Group Discussion 295 



state. So if you take the crude death rate over an interval of time, 

 then a fair measure of the expectation of hfe, in terms of that inter- 

 val, is the reciprocal of the crude death rate, provided that popula- 

 tion is stationary, or nearly stationary. If it is not nearly stationary, 

 you can probably make an approximate adjustment. 



The only other thing I want to say is that the actuary's use of life- 

 tables is very different from yours. The life-table is not an end 

 in itself for actuaries ; it is merely a step on the way from a set of 

 mortality rates to the calculation of premium rates, reserves, 

 bonuses and surrender values. 



Sacher: Logarithms are not introduced to mystify. They are 

 actually a great convenience for computation. The classical law of 

 allometry is that one dimension of an organism is related to another 

 as [F = AX'^'] so that one of them varies as a power of the other; 

 these allometric relations are almost always presented graphically 

 on a double logarithmic scale. When you take the log of Y and the 

 log of X there is then a linear relationship between these values. 

 There are great advantages in using logarithms to fit a power func- 

 tion by least squares. All of these considerations apply to brain 

 weight, body weight, and lifespan as I have analysed them here. The 

 index of cephalization is a pure number. It is the logarithm of the 

 ratio of the actual brain weight of a species to the brain weight that 

 is predicted by the overall regression of log brain weight on log body 

 weight. 



Logarithms are also convenient in the present application because 

 they introduce the property that all of the observations have 

 approximately the same statistical weight in terms of the logarithmic 

 transform. The lifespan, brain weight and body weight measurement 

 all have about the same percentage error from mice to elephants and 

 therefore the error in logarithmic units is roughly constant, even 

 though the original absolute values have a million-fold range of 

 variation — from 5 grams or so to 5,000 kilograms. 



Perks: I agree that if you have reason for a relationship of that 

 form, then logarithms may ease the arithmetical processes. 



Sacher: There is no reason, in the sense of a general theory of 

 growth and of the relationship between parts of an organism, that is 

 capable of explaining why the allometric relationships should be of 

 this form. It is, however, a fact of observation that the power 

 function does describe these relations, and no other function does it 

 as well. 



Comfort: It is also true that in drawing the survival curves of birds 

 and small mammals, where over a large part of their lifespan their 

 mortality is so high that it is almost age-independent, most people 



