Turning a handful of sightings into an orbit
Ceres did not appear in a continuous track across the sky. Astronomers had a series of dated observations—where the object seemed to be at particular moments—and needed to infer the path connecting them. The challenge was that many different orbits could pass near one sighting. A useful orbit had to account for the whole set and predict where Ceres would appear next.
Gauss treated an orbit as a mathematical model with adjustable quantities, such as its shape and orientation. For each proposed orbit, he could calculate the positions it would predict at the observation times, then compare those predictions with the recorded positions.
Why square the differences?
The gap between a predicted and measured position is a residual. Least squares chooses the orbit that makes the sum of the squared residuals as small as possible. Squaring prevents positive and negative errors from canceling out, and it makes a large miss count disproportionately more than a small one: a residual twice as large contributes four times as much to the total.
That emphasis is useful when a proposed path fits most observations but badly misses one. It also means that an unusually inaccurate measurement can have substantial influence, so the method depends on the observations being trustworthy enough to fit.
A prediction that could be checked
The calculation mattered because it turned scattered past sightings into a forward prediction. Once Gauss had an orbit that fit the available observations, astronomers could search for Ceres where that orbit said it should be—even after the object had become difficult to observe. The test was concrete: either the predicted path led back to Ceres, or it did not.
