194 THE MATHEMATICAL THEORIES OF THE EAKTH. 



peratiire gradient, from ihe surface towards tlie center of the sphere 

 can be computed. It is tolerably certain that the heat conducted from 

 the interior to the surface of the Earth does not set up any reaction 

 which iu any sensible degree retards the process of cooling. It escapes 

 so freely that, for practical purposes, we may say it is instantly dis- 

 sipated. Hence, if we can assume that the Earth had a specified uni- 

 form temperature at the initial epoch, and can assume its dittusivity to 

 remain constant, the whole history of cooling is known so soon as we 

 determine the ditfusivity and the temperature gradient at any point. 

 Now, Sir William Thomson determined a value for the diff'usivity from 

 measurements of the seasonal variations of under- ground temperatures, 

 and numerous 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 temi)era- 

 ture of 7000O Fahr., 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 1 think we may with much i)robability say that the con- 

 solidation can not have taken place less than 20 million years ago, or 

 we should have more underground heat than we actually have, nor more 

 than 400 million years ago, or we should not have so much as the least 

 observed underground increment of temperature. That is to say, I con- 

 clude that Leibnitz's epoch of emergence of the con.nstentior status was 

 probably between those dates." These conclusions were announced 

 twenty-seven years ago and were re-published without modification iu 

 1883. Recently, also, Professor Tait, reasoning from the same basis, 

 has insisted with equal confidence on cuttitig down the upper limit of 

 geologic time to some such figures as ten million or fifteen million years.* 

 As mathematicians and astronomers, we must all confess to a deep inter- 

 est in these conclusions and the hypothesis from which they flow. They 

 are very important if true. But what are the probabilities? Having 

 been at some pains to look into this matter, I feel bound to state that, 

 although the hypothesis appears to be the best which can be formulated 

 at present, the odds are against its correctness. Its weak links are the 

 unverified assumptions of an initial uniform temperature and a constant 

 dififusivity. Very likely these are approximations, but of what order 

 we can not decide. Futhermore, if we accept the hypothesis, the odds 

 appear to be against the present attainment of trustworthy numerical 

 results, since the data for calculation, obtained mostly from observa- 

 tions on continental areas, are far too meagre to give satisfactory' aver- 

 age values for the entire mass of the earth. In short, tliis phase of the 

 case seems to stand about where it did twenty years ago, when Huxley 

 warned us that the perfection of our mathematical mill is no guaranty 

 of the quality of the grist, adding that, " as the grandest mill will not 



* Recent Advances in Physical Science, London, X876. 



