Joint review of

When you think of geometry at the beginning of the twentieth century, it is Hilbert’s *Grundlagen der Geometrie* that first comes to mind. There geometry is revealed through axiomatic methods as unclad as possible, each set of axioms featuring a fundamental ideas from which the subsequent geometry would emerge, the musculature, skeleton, etc. Hadamard’s book was prepared for the *lycée*, for high school students. His goal is similar; to introduce beginners to “Euclidean methods,” rigorous arguments and challenging exercises that develop careful thinking in a diligent student. The contrast between the two texts is revealing, the Göttingen axiomatics set against Hadamard’s Euclidean approach, Hilbert’s precision against Hadamard’s wealth of results. In the end, the goals are closely matched. The publication of Hadamard in English gives us another take on geometry from the time and from a master teacher.

Hadamard’s organization is first in books: Book I, on the straight line; II, on the circle; III, on similarity; IV, on areas. Within each book are chapters, each ending with well selected exercises. This organization recalls for me the grand volumes of George Salmon, *A Treatise on Conic Sections* and *A Treatise on the Analytic Geometry of Three Dimensions*. Salmon’s main chapters begin with “the point,” “the right line,” “the circle,” before launching into the conics. Hadamard’s introduction recalls Euclid’s definitions, but in reverse order, from a volume to a surface to a line to a point. He goes on to define straight lines, planes, and circles, deducing elementary properties immediately from the definition. When book I begins, the first topic is the angle. This choice allows him to obtain reflections and rotations fairly quickly toward the goal of studying congruences of the plane. Triangles follow, with the standard Euclidean results.

Parallels are treated in chapter V of book I, for which the angles provide criteria for being parallel. The parallel postulate of Euclid takes the form given by Playfair: through a point not on a given line there is only one parallel to given line. The theory of parallels allows the author to treat translations via parallelograms. Book I closes with construction of many of the special points in a triangle: the incenter, circumcenter, and the centroid.

Book II opens with the theorem that three non-collinear points determine a unique circle. Incidence relations follow. Hadamard uses symmetry to define a diameter, from which many nice properties follow. After chapters on the intersections of circles and inscribed angles, comes an unusual chapter on *constructions*. He begins with the standard Euclidean constructions but ends with the construction of circles tangent to three given lines. The theme of tangent circles plays out later in his discussion of Gergonne’s solution to the problem of finding a circle tangent to three given circles. Book II ends with a discussion of congruences, that is, the motions of figures in the plane given by translations, rotations, reflections and the relations among them.

Book III opens with proportion. The aim is to establish metric relations in a triangle, and among segments in a circle. Similarity as a transformation is described by the notion of *homothecy*. Lines pass to lines under homothecy, and circles to circles, making homothecy the basis for pantograph, an instrument to reproduce figures with enlargement, described by Hadamard. Constructions follow including the construction of regular polygons and the approximation of π.

The most unusual part of the book is the complement to Book III that treats inversive geometry in detail. Hadamard begins with directed segments, which provides a home for Menelaus’s theorem. He follows with cross ratios and poles and polars with respect to a circle, leading to inversive transformations. With these tools, he can treat the problem with tangent circles and describe Peucellier’s mechanical inverter through a discussion of cyclic quadrilaterals.

Book IV treats areas, based on ratios to a unit area. Hadamard follows Euclid’s proof of the Pythagorean theorem. He ends with the area of a circle.

Appendices follow. The first, Note A, is a beautiful discussion of the strategy of proof. Hadamard does not shy away from giving detailed advice, supported by many examples from the book. His point of view is revealed: “All geometric methods can be legitimately called ‘transformation methods.’” And he goes on to identify the groups underlying geometry.

The subsequent notes treat non-Euclidean geometry, including a reference to its role in the then new theory of relativity, more on Gergonne’s solution to the problem of tangent circles, and area. Miscellaneous problems are gathered including some from the General Competition of Lycées and Colleges. These additional problems bring the total number of problems posed in the book to 422, a bounty. The book closes with an appendix on Malfatti’s problem.

This brings me to Mark Saul’s book. In it he has given solutions to exercises 1 through 342 of Hadamard. Like Hadamard, Saul aims to indicate possible motivation for difficult arguments. He has directed the book at high school teachers, but really it is for all lovers of synthetic geometry. Hadamard’s exercises are often challenging, sometimes leaving out certain results. Saul has added lemmas that fill in these gaps, and enrich Hadamard’s text wisely.

Answers to exercises 343–422 can be found on the website for Saul’s book.

André Weil remarked that Hadamard’s *Leçons de Géomtrie élémentaire* was “remarkable’ and most suitable for “the teachers and best students.” Hadamard’s prose is clean and clear, focused and without frills. Obviously, the problems will be where a student truly connects with the author’s intentions. Together these books offer a rich experience to everyone who loves geometry.

John McCleary is Professor of Mathematics at Vassar College.