MarkT:
The Arduino deals in voltages between 0 and +5V, the circuit you are trying to switch is -15V to 0V, so there is no overlap.
To switch between -15V and 0V controlled from the 0V to +5V signal out of the Arduino, use a PNP transistor in the common base configuration, if the current to be switched is below the 40mA rating of the Arduino.