A well-written book invites the reader into the mind of the author. Novels, sto-
ries, books of poetry do this on an unconscious level, sweeping the reader along with
the words. A book of mathematics does it differently: well-written mathematics
impels the reader to take a pencil in hand and have paper at the ready. The present
volume is offered as an enhancement to just such a reading of Jacques Hadamard:
Lessons in Geometry, I. Plane Geometry, by the American Mathematical Society,
Providence, RI, and Education Development Center, Inc., Newton, MA, 2008.
Jacques Hadamard was among the greatest mathematicians of the twentieth
century. He made signal contributions to a number of fields, including number
theory, differential geometry, and differential equations. But his mind could not
be confined to the upper reaches of mathematical thought. His legacy includes a
in which he reflects on the process of creating mathematics, both his own
and others. He was active in the Dreyfuss Affair (his wife was related to Alfred
Dreyfuss), and held and expressed strong political and philosophical views all his
And he was a teacher. For several years, as a graduate student, he worked
in a lyc´ ee, teaching elementary mathematics. Later, the mathematician Gaston
Darboux, involved in rewriting the French school mathematics program, thought of
Hadamard’s experience, and asked him to write a book for teachers on elementary
synthetic geometry. The result was a massive two-volume work on plane and solid
geometry. The present book is a reader’s companion to the translation of the first
volume of this work referred to above.
As might be expected of a great mathematician, Hadamard was a master poser
of problems. Although termed “exercises”, the problems in his Geometry are an
integral part of the plan of the book. Indeed, the text can be read as a minimal
exposition, providing the mathematics that will support the solution of the ensuing
That is, the problems interpret the text, in the way that the harmony interprets
the melody in a well-composed piece of music.
The problems are rich and complex. Some of them embroider the text, digging
deep into the intuitions behind Hadamard’s theorems and lemmas. Others, such
as the problems about the Simson line or the nine-point circle, extend the text in
various directions. Often these give results which are important in their own right.
Very few of them are in any way routine: rarely will the solver read the problem
and know immediately how to approach it.
Psychology of Invention in the Mathematical Field, Princeton University Press, 1945.
many more interesting details of Hadamard’s life, see Vladimir Mazya, Tatyana Sha-
poshnikova, Jacques Hadamard, A Universal Mathematician, American Mathematical Society,
Providence, Rhode Island, 1998.