Page:Lectures on Ten British Physicists of the Nineteenth Century.djvu/69

in store at any time after the commencement of the present order of things, and has been therefore very slowly diminishing from age to age. I have endeavored to prove this for the Sun's heat, in an article recently published in Macmillan's Magazine (March, 1862), where I have shown that most probably the Sun was sensibly hotter a million years ago than he is now. Hence, geological speculation, assuming somewhat greater extremes of heat, more violent storms and floods, more luxuriant vegetation, and harder and coarser grained plants and animals, in remote antiquity, are more probable than those of the extreme quietist, or "uniformitarian" school. A middle path, not generally safest in scientific speculation, seems to be so in this case. It is probable that hypotheses of grand catastrophes destroying all life from the Earth, and ruining its whole surface at once, are greatly in error; it is impossible that hypotheses assuming an equability of sun and storms for 1,000,000 years can be wholly true."

He proceeded in the paper cited, to apply Fourier's results to deduce a limit to the age of the Earth. Suppose a solid slab of uniform thickness and of great lateral dimensions to be originally heated to a temperature $V^{\circ }$, one side to be kept exposed to a temperature $O^{\circ }$, and the other to be kept exposed to a temperature $V$. Let $k$ denote the conductivity of the solid, when measured in terms of the thermal capacity of the unit of volume; and let $v$ denote the temperature at any distance $x$ from the surface at any time $t$ from the beginning of the cooling. Fourier showed that under these conditions,

${\frac {\;dv}{\;dx}}={\frac {V}{\sqrt {\pi kt}}}e^{-x^{2}/4kt}$

Here ${\frac {\;dv}{\;dx}}$ means the gradient of temperature, along a line normal to the face; it is the rate of change of the temperature as you go along the direction of $x$. This formula does not apply to any time prior to the beginning of the cooling, for then $t$ will be negative and the formula involves the square root of $t$.

But what application has this result to the case of the Earth?