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On the number of conditions determining geometrical figures

Published online by Cambridge University Press:  20 January 2009

A. Y. Fraser
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This paper is an attempt (I) to deduce from first principles the number of conditions required to determine a plane polygon of n sides; (II) thence to deduce the numbers for special cases; and (III) to discuss the effects of a redundancy and a deficiency in the number of conditions. An investigation of this kind should form an important as well as interesting accompaniment to the ordinary study of elementary geometry.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 3 , February 1884 , pp. 22 - 29
Copyright
Copyright © Edinburgh Mathematical Society 1884

References

* In crystallography, for example, angular relations are the all-important thing, not linear relations.

* It may be of interest to mention that Bonnyoastle's Elements of Geometry, sixth edition (1818), pp. 416–431, contains a list of one hundred and seventy-one cases of triangles determined by sets of three conditions, and this list might now be indefinitely increased.

* Thronghout this paper the assumption is made that if a set of m conditions determine a figure, any other set of m conditions will determine it. The whole investigation goes to support this assumption; but pending the production of a rigid proof, some considerations in its favour may be offered. Every plane figure is determined by its n angular points, and each of these has two degrees of freedom. Hence a plane figure has 2n degrees of freedom. And every condition reduces that number by one, and by one only. Thus for any figure the number of determining conditions is constant.

* “ A datum is any quantity, condition, or other mathematical premiss which is given in a particular problem.”—De Morgan, in The Penny Cyclopedia, voL viii, p. 313.