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):

"Upon the continent the individuals of the genera by spreading far, form, through differences of the locality, food and soil, varieties which finally become constant as new species, since owing to the distances they could never be crossed with other varieties and thus be brought back to the main type. 

    Next they may again, perhaps upon different roads, return to the old home where they find the old type likewise changed, both having become so different that they can interbreed no longer. 

    Not so upon islands, where the individuals shut up in narrow valleys or within narrow districts, can always meet one another and thereby destroy every new attempt towards the fixing of a new variety." 

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.

Donald Forsdyke


Tree snail from the island of Oahu. Achatinellae. By virtue of seed dispersal the environment (trees) became uniformly dispersed and uniform, but by virtue of their "snail's pace" progression, groups of snails tended to become reproductively isolated and non-uniform.

On Diversity of Evolution under One Set of External Conditions


Journal of the Linnean Society (Zoology) 11, 496-505.
(With copyright permission from Academic Press.)

[Read before the Linnean Society of London, November 21st 1872.]


Facts Throwing Light on the Subject

External Conditions not the Cause

Separation and Variation are Correlative Factors in the Evolution of Species

Migration and Variation are Opposing Factors in the Limitation of Areas

Stability of Type at Affected by Cultivation

The Natural Selection that Prevents Variation

Stability of Type in Island Fauna may be Impaired

Imaginary Case, Illustrating Evolution without Change in the External Conditions

Changes Produced by the Introduction of Enemies

Recapitulation and Conclusion


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 : --

  • Ist. Because in different valleys, on the same side of the mountain, where food, climate, and enemies are the same, there is still a difference in the species.

  • 2nd. Because we find no greater difference in the species when we pass from the more rainy to the drier side, than when we compare the forms from valleys on the same side of the mountain, separated by an equal distance.

  • 3rd. Because if, failing to find a reason in the more manifest conditions, we attribute the difference in the species to occult influences, such as magnetic currents, we must suppose that there are important differences in these hidden conditions for each successive mile, and that their power at the Sandwich Islands is a thousand times greater than in most countries.


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.

To state the conditions more fully: --

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.


Recapitulation and Conclusion

   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.



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.)

malefema.gif (201 bytes)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.

IV. Solution 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:

"The true basis of Darwin's Theory is the extrapolation from artificial selection to natural selection; this establishes how blind nature by the chance combination of circumstances achieves the same result as man achieves through design."

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.

"Let us suppose, as a fact, that in hot countries skin colour is an advantage which makes the inhabitants more able to bear the rigor of the climate; let us suppose further that in one of these lands there would be only white people and, at a given moment a black man accidentally came into being. Clearly, he would have an advantage over his compatriots; if you wish, assume that he would live longer.

    Should the man marry, what type of women would he take? Without doubt a white woman, since there would have been no other choice. Would the child who resulted from this union be black? Without doubt it would be mulatto. The child of this child would be a further one tone paler, and so the accidental colour would after some generations disappear and lose itself in the common characteristics of the race. Thus, although the dark colour would be an advantage, it would never have been able to perpetuate so as to create a new variety better adapted to the climate, and which could displace the whites in the struggle for survival."

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:

"In his book, The Influence of Selection, Wallace cites a very curious case of this type (a case of mimicry, that is to say, imitation), concerning a species of South American butterflies related to our own cabbage-white butterflies (pierides), members of the species Leptalis. In general, birds are very fond of pierides. In contrast, they almost never attack the butterflies of the family Heliconidae, which are represented by the species Ithomia in South America. The reason for this disdain is that the Heliconidae release a nauseating liquid when they sense danger which makes them the most unpleasant of foods. Now it so happens that certain types of Leptalis, without losing any of their essential characteristics, take on exactly the colour which would cause an untrained eye to confused them with the real Ithomia. With such a disguise, they escape the pursuit of their enemy much easier than their white coloured relatives. heliconidae species. Courtesy of William T.Hark, M.D.(93672 bytes) Wallace attributes to natural selection the creation of this protective form of Leptalis. This conclusion is attacked by Mr. Bennet using a line of reasoning which appears to us most rigorous. "It is clear," he says, "that, in order to pass from their usual form to the protective form, Leptalis must undergo a series of gradual transformations, and one can hazard a guess that no less than a thousand intermediary forms must have occurred between the original deviation and the ultimate form. On the other hand, it is clear that the first deviating Leptalis was not different enough from it's sisters to confuse the appetites of interested birds, who could see through their disguise. It is reasonable to believe that during the first fiftieth of the transformation period, supposing it continuous, the birds would not have been misled. If this is so, then the butterflies would not be at all preserved by their changed appearance, the basis for selection would disappear, and the continuation of the transformation process would be a matter of chance. These chances can be very approximately calculated.

    For example, let us take a pair of Leptalis and suppose that the species had the tendency to vary in twenty different directions of which only one tended towards that of Ithomia. In the first generation the chances of a favorable deviation would be represented by the fraction 1/20 and this evaluation is favorable to the Wallace hypotheses, because among the numerous progeny of a pair of butterflies, one would certainly find more than 20 forms, even if only just a little different, which would distinguish themselves from the previous form.

    In the second generation the forms that already had the tendency to deviate towards the Ithomia form would have no reason to regress and it would be just among this 20th of the descendants of the original couple that we can reasonably expect to find forms that more of less approach the protective form. But for this 20th selection is still not effective, and it is still chance that determines the pressure towards the form in question; only 1 in 20 of the offspring of the new descendants will assume this form; but these will only represent a 20th of the 20th of the offspring of the original couple. The chances of finding useful forms in this second generation can be represented by the fraction [1/20]2 , or 1/400. After ten generations, the chance will reduce to [1/20]10; this means that in every ten billion individuals just one will conserve traces of the original deviation, and we are not even halfway through the generations which compose the first 50th of the transformation period. This being so, when we apply this calculation to the total population of a district that we estimate at one million, we find only a single individual of the species Leptalis among 10 millions which after 10 generations have elapsed since the first deviation, can present a few characteristics similar to those of Ithomia: this is an absolutely negative result and forces one to reject completely the hypothesis of selection, since, before this could have acquired any validity, the original chance variation that was favourable to the survival of the individual would have completely disappeared among the mass of opposing variations.

     This conclusion carries even more weight in the situation of deviations tending to approximate the form of an animal to one very distant to itself, or even to inanimate objects. One must therefore look elsewhere for the cause of the phenomenon of mimicry and one might, as Mr. Bennet suggests, find it in instinct itself. --
 Mr. Bennet, without being opposed to natural selection, precisely defines the boundaries of its influence; it can do much for the transformation and especially for the fixation of species; but it is not capable of everything and the most enlightened advocates of Darwinism cannot deny this viewpoint".

    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.

    When one deals with mathematical formulas, it is necessary to be aware of the nature of the problem and its associated facts. A mistake in the case of equations carries serious consequences and such a mistake was made here. This is all the more frustrating since much mathematical reasoning assumes an air of infallibility, and he who uses it deserves some deference due to the quality of absolute certainty associated with arithmetic and algebra. One does not dare to contradict when one is placed in his territory and further find that all the advantages are in his favour. But, before we directly tackle the subject, let us complete the set of objections which, in this scheme, oppose transformation.

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 orangutan is alone among the primates in having no toe-nail on its big toe. I ask the Darwinians how this bizarre character could have been produced. They answer that one day a certain monkey was brought into the world without a big toe-nail, and that this variety of individual is perpetuated among its descendants.
   orangutan. Courtesy of Orangutan Foundation International, Los Angeles. (19968 bytes) For more clarity, let us give a name to this 'big toe-nailless' pythecian ancestor; since he was the origin of the Satyrus species, let us name him Prosatyrus 1st, giving him a number to signify his role as founder of a dynasty. Prosatyrus the 1st had a certain number of offspring of which a few doubtless resembled their other ancestors in having, like them, a nail on every toe. However, by virtue of the law of immediate inheritance, one or many of them would have been, like their father, born without a nail on their big toes; then, thanks to natural selection, this character became more and more frequent among the descendants of Prosatyrus the 1st, and eventually a time arrived when it had became a constant character.
    I ask myself, indeed how it could be that the absence of a big toe nail could have been given preference by natural selection; I do not see how this negative characteristic which did not improve any function, could have given to individuals so endowed an advantage in the struggle for existence. I would rather assume the contrary. I therefore cannot explain the success of the line of Prosatyrus the 1st; but one cannot understand everything, and I would like to attribute to natural selection the merit of having fixed this characteristic among the ancestors of our orangutans.
    However the orangutan distinguishes itself further from all other primates, living as well as fossils, by the absence of the round ligament of the hip. This odd ligament, which has no analog in any of the other joints, is not only found in all the primates, but also in most mammals, and it's absence in orangutans can be considered an anomaly. The Darwinians could, with some appearance of reason, attribute the appearance of this character to an individual deviation which came about by chance in the ancestors of the orangutan, and eventually became a fixed through natural selection.
    I continue to ask myself how natural selection and the struggle for existence could allow the endurance of a disorder that is more harmful than helpful to the operation of the coxo-femoral joint? Nonetheless, I continue to answer by saying that one cannot explain everything, and I confine myself to asking the following question: At which point in time did the absence of this ligament reveal itself among the ancestors of the orangutan species? Was it before or after the existence of the one I have named Prosatyrus the 1st?
     Let us first see if this first monkey without round ligaments was one of the descendants of Prosatyrus the 1st. If it was thus, then it would be appropriate to give the name "Prosatyrus II" to the one who, among the big toe-nailless monkeys, inaugurated this seconded distinctive character of the orangutan species.
     By the time that Prosatyrus II came into the world without his round ligament, a certain number of generations had already passed since the big toe nail had disappeared. It is by hundreds that one counts the descendants of Prosatyrus the 1st who, like him, were missing their big toe-nails but still bore round ligaments.
     It was from this large cohort of individuals resembling Prosatyrus the 1st that Prosatyrus II emerged successful in the struggle for existence. He was different from them only in lacking the round ligaments, a feature which was in no way an advantageous to him. I willingly admit that despite this defect, he was able to live until an adult age and was able to conceive of some offspring resembling himself, and that these, mating among themselves (I don't know how), created a variety characterized by the lack both of big toe-nails and of round ligaments, but there is no reason why this variety should have displaced the other, no reason why the numerous representatives of the variety of Prosatyrus the 1st should have lost their right to exist.
     Let us suppose that there were only a thousand or even just a hundred at the time of the birth of Prosatyrus II; these were dispersed over a wide area, mostly situated beyond the area where Prosatyrus II lived, and all would have had more or less the same chances of reproducing as Prosatysus II. They would have had descendants resembling themselves and, since the variety Prosatyrus II was able to maintain its existence despite its imperfection, the variety Prosatyrus the 1st would have been a hundred, even a thousand times more numerous, and, do not forget, better constituted, a strong reason for their survival. There must therefore have been co-existing with the orangutans lacking both the big toe-nails and the round ligaments, another variety, which, while similarly lacking the big toe-nail, still possessed the ligaments. Of the variety that would have existed if the round ligament had disappeared after the big toe-nail, there is no trace. This intermediary species does not exist; consequently it is not possible to agree that the round ligament was lost for the first time among the descendants of Prosatysus the 1st."

    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:

"a complete instantaneous transfiguration, contrary to all laws, Darwinian or otherwise; let us not shrink from the word, it is a supernatural act, equivalent to an act of creation."

    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:

  • perfect hermaphrodism (allowing an individual to produce, without the help of another, an individual resembling itself), then
  • imperfect hermaphrodism requiring a mutual pairing for reproduction, and eventually the
  • absolute separation of the sexes.


    At first glance, one would be inclined to conclude that the most favourable condition for the continuation of the species would be that in which the individuals of the species reproduce themselves by fission or, at least, that they are perfect hermaphrodites. However, actually perfect hermaphrodism is the exception and sexuality is the rule. By virtue of which law did the separations of the sexes become predominant in nature, when, contrarily, everything seemed to be opposing its expansion? How did seemingly advantaged species cede their positions to other seemingly most disadvantaged species? Above all, this is where the principle of Natural Selection fails.

It is now necessary to show that this problem has a purely mathematical side and that the law which applies to it, once established, will cast unanticipated clarity upon the solution which it admits. Moreover, does the necessity for the struggle for existence depend already in the property of progression? [Not clear here] From this inescapable fact does it not follow that, from the moment that a couple brings more than two descendants into the world, the posteriority that will consequently be born will one day cover the earth, at least when there is no permanent force of destruction to stop the expansion? I now would like to demonstrate that the definite predominance of the number of transformed individuals over those who preserve the primitive type is an obligatory consequence of the persistence of the cause, as weak as it may be, that leads to the first variation.

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.

    To clarify my exposition, I will have recourse to an image. Would the reader be so good as to imagine the perfect hermaphrodite as a capital U; the left side, if you will, will represent the male character, and the right side the female character. Hermaphrodism stops being perfect as soon as one of the two sides exceeds, no matter by how little the amount, the other in length. Let us also note that hermaphrodism can alternate, in the sense that the same individual can, in the course of time, function as male or female: it is sufficient for present purposes to say that the development of the two sides, although of equal length in the final analysis, manifest at different times a shortening or lengthening. Let us not dwell on this particular case. One can say that the separation of the sexes did occur, that is the individual has become either masculine or feminine, as soon as the difference in length of the two branches had attained a certain value. If I designate by "A" the perfect hermaphrodite, then I could represent the perfect male as A+m, the perfect female by A-m; and the intermediate stages may be represented by A+ or - 1, A+ or - 2 etc--  This image to which I otherwise attribute no precise scientific significance, I present after reading the work of my learned friend Ed. Van Beneden on the Hydractines. In these animals the testicles are a formation of ectoderm and the ovaries of endoderm. By bold induction, the young professor discovered a law, valid for this species of polyp, that applies to the entire animal kingdom. I do not know at what point such a generalization will be later confirmed, but for the particular case with which we are dealing, nothing stops us from assuming its legitimacy if only to give our imaginations a point of departure.

    Given the above, the question reduces to the following: Assuming that a perfect hermaphrodite brings into the world, for example, a thousand individuals resembling itself, and only a few deviate towards the paternal type, and provided further that the descendants multiply according to the same law, how is it possible for the earth not to end up populated by only hermaphrodites? Put in such a way, the question implies, as you have undoubtedly already noticed, the permanence of the cause which endows certain descendants with an ancestrally determined character. In the final analysis there are two forces in operation, one promoting uniformity, the other diversity. This is the point which neither Janet, nor Bennett, nor Broca have noticed. They have gone from the assumption that a white person would become black accidentally, or that a Leptalis would accidentally take on a part of the colouration of Ithomia, or that a PithEcien would accidentally lose the first toe-nails or the round ligaments. But the cause can only be called accidental in the sense that it only affects one in twenty, a hundred, a thousand individuals; however, in the following generation it will affect a proportional number of individuals, not only among the descendants of this black person, or of this Leptalis, or of this altered monkey, but also among the descendants of the white persons, as well as the descendants of the unaltered butterflies and monkeys. This is an element which these learned authors did not take into account. The word accident is used by them in the popular sense, as a fortuitous example, an exception, while in the scientific sense it is taken as a rare coincidence, an infrequent event {fait rare, peu frEquent}.

    If among a thousand and one balls all save one are white, it would be, if you wish, accidentally that I will select a black ball; but it will have nothing to do with an exception or a special chance happening, when among 1001 selections the black ball, on average, will have been selected once. The so-called accidental cause has nevertheless a quality of absolute permanence. I will now demonstrate this seemingly paradoxical, if not absurd, proposition that, as powerful as the general force sustaining ancestral similarity may be, and as weak as the diversifying force may be, the latter will triumph.

    Here, in other words, is what I am trying to show: no matter how large the number of beings resembling him, nor how little the number of beings not resembling him, for one who brings an isolated individual into the world, assuming that diverse generations continue to reproduce in the same fashion, there will always be some number of generations at the end of which the total of altered beings must outnumber the total of unaltered being.

    To make my ideas more understandable I will have recourse to numbers: If one hermaphrodite brings a thousand or a million individuals resembling itself into the world, and just two among them differ in that one is a little more masculine and the other is a little more feminine than the others; if each of these descendants again leaves behind the same number of individuals resembling their parents and just two, in the same sense not resembling their parents; and if the same rule stays in effect for all the successive generations, I say that one can predict the number of the generation at which there will be fewer primitive hermaphrodites than variant individuals, and, irrespective of degree of the variation, even the generation at which both numbers turn out to be the same. Thus, in this example, long before the thousandth or millionth generation, the number of changed individuals will exceed the number of individuals retaining the pure type, and the number of changed individuals in the first stage of variation (who will have acquired but a slight degree of variation) will be the same or almost the same as that of the original type. From this moment the variants grow at a relatively increasing speed. Presented in this way, the proposition takes on a general character: it does not account just for the replacement of perfect hermaphrodism by more or less imperfect hermaphrodism, but for all deviations from the original type, whether favorable or unfavorable. It follows from this law that, from the moment a constant pressure begins to alter an original form, in as weak an amount as one likes, the path leading to the defeat of the original by the variant is clear.

    Far from me, however, is the notion of wanting to replace the laws of Darwin with the laws of mathematics! But, because they act inevitably and necessarily, the latter must certainly intervene and lend their support. They alone explain why the primitive types get more and more rare, and incline towards complete disappearance, because rarity is a disadvantage; and, as with all species, both past and present, represented according to their descent, one recognizes that some must die, and that others are certainly destined to die in their turn if they do not possess certain characteristic assuring their survival (existence eternelle).

    This preamble was necessary to rouse interest in an extremely curious type of [evolutionary] progression, which leads to unexpected results. The problem itself belongs to the area of higher mathematics and differential calculus. I have not completely solved it myself; it involves an equation which I am not able to integrate. Maybe an analyst will become interested in the problem and will find the general formula. But, for the moment we place definite numbers in the place of algebraic numbers, and explanations become easy to follow and only require elementary knowledge and a moderate degree of attention to be understood. The impatient reader could even become satisfied with these preliminary explanations and be assured of the law by the examination of table II.

IV. Solution of the Problem

    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.)


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        
  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.

    To abbreviate the language let us say that the individuals having the characters A, A+1, A-1,--  A+ or - m, belong to the variety A, A+1, A-1,-- ..A+ or - m. Of course the word "variety" does not carry the scientific meaning here. *[Footnote in German translation: For this reason this expression "variety" will not, in our continued discussion, be alluded to as "type" or "species", but rather as "class".]  No demonstration is necessary to establish that the increase in number of individuals of the class A-1, A-2,--  A-m,, must be equal to that of the class A+1, A+2,-- .A+m; ; for this reason the table was not extended to the left of the first three classes, the region to the right of A being sufficient. The order of the generations is marked in the first column to the left. Thus provided, we see that in the first generation we will have n individuals of class A, and 1 individual from both the classes A-1 and A+1.  In the second generation each of the n individuals of class A will produce n individuals of the same class for a total of n2, and an additional 1 of the class A-1 for a total of n, as well as 1 of the class A+1, also a total of nThese numbers n2, n and n are the first of column A and the second of columns A-1 and A+1 (second generation). [Comment: This confusing sentence can be safely ignored.]

    On its part, the unique individual of class A-1 will give the world n individuals of its own class (the first number of column A-2, second generation), 1 individual of class A-2 and 1 individual of the original class A. The unique individual of class A+1 will behave in the same way, so that by the second generation there will be n2+2 individuals of class A, 2n individuals of classes A-1 and A+1, and finally 1 individual of classes A-2 and A+2. These total sums are given at the bottoms of the squares containing the numbers of which they are constituted.

    By the simple consideration of these first results, one can already recognize the effects of the law. Actually, the ratio between the number of class A+ or - 1 and A individuals in the first generation is 1 : n, and in the second generation is 2n : n2+2, so when n is big enough, the ratio is near to 2 : n. . It is easy to see the reason. For the relation 1 : n to hold, it would be necessary that class A+ or - 1 recruited only its own type; but it derives a part of its prosperity from class A. Certainly, class A derives in turn its composition from class A+ or - 1; but since the number of individuals of this last category is much smaller, the growth of A+ or - 1 is very great in an absolute sense and even greater in a relative sense. This becomes much clearer when one puts a number in the place of n, for example 1000. Then, in the first generation one has 1000 for class A and one for the class A+ or - 1; in the second generation, each of these receives from A an increase of 1000 individuals above the 1000 which they contain, while they furnish only two individuals to the 1,000,000 which they already possess. One sees here the error in which the authors, extracts of whose work I have cited above, have fallen.

    In the third generation, the number of individuals of class A has climbed to n3 +6n, coming from ( n2+2)n individuals produced by the n2+2 individuals of the preceding generation, plus, on one hand, 2n individuals coming from the class A-1 who partially restore the original class A, and on the other hand, 2n individuals of the class A+1. If n = 1 000, class A now consists of 1,000,000,000 individuals.

    One will notice, once and for all, that the number of class A individuals in any single generation is always determined by n times the number of individuals of the same type from the preceding generation, augmented by the number of individuals of classes A-1 and A+1, also from the preceding generation. And, since the classes A-1 and A+1 contain the same number of individuals, one can content oneself by doubling the number of one of them; as is done in the continuation of the table.

    If we proceed to class A-1, we see that the number of its members must be 3n2 +3 deriving, namely 2n2, from the 2n individuals of the original class A-1: further n2 +2 from the n 2 + 2` individuals of class A, and finally 1 individual from the class A-2, stemming from the original A-1. That which has been said about class A-1 also applies to class A+1, so we do not have to mention this again. If n equals 1,000, the number is now 3,000,003: in other words the relationship of the new form to the old form is now almost 3 : n.   We see that the total sum 3n2 +3 is obtained by multiplying by n the number of individuals (2n) of the previous generation of class A+ or - 1, and then by adding the number of individuals of classes A and A+ or - 2, also of this generation.

    Thus, in general terms, the numbers of the class A+ or - 1 will form themselves in the same way in all generations, in other words, from they will derive from the number of the preceding generation multiplied by n, and augmented by addition of numbers from the classes A and A+ or - 2 (in consequence of the left and right parts of Table 1), also from the previous generation.  We also remark at the same time here that the number of the classes A+ or - 1 will never equal the number of class A, because, as it increases, class A receives more and more reinforcement from classes A+ or - 1.

    Let us now examine the increase of A+ or - 2. It is easy to see that the rule which we have just stated, also applies to the results of this column. Thus, the results of the fourth generation (6n 2 +4) derives from the total of the third generation: 3n multiplied by n, plus the numbers 3n 2+3 and 1, all quantities furnished by the third generation. Class A+ or - 3 is in accordance with this point, as are all the following classes, as simple inspection of the table will reveal to the reader who takes the trouble. We can therefore formulate the general rule in the following way:   The number of individuals of class A+ or - m after generation p is equal to the product of n and the number of individuals of the same class after the p - 1st generation, augmented by the number of individuals of classes A+ or - (m-1) and A+ or - (m+1) after the same p-1st generation. For class A this general rule gives us with reason to observe that the numbers to be added to the [above] product of n, which are contributed by classes A+1 and A-1, are equal to one another.

    A brief glance at the first results for any species will immediately show that the number of individuals of the same class grow in a more rapid progression than those of the unmodified classes. Thus the class A+ or - 3, which in the third generation contains only one individual, will in the next generation have 4n, in the fifth 10n2+5, in the sixth 20n3 +30n, and so on, while the corresponding numbers of class A+ or - 2 are: 3n; 6n2+4; 10n3+20n; 15n4+60n2+15; and those of class A+ or - 1 are respectively: 3n2+3; 4n3+12n; 5n4+30n2+10; 6n5+60n3+60n and so on; and those of class A: n3+6n; n4+12n2+6; n5+20n3+30n, and n6+30n4+90n2+20, all progressions in which the rate of advance is always slower and slower.

    If we say that n=10, that is to say if the generative power is 12, we obtain table 2 (following). This table shows clearly that the progressive growth of classes is rapider the more they distance themselves from the original form. We see further that, already by the fourth generation, the number of changed individuals almost equals that of the individuals which have remained true to the pure type. In fact, the total number of the changed individuals of classes A+1, A-1, A+2, and A-2, etc., is 2(4,765+604+40+1)=10, 820, a number which is not very far from 11,206, the number of unchanged individuals. But by the fifth generation the ratio has already become 151,044: 120 300. Moreover, by the eighth generation the number of individuals from classes A+1, A+2 --  A+m, as from classes A-1, A-2 --  A-m (for example, more masculine or feminine) surpass the number of individuals which continue to follow/resemble the original form (perfect hermaphrodites). The ratio is actually 172,362,826: 160,256,070. As we see, around the number of generations close to half the generative power, the pure form already finds itself in the minority, and after an equal number of further generations, it comprises less than a third of the total number of individuals [footnote for esoteric solution].


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.

    This special relationship between the numerical progression of the original form and of any derived form permits the resolution of a difficulty which naturally comes to mind: If there is a certain tendency on the part of the hermaphrodites to develop separate sexes, or for white people to become black, and if we admit, on the other hand, that a similar tendency exists in which the pressures the variants back to the original form, how is it then possible that at a certain point of time when the variants exceed the original form in numbers, that this same tendency does not bring into predominance the original form again? This is so because each variant class is numerically inferior to the original form. The constant cause does detach a part of class A+ or - 1 in order to reattach it to the original class A, but what this delivers to the form of the first class is always numerically considerable. In the same fashion, class A+ or - 2 supplies class A+ or - 1 well, but this gives back more than it receives, and so on. Each degree of variation has numerically less members than the original form, but as this difference tends to zero, members from any two levels added to each other will end up predominant.

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).]

One point, however, is definitely established. This is that the proposition, which we have described above as paradoxical, is rigorously true: a constant cause of variation, weak as it may be, gradually changes uniformity, and diversifies it, ad infinitum. From uniformity left to itself only uniformity can emerge; but, if we perturb (add a light ferment to) the uniformity, then the uniformity will at some point be broken; the differentiation will propagate itself everywhere, infiltrating every region, and, after a time, admittedly infinite, it will have overrun it (l'aura envahie) entirely.

   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).

    These sorts of causes can only cease operating when nutrition is impaired. Their goal is infinite. The rectilinear and indiscriminate movement of a body can provide a gripping, although imprecise, image: where is this body, which always goes at the same speed in the same direction for all eternity, going to? Among the causes of transformation there are ones which seem to change always and incessantly just for the sake of change alone, which do not work towards a goal, yet strive restlessly with increasing strength. In general one can say that evolution in nature, understood in a certain way, has in principle a similar cause. If to evolve is to realize a more perfect state, then, since there always exists a more perfect state, the mind conceives of no limit to evolution.

The poet said:

"La gaite manque au grant roi sans amours;
La goutte d'eau manque au desert immense;
L'homme est un puits ou le vide toujours,

This thirst which is quenched by nothing and which presses us without respite, seems to be felt by all of nature.

"Elle n'a qu'un desir, la maratre immortelle,
C'est d'enfanter toujours, sans fin, sans treve, encor."

Individuals are nothing; they appear and disappear; but life does not extinguish itself.

"Tous les etres, formant une chaine eternelle,
Se passent, en courant, le flambeau de l'amour,
Chacun rapidement prend la torche immortelle,
Et la rend a son tour."

      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.

     Now let us go further. From the moment when among this ensemble of homogeneous concentrical spheres, each operating independently, one adds a second point of excentric attraction {?g and from this union allows a similarly shaped concentrical sphere to come into effect}, a new variant will occur. The ray which has taken on the direction of this point, will assume a particular appearance; the neighbouring rays will modify their composition and, in the end, the nebula /will take on the appearance of a rotating body whose parts will only offer uniformity in certain circles{G}/ will take on the form of a surface of revolution and will offer no more uniformity than along parallel circles whose plane will be perpendicular to its' axis {F}. This will cease to exist, if a third center of gravity appears outside of this axis.

    One can say that, at each of these moments, the globe tends towards a fixed equilibrium position, and that, once having achieving this, will be condemned to eternal immobility. It may well be that an infinite time is necessary to bring about this state; but this circumstance alone will not be sufficient to let us to recognize in these centers of attraction a principle of evolution.
    That which has just been said about inert matter, reduced to its most simple form, is applicable to a certain extent to living matter. Let us imagine this matter, homogenous and evenly distributed about the globe, and additionally endowed with a certain energy for transformation which guides it from birth to death, regenerating itself periodically. If the surface of the earth is everywhere identical in composition, the face of nature will be different during the phase of the period which we will consider, but each of these phases will present one single and even face.

    There will be, if you will, but one flower formed from a single species. The different phases of its development will not resemble one another, but a single phase will show no diversity. If one further assumes that a single particle of this living matter, like a single root of a plant, for example, deviates from the general rule, soon variety will replace monotony, infinite growth trends emerge and Nature will become progressively diversified in the finest detail. However, even in this case, we can be certain that she will strive towards a fixed goal which, once realized, will not be subjected to change; that is because all possible growth forms will have occurred. Mathematically speaking, an infinite length of time will be necessary for their full realization; but strictly speaking, an infinite time is also necessary for a hot body placed in a cold room to take on the temperature of the room.

    Whether these two chosen examples concern inert matter or living matter, the diversifying cause is limited. In the scale that it produces its effects, it exhausts itself, loses its intensity, and is therefore not constant in the mathematical sense of the word. Differentiation, even undefined, is therefore not the same thing as evolution. Evolution, in the natural sense of the word, does not simply mean transformation, but rather a transformation towards improvement, a progressive development towards ever more perfect forms. That which constitutes progress is not easy to define, but it remains nevertheless an indisputable fact. One cannot reasonably deny that between primitive species and present species there are enormous differences, certainly from the twin perspectives of the perfection of organs and the quality of intelligence. Man is indeed superior to the "moneres" [? monkey].

     That the lines of direction along which humans develop and perfect themselves have stops and turns along the way is undeniable; but it is not less proved that the general advance of certain of these lines indicates a constant tendency to maintain the same direction and strive towards a certain goal, more or less definite, and more of less distant, and this goal seems to be a certain ideal of perfection. What perfection is, I repeat, is not easily explained. It does not consist uniquely in the acquisition of the means to the goal: the wing of the bat is, in this instance, worth that of the bird. Nevertheless, the wing of the bird is more perfect that the wing of the bat: there is in the combination of diverse elements a much higher art than that which is revealed in the layout of the wing in the chiroptera.

    No more does perfection exists in the complexity of the combined parts. Complexity without co-ordination and economy of effort is nothing but waste; and, on the other hand, sometimes simplicity surprises us by its splendour. What man does not stand confounded at a honey-comb, a spider's web, and the nest of certain birds? And yet what is not poverty relative to the tools employed! Whatever it may be, we judge perfection of an organ or a being by comparing the end to the means, the organization of parts, the diversity and the unity. The more unity is complete and the more diversity is great, the more we are inclined to say that it is perfect. From these diverse points of view, the eye is an instrument without equal. But what degrees of difference between the eyes of a snail and those of a condor!

Evolution and progress are therefore almost synonymous terms. It is true that sometimes we speak of evolution advancing or regressing. This phenomenon and this contradiction is made to produce confusion of the mind. When a diurnal animal, which has eyes much suited to recognizing and pursuing its prey, placed in new conditions, digs in the ground, or penetrates obscure caverns and which, adapting to its new way of living, ends up losing its organ of sight, can we see evolution in this event? Would it not be better to call this a revolution? The animal has without doubt made some progress, in that it acquired new methods of survival at a particular time when its ancestors were failing. In certain respects, when we deal with excavating earth, the front paws of the mole are more suitable than those of the field mouse. But are they more perfect? One can even say then that they are more perfect than the hands of man. Now if the mole ended up losing its sight which is of no use to it, should we not also view this as a perfect adaptation? Certainly not. There are phenomena of adaptation and of accommodation. These phenomena can be attributed to the force of evolution only in the sense that all beings incapable of progressive development, are incapable of adapting to a new environment, but it is necessary to use the term evolution only for progressive evolution, and coin a new word to designate this type of reversion.

    These considerations are indispensable for clarifying the issues: What can be the cause that makes certain species approach more and more closely to perfection? That which was previously said reveals the evidence that these causes cannot uniquely be adaptation. Adaption has its natural limits and loses its right to exist as soon as it has reached its goal. Even if we admit that a cause for variation is as much active in the realm of inert matter as in living nature, if we do not otherwise specify this cause, we will not be able to derive from it a gradual and progressive improvement of living things. Undoubtedly, if one assumes that the physical environments, where living individuals are found, are constantly changing, specific types will never become permanent because the adaptation will always be provisional. In this case the ability to adapt could be an unlimited cause of variation, but it is impossible to see here a cause of progress; supposing at the very least that the change in physical nature is such that it necessarily leads to progressive development. Now this would divert the difficulty, not resolve it.

    Would this cause emerge in the law of the struggle for existence and survival of the fittest? As plausible as this solution seems at first glance, after a little reflection, one does not hesitate to recognize that this law can only hasten adaptation; it is insufficient to account for unlimited and constant evolution; it is not able to justify this induction, however legitimate, that nature, sensitive and intelligent, has not yet said its final word, and that the human species, among others, are reserved the highest of destinies, the greatest of futures.

What is in fact necessary for a species to advance at a steady pace along the path towards a constant improvement? When one has fully understood the significance of the mathematical law, one sees that there will be unlimited variation if there is a continuous cause of transformation, and one would be able to find it in the incessant fluctuation of physical nature; there will be greater and greater complexity provided that the organism is subjected to a permanent pressure in this direction, and the adaptation can be considered as coming to add continuously new characters to the ancestral type; but in order for a gradual improvement, for an evolution in the special sense which we have assigned to this word, it is necessary and sufficient that among the children of the same family there will always be at least one superior to the parents; be it only one in a hundred, a thousand, in a million. If, on the contrary, the rule determines that the best of them does not resemble/come equal to [French uses the word 'vaille' here] its father, instead of a progressive evolution, degeneration will occur. The species can also remain stationary. In this way the law of adaptation turns out to be a special case of the law of evolution. The most able is, in a certain sense, the best.

    If you now ask me what this superiority consists of, I would say that in my opinion it can only be related to the mental qualities which are particular to individuals. If the most intelligent survive and reproduce their kind and if, in this relation, they surpass the authors of their existence, the species will progressively improve, and in this way I explain more and more perfect adaptations towards goals, the appearance and co-ordination of diverse motor and sensory organs, in a word, the final state of all organisms. It is therefore to intelligence that I assign the first cause of evolution. Intelligence takes as "motif" the sensory input which informs a living being whether the environment where it finds itself does or does not match its needs, and sets it to search and to locate the cause of its well-being or of its discomfort; and, as "moyen", the voluntary and free mobility, which guides it in this research and allows it, all things otherwise equal, to flee the place where the origin of its discomfort lies and to reside where it finds pleasure. It is without doubt important that he be, to a certain extent, better endowed physically, but the dominion that man of our race exercises over all created beings shows well enough that it is neither the speed of the race, nor muscular strength, nor perfection of the senses, which grants him the sceptre.

    Maybe a time will come when the earth will have no other inhabitants but man, and the animals of use to him. In our day we see wild races gradually disappearing before the civilized races and, among these [the civilized races] it is mostly the families containing the most able members which assure the best to their posterity. All are called, but few are chosen. In nature the right of the firstborn does not count; other analogous rights have strength and vigour. The future belongs to intelligence. The cause of evolution, which we have pointed out must be unlimited, also implies: that among all children within the same family there will be intellectual differences and one will necessarily dominate over the others and its parents. Here we see the first ferment. The impulse is given. The animal form, must incessantly head towards perfection in certain of its branches, until it incarnates the man of our race, producing marvels of art, of science and of industry. The universe, in its initial state, would therefore contain, at least in potential, sensibility, intelligence, liberty, as much as it would contain matter and movement.


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)


     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.
     I now return to my subject. If in the formula (a) we replace yx,z by the correct formula of Lacroix, it becomes:


Tm,p = (np-m-2t )((t+1)(t+2)-- -- -- -- ..p)/(1.2.3-- (p-m-2t)(1.2.3-- (M+ttt))) (b)


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:


Tm,p = np-2t (t+1)(t+2)-- -- -- -- p)/(1.2.3-- (p-2t)(1.2.3-- ..t)) (c)


    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:


2 Tm,p


It is now necessary to resolve by replacing for p the following equation:


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-- ..


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 --


  • (1) The fact that varieties produced by artificial selection, however divergent, are always fertile among themselves, while species supposed to have been produced naturally by an analogous process are often not mutually fertile even when slightly divergent; and

  • (2) The fact that species evidently derived from a common ancestor, and differing only in small points of marking, though not fertile with one another, are often found side by side in places where it would seem that cross-breeding must prevent any division of the ancestral species into divergent branches.


    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:--


  • (i) If this theory is true we ought to find species (incipient) mutually infertile, but not otherwise distinguishable; and

  • (ii) We ought to find that island and other isolated species which have arisen, not by limited fertility, but by geographical instead of physiological separation, are often mutually fertile even when as widely divergent as the artificial varieties of dogs and pidgeons.


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