Author’s Prefaces 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 become superfluous. 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 geometry3 is inversion and its applications, which corresponds to Chapters V–VII of our Complements. 3 I note in this regard, that I have never attempted—despite the advocacy of such a step by 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. xi
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