12 COSMOS. 



of the process involved in the measurement of a degree is 

 strictly astronomical, since the altitudes of stars determine 

 the curvature of the arc, whose length is found by the solution 

 of a series of triangles. The higher departments of mathe- 

 matics have succeeded, from given numerical data, in solving 

 the difficult problems of the figure of the earth, and the sur- 

 face of equilibrium of a fluid homogeneous, or dense shell-like 

 heterogeneous mass, which rotates uniformly round a solid 

 axis. Since the time of Newton and Huygens, the most dis- 

 tinguished geometricians of the eighteenth century have de- 

 voted themselves to the solution of these problems. It is well 

 that we should bear in mind that all the great results which 

 have been attained by intellectual labour and by mathe- 

 matical combinations of ideas, derive their importance not 

 only from that which they have discovered and which has 

 been appropriated by science, but more especially from the 

 influence which they have exerted on the development and 

 improvement of analytical methods. 



" The geometrical figure of the earth, in contradistinction 

 to the physical, 3 determines the surface which the super- 

 ficies of water would assume in passing through a net-work 

 of canals, connected with the ocean, and covering and inter- 

 secting the earth in every direction. The geometrical surface 

 intersects the directions of the forces vertically, and these 

 forces are composed of all the attractions emanating from the 

 individual particles of the earth, combined with the centri- 

 fugal force, which corresponds with its velocity of rota- 

 tion. 4 This surface must be generally considered as approxi- 

 mating very closely to an oblate spheroid, for irregularities 

 in the distribution of the masses in the interior of the earth 

 will also, where the local density is altered, give rise to irre- 

 gularity in the geometrical surface, which is the product 

 of the co-operation of unequally distributed elements. The 

 physical surface is the direct product of the surface of 



3 Gauss, Bestimmwng des JSreitenunterschiedes zwisclien den Sternwarten 

 von Gottingen und Altona, 1828, s. 73. (These two observatories, 

 by a singular chance, are situated within a few yards of the same 

 meridian.) 



4 Bessel, Ueber den Einflms der Unregelmassigkeiten der Figur der Erdt 

 auf geoddtisclie Arbeiten und Hire Vergleichung mit astronomischen 

 Bestimmungen, in Schumachers Astron. Nachr. Bd. xiv, No. 329, s. 270, 

 and Bessel and Baeyer, Gradmtysung in Ostpreussen, 1838, s. 427 442. 



