An applying aspect is one in which the faster of the two planets is still moving toward the exact angle. A separating aspect is one in which the faster planet has already passed the exact angle and is moving away. The distinction matters because applying aspects are usually interpreted as gathering force or building toward an event, while separating aspects are interpreted as releasing or already in motion.
In horary astrology (the branch that answers specific questions from a chart cast at the moment of asking), the applying/separating distinction is critical. An applying aspect between the relevant significators usually answers the question affirmatively; a separating one usually answers it negatively. The convention varies by school, but the principle is widely shared.
In natal astrology, applying aspects in the chart are sometimes interpreted as life themes the person is still working toward, while separating aspects are themes that are already in motion or have been resolved before the chart was set. The interpretation is subtle and not all astrologers use it.
In transit work, the applying phase of a transit is often where the felt intensity builds. A Saturn transit applying to your natal Sun produces growing pressure as Saturn approaches the exact degree, peaks at exact, and then begins to release. Tracking this approach-perfection-separation cycle is one of the practical uses of an ephemeris.
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