Form of the Earth and Planets—Figure of a Homogeneous Spheroid in
Rotation—Figure of a Spheroid of variable Density—Figure of the Earth,
supposing it to be an Ellipsoid of Revolution—Mensuration of a Degree
of the Meridian—Compression and Size of the Earth from Degrees of
Meridian—Figure of Earth from the Pendulum.
THE theoretical investigation of the figure of the earth and planets is
so complicated, that neither the geometry of Newton, nor the refined
analysis of La Place, has attained more than an approximation. The
solution of that difficult problem has been accomplished by our
distinguished countryman Mr. Ivory. The investigation has been conducted
by successive steps, beginning with a simple case, and then proceeding
to the more difficult. But, in all, the forces which occasion the
revolutions of the earth and planets are omitted, because, by acting
equally upon all the particles, they do not disturb their mutual
relations. A fluid mass of uniform density, whose particles mutually
gravitate to each other, will assume the form of a sphere when at rest.
But, if the sphere begins to revolve, every particle will describe a
circle (N. 117), having its centre in the axis of revolution. The planes
of all these circles will be parallel to one another and perpendicular
to the axis, and the particles will have a tendency to fly from that
axis in consequence of the centrifugal force arising from the velocity
of rotation. The force of gravity is everywhere perpendicular to the
surface (N. 118), and tends to the interior of the fluid mass; whereas
the centrifugal force acts perpendicularly to the axis of rotation, and
is directed to the exterior. And, as its intensity diminishes with the
distance from the axis of rotation, it decreases from the equator to the
poles, where it ceases. Now it is clear that these two forces are in
direct opposition to each other in the equator alone, and that gravity
is there diminished by the whole effect of the centrifugal force,
whereas, in every other part of the fluid, the centrifugal force is
resolved into two parts, one of which, being perpendicular to the
surface, diminishes the force of gravity; but the other, being at a
tangent to the surface, urges the particles towards the equator, where
they accumulate till their numbers compensate the diminution of gravity,
which makes the mass bulge at the equator, and become flattened at the
poles. It appears, then, that the influence of the centrifugal force is
most powerful at the equator, not only because it is actually greater
there than elsewhere, but because its whole effect is employed in
diminishing gravity, whereas, in every other point of the fluid mass, it
is only a part that is so employed. For both these reasons, it gradually
decreases towards the poles, where it ceases. On the contrary, gravity
is least at the equator, because the particles are farther from the
centre of the mass, and increases towards the poles, where it is
greatest. It is evident, therefore, that, as the centrifugal force is
much less than the force of gravity—gravitation, which is the difference
between the two, is least at the equator, and continually increases
towards the poles, where it is a maximum. On these principles Sir Isaac
Newton proved that a homogeneous fluid (N. 119) mass in rotation assumes
the form of an ellipsoid of revolution (N. 120), whose compression is
1/230. Such, however, cannot be the form of the earth, because the
strata increase in density towards the centre. The lunar inequalities
also prove the earth to be so constructed; it was requisite, therefore,
to consider the fluid mass to be of variable density. Including this
condition, it has been found that the mass, when in rotation, would
still assume the form of an ellipsoid of revolution (N. 120); that the
particles of equal density would arrange themselves in concentric
elliptical strata (N. 121), the most dense being in the centre; but that
the compression or flattening would be less than in the case of the
homogeneous fluid. The compression is still less when the mass is
considered to be, as it actually is, a solid nucleus, decreasing
regularly in density from the centre to the surface, and partially
covered by the ocean, because the solid parts, by their cohesion, nearly
destroy that part of the centrifugal force which gives the particles a
tendency to accumulate at the equator, though not altogether; otherwise
the sea, by the superior mobility of its particles, would flow towards
the equator and leave the poles dry. Besides, it is well known that the
continents at the equator are more elevated than they are in higher
latitudes. It is also necessary for the equilibrium of the ocean that
its density should be less than the mean density of the earth, otherwise
the continents would be perpetually liable to inundations from storms
and other causes. On the whole, it appears from theory, that a
horizontal line passing round the earth through both poles must be
nearly an ellipse, having its major axis in the plane of the equator,
and its minor axis coincident with the axis of the earth’s rotation
(N. 122). It is easy to show, in a spheroid whose strata are elliptical,
that the increase in the length of the radii (N. 123), the decrease of
gravitation, and the increase in the length of the arcs of the meridian,
corresponding to angles of one degree, from the poles to the equator,
are all proportional to the square of the cosine of the latitude
(N. 124). These quantities are so connected with the ellipticity of the
spheroid, that the total increase in the length of the radii is equal to
the compression or flattening, and the total diminution in the length of
the arcs is equal to the compression, multiplied by three times the
length of an arc of one degree at the equator. Hence, by measuring the
meridian curvature of the earth, the compression, and consequently its
figure, become known. This, indeed, is assuming the earth to be an
ellipsoid of revolution; but the actual measurement of the globe will
show how far it corresponds with that solid in figure and constitution.
The courses of the great rivers, which are in general navigable to a
considerable extent, prove that the curvature of the land differs but
little from that of the ocean; and, as the heights of the mountains and
continents are inconsiderable when compared with the magnitude of the
earth, its figure is understood to be determined by a surface at every
point perpendicular to the direction of gravitation, or of the
plumb-line, and is the same which the sea would have if it were
continued all round the earth beneath the continents. Such is the figure
that has been measured in the following manner:—
A terrestrial meridian is a line passing through both poles, all the
points of which have their noon contemporaneously. Were the lengths and
curvatures of different meridians known, the figure of the earth might
be determined. But the length of one degree is sufficient to give the
figure of the earth, if it be measured on different meridians, and in a
variety of latitudes. For, if the earth were a sphere, all degrees would
be of the same length; but, if not, the lengths of the degrees would be
greater, exactly in proportion as the curvature is less. A comparison of
the length of a degree in different parts of the earth’s surface will
therefore determine its size and form.
An arc of the meridian may be measured by determining the latitude of
its extreme points by astronomical observations (N. 125), and then
measuring the distance between them in feet or fathoms. The distance
thus determined on the surface of the earth, divided by the degrees and
parts of a degree contained in the difference of the latitudes, will
give the exact length of one degree, the difference of the latitudes
being the angle contained between the verticals at the extremities of
the arc. This would be easily accomplished were the distance
unobstructed and on a level with the sea. But, on account of the
innumerable obstacles on the surface of the earth, it is necessary to
connect the extreme points of the arc by a series of triangles (N. 126),
the sides and angles of which are either measured or computed, so that
the length of the arc is ascertained with much laborious calculation. In
consequence of the irregularities of the surface each triangle is in a
different plane. They must therefore be reduced by computation to what
they would have been had they been measured on the surface of the sea.
And, as the earth may in this case be esteemed spherical, they require a
correction to reduce them to spherical triangles. The officers who
conducted the trigonometrical survey, in measuring 500 feet of a base in
Ireland twice over, found that the difference in the two measurements
did not amount to the 800th part of an inch; and in the General Survey
of Great Britain, five bases were measured from 5 to 7 miles long, and
some of them 400 miles apart, yet, when connected by series of
triangles, the measured and computed lengths did not differ by more than
3 inches, an unparalleled degree of accuracy; but such is the accuracy
with which these operations are conducted.
Arcs of the meridian have been measured in a variety of latitudes in
both hemispheres, as well as arcs perpendicular to the meridian. From
these measurements it appears that the length of the degrees increases
from the equator to the poles, nearly in proportion to the square of the
sine of the latitude (N. 127). Consequently, the convexity of the earth
diminishes from the equator to the poles.
Were the earth an ellipsoid of revolution, the meridians would be
ellipses whose lesser axes would coincide with the axis of rotation, and
all the degrees measured between the pole and the equator would give the
same compression when combined two and two. That, however, is far from
being the case. Scarcely any of the measurements give exactly the same
results, chiefly on account of local attractions, which cause the
plumb-line to deviate from the vertical. The vicinity of mountains
produces that effect. One of the most remarkable anomalies of this kind
has been observed in certain localities of northern Italy, where the
action of some dense subterraneous matter causes the plumb-line to
deviate seven or eight times more than it did from the attraction of
Chimborazo, in the observations of Bouguer, while measuring a degree of
the meridian at the equator. In consequence of this local attraction,
the degrees of the meridian in that part of Italy seem to increase
towards the equator through a small space, instead of decreasing, as if
the earth was drawn out at the poles, instead of being flattened.
Many other discrepancies occur, but from the mean of the five principal
measurements of arcs in Peru, India, France, England, and Lapland, Mr.
Ivory has deduced that the figure which most nearly follows this law is
an ellipsoid of revolution whose equatorial radius is 3962·824 miles,
and the polar radius 3949·585 miles. The difference, or 13·239 miles,
divided by the equatorial radius, is 1/299 nearly[3] (N. 128). This
fraction is called the compression of the earth, and does not differ
much from that given by the lunar inequalities. Since the preceding
quantities were determined, arcs of the meridian have been measured in
various parts of the globe, of which the most extensive are the Russian
arc of 25° 20ʹ between the Glacial Sea and the Danube, conducted under
the superintendence of M. Struve, and the Indian arc extended to 21°
21ʹ, by Colonel Everest. The compression deduced by Bessel from the sum
of ten arcs is 298-3/4, the equatorial radius 3962·802, and the polar
3949·554 miles, whilst Mr. Airy arrives at an almost identical result
(3962·824, 3949·585, and 298-83/100) from a consideration of all the
arcs, measured up to 1831, including the great Indian and Russian ones.
If we assume the earth to be a sphere, the length of a degree of the
meridian is 69-14/100 English miles. Therefore 360 degrees, or the whole
equatorial circumference of the globe, is 24,899 English miles.
Eratosthenes, who died 194 years before the Christian era, was the first
to give an approximate value of the earth’s circumference, by the
measurement of an arc between Alexandria and Syene.
There is another method of finding the figure of the earth, totally
different from the preceding, solely depending upon the increase of
gravitation from the equator to the poles. The force of gravitation at
any place is measured by the descent of a heavy body during the first
second of its fall. And the intensity of the centrifugal force is
measured by the deflection of any point from the tangent in a second.
For, since the centrifugal force balances the attraction of the earth,
it is an exact measure of the gravitating force. Were the attraction to
cease, a body on the surface of the earth would fly off in the tangent
by the centrifugal force, instead of bending round in the circle of
rotation. Therefore, the deflection of the circle from the tangent in a
second measures the intensity of the earth’s attraction, and is equal to
the versed sine of the arc described during that time, a quantity easily
determined from the known velocity of the earth’s rotation. Whence it
has been found that at the equator the centrifugal force is equal to the
289th part of gravity. Now, it is proved by analysis that, whatever the
constitution of the earth and planets may be, if the intensity of
gravitation at the equator be taken equal to unity, the sum of the
compression of the ellipsoid, and the whole increase of gravitation from
the equator to the pole, is equal to five halves of the ratio of the
centrifugal force to gravitation at the equator. This quantity with
regard to the earth is 5/2 of 1/289 or 1/115·2. Consequently, the
compression of the earth is equal to 1/115·2 diminished by the whole
increase of gravitation. So that its form will be known, if the whole
increase of gravitation from the equator to the pole can be determined
by experiment. This has been accomplished by a method founded upon the
following considerations:—If the earth were a homogeneous sphere without
rotation, its attraction on bodies at its surface would be everywhere
the same. If it be elliptical and of variable density, the force of
gravity, theoretically, ought to increase from the equator to the pole,
as unity _plus_ a constant quantity multiplied into the square of the
sine of the latitude (N. 127). But for a spheroid in rotation the
centrifugal force varies, by the laws of mechanics, as the square of the
sine of the latitude, from the equator, where it is greatest, to the
pole, where it vanishes. And, as it tends to make bodies fly off the
surface, it diminishes the force of gravity by a small quantity. Hence,
by gravitation, which is the difference of these two forces, the fall of
bodies ought to be accelerated from the equator to the poles
proportionably to the square of the sine of the latitude; and the weight
of the same body ought to increase in that ratio. This is directly
proved by the oscillations of the pendulum (N. 129), which, in fact, is
a falling body; for, if the fall of bodies be accelerated, the
oscillations will be more rapid: in order, therefore, that they may
always be performed in the same time, the length of the pendulum must be
altered. By numerous and careful experiments it is proved that a
pendulum, which oscillates 86,400 times in a mean day at the equator,
will do the same at every point of the earth’s surface, if its length be
increased progressively to the pole, as the square of the sine of the
latitude.
From the mean of these it appears that the whole decrease of gravitation
from the poles to the equator is 0·0051449, which, subtracted from
1/115·2, shows that the compression of the terrestrial spheroid is about
1/285·26. This value has been deduced by the late Mr. Baily, president
of the Astronomical Society, who devoted much attention to this subject;
at the same time, it may be observed that no two sets of pendulum
experiments give the same result, probably from local attractions. The
compression obtained by this method does not differ much from that given
by the lunar inequalities, nor from the arcs in the direction of the
meridian, and those perpendicular to it. The near coincidence of these
three values, deduced by methods so entirely independent of each other,
shows that the mutual tendencies of the centres of the celestial bodies
to one another, and the attraction of the earth for bodies at its
surface, result from the reciprocal attraction of all their particles.
Another proof may be added. The nutation of the earth’s axis and the
precession of the equinoxes (N. 146) are occasioned by the action of the
sun and moon on the protuberant matter at the earth’s equator. And,
although these inequalities do not give the absolute value of the
terrestrial compression, they show that the fraction expressing it is
comprised between the limits 1/279 and 1/573.
It might be expected that the same compression should result from each,
if the different methods of observation could be made without error.
This, however, is not the case; for after allowance has been made for
every cause of error, such discrepancies are found, both in the degrees
of the meridian and in the length of the pendulum, as show that the
figure of the earth is very complicated. But they are so small, when
compared with the general results, that they may be disregarded. The
compression deduced from the mean of the whole appears not to differ
much from 1/300; that given by the lunar theory has the advantage of
being independent of the irregularities of the earth’s surface and of
local attractions. The regularity with which the observed variation in
the length of the pendulum follows the law of the square of the sine of
the latitude proves the strata to be elliptical, and symmetrically
disposed round the centre of gravity of the earth, which affords a
strong presumption in favour of its original fluidity. It is remarkable
how little influence the sea has on the variation of the lengths of the
arcs of the meridian, or on gravitation; neither does it much affect the
lunar inequalities, from its density being only about a fifth of the
mean density of the earth. For, if the earth were to become fluid, after
being stripped of the ocean, it would assume the form of an ellipsoid of
revolution whose compression is 1/304·8, which differs very little from
that determined by observation, and proves, not only that the density of
the ocean is inconsiderable, but that its mean depth is very small.
There are profound cavities in the bottom of the sea, but its mean depth
probably does not much exceed the mean height of the continents and
islands above its level. On this account, immense tracts of land may be
deserted or overwhelmed by the ocean, as appears really to have been the
case, without any great change in the form of the terrestrial spheroid.
The variation in the length of the pendulum was first remarked by
Richter in 1672, while observing transits of the fixed stars across the
meridian at Cayenne, about five degrees north of the equator. He found
that his clock lost at the rate of 2^m 28^s daily, which induced him
to determine the length of a pendulum beating seconds in that latitude;
and, repeating the experiments on his return to Europe, he found the
seconds’ pendulum at Paris to be more than the twelfth of an inch longer
than that at Cayenne. The form and size of the earth being determined, a
standard of measure is furnished with which the dimensions of the solar
system may be compared.