When I was a freshman (or was it even before that?) I discovered what was soon to become my favorite biography of a mathematical titan, and has continued to occupy that position over the decades, namely Constance Reid’s *Hilbert*. I devoured the book, reread it, and reread it, and the figures populating its pages came to life to me and have stayed alive in my imagination over the years. They have figured prominently in my teaching of courses in the history of mathematics, and it is my hope that due to my exposing my students to these scholars, who are at the same time classical heroes and modern mathematicians, these young charges of mine might develop the same affection and affinity for them. After all, our subject has a marvelous history, and being aware of it, indeed, being able to relate to it and see its effects in what we face in our scholarly work today, is an incomparable advantage and all but a nonnegotiable element of good scholarship.

For this reason alone, then, it is highly desirable to be able to discern the traces of the past in the mathematics of the present, and this should really involve, at a foundational level, an awareness of how notions and concepts we take for granted, particularly the so-called elementary ones, fit into the bigger mathematical picture. And then we discover, of course, that “elementary” is not antithetical to “deep” at all — well, consider, for example the incomparable example of Euclidean geometry. Is it not stunning indeed, even after all these centuries, in fact after a couple of millennia, to observe that these ancients knew that parallel postulate were not of the same nature and status as the other postulates? And eventually, following that thread through history, we encounter none other than Gauss, Bolyai, Lobachevsky, and Riemann, and then, indeed, we meet Klein, Minkowski, and Einstein, all three of whom are prominent in the pages of Reid’s aforementioned book. For the purposes of the present review, then, our focus falls on Felix Klein, the great interpreter of Riemann and the mathematician responsible for Göttingen’s golden age, spanning approximately the first three decades of the 20th century, bringing Hilbert and Minkowski there early on, and then drawing in an unparalleled constellation of mathematical masters including, to name a few, Artin, Bernays, Hecke, Noether, van der Waerden, von Neumann, Weyl, and Zermelo.

The books under review are a time-honored trio which, when they appeared at the turn of the century (19th into 20th, of course), presented something altogether new. Klein sought to cover a wide panorama of elementary mathematics from the indicated advanced standpoint that he wished to assume, including (I) Arithmetic, Algebra, and Analysis, (II) Geometry, and (III) Precision Mathematics and Approximation Mathematics.

Well, let’s start with the obvious question: what does the latter mean? The contents of this third volume include such themes as “Remarks on the Single Independent Variable *x*,” “Functions *f*(*x*) of a Real Variable *x*,” and “The Approximated Representation of Functions”; then, concerning “Free Geometry of Plane Curves,” Klein contrasts “precision geometry” (in the plane) to “practical geometry,” including under the latter heading such things as geodesy and “drawing geometry.” Lest this be altogether misunderstood in the present day and age, in Klein’s hands drawing geometry includes consideration of, e.g., “Postulation of a Theory of Errors for Drawing Geometry, Explained Using a Graphic Representation of Pascal’s Theorem,” and “The Seven Kinds of Singular Points of Space Curves.” Obviously we are operating from a very advanced standpoint.

To us moderns, the subjects treated in Volumes I and II are more familiar and accessible, and it serves us to consider a few samples from each, in order to get an idea of what Klein is up to. Under “Arithmetic” we encounter, for example, a discussion of continued fractions, immediately followed by one concerning Pythagorean triples and Fermat’s Last Theorem. Both of the according sections contain beautiful and non-trivial material: Klein discusses, for instance, the continued fraction for \(\pi\) and uses it to provide successive decimal approximations to it, and then offers the gambit: “Let us now enliven these considerations with geometric pictures.” This angle on continued fractions culminates in a beautiful result in lattice theory characterizing alternating approximants to the irrational number being represented by a continued fraction. And, then, immediately on the heels of this marvelous excursion, he uses geometry (a well-chosen circle and a ray from its center) to produce the famous algorithm for producing all Pythagorean triples. But Klein does not let things rest there: he races on to consider the generalization of the Pythagorean Diophantine equation to Fermat’s, and gives a concise introduction to Kummer’s work. (After all, Wiles’ resolution of the problem, and his winning of the Wolfskehl Prize — cf. p. 51 of Klein’s Part I — was still almost a century in the future: in this regard see p. 1299 of http://www.ams.org/notices/199710/barner.pdf .)

So the flavor of Klein’s approach to his “elementary questions” is becoming evident: a true past master of mathematics, possessed of incredible breadth (and depth), is at work here. Thus, in connection with Algebra we find, to name but one example, a discussion of the Fundamental Theorem of Algebra which entails a good deal of evocative geometry (involving exploiting De Moivre’s Theorem) in the service of nothing less than Gauss’s proof of 1799. Regarding Klein’s discussion of Analysis, I want to mention something presented in the Supplement to this chapter: Klein’s discussion of the proofs of the transcendence of \(e\) and \(\pi\). Concerning the former, he says that he “will follow the simplified method given by Hilbert [as compared with Hermite’s original proof] in Volume 43 of the *Mathematische Annalen* (1893)”; we should note that Hilbert’s article was in fact double-barrelled: its title was *Über die Transcendenz der Zahlen *\(e\)* und *\(\pi\). And, to be sure, regarding \(\pi\) Klein states that he “shall follow again, in the main, Hilbert’s proof … which is an exact generalization of the discussion which we have given for \(e\).” It’s off to the races in both of these sections, which deal, to be sure, with two favorite themes in number theory, showing off the magnificent power of analysis. And yes, indeed, the whole set of books is like this!

Finally a word or two about Klein’s Volume II, on Geometry, which was after all the subject closest to his heart, what with his aforementioned association to Riemann (and the latter’s theory of manifolds and complex geometry) and of course the famous *Erlanger Programm*. Permit me here to start with a sampling of chapter headings: Part I, Section II: “The Grassmannian Determinant Principle for the Plane” (and then the next section extends it to space); Part I, Section IV: “Classification of the Elementary Configurations of Space According to Their Behaviour Under Transformations of Rectangular Coordinates” (yes, shades of Erlangen); Part II is simply titled “Geometric Transformations,” so nothing more needs to be said: if this isn’t Klein’s bailiwick, nothing is; and Part III, Section II: “Foundations of Geometry” — and the parallel postulate of Euclid appears with some prominence in subsection 2. After this long preceding section (how could it be otherwise?), Klein proceeds to topics that should resonate loudly in today’s pedagogy-friendly academic environment: the Final Chapter of Volume II is titled “Observations About the Teaching of Geometry.” Klein singles out the approaches taken in England, France, Italy, and Germany for explicit commentary and analysis.

It is clear, then, that even in our age, on the order of a century after Klein’s reign at the Georg August Universität in Göttingen (where, as Reid tells us, his nickname was “the Divine Felix”), the three volumes of *Elementary Mathematics from a Higher Standpoint* are well worth reading. The mathematics discussed is well-chosen, exceptionally interesting (possibly excepting what has become dated), and is dealt with brilliantly. On top of this, Klein is exceptionally clear in his prose, and equally incisive. These books have aged very well.

See also the pages for volumes II and III.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.