THE TEN PILLARS 17 



Sometimes one or more independent variables (y, z) are kept as constants 

 of the system while another (x) is varied. The rate of change of the dependent 

 variable, 0, as x changes, is expressed as an incomplete or partial derivative. 

 To emphasize the partial character, a rounded operator, d, is used; and the 

 constants of the system are stated as subscripts outside parentheses which 

 enclose the partial derivative. Thus: 



(d(f>/dx) y>i 



expresses the rate at which changes as x is changed, when y and z are 

 kept constant. 



The second-order partial derivative, the "acceleration," is expressed as 

 before: 



(d 2 <i>/dx 2 ) M 



This notation is used in all heat and mass transfer- considerations. For 

 instance, note the Haldane quotation which introduced this chapter. 



At this stage of development of biophysics (1962), the terminology of the 

 calculus is being used in published work, hence the need for introduction to 

 the bases and terminology of the subject. But explicit descriptions of most 

 biophysical phenomena are very rare; hence there seems to be no need to in- 

 troduce the operational calculus into an introductory book on biophysics at 

 this time. Therefore no attempt has been made to display the actual opera- 

 tions by which either differentiation or integration is accomplished. Opera- 

 tional calculus is treated in detail in many standard textbooks. 



7. Distribution of Observations 



A great many biological phenomena lend themselves to statistical meth- 

 ods of expression, i.e., age, height, weight, bloodcount, sugar analysis, etc. 

 This is so true that the "average value" over a large number is considered 

 the "normal" value, describing the "normal man." Hence it is instructive 

 to examine some of the methods of statistical expression, and to discuss their 

 reliability. 



Statistics has come a long way since the publication in 1662 of John 

 Graunt's "Natural and Political Observations Made upon the Bills of 

 Mortality," a study based on the records kept during the Black Plague in 

 London; and since Sir Edmund Halley (of "Comet" fame) wrote his basic 

 paper on life insurance, which appeared 30 years later. In the 20th century 

 statistical methods have penetrated nearly every field of learning in which 

 numerical measurement is possible. Moroney's book 4 gives a delightful in- 

 troduction to the subject. 



First of all, there are two factors which will result in a distribution in a 

 number of observations. One is errors in measurement; the other is a real 



