“The Riddle of the Ages”: James Joseph Sylvester and the Transcendence of <i>π</i> (Q6554389)

The subject of this paper is the proof of the transcendence of \(\pi\) that was published by J. J. Sylvester in [C. R. Acad. Sci., Paris 111, 866--871 (1890; JFM 22.0437.05)]. Curiously, this proof has only rarely been cited, for reasons that are explored here. The paper under review begins with a survey of previous irrationality and transcendence proofs for \(e\) and \(\pi\) by such figures as Leonhard Euler, Johann Lambert, Adrien-Marie Legendre, Joseph Fourier, Joseph Liouville, Charles Hermite, and Ferdinand Lindemann. Sylvester appears next, and his passion for what he called the `riddle of the ages' (the transcendence of \(\pi\)) is discussed. An outline of Sylvester's purported proof follows, and reasons for its neglect are considered. These include errors in Sylvester's method, his avoidance of Weierstrassian analysis, and the tact of the wider mathematical community in not drawing attention to the errors of one of the elderly statesmen of the discipline.