During recent years several attempts have been made to examine the processes of organic growth from the standpoint given by a chemical reaction of an autocatalytic nature. It is the purpose of this communication to extend the examination of this concept of metazoan size limitation in the light of new data on the growth of giant and pigmy rabbits.

The value of an hypothesis is properly assessed by observing the fidelity of its adherence to observations, and by an inquiry concerning the reasonableness of its assumptions. A theory of growth regulation may be rejected if it proves inconsistent with quantitative data, or if its premises are not in harmony with established principles. The quantitative aspect of the equations of Robertson and Crozier will therefore be considered in some detail.

It has been found that the growth of organisms and of populations generally exhibit an early positive acceleration culminating at an inflection point, after which the velocity of growth shows a negative acceleration. Similarly, certain chemical reactions take place at first slowly then with accelerated velocity and later still as slowly as they began. On the basis of this analogy the accumulation of material during growth has been considered as governed by a first order process in which one of the products of the transformation acts as a catalyst (Errera 1899–1900, Ostwald 1902, Chodat 1905, Robertson 1908, etc.).

T. B. Robertson (1923) has notably enlarged the autocatalytic interpretation, considering the velocity of development to be regulated by a succession of “master reactions.” That is to say, if the growth of an organism be regarded as a system of interdependent chemical reactions, the specifically slowest sets the pace for all the rest. And if each chemical reaction derives its substrata from the preceding reaction and in its turn supplies new materials for the reaction which follows it, the specifically slowest reaction in the series will become the “master reaction.” If these master reactions were first-order, self-accelerated chemical processes, the growth of the whole organism would be proportional to their velocity and so of the autocatalysed form.

The termination of the growth process, by analogy with well-known chemical processes, has been interpreted as due either to the exhaustion of the substrata of a master reaction, or to an acceleration of the reverse reaction by the accumulation of end-products from the forward reaction.

To represent such an autocatalysed process Robertson has used the formula :
formula
where A signifies the initial endowment of growth precursor and is proportional to the maximum growth attainable, whereas x represents the amount formed after time t. In the reaction A→ x, or A → x + y, the catalyst formed is proportional in amount to x, and K is the velocity constant for such a reaction. In this formula it is assumed that the cessation of growth is due to the exhaustion of the initial endowment A. From the nature of this equation it follows that for the integral curve the point of inflection is located at x = A/2, SO that about this mid point the curve of growth must be symmetric. If the activity of a reverse reaction were responsible for the termination of the growth process the formula should be written :
formula
K1 is the velocity constant of the forward reaction and K2 of the reverse reaction. The symbols A, x and t have the same meaning as above. When integrated, this formula is indistinguishable in form from equation (1) (unless the reverse reaction be multimolecular, in which case the curve expressing mass of the organism in relation to time may be asymmetric).

Several difficulties are connected with the application of the autocatalytic growth curve to actual data. Robertson’s explanation of observed deviations (by assuming a succession of “growth cycles”) has met with but partial success. For example, the early optimism as to the possibility of resolving the growth curve of man into three growth cycles superimposed one upon another has been found unsatisfactory. The existence of growth aside from specific cycles governed by first order chemical processes has been recently admitted by Robertson (1926, pp. 469–473), who introduces the idea and the term of “linear increment.” In the mouse he conceives this to begin about ten weeks after birth and to increase slowly to one hundred and forty weeks or later. This “linear increment” conception was necessitated by the fact that growth continues very slowly for a long time after the attainment of sexual maturity and of dimensions which might readily be supposed to be adult and therefore maximal. It is possible that a similar linear accretion is occurring in other animals and has escaped attention for lack of data concerning their later growth.

In recognition of these difficulties Crozier (1926–27) has advanced a modification of Robertson’s formula for which he claims a greater degree of consonance with chemical theory. Instead of interpreting growth as regulated by a succession of first order processes it is postulated that but one master reaction is concerned. This forward reaction A → x, being catalysed by x, is conceived to have not only a specific velocity constant K1 proper to the reaction in the absence of catalysis, but also a second velocity constant K2 representing the acceleration due to the presence of x. The relative magnitudes of these two velocities would determine the degree of asymmetry in the curve. Since many if not all growth cycles are asymmetric (Brody 1926), this formula offers a more satisfactory description of the normal curve. Deviations from the theoretical may be interpreted as minus aberrations due to the coincident activity of some retarding process (e.g. reproduction, Crozier 1926–27, Brown and Crozier 1927–28).

The formula is derived thus Crozier (1926–27) :

Let it be supposed that at the beginning of a developmental cycle there is available an unrenewable quantity A, of a substance giving rise to another, x, which determines the velocity growth. We will suppose that the material A gives rise to x by a first order reaction, and that x serves as a catalyst for this change. The reaction A → x will therefore be governed by a velocity constant (K1) proper to it in the absence of the influence of x, and also by the velocity constant (K2) due to catalysis by x. The decomposition of A must therefore be conceived as made up of two parallel reactions, and its differential equation is then :
formula
In support of this modification it is pointed out that temperature changes tend to modify the shape of the growth curve (Bliss 1926). This may be explained if K1 and K2 represent processes having different temperature characteristics, in which case a specific change of temperature would affect one velocity more than the other. Thus the shape of the curve would be affected and, as actually occurs, the graphs at two temperatures are not superimposable Crozier (1926–27). This argument from a temperature effect is a valid objection to Robertson’s equation on the condition that only a single regulatory process is concerned. Then, the conception of a single velocity constant would be inadequate to explain temperature distortion of the growth curve. But Robertson has postulated for mammals a superposition of growth cycles ; in such a case, the velocities proper to each cycle would be diversely accelerated and a comparable change of curve would ensue from change in temperature.

The following analysis of post-natal body growth in the rabbit is based on recent unpublished observations of Prof. W. E. Castle to whom I am deeply indebted for access to this material. In his studies of size inheritance, Dr Castle has employed two races of rabbits differing in velocity of growth and in ultimate body size. Whereas the Polish rabbits rarely attain a weight of fifteen hundred grams the Flemish Giants may exceed six kilograms. These races differ also with respect to the age at which sexual maturity is attained. From the evidence of breeding experiments Dr Castle has observed that the Polish males are sexually active at the age of one hundred and fifty days, or even earlier. Reciprocal hybrid Flemish-Polish males are found to be mature at about the same age. In the Flemish Giant males this adult condition generally arises between the age of two hundred and two hundred and fifty days.

These data comprise the observations made by W. E. Castle upon two hundred and four animals during a considerable period of time. In this group there were one hundred and ten Flemish Giants (forty males and seventy females), seventyone Polish (twenty-four males and forty-seven females), and twenty-three F1 hybrids from Flemish does by Polish bucks, of which ten were male and thirteen female. Weighings were made at intervals of two to seven days and recorded to the nearest ten grams. Only in very young animals would this method involve an error of more than 1 per cent. The data are not altogether homogeneous in that a different group of animals supplied data for the first thirty days. Also, the number of males observed after the onset of sexual maturity diminished to one-half. Allowance has been made for these circumstances by calculating the standard error of random sampling.

The greatest source of error lies in the fact that the rabbit is an herbivore. A two kilogram rabbit regularly transports through its alimentary tract about five hundred grams of foreign material. For this reason a true body weight (and likewise a true evaluation of basal metabolism), cannot be obtained in the living rabbit. A more accurate conception of true body weight may be obtained by using “cleaned body weight” which is obtained by removing the intestinal tract and subtracting its weight from the “live body weight.” If the tissues of the alimentary tract be discarded this method introduces an error up to 5 per cent, varying with the age of the animal (Robb 1928). It is particularly worthy of note that the ratio of extraneous matter to body weight does not remain constant during life. In the Polish rabbit at three months of age the gut contents constitute 20 per cent, of the observed live weight, whereas in an animal aged eight months this is reduced to about 10 per cent. In the Flemish rabbit this percentage does not change so rapidly in early life but during the second year a comparable reduction occurs. This fact detracts from the usefulness of the live body weight as a measure of body growth for all herbivorous animals, and probably, though to a lesser extent, for all animal data published.

The environment of these animals during growth has been quite constant. Seasonal temperature fluctuations were largely obviated by the use of a hot water heating system in the winter, and in the summer by the natural coolness characteristic of a stone building. Any seasonal tendency to distort the growth curve has been, we believe, counterbalanced by the fact that these animals were born during every month in the year. Nutritional disturbances have been likewise reduced to a minimum by expert care and the experience gained from many years of animal breeding in this Institution.

If the theory of autocatalysis be applied to the growth of the rabbit, it can be shown that the accumulation of material during the entire post-natal growth of the rabbit may be described with considerable accuracy by a single equation for an autocatalysed first order process. This agreement of the observed with the theoretical, as calculated by the formula used by Crozier, represents the most complete approximation hitherto published concerning any animal. The equation considered in detail above. It is assumed that the weight of the animal reflects the amount of growth determining material x, produced in the reaction A → x ; that the value of A is given by the maximum weight, and that the cycle starts at birth. The data for the growth of the Flemish Giant, hybrid, and Polish male rabbits have been plotted on Fig. 1. These values represent the arithmetic means of the observed weights. The standard deviation of the population x) has been graphically estimated as that deviation within which lie two-thirds of all the observations.

Fig. 1.

Observed growth in body weight of Flemish Giant, Flemish-Polish hybrid and Polish male rabbits represented by points. The solid lines represent theoretical curves expressing autocatalysis (see text and Table I).

Fig. 1.

Observed growth in body weight of Flemish Giant, Flemish-Polish hybrid and Polish male rabbits represented by points. The solid lines represent theoretical curves expressing autocatalysis (see text and Table I).

The standard error of random sampling for the above estimates of the mean has been derived by the formula
formula
This value (σm) has been used as approximately the radius given to the symbols in Figs, 1 and 2. The smooth curves in these Figs, are drawn from the equation:
formula
for which the necessary constants are given in Table I.
Table I.

Constants used in the application of the modified autocatalytic formula to the growth curves of Flemish Giant, Polish and Flemish-Polish hybrid male rabbits

Constants used in the application of the modified autocatalytic formula to the growth curves of Flemish Giant, Polish and Flemish-Polish hybrid male rabbits
Constants used in the application of the modified autocatalytic formula to the growth curves of Flemish Giant, Polish and Flemish-Polish hybrid male rabbits

The observed and theoretical mean weights are given in Tables II, III and IV, together with the standard deviation (σV) of the data. The general adherence of the data to a theoretical curve implies that a statistically significant divergence at any point might be correlated with some specific disturbing factor. There are two fairly obvious discrepancies of this nature, one at puberty in the data for the Polish males only, and the other occurring in all the data for a brief period after birth. These, being minus deviations, will be termed respectively the pubertal and the post-natal growth depressions. Unfortunately for their interpretation, the data for each of these extremes are not homogeneous with the data for the bulk of the curve.

Table II.

Growth of Flemish Giant male rabbits

Growth of Flemish Giant male rabbits
Growth of Flemish Giant male rabbits
Table III.

Growth of Flemish-Polish hybrid male rabbits

Growth of Flemish-Polish hybrid male rabbits
Growth of Flemish-Polish hybrid male rabbits
Table IV.

Growth of Polish male rabbits

Growth of Polish male rabbits
Growth of Polish male rabbits

The available data regarding a pubertal growth depression suggest that after the fifth month of post-natal development there occurs a retardation of growth in the Polish male rabbit. For the remainder of the first year their weights fall short of the expectation (Fig. 1). The significance of this observation may be questioned, in view of the fact that the number of animals observed has been reduced to one-half of the original twenty-four. No corresponding disturbance is reflected in the growth of the Flemish Giants nor in the hybrids, therefore it is not a phenomenon characteristic of all male rabbits. Nor can this be regarded as a racial characteristic. As shown (in Fig. 2), the Polish females closely approximate the theoretical curve with respect to their growth at this period. It is interesting to note that a comparable discrepancy of the rat growth curve is indicated by Crozier (1926–27) analysis of Donaldson’s data. If the pubertal depression of growth in Polish males, suggested by Dr Castle’s data, constitute a real phenomenon, it may be associated with the exceptional enlargement of adrenals and testes in these animals (Table V). Space has been given to a consideration of this non-conformity in the data for Polish male rabbits because of its possible significance. There are comparable irregularities in the growth of other animals (e.g. the rat, and man), for which adequate explanation has not yet been forthcoming.

Table V.

A correlation of adrenal and testes size with the degree of a post-pubertal growth depression

A correlation of adrenal and testes size with the degree of a post-pubertal growth depression
A correlation of adrenal and testes size with the degree of a post-pubertal growth depression
Fig. 2.

The observed growth in body weight of female rabbits is here compared with the theoretical curve for males as given in Fig. 1.

Fig. 2.

The observed growth in body weight of female rabbits is here compared with the theoretical curve for males as given in Fig. 1.

The post-natal growth retardation constitutes a further statistically significant divergence of the actual data from the theoretical. The observed weights from birth to the thirtieth day are consistently low for the Flemish Giant, the hybrid and Polish rabbits, to the extent of about 6σm (σm being the standard error of the mean due to random sampling). This post-natal growth “depression” may be, as Robertson would suggest, merely the termination of the first or pre-natal growth cycle. Or, one might interpret it as a general and nutritional disturbance initiated by the change of environment at birth.

In the development of viviparous animals the event of parturition is characteristically associated with a diminution of growth velocity. This generalisation is sustained by a large amount of evidence (e.g. Robertson 1923, Davenport 1926, Brody 1927). It is worthy of note that this episode separates the pre-natal and post-natal “growth cycles.” A discordant observation was made by Read (1913),t0 the effect that in the guinea pig the “pre-natal cycle” has terminated prior to birth and that the “second cycle” has already begun before birth. This observation has been discredited by the excellent data of Draper (1920), Ibsen and von Hensen (1868). These have been aligned by MacDowell et al. (1927) and clearly demonstrate that pre-natal growth in the guinea pig proceeds in one continuous curve. Thus, it is the concensus of numerous observations that, as Ostwald affirmed in 1908, the retardation of growth after the event of birth does not constitute an intrinsic tendency of the organism accidentally coinciding with this event, but that the observed fluctuation of growth velocity is an effect of the change in environment then experienced. The essential disturbance at this time may be one of heat regulation. The effect of a reduction of body temperature at birth would be identical with that observed on this occasion. Another contributing factor is the interruption of nutrition with the severance of the placental circulation. It is therefore maintained that the brief post-natal deviation from the theoretical recognised in these data are adequately interpreted as the sequellae of birth.

There remains for our consideration an apparent depression of growth prior to the onset of adolescence. This suggests a separation of the post-natal growth into two growth cycles and may properly be included in an exposition of Robertson’s views on the subject. His teachings differ from those of Crozier in one fundamental respect, namely, Crozier generalises, viewing the entire growth curve as a unit from which deviations may be effected by special causes. Robertson particularises, regarding the growth of the whole as a summation of integers, and by an arbitrary calculation of components has thought to reconstruct the whole. The interpretation favoured by Robertson is in part explained by the rigidity of that formula which he took to represent the phenomenon of autocatalysis. Thus in the application of his theory Robertson found his formula to be an inadequate description of the entire growth process. It could only apply to small portions of the growth curve which were dignified by the name of “growth cycles.” Thus the growth process was regarded as the summation of real component processes.

Although Robertson and others have found evidence of two post-natal cycles in the growth of most animals, it has generally been believed that in the rabbit only one post-natal cycle was present. These data exhibit a distinct period of growth depression which may correspond with that observed in other animals at the onset of the second cycle. The depression is consistently present in the development of Giant, hybrid and Polish rabbits and is characteristic of both sexes. In Figs. 3 and 4 are plotted the absolute increments of body weight during successive ten-day periods. It is apparent that the velocity of growth tends to show two post-natal maxima, one at forty days after birth and the other at about one hundred days.

Fig. 3.

The velocity of growth in Flemish Giant, Flemish-Polish hybrid and Polish male rabbits. Increments of body weight during ten-day periods are plotted. Note that the juvenile depression commencing at the fortieth day produces so-called growth cycles.

Fig. 3.

The velocity of growth in Flemish Giant, Flemish-Polish hybrid and Polish male rabbits. Increments of body weight during ten-day periods are plotted. Note that the juvenile depression commencing at the fortieth day produces so-called growth cycles.

Fig. 4.

The velocity of growth in Flemish Giant and Polish female rabbits. Increments of body weight during ten day periods are plotted against age. Note the onset of a juvenile growth depression at about the same age as in the males.

Fig. 4.

The velocity of growth in Flemish Giant and Polish female rabbits. Increments of body weight during ten day periods are plotted against age. Note the onset of a juvenile growth depression at about the same age as in the males.

Between these maxima there occurs the “juvenile” depression, for which no sufficient cause has been discovered. One is faced with a dilemma. These maxima may be construed as accelerations of growth velocity, cycles reflecting perhaps the activity of some autocatalysed master-reaction. This is Robertson’s thesis, and the widespread occurrence of a similar condition in birds and domestic animals (Robertson 1923, Brody 1926), constitutes the strongest argument in its favour.

The juvenile growth depression in the rabbit may be correlated with the activity of definite processes. As shown in Fig. 6 and Table VI, the percentage growth rate decreases rapidly throughout post-natal life. This decrease is apparently accentuated with the onset of gonad development. Full data will be given in a subsequent communication. Here it may be remarked that the juvenile growth depression commences about the fortieth day (see Figs. 3 and 4). At the same time thymus growth is retarded and the quantitative relationships of adrenal cortex pituitary and gonad are strikingly altered. These circumstances give rise to an interesting question, the chemical significance of weaning in relation to the endocrine manifestations observed a week later in the rabbit concurrently with the retardation of body growth.

Table VI.

Daily increase in weight of male rabbits

Daily increase in weight of male rabbits
Daily increase in weight of male rabbits
Table VII.

Analysis of growth of Flemish Giant, hybrid and Polish male rabbits during the first three months of post-natal life

Analysis of growth of Flemish Giant, hybrid and Polish male rabbits during the first three months of post-natal life
Analysis of growth of Flemish Giant, hybrid and Polish male rabbits during the first three months of post-natal life

The growth curve of man is strictly comparable to that in the rabbit but the phases are relatively more pronounced. The post-natal growth retardation is very marked, or, in the terminology of Davenport (1926) the first growth cycle has its maximum at birth. The juvenile growth retardation merges gradually with the former, and extends from the second to the fourteenth year. Thereafter there has been observed a great increase in the velocity of growth. If this apparent cycle does not represent any increase of specific growth rate, i.e., any increase of synthetic activity in the body tissues, it may be concluded that the theory of successive growth cycles is based on a misinterpretation of the juvenile depression as normal and therefore of the consequent maximum as intrinsically, rather than relatively, aberrant. Pertinent evidence is given by Brody (1927, No. 107, p. 15).

The second maximum is referred to in the literature as the “pre-pubertal acceleration.” This acceleration cannot, however, be said to constitute a universal feature of the growth curve (in man). This pre-pubertal acceleration appears to be quantitatively related to the percentage rate of growth between four and ten years. If the percentage rate of growth for a given group of children is relatively low during this earlier period, then there is usually an acceleration between twelve and fifteen years ; if it is high during this early period there is no pre-pubertal (percentage) acceleration. It appears that the pre-pubertal acceleration is an expression of compensating growth in children who were relatively undernourished during the earlier years.

This is, then, the first point in logically establishing the unity of post-natal growth. Even in the case of man, one of the most pronounced examples of growth curve aberration known, the apparent acceleration of growth velocity in the so-called “second cycle” is not unquestionably a true acceleration of specific growth rate.

In further support of this contention a theoretical growth curve has been constructed, exhibiting the effect of a decline in growth rate upon the growth velocity curve. The sigmoid curve, prior to its inflection point, may be approximated to an equation of the general type :
formula
where y represents the body weight at the time t, b is the constant for the initial value of y when t = o, and a is the natural logarithm of the instantaneous growth rate r (Blackman 1919). In the simplest case let b = 1, r = 2, and thus a = loge 2. This equation is plotted in Fig. 5, giving the curve A – A′. The velocity of growth is indicated in this instance by the increment curve X – X′. Let us suppose a reduction in the rate of growth to occur at ty, when y = 16, so that r assumes a new value of 1 · 25. The velocity of growth will likewise show a reduction to W, and thereafter gradually increase. In a second case let growth proceed at the original rate until y = 64, then let r become 1 · 25. Subsequent growth would be described by the curve C and the velocity of growth would fall to Z, and thereafter increase. The similarity of these theoretical increment curves to those observed for Polish and Flemish Giant rabbits will be apparent (Figs. 3 and 4).
Fig. 5.

Theoretical integral and velocity growth curves. Observe the effect of a decrease in the rate of acceleration in the integral curves (upper) upon the slope of the velocity curves (lower). In each pair of curves the divergence is due not to a difference in decrement but to a difference in the time at which it occurs.

Fig. 5.

Theoretical integral and velocity growth curves. Observe the effect of a decrease in the rate of acceleration in the integral curves (upper) upon the slope of the velocity curves (lower). In each pair of curves the divergence is due not to a difference in decrement but to a difference in the time at which it occurs.

Fig. 5 demonstrates that the existence of a second growth increment maximum is not conclusive evidence of an increased growth rate (as Robertson’s theory implied). On the contrary, a rapid decrease in the growth rate has been shown to effect a visible depression in the growth increment curve. That such a decrease is what actually causes the phenomenon of a subsequent “growth cycle” is attested by three forms of evidence; decrease of percentage growth rate at this period, consistent breaks in all the rabbit growth curves, and consequent depressions in the growth velocity curves.

The instantaneous growth rate k has been calculated from the equation :
formula
where W1 and W2 represent consecutive body weights separated by t units of time. The values of k for Flemish Giant, hybrid and Polish male rabbits are given in Table VI and recorded in Fig. 6. It is apparent that although these groups differ greatly in their birth weights their relative post-natal growth rates are quite similar. These growth rates are compared in Table VII and the velocity of growth retardation has been computed from consecutive values of k. After the fortieth day an increased growth depression is distinctly apparent.
Fig. 6.

The percentage growth rates of Flemish Giant, hybrid and Polish male rabbits plotted against age. Note that the same data given in Fig. 3, replotted here as percentages, indicating growth rates from birth until puberty, are identical for each race.

Fig. 6.

The percentage growth rates of Flemish Giant, hybrid and Polish male rabbits plotted against age. Note that the same data given in Fig. 3, replotted here as percentages, indicating growth rates from birth until puberty, are identical for each race.

Further confirmation is found in the shape of the integral curve. In Fig. 5 a change of slope is quite evident between curves A and B, likewise between curves A′ and C. It will be noted that similar changes of slopes are presented by the rabbit data at thirty days, as shown in Figs. 1 and 2. Furthermore this break in the integral growth curve immediately precedes the onset of a pronounced break in the growth velocity curve (Figs. 3 and 4), and since these breaks in both integral and velocity curves are consistently present in the data for Flemish Giant, hybrid and Polish rabbits both male and female, their validity seems assured. Hence it may be concluded that the existence of a second growth velocity maximum is not presumptive evidence of a new growth acceleration, as Robertson has supposed. On the contrary, it is maintained that the so-called “growth cycles” are manifestations of specific depressions in growth rate. If it be admitted that the growth cycle does not represent an increase of synthetic activity in the animal body but only a less activity of a greater mass, then it follows that the conception that the velocity of mammalian growth is regulated by a series of autocatalysed master reactions is bereft of the data which it sought to explain.

Brody (1926) has introduced into the consideration of animal growth the “principle of discontinuity.” The theory is clearly exemplified by his representation of growth in the unmated female white rat. The first hundred days of life have been pictured as four distinct growth epochs, each characterised by a specific level of synthetic activity. The rate of growth throughout each of these four periods is alleged to be constant ; the data given for the percentage rates of growth per day are, respectively, 52,11,4 ·7 and 3 ·1 per cent. The growth rate, as Minot (1908) observed, diminishes rapidly after birth and no one value is even approximately valid for more than a brief period. In seeking a biological significance for these values Brody has dismembered the growth curve, postulating for each section a specific, constant growth rate. The rabbit body weight data smoothed once, have been dealt with by the equation that Brody recommends. Values of k, the instantaneous rate of increase, have been given in Table VI and plotted in Fig. 6. No series of abruptly delineated periods of uniform increase, such as Brody postulates, are to be found in the post-natal history of Flemish Giant, hybrid or Polish male rabbits. The general application of Brody’s interpretation cannot be granted. His conclusions and the method by which they were obtained have been rigorously criticised by MacDowell (1927), with whom one must perforce concur. In the analysis of growth phenomena the usefulness of Brody’s hypothesis seems rather limited.

  1. New data are recorded for the post-natal growth in body weight of Flemish Giant and diminutive Polish rabbits, and of their F1 hybrids.

  2. The specific or percentage growth rate is identical for Flemish, hybrid and Polish males from the time of birth to the onset of puberty.

  3. The process of development in the Flemish Giants differs from that of the Polish in that the latter are but half as large at the time of birth. The fact of their shorter gestation period (thirty days in contrast to thirty-one) does not entirely account for the difference observed. Hence the existence of some growth depressant acting upon the young of the small race (or growth accelerator acting upon the young of the large race) in utero may be surmised.

  4. Hybrid animals from Giant does by Polish bucks and from the reciprocal cross are not so much affected, being at birth almost as large as the pure bred Giants. Accordingly the growth curves of hybrids and Giants closely approximate each other until the fourth month of post-natal existence.

  5. During the fourth month Polish and hybrid animals undergo a more rapid retardation of growth than do the Flemish Giants. Their growth rate values, which have been indistinguishable hitherto, diverge at this time. The lesser size of the Polish adult can be ascribed to two factors, one effective during the placental period and the other at a more advanced age. In the last analysis both may have a common cause.

  6. The growth curves of these rabbits have been described by the equation for autocatalysis sponsored by Crozier. These examples show the most extensive approximation to the theoretical of any mammalian data to which this equation has been hitherto applied.

  7. In the equation two velocity constants are employed, each having its own magnitude in the large and small races. Although the hybrid offspring are intermediate in adult size their growth curves do not lie midwav between those of their parents. Prior to its inflection point the hybrid curve is similar in magnitude to that of the Flemish Giants, thereafter the early retardation of growth resembles that of the Polish. Moreover, the hybrid values of K1 and K2 are not both intermediate between those of the parents—K1 being practically identical to that characteristic of the Polish parent. These facts invite extended investigation since they portend that in the inheritance of growth regulating processes there occurs a Mendelian segregation of unit characters.

  8. Two small deviations of the observed from the theoretical values are consistently present and have been subject to analysis. The first growth, aberration accompanies the circulatory, thermal and nutritional disturbances inevitably associated with birth. The second or juvenile deviation is concurrent with an endocrine reorganisation, in which the thymus, testes and suprarenal cortex are concerned.

  9. The natal and juvenile growth retardations tend to divide the animal growth curve into three periods (one pre-natal and two post-natal) which have been dignified by the name of “growth cycles.” Such a “cycle” does not represent an increase of the percentage growth rate but merely an increase of velocity due to the enlargement of a greater mass at a lesser rate. If one admit the validity of these observations the existence of disturbing factors is an adequate explanation of the three apparent cycles. Therefore it is maintained that to postulate a succession of autocatalysed master reactions is superfluous as well as inadequate.

  10. Brody’s conception of “five or more abruptly delimited periods of uniform increase” is not sustained by this material.

  11. The adherence of these data to an integral curve descriptive of autocatalysis does not assure the validity of an analogy but the utility of that hypothesis has been enlarged.

This investigation has been made possible by the grant of an Overseas Research Scholarship to the author by the British Royal Commission for the Exhibition of 1851. The animal data were provided by Prof. W. E. Castle to whom the Carnegie Institution of Washington has made a grant for the study of heredity in mammals.

It is a pleasure to acknowledge my indebtedness to Prof. W. E. Castle and to Prof. W. J. Crozier for guidance, suggestions and criticism during the progress of this study.

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