John Herschel
(1792-1871)
Sir John Frederick William Herschel was an English, mathematician, astronomer, chemist, and inventor, and the son of astronomer William Herschel.
His article in the
Edinburgh Review of July 1850 was a review of the work of
Adolphe Quételet on probabilities. It played an important role as the first appearance in English of a negative exponential function that would become a critical part of statistical mechanics in the future.
In his book
The Kind of Motion We Call Heat, volume 2, p.342, Stephen Brush reports a suggestion by C.C.Gillispie in 1972 that
James Clerk Maxwell had read this review by Herschel and used it to derive his famous distribution of velocities of gas molecules. The review does cite John Herschel as the author, but it was included in a book of Herschel's essays.
We excerpted a few pages from Herschel's review as a resource for those studying the origin of statistical physics.
Quételet on Probabilitities (Source,
Google Books.)
The most critical sentences are these (on p.20)...
Now, the
probability of any deviation depending solely on its magnitude,
and not on its direction, it follows that the probability of each
of these rectangular deviations must be the same function of its
square. And since the observed oblique deviation is equivalent
to the two rectangular ones, supposed concurrent, and is,
therefore, a compound event of which they are the simple constituents,
therefore its probability will be the product of their
separate probabilities. Thus the form of our unknown function
comes to be determined from this condition, viz. that the
product of such functions of two independent elements is equal
to the same function of their sum. But it is shown in every
work on algebra that this property is the peculiar characteristic
of, and belongs only to, the exponential or antilogarithmic
function. This, then, is the function of the square of the
error, which expresses the probability of committing that error.
That probability decreases, therefore, in geometrical progression,
as the square of the error increases in arithmetical. And
hence it further follows, that the probability of successively
committing any given system of errors on repetition of the
trial, being, by postulate 1, the product of their separate probabilities,
must be expressed by the same exponential function
of the sum of their squares however numerous, and is, therefore,
a maximum when that sum is a minimum.
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