[en] It is shown that a sensitive search for neutrino oscillations involving more flavors than just nu_e and nu_μ is provided by measurement of the ratio of the total interaction rates of upward- and downward-going cosmic-ray neutrinos within a massive (approx. 10 kiloton) detector. Assuming mixing between all pairs of nu_e, nu_μ and nu_tau, the experiment is capable of observing time-averaging probabilities < P_{e tau}>_t of magnitude set by mixing strengths corresponding to, e.g., the d- to s-quark mixing strength, and of reaching the limit δm_ij2 identical with vertical bar m_i2-m_j2 vertical bar approx. = 10^-₄ eV₂, where m_i, m_j are neutrino mass eigenstates, and P_{e tau} and p_{μ tau} are the probabilities for nu_e and nu_μ, respectively, to oscillate into nu_tau after traversing a distance L approx. = diameter of the earth