CHAPTER X.
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.