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sic work of Vochting； 〃Ueber Organbildung im Pflanzenreich〃； I。 Bonn； 1888； also 〃Bot。 Zeit。 1906； page 101； cf。 Goebel； 〃Experimentelle Morphologie〃； Leipzig and Berlin； 1908； Section V； Polaritat。） It is found； for example； that roots are formed from the base of a detached piece of stem and shoots from the apex。 Within the limits of this essay it is impossible to go into this difficult question； it is； however； important from the point of view of our general survey to emphasise the fact that the physiological distinctions between base and apex of pieces of stem are only of a quantitative kind； that is； they consist in the inhibition of certain phenomena or in favouring them。 As a matter of fact roots may be produced from the apices of willows and cuttings of other plants； the distinction is thus obliterated under the influence of environment。 The fixed polarity of cuttings from full grown stems cannot be destroyed； it is the expression of previous development。 Vochting speaks of polarity as a fixed inherited character。 This is an unconvincing conclusion； as nothing can be deduced from our present knowledge as to the causes which led up to polarity。 We know that the fertilised egg； like the embryo； is fixed at one end by which it hangs freely in the embryo…sac and afterwards in the endosperm。 From the first； therefore； the two ends have different natures； and these are revealed in the differentiation into root…apex and stem…apex。 A definite direction in the flow of food…substances is correlated with this arrangement； and this eventually leads to a polarity in the tissues。 This view requires experimental proof； which in the case of the egg…cells of flowering plants hardly appears possible； but it derives considerable support from the fact that in herbaceous plants； e。g。 Sempervivum （Klebs； 〃Variationen der Bluten〃； 〃Jahrb。 Wiss。 Bot。〃 1905； page 260。）； rosettes or flower…shoots are formed in response to external conditions at the base； in the middle； or at the apex of the stem； so that polarity as it occurs under normal conditions cannot be the result of unalterable hereditary factors。 On the other hand； the lower plants should furnish decisive evidence on this question； and the experiments of Stahl； Winkler； Kniep； and others indicate the right method of attacking the problem。
The relation of leaf…form to environment has often been investigated and is well known。 The leaves of bog and water plants （Cf。Goebel； loc。 cit。 chapter II。； also Gluck； 〃Untersuchungen uber Wasser… und Sumpfgewachse〃； Jena； Vols。 I。…II。 1905…06。） afford the most striking examples of modifications： according as they are grown in water； moist or dry air； the form of the species characteristic of the particular habitat is produced； since the stems are also modified。 To the same group of phenomena belongs the modification of the forms of leaves and stems in plants on transplantation from the plains to the mountains （Bonnier； 〃Recherches sur l'Anatomie experimentale des Vegetaux〃； Corbeil； 1895。） or vice versa。 Such variations are by no means isolated examples。 All plants exhibit a definite alteration in form as the result of prolonged cultivation in moist or dry air； in strong or feeble light； or in darkness； or in salt solutions of different composition and strength。
Every individual which is exposed to definite combinations of external factors exhibits eventually the same type of modification。 This is the type of variation which Darwin termed 〃definite。〃 It is easy to realise that indefinite or fluctuating variations belong essentially to the same class of phenomena； both are reactions to changes in environment。 In the production of individual variations two different influences undoubtedly cooperate。 One set of variations is caused by different external conditions； during the production； either of sexual cells or of vegetative primordia； another set is the result of varying external conditions during the development of the embryo into an adult plant。 The two sets of influences cannot as yet be sharply differentiated。 If； for purposes of vegetative reproduction； we select pieces of the same parent…plant of a pure species； the second type of variation predominates。 Individual fluctuations depend essentially in such cases on small variations in environment during development。
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 th