We derive predictions for the Nadir angle (theta(n)) dependence of the ratio N(mu)/N(e) of the rates of the mu-like and e-like multi-GeV events measured in water-Cerenkov detectors in the case of 3-neutrino oscillations of the atmospheric nu(e) (antinu(e)) and nu(mu) (antinu(mu)), driven by one neutrino mass squared difference, |Delta m2(31)| ~ (2.5 - 3.0) x 10^(-3) eV^2 >> Delta m2(21). This ratio is particularly sensitive to the Earth matter effects in the atmospheric neutrino oscillations, and thus to the values of sin^2(theta(13)) and sin^2(theta(23)), theta(13) and theta(23) being the neutrino mixing angle limited by the CHOOZ and Palo Verde experiments and that responsible for the dominant atmospheric nu(mu) -> nu(tau) (antinu(mu) -> antinu(tau)) oscillations. It is also sensitive to the type of neutrino mass spectrum which can be with normal (Delta m2(31) > 0) or with inverted (Delta m2(31) < 0) hierarchy. We show that for sin^2(theta(13)) > 0.01, sin^2(theta(23)) > 0.5 and at cos(theta(n)) > 0.4, the Earth matter effects modify substantially the theta(n)-dependence of the ratio N(mu)/N(e) and in a way which cannot be reproduced with sin^2(theta(13)) = 0 and a different value of sin^2(theta(23)). For normal hierarchy the effects can be as large as ~ 25% for cos(theta(n)) ~ (0.5 - 0.8), can reach ~ 35% in the Earth core bin cos(theta(n)) ~ (0.84 - 1.0), and might be observable. They are typically by ~ 10% smaller in the inverted hierarchy case. An observation of the Earth matter effects in the Nadir angle distribution of the ratio N(mu)/N(e) would clearly indicate that sin^2(theta(13)) > 0.01 and sin^2(theta(23)) > 0.50.