Texas,
Is there an audible "click" as you turn the Thermostat dial?.
If there isn't, it means the contacts in the Thermostat aren't closing and this is what is preventing the Compressor from starting.
Yeah, you're right about the humming Compressor, in some cases they can even blow the fuse/CB on the circuit that supplies the appliance.
I'd check the Thermostat and the terminal box first and then take it from there.
If you need any more help, just give me a yell!. [Linked Image]
Best of luck with the maths.