Mathematical Theories of the Earth. — Woodward. 271 
dimensious, and hence led to their too ready adoption. It seems 
also not improbable that the weight of Bessel's great name may have 
been too closely associated in the minds of his followers with the 
weight of his observations and results. The sanction of emi- 
nent authority, especially if there is added to it the stamp of an 
official seal, is sometimes a serious obstacle to real progress. We 
can not do less than accord to Bessel the first place among the astron- 
omers and geodesists of his day, but this is no adequate justification 
for the exaggerated estimate long entertained of the precision of the 
elements of his spheroid. 
The next step in the approximation was the important one of Clarke 
in 1866. His new values showed an increase over Bessel's of about 
half a mile in the equatorial semi-axis and about three-tenths of a 
mile in the polar semi- axis. Since 1866 general Clarke has kept pace 
with the accumulating data, and given us so many different elements 
for our spheroid that it is necessary to affix a date to any of his values 
we may use. The later values, however, differ but slightly from the 
earlier ones, so that the spheroid of 1866, which has come to be pretty 
generally adopted, seems likely to enjoy a justly greater celebrity than 
that of its immediate predecessor. The probable error of the axes of 
this spheroid is not much greater than the hundred thousandth part, 
and it is not likely that new data will change their lengths by more 
than a few hundred feet. 
In the present state of science, therefore, it may be said that the 
first order of approximation to the form and dimensions of the earth 
has been successfully attained. The question which follows naturally 
and immediately is, how much further can the approximation be car- 
ried? The answer to this question is not written, and the indications 
are not favorable for its speedy announcement. The first approxima- 
tion, as we have seen, requires no knowledge of the interior density 
and arrangement of the earth's mass ; it proceeds on the simple 
assumption that the sea surface is closely spheroidal. The second 
approximation, if it be more than a mere interpolation formula, 
requires a knowledge of both the density and arrangement of the con- 
stituents of the earth's mass, and especially of that part called the 
crust. "All astronomy," says Laplace, "rests on the stability of the 
earth's axis of rotation." In a similar sense we may say all geodesy 
rests on the direction of the plumb-line. The simple hypothesis of a 
spheroidal form assumes that the plumb-line is everywhere coincident 
with the normal to the spheroid, or that the surface of the spheroid 
coincides with the level of the sea. But this is not quite correct. The 
plumb-line is not in general coincident with the normal, and the 
actual sea level or geoid must be imagined to be an irregular surface 
lying partly above and partly below the ideal spheroidal surface. 
The deviations, it is true, are relatively small, but they are in general 
much greater than the unavoidable errors of observation, and they 
are the exact numerical expression of our ignorance in this branch 
of geodesy. It is well known, of course, that deflections of the plumb- 
