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The Treatment of Elementary Geometry by a Group-Calculus

Published online by Cambridge University Press:  03 November 2016

G. Thomsen
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Extract

There is no need to recommend to teachers in this country, familiar with Dobbs’ School Course and with later books of the same tendency, the free use of the ideas of displacement, rotation, and reflection. In the course of a paper read at the Zürich Congress last year, Prof. Thomsen, of Rostock, gave some striking examples of the effective introduction of the language and notation of the theory of groups into work of this kind, and afterwards he agreed willingly to contribute to the Gazette an article on the subject. For applications to their own problems we must refer readers elsewhere, since Prof. Thomsen has paid them the compliment of explaining a point of view that would be quite unsuitable in the school. To some of our readers we must apologise for replacing the original article by a translation, but since there are no familiar symbols or recognisable formulae to facilitate the grasping of a novel theory, we think that a substantial minority will be grateful for this assistance. In making his version, Prof. Neville has taken one or two liberties with the author’s notation.

Type
Research Article
Information
The Mathematical Gazette , Volume 17 , Issue 225 , October 1933 , pp. 230 - 242
Copyright
Copyright © Mathematical Association 1933

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References

page 230 note * G. Thomsen, “Über einen neuen Zweig geometrischer Axiomatik und eine neue Art von analytischer Geometrie,” Math. Zeitschrift, 34, pp. 668-720 (1932). See further: G. Thomsen, “Grundlagen der Elementargeometrie”, Hamburger mathematische Einzelschriften, 15 (Teubner, 1933).

page 230 note † The necessary elements of this calculus are set out in the paper to which reference has already been made.

page 230 note ‡ In the theory of groups, identity is usually denoted by the letter E, reserved for the purpose.

page 231 note * In the theory of equations ωn is commonly used to denote a primitive root of zn = 1, that is, a root which is not a root of a similar equation of lower degree. Two cycles of the same order are not necessarily identical, and therefore we use the logical symbol of inclusion, ∊, rather than a symbol of equality, which might prove misleading.

page 234 note * If either class is empty the geometry is uninteresting. Most geometrical theorems have assumptions which cannot be satisfied in this case, and according to the usual convention of formal logic the theorems are true theorems but there is nothing of which they are true.

page 238 note * Cf. Thomsen, Math. Zeitschrifi, 34, p. 712.

page 239 note * Cf. Thomsen, Math. Zeitschrift, 34, p. 682,

page 242 note * To appear in Math. Zeitschrift.