“The Riddle of the Ages”: James Joseph Sylvester and the Transcendence of π

Abstract

The first proof of the transcendence of π by Ferdinand Lindemann in 1882 prompted the publication of a wave of further proofs, all attempting to elucidate, simplify, or generalize this epoch-making result. Modifications by the likes of Weierstrass, Gordan, and Hilbert all entered the literature. But one such proof sank without a trace and was almost completely ignored. This is surprising as, not only did it appear in a major journal with a large international readership, but it was also written by one of the most famous mathematicians of the time—James Joseph Sylvester. This paper tells its story.

MSC:

• 01A55
• 11-03

Acknowledgment

The author wishes to thank Karen Parshall for valuable comments made on an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 For a discussion of this paper, see [7] and [8, p.185–190].

2 In [9, p.341], the author notes that “this proof was communicated to me by Mr. Poinsot, who told me that he got it from Mr. Fourier.”

3 His argument here is merely sketched, however, with Sylvester assuring the reader that “This will be developed at length in a subsequent Communication” [29, p.869]. This was necessary as the lemma as stated was not sufficient to prove this point. To do so, the lemma would have had to be modified to include the case where $m,m\prime ,m″,\dots$ are constant and $n,n\prime ,n″,\dots$ increase monotonically, so that the limit of the absolute value of J would be zero.

4 Borchardt used the French word géomètre here.

5 Sylvester was in fact 76 at this point.

6 Here, Hermite used the word géomètre.

7 Among Sylvester’s many health problems in his later years, from the late 1880s he was particularly troubled by issues relating to his eyesight, undergoing treatment in Germany in February 1889 and Sweden in August 1889, followed by surgery in Paris in December 1890. [30, p.311, 312, 319]. Dr. Xavier Galezowski was a Polish ophthalmologist practicing in France who was one of the first clinical practitioners of his specialty [45].

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Adrian Rice

Adrian Rice is the Dorothy and Muscoe Garnett Professor of Mathematics at Randolph-Macon College in Ashland, Virginia, where his research focuses on 19th-century and early 20th-century mathematics. In addition to papers on various aspects of the history of mathematics, his books include Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800–1945 (with Karen Hunger Parshall), Mathematics in Victorian Britain (with Raymond Flood and Robin Wilson), and Ada Lovelace: The Making of a Computer Scientist (with Christopher Hollings and Ursula Martin). He is a five-time recipient of awards for outstanding expository writing from the MAA. In his spare time, he enjoys music, travel, and spending time with his wife and son.