The Galilean transformation is based on the definition of *velocity*: *v* = *dx*/*dt*, which for constant velocity leads to

*x* = ∫ *v* *dt* = *x*_{0} + *vt*

So for two observers at constant velocity in relation to each other we have

*x′ = x + vt*

with their time coordinates unchanged: *t′* = *t* if their origins coincide. An independent variable such as time is the same for all observers, except perhaps for a constant difference adopted for convenience.

The dual Galilean transformation is based on the definition of *lenticity*: *w* = *dt*/*dx*, which for constant lenticity leads to

*t* = ∫ *w* *dx* = *t*_{0} + *wx*

So for two observers at constant lenticity in relation to each other we have

*t′ = t + wx*

with their length coordinates unchanged: *x′* = *x* if their origins coincide.

These transformations reinforce the proposition that time is not necessarily the independent variable, and so is best understood as measured by a stopwatch rather than a clock.