Science on the Farther Shore
Meanwhile, Back in Europe...
Pragmatic advice on how best to survey a farmer's field is not something you often find today in a journal of research mathematics, and the practical emphasis of Adrain's work might be taken as a sign of his provincialism. But in fact very similar problems in geodesy and astronomy motivated Gauss and Legendre as well. Indeed, both of them not only analyzed survey results; they also went out in the field and made measurements of their own.
Both Gauss and Legendre introduced the method of least squares in works on astronomy. Legendre was first to publish; he presented the technique (and also coined the name) in a book on comets published in 1805. The method is given as part of a recipe for determining an orbit from a set of observations, but Legendre offered no theoretical justification, and he did not mention probability at all.
Gauss's treatment of the subject appeared four years later, in his major work on celestial mechanics, Theoria Motus Corporum Coelestium. Based on dates of publication, Legendre would seem to have clear priority, but Gauss remarked that he had been using the method since 1795 and insisted the invention was his. Legendre complained in a bitter letter to Gauss: "There is no discovery that one cannot claim for oneself by saying that one had found the same thing some years previously...." They never made peace.
Although Gauss wasn't first to publish, he was certainly more thorough. His treatment was no ad hoc recipe for curve-fitting; he started from the premise that the arithmetic mean of several independent measurements "gives the most probable value, if not rigorously, yet very nearly, so that it is always most safe to hold onto it." He then derived the normal distribution from this premise, and showed how the distribution implies the method of least squares. (The arithmetic mean can be seen as a special case of the method of least squares.)
What Gauss could not establish was that the errors in real-world data—from land surveys or comet observations, say—actually follow a normal distribution. Gauss's "law of error" was widely accepted anyway. As Henri Poincaré quipped a century later: "Everyone believes in it, because the experimenters imagine that it is a mathematical theorem, and the mathematicians that it is an experimental fact." The actual scope of the "law" was not rigorously settled until a century later with the proof of the Central Limit Theorem.
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