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The Milankovitch theory of climate change proposes that glacial-interglacial cycles are driven by changes in summer insolation at high northern latitudes. The timing of climate change in the Southern Hemisphere at glacial-interglacial transitions (which are known as terminations) relative to variations in summer insolation in the Northern Hemisphere is an important test of this hypothesis. So far, it has only been possible to apply this test to the most recent termination, because the dating uncertainty associated with older terminations is too large to allow phase relationships to be determined. Here we present a new chronology of Antarctic climate change over the past 360,000 years that is based on the ratio of oxygen to nitrogen molecules in air trapped in the Dome Fuji and Vostok ice cores. This ratio is a proxy for local summer insolation, and thus allows the chronology to be constructed by orbital tuning without the need to assume a lag between a climate record and an orbital parameter. The accuracy of the chronology allows us to examine the phase relationships between climate records from the ice cores and changes in insolation. Our results indicate that orbital-scale Antarctic climate change lags Northern Hemisphere insolation by a few millennia, and that the increases in Antarctic temperature and atmospheric carbon dioxide concentration during the last four terminations occurred within the rising phase of Northern Hemisphere summer insolation. These results support the Milankovitch theory that Northern Hemisphere summer insolation triggered the last four deglaciations.