Ernst Zermelo is mostly known for his contributions to pure mathematics, particularly to the foundations of set theory, with the axiom of choice, together with Abraham Fraenkel. Such work is displayed in Volume 1 [1]. However, Zermelo also contributed to applied mathematics, and it is his collected works in this area that make up volume 2.

Zermelo’s applied works touch on a wide variety of subjects, ranging from the calculus of variations (which would influence many optimization techniques over time) and variational problems (such as the motion of an inextensible material string under the action of a potential W(x,y,z)), to the problem of the steering of a vessel under the action of a current of air or water. Zermelo also treated some important aspects of vortex motion, which are still the basis of many open questions, including the problem of the motion of a vortex on a surface more general than a sphere.

Other topics that appear in this volume are the probabilistic grading of chess players, and the witty, and perhaps a bit funny, but mathematically rigorous treatment of how a sugar cube may break under opposite tearing forces. Zermelo versus Bolztmann’s argument (across several papers and letters) over the validity of Boltzmann’s alleged mathematical proof that entropy must increase, and based on purely mechanical arguments, is, of course, included. It is interesting to note that in reading the papers exchanged between Boltzmann and Zermelo, the latter was concerned mostly with mathematical correctness, using, for example, Poincaré’s theorem of recurrence (that is, given enough time, any system must come to its original state of coordinates of position and momenta). Zermelo was not against the thermodynamically apparent fact that entropy does increase. Later in this volume, Zermelo also discusses Gibbs’ statistical mechanical approach to systems, and its possible connections to thermodynamics, making clear Zermelo’s inclination to mathematical rigor, rather than discussing physics by itself.

This edition of Zermelo’s collected works is not the first attempt to showcase an integrated collection of his mathematical developments, but after three-quarters of a century, it is the first successful one. It is worth mentioning that this edition includes the original manuscripts in German and authoritative English translations on the opposite pages. Experts in the respective subjects introduce each of Zermelo’s papers (some of these prefaces are by the editors themselves). Such prefaces provide explanations about Zermelo’s intentions for the paper discussed and its historical background, including Zermelo’s possible motivations and his interactions with other colleagues. Zermelo’s minor mistakes and linguistic nuances are always explained, and sometimes given a proper historical background. For example, Zermelo uses the German term “Wirbel” when referring to vortices in general, and “(einfacher) Strudel” or “Strudelpunkt” when the vorticity is concentrated at a single point. In order to preserve Zermelo’s terminology, but in contrast to the modern use of the term “vortex,” the editors took care to always translate “Wirbel” as “vortex” and “Strudel” as “whirl.”

This is the only authoritative version of Zermelo’s collected works with English translations. Therefore, it is a tour de force for anyone not only interested in Zermelo’s papers, but also the historical background of many research areas of applied mathematics. I highly recommend this volume to applied mathematicians, from advanced undergraduate students up to specialists, such as those of variational problems and functional analysis.