Chapter 41

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

Fluctuating variability may be regarded from two different points of view. The multiformity of a bed of flowers is often a desirable feature, and all means which widen the range of fluctuation are therefore used to enhance this feature, and variability affords specimens, which surpass the average, by yielding a better or larger product.

In the case of fruits and other cultivated forms, it is of course profitable to propagate from the better specimens only, and if possible only from the very best. Obviously the best are the extremes of the whole range of diverging forms, and moreover the extremes on one side of the group. Almost always the best for practical purposes is that in which some quality is strengthened. Cases occur however, in which it is desirable to diminish an injurious peculiarity as far as possible, and in these instances the opposite extreme is the most profitable one.

These considerations lead us to a discussion [743] of the results of the choice of extremes, which it may be easily seen is a matter of the greatest practical importance. This choice is generally designated as selection, but as with most of the terms in the domain of variability, the word selection has come to have more than one meaning. Facts have accumulated enormously since the time of Darwin, a more thorough knowledge has brought about distinctions, and divisions at a rapidly increasing rate, with which terminology has not kept pace. Selection includes all kinds of choice. Darwin distinguished between natural and artificial selection, but proper subdivisions of these conceptions are needed.

In the fourth lecture we dealt with this same question, and saw that selection must, in the first place, make a choice between the elementary species of the same systematic form. This selection of species or species-selection was the work of Le Couteur and Patrick Shirreff, and is now in general use in practice where it has received the name of variety-testing. This clear and unequivocal term however, can hardly be included under the head of natural selection. The poetic terminology of selection by nature has already brought about many difficulties that should be avoided in the future. On the other hand, the designation of the process as a natural [744] selection of species complies as closely as possible with existing terminology, and does not seem liable to any misunderstanding.

It is a selection between species. Opposed to it is the selection within the species. Manifestly the first should precede the second, and if this sequence is not conscientiously followed it will result in confusion. This is evident when it is considered that fluctuations can only appear with their pure and normal type in pure strains, and that each admixture of other units is liable to be shown by the form of the curves. More over, selection chooses single individuals, and a single plant, if it is not a hybrid, can scarcely pertain to two different species. The first choice therefore is apt to make the strain pure.

In contrasting selection between species with that within the species, of course elementary species are meant, including varieties. The terms would be of no consequence if only rightly understood. For the sake of clearness we might designate the last named process with the term of intra-specific selection, and it is obvious that this term is applicable both to natural and to artificial selection.

Having previously dealt with species-selection at sufficient length, we may now confine ourselves to the consideration of the intra-specific [745] selection process. In practice it is of secondary importance, and in nature it takes a very subordinate position. For this reason it will be best to confine further discussions to the experience of the breeders.

Two different ways are open to make fluctuating variability profitable. Both consist in the multiplication of the chosen extremes, and this increase may be attained in a vegetative manner, or by the use of seeds. Asexual and sexual propagation are different in many respects, and so they are also in the domain of variability.

In order to obtain a clear comprehension of this difference, it is necessary to start from the distinction between individual and partial fluctuations, as given in the last lecture. This distinction may be discussed more understandingly if the causes of the variability are taken into consideration. We have dealt with them at some length, and are now aware that inner conditions only, determine averages, while some fluctuation around them is allowable, as influenced by external conditions. These outward influences act throughout life. At the very first they impress their stamp on the whole organism, and incite a lasting change in distinct directions. This is the period of the development of the germ within the seed; it begins with the fusion of the sexual cells, and each of them may be influenced [746] to a noticeable degree before this union. This is the period of the determination of individual variability. As soon as ramifications begin, the external conditions act separately on every part, influencing some to a greater and others to a lesser degree. Here we have the beginning of partial variability. At the outset all parts may be affected in the same way and in the same measure, but the chances of such an agreement, of course, rapidly diminish. This is partly due to differences in exposure, but mainly to alterations of the sensibility of the organs themselves.

It is difficult to gain a clear conception of the contrast between individual and partial variability, and neither is it easy to appreciate their cooperation rightly. Perhaps the best way is to consider their activity as a gradual narrowing of possibilities. At the outset the plant may develop its qualities in any measure, nothing being as yet fixed. Gradually however, the development takes a definite direction, for better or for worse. Is a direction once taken, then it becomes the average, around which the remaining possibilities are grouped. The plant or the organ goes on in this way, until finally it reaches maturity with one of the thousands of degrees of development, between which at the beginning it had a free choice.

[747] Putting this discussion in other terms, we find every individual and every organ in the adult state corresponding with a single ordinate of the curve. The curve indicates the range of possibilities, the ordinate shows the choice that has been made. Now it is clear at once that this choice has not been made suddenly but gradually. Halfway of the development, the choice is halfway determined, but the other half is still undefined. The first half is the same for all the organs of the plant, and is therefore termed individual; the second differs in the separate members, and consequently is known as partial. Which of the two halves is the greater and which the lesser, of course depends on the cases considered.

Finally we may describe a single example, the length of the capsules of the evening-primrose. This is highly variable, the longest reaching more than twice the length of the smallest. Many capsules are borne on the same spike, and they are easily seen to be of unequal size. They vary according to their position, the size diminishing in the main from the base upwards, especially on the higher parts. Likewise the fruits of weaker lateral branches are smaller. Curves are easily made by measuring a few hundred capsules from corresponding parts of different plants, or even by limiting the [748] inquiry to a single individual. These curves give the partial variability, and are found to comply with Quetelet's law.