1911 Encyclopædia Britannica/Earth, Figure of the

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EARTH, FIGURE OF THE. The determination of the figure of the earth is a problem of the highest importance in astronomy, inasmuch as the diameter of the earth is the unit to which all celestial distances must be referred.


Reasoning from the uniform level appearance of the horizon, the variations in altitude of the circumpolar stars as one travels towards the north or south, the disappearance of a ship standing out to sea, and perhaps other phenomena, the earliest astronomers regarded the earth as a sphere, and they endeavoured to ascertain its dimensions. Aristotle relates that the mathematicians had found the circumference to be 400,000 stadia (about 46,000 miles). But Eratosthenes (c. 250 B.C.) appears to have been the first who entertained an accurate idea of the principles on which the determination of the figure of the earth really depends, and attempted to reduce them to practice. His results were very inaccurate, but his method is the same as that which is followed at the present day—depending, in fact, on the comparison of a line measured on the earth’s surface with the corresponding arc of the heavens. He observed that at Syene in Upper Egypt, on the day of the summer solstice, the sun was exactly vertical, whilst at Alexandria at the same season of the year its zenith distance was 7° 12′, or one-fiftieth of the circumference of a circle. He assumed that these places were on the same meridian; and, reckoning their distance apart as 5000 stadia, he inferred that the circumference of the earth was 250,000 stadia (about 29,000 miles). A similar attempt was made by Posidonius, who adopted a method which differed from that of Eratosthenes only in using a star instead of the sun. He obtained 240,000 stadia (about 27,600 miles) for the circumference. Ptolemy in his Geography assigns the length of the degree as 500 stadia.

The Arabs also investigated the question of the earth’s magnitude. The caliph Abdallah al Mamun (A.D. 814), having fixed on a spot in the plains of Mesopotamia, despatched one company of astronomers northwards and another southwards, measuring the journey by rods, until each found the altitude of the pole to have changed one degree. But the result of this measurement does not appear to have been very satisfactory. From this time the subject seems to have attracted no attention until about 1500, when Jean Fernel (1497-1558), a Frenchman, measured a distance in the direction of the meridian near Paris by counting the number of revolutions of the wheel of a carriage. His astronomical observations were made with a triangle used as a quadrant, and his resulting length of a degree was very near the truth.

Willebrord Snell[1] substituted a chain of triangles for actual linear measurement. He measured his base line on the frozen surface of the meadows near Leiden, and measured the angles of his triangles, which lay between Alkmaar and Bergen-op-Zoom, with a quadrant and semicircles. He took the precaution of comparing his standard with that of the French, so that his result was expressed in toises (the length of the toise is about 6.39 English ft.). The work was recomputed and reobserved by P. von Musschenbroek in 1729. In 1637 an Englishman, Richard Norwood, published a determination of the figure of the earth in a volume entitled The Seaman’s Practice, contayning a Fundamentall Probleme in Navigation experimentally verified, namely, touching the Compasse of the Earth and Sea and the quantity of a Degree in our English Measures. He observed on the 11th of June 1633 the sun’s meridian altitude in London as 62° 1′, and on the 6th of June 1635, his meridian altitude in York as 59° 33′. He measured the distance between these places partly with a chain and partly by pacing. By this means, through compensation of errors, he arrived at 367,176 ft. for the degree—a very fair result.

The application of the telescope to angular instruments was the next important step. Jean Picard was the first who in 1669, with the telescope, using such precautions as the nature of the operation requires, measured an arc of meridian. He measured with wooden rods a base line of 5663 toises, and a second or base of verification of 3902 toises; his triangulation extended from Malvoisine, near Paris, to Sourdon, near Amiens. The angles of the triangles were measured with a quadrant furnished with a telescope having cross-wires. The difference of latitude of the terminal stations was determined by observations made with a sector on a star in Cassiopeia, giving 1° 22′ 55″ for the amplitude. The terrestrial measurement gave 78,850 toises, whence he inferred for the length of the degree 57,060 toises.

Hitherto geodetic observations had been confined to the determination of the magnitude of the earth considered as a sphere, but a discovery made by Jean Richer (d. 1696) turned the attention of mathematicians to its deviation from a spherical form. This astronomer, having been sent by the Academy of Sciences of Paris to the island of Cayenne, in South America, for the purpose of investigating the amount of astronomical refraction and other astronomical objects, observed that his clock, which had been regulated at Paris to beat seconds, lost about two minutes and a half daily at Cayenne, and that in order to bring it to measure mean solar time it was necessary to shorten the pendulum by more than a line (about 112th of an in.). This fact, which was scarcely credited till it had been confirmed by the subsequent observations of Varin and Deshayes on the coasts of Africa and America, was first explained in the third book of Newton’s Principia, who showed that it could only be referred to a diminution of gravity arising either from a protuberance of the equatorial parts of the earth and consequent increase of the distance from the centre, or from the counteracting effect of the centrifugal force. About the same time (1673) appeared Christian Huygens’ De Horologio Oscillatorio, in which for the first time were found correct notions on the subject of centrifugal force. It does not, however, appear that they were applied to the theoretical investigation of the figure of the earth before the publication of Newton’s Principia. In 1690 Huygens published his De Causa Gravitatis, which contains an investigation of the figure of the earth on the supposition that the attraction of every particle is towards the centre.

Between 1684 and 1718 J. and D. Cassini, starting from Picard’s base, carried a triangulation northwards from Paris to Dunkirk and southwards from Paris to Collioure. They measured a base of 7246 toises near Perpignan, and a somewhat shorter base near Dunkirk; and from the northern portion of the arc, which had an amplitude of 2° 12′ 9″, obtained for the length of a degree 56,960 toises; while from the southern portion, of which the amplitude was 6° 18′ 57″, they obtained 57,097 toises. The immediate inference from this was that, the degree diminishing with increasing latitude, the earth must be a prolate spheroid. This conclusion was totally opposed to the theoretical investigations of Newton and Huygens, and accordingly the Academy of Sciences of Paris determined to apply a decisive test by the measurement of arcs at a great distance from each other—one in the neighbourhood of the equator, the other in a high latitude. Thus arose the celebrated expeditions of the French academicians. In May 1735 Louis Godin, Pierre Bouguer and Charles Marie de la Condamine, under the auspices of Louis XV., proceeded to Peru, where, assisted by two Spanish officers, after ten years of laborious exertion, they measured an arc of 3° 7′, the northern end near the equator. The second party consisted of Pierre Louis Moreau de Maupertuis, Alexis Claude Clairault, Charles Étienne Louis Camus, Pierre Charles Lemonnier, and Reginaud Outhier, who reached the Gulf of Bothnia in July 1736; they were in some respects more fortunate than the first party, inasmuch as they completed the measurement of an arc near the polar circle of 57′ amplitude and returned within sixteen months from the date of their departure.

The measurement of Bouguer and De la Condamine was executed with great care, and on account of the locality, as well as the manner in which all the details were conducted, it has always been regarded as a most valuable determination. The southern limit was at Tarqui, the northern at Cotchesqui. A base of 6272 toises was measured in the vicinity of Quito, near the northern extremity of the arc, and a second base of 5260 toises near the southern extremity. The mountainous nature of the country made the work very laborious, in some cases the difference of heights of two neighbouring stations exceeding 1 mile; and they had much trouble with their instruments, those with which they were to determine the latitudes proving untrustworthy. But they succeeded by simultaneous observations of the same star at the two extremities of the arc in obtaining very fair results. The whole length of the arc amounted to 176,945 toises, while the difference of latitudes was 3° 7′ 3″. In consequence of a misunderstanding that arose between De la Condamine and Bouguer, their operations were conducted separately, and each wrote a full account of the expedition. Bouguer’s book was published in 1749; that of De la Condamine in 1751. The toise used in this measure was afterwards regarded as the standard toise, and is always referred to as the Toise of Peru.

The party of Maupertuis, though their work was quickly despatched, had also to contend with great difficulties. Not being able to make use of the small islands in the Gulf of Bothnia for the trigonometrical stations, they were forced to penetrate into the forests of Lapland, commencing operations at Torneå, a city situated on the mainland near the extremity of the gulf. From this, the southern extremity of their arc, they carried a chain of triangles northward to the mountain Kittis, which they selected as the northern terminus. The latitudes were determined by observations with a sector (made by George Graham) of the zenith distance of α and δ Draconis. The base line was measured on the frozen surface of the river Torneå about the middle of the arc; two parties measured it separately, and they differed by about 4 in. The result of the whole was that the difference of latitudes of the terminal stations was 57′ 29″ .6, and the length of the arc 55,023 toises. In this expedition, as well as in that to Peru, observations were made with a pendulum to determine the force of gravity; and these observations coincided with the geodetic results in proving that the earth was an oblate and not prolate spheroid.

In 1740 was published in the Paris Mémoires an account, by Cassini de Thury, of a remeasurement by himself and Nicolas Louis de Lacaille of the meridian of Paris. With a view to determine more accurately the variation of the degree along the meridian, they divided the distance from Dunkirk to Collioure into four partial arcs of about two degrees each, by observing the latitude at five stations. The results previously obtained by J. and D. Cassini were not confirmed, but, on the contrary, the length of the degree derived from these partial arcs showed on the whole an increase with an increasing latitude. Cassini and Lacaille also measured an arc of parallel across the mouth of the Rhone. The difference of time of the extremities was determined by the observers at either end noting the instant of a signal given by flashing gunpowder at a point near the middle of the arc.

While at the Cape of Good Hope in 1752, engaged in various astronomical observations, Lacaille measured an arc of meridian of 1° 13′ 17″, which gave him for the length of the degree 57,037 toises—an unexpected result, which has led to the remeasurement of the arc by Sir Thomas Maclear (see Geodesy).

Passing over the measurements made between Rome and Rimini and on the plains of Piedmont by the Jesuits Ruggiero Giuseppe Boscovich and Giovanni Battista Beccaria, and also the arc measured with deal rods in North America by Charles Mason and Jeremiah Dixon, we come to the commencement of the English triangulation. In 1783, in consequence of a representation from Cassini de Thury on the advantages that would accrue from the geodetic connexion of Paris and Greenwich, General William Roy was, with the king’s approval, appointed by the Royal Society to conduct the operations on the part of England, Count Cassini, Méchain and Delambre being appointed on the French side. A precision previously unknown was attained by the use of Ramsden’s theodolite, which was the first to make the spherical excess of triangles measurable. The wooden rods with which the first base was measured were replaced by glass rods, which were afterwards rejected for the steel chain of Ramsden. (For further details see Account of the Trigonometrical Survey of England and Wales.)

Shortly after this, the National Convention of France, having agreed to remodel their system of weights and measures, chose for their unit of length the ten-millionth part of the meridian quadrant. In order to obtain this length precisely, the remeasurement of the French meridian was resolved on, and deputed to J. B. J. Delambre and Pierre François André Méchain. The details of this operation will be found in the Base du système métrique décimale. The arc was subsequently extended by Jean Baptiste Biot and Dominique François Jean Arago to the island of Iviza. Operations for the connexion of England with the continent of Europe were resumed in 1821 to 1823 by Henry Kater and Thomas Frederick Colby on the English side, and F. J. D. Arago and Claude Louis Mathieu on the French.

The publication in 1838 of Friedrich Wilhelm Bessel’s Gradmessung in Ostpreussen marks an era in the science of geodesy. Here we find the method of least squares applied to the calculation of a network of triangles and the reduction of the observations generally. The systematic manner in which all the observations were taken with the view of securing final results of extreme accuracy is admirable. The triangulation, which was a small one, extended about a degree and a half along the shores of the Baltic in a N.N.E. direction. The angles were observed with theodolites of 12 and 15 in. diameter, and the latitudes determined by means of the transit instrument in the prime vertical—a method much used in Germany. (The base apparatus is described in the article Geodesy.)

The principal triangulation of Great Britain and Ireland, which was commenced in 1783 under General Roy, for the more immediate purpose of connecting the observatories of Greenwich and Paris, had been gradually extended, under the successive direction of Colonel E. Williams, General W. Mudge, General T. F. Colby, Colonel L. A. Hall, and Colonel Sir Henry James; it was finished in 1851. The number of stations is about 250. At 32 of these the latitudes were determined with Ramsden’s and Airy’s zenith sectors. The theodolites used for this work were, in addition to the two great theodolites of Ramsden which were used by General Roy and Captain Kater, a smaller theodolite of 18 in. diameter by the same mechanician, and another of 24 in. diameter by Messrs Troughton and Simms. Observations for determination of absolute azimuth were made with those instruments at a large number of stations; the stars α, δ, and λ Ursae Minoris and 51 Cephei being those observed always at the greatest azimuths. At six of these stations the probable error of the result is under 0.4″, at twelve under 0.5″, at thirty-four under 0.7″: so that the absolute azimuth of the whole network is determined with extreme accuracy. Of the seven base lines which have been measured, five were by means of steel chains and two with Colby’s compensation bars (see Geodesy). The triangulation was computed by least squares. The total number of equations of condition for the triangulation is 920; if therefore the whole had been reduced in one mass, as it should have been, the solution of an equation of 920 unknown quantities would have occurred as a part of the work. To avoid this an approximation was resorted to; the triangulation was divided into twenty-one parts or figures; four of these, not adjacent, were first adjusted by the method explained, and the corrections thus determined in these figures carried into the equations of condition of the adjacent figures. The average number of equations in a figure is 44; the largest equation is one of 77 unknown quantities. The vertical limb of Airy’s zenith sector is read by four microscopes, and in the complete observation of a star there are 10 micrometer readings and 12 level readings. The instrument is portable; and a complete determination of latitude, affected with the mean of the declination errors of two stars, is effected by two micrometer readings and four level readings. The observation consists in measuring with the telescope micrometer the difference of zenith distances of two stars which cross the meridian, one to the north and the other to the south of the observer at zenith distances which differ by not much more than 10′ or 15′, the interval of the times of transit being not less than one nor more than twenty minutes. The advantages are that, with simplicity in the construction of the instrument and facility in the manipulation, refraction is eliminated (or nearly so, as the stars are generally selected within 25° of the zenith), and there is no large divided circle. The telescope, which is counterpoised on one side of the vertical axis, has a small circle for finding, and there is also a small horizontal circle. This instrument is universally used in American geodesy.

The principal work containing the methods and results of these operations was published in 1858 with the title “Ordnance Trigonometrical Survey of Great Britain and Ireland. Account of the observations and calculations of the principal triangulation and of the figure, dimensions and mean specific gravity of the earth as derived therefrom. Drawn up by Captain Alexander Ross Clarke, R.E., F.R.A.S., under the direction of Lieut.-Colonel H. James, R.E., F.R.S., M.R.I.A., &c.” A supplement appeared in 1862: “Extension of the Triangulation of the Ordnance Survey into France and Belgium, with the measurement of an arc of parallel in 52° N. from Valentia in Ireland to Mount Kemmel in Belgium. Published by ... Col. Sir Henry James.”

Extensive operations for surveying India and determining the figure of the earth were commenced in 1800. Colonel W. Lambton started the great meridian arc at Punnae in latitude 8° 9′, and, following generally the methods of the English survey, he carried his triangulation as far north as 20° 30′. The work was continued by Sir George (then Captain) Everest, who carried it to the latitude of 29° 30′. Two admirable volumes by Sir George Everest, published in 1830 and in 1847, give the details of this undertaking. The survey was afterwards prosecuted by Colonel T. T. Walker, R.E., who made valuable contributions to geodesy. The working out of the Indian chains of triangle by the method of least squares presents peculiar difficulties, but, enormous in extent as the work was, it has been thoroughly carried out. The ten base lines on which the survey depends were measured with Colby’s compensation bars.

The survey is detailed in eighteen volumes, published at Dehra Dun, and entitled Account of the Operations of the Great Trigonometrical Survey of India. Of these the first nine were published under the direction of Colonel Walker; and the remainder by Colonels Strahan and St G. C. Gore, Major S. G. Burrard and others. Vol. i., 1870, treats of the base lines; vol. ii., 1879, history and general descriptions of the principal triangulation and of its reduction; vol. v., 1879, pendulum operations (Captains T. P. Basevi and W. T. Heaviside); vols. xi., 1890, and xviii., 1906, latitudes; vols. ix., 1883, x., 1887, xv., 1893, longitudes; vol. xvii., 1901, the Indo-European longitude-arcs from Karachi to Greenwich. The other volumes contain the triangulations.

In 1860 Friedrich Georg Wilhelm Struve published his Arc du méridien de 25° 20′ entre le Danube et la Mer Glaciale mesuré depuis 1816 jusqu’en 1855. The latitudes of the thirteen astronomical stations of this arc were determined partly with vertical circles and partly by means of the transit instrument in the prime vertical. The triangulation, a great part of which, however, is a simple chain of triangles, is reduced by the method of least squares, and the probable errors of the resulting distances of parallels is given; the probable error of the whole arc in length is ± 6.2 toises. Ten base lines were measured. The sum of the lengths of the ten measured bases is 29,863 toises, so that the average length of a base line is 19,100 ft. The azimuths were observed at fourteen stations. In high latitudes the determination of the meridian is a matter of great difficulty; nevertheless the azimuths at all the northern stations were successfully determined,—the probable error of the result at Fuglenaes being ± 0″.53.

Before proceeding with the modern developments of geodetic measurements and their application to the figure of the earth, we must discuss the “mechanical theory,” which is indispensable for a full understanding of the subject.

Mechanical Theory.

Newton, by applying his theory of gravitation, combined with the so-called centrifugal force, to the earth, and assuming that an oblate ellipsoid of rotation is a form of equilibrium for a homogeneous fluid rotating with uniform angular velocity, obtained the ratio of the axes 229:230, and the law of variation of gravity on the surface. A few years later Huygens published an investigation of the figure of the earth, supposing the attraction of every particle to be towards the centre of the earth, obtaining as a result that the proportion of the axes should be 578 : 579. In 1740 Colin Maclaurin, in his De causa physica fluxus et refluxus maris, demonstrated that the oblate ellipsoid of revolution is a figure which satisfies the conditions of equilibrium in the case of a revolving homogeneous fluid mass, whose particles attract one another according to the law of the inverse square of the distance; he gave the equation connecting the ellipticity with the proportion of the centrifugal force at the equator to gravity, and determined the attraction on a particle situated anywhere on the surface of such a body. In 1743 Clairault published his Théorie de la figure de la terre, which contains a remarkable theorem (“Clairault’s Theorem”), establishing a relation between the ellipticity of the earth and the variation of gravity from the equator to the poles. Assuming that the earth is composed of concentric ellipsoidal strata having a common axis of rotation, each stratum homogeneous in itself, but the ellipticities and densities of the successive strata varying according to any law, and that the superficial stratum has the same form as if it were fluid, he proved that

\frac{g' - g}{g} + e {{=}} \frac{5}{2}m,\!

where g, g'\! are the amounts of gravity at the equator and at the pole respectively, e\! the ellipticity of the meridian (or “flattening”), and m\! the ratio of the centrifugal force at the equator to g\!. He also proved that the increase of gravity in proceeding from the equator to the poles is as the square of the sine of the latitude. This, taken with the former theorem, gives the means of determining the earth’s ellipticity from observation of the relative force of gravity at any two places. P. S. Laplace, who devoted much attention to the subject, remarks on Clairault’s work that “the importance of all his results and the elegance with which they are presented place this work amongst the most beautiful of mathematical productions” (Isaac Todhunter’s History of the Mathematical Theories of Attraction and the Figure of the Earth, vol. i. p. 229).

The problem of the figure of the earth treated as a question of mechanics or hydrostatics is one of great difficulty, and it would be quite impracticable but for the circumstance that the surface differs but little from a sphere. In order to express the forces at any point of the body arising from the attraction of its particles, the form of the surface is required, but this form is the very one which it is the object of the investigation to discover; hence the complexity of the subject, and even with all the present resources of mathematicians only a partial and imperfect solution can be obtained.

We may here briefly indicate the line of reasoning by which some of the most important results may be obtained. If \mathrm{X}, \mathrm{Y}, \mathrm{Z}\! be the components parallel to three rectangular axes of the forces acting on a particle of a fluid mass at the point x, y, z\!, then, p\! being the pressure there, and \rho\! the density,

dp {{=}} \rho (\mathrm{X}dx + \mathrm{Y}dy + \mathrm{Z}dz);\!

and for equilibrium the necessary conditions are, that \rho (\mathrm{X}dx + \mathrm{Y}dy + \mathrm{Z}dz)\! be a complete differential, and at the free surface \mathrm{X}dx + \mathrm{Y}dy + \mathrm{Z}dz {{=}} 0\!. This equation implies that the resultant of the forces is normal to the surface at every point, and in a homogeneous fluid it is obviously the differential equation of all surfaces of equal pressure. If the fluid be heterogeneous then it is to be remarked that for forces of attraction according to the ordinary law of gravitation, if \mathrm{X}, \mathrm{Y}, \mathrm{Z}\! be the components of the attraction of a mass whose potential is \mathrm{V}\!, then

\mathrm{X}dx + \mathrm{Y}dy + \mathrm{Z}dz {{=}} \frac{d\mathrm{V}}{dx}dx + \frac{d\mathrm{V}}{dy}dy + \frac{d\mathrm{V}}{dz}dz,\!

which is a complete differential. And in the case of a fluid rotating with uniform velocity, in which the so-called centrifugal force enters as a force acting on each particle proportional to its distance from the axis of rotation, the corresponding part of \mathrm{X}dx + \mathrm{Y}dy + \mathrm{Z}dz\! is obviously a complete differential. Therefore for the forces with which we are now concerned \mathrm{X}dx + \mathrm{Y}dy + \mathrm{Z}dz {{=}} d\mathrm{U}\!, where \mathrm{U}\! is some function of x, y, z\!, and it is necessary for equilibrium that dp {{=}} \rho d\mathrm{U}\! be a complete differential; that is, \rho\! must be a function of \mathrm{U}\! or a function of p\!, and so also p\! a function of \mathrm{U}\!. So that d\mathrm{U} {{=}} 0\! is the differential equation of surfaces of equal pressure and density.

We may now show that a homogeneous fluid mass in the form of an oblate ellipsoid of revolution having a uniform velocity of rotation can be in equilibrium. It may be proved that the attraction of the ellipsoid x^2 + y^2 + z^2(1 + \epsilon^2) {{=}} c^2(1 + \epsilon^2)\!; upon a particle \mathrm{P}\! of its mass at x, y, z\! has for components

\mathrm{X} {{=}} - \mathrm{A}x, \mathrm{Y} {{=}} - \mathrm{A}y, \mathrm{Z} {{=}} - \mathrm{C}z,\!


\mathrm{A} {{=}} 2\pi k^2\rho \left(\frac{1 + \epsilon^2}{\epsilon^3}\tan^{-1}\epsilon -\frac{1}{\epsilon^2}\right),\!

\mathrm{C} {{=}} 4\pi k^2\rho \left(\frac{1 + \epsilon^2}{\epsilon^2} - \frac{1 + \epsilon^2}{\epsilon^3}\tan^{-1}\epsilon \right),\!

and k^2\! the constant of attraction. Besides the attraction of the mass of the ellipsoid, the centrifugal force at \mathrm{P}\! has for components + x\omega^2, + y\omega^2, 0\!; then the condition of fluid equilibrium is

(\mathrm{A} - \omega^2)xdx + (\mathrm{A} - \omega^2)ydy + \mathrm{C}zdz {{=}} 0,\!

which by integration gives

(\mathrm{A} - \omega^2)(x^2 + y^2) + \mathrm{C}z^2 {{=}} \text{constant}.\!

This is the equation of an ellipsoid of rotation, and therefore the equilibrium is possible. The equation coincides with that of the surface of the fluid mass if we make

\mathrm{A} - \omega^2 {{=}} \mathrm{C} / (1 + \epsilon^2),\!

which gives

\frac{\omega^2}{2\pi k^2\rho} {{=}} \frac{3 + \epsilon^2}{\epsilon^3}\tan^{-1}\epsilon - \frac{3}{\epsilon^2}.\!

In the case of the earth, which is nearly spherical, we obtain by expanding the expression for \omega^2\! in powers of \epsilon^2\!, rejecting the higher powers, and remarking that the ellipticity e {{=}} \tfrac{1}{2}\epsilon^2\!,

\omega^2 / 2\pi k^2\rho {{=}} 4\epsilon^2 / 15 {{=}} 8e / 15.\!

Now if m\! be the ratio of the centrifugal force to the intensity of gravity at the equator, and a {{=}} c(1 + e)\!, then

m {{=}} a\omega^2 / \frac{4}{3}\pi k^2\rho a, \therefore \omega^2 / 2\pi k^2\rho {{=}} \frac{2}{3}m.\!

In the case of the earth it is a matter of observation that m {{=}} 1/289\!, hence the ellipticity

e {{=}} 5m / 4 {{=}} 1/231,\!

so that the ratio of the axes on the supposition of a homogeneous fluid earth is 230 : 231, as stated by Newton.

Now, to come to the case of a heterogeneous fluid, we shall assume that its surfaces of equal density are spheroids, concentric and having a common axis of rotation, and that the ellipticity of these surfaces varies from the centre to the outer surface, the density also varying. In other words, the body is composed of homogeneous spheroidal shells of variable density and ellipticity. On this supposition we shall express the attraction of the mass upon a particle in its interior, and then, taking into account the centrifugal force, form the equation expressing the condition of fluid equilibrium. The attraction of the homogeneous spheroid x^2 + y^2 + z^2(1 + 2e) {{=}} c^2(1 + 2e)\!, where e\! is the ellipticity (of which the square is neglected), on an internal particle, whose co-ordinates are x {{=}} f, y {{=}} 0, z {{=}} h\!, has for its x\! and z\! components

\mathrm{X}' {{=}} -\frac{4}{3}\pi k^2\rho f(1 - \frac{2}{5}e), \qquad \mathrm{Z}' {{=}} -\frac{4}{3}\pi k^2\rho h(1 + \frac{4}{5}e),

the \mathrm{Y}\! component being of course zero. Hence we infer that the attraction of a shell whose inner surface has an ellipticity e\!, and its outer surface an ellipticity e + de\!, the density being \rho\!, is expressed by

d\mathrm{X}' {{=}} \frac{4}{3} \cdot \frac{2}{5}\pi k^2\rho f de, \qquad d\mathrm{Z}' {{=}} -\frac{4}{3} \cdot \frac{4}{5}\pi k^2\rho h de.\!

To apply this to our heterogeneous spheroid; if we put c_{1}\! for the semiaxis of that surface of equal density on which is situated the attracted point \mathrm{P}\!, and c_{0}\! for the semiaxis of the outer surface, the attraction of that portion of the body which is exterior to \mathrm{P}\!, namely, of all the shells which enclose \mathrm{P}\!, has for components

\mathrm{X}_{0} {{=}} \frac{8}{15}\pi k^2f \int^{c0}_{c1} \rho \frac{de}{dc}dc, \qquad \mathrm{Z}_{0} {{=}} \frac{16}{15}\pi k^2h \int^{c0}_{c1} \rho \frac{de}{dc}dc,\!

both e\! and \rho\! being functions of c\!. Again the attraction of a homogeneous spheroid of density \rho\! on an external point f, h\! has the components

\mathrm{X}'' {{=}} -\frac{4}{3}\pi k^2\rho fr^{-3} {c^3(1 + 2e) - \lambda ec^{5}},\!

\mathrm{Z}'' {{=}} -\frac{4}{3}\pi k^2\rho hr^{-3} {c^3(1 + 2e) - \lambda'ec^{5}},\!

\text{where }\lambda {{=}} \frac{3}{5}(4h^2 - f^2) / r^{4}, \qquad \lambda' {{=}} \frac{3}{5}(2h^2 - 3f^2) / r^{4}, \qquad\text{ and }r^2 {{=}} f^2 + h^2.\!

Now e\! being considered a function of c\!, we can at once express the attraction of a shell (density \rho\!) contained between the surface defined by c + dc, e + de\! and that defined by c, e\! upon an external point; the differentials with respect to c\!, viz. d\mathrm{X}'' d\mathrm{Z}''\!, must then be integrated with \rho\! under the integral sign as being a function of c\!. The integration will extend from c {{=}} 0\! to c {{=}} c_{1}\!. Thus the components of the attraction of the heterogeneous spheroid upon a particle within its mass, whose co-ordinates are f, 0, h\!, are

\mathrm{X} {{=}} -\frac{4}{3}\pi k^2f \left[\frac{1}{r^{3}}\int^{c1}_{0} \rho d{c^3(1 + 2e)} - \frac{\lambda}{r^{3}}\int^{c1}_{0} \rho d(ec^{5}) + \frac{2}{5} \int^{c0}_{c1} \rho de\right],

\mathrm{Z} {{=}} -\frac{4}{3}\pi k^2h \left[\frac{1}{r^{3}}\int^{c1}_{0} \rho d{c^3(1 + 2e)} - \frac{\lambda'}{r^{3}}\int^{c1}_{0} \rho d(ec^{5}) + \frac{4}{5} \int^{c0}_{c1} \rho de\right].

We take into account the rotation of the earth by adding the centrifugal force f\omega^2 {{=}} \mathrm{F}\! to \mathrm{X}\!. Now, the surface of constant density upon which the point f, 0, h\! is situated gives (1 - 2e) fdf + hdh {{=}} 0\!; and the condition of equilibrium is that (\mathrm{X} + \mathrm{F})df + Zdh {{=}} 0\!. Therefore,

(\mathrm{X} + \mathrm{F}) h {{=}} Zf (1 - 2e),\!

which, neglecting small quantities of the order e^2\! and putting \omega^2t^2 {{=}} 4\pi^2k^2\!, gives

\frac{2e}{r^3}\int^{c1}_{0} \rho d{c^3(1 + 2e)} - \frac{6}{5r^{5}}\int^{c1}_{0} \rho d(ec^{5}) - \frac{6}{5}\int^{c1}_{0} \rho de {{=}} \frac{3\pi}{t^2}.

Here we must now put c\! for c_{1}, c\! for r\!; and 1 + 2e\! under the first integral sign may be replaced by unity, since small quantities of the second order are neglected. Two differentiations lead us to the following very important differential equation (Clairault):

\frac{d^2e}{dc^2} + \frac{2\rho c^2}{\int \rho c^2 dc}\cdot\frac{de}{dc} + \left(\frac{2\rho c}{\int \rho c^2 dc} - \frac{6}{c^2}\right) e {{=}} 0.

When \rho\! is expressed in terms of c\!, this equation can be integrated. We infer then that a rotating spheroid of very small ellipticity, composed of fluid homogeneous strata such as we have specified, will be in equilibrium; and when the law of the density is expressed, the law of the corresponding ellipticities will follow.

If we put \mathrm{M}\! for the mass of the spheroid, then

\mathrm{M} {{=}} \frac{4\pi}{3}\int^{c}_0 \rho d{c^3(1 + 2e)}; \qquad\text{ and }m {{=}} \frac{c^3}{\mathrm{M}}\cdot\frac{4\pi^2}{t^2},

and putting c {{=}} c_{0}\! in the equation expressing the condition of equilibrium, we find

\mathrm{M}(2e - m) {{=}} \frac{4}{3}\pi \cdot\frac{6}{5c^2}\int^{c}_0 \rho d(ec^{5}).

Making these substitutions in the expressions for the forces at the surface, and putting r/c {{=}} 1 + e - e(h/c)^2\!, we get

\mathrm{G} \cos \phi {{=}} \frac{Mk^2}{ac}\left\{1 - e - \frac{3}{2}m + \left(\frac{5}{2}m - 2e\right)\frac{h^2}{c^2}\right\}\frac{f}{c}

\mathrm{G} \sin \phi {{=}} \frac{Mk^2}{ac}\left\{1 + e - \frac{3}{2}m + \left(\frac{5}{2}m - 2e\right)\frac{h^2}{c^2}\right\}\frac{h}{c}.

Here \mathrm{G}\! is gravity in the latitude \phi\!, and a\! the radius of the equator. Since

\sec \phi {{=}} (c/f){1 + e + (eh^2/c^2)},\!

\mathrm{G} {{=}} \frac{Mk^2}{ac}\left\{1 - \frac{3}{2}m +\left(\frac{5}{2}m - e\right) \sin^2 \phi \right\},

an expression which contains the theorems we have referred to as discovered by Clairault.

The theory of the figure of the earth as a rotating ellipsoid has been especially investigated by Laplace in his Mécanique celeste. The principal English works are:—Sir George Airy, Mathematical Tracts, a lucid treatment without the use of Laplace’s coefficients; Archdeacon Pratt’s Attractions and Figure of the Earth; and O’Brien’s Mathematical Tracts; in the last two Laplace’s coefficients are used.

In 1845 Sir G. G. Stokes (Camb. Trans. viii.; see also Camb. Dub. Math. Journ., 1849, iv.) proved that if the external form of the sea—imagined to percolate the land by canals—be a spheroid with small ellipticity, then the law of gravity is that which we have shown above; his proof required no assumption as to the ellipticity of the internal strata, or as to the past or present fluidity of the earth. This investigation admits of being regarded conversely, viz. as determining the elliptical form of the earth from measurements of gravity; if \mathrm{G}\!, the observed value of gravity in latitude \phi\!, be expressed in the form \mathrm{G} {{=}} g(1 + \beta \sin^2 \phi)\!, where g\! is the value at the equator and \beta\! a coefficient. In this investigation, the square and higher powers of the ellipticity are neglected; the solution was completed by F. R. Helmert with regard to the square of the ellipticity, who showed that a term with \sin^2 2\phi\! appeared (see Helmert, Geodäsie, ii. 83). For the coefficient of this term, the gravity measurements give a small but not sufficiently certain value; we therefore assume a value which agrees best with the hypothesis of the fluid state of the entire earth; this assumption is well supported, since even at a depth of only 50 km. the pressure of the superincumbent crust is so great that rocks become plastic, and behave approximately as fluids, and consequently the crust of the earth floats, to some extent, on the interior (even though this may not be fluid in the usual sense of the word). This is the geological theory of “Isostasis” (cf. Geology); it agrees with the results of measurements of gravity (vide infra), and was brought forward in the middle of the 19th century by J. H. Pratt, who deduced it from observations made in India.

The \sin^2 2\phi\! term in the expression for \mathrm{G}\!, and the corresponding deviation of the meridian from an ellipse, have been analytically established by Sir G. H. Darwin and E. Wiechert; earlier and less complete investigations were made by Sir G. B. Airy and O. Callandreau. In consequence of the \sin^2 2\phi\! term, two parameters of the level surfaces in the interior of the earth are to be determined; for this purpose, Darwin develops two differential equations in the place of the one by Clairault. By assuming Roche’s law for the variation of the density in the interior of the Earth, viz. \rho {{=}} \rho_{1} - k(c/c_{1})^2, k\! being a coefficient, it is shown that in latitude 45°, the meridian is depressed about 3¼ metres from the ellipse, and the coefficient of the term \sin^2\phi \cos^2\phi (= \tfrac{1}{4} \sin^22\phi )\! is -0.0000295. According to Wiechert the earth is composed of a kernel and a shell, the kernel being composed of material, chiefly metallic iron, of density near 8.2, and the shell, about 900 miles thick, of silicates, &c., of density about 3.2. On this assumption the depression in latitude 45° is 2¾ metres, and the coefficient of \sin^2\phi \cos^2\phi\! is, in round numbers, -0.0000280.[2] To this additional term in the formula for \mathrm{G}\!, there corresponds an extension of Clairault’s formula for the calculation of the flattening from \beta\! with terms of the higher orders; this was first accomplished by Helmert.

For a long time the assumption of an ellipsoid with three unequal axes has been held possible for the figure of the earth, in consequence of an important theorem due to K. G. Jacobi, who proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium.[3] It has been objected to the figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi’s theorem (c : a < 1/\sqrt{}2). Admitting this, it has to be noted, on the other hand, that Jacobi’s theorem contemplates a homogeneous fluid, and this is certainly far from the actual condition of our globe; indeed the irregular distribution of continents and oceans suggests the possibility of a sensible divergence from a perfect surface of revolution. We may, however, assume the ellipsoid with three unequal axes to be an interpolation form. More plausible forms are little adapted for computation.[4] Consequently we now generally take the ellipsoid of rotation as a basis, especially so because measurements of gravity have shown that the deviation from it is but trifling.

Local Attraction.

In speaking of the figure of the earth, we mean the surface of the sea imagined to percolate the continents by canals. That this surface should turn out, after precise measurements, to be exactly an ellipsoid of revolution is a priori improbable. Although it may be highly probable that originally the earth was a fluid mass, yet in the cooling whereby the present crust has resulted, the actual solid surface has been left most irregular in form. It is clear that these irregularities of the visible surface must be accompanied by irregularities in the mathematical figure of the earth, and when we consider the general surface of our globe, its irregular distribution of mountain masses, continents, with oceans and islands, we are prepared to admit that the earth may not be precisely any surface of revolution. Nevertheless, there must exist some spheroid which agrees very closely with the mathematical figure of the earth, and has the same axis of rotation. We must conceive this figure as exhibiting slight departures from the spheroid, the two surfaces cutting one another in various lines; thus a point of the surface is defined by its latitude, longitude, and its height above the “spheroid of reference.” Calling this height \mathrm{N}\!, then of the actual magnitude of this quantity we can generally have no information, it only obtrudes itself on our notice by its variations. In the vicinity of mountains it may change sign in the space of a few miles; \mathrm{N}\! being regarded as a function of the latitude and longitude, if its differential coefficient with respect to the former be zero at a certain point, the normals to the two surfaces then will lie in the prime vertical; if the differential coefficient of \mathrm{N}\! with respect to the longitude be zero, the two normals will lie in the meridian; if both coefficients are zero, the normals will coincide. The comparisons of terrestrial measurements with the corresponding astronomical observations have always been accompanied with discrepancies. Suppose \mathrm{A}\! and \mathrm{B}\! to be two trigonometrical stations, and that at \mathrm{A}\! there is a disturbing force drawing the vertical through an angle \delta\!, then it is evident that the apparent zenith of \mathrm{A}\! will be really that of some other place \mathrm{A}'\!, whose distance from \mathrm{A}\! is r\delta\!, when r\! is the earth’s radius; and similarly if there be a disturbance at \mathrm{B}\! of the amount \delta'\!, the apparent zenith of \mathrm{B}\! will be really that of some other place \mathrm{B}'\!, whose distance from \mathrm{B}\! is r\delta'\!. Hence we have the discrepancy that, while the geodetic measurements deal with the points \mathrm{A}\! and \mathrm{B}\!, the astronomical observations belong to the points \mathrm{A}', \mathrm{B}'\!. Should \delta , \delta'\! be equal and parallel, the displacements \mathrm{AA}', \mathrm{BB}'\! will be equal and parallel, and no discrepancy will appear. The non-recognition of this circumstance often led to much perplexity in the early history of geodesy. Suppose that, through the unknown variations of \mathrm{N}\!, the probable error of an observed latitude (that is, the angle between the normal to the mathematical surface of the earth at the given point and that of the corresponding point on the spheroid of reference) be \epsilon\!, then if we compare two arcs of a degree each in mean latitudes, and near each other, say about five degrees of latitude apart, the probable error of the resulting value of the ellipticity will be approximately \pm\tfrac{1}{500}\epsilon, \epsilon\! being expressed in seconds, so that if \epsilon\! be so great as 2″ the probable error of the resulting ellipticity will be greater than the ellipticity itself.

It is necessary at times to calculate the attraction of a mountain, and the consequent disturbance of the astronomical zenith, at any point within its influence. The deflection of the plumb-line, caused by a local attraction whose amount is k^2\mathrm{A}\delta\!, is measured by the ratio of k^2\mathrm{A}\delta\! to the force of gravity at the station. Expressed in seconds, the deflection \Lambda\! is

\Lambda {{=}} 12''.447A\delta / \rho ,\!

where \rho\! is the mean density of the earth, \delta\! that of the attracting mass, and A {{=}} f s^{-3}xdv\!, in which dv\! is a volume element of the attracting mass within the distance s\! from the point of deflection, and x\! the projection of s\! on the horizontal plane through this point, the linear unit in expressing \mathrm{A}\! being a mile. Suppose, for instance, a table-land whose form is a rectangle of 12 miles by 8 miles, having a height of 500 ft. and density half that of the earth; let the observer be 2 miles distant from the middle point of the longer side. The deflection then is 1″.472; but at 1 mile it increases to 2″.20.

At sixteen astronomical stations in the English survey the disturbance of latitude due to the form of the ground has been computed, and the following will give an idea of the results. At six stations the deflection is under 2″, at six others it is between 2″ and 4″, and at four stations it exceeds 4″. There is one very exceptional station on the north coast of Banffshire, near the village of Portsoy, at which the deflection amounts to 10″, so that if that village were placed on a map in a position to correspond with its astronomical latitude, it would be 1000 ft. out of position! There is the sea to the north and an undulating country to the south, which, however, to a spectator at the station does not suggest any great disturbance of gravity. A somewhat rough estimate of the local attraction from external causes gives a maximum limit of 5″, therefore we have 5″ which must arise from unequal density in the underlying strata in the surrounding country. In order to throw light on this remarkable phenomenon, the latitudes of a number of stations between Nairn on the west, Fraserburgh on the east, and the Grampians on the south, were observed, and the local deflections determined. It is somewhat singular that the deflections diminish in all directions, not very regularly certainly, and most slowly in a south-west direction, finally disappearing, and leaving the maximum at the original station at Portsoy.

The method employed by Dr C. Hutton for computing the attraction of masses of ground is so simple and effectual that it can hardly be improved on. Let a horizontal plane pass through the given station; let r, \theta\! be the polar co-ordinates of any point in this plane, and r, \theta, z\!, the co-ordinates of a particle of the attracting mass; and let it be required to find the attraction of a portion of the mass contained between the horizontal planes z {{=}} 0, z {{=}} h\!, the cylindrical surfaces r {{=}} r_{1}, r {{=}} r_{2}\!, and the vertical planes \theta {{=}} \theta_{1}, \theta {{=}} \theta_{2}\!. The component of the attraction at the station or origin along the line \theta {{=}} 0\! is

k^2\delta \int^{r2}_{r1} \int^{\theta 2}_{\theta 1} \int^{h}_{0}\frac{r^2\cos \theta}{(r^2 + z^2)^{\frac{3}{2}}}dr d\theta dz {{=}} k^2\delta h (\sin \theta_{2} - \sin \theta_{1}) log{r_{2} + (r_{2}^2 + h^2)^{\frac{1}{2}} / r_{1} + (r_{1}^2 + h^2)^{\frac{3}{2}}}.

By taking r_{2} - r_{1}\!, sufficiently small, and supposing h\! also small compared with r_{1} + r_{2}\! (as it usually is), the attraction is

k^2\delta (r_{2} - r_{1}) (\sin \theta_{2} - \sin \theta_{1}) h/r,\!

where r {{=}} \tfrac{1}{2} (r_{1} + r_{2})\!. This form suggests the following procedure. Draw on the contoured map a series of equidistant circles, concentric with the station, intersected by radial lines so disposed that the sines of their azimuths are in arithmetical progression. Then, having estimated from the map the mean heights of the various compartments, the calculation is obvious.

In mountainous countries, as near the Alps and in the Caucasus, deflections have been observed to the amount of as much as 30″, while in the Himalayas deflections amounting to 60″ were observed. On the other hand, deflections have been observed in flat countries, such as that noted by Professor K. G. Schweizer, who has shown that, at certain stations in the vicinity of Moscow, within a distance of 16 miles the plumb-line varies 16″ in such a manner as to indicate a vast deficiency of matter in the underlying strata; deflections of 10″ were observed in the level regions of north Germany.

Since the attraction of a mountain mass is expressed as a numerical multiple of \delta : \rho\! the ratio of the density of the mountain to that of the earth, if we have any independent means of ascertaining the amount of the deflection, we have at once the ratio \rho : \delta\!, and thus we obtain the mean density of the earth, as, for instance, at Schiehallion, and afterwards at Arthur’s Seat. Experiments of this kind for determining the mean density of the earth have been made in greater numbers; but they are not free from objection (see Gravitation).

Let us now consider the perturbation attending a spherical subterranean mass. A compact mass of great density at a small distance under the surface of the earth will produce an elevation of the mathematical surface which is expressed by the formula

y {{=}} a\mu {(1 - 2u \cos \theta + u^2)^{-1/2} - 1},\!

where a\! is the radius of the (spherical) earth, a (1 - u)\! the distance of the disturbing mass below the surface, \mu\! the ratio of the disturbing mass to the mass of the earth, and a\theta\! the distance of any point on the surface from that point, say \mathrm{Q}\!, which is vertically over the disturbing mass. The maximum value of y\! is at \mathrm{Q}\!, where it is y {{=}} a\mu u (1 - u)\!. The deflection at the distance a\theta\! is \Lambda {{=}} \mu u \sin \theta (1 - 2u \cos \theta + u^2)^{-\frac{3}{2}}\!, or since \theta\! is small, putting h + u {{=}} 1\!, we have \Lambda {{=}} \mu \theta (h^2 + \theta^2)^{-\frac{3}{2}}\!. The maximum deflection takes place at a point whose distance from \mathrm{Q}\! is to the depth of the mass as 1 : \sqrt{2}\!, and its amount is 2\mu /3 \sqrt{3h^2}\!. If, for instance, the disturbing mass were a sphere a mile in diameter, the excess of its density above that of the surrounding country being equal to half the density of the earth, and the depth of its centre half a mile, the greatest deflection would be 5″, and the greatest value of y\! only two inches. Thus a large disturbance of gravity may arise from an irregularity in the mathematical surface whose actual magnitude, as regards height at least, is extremely small.

The effect of the disturbing mass \mu\! on the vibrations of a pendulum would be a maximum at \mathrm{Q}\!; if v\! be the number of seconds of time gained per diem by the pendulum at \mathrm{Q}\!, and \sigma\! the number of seconds of angle in the maximum deflection, then it may be shown that v/\sigma {{=}} \pi \sqrt{3}/10\!.

The great Indian survey, and the attendant measurements of the degree of latitude, gave occasion to elaborate investigations of the deflection of the plumb-line in the neighbourhood of the high plateaus and mountain chains of Central Asia. Archdeacon Pratt (Phil. Trans., 1855 and 1857), in instituting these investigations, took into consideration the influence of the apparent diminution of the mass of the earth’s crust occasioned by the neighbouring ocean-basins; he concluded that the accumulated masses of mountain chains, &c., corresponded to subterranean mass diminutions, so that over any level surface in a fixed depth (perhaps 100 miles or more) the masses of prisms of equal section are equal. This is supported by the gravity measurements at Moré in the Himalayas at a height of 4696 metres, which showed no deflection due to the mountain chain (Phil. Trans., 1871); more recently, H. A. Faye (Compt. rend., 1880) arrived at the same conclusion for the entire continent.

This compensation, however, must only be regarded as a general principle; in certain cases, the compensating masses show marked horizontal displacements. Further investigations, especially of gravity measurements, will undoubtedly establish other important facts. Colonel S. G. Burrard has recently recalculated, with the aid of more exact data, certain Indian deviations of the plumb-line, and has established that in the region south of the Himalayas (lat. 24°) there is a subterranean perturbing mass. The extent of the compensation of the high mountain chains is difficult to recognize from the latitude observations, since the same effect may result from different causes; on the other hand, observations of geographical longitude have established a strong compensation.[5]

Meridian Arcs.

The astronomical stations for the measurement of the degree of latitude will generally lie not exactly on the same meridian; and it is therefore necessary to calculate the arcs of meridian \mathrm{M}\! which lie between the latitude of neighbouring stations. If \mathrm{S}\! be the geodetic line calculated from the triangulation with the astronomically determined azimuths \alpha_{1}\! and \alpha_{2}\!, then

\mathrm{M} {{=}} \mathrm{S}\frac{\cos \alpha}{\cos \tfrac{1}{2}\Delta \alpha}\left\{1 + \tfrac{1}{12}\frac{\mathrm{S}^2}{\alpha^2}\sin^2\alpha \ldots \right\},

in which 2\alpha {{=}} \alpha_{1} + \alpha_{2} - 180^{\circ}, \Delta \alpha {{=}} \alpha_{2} - \alpha_{1} - 180^{\circ}\!.

The length of the arc of meridian between the latitudes \phi_{1}\! and \phi_{2}\! is

\mathrm{M} {{=}} \int^{\phi 2}_{\phi 1} \rho d\phi {{=}} \alpha \int^{\phi 2}_{\phi 1}\frac{(1 - e^2) d\phi}{(1 - e^2 \sin^2\phi )^{\frac{3}{2}}}

where a^2e^2 {{=}} a^2 - b^2\!; instead of using the eccentricity e\!, put the ratio of the axes b : a {{=}} 1 - n : 1 + n\!, then

\mathrm{M} {{=}} \int^{\phi 2}_{\phi 1}\frac{b (1 + n) (1 - n^2) d\phi}{(1 + 2n \cos 2\phi + n^2)^{\frac{3}{2}}}.

This, after integration, gives

\mathrm{M}/b {{=}} \left(1 + n + \frac{5}{4}n^2 + \frac{5}{4}n^3\right)\alpha_{0} - \left(3n + 3n^2 + \frac{21}{8}n^3\right)\alpha_{1} + \left(\frac{15}{8}n^2 + \frac{15}{8}n^3\right)\alpha_{2} - \left(\frac{35}{24}n^3\right)\alpha_{3},


\begin{align}&\alpha_{0} {{=}} \phi_{2} - \phi_{1}\\ &\alpha_{1} {{=}} \sin (\phi_{2} - \phi_{1}) \cos (\phi_{2} + \phi_{1})\\ &\alpha_{2} {{=}} \sin 2(\phi_{2} - \phi_{1}) \cos 2(\phi_{2} + \phi_{1})\\ &\alpha_{3} {{=}} \sin 3(\phi_{2} - \phi_{1}) \cos 3(\phi_{2} + \phi_{1}).\end{align}\!

The part of \mathrm{M}\! which depends on n^3\! is very small; in fact, if we calculate it for one of the longest arcs measured, the Russian arc, it amounts to only an inch and a half, therefore we omit this term, and put for \mathrm{M}/b\! the value

\left(1 + n + \frac{5}{4}n^2\right)\alpha_{0} - \left(3n + 3n^2\right)\alpha_{1} + \left(\frac{15}{8}n^2\right)\alpha_{2}.

Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are \phi_{1} + x_{1}, \phi_{2} + x_{2}\!; and if further we suppose that n_{1} + dn\! is the true value of a - b : a + b\!, and that n_{1}\! itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the corrections x\! on the position of the arc in latitude, i.e. on \phi_{1} + \phi_{2}\!,

\begin{align}\mathrm{M}/b &{{=}} \left(1 + n_{1} + \frac{5}{4}n_{1}^2\right)\alpha_{0} - \left(3n_{1} + 3n_{1}^2\right)\alpha_{1} + \left(\frac{15}{8}n_{1}^2\right)\alpha_{2}\\ &+ \left\{ \left(1 + \frac{5}{2}n_{1}\right)\alpha_{0} - \left(3 + 6n_{1}\right)\alpha_{1} + \left(\frac{15}{4}n_{1}\right)\alpha_{2} \right\}dn\\ &+ \left\{1 + n_{1} - 3n\frac{d\alpha_{1}}{d\alpha_{0}}\right\}d\alpha_{0};\end{align}

here d\alpha_{0} {{=}} x_{2} - x_{1}\!; and as b\! is only known approximately, put b {{=}} b_{1}(1 + u)\!; then we get, after dividing through by the coefficient of d\alpha_{0}\!, which is {{=}} 1 + n_{1} - 3n_{1} \cos (\phi_{2} - \phi_{1}) \cos (\phi_{2} + \phi_{1})\!, an equation of the form x_{2} {{=}} x_{1} + h + fu + gv\!, where for convenience we put v\! for dn\!.

Now in every measured arc there are not only the extreme stations determined in latitude, but also a number of intermediate stations so that if there be i + 1\! stations there will be i\! equations

\begin{align}x_{2} {{=}} &x_{1} + f_{1}u + g_{1}v + h_{1}\\ x_{3} {{=}} &x_{1} + f_{2}u + g_{2}v + h_{2}\\ \vdots \quad &\vdots \qquad \qquad \qquad \quad \vdots\\ x_{i} {{=}} &x_{1} + f_{i}u + g_{i}v + h_{i}\\ \end{align}\!

In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in u\! and v\! and the corrections to its observed latitudes. Then, according to the method of least squares, those values of u\! and v\! are the most probable which render the sum of the squares of all the errors x\! a minimum. The corrections x\! which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x\! to the observed latitudes are attributable to local attraction. Archdeacon Pratt objected to this mode of applying least squares first used by Bessel; but Bessel was right, and the objection is groundless. Bessel found, in 1841, from ten meridian arcs with a total amplitude of 50°.6:

\begin{align}&a {{=}} 3272077\text{ toises} {{=}} 6377397\text{ metres}.\\&e\text{  (ellipticity)} {{=}} (a - b) / a = 1/299.15\text{  (prob. error ± 3.2)}.\\\end{align}\!

The probable error in the length of the earth’s quadrant is ± 336 m.

We now give a series of some meridian-arcs measurements, which were utilized in 1866 by A. R. Clarke in the Comparisons of the Standards of Length, pp. 280-287; details of the calculations are given by the same author in his Geodesy (1880), pp. 311 et seq.

The data of the French arc from Formentera to Dunkirk are—

Stations. Astronomical
Distance of
  ° Ft.
Formentera 38 39 53.17 · ·
Mountjouy 41 21 44.96 982671.04
Barcelona 41 22 47.90 988701.92
Carcassonne 43 12 54.30 1657287.93
Pantheon 48 50 47.98 3710827.13
Dunkirk 51   2   8.41 4509790.84

The distance of the parallels of Dunkirk and Greenwich, deduced from the extension of the triangulation of England into France, in 1862, is 161407.3 ft., which is 3.9 ft. greater than that obtained from Captain Kater’s triangulation, and 3.2 ft. less than the distance calculated by Delambre from General Roy’s triangulation. The following table shows the data of the English arc with the distances in standard feet from Formentera.

  ° Ft.
Formentera · · · ·
Greenwich 51 28 38.30 4671198.3
Arbury 52 13 26.59 4943837.6
Clifton 53 27 29.50 5394063.4
Kellie Law 56 14 53.60 6413221.7
Stirling 57 27 49.12 6857323.3
Saxavord 60 49 37.21 8086820.7

The latitude assigned in this table to Saxavord is not the directly observed latitude, which is 60° 49′ 38.58″, for there are here a cluster of three points, whose latitudes are astronomically determined; and if we transfer, by means of the geodesic connexion, the latitude of Gerth of Scaw to Saxavord, we get 60° 49′ 36.59″; and if we similarly transfer the latitude of Balta, we get 60° 49′ 36.46″. The mean of these three is that entered in the above table.

For the Indian arc in long. 77° 40′ we have the following data:—

  ° Ft.
Punnea 8 9 31.132 · ·
Putchapolliam 10 59 42.276 1029174.9
Dodagunta 12 59 52.165 1756562.0
Namthabad 15 5 53.562 2518376.3
Daumergida 18 3 15.292 3591788.4
Takalkhera 21 5 51.532 4697329.5
Kalianpur 24 7 11.262 5794695.7
Kaliana 29 30 48.322 7755835.9

The data of the Russian arc (long. 26° 40′) taken from Struve’s work are as below:—

  ° Ft.
Staro Nekrasovsk 45 20 2.94 · ·
Vodu-Luy 47 1 24.98 616529.81
Suprunkovzy 48 45 3.04 1246762.17
Kremenets 50 5 49.95 1737551.48
Byelin 52 2 42.16 2448745.17
Nemesh 54 39 4.16 3400312.63
Jacobstadt 56 30 4.97 4076412.28
Dorpat 58 22 47.56 4762421.43
Hogland 60 5 9.84 5386135.39
Kilpi-maki 62 38 5.25 6317905.67
Torneå 65 49 44.57 7486789.97
Stuor-oivi 68 40 58.40 8530517.90
Fuglenaes 70 40 11.23 9257921.06

From the are measured in Cape Colony by Sir Thomas Maclear in long. 18° 30′, we have

  ° Ft.
North End 29 44 17.66 · ·
Heerenlogement Berg 31 58 9.11 811507.7
Royal Observatory 33 56 3.20 1526386.8
Zwart Kop 34 13 32.13 1632583.3
Cape Point 34 21 6.26 1678375.7

And, finally, for the Peruvian arc, in long. 281° 0′,

  ° Ft.
Tarqui 3 4 32.068 · ·
Cotchesqui 0 2 31.387 1131036.3

Having now stated the data of the problem, we may seek that oblate ellipsoid (spheroid) which best represents the observations. Whatever the real figure may be, it is certain that if we suppose it an ellipsoid with three unequal axes, the arithmetical process will bring out an ellipsoid, which will agree better with all the observed latitudes than any spheroid would, therefore we do not prove that it is an ellipsoid; to prove this, arcs of longitude would be required. The result for the spheroid may be expressed thus:—

\begin{array}{c}a {{=}} 20926062\text{ ft. }{{=}} 6378206.4\text{ metres.}\\ b {{=}} 20855121\text{ ft. }{{=}} 6356583.8\text{ metres.}\\ b : a {{=}} 293.98 : 294.98.\end{array}\!

As might be expected, the sum of the squares of the 40 latitude corrections, viz. 153.99, is greater in this figure than in that of three axes, where it amounts to 138.30. For this case, in the Indian arc the largest corrections are at Dodagunta, + 3.87″, and at Kalianpur, - 3.68″. In the Russian arc the largest corrections are + 3.76″, at Torneå, and - 3.31″, at Staro Nekrasovsk. Of the whole 40 corrections, 16 are under 1.0″, 10 between 1.0″ and 2.0″, 10 between 2.0″ and 3.0″, and 4 over 3.0″. The probable error of an observed latitude is ± 1.42″; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection.

If \rho\! be the radius of curvature of the meridian in latitude \phi, \rho'\! that perpendicular to the meridian, \mathrm{D}\! the length of a degree of the meridian, \mathrm{D}'\! the length of a degree of longitude, r\! the radius drawn from the centre of the earth, \mathrm{V}\! the angle of the vertical with the radius-vector, then

\begin{array}{rllll} & &\qquad \text{Ft.}\\ &\rho &{{=}} 20890606.6 &- 106411.5\;\;\cos 2\phi &+ 225.8 \cos 4\phi \\ &\rho' &{{=}} 20961607.3 &- \;\;35590.9\;\;\cos 2\phi &+ 45.2\;\; \cos 4\phi \\ &D &{{=}} \quad 364609.87 &- \quad 1857.14 \cos 2\phi &+ \;\; 3.94 \cos 4\phi \\ &D' &{{=}} \quad 365538.48 \cos \phi &- \quad\;\;310.17 \cos 3\phi &+ \;\; 0.39 \cos 5\phi \\ \text{Log}&r/a &{{=}} 9.9992645 &+ \;\;.0007374 \cos 2\phi &-.0000019 \cos 4\phi \\ &V &{{=}} 700.44'' \sin 2\phi &- 1.19'' \sin 4\phi .\end{array}

A. R. Clarke has recalculated the elements of the ellipsoid of the earth; his values, derived in 1880, in which he utilized the measurements of parallel arcs in India, are particularly in practice. These values are:—

\begin{array}{c}a {{=}} 20926202\text{ ft. }{{=}} 6378249\text{ metres,}\\ b {{=}} 20854895\text{ ft. }{{=}} 6356515\text{ metres,}\\ b : a {{=}} 292.465 : 293.465.\end{array}\!

The calculation of the elements of the ellipsoid of rotation from measurements of the curvature of arcs in any given azimuth by means of geographical longitudes, latitudes and azimuths is indicated in the article Geodesy; reference may be made to Principal Triangulation, Helmert’s Geodasie, and the publications of the Kgl. Preuss. Geod. Inst.:—Lotabweichungen (1886), and Die europ. Längengradmessung in 52° Br. (1893). For the calculation of an ellipsoid with three unequal axes see Comparison of Standards, preface; and for non-elliptical meridians, Principal Triangulation, p. 733.


According to Clairault’s theorem (see above) the ellipticity e\! of the mathematical surface of the earth is equal to the difference \tfrac{5}{2}m - \beta\!, where m\! is the ratio of the centrifugal force at the equator to gravity at the equator, and \beta\! is derived from the formula \mathrm{G} {{=}} g(1 + \beta \sin^2\phi)\!. Since the beginning of the 19th century many efforts have been made to determine the constants of this formula, and numerous expeditions undertaken to investigate the intensity of gravity in different latitudes. If m\! be known, it is only necessary to determine \beta\! for the evaluation of e; consequently it is unnecessary to determine \mathrm{G}\! absolutely, for the relative values of \mathrm{G}\! at two known latitudes suffice. Such relative measurements are easier and more exact than absolute ones. In some cases the ordinary thread pendulum, i.e. a spherical bob suspended by a wire, has been employed; but more often a rigid metal rod, bearing a weight and a knife-edge on which it may oscillate, has been adopted. The main point is the constancy of the pendulum. From the formula for the time of oscillation of the mathematically ideal pendulum, t {{=}} 2\pi \sqrt{l/\mathrm{G}}\!, l\! being the length, it follows that for two points \mathrm{G}_{1} / \mathrm{G}_{2} {{=}} t_{2}^2 / t_{1}^2\!.

In 1808 J. B. Biot commenced his pendulum observations at several stations in western Europe; and in 1817-1825 Captain Louis de Freycinet and L. I. Duperrey prosecuted similar observations far into the southern hemisphere. Captain Henry Kater confined himself to British stations (1818-1819); Captain E. Sabine, from 1819 to 1829, observed similarly, with Kater’s pendulum, at seventeen stations ranging from the West Indies to Greenland and Spitsbergen; and in 1824-1831, Captain Henry Foster (who met his death by drowning in Central America) experimented at sixteen stations; his observations were completed by Francis Baily in London. Of other workers in this field mention may be made of F. B. Lütke (1826-1829), a Russian rear-admiral, and Captains J. B. Basevi and W. T. Heaviside, who observed during 1865 to 1873 at Kew and at 29 Indian stations, particularly at Moré in the Himalayas at a height of 4696 metres. Of the earlier absolute determinations we may mention those of Biot, Kater, and Bessel at Paris, London and Königsberg respectively. The measurements were particularly difficult by reason of the length of the pendulums employed, these generally being second-pendulums over 1 metre long. In about 1880, Colonel Robert von Sterneck of Austria introduced the half-second pendulum, which permitted far quicker and more accurate work. The use of these pendulums spread in all countries, and the number of gravity stations consequently increased: in 1880 there were about 120, in 1900 there were about 1600, of which the greater number were in Europe. Sir E. Sabine[6] calculated the ellipticity to be 1/288.5, a value shown to be too high by Helmert, who in 1884, with the aid of 120 stations, gave the value 1/299.26,[7] and in 1901, with about 1400 stations, derived the value 1/298.3.[8] The reason for the excessive estimate of Sabine is that he did not take into account the systematic difference between the values of \mathrm{G}\! for continents and islands; it was found that in consequence of the constitution of the earth’s crust (Pratt) \mathrm{G}\! is greater on small islands of the ocean than on continents by an amount which may approach to 0.3 cm. Moreover, stations in the neighbourhood of coasts shelving to deep seas have a surplus, but a little smaller. Consequently, Helmert conducted his calculations of 1901 for continents and coasts separately, and obtained \mathrm{G}\! for the coasts 0.036 cm. greater than for the continents, while the value of \beta\! remained the same. The mean value, reduced to continents, is

\mathrm{G} {{=}} 978.03 (1 + 0.005302 \sin^2\phi - 0.000007 \sin^22\phi )\text{ cm/sec}^2.\!

The small term involving \sin^2 2\phi\! could not be calculated with sufficient exactness from the observations, and is therefore taken from the theoretical views of Sir G. H. Darwin and E. Wiechert. For the constant g {{=}} 978.03\! cm. another correction has been suggested (1906) by the absolute determinations made by F. Kühnen and Ph. Furtwängler at Potsdam.[9]

A report on the pendulum measurements of the 19th century has been given by Helmert in the Comptes rendus des séances de la 13e conférence générale de l’Association Géod. Internationale à Paris (1900), ii. 139-385.

A difficulty presents itself in the case of the application of measurements of gravity to the determination of the figure of the earth by reason of the extrusion or standing out of the land-masses (continents, &c.) above the sea-level. The potential of gravity has a different mathematical expression outside the masses than inside. The difficulty is removed by assuming (with Sir G. G. Stokes) the vertical condensation of the masses on the sea-level, without its form being considerably altered (scarcely 1 metre radially). Further, the value of gravity (g) measured at the height \mathrm{H}\! is corrected to sea-level by + 2g\mathrm{H}/\mathrm{R}\!, where \mathrm{R}\! is the radius of the earth. Another correction, due to P. Bouguer, is - \tfrac{3}{2}g\delta \mathrm{H}/\rho \mathrm{R}\!, where \delta\! is the density of the strata of height \mathrm{H}\!, and \rho\! the mean density of the earth. These two corrections are represented in “Bouguer’s Rule”: g_{\mathrm{H}} {{=}} g_{s} (1 - 2\mathrm{H}/\mathrm{R} + 3\delta \mathrm{H} / 2\rho \mathrm{R})\!, where g_{\mathrm{H}}\! is the gravity at height \mathrm{H}\!, and g_{s}\! the value at sea-level. This is supposed to take into account the attraction of the elevated strata or plateau; but, from the analytical method, this is not correct; it is also disadvantageous since, in general, the land-masses are compensated subterraneously, by reason of the isostasis of the earth’s crust.

In 1849 Stokes showed that the normal elevations \mathrm{N}\! of the geoid towards the ellipsoid are calculable from the deviations \Delta g\! of the acceleration of gravity, i.e. the differences between the observed g\! and the value calculated from the normal \mathrm{G}\! formula. The method assumes that gravity is measured on the earth’s surface at a sufficient number of points, and that it is conformably reduced. In order to secure the convergence of the expansions in spherical harmonics, it is necessary to assume all masses outside a surface parallel to the surface of the sea at a depth of 21 km. (= \mathrm{R}\! × ellipticity) to be condensed on this surface (Helmert, Geod. ii. 172). In addition to the reduction with 2g\mathrm{H}/\mathrm{R}\!, there still result small reductions with mountain chains and coasts, and somewhat larger ones for islands. The sea-surface generally varies but very little by this condensation. The elevation (\mathrm{N}\!) of the geoid is then equal to

\mathrm{N} {{=}} \mathrm{R} \int^{\pi} \mathrm{FG}^{-1}\Delta g_{\psi}\psi ,\!

where \psi\! is the spherical distance from the point \mathrm{N}\!, and \Delta g_{\psi}\! denotes the mean value of \Delta g\! for all points in the same distance \psi\! around; \mathrm{F}\! is a function of \psi\!, and has the following values:—

Ψ = 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180°
F = 1 1.22 0.94 0.47 -0.06 -0.54 -0.90 -1.08 -1.08 -0.91 -0.62 -0.27 +0.08 0.36 0.53 0.56 0.46 0.26 0

H. Poincaré (Bull. Astr., 1901, p. 5) has exhibited \mathrm{N}\! by means of Lamé’s functions; in this case the condensation is effected on an ellipsoidal surface, which approximates to the geoid. This condensation is, in practice, the same as to the geoid itself.

If we imagine the outer land-masses to be condensed on the sea-level, and the inner masses (which, together with the outer masses, causes the deviation of the geoid from the ellipsoid) to be compensated in the sea-level by a disturbing stratum (which, according to Gauss, is possible), and if these masses of both kinds correspond at the point \mathrm{N}\! to a stratum of thickness \mathrm{D}\! and density \delta\!, then, according to Helmert (Geod. ii. 260) we have approximately

 \Delta g {{=}}  \frac{3}{2} \cdot \frac{g}{\mathrm{R}} \left( \frac{\delta \mathrm{D}}{\rho}  - \mathrm{N}\right).

Since \mathrm{N}\! slowly varies empirically, it follows that in restricted regions (of a few 100 km. in diameter) \Delta g\! is a measure of the variation of \mathrm{D}\!. By applying the reduction of Bouguer to g\!, \mathrm{D}\! is diminished by \mathrm{H}\! and only gives the thickness of the ideal disturbing mass which corresponds to the perturbations due to subterranean masses. \Delta g\! has positive values on coasts, small islands, and high and medium mountain chains, and occasionally in plains; while in valleys and at the foot of mountain ranges it is negative (up to 0.2 cm.). We conclude from this that the masses of smaller density existing under high mountain chains lie not only vertically underneath but also spread out sideways.

The European Arc of Parallel in 52° Lat.

Many measurements of degrees of longitudes along central parallels in Europe were projected and partly carried out as early as the first half of the 19th century; these, however, only became of importance after the introduction of the electric telegraph, through which calculations of astronomical longitudes obtained a much higher degree of accuracy. Of the greatest moment is the measurement near the parallel of 52° lat., which extended from Valentia in Ireland to Orsk in the southern Ural mountains over 69° long, (about 6750 km.). F. G. W. Struve, who is to be regarded as the father of the Russo-Scandinavian latitude-degree measurements, was the originator of this investigation. Having made the requisite arrangements with the

governments in 1857, he transferred them to his son Otto, who, in 1860, secured the co-operation of England. A new connexion of England with the continent, via the English Channel, was accomplished in the next two years; whereas the requisite triangulations in Prussia and Russia extended over several decennaries. The number of longitude stations originally arranged for was 15; and the determinations of the differences in longitude were uniformly commenced by the Russian observers E. I. von Forsch, J. I. Zylinski, B. Tiele and others; Feaghmain (Valentia) being reserved for English observers. With the concluding calculation of these operations, newer determinations of differences of longitudes were also applicable, by which the number of stations was brought up to 29. Since local deflections of the plumb-line were suspected at Feaghmain, the most westerly station, the longitude (with respect to Greenwich) of the trigonometrical station Killorglin at the head of Dingle Bay was shortly afterwards determined.

The results (1891-1894) are given in volumes xlvii. and l. of the memoirs (Zapiski) of the military topographical division of the Russian general staff, volume li. contains a reconnexion of Orsk. The observations made west of Warsaw are detailed in the Die europ. Längengradmessung in 52° Br., i. and ii., 1893, 1896, published by the Kgl. Preuss. Geod. Inst.

The following figures are quoted from Helmert’s report “Die Grösse der Erde” (Sitzb. d. Berl. Akad. d. Wiss., 1906, p. 535):—

Easterly Deviation of the Astronomical Zenith.

Name. Longitude.
Feaghmain -10 21 -3.3
Killorglin -9 47 +2.8
Haverfordwest -4 58 +1.6
Greenwich 0 0 +1.5
Rosendaël-Nieuport +2 35 -1.7
Bonn +7 6 -4.4
Göttingen +9 57 -2.4
Brocken +10 37 +2.3
Leipzig +12 23 +2.7
Rauenberg-Berlin +13 23 +1.7
Grossenhain +13 33 -2.9
Schneekoppe +15 45 +0.1
Springberg +16 37 +0.8
Breslau-Rosenthal +17 2 +3.5
Trockenberg +18 53 -0.5
Schönsee +18 54 -2.9
Mirov +19 18 +2.2
Warsaw +21 2 +1.9
Grodno +23 50 -2.8
Bobruisk +29 14 +0.5
Orel +36 4 +4.4
Lipetsk +39 36 +0.2
Saratov +46 3 +6.4
Samara +50 5 -2.6
Orenburg +55 7 +1.7
Orsk +58 34 -8.0

These deviations of the plumb-line correspond to an ellipsoid having an equatorial radius (a) of nearly 6,378,000 metres (prob. error ± 70 metres) and an ellipticity 1/299.15. The latter was taken for granted; it is nearly equal to the result from the gravity-measurements; the value for a\! then gives \sum \eta^2\! a minimum (nearly). The astronomical values of the geographical longitudes (with regard to Greenwich) are assumed, according to the compensation of longitude differences carried out by van de Sande Bakhuyzen (Comp. rend, des séances de la commission permanente de l’Association Géod. Internationale à Genève, 1893, annexe A.I.). Recent determinations (Albrecht, Astr. Nach., 3993/4) have introduced only small alterations in the deviations, a being slightly increased.

Of considerable importance in the investigation of the great arc was the representation of the linear lengths found in different countries, in terms of the same unit. The necessity for this had previously occurred in the computation of the figure of the earth from latitude-degree-measurements. A. R. Clarke instituted an extensive series of comparisons at Southampton (see Comparisons of Standards of Length of England, France, Belgium, Prussia, Russia, India and Australia, made at the Ordnance Survey Office, Southampton, 1866, and a paper in the Philosophical Transactions for 1873, by Lieut.-Col. A. R. Clarke, C.B., R.E., on the further comparisons of the standards of Austria, Spain, the United States, Cape of Good Hope and Russia) and found that 1 toise = 6.39453348 ft., 1 metre = 3.28086933 ft.

In 1875 a number of European states concluded the metre convention, and in 1877 an international weights-and-measures bureau was established at Breteuil. Until this time the metre was determined by the end-surfaces of a platinum rod (mètre des archives); subsequently, rods of platinum-iridium, of cross-section Britannica E 9.jpg, were constructed, having engraved lines at both ends of the bridge, which determine the distance of a metre. There were thirty of the rods which gave as accurately as possible the length of the metre; and these were distributed among the different states (see Weights and Measures). Careful comparisons with several standard toises showed that the metre was not exactly equal to 443,296 lines of the toise, but, in round numbers, 1/75000 of the length smaller. The metre according to the older relation is called the “legal metre,” according to the new relation the “international metre.” The values are (see Europ. Längengradmessung, i. p. 230):—

Legal metre = 3.28086933 ft., International metre = 3.2808257 ft.

The values of a\! given above are in terms of the international metre; the earlier ones in legal metres, while the gravity formulae are in international metres.

The International Geodetic Association (Internationale Erdmessung).

On the proposition of the Prussian lieutenant-general, Johann Jacob Baeyer, a conference of delegates of several European states met at Berlin in 1862 to discuss the question of a “Central European degree-measurement.” The first general conference took place at Berlin two years later; shortly afterwards other countries joined the movement, which was then named “The European degree-measurement.” From 1866 till 1886 Prussia had borne the expense incident to the central bureau at Berlin; but when in 1886 the operations received further extension and the title was altered to “The International Earth-measurement” or “International Geodetic Association,” the co-operating states made financial contributions to this purpose. The central bureau is affiliated with the Prussian Geodetic Institute, which, since 1892, has been situated on the Telegraphenberg near Potsdam. After Baeyer’s death Prof. Friedrich Robert Helmert was appointed director. The funds are devoted to the advancement of such scientific works as concern all countries and deal with geodetic problems of a general or universal nature. During the period 1897-1906 the following twenty-one countries belonged to the association:—Austria, Belgium, Denmark, England, France, Germany, Greece, Holland, Hungary, Italy, Japan, Mexico, Norway, Portugal, Rumania, Russia, Servia, Spain, Sweden, Switzerland and the United States of America. At the present time general conferences take place every three years.[10]

Baeyer projected the investigation of the curvature of the meridians and the parallels of the mathematical surface of the earth stretching from Christiania to Palermo for 12 degrees of longitude; he sought to co-ordinate and complete the network of triangles in the countries through which these meridians passed, and to represent his results by a common unit of length. This proposition has been carried out, and extended over the greater part of Europe; as a matter of fact, the network has, with trifling gaps, been carried over the whole of western and central Europe, and, by some chains of triangles, over European Russia. Through the co-operation of France, the network has been extended into north Africa as far as the geographical latitude of 32°; in Greece a network, united with those of Italy and Bosnia, has been carried out by the Austrian colonel, Heinrich Hartl; Servia has projected similar triangulations; Rumania has begun to make the triangle measurements, and three base lines have been measured by French officers with Brunner’s apparatus. At present, in Rumania, there is being worked a connexion between the arc of parallel in lat. 47°/48° in Russia (stretching from Astrakan to Kishinev) with Austria-Hungary. In the latter country and in south Bavaria the connecting triangles for this parallel have been recently revised, as well as the French chain on the Paris parallel, which has been connected with the German net by the co-operation of German and French geodesists. This will give a long arc of parallel, really projected in the first half of the 19th century. The calculation of the Russian section gives, with an assumed ellipticity of 1/299.15, the value a\! = 6377350 metres; this is rather uncertain, since the arc embraces only 19° in longitude.

We may here recall that in France geodetic studies have recovered their former expansion under the vigorous impulse of Colonel (afterwards General) François Perrier. When occupied with the triangulation of Algeria, Colonel Perrier had conceived the possibility of the geodetic junction of Algeria to Spain, over the Mediterranean; therefore the French meridian line, which was already connected with England, and was thus produced to the 60th parallel, could further be linked to the Spanish triangulation, cross thence into Algeria and extend to the Sahara, so as to form an arc of about 30° in length. But it then became urgent to proceed to a new measurement of the French arc, between Dunkirk and Perpignan. In 1869 Perrier was authorized to undertake that revision. He devoted himself to that work till the end of his career, closed by premature death in February 1888, at the very moment when the Dépôt de la guerre had just been transformed into the Geographical Service of the Army, of which General F. Perrier was the first director. His work was continued by his assistant, Colonel (afterwards General) J. A. L. Bassot. The operations concerning the revision of the French arc were completed only in 1896. Meanwhile the French geodesists had accomplished the junction of Algeria to Spain, with the help of the geodesists of the Madrid Institute under General Carlos Ibañez (1879), and measured the meridian line between Algiers and El Aghuat (1881). They have since been busy in prolonging the meridians of El Aghuat and Biskra, so as to converge towards Wargla, through Ghardaïa and Tuggurt. The fundamental co-ordinates of the Panthéon have also been obtained anew, by connecting the Panthéon and the Paris Observatory with the five stations of Bry-sur-Marne, Morlu, Mont Valérien, Chatillon and Montsouris, where the observations of latitude and azimuth have been effected.[11]

According to the calculations made at the central bureau of the international association on the great meridian arc extending from the Shetland Islands, through Great Britain, France and Spain to El Aghuat in Algeria, a\! = 6377935 metres, the ellipticity being assumed as 1/299.15. The following table gives the difference: astronomical-geodetic latitude. The net does not follow the meridian exactly, but deviates both to the west and to the east; actually, the meridian of Greenwich is nearer the mean than that of Paris (Helmert, Grösse d. Erde).

West Europe-Africa Meridian-arc.[12]

Name. Latitude. A.-G.
Saxavord 60 49.6 -4.0
Balta 60 45.0 -6.1
Ben Hutig 58 33.1 +0.3
Cowhythe 57 41.1 +7.3
Great Stirling 57 27.8 -2.3
Kellie Law 56 14.9 -3.7
Calton Hill 55 57.4 +3.5
Durham 54 46.1 -0.9
Burleigh Moor 54 34.3 +2.1
Clifton Beacon 53 27.5 +1.3
Arbury Hill 52 13.4 -3.0
Greenwich 51 28.6 -2.5
Nieuport 51 7.8 -0.4
Rosendaël 51 2.7 -0.9
Lihons 49 49.9 +0.5
Panthéon 48 50.8 -0.0
Chevry 48 0.5 +2.2
Saligny le Vif 47 2.7 +3.0
Arpheuille 46 13.7 +6.3
Puy de Dôme 45 46.5 +7.0
Rodez 44 21.4 +1.7
Carcassonne 43 13.3 +0.7
Rivesaltes 42 45.2 -0.7
Montolar 41 38.5 +3.6
Lérida 41 37.0 -0.2
Javalon 40 13.8 -0.2
Desierto 40 5.0 -4.5
Chinchilla 38 55.2 +2.2
Mola de Formentera 38 39.9 -1.2
Tetíca 37 15.2 +3.5
Roldan 36 56.6 -6.0
Conjuros 36 44.4 -12.6
Mt. Sabiha 35 39.6 +6.5
Nemours 35 5.8 +7.4
Bouzaréah 36 48.0 +2.9
Algiers (Voirol) 36 45.1 -9.1
Guelt ès Stel 35 7.8 -1.0
El Aghuat 33 48.0 -2.8
Britannica Figure of the Earth.jpg

While the radius of curvature of this arc is obviously not uniform (being, in the mean, about 600 metres greater in the northern than in the southern part), the Russo-Scandinavian meridian arc (from 45° to 70°), on the other hand, is very uniformly curved, and gives, with an ellipticity of 1/299.15, a = 6378455 metres; this arc gives the plausible value 1/298.6 for the ellipticity. But in the case of this arc the orographical circumstances are more favourable.

The west-European and the Russo-Scandinavian meridians indicate another anomaly of the geoid. They were connected at the Central Bureau by means of east-to-west triangle chains (principally by the arc of parallel measurements in lat. 52°); it was shown that, if one proceeds from the west-European meridian arcs, the differences between the astronomical and geodetic latitudes of the Russo-Scandinavian arc become some 4″ greater.[13]

The central European meridian, which passes through Germany and the countries adjacent on the north and south, is under review at Potsdam (see the publications of the Kgl. Preuss. Geod. Inst., Lotabweichungen, Nos. 1-3). Particular notice must be made of the Vienna meridian, now carried southwards to Malta. The Italian triangulation is now complete, and has been joined with the neighbouring countries on the north, and with Tunis on the south.

The United States Coast and Geodetic Survey has published an account of the transcontinental triangulation and measurement of an arc of the parallel of 39°, which extends from Cape May (New Jersey), on the Atlantic coast, to Point Arena (California), on the Pacific coast, and embraces 48° 46′ of longitude, with a linear development of about 4225 km. (2625 miles). The triangulation depends upon ten base-lines, with an aggregate length of 86 km. the longest exceeding 17 km. in length, which have been measured with the utmost care. In crossing the Rocky Mountains, many of its sides exceed 100 miles in length, and there is one side reaching to a length of 294 km., or 183 miles; the altitude of many of the stations is also considerable, reaching to 4300 metres, or 14,108 ft., in the case of Pike’s Peak, and to 14,421 ft. at Elbert Peak, Colo. All geometrical conditions subsisting in the triangulation are satisfied by adjustment, inclusive of the required accord of the base-lines, so that the same length for any given line is found, no matter from what line one may start.[14]

Over or near the arc were distributed 109 latitude stations, occupied with zenith telescopes; 73 azimuth stations; and 29 telegraphically determined longitudes. It has thus been possible to study in a very complete manner the deviations of the vertical, which in the mountainous regions sometimes amount to 25 seconds, and even to 29 seconds.

With the ellipticity 1/299.15, a\! = 6377897 ± 65 metres (prob. error); in this calculation, however, some exceedingly perturbed stations are excluded; for the employed stations the mean perturbation in longitude is ± 4.9″ (zenith-deflection east-to-west ± 3.8″).

The computations relative to another arc, the “eastern oblique arc of the United States,” are also finished.[15] It extends from Calais (Maine) in the north-east, to the Gulf of Mexico, and terminates at New Orleans (Louisiana), in the south. Its length is 2612 km. (1623 miles), the difference of latitude 15° 1′, and of longitude 22° 47′. In the main, the triangulation follows the Appalachian chain of mountains, bifurcating once, so as to leave an oval space between the two branches. It includes among its stations Mount Washington (1920 metres) and Mount Mitchell (2038 metres). It depends upon six base-lines, and the adjustment is effected in the same manner as for the arc of the parallel. The astronomical data have been afforded by 71 latitude stations, 17 longitude stations, and 56 azimuth stations, distributed over the whole extent of the arc. The resulting dimensions of an osculating spheroid were found to be

\begin{array}{c}a {{=}} 6378157\text{ metres }\pm 90\text{ (prob. error)},\\ e\text{ (ellipticity) }{{=}} 1/304.5 \pm 1.9\text{ (prob. error)}.\end{array}\!

With the ellipticity 1/399.15, a\! = 6378041 metres ± 80 (prob. er.).

During the years 1903-1906 the United States Coast and Geodetic Survey, under the direction of O. H. Tittmann and the special management of John F. Hayford, executed a calculation of the best ellipsoid of rotation for the United States. There were 507 astronomical determinations employed, all the stations being connected through the net-work of triangles. The observed latitudes, longitude and azimuths were improved by the attractions of the earth’s crust on the hypothesis of isostasis for three depths of the surface of 114, 121 and 162 km., where the isostasis is complete. The land-masses, within the distance of 4126 km., were taken into consideration. In the derivation of an ellipsoid of rotation, the first case proved itself the most favourable, and there resulted:—

a {{=}} 6378283\text{ metres }\pm 74\text{ (prob. er.), ellipticity }{{=}} 1/297.8 \pm 0.9\text{ (prob. er.)}.\!

The most favourable value for the depth of the isostatic surface is approximately 114 km.

The measurement of a great meridian arc, in long. 98° W., has been commenced; it has a range of latitude of 23°, and will extend over 50° when produced southwards and northwards by Mexico and Canada. It may afterwards be connected with the arc of Quito. A new measurement of the meridian arc of Quito was executed in the years 1901-1906 by the Service géographique of France under the direction of the Académie des Sciences, the ground having been previously reconnoitred in 1899. The new arc has an amplitude in latitude of 5° 53′ 33″, and stretches from Tulcan (lat. 0° 48′ 25″) on the borders of Columbia and Ecuador, through Columbia to Payta (lat. - 5° 5′ 8″) in Peru. The end-points, at which the chain of triangles has a slight north-easterly trend, show a longitude difference of 3°. Of the 74 triangle points, 64 were latitude stations; 6 azimuths and 8 longitude-differences were measured, three base-lines were laid down, and gravity was determined from six points, in order to maintain indications over the general deformation of the geoid in that region. Computations of the attraction of the mountains on the plumb-line are also being considered. The work has been much delayed by the hardships and difficulties encountered. It was conducted by Lieut.-Colonel Robert Bourgeois, assisted by eleven officers and twenty-four soldiers of the geodetic branch of the Service géographique. Of these officers mention may be made of Commandant E. Maurain, who retired in 1904 after suffering great hardships; Commandant L. Massenet, who died in 1905; and Captains I. Lacombe, A. Lallemand, and Lieut. Georges Perrier (son of General Perrier). It is conceivable that the chain of triangles in longitude 98° in North America may be united with that of Ecuador and Peru: a continuous chain over the whole of America is certainly but a question of time. During the years 1899-1902 the measurement of an arc of meridian was made in the extreme north, in Spitzbergen, between the latitudes 76° 38′ and 80° 50′, according to the project of P. G. Rosén. The southern part was determined by the Russians—O. Bäcklund, Captain D. D. Sergieffsky, F. N. Tschernychev, A. Hansky and others—during 1899-1901, with the aid of 1 base-line, 15 trigonometrical, 11 latitude and 5 gravity stations. The northern part, which has one side in common with the southern part, has been determined by Swedes (Professors Rosén, father and son, E. Jäderin, T. Rubin and others), who utilized 1 base-line, 9 azimuth measurements, 18 trigonometrical, 17 latitude and 5 gravity stations. The party worked under excessive difficulties, which were accentuated by the arctic climate. Consequently, in the first year, little headway was made.[16] Sir David Gill, when director of the Royal Observatory, Cape Town, instituted the magnificent project of working a latitude-degree measurement along the meridian of 30° long. This meridian passes through Natal, the Transvaal, by Lake Tanganyika, and from thence to Cairo; connexion with the Russo-Scandinavian meridian arc of the same longitude should be made through Asia Minor, Turkey, Bulgaria and Rumania. With the completion of this project a continuous arc of 105° in latitude will have been measured.[17]

Extensive triangle chains, suitable for latitude-degree measurements, have also been effected in Japan and Australia.

Besides, the systematization of gravity measurements is of importance, and for this purpose the association has instituted many reforms. It has ensured that the relative measurements made at the stations in different countries should be reduced conformably with the absolute determinations made at Potsdam; the result was that, in 1906, the intensities of gravitation at some 2000 stations had been co-ordinated. The intensity of gravity on the sea has been determined by the comparison of barometric and hypsometric observations (Mohn’s method). The association, at the proposal of Helmert, provided the necessary funds for two expeditions:—English Channel—Rio de Janeiro, and the Red Sea—Australia—San Francisco—Japan. Dr O. Hecker of the central bureau was in charge; he successfully overcame the difficulties of the work, and established the tenability of the isostatic hypothesis, which necessitates that the intensity of gravity on the deep seas has, in general, the same value as on the continents (without regard to the proximity of coasts).[18]

As the result of the more recent determinations, the ellipticity, compression or flattening of the ellipsoid of the earth may be assumed to be very nearly 1/298.3; a value determined in 1901 by Helmert from the measurements of gravity. The semi-major axis, a, of the meridian ellipse may exceed 6,378,000 inter. metres by about 200 metres. The central bureau have adopted, for practical reasons, the value 1/299.15, after Bessel, for which tables exist; and also the value a = 6377397.155 (1 + 0.0001).

The methods of theoretical astronomy also permit the evaluation of these constants. The semi-axis a is calculable from the parallax of the moon and the acceleration of gravity on the earth; but the results are somewhat uncertain: the ellipticity deduced from lunar perturbations is 1/297.8 ± 2 (Helmert, Geodäsie, ii. pp. 460-473); William Harkness (The Solar Parallax and its related Constants, 1891) from all possible data derived the values: ellipticity = 1/300.2 ± 3, a = 6377972 ± 125 metres. Harkness also considered in this investigation the relation of the ellipticity to precession and nutation; newer investigations of the latter lead to the limiting values 1/296, 1/298 (Wiechert). It was clearly noticed in this method of determination that the influence of the assumption as to the density of the strata in the interior of the earth was but very slight (Radau, Bull. astr. ii. (1885) 157). The deviations of the geoid from the flattened ellipsoid of rotation with regard to the heights (the directions of normals being nearly the same) will scarcely exceed ± 100 metres (Helmert).[19]

The basis of the degree- and gravity-measurements is actually formed by a stationary sea-surface, which is assumed to be level. However, by the influence of winds and ocean currents the mean surface of the sea near the coasts (which one assumes as the fundamental sea-surface) can deviate somewhat from a level surface. According to the more recent levelling it varies at the most by only some decimeters.[20]

It is well known that the masses of the earth are continually undergoing small changes; the earth’s crust and sea-surface reciprocally oscillate, and the axis of rotation vibrates relatively to the body of the earth. The investigation of these problems falls in the programme of the Association. By continued observations of the water-level on sea-coasts, results have already been obtained as to the relative motions of the land and sea (cf. Geology); more exact levelling will, in the course of time, provide observations on countries remote from the sea-coast. Since 1900 an international service has been organized between some astronomical stations distributed over the north parallel of 39° 8′, at which geographical latitudes are observed whenever possible. The association contributes to all these stations, supporting four entirely: two in America, one in Italy, and one in Japan; the others partially (Tschardjui in Russia, and Cincinnati observatory). Some observatories, especially Pulkowa, Leiden and Tokyo, take part voluntarily. Since 1906 another station for South America and one for Australia in latitude - 31° 55′ have been added. According to the existing data, geographical latitudes exhibit variations amounting to ± 0.25″, which, for the greater part, proceed from a twelve- and a fourteen-month period.[21]

(A. R. C; F. R. H.)

  1. Eratosthenes Batavus, seu de terrae ambitus vera quantitate suscitatus, a Willebrordo Snellio, Lugduni-Batavorum (1617).
  2. O. Callandreau, “Mémoire sur la théorie de la figure des planètes,” Ann. obs. de Paris (1889); G. H. Darwin, “The Theory of the Figure of the Earth carried to the Second Order of Small Quantities,” Mon. Not. R.A.S., 1899; E. Wiechert, “Über die Massenverteilung im Innern der Erde,” Nach. d. kön. G. d. W. zu Gött., 1897.
  3. See I. Todhunter, Proc. Roy. Soc., 1870.
  4. J. H. Jeans, “On the Vibrations and Stability of a Gravitating Planet,” Proc. Roy. Soc. vol. 71; G. H. Darwin, “On the Figure and Stability of a liquid Satellite,” Phil. Trans. 206, p. 161; A. E. H. Love, “The Gravitational Stability of the Earth,” Phil. Trans. 207, p. 237; Proc. Roy. Soc. vol. 80.
  5. Survey of India, “The Attraction of the Himalaya Mountains upon the Plumb Line in India” (1901), p. 98.
  6. Account of Experiments to Determine the Figure of the Earth by means of a Pendulum vibrating Seconds in Different Latitudes (1825).
  7. Helmert, Theorien d. höheren Geod. ii., Leipzig, 1884.
  8. Helmert, Sitzber. d. kgl. preuss. Ak. d. Wiss. zu Berlin (1901), p. 336.
  9. “Bestimmung der absoluten Grösse der Schwerkraft zu Potsdam mit Reversionspendeln” (Veröffentlichung des kgl. preuss. Geod. Inst., N.F., No. 27).
  10. Die Königl. Observatorien für Astrophysik, Meteorologie und Geodäsie bei Potsdam (Berlin, 1890); Verhandlungen der I. Allgemeinen Conferenz der Bevollmächtigten zur mitteleurop. Gradmessung, October, 1864, in Berlin (Berlin, 1865); A. Hirsch, Verhandlungen der VIII. Allg. Conf. der Internationalen Erdmessung, October, 1886, in Berlin (Berlin, 1887); and Verhandlungen der XI. Allg. Conf. d. I. E., October, 1895, in Berlin (1896).
  11. Ibañez and Perrier, Jonction géod. et astr. de l’Algérie avec l’Espagne (Paris, 1886); Mémorial du dépôt général de la guerre, t. xii.: Nouvelle méridienne de France (Paris, 1885, 1902, 1904); Comptes rendus des séances de la 12e-19e conférence générale de l’Assoc. Géod. Internat., 1898 at Stuttgart, 1900 at Paris, 1903 at Copenhagen, 1906 at Budapest (Berlin, 1899, 1901, 1904, 1908); A. Ferrero, Rapport sur les triangulations, prés. à la 12e conf. gén. 1898.
  12. R. Schumann, C. r. de Budapest, p. 244.
  13. O. and A. Börsch, “Verbindung d. russ.-skandinav. mit der franz.-engl. Breitengradmessung” (Verhandlungen der 9. Allgem. Conf. d. I. E. in Paris, 1889, Ann. xi.).
  14. U.S. Coast and Geodetic Survey; H. S. Pritchett, superintendent. The Transcontinental Triangulation and the American Arc of the Parallel, by C. A. Schott (Washington, 1900).
  15. U.S. Coast and Geodetic Survey; O. H. Tittmann, superintendent. The Eastern Oblique Arc of the United States, by C. A. Schott (1902).
  16. Missions scientifiques pour la mesure d’un arc de méridien au Spitzberg entreprises en 1899-1902 sous les auspices des gouvernements russe et suédois. Mission russe (St Pétersbourg, 1904); Mission suédoise (Stockholm, 1904).
  17. Sir David Gill, Report on the Geodetic Survey of South Africa, 1833-1892 (Cape Town, 1896), vol. ii. 1901, vol. iii. 1905.
  18. O. Hecker, Bestimmung der Schwerkraft a. d. Atlantischen Ozean (Veröffentl. d. Kgl. Preuss. Geod. Inst. No. 11), Berlin, 1903.
  19. F. R. Helmert. “Neuere Fortschritte in der Erkenntnis der math. Erdgestalt” (Verhandl. des VII. Internationalen Geographen-Kongresses, Berlin, 1899), London, 1901.
  20. C. Lallemand, “Rapport sur les travaux du service du nivellement général de la France, de 1900 à 1906″ (Comp. rend. de la 14e conf. gén. de l’Assoc. Géod-Intern., 1903, p. 178).
  21. T. Albrecht, Resultate des internat. Breitendienstes, i. and ii. (Berlin, 1903 and 1906); F. Klein and A. Sommerfeld, Über die Theorie des Kreisels, iii. p. 672; R. Spitaler, “Die periodischen Luftmassenverschiebungen und ihr Einfluss auf die Lagenänderung der Erdaxe” (Petermanns Mitteilungen, Ergänzungsheft, 137); S. Newcomb, “Statement of the Theoretical Laws of the Polar Motion” (Astronomical Journal, 1898, xix. 158); F. R. Helmert, “Zur Erklärung der beobachteten Breitenänderungen” (Astr. Nachr. No. 3014); J. Weeder, “The 14-monthly period of the motion of the Pole from determinations of the azimuth of the meridian marks of the Leiden observatory” (Kon. Ak. van Wetenschappen to Amsterdam, 1900); A. Sokolof, “Détermination du mouvement du pôle terr. au moyen des mires méridiennes de Poulkovo” (Mél. math. et astr. vii., 1894); J. Bonsdorff, “Beobachtungen von \delta Cassiopejae mit dem grossen Zenitteleskop” (Mitteilungen der Nikolai-Hauptsternwarte zu Pulkowo, 1907); J. Larmor and E. H. Hills, “The irregular movement of the Earth’s axis of rotation: a contribution towards the analysis of its causes” (Monthly Notices R.A.S., 1906, lxvii. 22); A. S. Cristie, “The latitude variation Tide” (Phil. Soc. of Wash., 1895, Bull. xiii. 103); H. G. van de Sande Bakhuysen, “Über die Änderung der Polhöhe” (Astr. Nachr. No. 3261); A. V. Bäcklund, “Zur Frage nach der Bewegung des Erdpoles” (Astr. Nachr. No. 3787); R. Schumann, “Über die Polhöhenschwankung” (Astr. Nachr. No. 3873); “Numerische Untersuchung” (Ergänzungshefte zu den Astr. Nachr. No. 11); Weitere Untersuchungen (No. 4142); Bull. astr., 1900, June, report of different theoretical memoirs.