's at infinity; and if the time-changes of the gravitational field are neglected) the well-known equations of Newtonian mechanics emerge out of the differential equations of Einstein's theory, which were obtained from perfectly general beginnings.
[Note 27] (p. 53). The theory of surfaces, i.e. the study of geometry upon surfaces, makes it immediately apparent that the theorems, which have been established for any surface, also hold for any surface which can be generated by distorting the first without tearing. For if two surfaces have a point-to-point correspondence, such that the line-elements are equal at corresponding points, then corresponding finite arcs, angles, and areas, etc., will be equal. One thus arrives at the same planimetrical theorems for the two surfaces. Such surfaces are called "deformable" surfaces. The necessary and sufficient condition that surfaces be continuously deformable is that the expression for the line-element of the one surface
can be transformed into that for the other,
According to Gauss, it is necessary that both surfaces have equal measures of curvature. If the latter is constant over the whole surface, as e.g. in the case of a cylinder or a plane, all conditions for the deformability of the surfaces are fulfilled. In other cases, special equations offer a criterion as to whether surfaces, or portions of surfaces, are deformable into one another. The numerous subsidiary problems, which result out of these questions, are discussed at length in every book dealing with differential geometry (e.g. Bianchi-Lukat).[17] This branch of training, which was hitherto of interest only to mathematicians, now assumes very considerable importance for the physicist too.
[17]Forsyth's "Differential Geometry."—H. L. B.
[Note 28] (p. 61). One must avoid being deceived into the belief that Newton's fundamental law is in any way to be regarded as an explanation of gravitation. The conception of attractive force is borrowed from our muscular sensations, and has therefore no meaning when applied to dead matter. C. Neumann, who took great pains to place Newton's mechanics on a solid basis, glosses upon this point himself in a drastic fashion, in the following narrative, which shows up the weaknesses of the former view:
"Let us suppose an explorer to narrate to us his experiences in yonder mysterious ocean. He had succeeded in gaining access to it, and a remarkable sight had greeted his eyes. In the middle of the sea he had observed two floating icebergs, a larger and a smaller one, at a considerable distance from one another. Out of the interior of the larger one, a voice had resounded, issuing the following command in a peremptory tone: 'Ten feet nearer!' The little iceberg had immediately carried out the order, approaching ten feet nearer the larger one. Again, the larger gave out the order: 'Six feet nearer!' The other had again immediately executed it. And in this manner order after order had echoed out: and the little iceberg had continually been in motion, eager to put every command immediately and implicitly into action.