Preface to the Second Edition
Since the appearance of this work, the teaching of mathematics, and partic-
ularly of geometry, has undergone some profound modiﬁcations, not just in its
details, but in its whole spirit, changes which have been awaited for a long time
and are universally desired. In working with beginners, we now tend to rely on
practice and intuition, rather than on the Euclidean method, whose utility they are
incapable of understanding.
On the other hand, it is clear that we must return to this method when we
revisit these early starts, and complete them. It is to this stage of education that
our book corresponds, and thus we have not had to change its character.
But even in the area of rigorous logic, the classical exposition was uselessly
complicated and scholastic in its ﬁrst chapter, the one devoted to angles. The
convention—unchanged up to the present—which does not permit us to talk of
circles in the ﬁrst book, renders matters obscure which, in themselves, are perfectly
clear and natural. Thus this is a place which we have been able to notably simplify
things, by introducing arcs of circles into the discussion of angles. We had already
departed from the traditional considerations of continuity on which the existence
of perpendiculars is often based; the simple artiﬁce which replaced it has itself now
In the same way, the measure of the central angle is naturally integrated into
the theory of angles, its correct logical place.
The second book gains no less than the ﬁrst by this change in order. The
fundamental property of the inscribed angle, indeed, is no longer connected to angle
measure, a connection which gives one an idea of this property and its signiﬁcance
which is as false as could be.
With this exception, the plan of the work as a whole has been preserved. In
fact, the complementary materials introduced by the program of 1902 had been
already covered in our ﬁrst edition. The program of 1905, which has reduced the
importance of these materials, has not until now obliged us to do any essential
revision. It requires only a single addition: the inverter of Peaucellier. Having
made this addition, the only complementary material remaining in this revision,
at least in plane
is inversion and its applications, which corresponds to
Chapters V–VII of our Complements.
note in this regard, that I have never attempted—despite the advocacy of such a step by
M. M´ eray, whose initiative has proved so fertile and so fortunate in the teaching of geometry—to
mix plane and solid geometry together. As this is preferable from a purely logical point of view,
I would like very much to do this. But it seems to me that from a pedagogical point of view, we
must think, ﬁrst and foremost, of dividing up the diﬃculties. That of “spatial visualization” is so
serious in and of itself, that I haven’t considered adding it to the other diﬃculties initially.