Foreword

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

book1

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

life.2

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

problems.

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.

1The

Psychology of Invention in the Mathematical Field, Princeton University Press, 1945.

2For

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.

vii