170 GROWTH PRINCIPLES AND THEORY 2 



(b) General co?isiderations on models and laws of growth 



Because of the possibility quantitatively to express the relevant variables, 

 growth belongs to those biological phenomena which for a considerable time 

 have invited mathematical treatment. In these attempts, a lack of understanding 

 the meaning of theory and laws in science can often be noticed. Hence a brief 

 epistemological discussion appears to be in place. 



Biologists are apt to expect too much and too little from a mathematics of 

 growth: too much when looking forward to a "wonder formula" which would 

 represent any and all observed growth data, and being disappointed if such prodigy 

 cannot be produced; too little if correct approximation of a smaller or larger 

 set of data is considered sufficient proof that a "law of growth" has been dis- 

 covered. Furthermore, many biologists, believing in a somewhat mythical ritual 

 called "the experimental method", are still unfamiliar with the procedure applied 

 in theoretical science, even though this is commonplace in physics and since has 

 diffused into the behavioral and social sciences. 



The approach toward mathematical analysis of empirical data may be illus- 

 trated by a fictitious example. Suppose the problem is to establish a mathematical 

 expression to describe the relation between time and distances passed by free- 

 falling bodies. Two investigators may proceed in two essentially different ways. 

 The one will let fall different objects such as cannon-balls, stones, feathers, cats 

 and dogs; will measure the distances travelled; and tabulate or plot the results. 

 Then he may develop distance as a function of time by way of a series : 



S = a^ + a^t + a^t^ + . . . (5.7) 



and will calculate the coefficients. This procedure is infallible and will allow 

 description of any observation with any precision desired, if an arbitrary number 

 of terms is permitted. However, the terms and coefficients have no physical 

 meaning, and will be different from one case to another. 



The other investigator who, in history, was Galileo and his followers, proceeds 

 in an essentially different fashion. By way of some intuitive process he has arrived 

 at the concepts of "force "and "acceleration", constant force producing constant 

 acceleration. If acceleration is constant, the velocity reached by the falling body 

 will be proportional to the time passed: v = ct, and the distance traveled will be 

 average velocity during the time interval t multiplied by time elapsed, i.e. S = ct'^J2, 

 which is Galileo's law. 



Our investigator does not bother to fit all observations made or feasible, but 

 has discovered a fundamental "law of nature". This law represents observations 

 where the relevant relation is well isolated from others as, according to legend, 

 in the cannon-balls Galileo dropped from the leaning tower of Pisa. The law is a 

 poor approximation of many other observed phenomena but this does not disturb 

 our physicist who answers, with Galileo's student, Torricelli: "If balls of lead, 

 iron and stone do not obey the law, so much the worse for them; then we say 

 that we do not speak of them". 



However, it turns out that this seemingly arbitrary procedure is eminently 

 successful. If Galileo's law is taken for granted, seeming deviations eventually 



