278 Mathematical Theories of the Earth. — Woodward. 
interesting to geologists and palaeontologists. Whether we are pre- 
pared to accept his conclusions or not, we must all acknowledge our 
indebtedness to the contributions of his master hand in this field as 
well as in most other fields of terrestrial physics. The contribution of 
special interest to us in this connection is his remarkable memoir on 
the secular cooling of the earth. In this memoir he adopts the simple 
hypothesis of a solid sphere whose thermal properties remain invaria- 
ble while it cools by conduction from an initial state of uniform tem- 
perature, and draws therefrom certain striking limitations on geologic 
time. Many geologists were startled by these limitations, and geo- 
logic thought and opinion have since been widely influenced by them. 
It will be of interest, therefore, to state a little more fully and clearly 
the grounds from which his arguments proceed. Conceive a sphere 
having a uniform temperature initially, to cool in a medium which 
instantly dissipates all heat brought by conduction to its surface, thus 
keeping the surface at a constant temperature. Suppose we have 
given the initial excess of the sphere's temperature over that of the 
medium. Suppose also that the capacity of the mass of the sphere for 
diffusion of heat is known, and known to remain invariable during the 
process of cooling. This capacity is called diflfusivity, and is a con- 
stant which can be observed. Then from these data the distribution 
of temperature at any future time can be assigned, and hence also the 
rate of temperature increase, or the temperature gradient, from the 
surface towards the center of the sphere can be computed. It is tol- 
erably certain that the heat conducted from the interior to the surface 
of the earth does not set up any reaction which in any sensible degree 
retards the process of cooling. It escapes so freely that, for practical 
purposes we may say it .is^instantly dissipated. Hence if we can 
assume that the earth had a specified uniform temperature at the 
initial epoch, and can assume its diflTusivity to remain constant, the 
whole history of cooling is known as soon as we determine thediffus- 
ivity and the temperature gradient at any point. Now Sir William 
Thomson determined a value for the difi"usivity from measurements of 
the seasonal variations of underground temperatures, and numer- 
ous observations of the increase of temperature with depth below the 
earth's surface gave an average value for the temperature gradient. 
From these elements and from an assumed initial temperature of 
7,000°, he infers that geologic time is limited to something between 
twenty million and four hundred million years. He says : "We must 
allow very wide limits in such an estimate as I have attempted to 
make ; but I think we may with much probability say that the con- 
solidation can not have taken place less than twenty million years ago, 
or we should have more underground heat than we actually have, nor 
more than four hundred million years ago, or we should not have so 
much as the least observed underground increment of temperature. 
That is to say, I conclude that Leibnitz's epoch of emergence of the 
consistentior status was probably between those dates." These conclu- 
