Thirdly, the calculation must further be based on the fact of a peculiar circumstance in the movement of solid bodies, a fact so familiar to us that but for this inquiry it might never have been thought of as something that need not be. When in our space of three dimensions two points of a solid body are kept fixed, its movements are limited to rotations round the straight line connecting them. If we turn it completely round once, it again occupies exactly the position it had at first. This fact, that rotation in one direction always brings a solid body back into its original position, needs special mention. A system of geometry is possible without it. This is most easily seen in the geometry of a plane. Suppose that with every rotation of a plane figure its linear dimensions increased in proportion to the angle of rotation, the figure after one whole rotation through 360 degrees would no longer coincide with itself as it was originally. But any second figure that was congruent with the first in its original position might be made to coincide with it in its second position by being also turned through 360 degrees. A consistent system of geometry would be possible upon this supposition, which does not come under Riemann’s formula. On the other hand I have shown that the three assumptions taken together form a sufficient basis for the starting-point of Riemann’s investigation, and thence for all his further results relating to the distinction of different spaces according to their measure of curvature. It still remained to be seen whether the laws of motion, as dependent on moving forces, could also be consistently transferred to spherical or pseudospherical space. This investigation has been carried out by Professor Lipschitz of Bonn.[14] It is found that the comprehensive expression for all the laws of dynamics, Hamilton’s principle, may be directly transferred to spaces of which the measure of curvature is other than zero. Accordingly, in this respect also, the disparate systems of geometry lead to no contradiction. [Footnote 14: ‘Untersuchungen über die ganzen homogenen Functionen von _n_ Differentialen’ (Borchardt’s _Journal für Mathematik_, Bd. lxx. 3, 71; lxxiii. 3,1); ‘Untersuchung eines Problems der Variationsrechnung’ (_Ibid._ Bd. lxxiv.).]