Inutility of Characters (1872)
On Diversity of Evolution under One Set of External Conditions (Gulick) Natural and Artificial Selection (Belt) Mathematics of Biological Transformation (Delboeuf) An Unnoticed Factor in Evolution (Catchpool)
We begin with an early reference to "peripatric speciation;" i.e speciation that initiates at the periphery of the range of the parental species, otherwise known as the "founder principle". An even earlier reference is L. von Buch (1825) who remarks in his studies of the Canary Islands (Physikalische Beschreibung der Canarische Inseln):
However, in the case of snails, Gulick showed that the narrow valleys of the Sandwich islands (now Howaii) acted as sub-islands that sufficed to prevent crossing with other varieties.
On Diversity of Evolution under One Set of External Conditions By Rev. JOHN T. GULICK Journal of the Linnean Society (Zoology)
11, 496-505. [Read before the Linnean Society of London, November 21st 1872.]
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The terms "Natural Selection" and "Survival of the Fittest" present different phases of a law which can act only where there is variation. The words in which the law is expressed imply that there are variations which may be accumulated in different proportions according to the differing demands of external conditions. What, then, is the effect of these variations when the external conditions remain the same? Or can it be shown that there is no change in organisms that is not the result of change in external conditions? Again, if the initiation of change in the organism is through change in the "Environment," by what law is the cessation of change determined? If change continues in the organism long after the essential conditions of the "Environment" have become stationary, how do we know that it [i.e. change] is not perpetual? Does the change, whether transitory or continuous, expend itself in producing from each species placed in the new "Environment" just one new species completely fitted to the conditions? Or may it produce from one stock many that are equally fitted? If the latter, what is the law or condition that determines their number, their affinities, and the size and position of their respective areas, as related to each other and to the whole available area? |
Facts Throwing Light on the Subject
I believe that in the relations of species to each other as distributed in nature, we shall find light on the subject. I call attention at this time to the variation and distribution of terrestrial mollusks, more especially those found on the Sandwich Islands; but similar facts are not wanting elsewhere. The land-shells of the Sandwich Islands not only differ in species from those of other countries, but they belong, for the most part, to a group of genera found nowhere else. These are the Achatinellinae, of which there are seven arboreal genera (Achatinella, Bulimella, Helicterella, Laminella, Partulina, Newcombia, and Auriculella), and three ground-genera (Carelia, Amastra, and Leptachatina). Some of these genera are confined, in their distribution, to a single island. The average range of each species is five or six miles, while some are restricted to but one or two square miles, and only a very few have the range of a whole island. The forest-region that covers one of the mountain-ranges of Oahu is about forty miles in length and five or six miles in breadth. This small territory furnishes about 175 species, representing by 700 or 800 varieties. The fall of rain on the north-east side of the mountain is somewhat heavier than on the opposite side, and the higher ridges of the mountains are cooler than the valleys; but the valleys on one side of the range have a climate the same in every respect. The vegetation in the valleys differs somewhat from that on the ridges; but the vegetation of the different valleys is much the same; the birds, insects, and larger animals are the same. Though, as far as we can observe, the conditions are the same in the valleys on one side of the range, each has a molluscan fauna differing in some degree from that of any other. We frequently find a genus represented in several successive valleys by allied species, sometimes feeding on the same, sometimes on different plants. In every such case, the valleys that are nearest to each other furnish the most nearly allied forms; and a full set of the varieties of each species presents a minute gradation of forms between the more divergent types found in the more widely separated localities. No theory is satisfactory that does not account, first, for their being distributed according to their affinities in adjoining areas more or less distinctly defined, and, second, for their being restricted to very small areas. |
External Conditions not the Cause
I think the evolution of these different forms cannot be attributed to difference in their external conditions : --
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Separation and Variation are Correlative Factors in the Evolution of Species
If we would account for the difference and for the limited distribution of these allied forms on the hypothesis of Evolution from one original species, it seems to me necessary to suppose two conditions, both of which relate to the state of the species -- namely, Separation and Variation. I regard Separation as a condition of the species and not of surrounding nature, because it is a state of division in the stock which does not necessarily imply any external barriers, or even the occupation of separate districts. This may be illustrated by the separation between the castes of India or between different genera occupying the same locality.
Ist. We must suppose that they possess or have possessed an inherent tendency to variation, so strong that all that is necessary to secure a divergence of types in the descendants of one stock is to prevent, through a series of generations, their intermingling with each other to any great degree. This supposition is not at variance, but rather in accordance, with facts that are observed in analogous cases in the history of man and of domestic animals of one original stock, that are kept entirely apart. But this condition alone would not be enough to account for the species of Achatinellinae being confined to areas so much smaller than usual; for if this tendency has produced such results in the distribution of one family, why does it not in all? |
Migration and Variation are Opposing Factors in the Limitation of Areas
2nd. To account, therefore, for the small areas, we must further suppose that, as compared with other families, there is a disproportion between the tendency to variation and the tendency and opportunities to migrate. Either the tendency to variation in this family is very much greater than usual, or their tendency to migrate is weaker and their opportunities fewer than usual. According to a priori reasoning, the areas occupied must vary directly, as the tendency, power, and opportunities for migrating, but inversely as the tendency to variation. If the amount of migration is greatly expanded in proportion to the tendency to variation, the areas must be expanded; if, on the other hand, the tendency to variation is expanded as compared with the amount and extent of migration, the areas occupied by the different species must be correspondingly contracted. If the power of migrating and the opportunities for being transported are very limited in any family of creatures, we may expect that the areas occupied by the different species and varieties of that family will be more restricted than the areas occupied by the species of other families that have greater opportunities for migrating but the same tendency to variation. When we find that in Europe and North America nearly every species of Helix occupies an area many thousand times as large as the area occupied by any Achatinella, we naturally ask whether the difference can be accounted for by circumstances that limit the dispersion of the latter, or whether the results are to be attributed to a stronger tendency to variation. It is evident that to the forest species, that live on trees found chiefly in the valleys, the mountain-ridges separating the valleys must be partial barriers; but the valleys cannot be barriers to the species occupying the ridges, for the ridges rising between the valleys are all spurs from the one central range that forms the backbone of the island. In accordance with these facts we find that the distances over which the ridge species are distributed are usually somewhat greater than those reached by the valley species. But even the ridge species are limited in their distribution to very small areas. Few have a range of territory more than six or eight miles in length and three or four miles in breadth; and many are restricted to half that area. Though some of the groups of species are found both in the valleys and on the ridges, so that no barriers intervene to break the continuity of their intercourse, we still find them distributed over small areas, and these areas again divided amongst subordinate varieties. The streams that flow through these valleys cannot serve in carrying the shells from one valley to another; but the separation from this cause can be no greater than that which is experienced by mollusks inhabiting mountain valleys in other countries. It therefore appears that the limited range of the species of this family receives but slight explanation from the nature of the country. Neither can we suppose that the power of locomotion in this family is so immeasurably below that possessed by the Helices of Europe and America, and by, the Achatinae of Africa, as to account for the excessive disproportion in the areas occupied, as well as in the amount of divergence between the types found in any locality and those found at given distances. In Africa some of the species of Achatina have a range of more than a thousand miles, while on the island of Oahu the most widely diffused species of the arboreal genus Achatinella is restricted to about ten miles, and the utmost limit gained by any species of the ground-genus Amastra is about twenty miles. Again, the difference of type is quite as great between the species of Achatinella found in the mountains near the eastern end of Oahu and those found forty miles distant, on the other end of the same range of mountains, as the difference between the species of Achatina found in Sierra Leone and those in the region of Port Natal, nearly four thousand miles distant. The birds that prey upon these snails are probably few; but the forests are populous with fruit- and nectar-feeding birds, that might be supposed to give as effectual means of transportation as could be given by any. The number of species represented by these birds is no doubt less than would in most cases be found in an equal extent of continental forest; but the number of individuals is probably greater than the average number inhabiting equal areas in other parts of the world. If we find no reason for attributing the small areas occupied by these species to deficient means of transportation, may we not believe that rapidity of variation has had influence in determining the result? |
Stability of Type as Affected by Cultivation
It is known that there is a great difference in the stability of type in different species of plants and animals that have been subjected to cultivation. One produces striking varieties in a single generation; another requires careful selection of certain characters for many generations before well-marked varieties can be secured. We also know that continued cultivation will, in many instances, break down the stability of type in a species that, in the first place, adhered with great persistency to one form. It often happens that when the stability has once been disturbed, a wide range of variation may afterwards be obtained with comparative rapidity. Is it not possible that similar changes may sometimes take place in species in their wild state? Two important elements of the cultivation which tends to develop varieties are the removal of competitors and enemies, and the abundant supply of nourishment; but both these conditions may sometimes be furnished by nature without the intervention of man. |
The Natural Selection that Prevents Variation
The more severe the competition the more rigidly does Natural Selection adhere to the one form that is best suited to meet that competition, or, according to the language in which Professor Owen has stated the doctrine, the more certainly does the "Battle of Life" extinguish all variations from that one form. When a species is subjected to severe competition of the same kind for countless generations, we may well believe that it gains a stability of type that is not found in one that has during the same time been, either comparatively free from competition, or under the influence of a succession of different competitors and enemies*. {Footnote: The only terrestrial mollusks with which the Achatinellinae have to compete are a few Helices much inferior in size, and not arboreal in their habits.} |
Stability of Type in Island Fauna may be Impaired
1st. By Freedom from the Competition that Limits Variation. -- We can see that when animal life commences upon an island where vegetation has already become abundant, the first species that appears on the arena, unless immediately followed by other creatures capable of being either friends or foes, will enjoy for a time complete freedom from competition. If the vegetation is suited, it will also have an abundance of food. Under these circumstances every variation that occurs, unless decidedly malformed, will have a chance of living and exerting an influence upon the final result. 2nd. By Competition Accelerating Variation.-- If the introduction of competitive animals is long delayed, the first struggle for life will occur between the members of the one stock. But competition of this kind does not tend to prevent variation, but rather to accelerate it, by driving portions of the race into new spheres. Supposing the animals first inhabiting the island to be a species of arboreal mollusks, there would soon be an excess of occupants on the trees best suited to them in the region where they first appeared. The portion of the population that would survive this exigency would, in the first place, be those that found sustenance on trees of other kinds, Some of these would either themselves, or through their descendants, reach localities where the trees are again found on which the stock commenced its career. Those that, in this way, returned to the original trees, would have acquired some new tendencies to variation through the ordeal through which they had passed; and those that remained upon the other kinds of trees would rapidly develop new characters: in either case, there would be no outside competition limiting them to one definite form. New forms of variation would have an opportunity of being preserved. New shades of colour, for example, would not expose the owners to the attacks of enemies. Variations of shape, if not inconsistent with the pursuit of food, would be no disadvantage. 3rd. By Continual Change in the Character of the Natural Selection. -- Still further, we can see that when competition arises from the gradual introduction of animals, either friendly or hurtful to the first occupants, the character of the Natural Selection, to which they would thus be subjected would be continually changing; no one set of characters would have constant advantage through a long series of successive generations. In these ways the persistence of form might be impaired, and the variability which we may believe exists in some degree in all organisms might be greatly increased beyond what is usually found. This tendency to comparatively rapid variation having been established, the evolution of species would be correspondingly rapid, and the areas of each proportionately limited. |
Imaginary Case, Illustrating Evolution without Change in the External Conditions
If a bird should carry a leaf bearing two individuals of some species and drop it a mile beyond the limits already reached by others of that species, they might there find the same trees to which they were accustomed, and multiply for some tens of years before the first scattering individuals from the slowly advancing wave of migration would reach them. They might, by this time, have increased to many thousands; and having been entirely separated from the original stock for a considerable number of generations, with a preexisting tendency to rapid variation, a certain variety of form and colour might have partially established itself amongst them. The arrival of a few individuals representing the old stock would, amongst the multitudes of the new variety, have no influence in bringing back the succeeding generations to the original form. The new characters would become from year to year more distinctly set. Owing to an intervening ridge acting as a partial barrier, the number of individuals of the original stock coming amongst them might be always restricted; and even if no such barrier existed, the individuals arriving from abroad could never be more than a very small number compared with those produced on the spot and possessing the local characteristics. |
Changes Produced by the Introduction of Enemies
At this point one other inquiry naturally arises: -- If the multitude of varieties and the restricted distribution of both varieties and species is in any degree due to freedom from severe competition, what would be the effect if, by degrees, many birds and insects, hostile to these snails, should find their way to the Sandwich Islands and become numerous in those mountain-regions? One of the first effects would naturally be the disappearance of many varieties and species by which the different forms of each genus are now so minutely gradationed together. Certain protective colours would be made to prevail, to the partial exclusion of some of the brilliant contrasts of colour. The same enemies being found in all the valleys of an island, the forms that proved to be best fitted to survive in one valley would have the advantage everywhere, and therefore gradually spread from valley to valley. The distribution of species and their separation from each other by distinct forms would thus become similar to what is found in the case of continental species. The destruction of forests by the introduction of cattle and goats is now causing the extinction of some of the species. |
A comparison of the distribution of island mollusks with the widely contrasted distribution of continental species, leads me to believe that the evolution of many different species may take place without any difference in the food, climate, or enemies that surround them. The rapidity of evolution, or the time within which a certain amount of change is effected, must depend upon the average amount of change in one direction in a single generation, and the rapidity of succession in the generations. Ten thousand years would make but little difference in a species of cedar, in which the life of a single tree might count a third of that period. But in the case of some species of insects the same period might cover ten thousand generations; and though the change in each generation might be as imperceptible as in the cedar, the aggregate of change for the whole period might be very apparent. We must also bear in mind, the Natural Selection arising from severe competition with species that have a wide range tends to prevent variation and give a wider diffusion to forms that would otherwise be limited in their range and variable in their type. Natural Selection is as efficient in producing permanence of type in some cases as in accelerating variations in other cases. It we suppose separation without a difference of external circumstances is a condition sufficient to ensure variation, it renders intelligible the fact that, in nearly allied forms on the same island, the degree of divergence in type is in proportion to the distance in space by which they are separated. The difference between two miles and ten miles makes no change in climate; but it is easy to believe that it is the measure of a corresponding difference in the time of separation. In forms that differ more essentially, the separation may have been as complete and as long-continued in the case of those which now inhabit one valley as in the case of those which are separated by the length of an island. When a wide degree of divergence has been established, hybridation would be precluded. We accordingly find that the difference between species of different genera or subgenera is in most instances equally great whether we take for comparison those from the same or from different valleys. If, on the other hand, we suppose that a difference in the external conditions is necessary to the evolution of distinct forms, these and other similar facts remain unexplained. |
Natural and Artificial
Selection
From Chapter 11 of The Naturalist in Nicaragua, pages 206-209 by THOMAS BELT. Published by John Murray, London (1874).[Words in square brackets by DRF] Tschudi makes [describes] two races of indigenous dogs in tropical America. 1. The Canis caraibicus (Lesson), without hair, and which does not bark. 2. The Canis ingae (Tschudi), the common hairy dog, which has pointed nose and ears, and barks.* {Footnote: J. J. von Tschudi, quoted by Humboldt, [in] "Aspects of Nature", English edition, vol. i, p. 111.} The small eatable dog of the Mexicans was called by them Techichi; and Humboldt derives the name from Tetl, a stone, and says that it means "a dumb dog," but this appears rather a forced derivation. Chichi is Aztec for "to suck"; and it seems to me more probable that the little dogs they eat, and which are spoken of by the Spaniards as making very tender and delicate food, were the puppies of the Xoloitzcuintli, and that Techichi meant "a sucker". Whether the hairless dog was, or was not, the Techichi of which the Mexicans made such savoury dishes is an open question, but there can be no doubt that the former was found in tropical America by the Spanish conquerors, and that it has survived to the present time, with little or no change. That it should not have intermixed with the common haired variety, and lost its distinctive characters, is very remarkable. It has not been artificially preserved, for instead of being looked on with favour by the Indians, Humboldt states that in Peru, where it is abundant, it is despised and ill-treated. Under such circumstances, the variety can only have been preserved through not interbreeding with the common form, either from a dislike to such unions, or by some amount of sterility when they are formed. This is, I think, in favour of the inference that the variety has been produced by natural and not by artificial selection, for diminished fertility is seldom or never acquired between artificial varieties. Man isolates varieties, and breeds from them, and continuing to separate those that vary in the direction he wishes to follow, a very great difference is, in a comparatively short time, produced. But these artificial varieties, though often more different from each other than some natural species, readily interbreed, and if left to themselves, rapidly revert to the common type. In natural selection there is a great and fundamental difference. The varieties that arise can seldom be separated from the parent form and from other varieties, until they vary also in the elements of reproduction. Thousands of varieties probably revert to the parent type, but if at last one is produced that breeds only with its own form, we can easily see how a new species might be segregated. As long as varieties interbreed together and with the parent form, it does not seem possible that a new species can be formed by natural selection, excepting in cases of geographical isolation. All the individuals might vary in some one direction, but they could not split up into distinct species whilst they occupied the same area and interbred without difficulty. Before a variety can become permanent, it must either be separated from the others, or have acquired some disinclination or inability to breed with them. As long as they interbreed together, the possible divergence is kept within narrow limits, but whenever a variety is produced the individuals of which have a a partiality for interbreeding, and some amount of sterility when crossed with another form, the tie that bound it to the central stock is loosened, and the foundation is laid for the formation of a new species. Further divergence would be unchecked, or slightly checked, and the elements of reproduction having begun to vary, would probably continue to diverge from the parent form, for Darwin has shown that any organ in which a species has begun to vary, is liable to further change in the same direction.* {Footnote: See "Animals and Plants under Domestication," vol. ii. p. 241. Thus one of the best tests of the specific difference of two allied forms living together [i.e. whether they are distinct species], is their sterility when crossed, and nearly allied species separated by geographical barriers [not necessarily associated with a sterility barrier] are more likely to [successfully] interbreed than those inhibiting the same area [where a sterility barrier would be expected]. Artificial selection is more rapid in its results, but less stable than that of nature, because the barriers that man raises to prevent intermingling of varieties are temporary and partial, whilst that which nature fixes when sterility arises is permanent and complete. For these reasons I think the fact that the hairless dog of tropical America has not interbred with the common form, and regained its hairy coat, is in favour of the inference that the variety has been produced by natural and not by artificial selection. By this I do not mean that it has arisen as a wild variety, for it is probable that its domestication was an important element amongst the causes that led to its formation, but that it has not been produced by man selecting the individuals to breed from [amongst those] that had the least covering of hairs. I cannot agree with some eminent naturalists that the loss of a hairy covering would always be disadvantageous. My experience in tropical countries has led me to the conclusion that in such parts at least there is one serious drawback to the advantages of having the skin covered with hair. It affords cover for parasitical insects, which, if the skin were naked, might more easily be got rid of. |
INUTILITY OF CHARACTERS. PARADOX OF SEX. RANDOM DRIFT A translation from French and German by Polly V. Forsdyke and Donald R. Forsdyke of a foray into evolution of the polymath Joseph-Remi-Leopold Delboeuf (1831-1896), best remembered for his contributions to neurophysiology (see D. J. Murray (1993) A perspective for viewing the history of psychophysics. Behavioural and Brain Sciences 16, 115-186; S. Nicolas, D. J. Murray & B. Farahmand (1997) The psychophysics of J-R-L Delboeuf. Perception 26, 1297-1315.) A Law of Mathematics Applicable to the Theory of [Biological] Transformation By J. Delboeuf, Professor at the University of LiEge, Belgium. La Revue Scientifique de la France et de l'Etranger. (1877) Second series-Volume 12 (Volume 19 of the collection). pp. 669-680. Also published in Kosmos (1877) Volume 2, pp. 105-127. I. First statement of the problem: How can an advantageous transformation occur? II. Second statement of the problem: How does disadvantageous transformation occur? III. General Statement of the Problem. V. Conclusion and Further Observations I. First statement of the problem: How can an advantageous transformation occur? Of the various criticisms which can be applied to a new doctrine, without doubt the most penetrating are those resting on mathematical grounds. The transformation doctrine invites criticisms of this nature. To introduce the problem, I can do no better than relate the views of Mr. Paul Janet, upon which my argument is based. In his beautiful book, Final Causes, a work of distinct historical insight, he expresses his doubts and reservations as follows:
Man selects the factors of reproduction; for example, he selects the male and female who possess a particular character he desires to fix; but how exactly, in nature, can the male discover the female who possesses the same endowment [genre de superiorite] as himself? Without hesitation, Darwin invokes the struggle for existence in which only the strongest and most able survive. However, this new principle is not a sufficient explanation.
Every radical departure from the normal, according to Mr. de Quatrefages, will lose something of its influence in every successive generation due to it's fusion with the normal. Mr. Paul Janet next cites a passage from an article which appeared in the Revue Scientifique. Here, an English scholar, Mr. Bennett, applies this line of reasoning to another example. I can do not better than reproduce the passage in its entirety: Here, an English scholar, Mr. Bennett, applies this line of reasoning to another example. I can do not better than reproduce the passage in its entirety:
One sees that the conclusion of Mr.
Bennet is exactly that of Mr. Paul Janet. In his discourse, referred to above, Mr. Broca
presents similar objections. They will be dealt with shortly. Not wishing at this time to
comment on the merits of the problem, nor to defend Mr. Wallace's theory, I will remark
that the reasoning is not so peremptory as it may seem. On the same premises, I will later
come to completely opposite conclusions. II. Second statement of the problem: How does disadvantageous transformation occur? If one has understanding well the implication of the observations just described, one would have noticed that they tend to outline the impossibility of conceiving species transformation from a given trait to a more advantageous trait. However in living nature there are also transformations of the opposite sense, and those which are more or less indifferent; for the soundest reasons they also seem incapable of explanation by natural selection. Mr. Broca states:
The author further notes the other possible assumption, that the round ligament disappeared before the first toe-nail, is no more admissible than the other. Consequently, they disappeared at the same time, and Prosatyrus the 1st, doubly defected, was born on one occasion without both round ligaments and the first toe-nail. But the orangutan possesses further unusual characteristics: his lungs are individual, that is, each of his lungs contains only one lobe; furthermore, alone among the primates, it has only sixteen dorso-lumbar vertebrae. Applying to these special characteristics the above line of reasoning, one comes to the conclusion that Prosatyrus I must have been born at an instant with all of the characters of the species Sartyrus; that is to say, that there was no transition, no progressive transformation, but:
Mr. Broca's argument is more or less plausible, and the Darwinists can hardly escape it without recourse to further hypothesis. Meanwhile, one could remark that he assigns to natural selection the insignificant characters created in Prosatyrus I, which he ignores as soon as he deals with the formation of Prostysus II. Moreover, the conclusion, perfectly legitimate in its essentials, contains debatable elements. There was not necessarily "a complete instantaneous transfiguration", it could also be that the transformation occurred slowly and acted collectively on the four distinguishing characteristics of the orangatan by virtue of the mysterious laws of the correlation of growth. But, I repeat, the main point remains the method of fixation of these seemingly indifferent characters. Let us now take this line of reasoning and apply it to the appearance of characters which are disadvantageous with respect to the struggle for existence; and let us choose, as example, a particular, but sufficiently general case. The completely inferior animals, which one can reasonably view as the closest to the primitive types, reproduce themselves by division or fission. This method of reproduction is followed in the more advanced species by more complicated methods which function either in concurrence with the former or in isolation from it. Among the myxomycetes, at a certain moment in their evolution, separate individuals reunite themselves in order to form a certain type of spore which, in turn, creates zoospores, that is, capsules from which new distinct individuals emerge. Among other living species, Botrydium for example, there are generally two individuals who unite in order to form a new being; and that is obviously where we find the first glimmer of sexuality. Thus, there appear successively:
III. General Statement of the Problem To establish the idea, let us remain with the problem that was just
presented: How did separation of the sexes (sexuality)
manage to displace perfect hermaphrodism? The questions posed by Janet,
Bennett and Broca are basically the same and differ only in their choice of example. Some preliminary observations are necessary. To simplify and at the same time generalize the problem, let us suppose that one individual gives to the world n individuals resembling itself, besides one deviating towards the plus side and one deviating towards the minus side; the quantity n+2 we will call the generative power (la puissance generatrice). This generative power can always be represented by an expression such as n+2. First, it is appropriate that the second term be even, because as a rule children resemble their parents and, if by chance a deviation in one sense turns up, then one must suppose a compensatory deviation in the other sense would occur. Now if the generative power is n'+2a, for example n'+6, one can, on dividing by a, return to the original n+2. After a given number of generations it would suffice to multiply n by a (i.e. in the above example by 3) in order to retrieve the actual number. To further simplify the calculation, let us suppose that an individual dies as soon as it has introduced its descendants to the world; so that at a given moment there exists only individuals who are distanced from the original stock by a number equal to the number of generations. Finally, we calculate as if the multiplication were indefinite, as if no obstacle opposed the expansion of the number of beings generated. And this line of reasoning is perfectly legitimate. If in fact, for example, there were only enough room for one million of these beings, while by virtue of the law, there ought to have been two million, then half of these two million would have to disappear at the moment of their birth; death would strike indiscriminately the homogeneous and heterogenous in proportion to their numbers, so that their ratio would remain constant. Therefore if, with free space, 800,000 beings resembling the father [homogeneous], and 1,200,000 not resembling him [heterogeneous], are brought into the world, after death has accomplished its mission (the death of a million), there would remain 400,000 from one side and 600,000 from the other. It would be exactly as if the generative power had been reduced by half. It is taken for granted of course that all the beings are born equal with respect to the chances of survival. In mathematics all units are equal. I now pass to the presentation of the equation of the problem; (see the attached table.)
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Generations | A-3 | A-2 | A-1 | A | A+1 | A+2 | A+3 | A+4 | A+5 |
0 | 1 | ||||||||
1 | 1 | N | 1 | ||||||
2 | 1 | N | N2 | N | 1 | ||||
N | 1 | N | |||||||
1 | |||||||||
Sums | 1 | 2N | N2 + 2 | 2N | 1 | ||||
3 | 1 | N | 2N2 | N(N2+2) | 2N2 | N | 1 | ||
2N | N2+2 | 2N | N2+2 | 2N | |||||
1 | 2N | 1 | |||||||
Sums | 1 | 3N | 3N2+3 | N3 +6N | 3N2+3 | 3N | 1 | ||
[*In Delboeuf's original paper the number of generations extends to eight.] |
Let us designate by A the group with the characteristics of the
original stock; according to what was said above, when one of them receives an
augmentation, we will designate by A+1 the new group so produced; if, on the contrary,
there is a diminution, we will employ the symbol A-1. In the same way, when a further
augmentation or diminution occurs, we will have a set of qualities represented by A+2 and
A-2; and, continuing in the same way, we will derive the symbols A+3 and A-3 and, in
general, after m variations, we will have a total of set of qualities which we may
designate as A+ or - m. |
Genera- tions |
A | A+ or - 1 | A+ or - 2 | A+ or - 3 | A+ or - 4 | A+ or - 5 | A+ or - 6 | A+ or - 7 | A+ or - 8 |
0 | 1 | ||||||||
1 | 10 | 1 | |||||||
2 | 102 | 20 | 1 | ||||||
3 | 1060 | 303 | 30 | 1 | |||||
4 | 11206 | 4765 | 604 | 40 | 1 | ||||
5 | 120300 | 64266 | 10200 | 1005 | 50 | 1 | |||
6 | 1,309,020 | 839,482 | 156,015 | 20300 | 1506 | 60 | 1 | ||
7 | 14,411,400 | 8,071,035 | 2,241,050 | 360,521 | 35,420 | 2,107 | 70 | 1 | |
8 | 160,256,070 | 134,862,813 | 30,842,056 | 5,881,680 | 716,828 | 56,560 | 2808 | 80 | 1 |
We have until now assumed that the tendency to deviate is unlimited, in
other words that it strives {g} tends {f} continually to transform new forms into even
newer forms. In this way, there is pressure on class A+ or - 3 to transform to
A+ or - 4, and from this to A+ or - 5, and, in general, from the
class A+ or - m to the class A+ or - (m+1). We can also make
another supposition and imagine the cause becomes limited in its effectiveness after the
production of a form of a given rank, say A+ or - 3, or A+
or - 4
or, in general, A+ or - m. The problem has a solution at all similar points.
However, this final form, although indefinitely increasing in size, never manages itself
to equal that of the original form. Equality can only be reached after an infinite time.
This conclusion comes from the examination of the second table which, in a sufficiently
approximate way, indicates the rate of the growth of the classes, even in this particular
case. V. Conclusion and Further Observations However, we cannot stop here. Now that we possess a definite result, it is appropriate that we look for and draw general conclusions. The solution of the problem which faces us has a greater significance than is first apparent. For this, we must certainly abandon the solid ground of the positive science to expose ourselves to the insecure ground of conjecture and speculation. [The following paragraph is translated in George J. Romanes' book Darwin, and After Darwin, Volume 3, 1897, p. 13; (perhaps with the assistance of Ethel Romanes, who was probably fluent in French).]
Taking everything into account, this transformation is only the reflection of a completely rational process. Doubtless, absolute and general uniformity strives to conserve itself; but each permanent cause which tends to destroy it, having started somewhere, works ceaselessly; everyday it tears off a small part and, as these small altered parts work, in turn, to disrupt their environment, the transformation process becomes faster and faster. Nevertheless it is important to define as precisely as possible the concept of permanent cause. Let us begin by distinguishing carefully between a limited cause and and unlimited cause. A limited cause is one which has a defined goal. This would be one which would tend to transform class A into a given class, for example A+ or - 10, or, in general, A+ or - m; or, returning to the image we used before, to give to the arms of a U a determined difference. This type of cause progressively loses its effectiveness as it produces its effects. One can satisfy it and, in consequence, nullify it. It strives towards an end to which it ceaselessly approaches. In general one can say that all disequilibriums belong to the category of limited causes, because each broken equilibrium reestablishes itself again little by little. The warming of a cold body by a hot body, the fall of water to the valley floor, are examples of this. All other causes are unlimited causes, which cannot be lost. One of these would be the cause tending to provoke a constantly increasing difference of length between the two arms of the U, or incessantly to transform class A+ or - 8 to class A+ or - 9, this to class A+ or - 10, and so in essence, class A+ or - m to class A+ or - (m+1).
The poet said:
This thirst which is quenched by nothing and which presses us without respite, seems to be felt by all of nature.
Individuals are nothing; they appear and disappear; but life does not extinguish itself.
Now, strictly speaking, the only causes of evolution are the
constant causes; the others, namely the limited causes, are only approximately permanent
causes. Let us consider the mechanisms of them both. Would the reader imagine our
nebula in its primitive state when inert matter was still dispersed in space and let us
begin by attributing the force of attraction to this matter. The mass of cloud begins to
condense, its molecules order themselves in concentrical layers around a nucleus. Here we
see the first cause of differentiation. These sphere-like layers are different from each
other, but each is itself perfectly homogenous at all points. The only changes of which we
can conceive would be represented in the spherical layering around the middle point. So
that along a ray of light material parts are differentiated, but all rays are similarly
differentiated. A being who would see the composition of one of these rays change, would
be absolutely certain that that of the other rays changes in exactly the way. Footnotes: 1. Pages 390 and following. 2. Compare the essay of M. Broca on transformation in the Revue
des cours scientifiques, 7th year, p. 365, where the same argument is
produced. 3. 1871m 2nd series, volume 1, p. 99, under the authorship of Edmond
Per ? 4. Text already cited, p. 256. 5. A more complete solution to the problem will only have interest for mathematicians. The
following is as far as I can extent the subject. If we represent by Tm,p the
number of individuals in the generation A+ or - m, after p generations, we have the formula: Tm,p = (np-m-2t)(yp+1,m+1-2t)(ym+1+2t,t+1) -- -- .(a) t=0 A the upper limit, t is the largest whole contained in (p-m)/2. The quantity yx,z is the number of the arithmetical triangle with abscissa c, and ordinate z. In his great treatise on the Differential and Integral Calculus, n 1086, Lacroix assigns as the value for yx,z the expression: (z(z+1)(z+2)-- -- .x)/(1.2.3-- .,(x-z+1)). This value is obviously approximate. It is necessary to remove the last two terms of this fraction and write: yx,z = (z(z+1)(z+2)-- (x-1))/(1.2.3-- .,(x-z)). I must say that, when I got into this problem, I would have used my
own notation, and I arrived at a general formula, without being in doubt that I had made
myself sick and at total loss, that I had discovered long known properties. It was thus
that I went to see my friend Folie, whose name is much admired in the mathematical
sciences. I wished to verify if my formula coincided numerically with that of Lacroix: and
it was thus that I had suspected an error and contacted professional mathematicians who
might attempt to repeat the calculations and the geometrical illustrations. t=(p-m)/2 Tm,p = (np-m-2t )((t+1)(t+2)-- -- -- -- ..p)/(1.2.3-- (p-m-2t)(1.2.3-- (M+ttt))) (b) t=0 Examination of the fraction reveals it capable of great simplification if m and p are replaced by numbers. If in the formula (b) we make m=0, it becomes: t=p/2 Tm,p = np-2t (t+1)(t+2)-- -- -- -- p)/(1.2.3-- (p-2t)(1.2.3-- ..t)) (c) t=0 This is the number of individuals of the initial class after generation p. To obtain that of the variant class, it suffices in formula (b) to substitute for m all the values between m=1 and m=p, to sum these and then multiply by 2, since there are as many individuals of class A+1 as of class A-1. Here is the sum: m=p 2 Tm,p m=1 It is now necessary to resolve by replacing for p the following equation: m=p 2 Tm,p = ou > To,p (d) m=1 As I have said, the general solution is beyond my powers. Only, by
applying numerical substitutions, can I achieve the result that p has for the upper limit
(n/2)+1. However, there is some ambiguity. Simple inspection of the equation shows that it
has always a positive root. In effect, if we develop the two members of the equation (d),
and we order by raising to the power p, it is easy to see that we can always put p as high
as the first term of the first member to be equal to the first term of the second member;
then, from this, so that the following terms taken two by two from the first member be
equal to the terms of the second member,one on one. To resolve the problem completely, one
must prove this proposition reasoning in general terms. 6. See in the issue of the 25th November 1876, the second part of the article by Haeckel
on "A Naturalist Philosophy". 7. In my Psychologie comme Science Naturelle (Paris, Germer Bailliere; Bruxelles, Muquardt), I arrived at the same conclusion-- ..
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An Unnoticed Factor in Evolution by EDMUND CATCHPOOL, The Grove, Totley, Sheffield, October 23rd.[Author of A Text-book of Sound, the Physicist Catchpool was a tutor for the University Correspondence College, and thus would have been a colleague of H. G. Wells.] Nature (1884) 31, 4.(November 6th) Two observed biological facts seem to oppose great difficulties to any explanation on evolution principles; difficulties admitted by evolutionists as well as their opponents. I mean --
The first seems to require that a period much greater than that of artificial selection should be necessary to produce sterility between descendents from the same ancestor; a supposition which would require an almost incredible period for evolution as a whole. The second seems to require that many species now intermixed should once have been geographically separated, sometimes in cases where this is very difficult to imagine. Both these difficulties are completely removed if we suppose mutual sterility to be not the result, but the cause, of divergence. As far as can be judged, "sports" [mutations] are as likely to occur in the generative elements (ova and spermatozoa) as in other parts of the body, and from their similarity in widely unlike groups it seems certain that a very slight variation in these elements would render their owner infertile with the rest of its species. Such a variation occurring in a small group (say the offspring of one pair) would render them as completely separate from the rest of their species as they would be on an island, and divergence (as Wallace has sufficiently shown) would begin. This divergence might progress to a great or a small extent, or even be imperceptible, but in any case the new species would be infertile with the species it sprang from. If this theory be admitted, we must distinguish between varieties and species by saying that the former arise by spontaneous variations in various parts of the body, and only gradually become mutually infertile (thus becoming species), while the latter arise sometimes in this way, but sometimes by spontaneous variations in the generative elements, and are in this case originally mutually infertile, but only gradually become otherwise divergent. I would suggest the following tests, and should be glad of any facts, from experience or from books, which can help in applying them:--
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