1. The laws according to which the nature of physical systems alter are independent of the manner in which these changes are referred to two co-ordinate systems which have a uniform translators motion relative to each other.
2. Every ray of light moves in the “stationary co-ordinate system” with the same velocity c, the velocity being independent of the condition whether this ray of light is emitted by a body at rest or in motion.[[6]] Therefore
velocity = Path of Light/Interval of time,
where, by ‘interval of time’ we mean time as defined in [§1].
Let us have a rigid rod at rest; this has a length l, when measured by a measuring rod at rest; we suppose that the axis of the rod is laid along the X-axis of the system at rest, and then a uniform velocity v, parallel to the axis of X, is imparted to it. Let us now enquire about the length of the moving rod; this can be obtained by either of these operations.—
(a) The observer provided with the measuring rod moves along with the rod to be measured, and measures by direct superposition the length of the rod:—just as if the observer, the measuring rod, and the rod to be measured were at rest.
(b) The observer finds out, by means of clocks placed in a system at rest (the clocks being synchronous as defined in [§1]), the points of this system where the ends of the rod to be measured occur at a particular time t. The distance between these two points, measured by the previously used measuring rod, this time it being at rest, is a length, which we may call the “length of the rod.”
According to the Principle of Relativity, the length found out by the operation a), which we may call “the length of the rod in the moving system” is equal to the length l of the rod in the stationary system.
The length which is found out by the second method, may be called ‘the length of the moving rod measured from the stationary system.’ This length is to be estimated on the basis of our principle, and we shall find it to be different from l.
In the generally recognised kinematics, we silently assume that the lengths defined by these two operations are equal, or in other words, that at an epoch of time t, a moving rigid body is geometrically replaceable by the same body, which can replace it in the condition of rest.