Light propagation comes out of Maxwell's equations, which describe the origins and interaction of Electric and Magnetic fields. Maxwell's equations led to the "discovery" of the Lorentz transformation. Newtonian mechanics are invarient under Galilean transformation, but Maxwell's equations are not. The Lorentz transformation was originally derived to create a coordinate transform that maintained the invarience of Maxwell's equations. It is considered the correct transform to apply because experimental obervation shows us that Maxwell's equations are indeed invarient under transformations between inertial reference frames.
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Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time ; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.
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We were led to that conflict by the considerations of Section 6, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance w per second relative to the carriage, traverses the same distance also with respect to the embankment in each second of time. But, according to the foregoing considerations, the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment (as reference-body). Hence it cannot be contended that the man in walking travels the distance w relative to the railway line in a time which is equal to one second as judged from the embankment.
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Here, he's speaking of time dilation, where time in one frame has a different meaning than time in another frame.