[2] Physics, bk. iv, ch. xiv.
[3] De Natura Rerum, bk. i, vv. 460 ff.
[4] It is assumed that the ship is not rolling or pitching, and that there is no vibration in the train.
[5] The best definition of the second that can be given is the following: it is the time which light takes to cover 186,000 miles in empty space and far from any strong gravitational field. This definition, the only strict definition, is further justified by the fact that there is no better means of regulating clocks than luminous or Hertzian (which have the same speed) signals.
[6] In the geometrical calculus or representation that may be substituted for this the hypotenuse of the triangle is the distance in time, each second being represented by 300,000 kilometres.
[7] As an example of an identical force acting during periods of time successively equal to 1, 2, or 3, we may take three guns of the same calibre, but of lengths equal to 1, 2, and 3, and of which the charges, or rather, their propulsive forces, are identical and constant. It is found that the initial velocities of the shells are, in relation to each other, 1, 2, and 3.
[8] De Natura Rerum, bk. ii, vv. 235-40.
[9] It is obvious that we assume the projectile to be without rotation: that is to say, the Columbia cannon must not, in our hypotheses, be rifled. This is indispensable, for if the projectile turned, there would be centrifugal effects which would greatly complicate both the phenomena and our argument.
[10] It goes without saying that in all this we assume that the luminous ray travels in a homogeneous medium.
[11] We are, of course, imagining the earth as perfectly circular, without irregularities.