The Problem of the Angle Bisectors a Dissertation Submitted to the Faculty of the Ogden Graduate School of Science of the University of Chicago in Candidacy, for the Degree of Doctor of Philosophy, (Department or Mathematics) (Classic Reprint)

Bücher

Angebote / Angebote:

Excerpt from The Problem of the Angle Bisectors a Dissertation Submitted to the Faculty of the Ogden Graduate School of Science of the University of Chicago in Candidacy, for the Degree of Doctor of Philosophy, (Department or Mathematics) The problem of constructing n triangle when the lengths of the bisectors of the angles are given has been an outstanding problem among geometers probably from the time of Pascal and certainly from the time of Euler. Brocard has summed up the literature, dealing almost entirely with special eases, of which the most extensive treatment appears to be the solution of the problem when one angle is a right angle due to Marcus Baker. This problem is of the sixth order. Among many special treatments that appear in the smaller journals the fundamental problem of determining the character of the algebraic irrationality involved is not mentioned. As a result apparently conflicting statements occur as to the order of the equation concerned, this depending on accidental choice of the parameter field. The only paper dealing with the general ease where the internal bisector formulas arc used is P. Barbarin's. The ease where the external formulas are to be used and the cast where two of the assigned bisectors refer to the same vertex is not treated in general in any paper known to the writer. Barbarin proved that the general internal problem could be solved by the solution of an algebraic equation of order not greater than twelve. The irreducibility and group of the equation arc not discussed, and as the equation itself is not set out explicitly further reduction of the order of the problem is not precluded. The method of attack used by Barbarin is to solve first the problem when an angle and two bisectors are given, and to use the result as a basis for attacking the general problem. The necessary sacrifice of symmetry prevents any explicit comparison with the solution given in this paper except at the cost of labor disproportionate to the result. The problem must have an extensive domestic history in the schools: Barbarin charges that E. Catalan was among those who have proposed it as an elementary exercise, and from a Russian scholar the writer learns that it has been there extensively used in the schools as a standard set-back for ambitious young geometers. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.