The TRACER instrument (Transition Radiation Array for Cosmic Energetic Radiation) has been developed for direct measurements of the heavier primary cosmic-ray nuclei at high energies. The instrument had a successful long- duration balloon flight in Antarctica in 2003. The detector system and measurement process are described, details of the data analysis are discussed, and the individual energy spectra of the elements O, Ne, Mg, Si, S, Ar, Ca, and Fe (nuclear charge image -26) are presented. The large geometric factor of TRACER and the use of a transition radiation detector make it possible to determine the spectra up to energies in excess of 10 super(14) eV per particle. A power-law fit to the individual energy spectra above 20 GeV amu super(-1) exhibits nearly the same spectral index for all elements, without noticeable dependence on the elemental charge Z.