John HerschelSir 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.Normal | Teacher | Scholar