Researches in Chemistry, etc._, 1859, p. 374; Tyndall, _Forms of Water_, p. 178, 1872; Tomlinson, C., On some effects of small Quantities of Foreign Matter on Crystallisation, _Phil. Mag._ (5) XXXI, p. 393, 1891, and other papers.
[439] A Study in Crystallisation, _J. of Soc. of Chem. Industry_, XXV, p. 143, 1906.
[440] _Ueber Zonenbildung in kolloidalen Medien_, Jena, 1913.
[441] _Verh. d. d. Zool. Gesellsch._ p. 179, 1912.
[442] _Descent of Man_, II, pp. 132–153, 1871.
[443] As a matter of fact, the phenomena associated with the development of an “ocellus” are or may be of great complexity, inasmuch as they involve not only a graded distribution of pigment, but also, in “optical” coloration, a symmetrical distribution of structure or form. The subject therefore deserves very careful discussion, such as Bateson gives to it (_Variation_, chap. xii). This, by the way, is one of the very rare cases in which Bateson appears inclined to suggest a purely physical explanation of an organic phenomenon: “The suggestion is strong that the whole series of rings (in _Morpho_) may have been formed by some one central disturbance, somewhat as a series of concentric waves may be formed by the splash of a stone thrown into a pool, etc.”
[444] Cf. also Sir D. Brewster, On optical properties of Mother of Pearl, _Phil. Trans._ 1814, p. 397.
[445] Biedermann, W., Ueber die Bedeutung von Kristallisationsprozessen der Skelette wirbelloser Thiere, namentlich der Molluskenschalen, _Z. f. allg. Physiol._ I, p. 154, 1902; Ueber Bau und Entstehung der Molluskenschale, _Jen. Zeitschr._ XXXVI, pp. 1–164, 1902. Cf. also Steinmann, Ueber Schale und Kalksteinbildungen, _Ber. Naturf. Ges. Freiburg i. Br_ IV, 1889; Liesegang, _Naturw. Wochenschr._ p. 641, 1910.
[446] Cf. Bütschli, Ueber die Herstellung künstlicher Stärkekörner oder von Sphärokrystallen der Stärke, _Verh. nat. med. Ver. Heidelberg_, V, pp. 457–472, 1896.
[447] _Untersuchungen über die Stärkekörner_, Jena, 1905.
[448] Cf. Winge, _Meddel. fra Komm. for Havundersögelse_ (_Fiskeri_), IV, p. 20, Copenhagen, 1915.
[449] The anhydrite is sulphate of lime (CaSO_{4}); the polyhalite is a triple sulphate of lime, magnesia and potash (2 CaSO_{4}. MgSO_{4}. K_{2}SO_{4} + 2 H_{2}O).
[450] Cf. van’t Hoff, _Physical Chemistry in the Service of the Sciences_, p. 99 seq. Chicago, 1903.
[451] Sphärocrystalle von Kalkoxalat bei Kakteen, _Ber. d. d. Bot. Gesellsch._ p. 178, 1885.
[452] Pauli, W. u. Samec, M., Ueber Löslichkeitsbeeinflüssung von Elektrolyten durch Eiweisskörper, _Biochem. Zeitschr._ XVII, p. 235, 1910. Some of these results were known much earlier; cf. Fokker in _Pflüger’s Archiv_, VII, p. 274, 1873; also Irvine and Sims Woodhead, _op. cit._ p. 347.
[453] Which, in 1000 parts of ash, contains about 840 parts of phosphate and 76 parts of calcium carbonate.
[454] Cf. Dreyer, Fr., Die Principien der Gerüstbildung bei Rhizopoden, Spongien und Echinodermen, _Jen. Zeitschr._ XXVI, pp. 204–468, 1892.
[455] In an anomalous and very remarkable Australian sponge, just described by Professor Dendy (_Nature_, May 18, 1916, p. 253) under the name of _Collosclerophora_, the spicules are “gelatinous,” consisting of a gel of colloid silica with a high percentage of water. It is not stated whether an organic colloid is present together with the silica. These gelatinous spicules arise as exudations on the outer surface of cells, and come to lie in intercellular spaces or vesicles.
[456] Lister, in Willey’s _Zoological Results_, pt IV, p. 459, 1900.
[457] The peculiar spicules of Astrosclera are now said to consist of spherules, or calcospherites, of aragonite, spores of a certain red seaweed forming the nuclei, or starting-points, of the concretions (R. Kirkpatrick, _Proc. R. S._ LXXXIV (B), p. 579, 1911).
[458] See for instance the plates in Théel’s Monograph of the Challenger Holothuroidea; also Sollas’s Tetractinellida, p. lxi.
[459] For very numerous illustrations of the triradiate and quadriradiate spicules of the calcareous sponges, see (_int. al._), papers by Dendy (_Q. J. M. S._ XXXV, 1893), Minchin (_P. Z. S._ 1904), Jenkin (_P. Z. S._ 1908), etc.
[460] Cf. again Bénard’s _Tourbillons cellulaires_, _Ann. de Chimie_, 1901, p. 84.
[461] Léger, Stolc and others, in Doflein’s _Lehrbuch d. Protozoenkunde_, 1911, p. 912.
[462] See, for instance, the figures of the segmenting egg of Synapta (after Selenka), in Korschelt and Heider’s _Vergleichende Entwicklungsgeschichte_ (Allgem. Th., 3^{te} Lief.), p. 19, 1909. On the spiral type of segmentation as a secondary derivative, due to mechanical causes, of the “radial” type of segmentation, see E. B. Wilson, Cell-lineage of Nereis, _Journ. of Morphology_, VI, p. 450, 1892.
[463] Korschelt and Heider, p. 16.
[464] _Chall. Rep. Hexactinellida_, pls. xvi, liii, lxxvi, lxxxviii.
[465] “Hierbei nahm der kohlensaure Kalk eine halb-krystallinische Beschaffenheit an, und gestaltete sich unter Aufnahme von Krystallwasser und in Verbindung mit einer geringen Quantität von organischer Substanz zu jenen individuellen, festen Körpern, welche durch die natürliche Züchtung als _Spicula_ zur Skeletbildung benützt, und späterhin durch die Wechselwirkung von Anpassung und Vererbung im Kampfe ums Dasein auf das Vielfältigste umgebildet und differenziert wurden.” _Die Kalkschwämme_, I, p. 377, 1872; cf. also pp. 482, 483.
[466] _Op. cit._ p. 483. “Die geordnete, oft so sehr regelmässige und zierliche Zusammensetzung des Skeletsystems ist zum grössten Theile unmittelbares Product der Wasserströmung; die characteristische Lagerung der Spicula ist von der constanten Richtung des Wasserstroms hervorgebracht; zum kleinsten Theile ist sie die Folge von Anpassungen an untergeordnete äussere Existenzbedingungen.”
[467] Materials for a Monograph of the Ascones, _Q. J. M. S._ XL. pp. 469–587, 1898.
[468] Haeckel, in his _Challenger Monograph_, p. clxxxviii (1887) estimated the number of known forms at 4314 species, included in 739 genera. Of these, 3508 species were described for the first time in that work.
[469] Cf. Gamble, _Radiolaria_ (Lankester’s _Treatise on Zoology_), vol. I, p. 131, 1909. Cf. also papers by V. Häcker, in _Jen. Zeitschr._ XXXIX, p. 581, 1905, _Z. f. wiss. Zool._ LXXXIII, p. 336, 1905, _Arch. f. Protistenkunde_, IX, p. 139, 1907, etc.
[470] Bütschli, Ueber die chemische Natur der Skeletsubstanz der Acantharia, _Zool. Anz._ XXX, p. 784, 1906.
[471] For figures of these crystals see Brandt, _F. u. Fl. d. Golfes von Neapel_, XIII, _Radiolaria_, 1885, pl. v. Cf. J. Müller, Ueber die Thalassicollen, etc. _Abh. K. Akad. Wiss. Berlin_, 1858.
[472] Celestine, or celestite, is SrSO_{4} with some BaO replacing SrO.
[473] With the colloid chemists, we may adopt (as Rhumbler has done) the terms _spumoid_ or _emulsoid_ to denote an agglomeration of fluid-filled vesicles, restricting the name _froth_ to such vesicles when filled with air or some other gas.
[474] Cf. Koltzoff, Zur Frage der Zellgestalt, _Anat. Anzeiger_, XLI, p. 190, 1912.
[475] _Mém. de l’Acad. des Sci., St. Pétersbourg_, XII, Nr. 10, 1902.
[476] The manner in which the minute spicules of Raphidiophrys arrange themselves round the bases of the pseudopodial rays is a similar phenomenon.
[477] Rhumbler, Physikalische Analyse von Lebenserscheinungen der Zelle, _Arch. f. Entw. Mech._ VII, p. 103, 1898.
[478] The whole phenomenon is described by biologists as a “surprising exhibition of constructive and selective activity,” and is ascribed, in varying phraseology, to intelligence, skill, purpose, psychical activity, or “microscopic mentality”: that is to say, to Galen’s τεχνικὴ φύσις, or “artistic creativeness” (cf. Brock’s _Galen_, 1916, p. xxix). Cf. Carpenter, _Mental Physiology_, 1874, p. 41; Norman, Architectural achievements of Little Masons, etc., _Ann. Mag. Nat. Hist._ (5), I, p. 284, 1878; Heron-Allen, Contributions ... to the Study of the Foraminifera, _Phil. Trans._ (B), CCVI, pp. 227–279, 1915; Theory and Phenomena of Purpose and Intelligence exhibited by the Protozoa, as illustrated by selection and behaviour in the Foraminifera, _Journ. R. Microscop. Soc._ pp. 547–557, 1915; _ibid._, pp. 137–140, 1916. Prof. J. A. Thomson (_New Statesman_, Oct. 23, 1915) describes a certain little foraminifer, whose protoplasmic body is overlaid by a crust of sponge-spicules, as “a psycho-physical individuality whose experiments in self-expression include a masterly treatment of sponge-spicules, and illustrate that organic skill which came before the dawn of Art.” Sir Ray Lankester finds it “not difficult to conceive of the existence of a mechanism in the protoplasm of the Protozoa which selects and rejects building-material, and determines the shapes of the structures built, comparable to that mechanism which is assumed to exist in the nervous system of insects and other animals which ‘automatically’ go through wonderfully elaborate series of complicated actions.” And he agrees with “Darwin and others [who] have attributed the building up of these inherited mechanisms to the age-long action of Natural Selection, and the survival of those individuals possessing qualities or ‘tricks’ of life-saving value,” _J. R. Microsc. Soc._ April, 1916, p. 136.
[479] Rhumbler, _Das Protoplasma als physikalisches System_, Jena, p. 591, 1914; also in _Arch. f. Entwickelungsmech._ VII, pp. 279–335, 1898.
[480] Verworn, _Psycho-physiologische Protisten-Studien_, Jena, 1889 (219 pp.).
[481] Leidy, J., _Fresh-water Rhizopods of N. America_, 1879, p. 262, pl. xli, figs. 11, 12.
[482] Carnoy, _Biologie Cellulaire_, p. 244, fig. 108; cf. Dreyer, _op. cit._ 1892, fig. 185.
[483] In all these latter cases we recognise a relation to, or extension of, the principle of Plateau’s _bourrelet_, or van der Mensbrugghe’s _masse annulaire_, of which we have already spoken (p. 297).
[484] Apart from the fact that the apex of each pyramid is interrupted, or truncated, by the presence of the little central cell, it is also possible that the solid angles are not precisely equivalent to those of Maraldi’s pyramids, owing to the fact that there is a certain amount of distortion, or axial asymmetry, in the Nassellarian system. In other words (to judge from Haeckel’s figures), the tetrahedral symmetry in Nassellaria is not absolutely regular, but has a main axis about which three of the trihedral pyramids are symmetrical, the fourth having its solid angle somewhat diminished.
[485] Cf. Faraday’s beautiful experiments, On the Moving Groups of Particles found on Vibrating Elastic Surfaces, etc., _Phil. Trans._ 1831, p. 299; _Researches in Chem. and Phys._ 1859, pp. 314–358.
[486] We need not go so far as to suppose that the external layer of cells wholly lacked the power of secreting a skeleton. In many of the Nassellariae figured by Haeckel (for there are many variant forms or species besides that represented here), the skeleton of the partition-walls is very slightly and scantily developed. In such a case, if we imagine its few and scanty strands to be broken away, the central tetrahedral figure would be set free, and would have all the appearance of a complete and independent structure.
[487] The “bourrelet” is not only, as Plateau expresses it, a “surface of continuity,” but we also recognise that it tends (so far as material is available for its production) to further lessen the free surface-area. On its relation to vapour-pressure and to the stability of foam, see FitzGerald’s interesting note in _Nature_, Feb. 1, 1894 (_Works_, p. 309).
[488] Of the many thousand figures in the hundred and forty plates of this beautifully illustrated book, there is scarcely one which does not depict, now patently, now in pregnant suggestion, some subtle and elegant geometrical configuration.
[489] They were known (of course) long before Plato: Πλάτων δὲ καὶ ἐν τούτοις πυθαγορίζει.
[490] If the equation of any plane face of a crystal be written in the form _h_ _x_ + _k_ _y_ + _l_ _z_ = 1, then _h_, _k_, _l_ are the indices of which we are speaking. They are the reciprocals of the parameters, or reciprocals of the distances from the origin at which the plane meets the several axes. In the case of the regular or pentagonal dodecahedron these indices are 2, 1 + √5, 0. Kepler described as follows, briefly but adequately, the common characteristics of the dodecahedron and icosahedron: “Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quinquangulis figuratur expresse, hoc triangulis quidem sed in quinquanguli formam coaptatis. Utriusque horum corporum ipsiusque adeo quinquanguli _structura perfici non potest sine proportione illa, quam hodierni geometrae divinam appellant_” (_De nive sexangula_ (1611), Opera, ed. Frisch, VII, p. 723). Here Kepler was dealing, somewhat after the manner of Sir Thomas Browne, with the mysteries of the quincunx, and also of the hexagon; and was seeking for an explanation of the mysterious or even mystical beauty of the 5-petalled or 3-petalled flower,—_pulchritudinis aut proprietatis figurae, quae animam harum plantarum characterisavit_.
[491] Cf. Tutton, _Crystallography_, p. 932, 1911.
[492] However, we can often recognise, in a small artery for instance, that the so-called “circular” fibres tend to take a slightly oblique, or spiral, course.
[493] The spiral fibres, or a large portion of them, constitute what Searle called “the rope of the heart” (Todd’s _Cyclopaedia_, II, p. 621, 1836). The “twisted sinews of the heart” were known to early anatomists, and have been frequently and elaborately studied: for instance, by Gerdy (_Bull. Fac. Med. Paris_, 1820, pp. 40–148), and by Pettigrew (_Phil. Trans._ 1864), and of late by J. B. Macallum (_Johns Hopkins Hospital Report_, IX, 1900) and by Franklin P. Mall (_Amer. J. of Anat._ XI, 1911).
[494] Cf. Bütschli, “Protozoa,” in Bronn’s _Thierreich_, II, p. 848, III, p. 1785, etc., 1883–87; Jennings, _Amer. Nat._ XXXV, p. 369, 1901; Pütter, Thigmotaxie bei Protisten, _Arch. f. Anat. u. Phys._ (_Phys. Abth. Suppl._), pp. 243–302, 1900.
[495] A great number of spiral forms, both organic and artificial, are described and beautifully illustrated in Sir T. A. Cook’s _Curves of Life_, 1914, and _Spirals in Nature and Art_, 1903.
[496] Cf. Vines, The History of the Scorpioid Cyme, _Journ. of Botany_ (n.s.), X, pp. 3–9, 1881.
[497] Leslie’s _Geometry of Curved Lines_, p. 417, 1821. This is practically identical with Archimedes’ own definition (ed. Torelli, p. 219); cf. Cantor, _Geschichte der Mathematik_, I, p. 262, 1880.
[498] See an interesting paper by Whitworth, W. A., “The Equiangular Spiral, its chief properties proved geometrically,” in the _Messenger of Mathematics_ (1), I, p. 5, 1862.
[499] I am well aware that the debt of Greek science to Egypt and the East is vigorously denied by many scholars, some of whom go so far as to believe that the Egyptians never had any science, save only some “rough rules of thumb for measuring fields and pyramids” (Burnet’s _Greek Philosophy_, 1914, p. 5).
[500] Euclid (II, def. 2).
[501] Cf. Treutlein, _Z. f. Math. u. Phys._ (_Hist. litt. Abth._), XXVIII, p. 209, 1883.
[502] This is the so-called _Dreifachgleichschenkelige Dreieck_; cf. Naber, _op. infra cit._ The ratio 1 : 0·618 is again not hard to find in this construction.
[503] See, on the mathematical history of the Gnomon, Heath’s _Euclid_, I, _passim_, 1908; Zeuthen, _Theorème de Pythagore_, Genève, 1904; also a curious and interesting book, _Das Theorem des Pythagoras_, by Dr. H. A. Naber, Haarlem, 1908.
[504] For many beautiful geometrical constructions based on the molluscan shell, see Colman, S. and Coan, C. A., _Nature’s Harmonic Unity_ (ch. ix, Conchology), New York, 1912.
[505] The Rev. H. Moseley, On the Geometrical Forms of Turbinated and Discoid Shells, _Phil. Trans._ pp. 351–370. 1838.
[506] It will be observed that here Moseley, speaking as a mathematician and considering the _linear_ spiral, speaks of _whorls_ when he means the linear boundaries, or lines traced by the revolving radius vector; while the conchologist usually applies the term _whorl_ to the whole space between the two boundaries. As conchologists, therefore, we call the _breadth of a whorl_ what Moseley looked upon as the _distance between two consecutive whorls_. But this latter nomenclature Moseley himself often uses.
[507] In the case of Turbo, and all other “turbinate” shells, we are dealing not with a plane logarithmic spiral, as in Nautilus, but with a “gauche” spiral, such that the radius vector no longer revolves in a plane perpendicular to the axis of the system, but is inclined to that axis at some constant angle (θ). The figure still preserves its continued similarity, and may with strict accuracy be called a logarithmic spiral in space. It is evident that its envelope will be a right circular cone; and indeed it is commonly spoken of as a logarithmic spiral _wrapped upon a cone_, its pole coinciding with the apex of the cone. It follows that the distances of successive whorls of the spiral measured on the same straight line passing through the apex of the cone, are in geometrical progression, and conversely just as in the former case. But the ratio between any two consecutive interspaces (i.e. _R__{3} − _R__{2}/_R__{2} − _R__{1}) is now equal to ε^{2π sin θ cot α}, θ being the semi-angle of the enveloping cone. (Cf. Moseley, _Phil. Mag._ XXI, p. 300, 1842.)
[508] As the successive increments evidently constitute similar figures, similarly related to the pole (_P_), it follows that their linear dimensions are to one another as the radii vectores drawn to similar points in them: for instance as _P_ _P__{1}, _P_ _P__{2}, which (in Fig. 264, 1) are radii vectores drawn to the points where they meet the common boundary.
[509] The equation to the surface of a turbinate shell is discussed by Moseley (_Phil. Trans._ tom. cit. p. 370), both in terms of polar coordinates and of the rectangular coordinates _x_, _y_, _z_. A more elegant representation can be given in vector notation, by the method of quaternions.
[510] J. C. M. Reinecke, _Maris protogaei Nautilos, etc._, Coburg, 1818. Leopold von Buch, Ueber die Ammoniten in den älteren Gebirgsschichten, _Abh. Berlin. Akad., Phys. Kl._ pp. 135–158, 1830; _Ann. Sc. Nat._ XXVIII, pp. 5–43, 1833; cf. Elie de Beaumont, Sur l’enroulement des Ammonites, _Soc. Philom., Pr. verb._ pp. 45–48, 1841.
[511] _Biblia Naturae sive Historia Insectorum_, Leydae, 1737, p. 152.
[512] Alcide D’Orbigny, _Bull. de la soc. géol. Fr._ XIII, p. 200, 1842; _Cours élém. de Paléontologie_, II, p. 5, 1851. A somewhat similar instrument was described by Boubée. in _Bull. soc. géol._ I, p. 232, 1831. Naumann’s Conchyliometer (_Poggend. Ann._ LIV, p. 544, 1845) was an application of the screw-micrometer; it was provided also with a rotating stage, for angular measurement. It was adapted for the Study of a discoid or ammonitoid shell, while D’Orbigny’s instrument was meant for the study of a turbinate shell.
[513] It is obvious that the ratios of opposite whorls, or of radii 180° apart, are represented by the square roots of these values; and the ratios of whorls or radii 90° apart, by the square roots of these again.
[514] For the correction to be applied in the case of the helicoid, or “turbinate” shells, see p. 557.
[515] On the Measurement of the Curves formed by Cephalopods and other Mollusks. _Phil. Mag._ (5), VI, pp. 241–263, 1878.
[516] For an example of this method, see Blake, _l.c._ p. 251.
[517] Naumann, C. F., Ueber die Spiralen von Conchylien, _Abh. k. sächs_. Ges. pp. 153–196, 1846; Ueber die cyclocentrische Conchospirale u. über das Windungsgesetz von _Planorbis corneus_, _ibid._ I, pp. 171–195, 1849; Spirale von Nautilus u. _Ammonites galeatus_, _Ber. k. sächs. Ges._ II, p. 26, 1848; Spirale von _Amm. Ramsaueri_, _ibid._ XVI, p. 21, 1864; see also _Poggendorff’s Annalen_, L, p. 223, 1840; LI, p. 245, 1841; LIV, p. 541, 1845, etc.
[518] Sandberger, G., Spiralen des _Ammonites Amaltheus_, _A. Gaytani_, und _Goniatites intumescens_, _Zeitschr. d. d. Geol. Gesellsch._ X, pp. 446–449, 1858.
[519] Grabau, A. H., _Ueber die Naumannsche Conchospirale_, etc. Inauguraldiss. Leipzig, 1872; _Die Spiralen von Conchylien_, etc. Programm, Nr. 502, Leipzig, 1882.
[520] It has been pointed out to me that it does not follow at once and obviously that, because the interspace _AB_ is a mean proportional between the breadths of the adjacent whorls, therefore the whole distance _OB_ is a mean proportional between _OA_ and _OC_. This is a corollary which requires to be proved; but the proof is easy.
[521] A beautiful construction: _stupendum Naturae artificium_, Linnaeus.
[522] English edition, p. 537, 1900. The chapter is revised by Prof. Alpheus Hyatt, to whom the nomenclature is largely due. For a more copious terminology, see Hyatt, _Phylogeny of an Acquired Characteristic_, p. 422 _seq._, 1894.
[523] This latter conclusion is adopted by Willey, _Zoological Results_, p. 747, 1902.
[524] See Moseley, _op. cit._ pp. 361 _seq._
[525] In Nautilus, the “hood” has somewhat different dimensions in the two sexes, and these differences are impressed upon the shell, that is to say upon its “generating curve.” The latter constitutes a somewhat broader ellipse in the male than in the female. But this difference is not to be detected in the young; in other words, the form of the generating curve perceptibly alters with advancing age. Somewhat similar differences in the shells of Ammonites were long ago suspected, by D’Orbigny, to be due to sexual differences. (Cf. Willey, _Natural Science_, VI, p. 411, 1895; _Zoological Results_, p. 742, 1902.)
[526] Macalister, Alex., Observations on the Mode of Growth of Discoid and Turbinated Shells, _P. R. S._ XVIII, pp. 529–532, 1870.
[527] See figures in Arnold Lang’s _Comparative Anatomy_ (English translation), II, p. 161, 1902.
[528] Kappers, C. U. A., Die Bildung künstlicher Molluskenschalen, _Zeitschr. f. allg. Physiol._ VII, p. 166, 1908.
[529] We need not assume a _close_ relationship, nor indeed any more than such a one as permits us to compare the shell of a Nautilus with that of a Gastropod.
[530] Cf. Owen, “These shells [Nautilus and Ammonites] are revolutely spiral or coiled over the back of the animal, not involute like Spirula”: _Palaeontology_, 1861, p. 97; cf. _Mem. on the Pearly Nautilus_, 1832; also _P.Z.S._ 1878, p. 955.
[531] The case of Terebratula or of Gryphaea would be closely analogous, if the smaller valve were less closely connected and co-articulated with the larger.
[532] “It has been suggested, and I think in some quarters adopted as a dogma, that the formation of successive septa [in Nautilus] is correlated with the recurrence of reproductive periods. This is not the case, since, according to my observations, propagation only takes place after the last septum is formed;” Willey, _Zoological Results_, p. 746, 1902.
[533] Cf. Woodward, Henry, On the Structure of Camerated Shells, _Pop. Sci. Rev._ XI, pp. 113–120, 1872.
[534] See Willey, Contributions to the Natural History of the Pearly Nautilus, _Zoological Results_, etc. p. 749, 1902. Cf. also Bather, Shell-growth in Cephalopoda, _Ann. Mag. N. H._ (6), I, pp 298–310, 1888; _ibid._ pp. 421–427, and other papers by Blake, Riefstahl, etc. quoted therein.
[535] It was this that led James Bernoulli, in imitation of Archimedes, to have the logarithmic spiral graven on his tomb, with the pious motto, _Eadem mutata resurgam_. On Goodsir’s grave the same symbol is reinscribed.
[536] The “lobes” and “saddles” which arise in this manner, and on whose arrangement the modern classification of the nautiloid and ammonitoid shells largely depends, were first recognised and named by Leopold von Buch, _Ann. Sci. Nat._ XXVII, XXVIII, 1829.
[537] Blake has remarked upon the fact (_op. cit._ p. 248) that in some Cyrtocerata we may have a curved shell in which the ornaments approximately run at a constant angular distance from the pole, while the septa approximate to a radial direction; and that “thus one law of growth is illustrated by the inside, and another by the outside.” In this there is nothing at which we need wonder. It is merely a case where the generating curve is set very obliquely to the axis of the shell; but where the septa, which have no necessary relation to the _mouth_ of the shell, take their places, as usual, at a certain definite angle to the _walls_ of the tube. This relation of the septa to the walls of the tube arises after the tube itself is fully formed, and the obliquity of growth of the open end of the tube has no relation to the matter.
[538] Cf. pp. 255, 463, etc.
[539] In a few cases, according to Awerinzew and Rhumbler, where the chambers are added on in concentric series, as in Orbitolites, we have the crystalline structure arranged radially in the radial walls but tangentially in the concentric ones: whereby we tend to obtain, on a minute scale, a system of orthogonal trajectories, comparable to that which we shall presently study in connection with the structure of bone. Cf. S. Awerinzew, Kalkschale der Rhizopoden, _Z. f. w. Z._ LXXIV, pp. 478–490, 1903.
[540] Rhumbler, L., Die Doppelschalen von Orbitolites und anderer Foraminiferen, etc., _Arch. f. Protistenkunde_, I, pp. 193–296, 1902; and other papers. Also _Die Foraminiferen der Planktonexpedition_, I, 1911, pp. 50–56.
[541] Bénard, H, Les tourbillons cellulaires, _Ann. de Chimie_ (8), XXIV, 1901. Cf. also the pattern of cilia on an Infusorian, as figured by Bütschli in Bronn’s _Protozoa_, III, p. 1281, 1887.
[542] A similar hexagonal pattern is obtained by the mutual repulsion of floating magnets in Mr R. W. Wood’s experiments, _Phil. Mag._ XLVI, pp. 162–164, 1898.
[543] Cf. D’Orbigny, Alc., Tableau méthodique de la classe des Céphalopodes, _Ann. des Sci. Nat._ (1), VII, pp. 245–315, 1826; Dujardin. Félix, Observations nouvelles sur les prétendus Céphalopodes microscopiques, _ibid._ (2), III, pp. 108, 109, 312–315, 1835; Recherches sur les organismes inférieurs, _ibid._ IV, pp. 343–377, 1835, etc.
[544] It is obvious that the actual _outline_ of a foraminiferal, just as of a molluscan shell, may depart widely from a logarithmic spiral. When we say here, for short, that the shell _is_ a logarithmic spiral, we merely mean that it is essentially related to one: that it can be inscribed in such a spiral, or that corresponding points (such, for instance, as the centres of gravity of successive chambers, or the extremities of successive septa) wall always be found to lie upon such a spiral.
[545] von Möller, V., Die spiral-gewundenen Foraminifera des russischen Kohlenkalks, _Mém. de l’Acad. Imp. Sci., St Pétersbourg_ (7), XXV, 1878.
[546] As von Möller is careful to explain, Naumann’s formula for the “cyclocentric conchospiral” is appropriate to this and other spiral Foraminifera, since we have in all these cases a central or initial chamber, approximately spherical, about which the logarithmic spiral is coiled (cf. Fig. 309). In species where the central chamber is especially large, Naumann’s formula is all the more advantageous. But it is plain that it is only required when we are dealing with diameters, or with radii; so long as we are merely comparing the breadths of _successive whorls_, the two formulae come to the same thing.
[547] Van Iterson, G., _Mathem. u. mikrosk.-anat. Studien über Blattstellungen, nebst Betrachtungen über den Schalenbau der Miliolinen_, 331 pp., Jena, 1907.
[548] Hans Przibram asserts that the linear ratio of successive chambers tends in many Foraminifera to approximate to 1·26, which = ∛2; in other words, that the volumes of successive chambers tend to double. This Przibram would bring into relation with another law, viz. that insects and other arthropods tend to moult, or to metamorphose, just when they double their weights, or increase their linear dimensions in the ratio of 1 : ∛2. (Die Kammerprogression der Foraminiferen als Parallele zur Häutungsprogression der Mantiden, _Arch. f. Entw. Mech._ XXXIV p. 680, 1813.) Neither rule seems to me to be well grounded.
[549] Cf. Schacko, G., Ueber Globigerina-Einschluss bei Orbulina, _Wiegmann’s Archiv_, XLIX, p. 428, 1883; Brady, _Chall. Rep._, p. 607, 1884.
[550] Cf. Brady, H. B., _Challenger Rep._, _Foraminifera_, 1884, p. 203, pl. XIII.
[551] Brady, _op. cit._, p. 206; Batsch, one of the earliest writers on Foraminifera, had already noticed that this whole series of ear-shaped and crozier-shaped shells was filled in by gradational forms; _Conchylien des Seesandes_, 1791, p. 4, pl. VI, fig. 15_a_–_f_. See also, in particular, Dreyer, _Peneroplis_; _eine Studie zur biologischen Morphologie und zur Speciesfrage_, Leipzig, 1898; also Eimer und Fickert, Artbildung und Verwandschaft bei den Foraminiferen, _Tübinger zool. Arbeiten_, III, p. 35, 1899.
[552] Doflein, _Protozoenkunde_, 1911, p. 263; “Was diese Art veranlässt in dieser Weise gelegentlich zu varüren, ist vorläufig noch ganz räthselhaft.”
[553] In the case of Globigerina, some fourteen species (out of a very much larger number of described forms) were allowed by Brady (in 1884) to be distinct; and this list has been, I believe, rather added to than diminished. But these so-called species depend for the most part on slight differences of degree, differences in the angle of the spiral, in the ratio of magnitude of the segments, or in their area of contact one with another. Moreover with the exception of one or two “dwarf” forms, said to be limited to Arctic and Antarctic waters, there is no principle of geographical distribution to be discerned amongst them. A species found fossil in New Britain turns up in the North Atlantic: a species described from the West Indies is rediscovered at the ice-barrier of the Antarctic.
[554] Dreyer, F., Principien der Gerüstbildung bei Rhizopoden, etc., _Jen. Zeitschr._ XXVI, pp. 204–468, 1892.
[555] A difficulty arises in the case of forms (like Peneroplis) where the young shell appears to be more complex than the old, the first formed portion being closely coiled while the later additions become straight and simple: “die biformen Arten verhalten sich, kurz gesagt. gerade umgekehrt als man nach dem biogenetischen Grundgesetz erwarten sollte,” Rhumbler, _op. cit._, p. 33 etc.
[556] “Das Festigkeitsprinzip als Movens der Weiterentwicklung ist zu interessant und für die Aufstellung meines Systems zu wichtig um die Frage unerörtert zu lassen, warum diese Bevorzügung der Festigkeit stattgefunden hat. Meiner Ansicht nach lautet die Antwort auf diese Frage einfach, weil die Foraminiferen meistens unter Verhältnissen leben, die ihre Schalen in hohem Grade der Gefahr des Zerbrechens aussetzen; es muss also eine fortwahrende Auslese des Festeren stattfinden,” Rhumbler, _op. cit._, p. 22.
[557] “Die Foraminiferen kiesige oder grobsandige Gebiete des Meeresbodens _nicht lieben_, u.s.w.”: where the last two words have no particular meaning, save only that (as M. Aurelius says) “of things that use to be, we say commonly that they love to be.”
[558] In regard to the Foraminifera, “die Palaeontologie lässt uns leider an Anfang der Stammesgeschichte fast gänzlich im Stiche,” Rhumbler, _op. cit._, p. 14.
[559] The evolutionist theory, as Bergson puts it, “consists above all in establishing relations of ideal kinship, and in maintaining that wherever there is this relation of, so to speak, _logical_ affiliation between forms, _there is also a relation of chronological succession between the species in which these forms are materialised_”: _Creative Evolution_, 1911, p. 26. Cf. _supra_, p. 251.
[560] In the case of the ram’s horn, the assumption that the rings are annual is probably justified. In cattle they are much less conspicuous, but are sometimes well-marked in the cow; and in Sweden they are then called “calf-rings,” from a belief that they record the number of offspring. That is to say, the growth of the horn is supposed to be retarded during gestation, and to be accelerated after parturition, when superfluous nourishment seeks a new outlet. (Cf. Lönnberg, _P.Z.S._, p. 689, 1900.)
[561] Cf. Sir V. Brooke, On the Large Sheep of the Thian Shan, _P.Z.S._, p. 511, 1875.
[562] Cf. Lönnberg, E., On the Structure of the Musk Ox, _P.Z.S._, pp. 686–718, 1900.
[563] St Venant, De la torsion des prismes, avec des considérations sur leur flexion, etc., _Mém. des Savants Étrangers_, Paris, XIV, pp. 233–560, 1856.
[564] This is not difficult to do, with considerable accuracy, if the clay be kept well wetted, or semi-fluid, and the smoothing be done with a large wet brush.
[565] The curves are well shewn in most of Sir V. Brooke’s figures of the various species of Argali, in the paper quoted on p. 614.
[566] _Climbing Plants_, 1865 (2nd edit. 1875); _Power of Movement in Plants_, 1880.
[567] Palm, _Ueber das Winden der Pflanzen_, 1827; von Mohl, _Bau und Winden der Ranken_, etc., 1827; Dutrochet, Mouvements révolutifs spontanés, _C.R._ 1843, etc.
[568] Cf. (e.g.) Lepeschkin, Zur Kenntnis des Mechanismus der Variationsbewegungen, _Ber. d. d. Bot. Gesellsch._ XXVI A, pp. 724–735, 1908; also A. Tröndle, Der Einfluss des Lichtes auf die Permeabilität des Plasmahaut, _Jahrb. wiss. Bot._ XLVIII, pp. 171–282, 1910.
[569] For an elaborate study of antlers, see Rörig, A., _Arch. f. Entw. Mech._ X, pp. 525–644, 1900, XI, pp. 65–148, 225–309, 1901; Hoffmann, C., _Zur Morphologie der rezenten Hirschen_, 75 pp., 23 pls., 1901: also Sir Victor Brooke, On the Classification of the Cervidae, _P.Z.S._, pp. 883–928, 1878. For a discussion of the development of horns and antlers, see Gadow, H., _P.Z.S._, pp. 206–222, 1902, and works quoted therein.
[570] Cf. Rhumbler, L., Ueber die Abhängigkeit des Geweihwachstums der Hirsche, speziell des Edelhirsches, vom Verlauf der Blutgefässe im Kolbengeweih, _Zeitschr. f. Forst. und Jagdwesen_, 1911, pp. 295–314.
[571] The fact that in one very small deer, the little South American Coassus, the antler is reduced to a simple short spike, does not preclude the general distinction which I have drawn. In Coassus we have the beginnings of an antler, which has not yet manifested its tendency to expand; and in the many allied species of the American genus Cariacus, we find the expansion manifested in various simple modes of ramification or bifurcation. (Cf. Sir V. Brooke, Classification of the Cervidae, p. 897.)
[572] Cf. also the immense range of variation in elks’ horns, as described by Lönnberg, _P.Z.S._ II, pp. 352–360, 1902.
[573] Besides papers referred to below, and many others quoted in Sach’s _Botany_ and elsewhere, the following are important: Braun, Alex., Vergl. Untersuchung über die Ordnung der Schuppen an den Tannenzapfen, etc., _Verh. Car. Leop. Akad._ XV, pp. 199–401, 1831; Dr C. Schimper’s Vorträge über die Möglichkeit eines wissenschaftlichen Verständnisses der Blattstellung, etc., _Flora_, XVIII, pp. 145–191, 737–756, 1835; Schimper, C. F., Geometrische Anordnung der um eine Axe peripherische Blattgebilde, _Verhandl. Schweiz. Ges._, pp. 113–117, 1836; Bravais, L. and A., Essai sur la disposition des feuilles curvisériées, _Ann. Sci. Nat._ (2), VII, pp. 42–110, 1837; Sur la disposition symmétrique des inflorescences, _ibid._, pp. 193–221, 291–348, VIII, pp. 11–42, 1838; Sur la disposition générale des feuilles rectisériées, _ibid._ XII, pp. 5–41, 65–77, 1839; Zeising, _Normalverhältniss der chemischen und morphologischen Proportionen_, Leipzig, 1856; Naumann, C. F., Ueber den Quincunx als Gesetz der Blattstellung bei Sigillaria, etc., _Neues Jahrb. f. Miner._ 1842, pp. 410–417; Lestiboudois, T., _Phyllotaxie anatomique_, Paris, 1848; Henslow, G., _Phyllotaxis_, London, 1871; Wiesner, Bemerkungen über rationale und irrationale Divergenzen, _Flora_, LVIII, pp. 113–115, 139–143, 1875; Airy, H., On Leaf Arrangement, _Proc. R. S._ XXI, p. 176, 1873; Schwendener, S., _Mechanische Theorie der Blattstellungen_, Leipzig, 1878; Delpino, F., _Causa meccanica della filotassi quincunciale_, Genova, 1880; de Candolle, C., _Étude de Phyllotaxie_, Genève, 1881.
[574] _Allgemeine Morphologie der Gewächse_, p. 442, etc. 1868.
[575] _Relation of Phyllotaxis to Mechanical Laws_, Oxford, 1901–1903; cf. _Ann. of Botany_, XV, p. 481, 1901.
[576] “The proposition is that the genetic spiral is a logarithmic spiral, homologous with the line of current-flow in a spiral vortex; and that in such a system the action of orthogonal forces will be mapped out by other orthogonally intersecting logarithmic spirals—the ‘parastichies’ ”; Church, _op. cit._ I, p. 42.
[577] Mr Church’s whole theory, if it be not based upon, is interwoven with, Sachs’s theory of the orthogonal intersection of cell-walls, and the elaborate theories of the symmetry of a growing point or apical cell which are connected therewith. According to Mr Church, “the law of the orthogonal intersection of cell-walls at a growing apex may be taken as generally accepted” (p. 32); but I have taken a very different view of Sachs’s law, in the eighth chapter of the present book. With regard to his own and Sachs’s hypotheses, Mr Church makes the following curious remark (p. 42): “Nor are the hypotheses here put forward more imaginative than that of the paraboloid apex of Sachs which remains incapable of proof, or his construction for the apical cell of Pteris which does not satisfy the evidence of his own drawings.”
[578] _Amer. Naturalist_, VII, p. 449, 1873.
[579] This celebrated series, which appears in the continued fraction (1/1) + (1/(1 + )) etc. and is closely connected with the _Sectio aurea_ or Golden Mean, is commonly called the Fibonacci series, after a very learned twelfth century arithmetician (known also as Leonardo of Pisa), who has some claims to be considered the introducer of Arabic numerals into christian Europe. It is called Lami’s series by some, after Father Bernard Lami, a contemporary of Newton’s, and one of the co-discoverers of the parallelogram of forces. It was well-known to Kepler, who, in his paper _De nive sexangula_ (cf. _supra_, p. 480), discussed it in connection with the form of the dodecahedron and icosahedron, and with the ternary or quinary symmetry of the flower. (Cf. Ludwig, F., Kepler über das Vorkommen der Fibonaccireihe im Pflanzenreich, _Bot. Centralbl._ LXVIII, p. 7, 1896). Professor William Allman, Professor of Botany in Dublin (father of the historian of Greek geometry), speculating on the same facts, put forward the curious suggestion that the cellular tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra. and that of the monocotyledons or endogens of icosahedra (_On the mathematical connexion between the parts of Vegetables_: abstract of a Memoir read before the Royal Society in the year 1811 (privately printed, _n.d._). Cf. De Candolle, _Organogénie végétale_, I, p. 534).
[580] _Proc. Roy. Soc. Edin._ VII, p. 391, 1872.
[581] The necessary existence of these recurring spirals is also proved, in a somewhat different way, by Leslie Ellis, On the Theory of Vegetable Spirals, in _Mathematical and other Writings_, 1853, pp. 358–372.
[582] _Proc. Roy. Soc. Edin._ VII, p. 397, 1872; _Trans. Roy. Soc. Edin._ XXVI, p. 505, 1870–71.
[583] A common form of pail-shaped waste-paper basket, with wide rhomboidal meshes of cane, is well-nigh as good a model as is required.
[584] _Deutsche Vierteljahrsschrift_, p. 261, 1868.
[585] _Memoirs of Amer. Acad._ IX, p. 389.
[586] _De avibus circa aquas Danubii vagantibus et de ipsarum Nidis_ (Vol. V of the _Danubius Pannonico-mysicus_), Hagae Com., 1726.
[587] Sir Thomas Browne had a collection of eggs at Norwich, according to Evelyn, in 1671.
[588] Cf. Lapierre, in Buffon’s _Histoire Naturelle_, ed. Sonnini, 1800.
[589] _Eier der Vögel Deutschlands_, 1818–28 (_cit._ des Murs, p. 36).
[590] _Traité d’Oologie_, 1860.
[591] Lafresnaye, F. de, Comparaison des œufs des Oiseaux avec leurs squelettes, comme seul moven de reconnaître la cause de leurs différentes formes, _Rev. Zool._, 1845, pp. 180–187, 239–244.
[592] Cf. Des Murs, p. 67: “Elle devait encore penser au moment où ce germe aurait besoin de l’espace nécessaire à son accroissement, à ce moment où ... il devra remplir exactement l’intervalle circonscrit par sa fragile prison, etc.”
[593] Thienemann, F. A. L., _Syst. Darstellung der Fortpflanzung der Vögel Europas_. Leipzig, 1825–38.
[594] Cf. Newton’s _Dictionary of Birds_, 1893, p. 191; Szielasko, Gestalt der Vogeleier, _J. f. Ornith._ LIII, pp. 273–297, 1905.
[595] Jacob Steiner suggested a Cartesian oval, _r_ + _mr′_ = _c_, as a general formula for all eggs (cf. Fechner, _Ber. sächs. Ges._, 1849, p. 57); but this formula (which fails in such a case as the guillemot), is purely empirical, and has no mechanical foundation.
[596] Günther, F. C., _Sammlung von Nestern und Eyern verschiedener Vögel_, Nürnb. 1772. Cf. also Raymond Pearl, Morphogenetic Activity of the Oviduct, _J. Exp. Zool._ VI, pp. 339–359, 1909.
[597] The following account is in part reprinted from _Nature_, June 4, 1908.
[598] In so far as our explanation involves a shaping or moulding of the egg by the uterus or “oviduct” (an agency supplemented by the proper tensions of the egg), it is curious to note that this is very much the same as that old view of Telesius regarding the formation of the embryo (_De rerum natura_, VI, cc. 4 and 10), which he had inherited from Galen, and of which Bacon speaks (_Nov. Org._ cap. 50; cf. Ellis’s note). Bacon expressly remarks that “Telesius should have been able to shew the like formation in the shells of eggs.” This old theory of embryonic modelling survives only in our usage of the term “matrix” for a “mould.”
[599] _Journal of Tropical Medicine_, 15th June, 1911. I leave this paragraph as it was written, though it is now once more asserted that the terminal and lateral-spined eggs belong to separate and distinct species of Bilharzia (Leiper, _Brit. Med. Journ._, 18th March, 1916, p. 411).
[600] Cf. Bashforth and Adams, _Theoretical Forms of Drops, etc._, Cambridge, 1883.
[601] Woods, R. H., On a Physical Theorem applied to tense Membranes, _Journ. of Anat. and Phys._ XXVI, pp. 362–371, 1892. A similar investigation of the tensions in the uterine wall, and of the varying thickness of its muscles, was attempted by Haughton in his _Animal Mechanics_, pp. 151–158, 1873.
[602] This corresponds with a determination of the normal pressures (in systole) by Krohl, as being in the ratio of 1 : 6·8.
[603] Cf. Schwalbe, G., Ueber Wechselbeziehungen und ihr Einfluss auf die Gestaltung des Arteriensystem, _Jen. Zeitschr._ XII, p. 267, 1878, Roux, Ueber die Verzweigungen der Blutgefässen des Menschen, _ibid._ XII, p. 205, 1878; Ueber die Bedeutung der Ablenkung des Arterienstämmen bei der Astaufgabe, _ibid._ XIII, p. 301, 1879; Hess, Walter, Eine mechanisch bedingte Gesetzmässigkeit im Bau des Blutgefässsystems, _A. f. Entw. Mech._ XVI, p. 632, 1903; Thoma, R., _Ueber die Histogenese und Histomechanik des Blutgefässsystems_, 1893.
[604] _Essays_, etc., edited by Owen, I, p. 134, 1861.
[605] On the Functions of the Heart and Arteries, _Phil. Trans._ 1809, pp. 1–31, cf. 1808, pp. 164–186; _Collected Works_, I, pp. 511–534, 1855. The same lesson is conveyed by all such work as that of Volkmann, E. H. Weber and Poiseuille. Cf. Stephen Hales’ _Statical Essays_, II, _Introduction_: “Especially considering that they [i.e. animal Bodies] are in a manner framed of one continued Maze of innumerable Canals, in which Fluids are incessantly circulating, some with great Force and Rapidity, others with very different Degrees of rebated Velocity: Hence, _etc._”
[606] “Sizes” is Owen’s editorial emendation, which seems amply justified.
[607] For a more elaborate classification, into colours cryptic, procryptic, anticryptic, apatetic, epigamic, sematic, episematic, aposematic, etc., see Poulton’s _Colours of Animals_ (Int. Scientific Series, LXVIII), 1890; cf. also Meldola, R., Variable Protective Colouring in Insects, _P.Z.S._ 1873, pp. 153–162, etc.
[608] Dendy, _Evolutionary Biology_, p. 336, 1912.
[609] Delight in beauty is one of the pleasures of the imagination; there is no limit to its indulgence, and no end to the results which we may ascribe to its exercise. But as for the particular “standard of beauty” which the bird (for instance) admires and selects (as Darwin says in the _Origin_, p. 70, edit. 1884), we are very much in the dark, and we run the risk of arguing in a circle: for wellnigh all we can safely say is what Addison says (in the 412th _Spectator_)—that each different species “is most affected with the beauties of its own kind .... Hinc merula in nigro se oblectat nigra marito; ... hinc noctua tetram Canitiem alarum et glaucos miratur ocellos.”
[610] Cf. Bridge, T. W., _Cambridge Natural History_ (Fishes), VII, p. 173, 1904; also Frisch, K. v., Ueber farbige Anpassung bei Fische, _Zool. Jahrb._ (_Abt. Allg. Zool._), XXXII, pp. 171–230, 1914.
[611] _Nature_, L, p. 572; LI, pp. 33, 57, 533, 1894–95.
[612] They are “wonderfully fitted for ‘vanishment’ against the flushed, rich-coloured skies of early morning and evening .... their chief feeding-times”; and “look like a real sunset or dawn, repeated on the opposite side of the heavens,—either east or west as the case may be”: Thayer, _Concealing-coloration in the Animal Kingdom_, New York, 1909, pp. 154–155. This hypothesis, like the rest, is not free from difficulty. Twilight is apt to be short in the homes of the flamingo: and moreover, Mr Abel Chapman, who watched them on the Guadalquivir, tells us that they _feed by day_.
[613] Principal Galloway, _Philosophy of Religion_, p. 344, 1914.
[614] Cf. Professor Flint, in his Preface to Affleck’s translation of Janet’s _Causes finales_: “We are, no doubt, still a long way from a mechanical theory of organic growth, but it may be said to be the _quaesitum_ of modern science, and no one can say that it is a chimaera.”
[615] Cf. Sir Donald MacAlister, How a Bone is Built, _Engl. Ill. Mag._ 1884.
[616] Professor Claxton Fidler, _On Bridge Construction_, p. 22 (4th ed.), 1909; cf. (_int. al._) Love’s _Elasticity_, p. 20 (_Historical Introduction_), 2nd ed., 1906.
[617] In preparing or “macerating” a skeleton, the naturalist nowadays carries on the process till nothing is left but the whitened bones. But the old anatomists, whose object was not the study of “comparative” morphology but the wider theme of comparative physiology, were wont to macerate by easy stages; and in many of their most instructive preparations, the ligaments were intentionally left in connection with the bones, and as part of the “skeleton.”
[618] In a few anatomical diagrams, for instance in some of the drawings in Schmaltz’s _Atlas der Anatomie des Pferdes_, we may see the system of “ties” diagrammatically inserted in the figure of the skeleton. Cf. Gregory, On the principles of Quadrupedal Locomotion, _Ann. N. Y. Acad. of Sciences_, XXII, p. 289, 1912.
[619] Galileo, _Dialogues concerning Two New Sciences_ (1638), Crew and Salvio’s translation, New York, 1914, p. 150; _Opere_, ed. Favaro, VIII, p. 186. Cf. Borelli, _De Motu Animalium_, I, prop. CLXXX, 1685. Cf. also Camper, P., La structure des os dans les oiseaux, _Opp._ III, p. 459, ed. 1803; Rauber, A., Galileo über Knochenformen, _Morphol. Jahrb._ VII, pp. 327, 328, 1881; Paolo Enriques, Della economia di sostanza nelle osse cave, _Arch. f. Ent. Mech._ XX, pp. 427–465, 1906.
[620] _Das mechanische Prinzip. im anatomischen Bau der Monocotylen_, Leipzig, 1874.
[621] For further botanical illustrations, see (_int. al._) Hegler, Einfluss der Zugkraften auf die Festigkeit und die Ausbildung mechanischer Gewebe in Pflanzen, _SB. sächs. Ges. d. Wiss._ p. 638, 1891; Kny, L., Einfluss von Zug und Druck auf die Richtung der Scheidewande in sich teilenden Pflanzenzellen, _Ber. d. bot. Gesellsch._ XIV, 1896; Sachs, Mechanomorphose und Phylogenie, _Flora_, LXXVIII, 1894; cf. also Pflüger, Einwirkung der Schwerkraft, etc., über die Richtung der Zelltheilung, _Archiv_, XXXIV, 1884.
[622] Among other works on the mechanical construction of bone see: Bourgery, _Traité de l’anatomie_ (_I. Ostéologie_), 1832 (with admirable illustrations of trabecular structure); Fick, L., _Die Ursachen der Knochenformen_, Göttingen, 1857; Meyer, H., Die Architektur der Spongiosa, _Archiv f. Anat. und Physiol._ XLVII, pp. 615–628, 1867; _Statik u. Mechanik des menschlichen Knochengerüstes_, Leipzig, 1873; Wolff, J., Die innere Architektur der Knochen, _Arch. f. Anat, und Phys._ L, 1870; _Das Gesetz der Transformation bei Knochen_, 1892; von Ebner, V., Der feinere Bau der Knochensubstanz, _Wiener Bericht_, LXXII, 1875; Rauber, Anton, _Elastizität und Festigkeit der Knochen_, Leipzig, 1876; O. Meserer, _Elast, u. Festigk. d. menschlichen Knochen_, Stuttgart, 1880; MacAlister, Sir Donald, How a Bone is Built, _English Illustr. Mag._ pp. 640–649, 1884; Rasumowsky, Architektonik des Fussskelets, _Int. Monatsschr. f. Anat._ p. 197, 1889; Zschokke, _Weitere Unters. über das Verhältniss der Knochenbildung zur Statik und Mechanik des Vertebratenskelets_, Zürich, 1892; Roux, W., _Ges. Abhandlungen über Entwicklungsmechanik der Organismen, Bd. I, Funktionelle Anpassung_, Leipzig, 1895; Triepel, H., Die Stossfestigkeit der Knochen, _Arch. f. Anat. u. Phys._ 1900; Gebhardt, Funktionell wichtige Anordnungsweisen der feineren und gröberen Bauelemente des Wirbelthierknochens, etc., _Arch. f. Entw. Mech._ 1900–1910; Kirchner. A., Architektur der Metatarsalien, _A. f. E. M._ XXIV, 1907; Triepel, Herm., Die trajectorielle Structuren (in _Einf. in die Physikalische Anatomie_, 1908); Dixon, A. F., Architecture of the Cancellous Tissue forming the Upper End of the Femur, _Journ. of Anat. and Phys._ (3) XLIV, pp. 223–230, 1910.
[623] Sédillot, De l’influence des fonctions sur la structure et la forme des organes; _C. R._ LIX, p. 539, 1864; cf. LX, p. 97, 1865, LXVIII. p. 1444. 1869.
[624] E.g. (1) the head, nodding backwards and forwards on a fulcrum, represented by the atlas vertebra, lying between the weight and the power; (2) the foot, raising on tip-toe the weight of the body against the fulcrum of the ground, where the weight is between the fulcrum and the power, the latter being represented by the _tendo Achillis_; (3) the arm, lifting a weight in the hand, with the power (i.e. the biceps muscle) between the fulcrum and the weight. (The second case, by the way, has been much disputed; cf. Haycraft in Schäfer’s _Textbook of Physiology_, p. 251, 1900.)
[625] Our problem is analogous to Dr Thomas Young’s problem of the best disposition of the timbers in a wooden ship (_Phil. Trans._ 1814, p. 303). He was not long of finding that the forces which may act upon the fabric are very numerous and very variable, and that the best mode of resisting them, or best structural arrangement for ultimate strength, becomes an immensely complicated problem.
[626] In like manner, Clerk Maxwell could not help employing the term “skeleton” in defining the mathematical conception of a “frame,” constituted by points and their interconnecting lines: in studying the equilibrium of which, we consider its different points as mutually acting on each other with forces whose directions are those of the lines joining each pair of points. Hence (says Maxwell), “in order to exhibit the mechanical action of the frame in the most elementary manner, we may draw it as a _skeleton_, in which the different points are joined by straight lines, and we may indicate by numbers attached to these lines the tensions or compressions in the corresponding pieces of the frame” (_Trans. R. S. E._ XXVI, p. 1, 1870). It follows that the diagram so constructed represents a “diagram of forces,” in this limited sense that it is geometrical as regards the position and direction of the forces, but arithmetical as regards their magnitude. It is to just such a diagram that the animal’s skeleton tends to approximate.
[627] When the jockey crouches over the neck of his race-horse, and when Tod Sloan introduced the “American seat,” the object in both cases is to relieve the hind-legs of weight, and so leave them free for the work of propulsion. Nevertheless, we must not exaggerate the share taken by the hind-limbs in this latter duty; cf. Stillman, _The Horse in Motion_, p. 69, 1882.
[628] This and the following diagrams are borrowed and adapted from Professor Fidler’s _Bridge Construction_.
[629] The method of constructing _reciprocal diagrams_, in which one should represent the outlines of a frame, and the other the system of forces necessary to keep it in equilibrium, was first indicated in Culmann’s _Graphische Statik_; it was greatly developed soon afterwards by Macquorn Rankine (_Phil. Mag._ Feb. 1864, and _Applied Mechanics_, passim), to whom is mainly due the general application of the principle to engineering practice.
[630] _Dialogues concerning Two New Sciences_ (1638): Crew and Salvio’s translation, p. 140 _seq._
[631] The form and direction of the vertebral spines have been frequently and elaborately described; cf. (e.g.) Gottlieb, H., Die Anticlinie der Wirbelsäule der Säugethiere, _Morphol. Jahrb._ LXIX, pp. 179–220, 1915, and many works quoted therein. According to Morita, Ueber die Ursachen der Richtung und Gestalt der thoracalen Dornfortsätze der Säugethierwirbelsäule (_ibi cit._ p. 201), various changes take place in the direction or inclination of these processes in rabbits, after section of the interspinous ligaments and muscles. These changes seem to be very much what we should expect, on simple mechanical grounds. See also Fischer, O., _Theoretische Grundlagen für eine Mechanik der lebenden Körper_, Leipzig, pp. 3, 372, 1906.
[632] I owe the first four of these determinations to the kindness of Dr Chalmers Mitchell, who had them made for me at the Zoological Society’s Gardens; while the great Clydesdale carthorse was weighed for me by a friend in Dundee.
[633] This pose of Diplodocus, and of other Sauropodous reptiles, has been much discussed. Cf. (_int. al._) Abel, O., _Abh. k. k. zool. bot. Ges. Wien_, V. 1909–10 (60 pp.); Tornier, _SB. Ges. Naturf. Fr. Berlin_, pp. 193–209, 1909; Hay, O. P., _Amer. Nat._ Oct. 1908; _Tr. Wash. Acad. Sci._ XLII, pp. 1–25, 1910; Holland, _Amer. Nat._ May, 1910, pp. 259–283; Matthew, _ibid._ pp. 547–560; Gilmore, C. W. (_Restoration of Stegosaurus_). _Pr. U.S. Nat. Museum_, 1915.
[634] The form of the cantilever is much less typical in the small flying birds, where the strength of the pelvic region is insured in another way, with which we need not here stop to deal.
[635] The motto was Macquorn Rankine’s.
[636] John Hunter was seldom wrong; but I cannot believe that he was right when he said (_Scientific Works_, ed. Owen, I, p. 371), “The bones, in a mechanical view, appear to be the first that are to be considered. We can study their shape, connexions, number, uses, etc., _without considering any other part of the body_.”
[637] _Origin of Species_, 6th ed. p. 118.
[638] _Amer. Naturalist_, April, 1915, p. 198, etc. Cf. _infra_, p. 727.
[639] Driesch sees in “Entelechy” that something which differentiates the whole from the sum of its parts in the case of the organism: “The organism, we know, is a system the single constituents of which are inorganic in themselves; only the whole constituted by them in their typical order or arrangement owes its specificity to ‘Entelechy’ ” (_Gifford Lectures_, p. 229, 1908): and I think it could be shewn that many other philosophers have said precisely the same thing. So far as the argument goes, I fail to see how _this_ Entelechy is shewn to be peculiarly or specifically related to the _living_ organism. The conception that the whole is _always_ something very different from its parts is a very ancient doctrine. The reader will perhaps remember how, in another vein, the theme is treated by Martinus Scriblerus: “In every Jack there is a _meat-roasting_ Quality, which neither resides in the fly, nor in the weight, nor in any particular wheel of the Jack, but is the result of the whole composition; etc., etc.”
[640] “There can be no doubt that Fraas is correct in regarding this type (_Procetus_) as an annectant form between the Zeuglodonts and the Creodonta, but, although the origin of the Zeuglodonts is thus made clear, it still seems to be by no means so certain as that author believes, that they may not themselves be the ancestral forms of the Odontoceti”; Andrews, _Tertiary Vertebrata of the Fayum_, 1906, p. 235.
[641] Reprinted, with some changes and additions, from a paper in the _Trans. Roy. Soc. Edin._ L, pp. 857–95, 1915.
[642] M. Bergson repudiates, with peculiar confidence, the application of mathematics to biology. Cf. _Creative Evolution_, p. 21, “Calculation touches, at most, certain phenomena of organic destruction. Organic creation, on the contrary, the evolutionary phenomena which properly constitute life, we cannot in any way subject to a mathematical treatment.”
[643] In this there lies a certain justification for a saying of Minot’s, of the greater part of which, nevertheless, I am heartily inclined to disapprove. “We biologists,” he says, “cannot deplore too frequently or too emphatically the great mathematical delusion by which men often of great if limited ability have been misled into becoming advocates of an erroneous conception of accuracy. The delusion is that no science is accurate until its results can be expressed mathematically. The error comes from the assumption that mathematics can express complex relations. Unfortunately mathematics have a very limited scope, and are based upon a few extremely rudimentary experiences, which we make as very little children and of which no adult has any recollection. The fact that from this basis men of genius have evolved wonderful methods of dealing with numerical relations should not blind us to another fact, namely, that the observational basis of mathematics is, psychologically speaking, very minute compared with the observational basis of even a single minor branch of biology .... While therefore here and there the mathematical methods may aid us, _we need a kind and degree of accuracy of which mathematics is absolutely incapable_ .... With human minds constituted as they actually are, we cannot anticipate that there will ever be a mathematical expression for any organ or even a single cell, although formulae will continue to be useful for dealing now and then with isolated details...” (_op. cit._, p. 19, 1911). It were easy to discuss and criticise these sweeping assertions, which perhaps had their origin and parentage in an _obiter dictum_ of Huxley’s, to the effect that “Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation” (_cit._ Cajori, _Hist of Elem. Mathematics_, p. 283). But Gauss called mathematics “a science of the eye”; and Sylvester assures us that “most, if not all, of the great ideas of modern mathematics have had their origin in observation” (_Brit. Ass. Address_, 1869, and _Laws of Verse_, p. 120, 1870).
[644] _Historia Animalium_ I, 1.
[645] Cf. _supra_, p. 714.
[646] Cf. Osborn, H. F., On the Origin of Single Characters, as observed in fossil and living Animals and Plants, _Amer. Nat._ XLIX, pp. 193–239, 1915 (and other papers); _ibid._ p. 194, “Each individual is composed of a vast number of somewhat similar new or old characters, each character has its independent and separate history, each character is in a certain stage of evolution, each character is correlated with the other characters of the individual .... The real problem has always been that of the origin and development of characters. Since the _Origin of Species_ appeared, the terms variation and variability have always referred to single characters; if a species is said to be variable, we mean that a considerable number of the single characters or groups of characters of which it is composed are variable,” etc.
[647] Cf. Sorby, _Quart. Journ. Geol. Soc._ (_Proc._), 1879, p. 88.
[648] Cf. D’Orbigny, Alc., _Cours élém. de Paléontologie_, etc., I, pp. 144–148, 1849; see also Sharpe, Daniel, On Slaty Cleavage, _Q.J.G.S._ III, p. 74, 1847.
[649] Thus _Ammonites erugatus_, when compressed, has been described as _A. planorbis_: cf. Blake, J. F., _Phil. Mag._ (5), VI, p. 260, 1878. Wettstein has shewn that several species of the fish-genus _Lepidopus_ have been based on specimens artificially deformed in various ways: Ueber die Fischfauna des Tertiären Glarnerschiefers, _Abh. Schw. Palaeont. Gesellsch._ XIII, 1886 (see especially pp. 23–38, pl. I). The whole subject, interesting as it is, has been little studied: both Blake and Wettstein deal with it mathematically.
[650] Cf. Sir Thomas Browne, in _The Garden of Cyrus_: “But why ofttimes one side of the leaf is unequall unto the other, as in Hazell and Oaks, why on either side the master vein the lesser and derivative channels stand not directly opposite, nor at equall angles, respectively unto the adverse side, but those of one side do often exceed the other, as the Wallnut and many more, deserves another enquiry.”
[651] Where gourds are common, the glass-blower is still apt to take them for a prototype, as the prehistoric potter also did. For instance, a tall, annulated Florence oil-flask is an exact but no longer a conscious imitation of a gourd which has been converted into a bottle in the manner described.
[652] Cf. _Elsie Venner_, chap. ii.
[653] This significance is particularly remarkable in connection with the development of speed, for the metacarpal region is the seat of very important leverage in the propulsion of the body. In the Museum of the Royal College of Surgeons in Edinburgh, there stand side by side the skeleton of an immense carthorse (celebrated for having drawn all the stones of the Bell Rock Lighthouse to the shore), and a beautiful skeleton of a racehorse, which (though the fact is disputed) there is good reason to believe is the actual skeleton of Eclipse. When I was a boy my grandfather used to point out to me that the cannon-bone of the little racer is not only relatively, but actually, longer than that of the great Clydesdale.
[654] Cf. Vitruvius, III, 1.
[655] _Les quatres livres d’Albert Dürer de la proportion des parties et pourtraicts des corps humains_, Arnheim, 1613, folio (and earlier editions). Cf. also Lavater, _Essays on Physiognomy_, III, p. 271, 1799.
[656] It was these very drawings of Dürer’s that gave to Peter Camper his notion of the “facial angle.” Camper’s method of comparison was the very same as ours, save that he only drew the axes, without filling in the network, of his coordinate system; he saw clearly the essential fact, that the skull _varies as a whole_, and that the “facial angle” is the index to a general deformation. “The great object was to shew that natural differences might be reduced to rules, of which the direction of the facial line forms the _norma_ or canon; and that these directions and inclinations are always accompanied by correspondent form, size and position of the other parts of the cranium,” etc.; from Dr T. Cogan’s preface to Camper’s work _On the Connexion between the Science of Anatomy and the Arts of Drawing, Painting and Sculpture_ (1768?), quoted in Dr R. Hamilton’s Memoir of Camper, in _Lives of Eminent Naturalists_ (_Nat. Libr._), Edin. 1840.
[657] The co-ordinate system of Fig. 382 is somewhat different from that which I drew and published in my former paper. It is not unlikely that further investigation will further simplify the comparison, and shew it to involve a still more symmetrical system.
[658] _Dinosaurs of North America_, pl. LXXXI, etc. 1896.
[659] _Mem. Amer. Mus. of Nat. Hist._ I, III, 1898.
[660] These and also other coordinate diagrams will be found in Mr G. Heilmann’s book _Fuglenes Afstamning_, 398 pp., Copenhagen, 1916; see especially pp. 368–380.
[661] Cf. W. B. Scott (_Amer. Journ. of Science_, XLVIII, pp. 335–374, 1894), “We find that any mammalian series at all complete, such as that of the horses, is remarkably continuous, and that the progress of discovery is steadily filling up what few gaps remain. So closely do successive stages follow upon one another that it is sometimes extremely difficult to arrange them all in order, and to distinguish clearly those members which belong in the main line of descent, and those which represent incipient branches. Some phylogenies actually suffer from an embarrassment of riches.”
[662] Cf. Dwight, T., The Range of Variation of the Human Scapula, _Amer. Nat._ XXI, pp. 627–638, 1887. Cf. also Turner, _Challenger Rep._ XLVII, on Human Skeletons, p. 86, 1886: “I gather both from my own measurements, and those of other observers, that the range of variation in the relative length and breadth of the scapula is very considerable in the same race, so that it needs a large number of bones to enable one to obtain an accurate idea of the mean of the race.”
[663] There is a paper on the mathematical study of organic forms and organic processes by the learned and celebrated Gustav Theodor Fechner, which I have only lately read, but which would have been of no little use and help to our argument had I known it before. (Ueber die mathematische Behandlung organischer Gestalten und Processe, _Berichte d. k. sächs. Gesellsch._, _Math.-phys. Cl._, Leipzig, 1849, pp. 50–64.) Fechner’s treatment is more purely mathematical and less physical in its scope and bearing than ours, and his paper is but a short one; but the conclusions to which he is led differ little from our own. Let me quote a single sentence which, together with its context, runs precisely on the lines of the discussion with which this chapter of ours began. “So ist also die mathematische Bestimmbarkeit im Gebiete des Organischen ganz eben so gut vorhanden als in dem des Unorganischen, und in letzterem eben solchen oder äquivalenten Beschränkungen unterworfen als in ersterem; und nur sofern die unorganischen Formen und das unorganische Geschehen sich einer einfacheren Gesetzlichkeit mehr nähern als die organischen, kann die Approximation im unorganischen Gebiet leichter und weiter getrieben werden als im organischen. Dies wäre der ganze, sonach rein relative, Unterschied.” Here in a nutshell, in words written some seventy years ago, is the gist of the whole matter.
An interesting little book of Schiaparelli’s (which I ought to have known long ago)—_Forme organiche naturali e forme geometriche pure_, Milano, Hoepli, 1898—has likewise come into my hands too late for discussion.
{780}
INDEX.
Abbe’s diffraction plates, 323
Abel, O., 706
Abonyi, A., 127
Acantharia, spicules of, 458
Acanthometridae, 462
Acceleration, 64
Aceratherium, 761
Achlya, 244
Acromegaly, 135
Actinomma, 469
Actinomyxidia, 452
Actinophrys, 165, 197, 264, 298
Actinosphaerium, 197, 266, 298, 468
Adams, J. C., 663
Adaptation, 670
Addison, Joseph, 671
Adiantum, 408
Adsorption, 192, 208, 241, 277, 357; orientirte, 440, 590; pseudo, 282
Agglutination, 201
Aglaophenia, 748
Airy, H., 636
Albumin molecule, 41
Alcyonaria, 387, 413, 424, 459
Alexeieff, A., 157, 165
Allmann, W., 643
Alpheus, claws of, 150
Alpine plants, 124
Altmann’s granules, 285
Alveolar meshwork, 170
Ammonites, 526, 530, 537, 539, 550, 552, 576, 583, 584, 728
Amoeba, 12, 165, 209, 212, 245, 255, 288, 463, 605
Amphidiscs, 440
Amphioxus, 311
Ampullaria, 560
Anabaena, 300
Anaxagoras, 8
Ancyloceras, 550
Andrews, G. F., 164; C. W., 716
Anhydrite, 433
Anikin, W. P., 130
Anisonema, 126
Anisotropy, 241, 357
Anomia, 565, 567
Antelopes, horns of, 614, 671
Antheridia, 303, 403, 405, 409
Anthoceros, spore of, 397
Anthogorgia, spicules of, 413
Anthropometry, 51
Anticline, 360
Antigonia, 750, 775
Antlers, 628
Apatornis, 757
Apocynum, pollen of, 396
Aptychus, 576
Arachnoidiscus, 387
Arachnophyllum, 325
Arcella, 323
Arcestes, 539, 540
Archaeopteryx, 757
Archimedes, 580; spiral of, 503, 524, 552
Argali, horns of, 617
Argiope, 561
Argonauta, 546, 561
Argus pheasant, 431, 631
Argyropelecus, 748
Aristotle, 3, 4, 5, 8, 15, 138, 149, 158, 509, 653, 714, 725, 726
Arizona trees, 121
Arrhenius, Sv., 28, 48, 171
Artemia, 127
Artemis, 561
Ascaris megalocephala, 180, 195
Aschemonella, 255
Assheton, R., 344
Asterina, 342
Asteroides, 423
Asterolampra, 386
Asters, 167, 174
Asthenosoma, 664
Astrorhiza, 255, 463, 587, 607
Astrosclera, 436
Asymmetric substances, 416
Asymmetry, 241
Atrypa, 569
Auerbach, F., 9
Aulacantha, 460
Aulastrum, 471
Aulonia, 468
Auricular height, 93
Autocatalysis, 131
Auximones, 135
Awerinzew, S., 589
Babak, E., 32
Babirussa, teeth of, 634
Baboon, skull of, 771
Bacillus, 39; B. ramosus, 133
Bacon, Lord, 4, 5, 51, 53, 131, 656, 716
Bacteria, 245, 250
Baer, K. E., von, 3, 55, 57, 155
Balancement, 714, 776
Balfour, F. M., 57, 348
Baltzer, Fr., 327
Bamboo, growth of, 77
Barclay, J., 334
Barfurth, D., 85
Barlow, W., 202
Barratt, J. O. W., 285
Bartholinus, E., 329
Bashforth, Fr., 663
Bast-fibres, strength of, 679
Baster, Job, 138
Bateson, W., 104, 431
Bather, F. A., 578
Batsch, A. J. G. K., 606
Baudrimont, A., and St Ange, 124
Baumann and Roos, 136
Bayliss, W. M., 135, 277
Beads or globules, 234
Beak, shape of, 632
Beal, W. J., 643
Beam, loaded, 674
Bee’s cell, 327, 779
Begonia, 412, 733
Beisa antelope, horns of, 616, 621
Bellerophon, 550
Bénard, H., 259, 319, 448, 590
Bending moments, 19, 677, 696
Beneden, Ed. van, 153, 170, 198
Bergson, H., 7, 103, 251, 611, 721
Bernard, Claude, 2, 13, 127
Bernoulli, James, 580; John, 30, 54
Berthold, G., 8, 234, 298, 306, 322, 346, 351, 357, 358, 372, 399
Bethe, A., 276
Bialaszewicz, K., 114, 125
Biedermann, W., 431
Bilharzia, egg of, 656
Binuclearity, 286
Biocrystallisation, 454
Biogenetisches Grundgesetz, 608
Biometrics, 78
Bird, flight of, 24; form of, 673
Bisection of solids, 352, etc.
Bishop, John 31
Bivalve shells, 561
Bjerknes, V. 186
Blackman, F. F. 108, 110, 114, 124, 131, 132
Blackwall, J. 234
Blake, J. F. 536, 547, 553, 578, 583, 728
Blastosphere, 56, 344
Blood-corpuscles, form of, 270; size of, 36
Blood-vessels, 665
Boas, Fr., 79
Bodo, 230, 269
Boerhaave, Hermann, 380
Bonanni, F., 318
Bone, 425, 435; repair of, 687; structure of, 673, 680
Bonnet, Ch., 108, 138, 334, 635
Borelli, J. A., 8, 27, 29, 318, 677, 690
Bosanquet, B., 5
Boscovich, Father R. J., S.J., 8
Bose, J. C., 87
Bostryx, 502
Bottazzi, F., 127
Bottomley, J. T., 135
Boubée, N., 529
Bourgery, J. M., 683
Bourne, G. C., 199
Bourrelet, Plateau’s, 297, 339, 446, 470, 477
Boveri, Th., 38, 147, 170, 198
Bowditch, H. P., 61, 79
Bower, F. O., 406
Bowman, J. H., 428
Boyd, R., 61
Boys, C. V., 233
Brachiopods, 561, 568, 577
Bradford, S. C., 428
Brady, H. B., 255, 606
Brain, growth of, 89; weight of, 90
Branchipus, 128, 342
Brandt, K., 459, 482
Brauer, A., 180
Braun, A., 636
Bravais, L. and A., 202, 502, 636
Bredig, G., 178
Brewster, Sir D., 209, 337, 350, 431
Bridge, T. W., 671
Bridge construction, 18, 691
Brine shrimps, 127
Brooke, Sir V., 614, 624, 628, 631
Browne, Sir T., 324, 329, 480, 650, 652, 733
Brownian movement, 45, 279, 421
Brücke, C., 160, 199
Buccinum, 520, 527
Buch, Leopold von, 528, 583
Buchner, Hans, 133
Budding, 213, 399
Buffon, on the bee’s cell, 333
Bühle, C. A., 653
Bulimus, 549, 556
Burnet, J., 509
Bütschli, O., 165, 170, 171, 204, 432, 434, 458, 492
Büttel-Reepen, H. von, 332
Byk, A., 419
Cactus, sphaerocrystals, in 434
Cadets, growth of German, 119
Calandrini, G. L., 636
Calcospherites, 421, 434
Callimitra, 472
Callithamnion, spore of, 396
Calman, T. W., 149
Calyptraea, 556
Camel, 703, 704
Campanularia, 237, 262, 747
Campbell, D. H., 302, 397, 402
Camper, P., 742
Camptosaurus, 754
Cannon bone, 730
Cantilever, 678, 694
Cantor, Moritz, 503
Caprella, 743
Caprinella, 567, 577
Carapace of crabs, 744
Cardium, 561
Cariacus, 629
Carlier, E. W., 211
Carnoy, J. B., 468
Carpenter, W. B., 45, 422, 465
Caryokinesis, 14, 157, etc.
Cassini, D., 329
Cassis, 559
Catabolic products, 435
Catalytic action, 130
Catenoid, 218, 223, 227, 252
Causation, 6
Cavolinia, 573
Cayley, A., 385
Celestite, 459
Cell-theory, 197, 199
Cells, forms of, 201; sizes of, 35
Cellular pathology, 200; tissue, artificial, 320
Cenosphaera, 470
Centres of force, 156, 196
Centrosome, 167, 168, 173
Cephalopods, 548, etc.; eggs of, 378
Ceratophyllum, growth of, 97
Ceratorhinus, 612
Cerebratulus, egg of, 189
Cerianthus, 125
Cerithium, 530, 557, 559
Chabrier, J., 25
Chabry, L., 30, 306, 415
Chaetodont fishes, 671, 749
Chaetopterus, egg of, 195
Chamois, horns of, 615
Chapman, Abel, 672
Chara, 303
Characters, biological, 196, 727
Chevron bones, 709
Chick, hatching of, 108
Chilomonas, 114
Chladni figures, 386, 475
Chlorophyll, 291
Choanoflagellates, 253
Chodat, R., 78, 132
Cholesterin, 272
Chondriosomes, 285
Chorinus, 744
Chree, C., 19
Chromatin, 153
Chromidia, 286
Chromosomes, 157, 173, 179, 181, 190, 195
Church, A. H., 639
Cicero, 62
Cicinnus, 502
Cidaris, 664
Circogonia, 479
Cladocarpus, 748
Claparède, E. R, 423
Clathrulina, 470
Clausilia, 520, 549
Claws, 149, 632
Cleland, John, 4
Cleodora, 570–575
Climate and growth, 121
Clio, 570
Close packing, 453
Clytia, 747
Coan, C. A., 514
Coassus, 629
Cod, otoliths of, 432; skeleton of, 710
Codonella, 248
Codosiga, 253
Coe, W. R., 189
Coefficient of growth, 153; of temperature, 109
Coelopleurus, 664
Cogan, Dr T., 742
Cohen, A., 110
Cohesion figures, 259
Collar-cells, 253
Colloids, 162, 178, 201, 279, 412, 421, etc.
Collosclerophora, 436
Collosphaera, 459
Colman, S., 514
Comoseris, 327
Compensation, law of, 714, 776
Conchospiral, 531, 539, 594
Conchyliometer, 529
Concretions, 410, etc.
Conjugate curves, 561, 613
Conklin, E. G., 36, 191, 310, 340, 377
Conostats, 427
Continuous girder, 700
Contractile vacuole, 165, 264
Conus, 557, 559, 560
Cook, Sir T. A., 493, 635, 639, 650
Co-ordinates, 723
Corals, 325, 388, 423
Cornevin, Ch., 102
Cornuspira, 594
Correlation, 78, 727
Corystes, 744
Cotton, A., 418
Cox, J., 46
Crane-head, 682
Crayfish, sperm-cells of, 273
Creodonta, 716
Crepidula, 36, 310, 340
Creseis, 570
Cristellaria, 515, 600
Crocodile, 704, 752
Crocus, growth of, 88
Crookes, Sir W., 32
Cryptocleidus, 755
Crystals, 202, 250, 429, 444, 480, 601
Ctenophora, 391
Cube, partition of, 346
Cucumis, growth of, 109
Culmann, Professor C., 682, 697
Cultellus, 564
Curlew, eggs of, 652
Cushman, J. A., 323
Cuvier, 727
Cuvierina, 258, 570
Cyamus, 743
Cyathophyllum, 325, 391
Cyclammina, 595, 596, 602
Cyclas, 561
Cyclostoma, 554
Cylinder, 218, 227, 377
Cymba, 559
Cyme, 502
Cypraea, 547, 554, 560, 561
Cyrtina, 569
Cyrtocerata, 583
Cystoliths, 412
Daday de Dees, E. v., 130
Daffner, Fr., 61, 118
Dalyell, Sir John G., 146
Danilewsky, B., 135
Darling, C. R., 219, 257, 664
D’Arsonval, A., 192, 281
Darwin, C., 4, 44, 57, 332, 431, 465, 549, 624, 671, 714
Dastre, A., 136
Davenport, C. B., 107, 123, 125, 126, 211
De Candolle, A., 108, 643; A. P., 20; C., 636
Decapod Crustacea, sperm-cells of, 273
Deer, antlers of, 628
Deformation, 638, 728, etc.
Degree, differences of, 586, 725
Delage, Yves, 153
Delaunay, C. E., 218
Delisle, 31
Dellinger, O. P., 212
Delphinula, 557
Delpino, F., 636
Democritus, 44
Dendy, A., 137, 436, 440, 671
Dentalium, 535, 537, 546, 555, 556, 561
Dentine, 425
Descartes, R., 185, 723
Des Murs, O., 653
Devaux, H., 43
De Vries, H., 108
Diatoms, 214, 386, 426
Diceras, 567
Dickson, Alex., 647
Dictyota, 303, 356, 474
Diet and growth, 134
Difflugia, 463, 466
Diffusion figures, 259, 430
Dimorphism of earwigs, 105
Dimorphodon, 756
Dinenympha, 252
Dinobryon, 248
Dinosaurs, 702, 704, 754
Diodon, 751, 777
Dionaea, 734
Diplodocus, 702, 706, 710
Disc, segmentation of a, 367
Discorbina, 602
Distigma, 246
Distribution, geographical, 457, 606
Ditrupa, 586
Dixon, A. F., 684
Dobell, C. C., 286
Dodecahedron, 336, 478, etc.
Doflein, F. J., 46, 267, 606
Dog’s skull, 773
Dolium, 526, 528, 530, 557, 559, 560
Dolphin, skeleton of, 709
Donaldson, H. H., 82, 93
Dorataspis, 481
D’Orbigny, Alc., 529, 555, 591, 728
Douglass, A. E., 121
Draper, J. W., 165, 264
Dreyer, F. R., 435, 447, 455, 468, 606, 608
Driesch, H., 4, 35, 157, 306, 310, 312, 377, 378, 714
Dromia, 275
Drops, 44, 257, 587
Du Bois-Reymond, Emil, 1, 92
Duerden, J. E., 423
Dufour, Louis, 219
Dujardin, F., 257, 591
Dunan, 7
Duncan, P. Martin, 388
Dupré, Athanase, 279
Durbin, Marion L., 138
Dürer, A., 55, 740, 742
Dutrochet, R. J. H., 212, 624
Dwight, T., 769
Dynamical similarity, 17
Earthworm, calcospheres in, 423
Earwigs, dimorphism in, 104
Ebner, V. von, 444, 683
Echinoderms, larval, 392; spicules of, 449
Echinus, 377, 378, 664
Eclipse, skeleton of, 739
Ectosarc, 281
Eel, growth of, 85
Efficiency, mechanical, 670
Efficient cause, 6, 158, 248
Eggs of birds, 652
Eiffel tower, 20
Eight cells, grouping of, 381, etc.
Eimer, Th., 606
Einstein formula, 47
Elastic curve, 219, 265, 271
Elaters, 489
Electrical convection, 187; stimulation of growth, 153
Elephant, 21, 633, 703, 704
Elk, antlers of, 629, 632
Ellipsolithes, 728
Ellis, R. Leslie, 4, 329, 647; M. M., 147, 656
Elodea, 322
Emarginula, 556
Emmel, V. E., 149
Empedocles, 8
Emperor Moth, 431
Encystment, 213, 283
Engelmann, T. W., 210, 285
Enriques, P., 4, 36, 64, 133, 134, 677
Entelechy, 4, 714
Entosolenia, 449
Enzymes, 135
Epeira, 233
Epicurus, 47
Epidermis, 314, 370
Epilobium, pollen of, 396
Epipolic force, 212
Equatorial plate, 174
Equiangular spiral, 50, 505
Equilibrium, figures of, 227
Equipotential lines, 640
Equisetum, spores of, 290, 489
Errera, Leo, 8, 40, 110, 111, 213, 306, 346, 348, 426
Erythrotrichia, 358, 372, 390
Ethmosphaera, 470
Euastrum, 214
Eucharis, 391
Euclid, 509
Euglena, 376
Euglypha, 189
Euler, L., 3, 208, 385, 484, 690
Eulima, 559
Eunicea, spicules of, 424
Euomphalus, 557, 559
Evelyn, John, 652
Evolution, 549, 610, etc.
Ewart, A. J., 20
Fabre, J. H., 64, 779
Facial angle, 742, 770, 772
Faraday, M., 163, 167, 428, 475
Farmer, J. B. and Digby, 190
Fatigue, molecular, 689
Faucon, A., 88
Favosites, 325
Fechner, G. T., 654, 777
Fedorow, E. S. von, 338
Fehling, H., 76, 126
Ferns, spores of, 396
Fertilisation, 193
Fezzan-worms, 127
Fibonacci, 643
Fibrillenkonus, 285
Fick, R., 57, 683
Fickert, C., 606
Fidler, Prof. T. Claxton, 691, 674, 696
Films, liquid, 215, 217, 426
Filter-passers, 39
Final cause, 3, 248, 714
Fir-cone, 635, 647
Fischel, Alfred, 88
Fischer, Alfred, 40, 172; Emil, 417, 418; Otto, 30, 699
Fishes, forms of, 748
Fission, multiplication by, 151
Fissurella, 556
FitzGerald, G. F., 158, 281, 323, 440, 477
Flagellum, 246, 267, 291
Flemming, W., 170, 172, 180
Flight, 24
Flint, Professor, 673
Fluid crystals, 204, 272, 485
Fluted pattern, 260
Fly’s cornea, 324
Fol, Hermann, 168, 194
Folliculina, 249
Foraminifera, 214, 255, 415, 495, 515
Forth Bridge, 694, 699, 700
Fossula, 390
Foster, M., 185
Fraas, E., 716
Frankenheim, M. L., 202
Frazee, O. E., 153
Frédéricq, L., 127, 130
Free cell formation, 396
Friedenthal, H., 64
Frisch, K. von, 671
Frog, egg of, 310, 363, 378, 382; growth of, 93, 126
Froth or foam, 171, 205, 305, 314, 322, 343
Froude, W., 22
Fucus, 355
Fundulus, 125
Fusulina, 593, 594
Fusus, 527, 557
Gadow, H. F., 628
Galathea, 273
Galen, 3, 465, 656
Galileo, 8, 19, 28, 562, 677, 720
Gallardo, A., 163
Galloway, Principal, 672
Gamble, F. A., 458
Ganglion-cells, size of, 37
Gans, R., 46
Garden of Cyrus, 324, 329
Gastrula, 344
Gauss, K. F., 207, 278, 723
Gebhardt, W., 430, 683
Gelatination, water of, 203
Generating curves and spirals, 526, 561, 615, 637, 641
Geodetics, 440, 488
Geoffroy St Hilaire, Et. de, 714
Geotropism, 211
Gerassimow, J. J., 35
Gerdy, P. N., 491
Geryon, 744
Gestaltungskraft, 485
Giard, A., 156
Gilmore, C. W., 707
Giraffe, 705, 730, 738
Girardia, 321, 408
Glaisher, J., 250
Glassblowing, 238, 737
Gley, E., 135, 136
Globigerina, 214, 234, 440, 495, 589, 602, 604, 606
Gnomon, 509, 515, 591
Goat, horns of, 613
Goat moth, wings of, 430
Goebel, K., 321, 397, 408
Goethe, 20, 38, 199, 714, 719
Golden Mean, 511, 643, 649
Goldschmidt, R., 286
Goniatites, 550, 728
Gonothyraea, 747
Goodsir, John, 156, 196, 580
Gottlieb, H., 699
Gourd, form of, 737
Grabau, A. H., 531, 539, 550
Graham, Thomas, 162, 201, 203
Grant, Kerr, 259
Grantia, 445
Graphic statics, 682
Gravitation, 12, 32
Gray, J., 188
Greenhill, Sir A. G., 19
Gregory, D. F., 330, 675
Greville, R. K., 386
Gromia, 234, 257
Gruber, A., 165
Gryphaea, 546, 576, 577
Guard-cells, 394
Gudernatsch, J. F., 136
Guillemot, egg of, 652
Gulliver, G., 36
Günther, F. C., 633, 654
Gurwitsch, A., 285
Häcker, V., 458
Haddock, 774
Haeckel, E., 199, 445, 454, 455, 457, 467, 480, 481
Hair, pigmentation of, 430
Hales, Stephen, 36, 59, 95, 669
Haliotis, 514, 527, 546, 547, 554, 555, 557, 561
Hall, C. E., 119
Haller, A. von, 2, 54, 56, 59, 64, 68
Hardesty, Irving, 37
Hardy, W. B., 160, 162, 172, 187, 287
Harlé, N., 28
Harmozones, 135
Harpa, 526, 528, 559
Harper, R. A., 283
Harpinia, 746
Harting, P., 282, 420, 426, 434
Hartog, M., 163, 327
Harvey, E. N. and H. W., 187
Hatai, S., 132, 135
Hatchett, C., 420
Hatschek, B., 180
Haughton, Rev. S., 334, 666
Haüy, R. J., 720
Hay, O. P., 707
Haycraft, J. B., 211, 690
Head, length of, 93
Heart, growth of, 89; muscles of, 490
Heath, Sir T., 511
Hegel, G. W. F., 4
Hegler, 680, 688
Heidenhain, M., 170, 212
Heilmann, Gerhard, 757, 768, 772
Helicoid, 230; cyme, 502, 605
Helicometer, 529
Helicostyla, 557
Heliolites, 326
Heliozoa, 264, 460
Helix, 528, 557
Helmholtz, H. von, 2, 9, 25
Henderson, W. P., 323
Henslow, G., 636
Heredity, 158, 286, 715
Hermann, F., 170
Hero of Alexandria, 509
Heron-Allen, E., 257, 415, 465
Herpetomonas, 268
Hertwig, O., 56, 114, 153, 199, 310; R., 170, 285
Hertzog, R. O., 109
Hess, W., 666, 668
Heteronymous horns, 619
Heterophyllia, 388
Hexactinellids, 429, 452, 453
Hexagonal symmetry, 319, 323, 471, 513
Hickson, S. J., 424
Hippopus, 561
His, W., 55, 56, 74, 75
Hobbes, Thomas, 159
Höber, R., 1, 126, 130, 172
Hodograph, 516
Hoffmann, C., 628
Hofmeister, F., 41; W., 87, 210, 234, 304, 306, 636, 639
Holland, W. J., 707
Holmes, O. W., 62, 737
Holothuroid spicules, 440, 451
Homonymous horns, 619
Homoplasy, 251
Hooke, Robert, 205
Hop, growth of, 118; stem of, 627
Horace, 44
Hormones, 135
Horns, 612
Horse, 694, 701, 703, 764
Houssay, F., 21
Huber, P., 332
Huia bird, 633
Humboldt, A. von, 127
Hume, David, 6
Hunter, John, 667, 669, 713, 715
Huxley, T. H., 423, 722, 752
Hyacinth, 322, 394
Hyalaea, 571–577
Hyalonema, 442
Hyatt, A., 548
Hyde, Ida H., 125, 163, 184, 188
Hydra, 252; egg of, 164
Hydractinia, 342
Hydraulics, 669
Hydrocharis, 234
Hyperia, 746
Hyrachyus, 760, 765
Hyracotherium, 766, 768
Ibex, 617
Ice, structure of, 428
Ichthyosaurus, 755
Icosahedron, 478
Iguanodon, 706, 708
Inachus, sperm-cells of, 273
Infusoria, 246, 489
Intussusception, 202
Inulin, 432
Invagination, 56, 344
Iodine, 136
Irvine, Robert, 414, 434
Isocardia, 561, 577
Isoperimetrical problems, 208, 346
Isotonic solutions, 130, 274
Iterson, G. van, 595
Jackson, C. M., 75, 88, 106
Jamin, J. C., 418
Janet, Paul, 5, 18, 673
Japp, F. R., 417
Jellett, J. H., 1
Jenkin, C. F., 444
Jenkinson, J. W., 94, 114, 170
Jennings, H. S., 212, 492; Vaughan, 424
Jensen, P., 211
Johnson, Dr S., 62
Joly, John, 9, 63
Jost, L., 110, 111
Juncus, pith of, 335
Jungermannia, 404
Kangaroo, 705, 706, 709
Kanitz, Al., 109
Kant, Immanuel, 1, 3, 714
Kappers, C. U. A., 566
Kellicott, W. E., 91
Kelvin, Lord, 9, 49, 188, 202, 336, 453
Kepler, 328, 480, 486, 643, 650
Kienitz-Gerloff, F., 404, 408
Kirby and Spence, 28, 30, 127
Kirchner, A., 683
Kirkpatrick, R., 437
Klebs, G., 306
Kny, L., 680
Koch, G. von, 423
Koenig, Samuel, 330
Kofoid, C. A., 268
Kölliker, A. von, 413
Kollmann, M., 170
Koltzoff, N. K., 273, 462
Koninckina, 570
Koodoo, horns of, 624
Köppen, Wladimir, 111
Korotneff, A., 377
Kraus, G., 77
Krogh, A., 109
Krohl, 666
Kühne, W., 235
Küster, E., 430
Lafresnaye, F. de, 653
Lagena, 251, 256, 260, 587
Lagrange, J. L., 649
Lalanne, L., 334
Lamarck, J. B. de, 549, 716
Lamb, A. B., 186
Lamellaria, 554
Lamellibranchs, 561
Lami, B., 296, 643
Laminaria, 315
Lammel, R., 100
Lanchester, F. W., 26
Lang, Arnold, 561
Lankester, Sir E. Ray, 4, 251, 348, 465
Laplace, P. S. de, 1, 207, 217
Larmor, Sir J., 9, 259
Lavater, J. C., 740
Law, Borelli’s, 29; Brandt’s, 482; of Constant Angle, 599; Errera’s, 213, 306; Froude’s, 22; Lamarle’s, 309; of Mass, 130; Maupertuis’s, 208; Müller’s, 481; of Optimum, 110; van’t Hoff’s, 109; Willard-Gibbs’, 280; Wolff’s, 3, 51, 155
Leaping, 29
Leaves, arrangement of, 635; form of, 731
Ledingham, J. C. G., 211
Leduc, Stéphane, 162, 167, 185, 219, 259, 415, 428, 431, 590
Leeuwenhoek, A. van, 36, 209
Leger, L., 452
Le Hello, P., 30
Lehmann, O., 203, 272, 440, 485, 590
Leibniz, G. W. von, 3, 5, 159, 385
Leidenfrost, J. G., 279
Leidy, J., 252, 468
Leiper, R. T., 660
Leitch, I., 112
Leitgeb, H., 305
Length-weight coefficient, 98–103, 775
Leonardo da Vinci, 27, 635; of Pisa, 643
Lepeschkin, 625
Leptocephalus, 87
Leray, Ad., 18
Lesage, G. L., 18
Leslie, Sir John, 163, 503
Lestiboudois, T., 636
Leucocytes, 211
Levers, Orders of, 690
Levi, G., 35, 37
Lewis, C. M., 280
Lhuilier, S. A. J., 330
Liesegang’s rings, 427, 475
Light, pressure of, 48
Lillie, F. R., 4, 147, 341; R. S., 180, 187, 192
Lima, 565
Limacina, 571
Lines of force, 163; of growth, 562
Lingula, 251, 567
Linnaeus, 28, 250, 547, 720
Lion, brain of, 91
Liquid veins, 265
Lister, Martin, 318; J. J., 436
Listing, J. B., 385
Lithostrotion, 325
Littorina, 524
Lituites, 546, 550
Llama, 703
Lobsters’ claws, 149
Locke, John, 6
Loeb, J., 125, 132, 135, 136, 147, 157, 191, 193
Loewy, A., 281
Logarithmic spiral, 493, etc.
Loisel, G., 88
Loligo, shell of, 575
Lo Monaco, 83
Lönnberg, E., 614, 632
Looss, A., 660
Lotze, R. H., 55
Love, A. E. H., 674
Lucas, F. A., 138
Luciani, L., 83
Lucretius, 47, 71, 137, 160
Ludwig, Carl, 2; F., 643; H. J., 342
Lupa, 744
Lupinus, growth of, 109, 112
Macalister, A., 557
MacAlister, Sir D., 673, 683
Macallum, A. B., 277, 287, 357, 395; J. B., 492
McCoy, F., 388
Mach, Ernst, 209, 330
Machaerodus, teeth of, 633
McKendrick, J. G., 42
McKenzie, A., 418
Mackinnon, D. L., 268
Maclaurin, Colin, 330, 779
Macroscaphites, 550
Mactra, 562
Magnitude, 16
Maillard, L., 163
Maize, growth of, 109, 111, 298
Mall, F. P., 492
Maltaux, Mlle, 114
Mammoth, 634, 705
Man, growth of, 61; skull of, 770
Maraldi, J. P., 329, 473
Marbled papers, 736
Marcus Aurelius, 609
Markhor, horns of, 619
Marsh, O. C., 706, 754
Marsigli, Comte L. F. de, 652
Massart, J., 114
Mastodon, 634
Mathematics, 719, 778, etc.
Mathews, A., 285
Matrix, 656
Matter and energy, 11
Matthew, W. D., 707
Matuta, 744
Maupas, M., 133
Maupertuis, 3, 5, 208
Maxwell, J. Clerk, 9, 18, 40, 44, 160, 207, 385, 691
Mechanical efficiency, 670
Mechanism, 5, 161, 185, etc.
Meek, C. F. U., 190
Melanchthon, 4
Melanopsis, 557
Meldola, R., 670
Melipona, 332
Mellor, J. W., 134
Melo, 525
Melobesia, 412
Melsens, L. H. F., 282
Membrane-formation, 281
Mensbrugghe, G. van der, 212, 298, 470
Meserer, O., 683
Mesocarpus, 289
Mesohippus, 766
Metamorphosis, 82
Meves, F., 163, 285
Meyer, Arthur, 432; G. H., 8, 682, 683
Micellae, 157
Michaelis, L., 277
Microchemistry, 288
Micrococci, 39, 245, 250
Micromonas, 38
Miliolidae, 595, 604
Milner, R. S., 280
Milton, John, 779
Mimicry, 671
Minchin, E. A., 267, 444, 449, 455
Minimal areas, 208, 215, 225, 293, 306, 336, 349
Minot, C. S., 37, 72, 722
Miohippus, 767
Mitchell, P. Chalmers, 703
Mitosis, 170
Mitra, 557, 559
Möbius, K., 449
Modiola, 562
Mohl, H. von, 624
Molar and molecular forces, 53
Mole-cricket, chromosomes of, 181
Molecular asymmetry, 416
Molecules, 41
Möller, V. von, 593
Monnier, A., 78, 132
Monticulipora, 326
Moore, B., 272
Morey, S., 264
Morgan, T. H., 126, 134, 138, 147
Morita, 699
Morphodynamique, 156
Morphologie synthétique, 420
Morphology, 719, etc.
Morse, Max, 136
Moseley, H., 8, 518, 521, 538, 553, 555, 592
Moss, embryo of, 374; gemma of, 403; rhizoids of, 356
Mouillard, L. P., 27
Mouse, growth of, 82
Mucor, sporangium of, 303
Müllenhof, K. von, 25, 332
Müller, Fritz, 3; Johannes, 459, 481
Mummery, J. H., 425
Munro, H., 323
Musk-ox, horns of, 615
Mya, 422, 561
Myonemes, 562
Naber, H. A., 511, 650
Nägeli, C., 124, 159, 210
Nassellaria, 472
Natica, 554, 557, 559
Natural selection, 4, 58, 137, 456, 586, 609, 651, 653
Naumann, C. F., 529, 531, 539, 550, 577, 594, 636; J. F., 653
Nautilus, 355, 494, 501, 515, 518, 532, 535, 546, 552, 557, 575, 577, 580, 592, 633; hood of, 554; kidney of, 425; N. umbilicatus, 542, 547, 554
Nebenkern, 285
Neottia, pollen of, 396
Nereis, egg of, 342, 378, 453
Nerita, 522, 555
Neumayr, M., 608
Neutral zone, 674, 676, 686
Newton, 1, 6, 158, 643, 721
Nicholson, H. A., 325, 327
Noctiluca, 246
Nodoid, 218, 223
Nodosaria, 262, 535, 604
Norman, A. M., 465
Norris, Richard, 272
Nostoc, 300, 313
Notosuchus, 753
Nuclear spindle, 170; structure, 166
Nummulites, 504, 552, 591
Nussbaum, M., 198
Oekotraustes, 550
Ogilvie-Gordon, M. M., 423
Oil-globules, Plateau’s, 219
Oithona, 742
Oken, L., 4, 635
Oliva, 554
Ootype, 660
Operculina, 594
Operculum of gastropods, 521
Oppel, A., 88
Optimum temperature, 110
Orbitolites, 605
Orbulina, 59, 225, 257, 587, 598, 604, 607
Organs, growth of, 88
Orthagoriscus, 751, 775, 777
Orthis, 561, 567
Orthoceras, 515, 548, 551, 556, 579, 735
Orthogenesis, 549
Orthogonal trajectories, 305, 377, 400, 640, 678
Orthostichies, 649
Orthotoluidene, 219
Oryx, horns of, 616
Osborn, H. F., 714, 727, 760
Oscillatoria, 300
Osmosis, 124, 287, etc.
Osmunda, 396, 406
Ostrea, 562
Ostrich, 25, 707, 708
Ostwald, Wilhelm, 44, 131, 426; Wolfgang, 32, 77, 82, 132, 277, 281
Otoliths, 425, 432
Ovis Ammon, 614
Owen, Sir R., 20, 575, 654, 669, 715
Ox, cannon-bone of, 730, 738; growth of, 102
Oxalate, calcium, 412, 434
Palaeechinus, 663
Palm, 624
Pander, C. H., 55
Pangenesis, 44, 157
Papillon, Fernand, 10
Pappus of Alexandria, 328
Parabolic girder, 693, 696
Parahippus, 767
Paralomis, 744
Paraphyses of mosses, 351
Parastichies, 640, 641
Passiflora, pollen of, 396
Pasteur, L., 416
Patella, 561
Pauli, W., 211, 434
Pearl, Raymond, 90, 97, 654
Pearls, 425, 431
Pearson, Karl, 36, 78
Peas, growth of, 112
Pecten, 562
Peddie, W., 182, 272, 344, 448
Pellia, spore of, 302
Pelseneer, P., 570
Pendulum, 30
Peneroplis, 606
Percentage-curves, Minot’s, 72
Pericline, 360
Periploca, pollen of, 396
Peristome, 239
Permeability, magnetic, 177, 182
Perrin, J., 43, 46
Peter, Karl, 117
Pettigrew, J. B., 490
Pfeffer, W., 111, 273, 688
Pflüger, E., 680
Phagocytosis, 211
Phascum, 408
Phase of curve, 68, 81, etc.
Phasianella, 557, 559
Phatnaspis, 482
Phillipsastraea, 327
Philolaus, 779
Pholas, 561
Phormosoma, 664
Phractaspis, 484
Phyllotaxis, 635
Phylogeny, 196, 251, 548, 716
Pike, F. H., 110
Pileopsis, 555
Pinacoceras, 584
Pithecanthropus, 772
Pith of rush, 335
Plaice, 98, 105, 117, 432, 710, 774
Planorbis, 539, 547, 554, 557, 559
Plateau, F., 30, 232; J. A. F., 192, 212, 218, 239, 275, 297, 374, 477
Plato, 2, 478, 720; Platonic bodies, 478
Plesiosaurs, 755
Pleurocarpus, 289
Pleuropus, 573
Pleurotomaria, 557
Plumulariidae, 747
Pluteus larva, 392, 415
Podocoryne, 342
Poincaré, H., 134
Poiseuille, J. L. M., 669
Polar bodies, 179; furrow, 310, 340
Polarised light, 418
Polarity, morphological, 166, 168, 246, 295, 284
Pollen, 396, 399
Polyhalite, 433
Polyprion, 749, 776
Polyspermy, 193
Polytrichum, 355
Pomacanthus, 749
Popoff, M., 286
Potamides, 554
Potassium, in living cells, 288
Potential energy, 208, 294, 601, etc.
Potter’s wheel, 238
Potts, R., 126
Pouchet, G., 415
Poulton, E. B., 670
Poynting, J. H., 235
Precocious segregation, 348
Preformation, 54, 159
Prenant, A., 163, 104, 189, 286, 289
Prévost, Pierre, 18
Pringsheim, N., 377
Probabilities, theory of, 61
Productus, 567
Protective colouration, 671
Protococcus, 59, 300, 410
Protoconch, 531
Protohippus, 767
Protoplasm, structure of, 172
Przibram, Hans, 16, 82, 107, 149, 204, 211, 418, 595; Karl, 46
Psammobia, 564
Pseuopriacauthus, 749
Pteranodon, 756
Pteris, antheridia of, 409
Pteropods of, 258, 570
Pulvinulina, 514, 595, 600, 602
Pupa, 530, 549, 556
Pütter, A., 110, 211, 492
Pyrosoma, egg of, 377
Pythagoras, 2, 509, 651, 720, 779
Quadrant, bisection of, 359
Quekett, J. T., 423
Quetelet, A., 61, 78, 93
Quincke, G. H., 187, 191, 279, 421
Rabbit, skull of, 764
Rabl, K., 36, 310
Radial co-ordinates, 730
Radiolaria, 252, 264, 457, 467, 588, 607
Rainey, George, 7, 420, 431, 434
Rainfall and growth, 121
Ram, horns of, 613–624
Ramsden, W., 282
Ramulina, 255
Rankine, W. J. Macquorn, 697, 712
Ransom’s waves, 164
Raphides, 412, 429, 434
Raphidiophrys, 460, 463
Rasumowsky, 683
Rat, growth of, 106
Rath, O. vom, 181
Rauber, A., 200, 305, 310, 380, 382, 398, 677, 683
Ray, John, 3
Rayleigh, Lord, 43, 44
Réaumur, R. A. de, 8, 108, 329
Reciprocal diagrams, 697
Rees, R. van, 374
Regeneration, 138
Reid, E. Waymouth, 272
Reinecke, J. C. M., 528
Reinke, J., 303, 305, 355, 356
Reniform shape, 735
Reticularia, 569
Reticulated patterns, 258
Réticulum plasmatique, 468
Rhabdammina, 589
Rheophax, 263
Rhinoceros, 612, 760
Rhumbler, L., 162, 165, 260, 322, 344, 465, 466, 589, 590, 595, 599, 608, 628
Rhynchonella, 561
Riccia, 372, 403, 405
Rice, J., 242, 273
Richardson, G. M., 416
Riefstahl, E., 578
Riemann, B., 385
Ripples, 33, 261, 323
Rivularia, 300
Roaf, H. C., 272
Robert, A., 306, 339, 348, 377
Roberts, C., 61
Robertson, T. B., 82, 132, 191, 192
Robinson, A., 681
Rörig, A., 628
Rose, Gustav, 421
Rossbach, M. J., 165
Rotalia, 214, 535, 602
Rotifera, cells of, 38
Roulettes, 218
Roux, W., 8, 55, 57, 157, 194, 378, 383, 666, 683
Ruled surfaces, 230, 270, 582
Ruskin, John, 20
Russow, ——, 73, 75
Ryder, J. A., 376
Sachs, J., 35, 38, 95, 108, 110, 111, 200, 360, 398, 399, 624, 635, 640, 651, 680
Sachs’s rule, 297, 300, 305, 347, 376
Saddles, of ammonites, 583
Sagrina, 263
St Venant, Barré de, 621, 627
Salamander, sperm-cells of, 179
Salpingoeca, 248
Salt, crystals of, 429
Salvinia, 377
Samec, M., 434
Samter, M. and Heymons, 130
Sandberger, G., 539
Sapphirina, 742
Saville Kent, W., 246, 247, 248
Scalaria, 526, 547, 554, 557, 559
Scale, effect of, 17, 438
Scaphites, 550
Scapula, human, 769
Scarus, 749
Schacko, G., 604
Schaper, A. A., 83
Schaudinn, F., 46, 286
Scheerenumkehr, 149
Schewiakoff, W., 189, 462
Schimper, C. F., 502, 636
Schmaltz, A., 675
Schmankewitsch, W., 130
Schmidt, Johann, 85, 87, 118
Schönflies, A., 202
Schultze, F. E., 452, 454
Schwalbe, G., 666
Schwann, Theodor, 199, 380, 591
Schwartz, Fr., 172
Schwendener, S., 210, 305, 636, 678
Scorpaena, 749
Scorpioid cyme, 502
Scott, E. L., 110; W. B., 768
Scyromathia, 744
Searle, H., 491
Sea urchins, 661; egg of, 173; growth of, 117, 147
Sebastes, 749
Sectio aurea, 511, 643, 649
Sedgwick, A., 197, 199
Sédillot, Charles E., 688
Segmentation of egg, 57, 310, 344, 382, etc.; spiral do., 371, 453
Segner, J. A. von, 205
Selaginella, 404
Semi-permeable membranes, 272
Sepia, 575, 577
Septa, 577, 592
Serpula, 603
Sexual characters, 135
Sharpe, D., 728
Shearing stress, 684, 730, etc.
Sheep, 613, 730, 738
Shell, formation of, 422
Sigaretus, 554
Silkworm, growth of, 83
Similitude, principle of, 17
Sims Woodhead, G., 414, 434
Siphonogorgia, 413
Skeleton, 19, 438, 675, 691, etc.
Snow crystals, 250, 480, 611
Soap-bubbles, 43, 219, 299, 307, etc.
Socrates, 8
Sohncke, L. A., 202
Solanum, 625
Solarium, 547, 554, 557, 559
Solecurtus, 564
Solen, 565
Sollas, W. J., 440, 450, 455
Solubility of salts, 434
Sorby, H. C., 412, 414, 728
Spallanzani, L., 138
Span of arms, 63, 93
Spangenberg, Fr., 342
Specific characters, 246, 380; inductive capacity, 177; surface, 32, 215
Spencer, Herbert, 18, 22
Spermatozoon, path of, 193
Sperm-cells of Crustacea, 273
Sphacelaria, 351
Sphaerechinus, 117, 147
Sphagnum, 402, 407
Sphere, 218, 225
Spherocrystals, 434
Spherulites, 422
Spicules, 282, 411, etc.
Spider’s web, 231
Spindle, nuclear, 169, 174
Spinning of protoplasm, 164
Spiral, geodetic, 488; logarithmic, 493, etc.; segmentation, 371, 453
Spireme, 173, 180
Spirifer, 561, 568
Spirillum, 46, 253
Spirochaetes, 46, 230, 266
Spirographis, 586
Spirogyra, 12, 221, 227, 242, 244, 275, 287, 289
Spirorbis, 586, 603
Spirula, 528, 547, 554, 575, 577
Spitzka, E. A., 92
Splashes, 235, 236, 254, 260
Sponge-spicules, 436, 440
Spontaneous generation, 420
Sporangium, 406
Spottiswoode, W., 779
Spray, 236
Stallo, J. B., 1
Standard deviation, 78
Starch, 432
Starling, E. H., 135
Stassfurt salt, 433
Stegocephalus, 746
Stegosaurus, 706, 707, 710, 754
Steiner, Jacob, 654
Steinmann, G., 431
Stellate cells, 335
Stentor, 147
Stereometry, 417
Sternoptyx, 748
Stillmann, J. D. B., 695
St Loup, R., 82
Stokes, Sir G. G., 44
Stolc, Ant., 452
Stomach, muscles of, 490
Stomata, 393
Stomatella, 554
Strasbürger, E., 35, 283, 409
Straus-Dürckheim, H. E., 30
Stream-lines, 250, 673, 736
Strength of materials, 676, 679
Streptoplasma, 391
Strophomena, 567
Studer, T., 413
Stylonichia, 133
Succinea, 556
Sunflower, 494, 635, 639, 688
Surface energy, 32, 34, 191, 207, 278, 293, 460, 599
Survival of species, 251
Sutures of cephalopods, 583
Swammerdam, J., 8, 87, 380, 528, 585
Swezy, Olive, 268
Sylvester, J. J., 723
Symmetry, meaning of, 209
Synapta, egg of, 453
Syncytium, 200
Synhelia, 327
Szielasko, A., 654
Tadpole, growth of, 83, 114, 138, 153
Tait, P. G., 35, 43, 207, 644
Taonia, 355, 356
Tapetum, 407
Tapir, 741, 763
Taylor, W. W., 277, 282, 426, 428
Teeth, 424, 612, 632
Telescopium, 557
Telesius, Bernardinus, 656
Tellina, 562
Temperature coefficient, 109
Terebra, 529, 557, 559
Terebratula, 568, 574, 576
Teredo, 414
Terni, T., 35
Terquem, O., 329
Tesch, J. J., 573
Tetractinellida, 443, 450
Tetrahedral symmetry, 315, 396, 476
Tetrakaidecahedron, 337
Tetraspores, 396
Textularia, 604
Thamnastraea, 327
Thayer, J. E., 672
Thecidium, 570
Thecosmilia, 325
Théel, H., 451
Thienemann, F. A. L., 653
Thistle, capitulum of, 639
Thoma, R., 666
Thomson, James, 18, 259; J. A., 465; J. J., 235, 280; Wyville, 466
Thurammina, 256
Thyroid gland, 136
Time-element, 51, 496, etc.; time-energy diagram, 63
Tintinnus, 248
Tissues, forms of, 293
Titanotherium, 704, 762
Tomistoma, 753
Tomlinson, C., 259, 428
Tornier, G., 707
Torsion, 621, 624
Trachelophyllum, 249
Transformations, theory of, 562, 719
Traube, M., 287
Trees, growth of, 119; height of, 19
Trembley, Abraham, 138, 146
Treutlein, P., 510
Trianea, hairs of, 234
Triangle, properties of, 508; of forces, 295
Triasters, 327
Trichodina, 252
Trichomastix, 267
Triepel, H., 683, 684
Triloculina, 595
Triton, 554
Trochus, 377, 557, 560; embryology of, 340
Tröndle, A., 625
Trophon, 526
Trout, growth of, 94
Trypanosomes, 245, 266, 269
Tubularia, 125, 126, 146
Turbinate shells, 534
Turbo, 518, 555
Turgor, 125
Turner, Sir W., 769
Turritella, 489, 524, 527, 555, 557, 559
Tusks, 515, 612
Tutton, A. E. H., 202
Twining plants, 624
Tyndall, John, 428
Umbilicus of shell, 547
Underfeeding, effect of, 106
Undulatory membrane, 266
Unduloid, 218, 222, 229, 246, 256
Unio, 341
Univalve shells, 553
Urechinus, 664
Vaginicola, 248
Vallisneri, Ant., 138
Van Iterson, G., 595
Van Rees, R., 374
Van’t Hoff, J. H., 1, 110, 433
Variability, 78, 103
Venation of wings, 385
Verhaeren, Emile, 778
Verworn, M., 198, 211, 467, 605
Vesque, J., 412
Vierordt, K., 73
Villi, 32
Vincent, J. H., 323
Vines, S. H., 502
Virchow, R., 200, 286
Vital phenomena, 14, 417, etc.
Vitruvius, 740
Volkmann, A. W., 669
Voltaire, 4, 146
Vorticella, 237, 246, 291
Wager, H. W. T., 259
Walking, 30
Wallace, A. R., 5, 432, 549
Wallich-Martius, 77
Warburg, O., 161
Warburton, C., 233
Ward, H. Marshall, 133
Warnecke, P., 93
Watase, S., 378
Water, in growth, 125
Watson, F. R., 323
Weber, E. H., 210, 259, 669; E. H. and W. E., 30; Max, 91
Weight, curve of, 64, etc.
Weismann, A., 158
Werner, A. G., 19
Wettstein, R. von, 728
Whale, affinities, 716; size, 21; structure, 708
Whipple, I. L., 123
Whitman, C. O., 157, 164, 193, 194, 199, 200
Whitworth, W. A., 506, 512
Wiener, A. F., 45
Wildeman, E. de, 307, 355
Willey, A., 425, 548, 555, 578
Williamson, W. C., 423, 609
Willughby, Fr., 318
Wilson, E. B., 150, 163, 173, 195, 199, 311, 341, 342, 398, 453
Winge, O., 433
Winter eggs, 283
Wissler, Clark, 79
Wissner, J., 636
Wöhler, Fr., 416, 420
Wolff, J., 683; J. C. F., 3, 51, 155
Wood, R. W., 590
Woods, R. H., 666
Woodward, H., 578; S. P., 554, 567
Worthington, A. M., 235, 254
Wreszneowski, A., 249
Wright, Chauncey, 335
Wright, T. Strethill, 210
Wyman, Jeffrey, 335
Yeast cell, 213, 242
Yield-point, 679
Yolk of egg, 165, 660
Young, Thomas, 9, 36, 669, 691
Zangger, H., 282
Zeising, A., 636, 650
Zeleny, C., 149
Zeuglodon, 716
Zeuthen, H. G., 511
Ziehen, Ch., 92
Zittel, K. A. von, 325, 327, 548, 584
Zoogloea, 282
Zschokke, F., 683
Zsigmondy, 39
Zuelzer, M., 165
CAMBRIDGE: PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PRESS
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