A new technique using radioactive tracers has been developed for accurate determination of the degree of melting of rocks. Trace amounts of a compound (CO2 is used here) or an element that preferentially enters the melt phase are determined. A glass of the starting material is used as standard. The proportion of partial melt equals the ratio of the amount of tracer in the standard and the unknown. Using this technique, two garnet peridotite nodules, samples 1611 (from a Lesotho kimberlite) and 66SAL-1 (from Hawaiian nephelinite tuff), were subjected to temperatures from their solidi to 1750 degrees C at 20 and 35 kbar pressure. Both peridotite nodules show similar melting behavior. The degree of melting was plotted against temperature, and such melting curves show a distinct change of slope wherever one or more phases disappear. At 20 kbar, olivine (Ol), clinopyroxene (Cpx), orthopyroxene (Opx), and spinel (Sp) coexist with tholeiitic liquid during the initial 25 percent melting of sample 1611 and 60 percent melting of 66SAL-1. The temperature interval of tholeiite melt + 2 pyroxenes + olivine is less than 40 degrees C in both cases. At 35 kbar, quality chemical analysis of the samples became impossible because of quenching problems. Nodule 1611 begins to melt at about 1625 degrees -1630 degrees C, apparently producing alkali picritic liquid coexisting with garnet, two pyroxenes, and olivine until about 25 percent melting has occurred and the temperature has risen to about 1660 degrees C, where garnet disappears. The flat slope of the melting curves in the lowest-temperature melting interval and the nearly constant bulk composition of the liquid suggest that the tholeiitic liquid in this melting range may be derived from a peridotitic parent under near isobaric, invariant conditions. In the lowest-temperature melting interval at 20 kbar two reaction relations between partial melt and crystalline residue are suggested: for 66SAL-1 (lowest temperature) the reaction is Sp + Opx + Cpx (Diss) --> Ol + Liq; for the higher temperature melting of 1611 the reaction is Ol + Cpx (pigeonite) --> Opx + Liq. The observations suggest that phase equilibria in simple systems such as MgSiO3-CaSiO3-Al2O3 and Mg2SiO4-CaMgSi2O6-SiO2 closely resemble those of natural peridotite in the upper mantle.