[Paper Review] A case of mathematical eponymy: the Vandermonde determinant

Bernard Ycart|arXiv (Cornell University)|Apr 20, 2012

History and Theory of Mathematics4 citations

TL;DR

This paper investigates the historical and sociological origins of the term 'Vandermonde determinant,' tracing how a mathematical object not fully developed by Vandermonde became widely named after him. Using statistical analysis of MathSciNet data, the study shows the denomination's growth rate significantly outpaces general mathematical literature, indicating a rising pedagogical and conventional adoption rather than increased research activity on the object itself.

ABSTRACT

We study the historical process that led to the worldwide adoption, throughout mathematical research papers and textbooks, of the denomination "Vandermonde determinant". The mathematical object can be related to two passages in Vandermonde's writings, of which one inspired Cauchy's definition of determinants. Influential citations of Cauchy and Jacobi may have initiated the naming process. It started during the second half of the 19 extsuperscript{th} century as a pedagogical practice in France. The spread in textbooks and research journals began during the first half of 20 extsuperscript{th} century, and only reached full acceptance after the 1960's. The naming process is still ongoing, in the sense that the volume of publications using the denomination grows significantly faster than the overall volume of the field.

Motivation & Objective

• To investigate the historical process behind the naming of the Vandermonde determinant.
• To determine why Vandermonde, despite not fully formulating the determinant, became associated with it.
• To analyze the role of pedagogical practice and influential citations in the spread of the eponym.
• To assess whether the growth in usage reflects increased research interest or conventional naming.

Proposed method

• Historical analysis of Vandermonde’s original works, particularly his 1771 and 1772 memoirs, to identify connections to the determinant.
• Examination of Cauchy’s and Jacobi’s works to trace how their definitions and citations may have initiated the naming process.
• Use of linear regression on logarithmic data from MathSciNet to compare growth rates of 'Vandermonde determinant' versus general 'determinant' or 'matrix' usage.
• Statistical comparison of exponential growth rates to detect whether the denomination's rise reflects research dynamism or naming convention.
• Analysis of textbook and research paper adoption patterns across the 19th and 20th centuries to map the timeline of naming acceptance.
• Use of p-values (e.g., 6.9 × 10⁻⁴) to test the significance of the difference in growth rates between the denomination and general terms.

Experimental results

Research questions

• RQ1 Why was the term 'Vandermonde determinant' adopted despite Vandermonde not fully formulating the object in general form?
• RQ2 How did Cauchy’s and Jacobi’s works contribute to the institutionalization of the eponym?
• RQ3 What role did pedagogical practice in France during the 19th century play in the initial spread of the name?
• RQ4 Why has the usage of the term grown faster than the overall volume of mathematical publications?
• RQ5 To what extent is the naming convention driven by pedagogical utility rather than mathematical discovery?

Key findings

• The term 'Vandermonde determinant' began spreading in the second half of the 19th century primarily as a teaching convention in France, not due to original discovery.
• No direct evidence of a mix-up between indices and exponents in Vandermonde’s work was found, though such a confusion may have inspired Cauchy’s determinant theory.
• The growth rate of the term 'Vandermonde determinant' in research literature is significantly higher than that of 'determinant' or 'matrix' overall, with a p-value of 6.9 × 10⁻⁴ for the difference in growth rates.
• The rise in usage is not due to increased research on the object itself, but rather to a growing convention of naming it explicitly in publications.
• The study concludes that the naming process is still ongoing, with the denomination being adopted more rapidly than the broader field of linear algebra.
• The most plausible explanation is that mathematicians increasingly use the standard name as a pedagogical shortcut, not because of new research on the object.