PREFACE TO THE FIRST EDITION xiii

exercising an undeniable influence on the activity of the mind. I have, ﬁrst of all,

sought to develop this influence in awakening and assisting the student’s initiative.

Thus it seemed to me necessary to increase the number of exercises as much as

the framework of the work would allow. This requirement has been, so to speak,

the only rule guiding me in this part of my work. I believed that I must pose

questions of very diﬀerent and gradually increasing diﬃculty. While the exercises

at the end of each chapter, and especially the ﬁrst few chapters, are very simple,

those which I have inserted after each book have solutions which are less immedi-

ate. Finally, I have postponed to the end of the volume the statement of problems

which are relatively diﬃcult. Certain questions have been borrowed from some

important theories—among these we note problems related to the theory of inver-

sion and to systems of circles, many of which come from the note On the relations

between groups of points, of circles, and of spheres in the plane and in space, of

M. Darboux.5 Others, on the contrary, have no pretensions other than to train the

mind in the rules of reason. I have been no less eclectic in the choice of sources I

have drawn on: alongside classic exercises which are immediate applications of the

theory, and whose absence in this sort of book would be almost astonishing, can

be found exercises which are borrowed from various authors and periodicals, both

French and foreign, and also a large number which are original.

I have also included, at the end of the work, a note in which I seek to sum-

marize the basic principles of the mathematical method, methods which students

must begin to understand starting from the ﬁrst year of instruction, and which we

ﬁnd poorly understood even by students in our schools of higher education. The

dogmatic form which I have had to adopt is not, it must be admitted, the one that

ﬁts this topic best: this sort of subject is best taught through a sort of dialogue

in which each rule intervenes at exactly the moment when it applies. I believed,

despite all, that I had to attempt this exposition, hoping to ﬁnd readers who are

indulgent of the fact that it is presented imperfectly. Let this essay, imperfect as it

is, perform a number of services and contribute to the infusion into the classroom

of ideas on whose importance we must not tire of insisting.

The other notes, also placed at the end of the volume, are more special in

character. Note B concerns Euclid’s postulate. The ideas of modern geometers on

this subject have assumed a form which is clear and well enough deﬁned that it is

possible to give an account of them even in an elementary work.

Note C concerns the problem of tangent circles. As M. Koenigs has noted,6

the known solution of Gergonne, even when completed by the synthesis neglected

by the author, leaves something to be desired. It is this gap that I seek to ﬁll.

Finally, Note D is devoted to the notion of area. The usual theory of area

presents, as we know, a serious logical fault. It supposes a priori that this quantity

is well-deﬁned and enjoys certain properties. The theory that I give in the note in

question, and in which we do without this postulatum , must be preferred, especially

if one realizes that it applies to space geometry without any signiﬁcant change.

In the text itself, various classical arguments might be modiﬁed to advantage,

sometimes to support more rigor, and sometimes in the interests of simplicity.

5Annales

Scientiﬁques de l’Ecole

´

Normale Superieure,

2nd

series, Vol. I, 1876. Exercise 401

(the construction of tangent circles) was provided to me by M. G´ erard, a teacher at the Lyc´ee

Amp`ere

6Le¸

cons de l’agr´ egation classique de Math´ ematiques, p. 92, Paris, Hermann, 1892.