Science on the Farther Shore
A $6 Problem
Adrain was among the contributors to the very first mathematical periodical published in the United States, the Mathematical Correspondent, begun in 1804 by George Baron, who had been the first mathematics instructor at West Point. When Baron abandoned the journal, Adrain took over, and then transformed the Correspondent into a magazine of his own, renamed The Analyst, or Mathematical Museum. This enterprise also foundered, after just four issues appeared, but not before Adrain published in its pages his one claim to lasting recognition.
Adrain's moment of inspiration came in 1808, in an article summarizing work on a problem posed by a reader, with the offer of a $6 prize. The prize was awarded to Nathaniel Bowditch (who is better remembered than Adrain, primarily for his American Practical Navigator). As editor, Adrain was disqualified from the prize competition, but his discussion of the problem goes deeper than Bowditch's.
The problem concerns a land surveyor who traces the boundary of a polygonal field, measuring each of its five sides by traversing a prescribed distance on a prescribed angular bearing. At the end, the survey should return to the starting point, forming a closed pentagon, but instead there is a small gap. The $6 problem is to adjust the end points of the five segments so that the path closes. Of course there are innumerable ways to achieve this goal; the idea is to choose from among all possible adjustments those that put the vertices in their most probable positions.
Adrain begins his analysis by simplifying and generalizing the problem, dispensing with the surveyor's vocabulary of perches and chains and bearings. He writes: "The question which I propose to resolve is this: Supposing AB to be the true value of any quantity, of which the measure by observation or experiment is Ab, the error being Bb; what is the expression of the probability that the error Bb happens in measuring AB?" A tiny diagram like the one in the margin here makes it clear that Adrain is thinking of AB as the length of a line segment, which the error Bb can either increase or decrease. He argues that such errors should have a particular distribution, based on the "evident principle" that the uncertainty in measuring the length of a segment is proportional to the length itself. Since the actual length AB is the unknown quantity in this problem, it is not much use as an error estimator; Adrain brushes aside this subtlety, taking the measured length Ab as the basis.
Now suppose there are two measured segments, with unknown individual errors a and b but a known total error c. If the errors are independent, then the probability of both occurring together is the product of the separate probabilities, Pr(a)Pr(b). (This is another trouble spot in the argument: The hypotheses of proportionality and independence appear to be inconsistent. But Adrain presses on.) The likeliest values of a and b are those that maximize the product Pr(a)Pr(b), subject to the constraints that a+b=c and that the errors are proportional to the measured lengths. Adrain proceeds by taking the derivative of the probability equation and setting it to zero, in the usual process for identifying a maximum or minimum. After several further manipulations—some of them a little murky—he emerges with a famous equation. In modernized notation it states:
This is the equation of the normal distribution, or density, which gives the probability of observing the result x as a function of the true result µ and the standard deviation s. For any given µ and s, Pr(x) takes on its maximum value when the expression (x??)2 is made as small as possible. This fact is the origin of the least-squares principle: The best predictor of a normally distributed variable is the one that minimizes the square of the difference between the observed and the predicted values.
Adrain went on to give a second derivation of the same distribution, based on a geometric argument. He also applied the least-squares method to four practical problems, including a version of the original prize question.