The collisionless dissipation wavenumber kd of a plasma fluctuation may be defined as the minimum value of wavenumber at which damping becomes significant. Linear Vlasov theory is used to determine kd for both magnetosonic and Alfv¿n fluctuations in a spatially homogeneous, collisionless plasma with Maxwellian velocity distributions. Alfv¿n mode properties at maximum dissipation are independent of Te/Tp, and the corresponding magnetosonic mode properties are weakly dependent on this temperature ratio, so theory predicts that the proton beta &bgr;p is the primary parameter determining kd. Theory yields kdc/&ohgr;p=Sk/&bgr;p&agr;k, where &ohgr;p is the proton plasma frequency and Sk and &agr;k are fitting parameters; for the Alfv¿n mode &agr;k≃0.3, whereas for the magnetosonic mode &agr;k≃0.8. This implies that Alfv¿n fluctuations are the more likely source of collisionless dissipation for long wavelength magnetic turbulence cascading to shorter wavelengths at &bgr;p≲1, and magnetosonic fluctuations are more likely to dissipate such turbulence at &bgr;p≳1. ¿ 1999 American Geophysical Union