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I have said that in the division of a cell we seem to see the problem in its simplest form, but it is important to observe that the problem of division may be presented by the bodies of animals and plants in forms which are independent of the divisions between cells. The existence of pattern implies a repetition of parts, and repetition of parts when developed in a material originally homogeneous can only be created by division. Cell-division is probably only a special case of a process similar to that by which the pattern of the skeleton is laid down in a unicellular body such as that of a Radiolarian or Foraminiferan. Attempts have lately been made to apply mathematical treatment to problems of biology. It has sometimes seemed to me that it is in the geometrical phenomena of life that the most hopeful field for the introduction of mathematics will be found. If anyone will compare one of our animal patterns, say that of a zebra's hide, with patterns known to be of purely mechanical production, he will need no argument to convince him that there must be an essential similarity between the processes by which the two kinds of patterns were made and that parts at least of the analysis applicable to the mechanical patterns are applicable to the zebra stripes also. Patterns mechanically produced are of many and very diverse kinds. One of the most familiar examples, and one presenting some especially striking analogies to organic patterns, is that provided by the ripples of a mackerel sky, or those made in a flat sandy beach by the wind or the ebbing tide. With a little search we can find among the ripple-marks, and in other patterns produced by simple physical means, the closest parallels to all the phenomena of striping as we see them in our animals. The forking of the stripes, the differentiation of two "faces," the deflections round the limbs and so forth, which in the body we know to be phenomena of division, are common both to the mechanical and the animal patterns. We cannot tell what in the zebra corresponds to the wind or the flow of the current, but we can perceive that in the distribution of the pigments, that is to say, of the chromogen-substances or of the ferments which act upon them, a rhythmical disturbance has been set up which has produced the pattern we see; and I think we are entitled to the inference that in the formation of patterns in animals and plants mechanical forces are operating which ought to be, and will prove to be, capable of mathematical analysis. The comparison between the striping of a living organism and the sand-ripples will serve us yet a little farther, for a pattern may either be formed by actual cell-divisions, and the distribution of differentiation coincidently determined, or—as visibly in the pigmentation of many animal and plant tissues—the pattern may be laid down and the pigment (for example) distributed through a tissue across or independently of the cell-divisions of the tissue. Our tissues therefore are like a beach composed of sands of different kinds, and different kinds of sands may show distinct and interpenetrating ripples. When the essential analogy between these various classes of phenomena is perceived, no one will be astonished at, or reluctant to admit, the reality of discontinuity in Variation, and if we are as far as ever from knowing the actual causation of pattern we ought not to feel surprised that it may arise suddenly or be suddenly modified in descent. Biologists have felt it easier to conceive the evolution of a striped animal like a zebra from a self-coloured type like a horse (or of the self-coloured from the striped) as a process involving many intergradational steps; but so far as the pattern is concerned, the change may have been decided by a single event, just as the multitudinous and ordered rippling of a beach may be created or obliterated at one tide.

Fig. 1. Tusk of Indian elephant, showing an abnormal segmentation.

This point is well illustrated by the tusk of an Indian elephant which I lately found in a London sale-room. This tusk is by some unknown cause, presumably a chronic inflammation, thrown up into thirteen well-marked ridges which closely simulate a series of segments (Fig. 1). Whatever the cause the condition shows how easily a normally unsegmented structure may be converted into a series of repeated parts.

The spread of segmentation through tissues normally unsegmented is very clearly exemplified in the skates' jaws shown in in Fig. 2. The right side of the upper figure shows the normal arrangement in the species Rhinoptera jussieui, but the structure on the left side is very different. The probable relations of the several rows of teeth to the normal rows is indicated by the lettering, but it is evident that by the appearance of new planes of division constituting separate centers of growth, the series has been recast. The pattern of the left side is so definite that had the variation affected the right side also, no systematist would have hesitated to give the specimen a new specific name. The other two drawings show similar variations of a less extensive kind, the nature of which is explained by the lettering of the rows of teeth.

Fig. 2. Jaws of Skates (Rhinoptera) showing meristic variation. (For a detailed discussion see Materials for the Study of Variation, p. 259.)

This power to divide is a fundamental attribute of life, and of that power cell-division is a special example. In regard to almost all the chief vital phenomena we can say with truth that science has made some progress. If I mention respiration, metabolism, digestion, each of these words calls to mind something more than a bare statement that such acts are performed by an animal or a plant. Each stands for volumes of successful experiment and research, But the expression cell-division, the fundamental act which typifies the rest, and on which they all depend, remains a bare name. We can see with the microscope the outward symptoms of division, but we have no surmise as to the nature of the process by which the division is begun or accomplished. I know nothing which to a man well trained in scientific knowledge and method brings so vivid a realisation of our ignorance of the nature of life as the mystery of cell-division. What is a living thing? The best answer in few words that I know is one which my old teacher, Michael Foster, used to give in his lectures introductory to biology. "A living thing is a vortex of chemical and molecular change." This description gives much, if not all, that is of the essence of life. The living thing is unlike ordinary matter in the fact that, through it, matter is always passing. Matter is essential to it; but, provided that the flow in and out is unimpeded, the life-process can go on so far as we know indefinitely. Yet the living "vortex" differs from all others in the fact that it can divide and throw off other "vortices," through which again matter continually swirls.

We may perhaps take the parallel a stage further. A simple vortex, like a smoke-ring, if projected in a suitable way will twist and form two rings. If each loop as it is formed could grow and then twist again to form more loops, we should have a model representing several of the essential features of living things.

It is this power of spontaneous division which most sharply distinguishes the living from the non-living. In the excellent book dealing with the problems of development, lately published by Mr. Jenkinson a special emphasis is very properly laid on the distinction between the processes of division, and those of differentiation. Too often in discussions of the developmental processes the distinction is obscured. He regards differentiation as the "central difficulty." "Growth and division of the nucleus and the cells," he tells us, are side-issues. This view is quite defensible, but I suspect that the division is the central difficulty, and that if we could get a rationale of what is happening in cell-division we should not be long before we had a clue to the nature of differentiation. It may be self-deception, but I do not feel it impossible to form some hypothesis as to the mode of differentiation, but in no mood of freest speculation are we ever able to form a guess as to the nature of the division. We see differentiations occurring in the course of chemical action, in some phenomena of vibration and so forth: but where do we see anything like the spontaneous division of the living cell? Excite a gold-leaf electroscope, and the leaves separate, but we know that is because they were double before. In electrolysis various substances separate out at the positive and negative poles respectively. Now if in cell-division the two daughter-cells were always dissimilar—that is to say, if differentiation always occurred—we could conceive some rough comparison with such dissociations. But we know the dissimilarity between daughter-cells is not essential. In the reproduction of unicellular organisms and many other cases, the products formed at the two poles are, so far as we can tell, identical. Any assumption to the contrary, if we were disposed to make it, would involve us in difficulties still more serious. At any rate, therefore, if differentiation be really the central difficulty in development, it is division which is the essential problem of heredity.

Sir George Darwin and Professor Jeans tell us that "gravitational instability" consequent on the condensation of gases is "the primary agent at work in the actual evolution of the universe," which has led to the division of the heavenly bodies. The greatest advance I can conceive in biology would be the discovery of the nature of the instability which leads to the continual division of the cell. When I look at a dividing cell I feel as an astronomer might do if he beheld the formation of a double star: that an original act of creation is taking place before me. Enigmatical as the phenomenon seems, I am not without hope that, if it were studied for its own sake, dissociated from the complications which obscure it when regarded as a mere incident in development, some hint as to the nature of division could be found. It is I fear a problem rather for the physicist than for the biologist. The sentiment may not be a popular one to utter before an assembly of biologists, but looking at the truth impersonally I suspect that when at length minds of first rate analytical power are attracted to biological problems, some advance will be made of the kind which we are awaiting.

The study of the phenomena of bodily symmetry offers perhaps the most hopeful point of attack. The essential fact in reproduction is cell-division, and the essential basis of hereditary resemblance is the symmetry of cell-division. The phenomena of twinning provide a convincing demonstration that this is so. By twinning we mean the production of equivalent structures by division. The process is one which may affect the whole body of an animal or plant, or certain of its parts. The term twin as ordinarily used refers to the simultaneous birth of two individuals. Those who are naturalists know that such twins are of two kinds, (1) twins that are not more alike than any other two members of the same family, and (2) twins that are so much alike that even intimate friends mistake them. These latter twins, except in imaginative literature, are always of the same sex.

It is scarcely necessary for me to repeat the evidence from which it has been concluded that without doubt such twins arise by division of the same fertilised ovum. There is a perfect series of gradations connecting them with the various forms of double monsters united by homologous parts. They have been shown several times to be enclosed in the same chorion, and the proofs of experimental embryology show that in several animals by the separation of the two first hemispheres of a dividing egg twins can be produced. Lastly we have recently had the extraordinarily interesting demonstration of Loeb, to which I may specially refer. Herbst some years ago found that in sea water, from which all lime salts had been removed, the segments of the living egg fall apart as they are formed. Using this method Loeb has shown that a temporary immersion in lime-free sea water may result in the production of 90 per cent. of twins. We are therefore safe in regarding the homologous or "identical" twins as resulting from the divisions of one fertilised egg, while the non-identical or "fraternal" twins, as they are called, arise by the fertilisation of two separate ova.[3]

In the resemblance of identical twins we have an extreme case of hereditary likeness[4] and a proof, if any were needed, that the cause of individual variation is to be sought in the differentiation of germ-cells. The resemblance of identical twins depends on two circumstances, First, since only two germ-cells take part in their production, difference between the germ cells of the same individual cannot affect them. Secondly the division of the fertilised ovum, the process by which they became two instead of one, must have been a symmetrical division. The structure of twins raises however one extremely significant difficulty, which as yet we cannot in any way explain. The resemblance between twins is a phenomenon of symmetry, like the resemblance between the two sides of a bilaterally symmetrical body. Not only is the general resemblance readily so interpreted, but we know also that in double monsters, namely unseparated twins, various anatomical abnormalities shown by the one half-body are frequently shown by the other half-also.[5] The two belong to one system of symmetry. How then does it happen that the body of one of a pair of twins does not show a transposition of viscera? We know that the relation of right and left implies that the one should be the mirror-image of the other. Such a relation of images may be maintained even in minute details. For example if the same pattern of finger-print is given by the fingers of the two hands, one is the reverse of the other. In double monsters, namely unseparated twins, there is evidence that an inversion of viscera does occur with some frequency. Evidence from such cases is not so clear and simple as might be expected, because as a matter of fact, the heart and stomach, upon which the asymmetry of the viscera chiefly depend, are usually common to the two bodies. Duplicity generally affects either the anterior end alone, or the posterior end alone. The division is generally from the heart forwards, giving two heads and two pairs of anterior limbs on a common trunk, or from the heart backwards, giving two pairs of posterior limbs with the anterior body common. In either case, though the bodies may be grouped in a common system of symmetry, neither can be proved to show definite reversal of the parts. To see that reversal recourse must be had to more extreme duplications, such as the famous Siamese Twins. They, as a matter of fact, were an excellent instance of the proposition that twins are related as mirror-images, for both of them had eleven pairs of ribs instead of the normal twelve, and one of them had a partial reversal of viscera.[6] (Küchenmeister, Verlagerung, etc., p. 204.)

If anyone could show how it is that neither of a pair of twins has transposition of viscera the whole mystery of division would, I expect, be greatly illuminated.[7] At present we have simply to accept the fact that twins, by virtue of their detachment from each other, have the power of resuming the polarity which is proper to any normal individual. It was nevertheless with great interest that I read Wilder's recent observation[8] that occasionally in identical twins the finger-print of one or both the index-fingers may be reversed, showing that there is after all some truth in the notion that reversal should occur in them.

There is another phenomenon by twinning which, if we could understand it, might help. I refer to the free-martin, the subject of one of John Hunter's masterpieces of anatomical description. In horned cattle twin births are rare, and when twins of opposite sexes are born, the male is perfect and normal, but the reproductive organs of the female are deformed and sterile, being known as a free-martin. The same thing occasionally happens in sheep, suggesting that in sheep also twins may be formed by the division of one ovum; for it is impossible to suppose that mere development in juxtaposition can produce a change of this character. I mention the free-martin because it raises a question of absorbing interest. It is conceivable that we should interpret it by reference to the phenomenon of gynandromorphism, seen occasionally in insects, and also in birds as a great rarity. In the gynandromorph one side of the body is male, the other female. A bullfinch for instance has been described with a sharp line of division down the breast between the red feathers of the cock on one side and the brown feathers of the hen on the other. (Poll, H., SB. Ges. Nat. Fr., Berlin, 1909, p. 338.) In such cases neither side is sexually perfect. If the halves of such a gynandromorph came apart, perhaps one would be a free-martin.

The behaviour of homologous twinning in heredity has been little studied. It does not exist as a normal feature in any animal which is amenable to experiment, and we cannot positively assert that a comparable phenomenon exists in plants; for in them—the Orange, for example—polyembryony may evidently be produced by a parthenogenetic development of nucellar tissue. It is possible that in Man twinning is due to a peculiarity of the mother, not of the father. It may and not rarely does descend from mother to daughter, but whether it can be passed on through a male generation to a daughter again, there is not sufficient evidence to show. The facts as far as they go are consistent with the inference which may be drawn from Loeb's experiment, that the twinning of a fertilized ovum may be determined not by the germ-cells which united to form it, but by the environment in which it begins to develop. The opinion that twinning may descend through the male directly has been lately expressed by Dr. J. Oliver in the Eugenics Review (1912), on the evidence of cases in which twins had occurred among the relations of fathers of twins, but I do not know of any comprehensive collection of evidence bearing on the subject.

Besides twinning of the whole body a comparable duplicity of various parts of the same body may occur. Such divisions affect especially those organs which have an axis of bilateral symmetry, such as the thumb, a cotyledon, a median petal, the frond of a fern or the anal fin of a fish. From the little yet known it is clear that the genetic analysis of these conditions must be very difficult, but evidence of any kind regarding them will be valuable. We want especially to know whether these divisions are due to the addition of some factor or power which enables the part to divide, or whether the division results from the absence of something which in the normal body prevents the part from dividing. Breeding experiments, so far as they go, suggest that the less divided state is usually dominant to the more divided.[9] The two-celled Tomato fruit is dominant to the many-celled type. The Manx Cat's tail, with its suppression of caudal segmentation is a partial dominant over the normal tail. The tail of the Fowl in what is called the "Rumpless" condition is at least superficially comparable with that of the Manx Cat, and though the evidence is not wholly consistent, Davenport obtained facts indicating that this suppressed condition of the caudal vertebrae is an imperfect dominant.[10]

Some evidence may also be derived from other examples of differences which at first sight appear to be substantive though they are more probably meristic in ultimate nature. The distinction between the normal and the "Angora" hair of the Rabbit is a case in point. We can scarcely doubt that one of the essential differences between these two types is that in the Angora coat the hair-follicles are more finely divided than they are in the normal coat, and we know that the normal, or less-divided condition, is dominant to the Angora, or more finely divided.

Fig. 3. I, II, III, various degrees of syndactyly affecting the medius and annularis in the hand; IV, syndactyly affecting the index and medius in the foot. (After Annandale.)

In the case of the solid-hoofed or "mule-footed" swine, the evidence shows, as Spillman has lately pointed out,[11] that the condition behaves as a dominant. The essential feature of this abnormality is that the digits III and IV are partially united. The union is greatest peripherally. Sometimes the third phalanges only are joined to form one bone, but the second and even the first phalanges may also be compounded together. Here the variation is obviously meristic and consists in a failure to divide, the normal separation of the median digits of the foot being suppressed.

Fig. 4. Case of complete syndactyly in the foot. II and III, digit apparently representing the index and medius. c2 + c3, bone apparently representing the middle and external cuneiform; cb, cuboid; c1, internal cuneiform. (After Gruber.)

Webbing between the digits, in at least some of its manifestations, is a variation of similar nature. The family recorded by Newsholme[12] very clearly shows the dominance of this condition. The case is morphologically of great interest and must undoubtedly have a bearing on the problems of the mechanics of Division. In discussing the phenomena of syndactylism I pointed out some years ago that the digits most frequently united in the human hand are III and IV, while in the foot, union most frequently takes place between II and III.[13] In Newsholme's family the union was always between II and III of the foot, except in the case of one male who had the digits III and IV of the right hand alone webbed together. There can be little doubt that the geometrical system on which the foot is planned has an axis of symmetry passing between the digits II and III, while the corresponding axis in the hand passes between III and IV. Union between such digits may therefore be regarded as comparable with any non-division or "coalescence" of lateral structures in a middle line, and when as in these examples such a condition is shown to be a dominant we cannot avoid the inference that some concrete factor has the power of suppressing or inhibiting this division. Figs. 3 and 4 illustrate degrees of union between digits in the human hand and foot.

It is not in question that various other forms of irregular webbing and coalescence of digits exist, and respecting the genetic behaviour of these practically nothing is as yet known. Such a case is described by Walker,[14] in which the first and second metacarpals of both feet were fused in mother and daughter, and several more are found in literature. Contrasted with these phenomena we have the curious fact that in the Pigeon, Staples-Browne found webbing of the toes a recessive character. The question thus arises whether this webbing is of the same nature as that shown to be a dominant in Man, and indeed whether the phenomenon in pigeons is really meristic at all. There is some difference perceptible between the two conditions; for in Man there is not so much a development of a special web-like skin uniting the digits as a want of proper division between the digits themselves, and in extreme cases two digits may be represented by a single one. In the Pigeon I am not aware that a real union of this kind has ever been observed, and though the web-like skin may extend the whole length of the digits and be so narrow as to prevent the spread of the toes, it may, I think, be maintained that the unity of the digits is unimpaired. For the present the nature of this variation in the pigeon's feet must be regarded as doubtful, and we should note that if it is actually an example of a more perfect division being dominant to a less perfect division, the case is a marked exception to the general rule that non-division is dominant to division.

Reference must also be made to the phenomenon of fasciation in the stems of plants. As Mendel showed in the case of Pisum this condition is often a recessive. The appearances suggest that the difference between a normal and a fasciated plant consists in the inability of the fasciated plant to separate its lateral branches. The nature of the condition is however very obscure and it is equally likely that some multiplication of the growing point is the essential phenomenon.[15]

Stockard's interesting experiments[16] illustrate this question. He showed that by treating the embryos of a fish (Fundulus heteroclitus) with a dilute solution of magnesium salts, various cyclopian monstrosities were frequently produced. These have been called cases of fusion of the optic vesicles. I would prefer to regard them as cases of a division suppressed or restricted by the control of the environment. Conversely, the splendid discovery of Loeb, that an unfertilised egg will divide and develop parthenogenetically without fertilisation, as a consequence of exposure to various media, may be interpreted as suggesting that the action of those media releases the strains already present in the ovum, though I admit that an interpretation based on the converse hypothesis, that the medium acts as a stimulus, is as yet by no means excluded.

In these cases we come nearest to the direct causation or the direct inhibition of a division, but the meaning of the evidence is still ambiguous. I incline to compare Loeb's parthenogenesis with the development (and of course accompanying cell-division) of dormant buds on stems which have been cut back.

It is interesting to note that sometimes as an abnormality, the faculty of division gets out of hand and runs a course apparently uncontrolled. A remarkable instance of this condition is seen in Begonia "phyllomaniaca", which breaks out into buds at any point on the stem, petioles, or leaves, each bud having, like other buds, the power of becoming a new plant if removed. We would give much to know the genetic properties of B. phyllomaniaca, and in conjunction with Mr. W. O. Backhouse I have for some time been experimenting with this plant. It proved totally sterile. Its own anthers produce no pollen, and all attempts to fertilise it with other species failed though the pollen of a great number of forms was tried.

Recently however we have succeeded in making plants which are in every respect Begonia phyllomaniaca, so far as the characters of stems and leaves are concerned. These plants, of which we have sixteen, were made by fertilising B. heracleifolia with B. polyantha. They are all beginning to break out in "phyllomania." As yet they have not flowered, but as they agree in all details with phyllomaniaca there can be little doubt that the original plant bearing that name was a hybrid similarly produced. The production of "phyllomania" on a hybrid Begonia has also been previously recorded by Duchartre.[17] In this case the cross was made between B. incarnata and lucida. The synonymy of the last species is unfortunately obscure, and I have not succeeded in repeating the experiment.