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 a.s.sertion 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 a.n.a.lyze 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 s.p.a.ce, 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 r.e.t.a.r.d 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 a.s.sertion 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 s.e.xual cells may be in widely different conditions at the moment of fusion, and perhaps this state of the s.e.xual 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 const.i.tuent 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 _Hyperic.u.m 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 b.u.t.tercups 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 a.s.serts 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 pa.s.sing 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 const.i.tute 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 gla.s.ses of the same width, each gla.s.s receiving only beans of equal length. It is clear that by this method the height to which beans fill the gla.s.ses is approximately a measure of their number. If now the gla.s.ses 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 gla.s.ses 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 gla.s.s. 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;

47 47 52 41 54 68 44 50 62 75 36 45 58 65 72 __ 99

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 ma.s.s 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 a.s.sume 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 a.n.a.logous 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 a.s.sumed to be innumerable, and the whole calculation is based on this a.s.sumption. 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, s.p.a.ce, 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 a.s.sume 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 a.s.sured 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 surpa.s.sed 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.