[24] _Proof_—According to the theory of Newton, the number of “lines of force” which come from infinity and terminate in a mass m is proportional to the mass _m_. If, on the average, the mass density ρ0 is constant throughout the universe, then a sphere of volume _V_ will enclose the average mass ρ0_V_. Thus the number of lines of force passing through the surface _F_ of the sphere into its interior is proportional to ρ0_V_. For unit area of the surface of the sphere the number of lines of force which enters the sphere is thus proportional to ρ0_V/F_ or to ρ0_R_. Hence the intensity of the field at the surface would ultimately become infinite with increasing radius _R_ of the sphere, which is impossible. This conception is in itself not very satisfactory. It is still less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into infinite space, never to return, and without ever again coming into interaction with other objects of nature. Such a finite material universe would be destined to become gradually but systematically impoverished. In order to escape this dilemma, Seeliger suggested a modification of Newton’s law, in which he assumes that for great distances the force of attraction between two masses diminishes more rapidly than would result from the inverse square law. In this way it is possible for the mean density of matter to be constant everywhere, even to infinity, without infinitely large gravitational fields being produced. We thus free ourselves from the distasteful conception that the material universe ought to possess something of the nature of a centre. Of course we purchase our emancipation from the fundamental difficulties mentioned, at the cost of a modification and complication of Newton’s law which has neither empirical nor theoretical foundation. We can imagine innumerable laws which would serve the same purpose, without our being able to state a reason why one of them is to be preferred to the others; for any one of these laws would be founded just as little on more general theoretical principles as is the law of Newton. XXXI. THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED” UNIVERSE But speculations on the structure of the universe also move in quite another direction. The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the _infiniteness_ of our space without coming into conflict with the laws of thought or with experience (Riemann, Helmholtz). These questions have already been treated in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them briefly here.