xii AUTHOR’S PREFACES
Another tendency has appeared in the teaching corps in the last few years,
which we have had the bad manners not to applaud. We speak here—and I hope
we begin to use it a bit—of the method of heuristics. The Note we have added
in 1898 to our ﬁrst edition (Note A) had exactly the goal of describing how, in
our view, this method might be understood: how it might be understood, at least,
from a theoretical point of view, since both are needed for the application of the
heuristic method. I hope that this Note might now be of some use in indicating, at
least, how these principles can be put to work.
I have already explained (Preface to Solid Geometry) that the method described
in Note C for tangent circles belongs in fact to M. Fouch´ e, or to Poncelet himself,
and that a solution to the question of areas of plane ﬁgures, diﬀerent, it is true, from
that in Note D, is due to M. G´erard. I seize this occasion to add that an objection
concerning the theory of dihedral angles has already been noted and refuted by M.
Preface to the Eighth Edition
The present addition contains no important changes from those that preceded
it. We must note, however, that our ideas about the Postulate of Euclid have been
modiﬁed considerably by recent progress in physics: I have had to recast the end
of Note B to take into account this scientiﬁc evolution.
A few modiﬁcation have been made in the present edition, intended to give a
bit more importance to properties of the most common articulated systems.
Preface to the Twelfth Edition
This edition diﬀers from the preceding only in the addition of several exercises.
The elegant Exercise 421b is due to M. Daynac, a teacher in the French School
of Cairo; the simple proof of Morley’s theorem which is given by Exercise 422,
and which brings into play only the ﬁrst two Books, is due to M.
the supplement added to exercise 107 is borrowed from an article of M. Lapierre
(Enseignement Scientiﬁque, November 1934), with certain modiﬁcations intended
to reduce the proof to properties of the inscribed angle; Exercise 314b is from
Preface to the First Edition
In editing these Lessons in Geometry, I have not lost sight of the very special
role played by this science in the area of elementary mathematics.
Placed at the entry point to the teaching of mathematics, it is in fact the
simplest and most accessible form of reasoning. The importance of its methods, and
their fecundity, are here more immediately tangible then in the relatively abstract
theories of arithmetic or algebra. Because of this, geometry reveals itself capable of
essentially equivalent proof was sent me by M. Gauthier, a student at the