Peano on Wronskians: A Translation

In 1882, Thomas Muir wrote the following passage in his Treatise on the theory of determinants ([TM, p. 224]), giving the definition of the Wronskian with which we are familiar today.

This passage was actually the first time that someone called these matrices Wronskians, after Jozef Maria Hoene-Wronski ([PP]). Muir continued ([TM, p. 234]):

Obviously, mathematicians accepted Muir’s suggestion with zeal, as evidenced by the fact that we continue associating Wronski’s name with these determinants today. Most students see the Wronskian in a differential equations class during the discussion of fundamental sets of solutions. 

We learn in basic linear algebra that if functions \( y_1,y_2,\dots,y_n \) are linearly dependent, then the determinant of the Wronskian will be zero. However, as noted by Bostan and Dumas in [BD], if one were to open any modern text in differential equations, one would find the following warning: “linearly independent functions may have an identically zero Wronskian!” 

The story behind this warning is quite interesting. For years, respected mathematicians took it for granted, and even provided proofs, that a zero Wronskian would imply linear dependence. The first person to realize that this was not true appears to have been Giuseppe Peano, but even after he provided an elementary counterexample, people had difficulty understanding the subtlety of the situation – just as our students often do!

Mark Krusemeyer explained, “This warning was first given by Peano ...; he actually had to give it twice, because an editor added a mistaken footnote that contradicted the main point of Peano’s first warning” [K]. The two papers to which Krusemeyer referred are short notes (each is less than two pages) published in 1889 in the French journal Mathesis. The articles appeared in the exact same issue, 25 pages apart. They can be seen in the original French in [P1] and [P2] and translated in their entirety in Appendix 2. In this paper we present a translation and analysis of these two notes by Peano, the mistaken footnote added by the editor, as well as the other papers Peano cites.

We will begin by translating Peano’s first note, Sur le déterminant Wronskien ([P1]), which contains his original warning, an example showing that linearly independent functions may have a zero Wronskian, and the footnote that contradicts his main point. We next translate this footnote, added by editor Paul Mansion, which makes reference to Mansion’s textbook on analysis. This textbook contains an incorrect proof that a zero Wronskian implies linear dependence.

We continue by translating Peano’s second note, Sur les Wronskiens ([P2]), which is an excerpt of a letter from Peano to the editors. It explains in no uncertain terms the problem with stating that a zero Wronskian provides a sufficient condition for equations to be linearly dependent. He also gave the example we typically see in an ordinary differential equations course today. At the end of this letter he implied that he had never seen this fact accurately expressed, and gave several citations of published incorrect propositions. Our last translations are of these cited works.

Our paper finishes by looking at ways in which this project could be used in a standard ordinary differential equations class. It is our hope to show how it can be a helpful addition to a differential equations course—either as a short point of interest, or as a starting point for homework, projects, or exam questions.