This thesis investigates the direct measurement of the thermal noise spectral distribution. Long base line gravitational wave detectors, being commissioned around the world, are limited in sensitivity in the intermediate frequencies by the thermal noise. These detectors are utilising suspended test mirrors for the detection of gravitational waves by measuring their relative displacement. One of the fundamental noise sources in these detectors is the thermally induced displacement of the suspension onto and within the mirrors. This thermally induced motion of the test mirrors limits the displacement sensitivity of the gravitational wave detectors. Knowledge of the spectral behavior of thermal noise over a wide frequency range will improve predictions and understanding of the behavior of the suspension and test mirrors. ¶ In this thesis the direct measurement of the thermal noise spectral distribution of a mechanical flexure resonator is described. The mechanical flexure resonator is an unidirectional ’wobbly table’ made from copper-beryllium, which hinges around four thin flexures 15 mm wide, 1 mm high and ~116 µm thick. The mechanical flexure resonator has a resonant frequency of 192 Hz, with a quality factor of ~3000. ¶ ...