THE EXACT SIZE AND SHAPE OF THE EARTH.
GEODESY.—We have already seen that the earth is a sphere, or of some
form which differs but little from a sphere, and a rough method of
determining its size, on this supposition, has been indicated. Now we
have to inquire more minutely into the size and shape of our planet,
for, as we shall see presently, a knowledge of these facts is essential
to the adequate explanation of the various movements of the heavenly
bodies, besides forming the basis of all our knowledge of the distances
which separate us from the other bodies which people space. As an
illustration of the importance of an exact knowledge of the size of the
earth, it may be remarked that Newton’s grand law of gravitation was
kept from the world for ten years, owing to an error in the generally
accepted value of the earth’s radius, which was afterwards rectified by
the labours of a French astronomer, Picard.
A great amount of labour has been expended in the endeavour to arrive at
the true size and shape of the earth, and the name _geodesy_ is given to
the science which deals with these operations. As a secondary object,
geodesy is concerned with the measurement and description of tracts of
country.
AN ARC OF MERIDIAN.—The measurement of the size of the earth is
accomplished by first measuring relatively small parts of its surface,
and then applying geometrical principles, in order to determine the
whole circumference. If the earth were a true sphere, and we could
measure the exact distance in miles between two places on the same
meridian, a subsequent determination of the difference of latitudes of
the two places would enable us to find the length of a degree, measured
on the earth’s circumference. As there are 360° in a circle, the
circumference would be 360 times the length of a degree, and the
diameter of the earth would be the length of the circumference divided
by 3·14159, this number expressing the constant ratio which exists
between the circumference and diameter of a circle of any size
whatsoever.
The determination of the size and shape of the earth thus involves two
distinct sets of operations; first, measures of distances; and second,
astronomical observations to determine the angular measurements of the
arcs on the earth’s surface comprised between stations separated by
known distances. When two such stations lie on the same meridian, the
arc measured in this way is called an _arc of meridian_. We have already
seen what means are available for finding the latitudes and longitudes
of places on the earth, and it now remains for us to apply a yard
measure, or its equivalent, to the precise measurement of the distance
between places which are many miles apart.
THE BASE LINE.—In the first instance a line of unimpeachable
straightness is measured with scrupulous accuracy. The measuring-rod
which has been most successfully employed is one consisting of a
combination of brass and steel bars, which automatically corrects itself
for changes of temperature in very much the same way that the
balance-wheel of a chronometer, or of a good watch, corrects itself so
as to perform its swing in equal periods at all temperatures. Several of
these compensated rods are used, and they are enclosed in wooden boxes
which are provided with levels and sights. When in use the outer boxes
rest on adjustable trestles, and instead of putting the rods end to end
they are placed a certain definite distance apart by the use of
microscopes, which are themselves mounted on compensating bars. The
first rod is put in position and levelled, and the others are
successively placed in line with it by means of the sights. As the
ground ceases to be perfectly flat it becomes necessary to raise the
level of succeeding bars, but they are kept in the same vertical plane.
Six bars are frequently employed in laying out a base line, and in order
to protect them from extremes of temperature they are usually kept
covered with long tents. In this way a distance of several miles can be
measured with no greater probable error than a couple of inches, and the
ends of such a measured base line are marked on metal plugs built in
columns of masonry. The chief base lines measured in connection with
British map construction were on the sandy shores of Lough Foyle in
Ireland, 41,614 feet in length, and on Salisbury Plain, 36,578 feet
long.
TRIANGULATION.—When a base line has been accurately measured in this
way, a distant object which is clearly visible from both ends is
observed with the aid of an instrument called the _theodolite_, and the
angles between the base line and the lines joining its ends with the
object are very carefully determined. Thus if A B in Fig. 36 represent
the base line, and C a conspicuous object several miles away, the angles
C A B and C B A are measured, and then it becomes easy to determine the
distances A C and B C by trigonometrical calculations. A check on the
accuracy of the observations is obtained by transferring the theodolite
to C and measuring the angle A C B. The sides of the triangle may then
be employed as new base lines for the measurement of other distances.
With the theodolite at C, another object, D, is sighted, and the angle D
C A is measured; similarly, with the theodolite at A, the angle C A D is
determined, and from these observations the distances of D from the
points A and C are easily computed. These distances again become
available for base lines, and so the triangulation can be extended
indefinitely.
[Illustration:
FIG. 36.—_Triangulation._
]
In a mountainous country, the sides of the triangles are often as much
as 100 miles in length. Signals on the Wicklow Mountains in Ireland have
been observed from Ben Lomond in Scotland and from Scafell in
Cumberland. The stations are chosen so that none of the angles to be
measured are very small, and in this way the chances of error are
greatly reduced. Hence the triangles in the immediate neighbourhood of
the base line are comparatively small, but the sides are gradually
extended as the survey proceeds.
The process of triangulation forms the basis of the construction of
accurate _maps_, and for this purpose the great triangles are subdivided
by a secondary triangulation, so that the exact situations of a very
great number of places are determined. These, again, serve for another
set of still smaller triangles, with sides perhaps a mile in length; and
finally the details are filled in by local chain surveys and
draughtsmanship.
There is another point of some importance in connection with these
triangulations when on a large scale. The larger triangles must be
corrected for the curvature of the earth’s surface. The construction of
the theodolite is such that two adjacent sides of any triangle, measured
from their intersection, are referred to the same horizon; but when the
instrument is transferred to another corner of the triangle, the
adjacent sides are referred to a new horizon. The sum of the three
angles of a triangle in these geodetical surveys thus exceed two right
angles, whereas in plane triangles they are always equal to two right
angles; the difference is called the _spherical excess_, and in the
computations the observed angles have to be corrected on this account.
Thus, after an extremely laborious survey, it becomes possible to
determine with great accuracy the distance between any two places
whatever, and so the number of miles between two places at the
extremities of an arc of meridian is ascertained. An arc of meridian
extending nearly 18° has been measured in India, and another over 25°
long extends from Hammerfest in Norway to the mouth of the Danube.
EXACT SHAPE AND SIZE OF THE EARTH.—From the facts which have been
gleaned by the measurements of arcs of meridian in different parts of
the world, it is found that the length of a degree of latitude as
measured on the earth’s circumference increases towards the Poles. In
latitude 66° N. a degree is about 3,000 feet longer than a degree near
the Equator. This means that the curvature of a meridional arc is
greatest at the Equator, whence it is concluded that the earth is
flattened at the Poles. The figure which best accords with the
observations is the ellipse, and thus it becomes possible to calculate
the polar diameter, although no arcs have been measured in the immediate
neighbourhood of the Poles.
Arcs of longitude, extending between two places which have the same
latitude, have also been measured and applied to the determination of
the figure of the earth, and, indeed, any arcs between two places of
known latitude and longitude can be utilised.
When all the facts are brought together it is found that the earth’s
polar diameter is about 26 miles shorter than the average equatorial
diameter, while an equatorial section of the earth is also elliptical,
the diameter passing through longitude 14° E, being two miles longer
than the one at right angles to it. According to the calculations of
Colonel Clarke, R.E., we have the following principal dimensions:
Earth’s mean equatorial semi-diameter = 3,963·296 miles.
„ „ polar „ = 3,950·738 „
Polar compression ¹⁄₂₉₃.₄₆
A solid which has a shape like that of the earth, with three axes of
unequal lengths, is called an _ellipsoid_.
A very important consequence of the ellipsoidal form of the earth is
that lines which are vertical—that is, perpendicular to the surface of
water—do not pass through the centre of the earth, unless they are at
the Poles or at certain points on the Equator.
There is every reason to suppose that at one time the earth was in a
molten condition, and in response to physical laws, such a mass of
matter could not retain a spherical form when set in rotation, although
the sphere would be its natural shape if at rest. This has been
demonstrated by a variety of experiments.
Thus, taking it generally, the shape of the earth is very intimately
associated with its rotation, and it will subsequently appear that the
same holds good for the sun and planets. Those bodies which have the
most rapid rotation show the greatest flattening in the direction of the
polar diameter.
In addition to direct measurements of the earth, there are other ways of
studying the shape of our planet. One of these depends upon observations
of the swing of a pendulum at different parts of the earth’s surface; as
the time of oscillation of a pendulum depends upon the force of gravity,
which itself varies with the distance from the earth’s centre, it is
evident that this method is a practicable one. It is true that the
matter is complicated in various ways, but after everything has been
taken into account, these pendulum observations indicate, not only that
the earth is flattened at the Poles, but they show further that the
amount of polar compression deduced from geodetical work is in all
probably very near the truth.
Again, the movement of the moon around the earth is found to be subject
to certain irregularities which would not exist if the earth were a
perfect sphere. These inequalities being deduced from observations of
the moon’s position, the amount of polar flattening necessary to produce
them can be calculated, and this is found to agree very closely with the
value derived from the measurements of arcs of meridian.
DIFFERENT KINDS OF LATITUDE.—If the earth were a smooth spherical body,
the latitude of a place would be simply equal to the angle made by a
line joining it to the earth’s centre with the plane of the Equator.
Owing to the bulging out of the earth in its equatorial part, however,
it becomes necessary to distinguish between different kinds of latitude.
If we adopt the definition given above, the name of _geocentric
latitude_ is given to the angular measurement. Taking the earth as a
smooth geometrical spheroid, and assuming it to have certain dimensions,
the angle which a line perpendicular to the surface makes with the plane
of the Equator determines the _geographical latitude_. As the line
perpendicular to the surface does not pass quite through the centre of
the earth, the geographical and geocentric latitude differ by as much as
11′ in mid-latitudes, although nearly agreeing at the Poles and on the
Equator.
As there are no direct means of finding the direction of a line passing
through the earth’s centre, or of one perpendicular to the imaginary
standard spheroid, geocentric and geographical latitudes must be
calculated from the _astronomical latitude_, which is determined by
observations of the elevation of the Pole, or its equivalent. The
astronomical latitude is the angle between the direction of gravity and
the Equator, and is therefore to a small extent dependent upon local
irregularities of the earth’s surface.
A knowledge of geocentric latitude is chiefly of use in making
corrections for parallax, in order that the data calculated for the
earth’s centre may be precisely corrected for the place of observation,
or _vice versâ_, as in the case of a lunar distance measured for the
determination of longitude, or in the calculation of a solar eclipse.
VARIATION OF LATITUDE.—For some years past a widespread interest has
been taken in the question of a possible change in the position of the
earth’s axis with regard to its surface. The subject is by no means a
new one, for as far back as two thousand years ago, such variations were
suspected. Changes amounting to several degrees were then believed to
have occurred, but it is now certain that the supposed variation was due
solely to the imperfection of the observations. As astronomical science
became more and more precise, even before the discovery of aberration,
it became evident that if any changes of latitude were taking place at
all, they must be very minute.
In its geological aspect, the possibility of great changes of latitude
having occurred in the past history of our globe is evidently well worth
serious investigation. Granted a sufficient change in the position of
the earth’s axis, the climate of London might become Arctic, or that of
Greenland tropical. From this point of view the subject has been
mathematically investigated by Professor G. H. Darwin, and it appears
that if only the varying distribution of land and sea indicated by the
geological records be taken into account, past changes of more than
about three degrees are very improbable. Admitting that at any time
during the life-history of our globe the earth was sufficiently plastic
to be deformed by earthquakes or other disturbances, it is possible that
changes amounting to 10° or 15° may have occurred.
Opinion is perhaps best reserved as to what has happened in the past. We
are on surer ground when we consider the variations of latitude which
are now going on.
Many competent observers have investigated the present movements of the
Pole, and it has been conclusively demonstrated that changes in the
position of the earth’s axis do really occur. Dr. Küstner, of Berlin,
commenced a series of observations for a different purpose in 1884, and
found that some anomalous results could only be explained by supposing
that the latitude of Berlin was from 0″·2 to 0″·3 greater from August to
November, 1884, than from March to May in 1884 and 1885. Great interest
was excited by this striking result, and steps were at once taken to
test its truth. Old observations were re-discussed and compared, and new
observations were made, with the final result that the movement of the
earth’s axis of rotation was placed beyond dispute. It was not until Dr.
Chandler attacked the problem, however, in 1891, that the nature of the
changes became clear. His masterly analysis indicated that the observed
variations in latitude arise from two periodic fluctuations superposed
upon each other; one of these has a period of 427 days, and a
semi-amplitude of 0″·12, while the other is an annual change which has
ranged between 0″·04 and 0″·20 during the last fifty years. The
resultant of the two movements produces changes which are seemingly very
irregular in amount and of varying period, but a cycle is completed
about every seven years. When the two sources of difference are at their
maximum at the same time, the total range reaches about two-thirds of a
second of arc. In consequence of the inequality of the annual part of
the change, the apparent average period between 1840 and 1855
approximated to 380 or 390 days; widely fluctuated from 1855 to 1865;
from 1865 to about 1885 was very nearly 427 days, afterwards increased
to near 440 days, and very recently fell to somewhat below 400 days.
[Illustration:
FIG. 37.—_Movements of the Earth’s Pole, 1890–95._
]
At the present time the variation of latitude is being very carefully
investigated by the International Geodetic Association, and the latest
results obtained are illustrated diagrammatically in Fig. 37. The mean
position of the Pole is at the centre of the diagram,[3] and the
horizontal line to the right of this point is directed towards
Greenwich. The remarkable spiral curve shows the wanderings of the Pole
about its mean position during five recent years. To simplify matters,
the amount of deviation is represented in feet instead of in angular
measure, and it will be seen that although the variation of latitude may
be of considerable interest and importance in astronomical matters, it
really does not amount to very much in matters terrestrial, the greatest
change in the position of the Pole not amounting to more than 20 yards.
Nevertheless, it is not inconceivable that it may yet have to be
reckoned with in questions relating to boundary lines which depend upon
latitude determinations.