316 Robertson, Explanatory Remarks concerning the Normal Rate of Growth etc. 



Bewegungen unzulanglich. Sie reicht eben nur bei reiner Vor- 

 wartsbewegung bin, jede Richtungsanderung zu erzielen, ein Ver- 

 mogen, das sich wohl stets bei einseitigem Ausfalle durch Kreis- 

 bewegung offenbart. 



Im ganzen finden wir also bei Schnecken und Krebsen Gleiches 

 ermogiicht, aber durch verschiedene Mittel, die sich der Verschieden- 

 artigkeit der Organisation beider Tiergruppen angepasst haben. 



Explanatory Remarks concerning the Normal Rate of 

 Growth of an Individual and its Biochemical Significance. 



By T. Brailsford Robertson. 



(From the Rudolph Spreckel's Physiological Laboratory of the University of 



California.) 



Recent publications by the author 1 ) on the Normal Rate of 

 Growth of an Individual and its Biochemical Significance have called 

 forth certain criticisms which, although for the most part devoid 

 of significance, are nevertheless such as might mislead those who 

 lack technical knowledge of the questions at issue, questions 

 which are, for the most part, mathematical in character. 



Enriques 2 ) in a recent number of this journal, has devoted 



some space to pointing out that other formulae besides my formula 



^ 

 log r - = K (t - - tj where x is the amount of growth after time 



t and A, K and t t are constants, might be applied to the growth 

 of organisms with equal success. Hence, he argues that there is 

 no reason why the formula which I suggest should be regarded as 

 representing the growth of organisms rather than one of the other 

 formulae which he quotes or suggests. He overlooks the well- 

 known mathematical fact that any regular (non-discontinuous) plane 

 curve can be represented with close approximation to accuracy by 

 a great number of very different formulae 3 ). Thus, for example, 

 any continuous plane curve whatever can be represented to any desired 

 order of accuracy by some formula of the type y = a -j- bx -j- ex 2 -j- dx 3 



-j- , provided, only, that we include in our equation a 



sufficient number of terms. Hence, were we to carry out En- 



1) T. Brailsford Robertson. ,,On^ the Normal Rate of Growth of an 

 Individual, and its Biochemical Significance." Arch. f. Eutwickelungsmech., 25 (1908), 

 p. 581. - - ,,Further Remarks on the Normal Rate of Growth of an Individual and 

 its Biochemical Significance." Ibid. 2(5 (1908). p. 108. 



2) Paolo Enriques. ,,Wachstum und seine analytische Darstelluug". Biol. 

 Centralbl., 29 (1909), p. 331. 



.'>) Although this number is exceedingly small compared with the infinite number 

 of formulae which will not represent a given curve. 



