The Trisection of the Angle by Plane Geometry: Verified by Trigonometry with Concrete Examples

Abstract

THE Paralogistes pseudomathematicus has become so rare, or possibly so shy, that it is a real pleasure to find that the species is not extinct. Alack! that De Morgan is not with us, to do justice to this latest attempt at solving one of the three famous problems that have been proved to be beyond the power of Euclidean constructions. The curious thing is that the author, in his introduction, gives two long quotations from De Morgan, in which he states the conditions of the problem with the utmost precision, except that he does not explicitly say that the tri-section must be performed by a finite number of operations. It is here that Dr. Whiteford has come to grief, for his method is nothing more or less than successive approximations, each of which involves a Euclidean construction. It is only fair to add that the author is no vulgar paradoxer, and- that his method, as an approximation, is sound, and leads to accurate values with a comparatively small number of trials; thus in his examples he works to seven places of decimals, and we have not noticed a case in which more than seven trials are required. The one, unfortunately fatal, objection, is that he has ignored the conditions of the problem; it is as though the value of π were found from the perimeter of a regular polygon of 2ⁿ sides. By taking n large enough, we can get by Euclidean construction a value as near π as we please; but it is needless to say that this is not what is meant by “squaring the circle” with rule and compass.

The Trisection of the Angle by Plane Geometry: Verified by Trigonometry with Concrete Examples.

By Dr. J. Whiteford. Pp. 169. (Greenock: J. McKelire and Sons, Ltd.; Edinburgh and Glasgow: J. Menzies and Co., Ltd.; Cambridge: Bowes and Bowes, 1911.)