Natstrack,
try this for an explanation and the relevant equations:
Machmeter - Wikipedia, the free encyclopedia
Put simply, the Machmeter does not need to know temperature. For subsonic speeds it simply uses the ratio of impact pressure to static pressure and then does some additional maths. The maths is simple enough to be implemented with a purely mechanical mechanism, which is exactly what was done in early Mach meters, prior to the invention of air data computers. You don't need to know local air density to calculate mach number - it does not appear in the equation at all (although you could ague it is implicit within impact pressure, but to determine it you would need to know your TAS first).
You DO need temperature to calculate TAS, because you need to calculate the local air density from the temperaure and static pressure. Air follows the perfect gas law well enough:
P = Rho * R * T therefore Rho = P/(RT)
thus if you need to calculate TAS your aicraft will amost certainly have a total temperaure probe, which will supply the information the air data computer needs to calculate local air density and thus derive TAS from EAS.
Total air temperature - Wikipedia, the free encyclopedia