If these measurements do not always accord in the curvatures of different meridians under the same degree of latitude, this very circumstance speaks in favor of the exactness of the instruments and the methods employed, and of the accuracy and the fidelity to nature of these partial results. The conclusion to be drawn from the increase of forces of attraction (in the direction from the equator to the poles) with respect to the figure of a planet is dependent on the distribution of density in its interior. Newton, from theoretical principles, and perhaps likewise prompted by Cassini's discovery, previously to 1666, of the compression of Jupiter,* determined, in his immortal work, 'Philosophiae Naturalis Principia', that the compression of the Earth, as a homogeneous mass, was 1/230th.
[footnote] *Brewster, 'Life of Sir Isaac Newton', 1831, p. 162. "The discovery of the spheroidal form of Jupiter by Cassini had probably directed the attention of Newton to the determination of its cause, and consequently, to the investigation of the true figure of the Earth." Although Cassini did not announce the amount of the compression of Jupiter (1/15th) till 1691 ('Anciens Memoires de l'Acad. des Sciences', t. ii., p. 108), yet we know from Lalande ('Astron.', 3me ed., t. iii., p. 335) that Moraldi possessed some printed sheets of a Latin work, "On the Spots of the Planets," commenced by Cassini, from which it was obvious that he was aware of the compression of Jupiter before the year 1666, and therefore at least twenty-one years before the publication of Newton's 'Principia'.
Actual mesurements, p 165 made by the aid of new and more perfect analysis, have, however, shown that the compression of the poles of the terrestrial spheroid, when the density of the strata is regarded as increasing toward the center, is very nearly 1/300th.
Three methods have been employed to investigate the curvature of the Earth's surface, viz., measurements of degrees, oscillations of the pendulum, and observations of the inequalities in the Moon's orbit. The first is a direct geometrical and astronomical method, while in the other two we determine from accurately observed movements the amount of the forces which occasion those movements, and from these forces we arrive at the cause from whence they have originated, viz., the compression of our terrestrial spheroid. In this part of my delineation of nature, contrary to my usual practice, I have instanced methods because their accuracy affords a striking illustration of the intimate connection existing among the forms and forces of natural phenomena, and also because their application has given occasion to improvements in the exactness of instruments (as those employed in the measurements of space) in optical and chronological observations; to greater perfection in the fundamental branches of astronomy and mechanics in respect to lunar motion and to the resistance experienced by the oscillations of the pendulum; and to the discovery of new and hitherto untrodden paths of analysis. With the exception of the investigations of the parallax of stars, which led to the discovery of aberration and nutation, the history of science presents no problem in which the object attained — the knowledge of the compression and of the irregular form of our planet — is so far exceeded in importance by the incidental gain which has accrued, through a long and weary course of investigation, in the general furtherance and improvement of the mathematical and astronomical sciences. The comparison of eleven measurements of degrees (in which are included three extra-European, namely, the old Peruvian and two East Indian) gives, according to the most strictly theoretical requirements allowed for by Bessel,* a compression p 166 of 1/299th.
[footnote] *According to Bessel's examination of ten measurements of degrees, in which the error discovered by Poissant in the calculation of the French measurements is taken into consideration (Schumacher, 'Astron. Nachr.', 1841, No. 438, s. 116), the semi-axis major of the elliptical spheroid of revolution to which the irregular figure of the Earth most closely approximates is 3,272,077.14 toises, or 20,924,774 feet; the semi-axis minor, 3,261,159,83 toises, or 20,854,821 feet; and the amount of compression or eccentricity 1/299.152d; the length of a mean degree of the meridian, 57,013.109 toises, or 364,596 feet, with an error of + 2.8403 toises, or 18.16 feet, whence the length of a geographical mile is 3807.23 toises, or 6086.7 feet. Previous combinations of measurements of degrees varied between 1/302d and 1/297th; thus Walbeck ('De Forma of Magnitudine telluris in demensis arcubus Meridiani definiendis', 1819) gives 1/30278th: Ed. Schmidt ('Lehrbuch der Mathem. und Phys. Geographie', 1829, s. 5) gives 1/20742d, as the mean of seven measures. Respecting the influence of great differences of longitude on the polar compression, see 'Bibliotheque Universelle', t. xxxiii., p. 181, and t. xxxv., p. 50: likewise 'Connaissance des Tems', 1829, p. 290. From the lunar inequalities alone, Laplace ('Exposition du Syst. du Monde', p. 229) found it, by the older tables of Burg, to be 1/3245th; and subsequently, from the lunar observations of Burckhardt and Bouvard, he fixed it at 1/299.1th ('Mecanique Celeste', t. v., p. 13 and 43).
In accordance with this, the polar radius is 10,938 toises (69,944 feet), or about 11 1/2 miles, shorter than the equatorial radius of our terrestrial spheroid. The excess at the equator in consequence of the curvature of the upper surface of the globe amounts, consequently, in the direction of gravitation, to somewhat more than 4 3/7th times the height of Mont Blanc, or only 2 1/2 times the probable height of the summit of the Chawalagiri, in the Himalaya chain. The lunar inequalities (perturbation in the moon's latitude and longitude) give according to the last investigations of Laplace, almost the same result for the ellipticity as the measurements of degrees, viz., 1/299th. The results yielded by the oscillation of the pendulum give, on the whole, a much greater amount of compression, viz., 1/288th.*
[footnote] *The oscillations of the pendulum give 1/288.7th as the general result of Sabine's great expedition (1822 and 1823, from the equator to 80 degrees north latitude); according to Freycinet, 1/286.2d, exclusive of the experiments instituted at the Isle of France, Guam, and Mowi (Mawi); according to Forster, 1/289.5th; according to Duperrey, 1/266.4th; and according to Lutke ('Partie Nautique', 1836, p. 232), 1/270th, calculated from eleven stations. On the other hand, Mathieu ('Connais. des Temps', 1816, p. 330) fixed the amount at 1/298.2d, from observations made between Formentera and Dunkirk; and Biot, at 1/304th, from observations between Formentera and the island of Ust. Compare Baily, 'Report on Pendulum Experiments', in the 'Memoirs of the Royal Astronomical Society', vol. vii., p. 96; also Borenius, in the 'Bulletin de l'Acad. de St. Petersbourg', 1843, t. i., p. 25. The first proposal to apply the length of the pendulum as a standard of measure, and to establish the third part of the seconds pendulum (then supposed to be every where of equal length) as a 'pes horarius', or general measure, that might be recovered at any age and by all nations, is to be found in Huygens's 'Horologium Oscillatorium', 1673, Prop. 25. A similar wish was afterward publicly expressed, in 1742, on a monument erected at the equator by Bouguer, La Condamine, and Godin. On the beautiful marble tablet which exists, as yet uninjured, in the old Jesuits' College at Quito, I have myself read the inscription, 'Penduli simplicis aequinoctialis unius minuti secundi archetypus, mensurae naturalis exemplar, utinam universalis!' From an observation made by La Condamine, in his 'Journal du Voyage a l'Equateur', 1751, p. 163, regarding parts of the inscription that were not filled up, and a slight difference between Bonguer and himself respecting the numbers, I was led to expect that I should find considerable discrepancies between the marble tablet and the inscription as it had been described in Paris; but, after a careful comparison, I merely found two "ex arca graduum plusquam trium," and the date of 1745 instead of 1742. The latter circumstance is singular, because La Condamine returned to Europe in November, 1744, Bouguer in June of the same year, and Godin had left South America in July, 1744. The most necessary and useful amendment to the numbers on this inscription would have been the astronomical longitude of Quito. (Humboldt, 'Recueil d'Observ. Astron.', t. ii., p. 319-354.) Nouet's latitudes, engraved on Egyptian monuments, offer a more recent example of the danger presented by the grave perpetuation of false or careless results.
Galileo, who first observed when a boy (having, probably, suffered his thoughts to wander from the service) that the height of the vaulted roof of a church might be measured by the time of the vibration of the chandeliers suspended at different altitudes, could hardly have anticipated that the pendulum would one day be carried from pole to pole, in order to determine the form of the Earth, or, rather, that the unequal density of the strata of the Earth affects the length of the seconds pendulum by means of intricate forces of local attraction, which are, however, almost regular in large tracts of land. These geognostic relations of an instrument intended for the measurement of time — this property of the pendulum, by which, like a sounding line, it searches unknown depths, and reveals in volcanic islands,* or in the declivity of elevated continental mountain chains,** dense masses of basalt and melaphyre instead of cavities, render it difficult, notwithstanding the admirable simplicity of the method, to arrive at any great result regarding the figure of the Earth from observation of the oscillations of the pendulum.
[footnote] *Respecting the augmented intensity of the attraction of gravitation in volcanic islands (St. Helena, Ualan, Fernando de Noronha, Isle of France, Guam, Mowe, and Galapagos), Rawak (Lutke, p. 240) being an exception, probably in consequence of its proximity to the highland of New Guinea, see Mathieu, in Delambre, 'Hist. de l'Astronomie, au 18me Siecle', p. 701.
[footnote] **Numerous observations also show great irregularities in the length of the pendulum in the midst of continents, and which are ascribed to local attractions. (Delambre, 'Mesure de la Meridienne', t. iii., p. 548; Biot, in the 'Mem. de l'Academie des Sciences', t. viii., 1829, p. 18 and 23.) In passing over the South of France and Lombardy from west to east, we find the minimum intensity of gravitation at Bordeaux; from thence it increases rapidly as we advance eastward, through Figeac, Clermont-Ferrand, Milan, and Padua; and in the last town we find that the intensity has attained its maximum. The influence of the southern declivities of the Alps is not merely t on the general size of their mass, but (much more), in the opinion of Elie de Beaumont ('Rech. sur les Revol. de la Surface du Globe', 1830, p. 729), on the rocks of melaphyre and serpentine, which have elevated the chain. On the declivity of Ararat, which with Caucasus may be said to lie in the center of gravity of the old continent formed by Europe, Asia, and Africa, the very exact pendulum experiments of Fedorow give indications, not of subterranean cavities, but of dense volcanic masses. (Parrot, 'Reise zum Ararat', bd. ii., s. 143.) In the geodesic operations of Carlini and Plana, in Lombardy, differences ranging from 20" to 47".8 have been found between direct observations of latitude and the results of these operations. (See the instances of Andrate and Mondovi, and those of Milan and Padua, in the 'Operations Geodes. et Astron. pour la Mesure d'un Arc du Parallele Moyen', t. ii., p. 347; 'Effemeridi Astron. di Milano', 1842, p. 57.) The latitude of Milan, deduced from that of Berne, according to the , is 45ºdegrees 27' 52", while, according to direct astronomical observations, it is 45 degrees 27' 35". As the perturbations extend in the plain of Lombardy to Parma, which is far south of the Po (Plana, 'Operat. Geod.', t. ii., p. 847), it is probable that there are deflecting causes 'concealed beneath the soil of the plain itself'. Struve has made similar experiments [with corresponding results] in the most level parts of eastern Europe. (Schumacher, 'Astron. Nachrichten', 1830, No. 164, s. 399.) Regarding the influence of dense masses supposed to lie at a small depth, equal to the mean height of the Alps, see the analytical expressions given by Hossard and Rozet, in the 'Comptes Rendus', t. xviii., 1844, p. 292, and compare them with Poisson, 'Traite de Mecanique' (2me ed., t. i., p. 482. The earliest observations on the influence which different kinds of rocks exercise on the vibration of the pendulum are those of Thomas Young, in the 'Philos. Transactions' for 1819, p. 70-96. In drawing conclusions regarding the Earth's curvature from the length of the pendulum, we ought not to overlook the possibility that its crust may have undergone a process of hardening previously to metallic and dense basaltic masses having penetrated from great depths, through open clefts, and approached near the surface.