In the present case P is for a unit square (taken relatively small and
so assumed as plane) of the shell, suppose a square mile, equal to the
effect of gravity upon that unit, r being the earth's radius, and if we
assume the unit square be also a unit in thickness, P is then the weight
of a cubic mile of its material; and if we take (roughly) the earth's
radius as 4,000 miles, the tangential pressure, T, is, on _each face_ of
the cubic mile, equal to
(4000/2) P,
or equal to the pressure of a column of the same material of 2,000 times
its weight.
If the cubic mile that we have thus supposed cut out of the earth's
crust at the surface were of the hardest known granite or porphyry, it
would be exposed to a crushing tangential pressure equal to between 400
and 500 times what it could withstand, and so must crush, even though
only left unsupported by the nucleus beneath, to the extent of 1/400 or
1/500 of its entire weight. And what is true here of a mile taken at the
surface, is true (neglecting some minute corrections for difference in
the co-efficient of gravity, etc.) if taken at any other depth within
the thick crust.[F]
The crust of our earth, then, as it now is, must crush, to follow down
after the shrinking nucleus--if so be that the globe be still cooling,
and constituted as it is; even to the limited extent to which we know
anything of its nature--it must crush unequally, both regarded
superficially and as to depth; generally the crushing lines being
confined to the planes or places of greatest weakness; and the crushing
will not be absolutely constant and uniform anywhere, or at any time, or
at any of those places of weakness to which it will be principally
confined, but will be more or less irregular, quasi-periodic, or
paroxysmal: as is, indeed, the way in which all known material
substances (more or less rigid) give way to a slow and constantly
increasing, steady pressure.
We have now to ask, _How much_ of this crushing is going on at present
year by year? And the answer to this depends upon what amount of heat
our world is losing into space year by year.
Geologists who have taken on trust the statement, that La Place has
proved that the world has lost no sensible amount of heat for the last
10,000 years seem generally to suppose that to be a fact; but in reality
La Place has _proved_ nothing of the sort, as those geological teachers
who have echoed the conclusion should have known, had they deciphered
the mathematical argument upon which it has been supposed to rest.
By application of Fourier's theorem (or definition) to the observed rate
of increment of heat in descending from the geothermal _couche_ of
invariable temperature, and the co-efficients of conductivity of the
rocks of our earth's crust, as given by the long-continued observations
made beneath the Observatories of Paris and of Edinburgh, it results
that the annual loss of heat into space of our globe at present is equal
to that which would liquefy into water, at 32° Fahr., about 777 cubic
miles of ice; and this is the measuring unit for the amount of
contraction of our globe now going on. The figures are not probably
exact, for the data are not on a basis sufficiently full or exactly
established as yet; but they are not very widely wrong, and their
precise exactness is not material here. Now, how is this annual loss of
heat (great or small, as we may please to view it) from the interior of
our globe disposed of?
What does it _do_ in the interior? We have already seen that it is
primarily disposed of by conversion into work; into the work of
diminishing the earth's volume as a whole, and in so doing crushing
portions of the solid surrounding shell.
But does the transformation of lost heat into the work of vertical
descent, and of the crush as it follows down after the shrinking
nucleus, end the cycle? No. A very large portion of the mechanical work
thus produced, and resolved, as we have seen, into tangential crushing
pressure, is retransformed into heat again in the very act of crushing
the solid material of the shell. If we see a cartload of granite
paving-stones shot out in the dark, we see fire and light produced by
their collision; if we rub two pieces of quartz together, and crush thus
their surfaces against each other, we find we heat the pieces and evolve
light.
The machinery used for crushing by steam-power, hard rocks into road
metal, gets so hot that the surfaces cannot be touched.
These are familiar instances of one result of what is now taking place
by the crushing of the rocky masses of our cooling and descending
earth's crust, every hour beneath our feet, only upon a vastly greater
scale. It is in this local transformation of work into heat that I find
the true origin of volcanic heat within our globe. But if we are to test
this, so as in the only way possible to decide is it a true solution of
this great problem, we must again ask the question, _How much?_ and to
answer this, we must determine _experimentally_ how much heat can be
developed by the crushing of a given volume, say a cubic mile, of such
rocky materials as we know must constitute the crust of our globe down
to the bottom of the known sedimentary strata, and extending to such
crystalloid rocks as we may presume underlie these. We must also obtain
at least approximately what are the co-efficients of _total contraction_
between fusion and atmospheric temperature of such melted rocks, basic
and acid silicates, as may be deemed representative of that co-efficient
for the range of volcanic fused products, basalts, trachytes, etc.,
which probably sufficiently nearly coincide with that of the whole
non-metallic mass of our globe.
The first I have determined experimentally by two different methods, but
principally by the direct one of the _work_ expended in crushing prisms
of sixteen representative classes of rock; the specific gravities and
specific heats of which I have also determined.
If H be the height of a prism of rock crushed to powder by a pressure,
P, applied to two opposite faces, which, when the prism has been
reduced to its volume in powder, has acted through a range of H - t,
then