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On the Constructions which are Possible by Euclid’s Methods.*
Published online by Cambridge University Press: 03 November 2016
Extract
That some advantages have been derived from the freedom now enjoyed in the teaching of Geometry will be denied by no one; but there are disadvantages equally obvious. The, confusion both of order and method, the neglect of the theoretical and deductive work as compared with the practical and experimental, and the failure to grasp the value of Euclid’s work as an educational discipline may be mentioned. Perhaps it is not too late to express the hope that one of our great mathematicians—or a representative group chosen from among them—may yet produce a Textbook of Geometry which will be to English-speaking people what Legendre’s Eléments has been, during more than 100 years, to so large a part of the Continent of Europe.
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- Copyright © Mathematical Association 1910
Footnotes
page 170 note *
The only reference which I can find in the English text-books to this subject is a short paragraph in Hardy’s Pure Mathematics (pp. 64-5).
Klein’s Vortrage über ausgewählte Fragen der Elementar-Geometrie (Leipzig, 1898) is out of print, and I have been unable to obtain a copy; but it is worthy of note that no one has done more for the improvement of the teaching of Elementary Mathematics in Germany than Klein, himself the most famous of living German mathematicians.
In Weber-Wellsteins Encyklopadie der Elementar-M athematik (Leipzig, 1905) the teacher will find a most valuable handbook. The articles on Kreisteilung (Bd. I. §§ 105-7) and Unmöglichkeitsbeueise (Bd. I. §§108-114) discuss at some length the subjects to which I shall refer.
Further, any one who can read Italian will find a complete and beautiful study of these and other questions of Elementary Geometry in the collection of monographs contained in Enriques’ volume, entitled Questioni riguardanti la Geometria Elementare (Bologna, 1900). A German translation of this work is in progress, of which the second volume has been issued, under the title, Enriques, Fragen der Elementar-Geometrie (Leipzig, 1907). The first volume is advertised to appear in 1910.
To the various papers in Enriques’ volume I am particularly indebted, and if it had been available in English, these pages would not have been written.
References
page 171 note * For the Theory of Parallels, Engel und Stäckel’s Theorie der Parallel-Linien con Euclid bis auf Gauss (Leipzig, 1895) is the standard book; but for an elementary historical treatment of this theory, and of the rise of the different Non-Euclidean Geometries, Bonola’s La Geometria non Euclidea (Bologna, 1906) is unique. A German translation by Liebmann of this little volume has already appeared; a Russian translation is in progress, and an English translation, which Professor Bonola has kindly permitted me to undertake, is now in the press.
page 171 note †Cf. Bull. Amer. Math. Soc, vol. xv., pp. 261-4 (1909). From a review of the report of the Commission appointed in 1904 by the Society of German Natural Scientists and Physicists to examine and report upon various proposed reforms in the teaching of mathematics and the natural sciences in Germany, we quote the following:
“The Commission also recommend emphatically that at the end of the general studies in pure mathematics a course be given organizing the entire mathematical material according to its essential inter-relation, and as fur as possible presenting the import of the higher branches for the different stages of school mathematics. For, in fact, experience teaches that without such a course of study, the majority of the students do not discover the inner bond that connects the various parts of mathematical science, and thus the prospective teacher loses what should be for him the real gist of his mathematical studies. To avoid misunderstanding, we add expressly that this course presupposes matured hearers, and should not be brought down to the level of those preparing to teach mathematics as a minor subject only.”
page 177 note * For the Trisection of an Angle by means of Conies, or other curves, or by mechanical means, cf. Enriques, loc. cit., p. 451, et seq.
page 178 note * Cf. Forsyth’s Theory of Functions, ch. iv., §37.
page 178 note † This discovery was made by Lindemann: cf. Math. Ann., Bd. XX., p. 213. See Klein, Lectures on Elementary Mathematics (The Evanston Colloquium), p. 51 (1894), and Enriques, loc. cit., p. 471, et seq.
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