Gauss
Among scattered measurements, which is closest to truth?
That time, that place
On the first day of 1801 a new object was found in Italy and named Ceres. It was tracked for about forty days, then moved toward the Sun and out of sight.
It would return — but no one knew where in the sky. The observations covered only a small arc of one orbit.
Why this question
A harder problem sits on top. Observations carry error: measure the same thing repeatedly and the numbers differ slightly.
Which is closest to the truth? Or rather — how should many measurements be combined to come closest?
What was found
Gauss did not try to eliminate error. He decided to handle it.
His method chose the orbit for which the sum of the squared discrepancies against all observations is smallest — penalizing large misses more heavily. This is the method of least squares, named for exactly what it does: choosing the answer that makes the sum of the squared misses least.
With it he predicted where Ceres would reappear, and that December it was found exactly there.
He also described how errors themselves scatter: a bell-shaped spread, high in the middle and tapering both ways. Error may be random, and still the way it scatters follows a rule.
Error was a flaw to be eliminated
Ceres recovered at the predicted position
The method and the bell-shaped distribution underpin nearly all measurement science today