The prevailing belief that slow and gradual, nearly invisible changes constitute the process of evolution in the animal and vegetable kingdom, did not offer a strong stimulus for experimental research. No appreciable response to any external agency was of course to be expected. Responses were supposed to be produced, but the corresponding outward changes would be too small to betray themselves to the investigator.
The direct observation of the mutations of the evening-primrose has changed the whole aspect of the problem at once. It is no longer a matter dealing with purely hypothetical conditions. Instead of the vague notions, uncertain hopes, and a priori conceptions, that have hitherto confused the investigator, methods of observation have been formulated, suitable for the attainment of definite results, the general nature of which is already known.
To my mind the real value of the discovery [687] of the mutability of the evening-primrose lies in its usefulness as a guide for further work. The view that it might be an isolated case, lying outside of the usual procedure of nature, can hardly be sustained. On such a supposition it would be far too rare to be disclosed by the investigation of a small number of plants from a limited area. Its appearance within the limited field of inquiry of a single man would have been almost a miracle.
The assumption seems justified that analogous cases will be met with, perhaps even in larger numbers, when similar methods of observation are used in the investigation of plants of other regions. The mutable condition may not be predicated of the evening-primroses alone. It must be a universal phenomenon, although affecting a small proportion of the inhabitants of any region at one time: perhaps not more than one in a hundred species, or perhaps not more than one in a thousand, or even fewer may be expected to exhibit it. The exact proportion is immaterial, because the number of mutable instances among the many thousands of species in existence must be far too large for all of them to be submitted to close scrutiny.
It is evident from the above discussion that next in importance to the discovery of the prototype of mutation is the formulation of methods [688] for bringing additional instances to light. These methods may direct effort toward two different modes of investigation. We may search for mutable plants in nature, or we may hope to induce species to become mutable by artificial methods. The first promises to yield results most quickly, but the scope of the second is much greater and it may yield results of far more importance. Indeed, if it should once become possible to bring plants to mutate at our will and perhaps even in arbitrarily chosen directions, there is no limit to the power we may finally hope to gain over nature.
What is to guide us in this new line of work? Is it the minute inspection of the features of the process in the case of the evening-primroses? Or are we to base our hopes and our methods on broader conceptions of nature's laws? Is it the systematic study of species and varieties, and the biologic inquiry into their real hereditary units? Or is the theory of descent to be our starting-point? Are we to rest our conceptions on the experience of the breeder, or is perhaps the geologic pedigree of all organic life to open to us better prospects of success?
The answer to all such questions is a very simple one. All possibilities must be considered, and no line of investigation ignored. For myself I have based my field-researches and my [689] testing of native plants on the hypothesis of unit-characters as deduced from Darwin's Pangenesis. This conception led to the expectation of two different kinds of variability, one slow and one sudden. The sudden ones known at the time were considered as sports, and seemed limited to retrograde changes, or to cases of minor importance. The idea that sudden steps might be taken as the principal method of evolution could be derived from the hypothesis of unit characters, but the evidence might be too remote for a starting point for experimental investigation.
The success of my test has given proof to the contrary. Hence the assertion that no evidence is to be considered as inadequate for the purpose under discussion. Sometime a method of discovering, or of producing, mutable plants may be found, but until this is done, all facts of whatever nature or direction must be made use of. A very slight indication may change forever the whole aspect of the problem.
The probabilities are now greatly in favor of our finding out the causes of evolution by a close scrutiny of what really happens in nature. A persistent study of the physiologic factors of this evolution is the chief condition of success. To this study field-observations may contribute as well as direct experiments, [690] microscopical investigations as well as extended pedigree-cultures. The cooperation of many workers is required to cover the field. Somewhere no doubt the desired principle lies hidden, but until it is discovered, all methods must be tried.
With this conception as the best starting point for further investigation, we may now make a brief survey of the other phase of the problem. We shall try to connect our observations on the evening-primroses with the theory of descent at large.
We start with two main facts. One is the mutability of Lamarck's primrose, and the second is the immutable condition of quite a number of other species. Among them are some of its near allies, the common and the small flowered evening-primrose, or Oenothera biennis and O. muricata.
From these facts, a very important question arises in connection with the theory of descent. Is the mutability of our evening-primroses temporary, or is it a permanent condition? A discussion of this problem will give us the means of reaching a definite idea as to the scope of our inquiries.
Let us consider the present first. If mutability is a permanent condition, it has of course no beginning, and moreover is not due to the [691] agency of external circumstances. Should this be granted for the evening-primrose, it would have to be predicated for other species found in a mutable state. Then, of course, it would be useless to investigate the causes of mutability at large, and we should have to limit ourselves to the testing of large numbers of plants in order to ascertain which are mutable and which not.
If, on the other hand, mutability is not a permanent feature, it must once have had a beginning, and this beginning itself must have had an external cause. The amount of mutability and its possible directions may be assumed to be due to internal causes. The determination of the moment at which they will become active can never be the result of internal causes. It must be assigned to some external factor, and as soon as this is discovered the way for experimental investigation is open.
In the second place we must consider the past. On the supposition of permanency all the ancestors of the evening-primrose must have been mutable. By the alternative view mutability must have been a periodic phenomenon, producing at times new qualities, and at other times leaving the plants unchanged during long successions of generations. The present mutable state must then have been preceded by an immutable [692] condition, but of course thousands of mutations must have been required to produce the evening-primroses from their most remote ancestors.
If we take the species into consideration that are not mutable at present, we may ask how we are to harmonize them with each of the two theories proposed. If mutability is permanent, it is manifest that the whole pedigree of the animal and vegetable kingdom is to be considered as built up of main mutable lines, and that the thousands of constant species can only be taken to represent lateral branches of the genealogic tree.
These lateral branches would have lost the capacity of mutating, possessed by all their ancestors. And as the principle of the hypothesis under discussion does not allow a resumption of this habit, they would be doomed to eternal constancy until they finally die out. Loss of mutability, under this conception, means loss of the capacity for all further development. Only those lines of the main pedigree which have retained this capacity would have a future; all others would die out without any chance of progression.
If, on the other hand, mutability is not permanent, but a periodic condition, all lines of the genealogic tree must be assumed to show alternatively [693] mutating and constant species. Some lines may be mutating at the present moment; others may momentarily be constant. The mutating lines will probably sooner or later revert to the inactive state, while the powers of development now dormant may then become awakened on other branches.
The view of permanency represents life as being surrounded with unavoidable death, the principle of periodicity, on the contrary, follows the idea of resurrection, granting the possibility of future progression for all living beings. At the same time it yields a more hopeful prospect for experimental inquiry.
Experience must decide between the two main theories. It demonstrates the existence of polymorphous genera, such as Draba and Viola and hundreds of others. They clearly indicate a previous state of mutability. Their systematic relation is exactly what would be expected, if they were the result of such a period. Perhaps mutability has not wholly ceased in them, but, might be found to survive in some of their members. Such very rich genera however, are not the rule, but are exceptional cases, indicating the rarity of powerful mutative changes.
On the other hand, species may remain in a state of constancy during long, apparently during indefinite, ages.
[694] Many facts plead in favor of the constancy of species. This principle has always been recognized by systematists. Temporarily the current form of the theory of natural selection has assumed species to be inconstant, ever changing and continuously being improved and adapted to the requirements of the life-conditions. The followers of the theory of descent believed that this conclusion was unavoidable, and were induced to deny the manifest fact that species are constant entities. The mutation theory gives a clew to the final combination of the two contending ideas. Reducing the changeability of the species to distinct and probably short periods, it at once explains how the stability of species perfectly agrees with the principle of descent through modification.
On the other hand, the hypothesis of mutative periods is by no means irreconcilable with the observed facts of constancy. Such casual changes can be proved by observations such as those upon the evening-primrose, but it is obvious that a disproof can never be given. The principle grants the present constancy of the vast majority of living forms, and only claims the exceptional occurrence of definite changes.
Proofs of the constancy of species have been given in different ways. The high degree of similarity of the individuals of most of our [695] species has never been denied. It is observed throughout extended localities, and during long series of years. Other proofs are afforded by those plants which have been transported to distant localities some time since, but do not exhibit any change as a result of this migration. Widely dispersed plants remain the same throughout their range, provided that they belong to a single elementary species. Many species have been introduced from America into Europe and have spread rapidly and widely. The Canadian horsetail (Erigeron canadensis), the evening-primrose and many other instances could be given. They have not developed any special European features after their introduction. Though exposed to other environmental conditions and to competition with other species, they have not succeeded in developing a new character. Such species as proved adequate to the new environment have succeeded, while those which did not have succumbed.
Much farther back is the separation of the species which now live both in arctic regions and on the summits of our highest mountaintops. If we compare the alpine flora with the arctic plants, a high degree of similarity at once strikes us. Some forms are quite identical; others are slightly different, manifestly representing elementary species of the same systematic [696] type. Still others are more distant or even belong to different genera. The latter, and even the diverging, though nearly allied, elementary species, do not yield adequate evidence in any direction.
They may as well have lived together in the long ages before the separation of the now widely distant floras, or have sprung from a common ancestor living at that time, and subsequently have changed their habits. After excluding these unreliable instances, a good number of species remain, which are quite the same in the arctic and alpine regions and on the summits of distant mountain ranges. As no transportation over such large distances can have brought them from one locality to the other, no other explanation is left than that they have been wholly constant and unchanged ever since the glacial period which separated them. Obviously they must have been subjected to widely changing conditions. The fact of their stability through all these outward changes is the best proof that the ordinary external conditions do not necessarily have an influence on specific evolution. They may have such a result in some instances, in others they obviously have not. Many arctic forms bearing the specific name of alpinus justify this conclusion. Astragalus alpinus, Phleum alpinum, Hieracium alpinum and [697] others from the northern parts of Norway may be cited as examples.
Thus Primula imperialis has been found in the Himalayas, and many other plants of the high mountains of Java, Ceylon and northern India are identical forms. Some species from the Cameroons and from Abyssinia have been found on the mountains of Madagascar. Some peculiar Australian types are represented on the summit of Kini Balu in Borneo. None of these species, of course, are found in the intervening lowlands, and the only possible explanation of their identity is the conception of a common post-glacial origin, coupled with complete stability. This stability is all the more remarkable as nearly allied but slightly divergent forms have also been reported from almost all of these localities. Other evidence is obtained by the comparison of ancient plants with their living representatives. The remains in tombs of ancient Egypt have always afforded strong support of the views of the adherents of the theory of stability, and to my mind they still do so. The cereals and fruits and even the flowers and leaves in the funeral wreaths of Rameses and Amen-Hotep are the same that are still now cultivated in Egypt. Nearly a hundred or more species have been identified. Flowers of Acacia, leaves of Mimusops, [698] petals of Nymphaea may be cited as instances, and they are as perfectly preserved as the best herbarium-specimens of the present time. The petals and stamens retain their original colors, displaying them as brightly as is consistent with their dry state.
Paleontologic evidence points to the same conclusion. Of course the remains are incomplete, and rarely adequate for a close comparison. The range of fluctuating variability should be examined first, but the test of elementary species given by their constancy from seed cannot, of course, be applied. Apart from these difficulties, paleontologists agree in recognizing the very great age of large numbers of species. It would require a too close survey of geologic facts to go into details on this point. Suffice it to say that in more recent Tertiary deposits many species have been identified with living forms. In the Miocene period especially, the similarity of the types of phanerogamic plants with their present offspring, becomes so striking that in a large number of cases specific distinctions rest in greater part on theoretical conceptions rather than on real facts. For a long time the idea prevailed that the same species could not have existed through more than one geologic period. Many distinctions founded on this belief have since had to be abandoned. [699] Species of algae belonging to the well-preserved group of the diatoms, are said to have remained unchanged from the Carboniferous period up to the present time.
Summing up the results of this very hasty survey, we may assert that species remain unchanged for indefinite periods, while at times they are in the alternative condition. Then at once they produce new forms often in large numbers, giving rise to swarms of subspecies. All facts point to the conclusion that these periods of stability and mutability alternate more or less regularly with one another. Of course a direct proof of this view cannot, as yet, be given, but this conclusion is forced upon us by a consideration of known facts bearing on the principle of constancy and evolution.
If we are right in this general conception, we may ask further, what is to be the exact place of our group of new evening-primroses in this theory? In order to give an adequate answer, we must consider the whole range of the observations from a broader point of view. First of all it is evident that the real mutating period must be assumed to be much longer than the time covered by my observations. Neither the beginning nor the end have been seen. It is quite obvious that Oenothera lamarckiana was in a mutating condition when I first [700] saw it, seventeen years ago. How long had it been so? Had it commenced to mutate after its introduction into Europe, some time ago, or was it already previously in this state? It is as yet impossible to decide this point. Perhaps the mutable state is very old, and dates from the time of the first importation of the species into Europe.
Apart from all such considerations the period of the direct observations, and the possible duration of the mutability through even more than a century, would constitute only a moment, if compared with the whole geologic time. Starting from this conception the pedigree of our mutations must be considered as only one small group. Instead of figuring a fan of mutants for each year, we must condense all the succeeding swarms into one single fan, as might be done also for Draba verna and other polymorphous species. In Oenothera the main stem is prolonged upwards beyond the fan; in the others the main stem is lacking or at least undiscernable, but this feature manifestly is only of secondary importance. We might even prefer the image of a fan, adjusted laterally to a stem, which itself is not interrupted by this branch.
On this principle two further considerations are to be discussed. First the structure of the [701] fan itself, and secondly the combination of succeeding fans into a common genealogic tree.
The composition of the fan as a whole includes more than is directly indicated by the facts concerning the birth of new species. They arise in considerable quantities, and each of them in large numbers of individuals, either in the same or in succeeding years. This multiple origin must obviously have the effect of strengthening the new types, and of heightening their chances in the struggle for life. Arising in a single specimen they would have little chance of success, since in the field among thousands of seeds perhaps one only survives and attains complete development. Thousands or at least hundreds of mutated seeds are thus required to produce one mutated individual, and then, how small are its chances of surviving! The mutations proceed in all directions, as I have pointed out in a former lecture. Some are useful, others might become so if the circumstances were accidentally changed in definite directions, or if a migration from the original locality might take place. Many others are without any real worth, or even injurious. Harmless or even slightly useless ones have been seen to maintain themselves in the field during the seventeen years of my research, as proved by Oenothera laevifolia and Oenothera [702] brevistylis. Most of the others quickly disappear.
This failure of a large part of the productions of nature deserves to be considered at some length. It may be elevated to a principle, and may be made use of to explain many difficult points of the theory of descent. If, in order to secure one good novelty, nature must produce ten or twenty or perhaps more bad ones at the same time, the possibility of improvements coming by pure chance must be granted at once. All hypotheses concerning the direct causes of adaptation at once become superfluous, and the great principle enunciated by Darwin once more reigns supreme.
In this way too, the mutation-period of the evening-primrose is to be considered as a prototype. Assuming it as such provisionally, it may aid us in arranging the facts of descent so as to allow of a deeper insight and a closer scrutiny. All swarms of elementary species are the remains of far larger initial groups. All species containing only a few subspecies may be supposed to have thrown off at the outset far more numerous lateral branches, out of which however, the greater part have been lost, being unfit for the surrounding conditions. It is the principle of the struggle for life between elementary species, followed by the survival of the [703] fittest, the law of the selection of species, which we have already laid stress upon more than once.
Our second consideration is also based upon the frequent repetition of the several mutations. Obviously a common cause must prevail. The faculty of producing nanella or lata remains the same through all the years. This faculty must be one and the same for all the hundreds of mutative productions of the same form. When and how did it originate? At the outset it must have been produced in a latent condition, and even yet it must be assumed to be continuously present in this state, and only to become active at distant intervals. But it is manifest that the original production of the characters of Oenothera gigas was a phenomenon of far greater importance than the subsequent accidental transition of this quality into the active state. Hence the conclusion that at the beginning of each series of analogous mutations there must have been one greater and more intrinsic mutation, which opened the possibility to all its successors. This was the origination of the new character itself, and it is easily seen that this incipient change is to be considered as the real one. All others are only its visible expressions.
Considering the mutative period of our evening-primrose [704] as one unit-stride section in the great genealogic tree, this period includes two nearly related, but not identical changes. One is the production of new specific characters in the latent condition, and the other is the bringing of them to light and putting them into active existence. These two main factors are consequently to be assumed in all hypothetic conceptions of previous mutative periods.
Are all mutations to be considered as limited to such periods? Of course not. Stray mutations may occur as well. Our knowledge concerning this point is inadequate for any definite statement. Swarms of variable species are easily recognized, if the remnants are not too few. But if only one or two new species have survived, how can we tell whether they have originated-alone or together with others. This difficulty is still more pronounced in regard to paleontologic facts, as the remains of geologic swarms are often found, but the absence of numerous mutations can hardly be proved in any case.
I have more than once found occasion to lay stress on the importance of a distinction between progressive and retrograde mutations in previous lectures. All improvement is, of course, by the first of these modes of evolution, but apparent losses of organs or qualities are [705] perhaps of still more universal occurrence. Progression and regression are seen to go hand in hand everywhere. No large group and probably even no genus or large species has been evolved without the joint agency of these two great principles. In the mutation-period of the evening-primroses the observed facts give direct support to this conclusion, since some of the new species proved, on closer inspection, to be retrograde varieties, while others manifestly owe their origin to progressive steps. Such steps may be small and in a wrong direction; notwithstanding this they may be due to the acquisition of a wholly new character and therefore belong to the process of progression at large.
Between them however, there is a definite contrast, which possibly is in intimate connection with the question of periodic and stray mutations. Obviously each progressive change is dependent upon the production of a new character, for whenever this is lacking, no such mutation is possible. Retrograde changes, on the other hand, do not require such elaborate preliminary work. Each character may be converted into the latent condition, and for all we know, a special preparation for this purpose is not at all necessary. It is readily granted that such special preparation may occur, because the [706] great numbers in which our dwarf variety of the Oenothera are yearly produced are suggestive of such a condition. On the other hand, the laevifolia and brevistylis mutations have not been repeated, at least not in a visible way.
From this discussion we may infer that it is quite possible that a large part of the progressive changes, and a smaller part of the retrograde mutations, are combined into groups, owing their origin to common external agencies. The periods in which such groups occur would constitute the mutative periods. Besides them the majority of the retrograde changes and some progressive steps might occur separately, each being due to some special cause. Degressive mutations, or those which arise by the return of latent qualities to activity, would of course belong with the latter group.
This assumption of a stray and isolated production of varieties is to a large degree supported by experience in horticulture. Here there are no real swarms of mutations. Sudden leaps in variability are not rare, but then they are due to hybridization. Apart from this mixture of characters, varieties as a rule appear separately, often with intervals of dozens of years, and without the least suggestion of a common cause. It is quite superfluous to go into details, as we have dealt with the horticultural [707] mutations at sufficient length on a previous occasion. Only the instance of the peloric toadflax might be recalled here, because the historic and geographic evidence, combined with the results of our pedigree-experiment, plainly show that peloric mutations are quite independent of any periodic condition. They may occur anywhere in the wide range of the toad-flax, and the capacity of repeatedly producing them has lasted some centuries at least, and is perhaps even as old as the species itself.
Leaving aside such stray mutations, we may now consider the probable constitution of the great lines of the genealogic tree of the evening primroses, and of the whole vegetable and animal kingdom at large. The idea of drawing up a pedigree for the chief groups of living organisms is originally due to Haeckel, who used this graphic method to support the Darwinian theory of descent. Of course, Haeckel's genealogic trees are of a purely hypothetic nature, and have no other purpose than to convey a clear conception of the notion of descent, and of the great lines of evolution at large. Obviously all details are subject to doubt, and many have accordingly been changed by his successors. These changes may be considered as partial improvements, and the somewhat picturesque form of Haeckel's pedigree might well be replaced by [708] more simple plans. But the changes have by no means removed the doubts, nor have they been able to supplant the general impression of distinct groups, united by broad lines. This feature is very essential, and it is easily seen to correspond with the conception of swarms, as we have deduced it from the study of the lesser groups.
Genealogic trees are the result of comparative studies; they are far removed from the results of experimental inquiry concerning the origin of species. What are the links which bind them together? Obviously they must be sought in the mutative periods, which have immediately preceded the present one. In the case of the evening-primrose the systematic arrangement of the allied species readily guides us in the delimitations of such periods. For manifestly the species of the large genus of Oenothera are grouped in swarms, the youngest or most recent of which we have under observation. Its immediate predecessor must have been the subgenus Onagra, which is considered by some authors as consisting of a single systematic species, Oenothera biennis. Its multifarious forms point to a common origin, not only morphologically but also historically. Following this line backward or downward we reach another apparent mutation-period, which includes the origin of [709] the group called Oenothera, with a large number of species of the same general type as the Onagra-forms, Still farther downward comes the old genus Oenothera itself, with numerous subgenera diverging in sundry characters and directions.
Proceeding still farther we might easily construct a main stem with numerous succeeding fans of lateral branches, and thus reach, from our new empirical point of view, the theoretical conclusion already formulated.
Paleontologic facts readily agree with this conception. The swarms of species and varieties are found to succeed one another like so many stories. The same images are repeated, and the single stories seem to be connected by the main stems, which in each tier produce the whole number of allied forms. Only a few prevailing lines are prolonged through numerous geologic periods; the vast majority of the lateral branches are limited each to its own storey. It is simply the extension of the pedigree of the evening-primroses backward through ages, with the same construction and the same leading features. There can be no doubt that we are quite justified in assuming that evolution has followed the same general laws through the whole duration of life on earth. Only a moment of their lifetime is disclosed to us, but it [710] is quite sufficient to enable us to discern the laws and to conjecture the outlines of the whole scheme of evolution.
A grave objection which has, often, and from the very outset, been urged against Darwin's conception of very slow and nearly imperceptible changes, is the enormously long time required. If evolution does not proceed any faster than what we can see at present, and if the process must be assumed to have gone on in the same slow manner always, thousands of millions of years would have been needed to develop the higher types of animals and plants from their earliest ancestors.
Now it is not at all probable that the duration of life on earth includes such an incredibly long time. Quite on the contrary the lifetime of the earth seems to be limited to a few millions of years. The researches of Lord Kelvin and other eminent physicists seem to leave no doubt on this point. Of course all estimates of this kind are only vague and approximate, but for our present purposes they may be considered as sufficiently exact.
In a paper published in 1862 Sir William Thomson (now Lord Kelvin) first endeavored to show that great limitation had to be put upon the enormous demand for time made by Lyell, Darwin and other biologists. From a consideration [711] of the secular cooling of the earth, as deduced from the increasing temperature in deep mines, he concluded that the entire age of the earth must have been more than twenty and less than forty millions of years, and probably much nearer twenty than forty. His views have been much criticised by other physicists, but in the main they have gained an ever-increasing support in the way of evidence. New mines of greater depth have been bored, and their temperatures have proved that the figures of Lord Kelvin are strikingly near the truth. George Darwin has calculated that the separation of the moon from the earth must have taken place some fifty-six millions of years ago. Geikie has estimated the existence of the solid crust of the earth at the most as a hundred million years. The first appearance of the crust must soon have been succeeded by the formation of the seas, and a long time does not seem to have been required to cool the seas to such a degree that life became possible. It is very probable that life originally commenced in the great seas, and that the forms which are now usually included in the plankton or floating-life included the very first living beings. According to Brooks, life must have existed in this floating condition during long primeval epochs, and evolved nearly all the main branches of the animal and vegetable kingdom [712] before sinking to the bottom of the sea, and later producing the vast number of diverse forms which now adorn the sea and land.
All these evolutions, however, must have been very rapid, especially at the beginning, and together cannot have taken more time than the figures given above.
The agency of the larger streams, and the deposits which they bring into the seas, afford further evidence. The amount of dissolved salts, especially of sodium chloride, has been made the subject of a calculation by Joly, and the amount of lime has been estimated by Eugene Dubois. Joly found fifty-five and Dubois thirty-six millions of years as the probable duration of the age of the rivers, and both figures correspond to the above dates as closely as might be expected from the discussion of evidence so very incomplete and limited.
All in all it seems evident that the duration of life does not comply with the demands of the conception of very slow and continuous evolution. Now it is easily seen, that the idea of successive mutations is quite independent of this difficulty. Even assuming that some thousands of characters must have been acquired in order to produce the higher animals and plants of the present time, no valid objection is raised. The demands of the biologists and the results of [713] the physicists are harmonized on the ground of the theory of mutation.
The steps may be surmised to have never been essentially larger than in the mutations now going on under our eyes, and some thousands of them may be estimated as sufficient to account for the entire organization of the higher forms. Granting between twenty and forty millions of years since the beginning of life, the intervals between two successive mutations may have been centuries and even thousands of years. As yet there has been no objection cited against this assumption, and hence we see that the lack of harmony between the demands of biologists and the results of the physicists disappears in the light of the theory of mutation.
Summing up the results of this discussion, we may justifiably assert that the conclusions derived from the observations and experiments made with evening-primroses and other plants in the main agree satisfactorily with the inferences drawn from paleontologic, geologic and systematic evidence. Obviously these experiments are wonderfully supported by the whole of our knowledge concerning evolution. For this reason the laws discovered in the experimental garden may be considered of great importance, and they may guide us in our further inquiries. Without doubt many minor [714] points are in need of correction and elaboration, but such improvements of our knowledge will gradually increase our means of discovering new instances and, new proofs.
The conception of mutation periods producing swarms of species from time to time, among which only a few have a chance of survival, promises to become the basis for speculative pedigree-diagrams, as well as for experimental investigations.

[715]

LECTURE XXV

GENERAL LAWS OF FLUCTUATION

The principle of unit-characters and of elementary species leads at once to the recognition of two kinds of variability. The changes of wider amplitude consist of the acquisition of new units, or the loss of already existing ones. The lesser variations are due to the degree of activity of the units themselves.
Facts illustrative of these distinctions were almost wholly lacking at the time of the first publication of Darwin's theories. It was a bold conception to point out the necessity for such distinction on purely theoretical grounds. Of course some sports were well known and fluctuations were evident, but no exact analysis of the details was possible, a fact that was of great importance in the demonstration of the theory of descent. The lack of more definite knowledge upon this matter was keenly felt by Darwin, [716] and exercised much influence upon his views at various times.
Quetelet's famous discovery of the law of fluctuating variability changed the entire situation and cleared up many difficulties. While a clear conception of fluctuations was thus gained, mutations were excluded from consideration, being considered as very rare, or non-existent. They seemed wholly superfluous for the theory of descent, and very little importance was attached to their study. Current scientific belief in the matter has changed only in recent years. Mendel's law of varietal hybrids is based upon the principle of unit-characters, and the validity of this conception has thus been brought home to many investigators.
A study of fluctuating or individual variability, as it was formerly called, is now carried on chiefly by mathematical methods. It is not my purpose to go into details, as it would require a separate course of lectures. I shall consider the limits between fluctuation and mutation only, and attempt to set forth an adequate idea of the principles of the first as far as they touch these limits. The mathematical treatment of the facts is no doubt of very great value, but the violent discussions now going on between mathematicians such as Pearson, Kapteyn and others should warn biologists to abstain [717] from the use of methods which are not necessary for the furtherance of experimental work.
Fortunately, Quetelet's law is a very clear and simple one, and quite sufficient for our considerations. It claims that for biologic phenomena the deviations from the average comply with the same laws as the deviations from the average in any other case, if ruled by chance only. The meaning of this assertion will become clear by a further discussion of the facts. First of all, fluctuating variability is an almost universal phenomenon. Every organ and every quality may exhibit it. Some are very variable, while others seem quite constant. Shape and size vary almost indefinitely, and the chemical composition is subject to the same law, as is well known for the amount of sugar in sugar-beets. Numbers are of course less liable to changes, but the numbers of the rays of umbels, or ray-florets in the composites, of pairs of blades in pinnate leaves, and even of stamens and carpels are known to be often exceedingly variable. The smaller numbers however, are more constant, and deviations from the quinate structure of flowers are rare. Complicated structures are generally capable of only slight deviations.
From a broad point of view, fluctuating variability [718] falls under two heads. They obey quite the same laws and are therefore easily confused, but with respect to questions of heredity they should be carefully separated. They are designated by the terms individual and partial fluctuation. Individual variability indicates the differences between individuals, while partial variability is limited to the deviations shown by the parts of one organism from the average structure. The same qualities in some cases vary individually and in others partially. Even stature, which is as markedly individual for annual and biennial plants as it is for man, becomes partially variant in the case of perennial herbs with numbers of stems. Often a character is only developed once in the whole course of evolution, as for instance, the degree of connation of the seed-leaves in tricotyls and in numerous cases it is impossible to tell whether a character is individual or partial. Consequently such minute details are generally considered to have no real importance for the hereditary transmission of the character under discussion.
Fluctuations are observed to take place only in two directions. The quality may increase or decrease, but is not seen to vary in any other way. This rule is now widely established by numerous investigations, and is fundamental to [719] the whole method of statistical investigation. It is equally important for the discussion of the contrast between fluctuations and mutations, and for the appreciation of their part in the general progress of organization. Mutations are going on in all directions, producing, if they are progressive, something quite new every time. Fluctuations are limited to increase and decrease of what is already available. They may produce plants with higher stems, more petals in the flowers, larger and more palatable fruits, but obviously the first petal and the first berry, cannot have originated by the simple increase of some older quality. Intermediates may be found, and they may mark the limit, but the demonstration of the absence of a limit is quite another question. It would require the two extremes to be shown to belong to one unit, complying with the simple law of Quetelet.
Nourishment is the potent factor of fluctuating variability. Of course in thousands of cases our knowledge is not sufficient to allow us to analyze this relation, and a number of phases of the phenomenon have been discovered only quite recently. But the fact itself is thoroughly manifest, and its appreciation is as old as horticultural science. Knight, who lived at the beginning of the last century, has laid great stress upon it, and it has since influenced practice in a [720] large measure. Moreover, Knight pointed out more than once that it is the amount of nourishment, not the quality of the various factors, that exercises the determinative influence. Nourishment is to be taken in the widest sense of the word, including all favorable and injurious elements. Light and temperature, soil and space, water and salts are equally active, and it is the harmonious cooperation of them all that rules growth.
We treated this important question at some length, when dealing with the anomalies of the opium-poppies, consisting of the conversion of stamens into supernumerary pistils. The dependency upon external influences which this change exhibited is quite the same as that shown by fluctuating variability at large. We inquired into the influence of good and bad soil, of sunlight and moisture and of other concurrent factors. Especial emphasis was laid upon the great differences to which the various individuals of the same lot may be exposed, if moisture and manure differ on different portions of the same bed in a way unavoidable even by the most careful preparation. Some seeds germinate on moist and rich spots, while their neighbors are impeded by local dryness, or by distance from manure. Some come to light on a sunny day, and increase their first leaves rapidly, while on [721] the following day the weather may be unfavorable and greatly retard growth. The individual differences seem to be due, at least in a very great measure, to such apparent trifles.
On the other hand partial differences are often manifestly due to similar causes. Considering the various stems of plants, which multiply themselves by runners or by buds on the roots, the assertion is in no need of further proof. The same holds good for all cases of artificial multiplication by cuttings, or by other vegetative methods. But even if we limit ourselves to the leaves of a single tree, or the branches of a shrub, or the flowers on a plant, the same rule prevails. The development of the leaves is dependent on their position, whether inserted on strong or weak branches, exposed to more or less light, or nourished by strong or weak roots. The vigor of the axillary buds and of the branches which they may produce is dependent upon the growth and activity of the leaves to which the buds are axillary.
This dependency on local nutrition leads to the general law of periodicity, which, broadly speaking, governs the occurrence of the fluctuating deviations of the organs. This law of periodicity involves the general principle that every axis, as a rule, increases in strength when [722] growing, but sooner or later reaches a maximum and may afterwards decrease.
This periodic augmentation and declination is often boldly manifest, though in other cases it may be hidden by the effect of alternate influences. Pinnate leaves generally have their lower blades smaller than the upper ones, the longest being seen sometimes near the apex and sometimes at a distance from it. Branches bearing their leaves in two rows often afford quite as obvious examples, and shoots in general comply with the same rule. Germinating plants are very easy of observation on this point. When they are very weak they produce only small leaves. But their strength gradually increases and the subsequent organs reach fuller dimensions until the maximum is attained. The phenomenon is so common that its importance is usually overlooked. It should be considered as only one instance of a rule, which holds good for all stems and all branches, and which is everywhere dependent on the relation of growth to nutrition.
The rule of periodicity not only affects the size of the organs, but also their number, whenever these are largely variable. Umbellate plants have numerous rays on the umbels of strong stems, but the number is seen to decrease and to become very small on the weakest lateral [723] branches. The same holds good for the number of ray-florets in the flower-heads of the composites, even for the number of stigmas on the ovaries of the poppies, which on weak branches may be reduced to as few as three or four. Many other instances could be given.
One of the best authenticated cases is the dependency of partial fluctuation on the season and on the weather. Flowers decline when the season comes to an end, become smaller and less brightly colored. The number of ray-florets in the flower-heads is seen to decrease towards the fall. Extremes become rarer, and often the deviations from the average seem nearly to disappear. Double flowers comply with this rule very closely, and many other cases will easily occur to any student of nature.
Of course, the relation to nourishment is different for individual and partial fluctuations. Concerning the first, the period of development of the germ within the seed is decisive. Even the sexual cells may be in widely different conditions at the moment of fusion, and perhaps this state of the sexual cells includes the whole matter of the decision for the average characters of the new individual. Partial fluctuation commences as soon as the leaves and buds begin to form, and all later changes in nutrition can only cause partial differences. All leaves, [724] buds, branches, and flowers must come under the influence of external conditions during the juvenile period, and so are liable to attain a development determined in part by the action of these factors.
Before leaving these general considerations, we must direct our attention to the question of utility. Obviously, fluctuating variability is a very useful contrivance, in many cases at least. It appears all the more so, as its relation to nutrition becomes manifest. Here two aspects are intimately combined. More nutrient matter produces larger leaves and these are in their turn more fit to profit by the abundance of nourishment. So it is with the number of flowers and flower-groups, and even with the numbers of their constituent organs. Better nourishment produces more of them, and thereby makes the plant adequate to make a fuller use of the available nutrient substances. Without fluctuation such an adjustment would hardly be possible, and from all our notions of usefulness in nature, we therefore must recognize the efficiency of this form of variability.
In other respects the fluctuations often strike us as quite useless or even as injurious. The numbers of stamens, or of carpels are dependent on nutrition, but their fluctuation is not known to have any attraction for the visiting insects.
[725] If the deviations become greater, they might even become detrimental. The flowers of the St. Johnswort, or Hypericum perforatum, usually have five petals, but the number varies from three to eight or more. Bees could hardly be misled by such deviations. The carpels of buttercups and columbines, the cells in the capsules of cotton and many other plants are variable in number. The number of seeds is thereby regulated in accordance with the available nourishment, but whether any other useful purpose is served, remains an open question. Variations in the honey-guides or in the pattern of color-designs might easily become injurious by deceiving insects, and such instances as the great variability of the spots on the corolla of some cultivated species of monkey-flowers, for instance, the Mimulus quinquevulnerus, could hardly be expected to occur in wild plants. For here the dark brown spots vary between nearly complete deficiency up to such predominancy as almost to hide the pale yellow ground-color.
After this hasty survey of the causes of fluctuating variability, we now come to a discussion of Quetelet's law. It asserts that the deviations from the average obey the law of probability. They behave as if they were dependent on chance only.
Everyone knows that the law of Quetelet can [726] be demonstrated the most readily by placing a sufficient number of adult men in a row, arranging them according to their size. The line passing over their heads proves to be identical with that given by the law of probability. Quite in the same way, stems and branches, leaves and petals and even fruits can be arranged, and they will in the main exhibit the same line of variability. Such groups are very striking, and at the first glance show that the large majority of the specimens deviate from the mean only to a very small extent. Wider deviations are far more rare, and their number lessens, the greater the deviation, as is shown by the curvature of the line. It is almost straight and horizontal in the middle portion, while at the ends it rapidly declines, going sharply downward at one extreme and upward at the other.
It is obvious however, that in these groups the leaves and other organs could conveniently be replaced by simple lines, indicating their size. The result would be quite the same, and the lines could be placed at arbitrary, but equal distances. Or the sizes could be expressed by figures, the compliance of which with the general law could be demonstrated by simple methods of calculation. In this manner the variability of different organs can easily be compared. Another method of demonstration consists in [727] grouping the deviations into previously fixed divisions. For this purpose the variations are measured by standard units, and all the instances that fall between two limits are considered to constitute one group. Seeds and small fruits, berries and many other organs may conveniently be dealt with in this way. As an example we take ordinary beans and select them according to their size. This can be done in different ways. On a small piece of board a long wedge-shaped slit is made, into which seeds are pushed as far as possible. The margin of the wedge is calibrated in such a manner that the figures indicate the width of the wedge at the corresponding place. By this device the figure up to which a bean is pushed at once shows its length. Fractions of millimeters are neglected, and the beans, after having been measured, are thrown into cylindrical glasses of the same width, each glass receiving only beans of equal length. It is clear that by this method the height to which beans fill the glasses is approximately a measure of their number. If now the glasses are put in a row in the proper sequence, they at once exhibit the shape of a line which corresponds to the law of chance. In this case however, the line is drawn in a different manner from the first. It is to be pointed out that the glasses may be replaced by lines indicating [728] the height of their contents, and that, in order to reach a more easy and correct statement, the length of the lines may simply be made proportionate to the number of the beans in each glass. If such lines are erected on a common base and at equal distances, the line which unites their upper ends will be the expression of the fluctuating variability of the character under discussion.
The same inquiry may be made with other seeds, with fruits, or other organs. It is quite superfluous to arrange the objects themselves, and it is sufficient to arrange the figures indicating their value. In order to do this a basal line is divided into equal parts, the demarcations corresponding to the standard-units chosen for the test. The observed values are then written above this line, each finding its place between the two demarcations, which include its value. It is very interesting and stimulating to construct such a group. The first figures may fall here and there, but very soon the vertical rows on the middle part of the basal line begin to increase. Sometimes ten or twenty measurements will suffice to make the line of chance appear, but often indentations will remain. With the increasing number of the observations the irregularities gradually [729] disappear, and the line becomes smoother and more uniformly curved.
This method of arranging the figures directly on a basal line is very convenient, whenever observations are made in the field or garden. Very few instances need be recorded to obtain an appreciation of the mean value, and to show what may be expected from a continuance of the test. The method is so simple and so striking, and so wholly independent of any mathematical development that it should be applied in all cases in which it is desired to ascertain the average value of any organ, and the measure of the attendant deviations.
I cite an instance, secured by counting the ray-florets on the flower-heads of the corn-marigold or Chrysanthemum segetum. It was that, by which I was enabled to select the plant, which afterwards showed the first signs of a double head. I noted them in this way;

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Of course the figures might be replaced in this work by equidistant dots or by lines, but experience teaches that the chance of making mistakes is noticeably lessened by writing down [730] the figures themselves. Whenever decimals are made use of it is obviously the best plan to keep the figures themselves. For afterwards it often becomes necessary to arrange them according to a somewhat different standard.
Uniting the heads of the vertical rows of figures by a line, the form corresponding to Quetelet's law is easily seen. In the main it is always the same as the line shown by the measurements of beans and seeds. It proves a dense crowding of the single instances around the average, and on both sides of the mass of the observations, a few wide deviations. These become more rare in proportion to the amount of their divergency. On both sides of the average the line begins by falling very rapidly, but then bends slowly so as to assume a nearly horizontal direction. It reaches the basal line only beyond the extreme instances.
It is quite evident that all qualities, which can be expressed by figures, may be treated in this way. First, of all the organs occurring in varying numbers, as for instance the ray-florets of composites, the rays of umbels, the blades of pinnate and palmate leaves, the numbers of veins, etc., are easily shown to comply with the same general rule. Likewise the amount of chemical substances can be expressed in percentage numbers, as is done on a large [731] scale with sugar in beets and sugar-cane, with starch in potatoes and in other instances. These figures are also found to follow the same law.
All qualities which are seen to increase and to decrease may be dealt with in the same manner, if a standard unit for their measurement can be fixed. Even the colors of flowers may not escape our inquiry.
If we now compare the lines, compiled from the most divergent cases, they will be found to exhibit the same features in the main. Ordinarily the curve is symmetrical, the line sloping down on both sides after the same manner. But it is not at all rare that the inclination is steep on one side and gradual on the other. This is noticeably the case if the observations relate to numbers, the average of which is near zero. Here of course the allowance for variation is only small on one side, while it may increase with out distinct limits on the alternate slope. So it is for instance with the numbers of ray-florets in the example given on p. 729. Such divergent cases, however, are to be considered as exceptions to the rule, due to some unknown cause.
Heretofore we have discussed the empirical side of the problem only. For the purpose of experimental study of questions of heredity this is ordinarily quite sufficient. The inquiry [732] into the phenomenon of regression, or of the relation of the degree of deviation of the progeny to that of their parents, and the selection of extreme instances for multiplication are obviously independent of mathematical considerations. On the other hand an important inquiry lies in the statistical treatment of these phenomena, and such treatment requires the use of mathematical methods.
Statistics however, are not included in the object of these lectures, and therefore I shall refrain from an explanation of the method of their preparation and limit myself to a general comparison of the observed lines with the law of chance. Before going into the details, it should be repeated once more that the empirical result is quite the same for individual and for partial fluctuations. As a rule, the latter occur in far greater number, and are thus more easily investigated, but individual or personal averages have also been studied.
Newton discovered that the law of chance can be expressed by very simple mathematical calculations. Without going into details, we may at once state that these calculations are based upon his binomium. If the form (a + b) is calculated for some value of the exponent, and if the values of the coefficients after development are alone considered, they yield the basis [733] for the construction of what is called the line or curve of probability. For this construction the coefficients are used as ordinates, the length of which is to be made proportionate to their value. If this is done, and the ordinates are arranged at equal distances, the line which unites their summits is the desired curve. At first glance it exhibits a form quite analogous to the curves of fluctuating variability, obtained by the measurements of beans and in other instances. Both lines are symmetrical and slope rapidly down in the region of the average, while with increasing distance they gradually lose their steep inclination, becoming nearly parallel to the base at their termination.
This similarity between such empirical and theoretical lines is in itself an empirical fact. The causes of chance are assumed to be innumerable, and the whole calculation is based on this assumption. The causes of the fluctuations of biological phenomena have not as yet been critically examined to such an extent as to allow of definite conceptions. The term nourishment manifestly includes quite a number of separate factors, as light, space, temperature, moisture, the physical and chemical conditions of the soil and the changes of the weather. Without doubt the single factors are very numerous, but whether they are numerous enough to be treated [734] as innumerable, and thereby to explain the laws of fluctuations, remains uncertain. Of course the easiest way is to assume that they combine in the same manner as the causes of chance, and that this is the ground of the similarity of the curves. On the other hand, it is manifestly of the highest importance to inquire into the part the several factors play in the determination of the curves. It is not at all improbable that some of them have a larger influence on individual, and others on partial, fluctuations. If this were the case, their importance with respect to questions of heredity might be widely different. In the present state of our knowledge the fluctuation-curves do not contribute in any large measure to an elucidation of the causes. Where these are obvious, they are so without statistics, exactly as they were, previous to Quetelet's discovery.
In behalf of a large number of questions concerning heredity and selection, it is very desirable to have a somewhat closer knowledge of these curves. Therefore I shall try to point out their more essential features, as far as this can be done without mathematical calculations.
At a first glance three points strike us, the average or the summit of the curve, and the extremes. If the general shape is once denoted by the results of observations or by the coefficients [735] of the binomium, all further details seem to depend upon them. In respect to the average this is no doubt the case; it is an empirical value without need of any further discussion. The more the number of the observations increases, the more assured and the more correct is this mean value, but generally it is the same for smaller and for larger groups of observations.
This however, is not the case with the extremes. It is quite evident that small groups have a chance of containing neither of them. The more the number of the observations increases, the larger is the chance of extremes. As a rule, and excluding exceptional cases, the extreme deviations will increase in proportion to the number of cases examined. In a hundred thousand beans the smallest one and the largest one may be expected to differ more widely from one another than in a few hundred beans of the same sample. Hence the conclusion that extremes are not a safe criterion for the discussion of the curves, and not at all adequate for calculations, which must be based upon more definite values.
A real standard is afforded by the steepness of the slope. This may be unequal on the two sides of one curve, and likewise it may differ for different cases. This steepness is usually measured by means of a point on the half curve and [736 ] for this purpose a point is chosen which lies exactly half way between the average and the extreme. Not however half way with respect to the amplitude of the extreme deviation, for on this ground it would partake of the uncertainty of the extreme itself. It is the point on the curve which is surpassed by half the number, and not reached by the other half of the number of the observations included in the half of the curve. This point corresponds to the important value called the probable error, and was designated by Galton as the quartile. For it is evident that the average and the two quartiles divide the whole of the observations into four equal parts.
Choosing the quartiles as the basis for calculations we are independent of all the secondary causes of error, which necessarily are inherent in the extremes. At a casual examination, or for demonstrative purposes, the extremes may be prominent, but for all further considerations the quartiles are the real values upon which to rest calculations.
Moreover if the agreement with the law of probability is once conceded, the whole curve is defined by the average and the quartiles, and the result of hundreds of measurements or countings may be summed up in three, or, in [737] the case of symmetrical curves, perhaps in two figures.
Also in comparing different curves with one another, the quartiles are of great importance. Whenever an empirical fluctuation-curve is to be compared with the theoretical form, or when two or more cases of variability are to be considered under one head, the lines are to be drawn on the same base. It is manifest that the averages must be brought upon the same ordinate, but as to the steepness of the line, much depends on the manner of plotting. Here we must remember that the mutual distance of the ordinates has been a wholly arbitrary one in all our previous considerations. And so it is, as long as only one curve is considered at a time. But as soon as two are to be compared, it is obvious that free choice is no longer allowed. The comparison must be made on a common basis, and to this effect the quartiles must be brought together. They are to lie on the same ordinates. If this is done, each division of the base corresponds to the same proportionate number of individuals, and a complete comparison is made possible.
On the ground of such a comparison we may thus assert that, fluctuations, however different the organs or qualities observed, are the same whenever their curves are seen to overlap one [738] another. Furthermore, whenever an empirical curve agrees in this manner with the theoretical one, the fluctuation complies with Quetelet's law, and may be ascribed to quite ordinary and universal causes. But if it seems to diverge from this line, the cause of this divergence should be inquired into.
Such abnormal curves occur from time to time, but are rare. Unsymmetrical instances have already been alluded to, and seem to be quite frequent. Another deviation from the rule is the presence of more than one summit. This case falls under two headings. If the ray florets of a composite are counted, and the figures brought into a curve, a prominent summit usually corresponds to the average. But next to this, and on both sides, smaller summits are to be seen. On a close inspection these summits are observed to fall on the same ordinates, on which, in the case of allied species, the main apex lies. The specific character of one form is thus repeated as a secondary character on an allied species. Ludwig discovered that these secondary summits comply with the rule discovered by Braun and Schimper, stating the relation of the subsequent figures of the series. This series gives the terms of the disposition of leaves in general, and of the bracts and flowers on the composite flower [739] heads in our particular case. It is the series to which we have already alluded when dealing with the arrangement of the leaves on the twisted teasels. It commences with 1 and 2 and each following figure is equal to the sum of its two precedents. The most common figures are 3, 5, 8, 13, 18, 21, higher cases seldom coming under observation. Now the secondary summits of the ray-curves of the composites are seen to agree, as a rule, with these figures. Other instances could readily be given.
Our second heading includes those cases which exhibit two summits of equal or nearly equal height. Such cases occur when different races are mixed, each retaining its own average and its own curve-summit. We have already demonstrated such a case when dealing with the origin of our double corn-chrysanthemum. The wild species culminates with 13 rays, and the grandiflorum variety with 21. Often the latter is found to be impure, being mixed with the typical species to a varying extent. This is not easily ascertained by a casual inspection of the cultures, but the true condition will promptly betray itself, if curves are constructed. In this way curves may in many instances be made use of to discover mixed races. Double curves may also result from the investigation [740] of true double races, or ever-sporting varieties. The striped snapdragon shows a curve of its stripes with two summits, one corresponding to the average striped flowers, and the other to the pure red ones. Such cases may be discovered by means of curves, but the constituents cannot be separated by culture-experiments.
A curious peculiarity is afforded by half curves. The number of petals is often seen to vary only in one direction from what should be expected to be the mean condition. With buttercups and brambles and many others there is only an increase above the typical five; quaternate flowers are wanting or at least are very rare. With weigelias and many others the number of the tips of the corolla varies downwards, going from five to four and three. Hundreds of flowers show the typical five, and determine the summit of the curve. This drops down on one side only, indicating unilateral variability, which in many cases is due to a very intimate connection of a concealed secondary summit and the main one. In the case of the bulbous buttercup, Ranunculus bulbosus, I have succeeded in isolating this secondary summit, although not in a separate variety, but only in a form corresponding to the type of ever-sporting varieties.
[741] Recapitulating the results of this too condensed discussion, we may state that fluctuations are linear, being limited to an increase and to a decrease of the characters. These changes are mainly due to differences in nourishment, either of the whole organism or of its parts. In the first case, the deviations from the mean are called individual; they are of great importance for the hereditary characters of the offspring. In the second case the deviations are far more universal and far more striking, but of lesser importance. They are called partial fluctuations.
All these fluctuations comply, in the main, with the law of probability, and behave as if their causes were influenced only by chance.

[742]

LECTURE XXVI

ASEXUAL MULTIPLICATION OF EXTREMES