Alexander Ross Clarke and Friedrich Robert Helmert

“ Since the attraction of a mountain mass is expressed as a numerical multiple of 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 , 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 ”

“ 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. ”

“ 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. ”

“ 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:—
The most favourable value for the depth of the isostatic surface is approximately 114 km. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 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. ”

“ 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. ”

“ By assuming Roche’s law for the variation of the density in the interior of the Earth, viz. being a coefficient, it is shown that in latitude 45°, the meridian is depressed about 31/4 metres from the ellipse, and the coefficient of the term 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. ”

“ 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 ”

“ 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. ”

“ 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 for continents and islands; it was found that in consequence of the constitution of the earth’s crust (Pratt) is greater on small islands of the ocean than on continents by an amount which may approach to 0·3 cm. ”

“ 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, =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) . ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ Suppose and to be two trigonometrical stations, and that at there is a disturbing force drawing the vertical through an angle , then it is evident that the apparent zenith of will be really that of some other place , whose distance from is , when is the earth’s radius ”

“ 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. ”

“ 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. ”

“ Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are ; and if further we suppose that is the true value of , and that itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the corrections on the position of the arc in latitude, i.e. on ,
here ”

“ With the ellipticity 1/299·15, =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. ”