An endmember extraction method was proposed based on the maximum N dimensional solid spectral angle theory, which was termed as MNSSA (Maximum N-dimensional Solid Spectral Angle) in this paper. By using the ability of calculating the solid spectral angle which constructed by N spectral vectors in the high-dimensional Euclid space for the method, the independences of the N spectral vectors were measured. In the assumption of linear unmixing model, the solid spectral angle constitutes the maximum value for N endmember spectral vectors. The MNSSA method is not restricted by the numbers of endmembers or the numbers of bands. More importantly, the MNSSA method is not sensitive with the amplitude variations as well as robustness with effects from shadow strength and illumination intensity. Experiment results on synthetic and real hyperspectral data collected by AVIRIS (Airborne Visible/Infrared Imaging Spectrometer) indicate that the method of MNSSA is better than the current mainstream used endmember extraction methods. The influences on the amplitude of the endmembers caused by the shadow factor are overcomed by the method of MNSSA. The method is with good anti-noise performance. Furthermore, the reconstruction errors for real hyperspectral data were reduced remarkably.