Studies in the History of Probability and Statistics. XXXIII Cauchy and the witch of Agnesi: An historical note on the Cauchy distribution

Abstract

The curve we commonly call the Cauchy density, or rather curves proportional to y = (a2 + x2)-1, have been appearing in mathematical works for over three hundred years. The curve seems to have first appeared in the works of Fermat in the middle of the seventeenth century, and was subsequently studied by such men and women as Newton, Leibniz, Huygens, Guido Grandi, and Maria Agnesi; by the nineteenth century it had acquired the singular name 'the witch of Agnesi', a name that is still commonly found in dictionaries and encyclopaedias. With its simple, symmetric form, it is surprising that the Cauchy density did not appear as a possible error distribution before 1824. When it finally did appear, it was in the role it plays most often today, as a counterexample to otherwise general theorems. The Cauchy distribution appeared naturally as a part of the move to greater rigour in mathematics during the nineteenth century, and was in fact discovered twice. It is interesting that the two discoverers had different motives in looking for it and interpreted its importance in two different ways. In the case of the second discovery, by Cauchy, the distribution appeared as part of an argument which persists today. Howimportant are regularity conditions in practical applications, and how much weight should be placed on large sample theory?