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These relations must be borne in mind if we wish to understand the results of statistical methods. Since the work of Quetelet, Galton, and others the statistical examination of individual differences in animals and plants has become a special science, which is primarily based on the consideration that the application of the theory of probability renders possible mathematical statement and control of the results. The facts show that any character, size of leaf, length of stem, the number of members in a flower, etc. do not vary haphazard but in a very regular manner. In most cases it is found that there is a value which occurs most commonly, the average or medium value, from which the larger and smaller deviations, the so-called plus and minus variations fall away in a continuous series and end in a limiting value. In the simpler cases a falling off occurs equally on both sides of the curve; the curve constructed from such data agrees very closely with the Gaussian curve of error. In more complicated cases irregular curves of different kinds are obtained which may be calculated on certain suppositions.

The regular fluctuations about a mean according to the rule of probability is often attributed to some law underlying variability. (de Vries, "Mutationstheorie", Vol. I. page 35, Leipzig, 1901.) But there is no such law which compels a plant to vary in a particular manner. Every experimental investigation shows, as we have already remarked, that the fluctuation of characters depends on fluctuation in the external factors.

The applicability of the method of probability follows from the fact that the numerous individuals of a species are influenced by a limited number of variable conditions. (Klebs, "Willkurl. Ent." Jena, 1903, page 141.) As each of these conditions includes within certain limits all possible values and exhibits all possible combinations, it follows that, according to the rules of probability, there must be a mean value, about which the larger and smaller deviations are distributed. Any character will be found to have the mean value which corresponds with that combination of determining factors which occurs most frequently. Deviations towards plus and minus values will be correspondingly produced by rarer conditions.

A conclusion of fundamental importance may be drawn from this conception, which is, to a certain extent, supported by experimental investigation.

(Klebs, "Studien uber Variation", "Arch. fur Entw." 1907.) There is no normal curve for a particular CHARACTER, there is only a curve for the varying combinations of conditions occurring in nature or under cultivation. Under other conditions entirely different curves may be obtained with other variants as a mean value. If, for example, under ordinary conditions the number 10 is the most frequent variant for the stamens of Sedum spectabile, in special circumstances (red light) this is replaced by the number 5. The more accurately we know the conditions for a particular form or number, and are able to reproduce it by experiment, the nearer we are to achieving our aim of rendering a particular variation impossible or of making it dominant.

In addition to the individual variations of a species, more pronounced fluctuations occur relatively rarely and sporadically which are spoken of as "single variations," or if specially striking as abnormalities or monstrosities. These forms have long attracted the attention of morphologists; a large number of observations of this kind are given in the handbooks of Masters (Masters, "Vegetable Teratology", London, 1869.) and Penzig (Penzig, "Pflanzen-Teratologie, Vols I. and II. Genua, 1890-94.)These variations, which used to be regarded as curiosities, have now assumed considerable importance in connection with the causes of form-development. They also possess special interest in relation to the question of heredity, a subject which does not at present concern us, as such deviations from normal development undoubtedly arise as individual variations induced by the influence of environment.

Abnormal developments of all kinds in stems, leaves, and flowers, may be produced by parasites, insects, or fungi. They may also be induced by injury, as Blaringhem (Blaringhem, "Mutation et traumatismes", Paris, 1907.) has more particularly demonstrated, which, by cutting away the leading shoots of branches in an early stage of development, caused fasciation, torsion, anomalous flowers, etc. The experiments of Blaringhem point to the probability that disturbances in the conditions of food-supply consequent on injury are the cause of the production of monstrosities.

This is certainly the case in my experiments with species of Sempervivum (Klebs, "Kunstliche Metamorphosen", Stuttgart, 1906.); individuals, which at first formed normal flowers, produced a great variety of abnormalities as the result of changes in nutrition, we may call to mind the fact that the formation of inflorescences occurs normally when a vigorous production of organic compounds, such as starch, sugar, etc. follows a diminution in the supply of mineral salts. On the other hand, the development of inflorescences is entirely suppressed if, at a suitable moment before the actual foundations have been laid, water and mineral salts are supplied to the roots. If, during the week when the inflorescence has just been laid down and is growing very slowly, the supply of water and salts is increased, the internal conditions of the cells are essentially changed.