This consideration can be readily elucidated with the help of Ammon's exposition, and especially of his graphic representation of the 'playground of variations.' If we think of the indifferent variations occurring in any character of a species as arranged in a series ascending from the smallest to the largest, this line may be regarded as the abscissal-axis, and from it ordinates may be drawn which express the frequency of the variation in question by the differences in their length. If the tips of these ordinates be united, we have the curve of frequency (Fig. 120, A), which according to Galton ought to be symmetrical, and in most cases really is so. Ammon calls the space between the smallest and the largest variations the 'variation-playground,' that is, the playground within which all variations are equally advantageous to the species. This is not co-extensive with the variation-area, for there may be more marked deviations below the beginning or above the upper end of the variation-playground, but these, being disadvantageous, fall under the shears of personal selection. The variation-playground may also be called the area of indulgence of variation, because the variations falling within it are spared from the eliminating activity of selection, or the variation-area of survivors, because on an average only those survive whose variations do not overstep the limits of this area.

Fig. 120. A, symmetrical, and B, asymmetrical curve of frequency; after Ammon. U, minimal, O, maximal limit of individual variation. U-O, the 'variation-playground.' M, the mean of variation. H, the greatest frequency or mode of variation.

This implies that variations below U (the lower limit of the area of exemption) and above O (the upper limit) can occur, but do not survive and leave descendants, and we can therefore easily understand why characters, of which different degrees arise with equal ease from the constitution of the species, must gradually develop a symmetrical curve of frequency because of the constant crossing. Obviously those individuals which stand just upon the borders of admissible variation will, other conditions being equal, leave behind them fewer descendants than those which approximate to the middle of the area of exemption; for as the characters concerned can vary in the offspring in both directions, there will always be at the lower end some of the descendants of a pair which will fall below the limits of exemption, and at the upper end some which will rise above it. This will happen even when pairing takes place between parents at the middle or at the other end of the abscissa, for there are always cases of the preponderance of one parent in heredity. A higher percentage of the descendants of individuals on the borderline will therefore be eliminated, and their frequency must therefore be less. Even if at the beginning of the series of observations a condition obtained in which all the ordinates of the area of exemption were equally high, those nearest the boundaries would of necessity very soon become lower, and this in proportion to their distance from the boundary, and the frequency-curve, which at first would be a straight line (according to our assumption, which of course does not tally with natural conditions), would become a symmetrical curve, highest in the middle and falling equally at either side.

Ammon has worked out the hypotheses on which the curve of frequency would become asymmetrical. Firstly, when the fertility is greater towards the upper or lower limit of the area of exemption; secondly, when germinal selection forces the variation in a particular direction, upwards or downwards; and thirdly, 'when natural selection intervenes diversely at the upper or lower limit.' Of these three possibilities the first two must be acknowledged as quite probable, but the third, it seems to me, could only cause a temporary asymmetry of the curve, lasting, that is, only until a state of equilibrium has again been reached; but that may in certain conditions take a long time.

Asymmetrical curves of frequency (Fig. 120, B) therefore arise, for instance, when the intra-germinal conditions (the 'constitution of the species') more easily and therefore more frequently produce extreme variations. In this case the area of exemption can only extend on one side, and must remain in this state. In Caltha palustris, the marsh marigold, we may find, according to De Vries, among a hundred flowers, those with five, six, seven, and eight petals, in the following proportions:—

Petals5678
Number of flowers722161

and thus there is an asymmetrical curve of frequency. But if we take the whole area of variation as the area of exemption, that is, if we assume that it is indifferent for the species whether the flowers have five, six, seven, or eight petals, the preponderance of the five-petalled flowers may have its reason in the fact that it is much easier for five than six or more petals to be produced because of the internal structure of the whole plant.

In this case the maximum of frequency lies at the lower limit of variation, but it may also lie at the upper. Thus, according to De Vries, the blossoms of Weigelia vary, in regard to the number of their petal-tips, in the following manner. Six-tipped corollas were not found, and among 1,145 flowers there were the following proportions:—

Tips of the corolla345
Number of flowers61196888