We report in detail that unlike other materials, carbon nanotubes are so small that changes in structure can affect the Young's modulus. The variation in modulus is attributed to differences in torsional strain, which is the dominant component of the total strain energy. Torsional strain, and correspondingly Young's modulus, increases significantly with decreasing tube diameter and increases slightly with decreasing tube helicity.  Journal of Applied Physics 84, 1939 (1998).