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How did Hipparchus calculate the displacement of the Sun’s orbit?

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Engraving of six astronomers, with Hipparchus seated at center and, from left to right, Galileo, Johannes Hevelius, Tycho Brahe, Nicolaus Copernicus, and Ptolemy standing, each holding scientific instruments. — Hipparchus “Six astronomers,” the frontispiece of *Astronomica Institutio* (1692) by Jan Luyts, engraving after drawing by Gerard Hoet. Hipparchus is seated at center with from left to right: Galileo, Johannes Hevelius, Tycho Brahe, Nicolaus Copernicus, and Ptolemy.

Hipparchus calculated the displacement of the Sun’s orbit by utilizing the observed dates of two equinoxes and a solstice. By applying his mathematical model to these measurements, he was able to determine both the size and the direction of the Sun’s orbital off-centredness.1

This calculation was necessary because the Sun’s passage through the seasons is not symmetrical, resulting in seasons of unequal length. Hipparchus used an eccentric circle model, which assumed the Sun traveled at a constant speed along a perfect circle that was slightly displaced from Earth. By basing his calculations on the measured length of the seasons, he determined that the off-centredness of the Sun’s orbit was approximately 4 percent of its radius. This model allowed for accurate predictions of the Sun's position as seen from Earth and remained a standard in celestial mechanics until the time of Johannes Kepler.12