Hilbert's nineteenth and twentieth problems proposed themes related to partial differential equations. The nineteenth problem asked whether certain variational problems, described as "regular", must always have analytic solutions. The twentieth problem posed very generally the issue of existence of solutions for boundary value problems; in stating it, Hilbert proposed the idea of generalized (or weak) solutions. These problems were taken on by Serge Bernstein, who studied both in Paris and in Göttingen. Bernstein assembled methods from the French context (Poincaré's continuity method) with Hilbert's ideas and made decisive contributions to the solution of both problems, in particular using a priori estimates in a way that was not fully appreciated until Schauder grasped the method in the 1930s. The paper will sketch these events.

One of the major developments in early 20th-century mathematics teaching is the inclusion of the function concept and its graphical representation. In this talk I will discuss the curious history of one particular kind of graph, the so-called string chart or train graph. I will trace the rise and fall of the string chart from its mid-19th century beginnings as a tool to scheduling trains via a public convenience to a mere application in a number of mathematics textbooks of the first decades of the 20th century. Emphasis will be on the striking differences in the reception of the train graph as a teaching tool between the European Continent and the English-speaking countries.

When most historians discuss "L'Hospital's Rule", they focus on the story of Bernoulli's letters to L'Hospital and of their "arrangement". When one actually looks at their work, however, it becomes clear that L'Hospital's "L'Hospital's Rule" is very different from what one finds in current books. There is no reference to limits, the proof does not involve the mean value theorem, and there is no "infinity over infinity" case. This talk is a preliminary report on joint work with Colby student Melissa Yosua investigating the post-L'Hospital history of the "rule".

It is a commonplace that among the great philosophers Plato assigned unusual significance to mathematics. I shall attempt an overview, taking into account both the intellectual context and the social milieu. My central theme will be the place of mathematics in the origin and subsequent career of the theory of Forms-a more complex and interesting tale than it might seem. As time allows I shall try to touch on related issues, especially Plato's conception of the role of mathematics in education.