A systematic study of rare earth and other trace elements in discrete diopsides residual abyssal peridotites sampled from 5000 km of ocean ridge demonstrates that they are the residues of variable degrees of melting in the garnet and spinel peridotite fields. Further, the data clearly demonstrate that the peridotites are the residues of near-fractional melting, not batch melting, and that typical abyssal basalt can evolve from aggregated fractional melts. Ion microprobe analyses of diopsides in abyssal peridotites from fracture zones along the American-Antarctica and Southwest Indian ridges reveal ubiquitous extreme fractionation of rare earth elements (REE) (n=0.002--0.05); depletion of Ti (300--1600 ppm), Zr (0.1--10 ppm), and Sr (0.1--10 ppm); and fractionation of Zr relative to Ti (Ti/Zr=250--4000). Ti and Zr in diopsides decrease with decreasing modal cpx in the peridotites, and samples dredged near hotspots are more depleted in incompatible elements than those dredged away from hotspots, consistent with higher degrees upper mantle-melting in the former. All studied samples exhibit marked negative anomalies in Ti and Zr relative to REE. Incompatible element concentrations in peridotite clinopyroxenes are well modeled by repeated melting and segregation in ≤0.1% increments to a total of 5--25% melting, a process very close to Rayleigh (fractional) melting; batch melting of a LREE-depleted source cannot account for the observed trace element concentrations in abyssal peridotites. The shapes of some REE patterns are consistent with variable degrees of melting initiated within the garnet stability field. Trace element concentrations in calculated integrated f fractional liquids approximate the composition of primitive ocean floor basalts, consistent with postsegregation aggregation of small increment melts produced over a depth and melting interval. ¿ American Geophysical Union 1990