A high resolving power mid-infrared, astronomical Fourier transform spectrometer has been developed for use at ground based telescopes. The spectrometer is designed to operate over a broad wavelength range from l0 to 500 Pm, (i.e. 1000 to 20 cm-1), the specific range being defined by the choice of beamsplitter, filters and detector. The spectrometer employs a folded optics configuration in which a double-sided mirror moves on a linear carriage to produce a factor of four multiplication of optical path difference. This results in a compact instrument in which an unapodised resolution of .01 cm-1 is achieved with a mirror translation of only 12.5 cm-1 and which can easily be mounted at the f/35 Cassegrain foci of infrared telescopes. A major feature of the system is the ability to compute, display and monitor any part of a 32,768 point spectrum of a phase corrected single-sided interferogram in real time. This is accomplished by efficient Fourier transform routines which exploit the inherent symmetry properties of the interferogram. These and other features of the complete spectrometer system are discussed.