Discrete transitions in the heat flux through fluids confined between horizontal plates were first observed by Malkus. The transitions, and the remarkable linear segments in a linear plot of Nusselt number times Rayleigh number versus Rayleigh number, are confirmed experimentally up to a Rayleigh number of Ra = 2.8 x 10⁶ in a convection chamber of different geometry and in fluids of different Prandtl numbers than those used by Malkus. Two transitions at Ra ≈ 8200 and 24,000 were always observed instead of the single transition suggested to occur at Ra ≈ 18,000. The first four transitions are explained by quasi-linear stability theory for the appearance of unstable vertical modes upon a distorted mean temperature gradient, ᵦ(z) , with rigid boundaries. The approximations of substituting a constant ᵦ in place of a realistic , ᵦ(z), and free surfaces for rigid boundaries, are shown to produce large errors in calculated critical Rayleigh numbers for the higher modes. Pre-publication review copy.