A new technique of x-ray spectroscopy of μ-mesonic atoms has been developed. The x-rays are produced when a μ- meson undergoes transitions between Bohr orbits about nuclei of various Z. The mesons are produced by the Columbia University 164-in. Nevis cyclotron. The x-rays are detected, and their energies are measured to better than 1 percent accuracy (for Z>=22) using a NaI crystal scintillation spectrometer. The 2p-->1s transition energies were measured to be 0.35, 0.41, 0.955, 1.55, 1.60, 3.50, 5.80, 6.02, and 6.02 Mev for Z=13, 14, 22, 29, 30, 51, 80, 82, and 83. Special attention was paid to the Pb spectrum, and it is believed that an 0.2-Mev fine structure splitting has been observed. This is the expected splitting if the μ- meson is a spin 1/2 Dirac "heavy electron" of 210 electron masses, having the expected Dirac magnetic moment and having no strong nonelectromagnetic interaction with nuclear matter. Since the μ- meson Bohr orbits are 210 times closer to the nucleus than the equivalent electron orbits, the x-ray energies are quite sensitive to nuclear size for medium and large Z. In the case of Pb, a 1 percent change in nuclear radius gives a 1 percent change in the calculated x-ray energy. Assuming constant proton density inside a spherical nucleus of radius R0=r0A13 and the above properties for the μ meson, we obtain r0=1.17, 1.21, 1.22, and 1.17×10-13 cm for Z=22, 29, 51, and 82. The significance of these results in relation to other nuclear size measurements is discussed.