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Bessel Functions - Attribution Debate
The central question in the historiography of Bessel functions is whether Friedrich Wilhelm Bessel deserves the eponymic credit commonly granted to him, or whether priority properly belongs to Daniel Bernoulli, Leonhard Euler, or both. The debate turns not only on who first encountered these functions, but on what counts as mathematical “discovery” - whether it is the initial encounter with a particular solution, the recognition of a general class of functions, or the systematic theoretical treatment that makes a concept mathematically productive. The competing positions reflect broader disagreements in the history of mathematics about the relationship between priority, depth of contribution, and the legitimacy of mathematical eponymy.
Bernoulli Has Prior Claim
Those who argue that Bernoulli deserves primary credit point to his 1732 work on the oscillations of a heavy chain suspended at one end, in which he employed what is now recognized as the Bessel function of zero order, J0(x).1) On this view, Bernoulli's encounter with the function predates Bessel's systematic work by nearly a century, and the conventional name obscures a clear historical priority. Some treatments in mathematical literature acknowledge this directly, noting that J0(x) “was first discovered by Daniel Bernoulli” before Bessel's name became attached to the broader family.2) Proponents of this view may also invoke Euler's 1764 extension of Bernoulli's work to functions of integer orders in connection with vibrating membranes, arguing that by the time Bessel entered the field, the essential mathematical objects had already been characterized by Swiss mathematicians of the first rank.3)
Bessel's Contribution Justifies the Name
Those who defend the conventional eponymy argue that the name reflects the qualitative character of Bessel's contribution rather than a bare priority race. On this account, Bernoulli and Euler each encountered particular instances of these functions in the course of solving specific physical problems, without recognizing them as members of a general class or developing their properties systematically. Bessel, by contrast, undertook a thorough analytical study of the functions in his 1824 memoir on planetary perturbations, characterizing them as coefficients in a series expansion for the indirect perturbation of a planet - work that brought mathematical clarity and general applicability that earlier appearances lacked.4) Defenders of the eponymy hold that mathematical discovery is better understood as the systematic unpacking of a concept than as the first incidental appearance of a formula, and that Bessel's treatment - generalizing to non-integer orders, establishing integral representations, and demonstrating broad applicability - constitutes a distinct and deeper contribution than his predecessors made. On this view, the name “Bessel functions” is not a misattribution but an accurate reflection of where the theory became mathematically tractable.
The Naming Convention Itself Is the Problem
A third position questions the framing of the debate altogether, arguing that mathematical eponymy systematically rewards synthesis and communication over original discovery, and that the Bessel case is simply one instance of a well-documented pattern. Historians of mathematics have observed - invoking what is sometimes called Boyer's Law or Stigler's Law of Eponymy - that mathematical results are almost never named after their original discoverers, and that the honors tend to accrue to those who present a concept effectively to a wider audience or embed it in a more prestigious theoretical context.5)6) On this view, neither Bernoulli's priority nor Bessel's synthesis gives a fully satisfying answer to the attribution question, because the question itself rests on a naive model of discovery. What matters historiographically is tracing the full developmental genealogy - Bernoulli (1732), Euler (1764), Bessel (1817-1824) - rather than assigning a single name as if mathematical objects spring fully formed from one mind.
Points of Agreement
- The basic chronology is not disputed: Bernoulli (1732) preceded Euler (1764), who preceded Bessel (1817-1824).
- There is general agreement that Bessel's 1824 memoir represented a systematic and general treatment that earlier work did not achieve.
- Historians across positions agree that the conventional name “Bessel functions” is now too entrenched in mathematical literature to be practically revisited, whatever its historical accuracy.
- All sides acknowledge that Bessel's contributions extended beyond these functions - his astronomical work, including the first measurement of stellar parallax, would secure his place in scientific history independently.