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Search - Principles of Geometry, Vol. 4 (Classic Reprint)

Principles of Geometry Vol 4 - Classic Reprint
Author: H. F. Baker

Excerpt from Principles of Geometry, Vol. 4 — The present chapter consists of various examples of the interest and importance of the comparison of the geometry of spaces of different dimensions. The first section (pp. 1-32) is concerned with relations between theorems in to and in three dimensions. The second section (pp. 32-40) deals with the re... more »presentation in four dimensions of some results belonging to ordinary space of three dimensions. The last section (pp. 40-64) deals with the employment of space of five dimensions for the consideration of properties arising both in three and in two dimensions. Some few references occur to space of any number of dimensions.

Section I. Theorems Of Two And Three Dimensions

The conics touching the fives from six arbitrary lines of a plane. Let three lines, p, q, r, be given in a plane, as well as a fourth line containing two points, I, J; let any conic be drawn touching the four lines; let ? be the conic, through the points I, J, which contains the three intersections of the lines p, q, r; then this conic ? passes through the point, S, in which intersect the tangents from I, J to the former conic. Or, in other words (Vol. II, p. 81), the circle through the intersections of three tangents of a parabola contains the focus of this parabola. Thus if four lines be given, beside the line which contains the points, J, the conic touching the five lines being then definite, the four conics, all through I, J, each containing the intersections of three of the four given lines, meet in a point, namely, the point, S, in which the tangents from I, J to the former conic intersect (Vol. II, p.82); namely, these are four circles meeting in the focus of the parabola. If now, finally, five lines be given, beside the line containing the points I, J, there will be five parabolas, each a conic touching the last line and four of the others, and five foci, S1, S2, ..., S5. It is the case that the circle containing any three of these foci passes through the other two, that is, that the seven points S1, ... , S5, I, J lie on a conic. Of this theorem a proof was given by Clifford ("A synthetic proof of Miquel's theorem," Math. Papers, 1882, p. 38), with the help of certain particular cubic curves.

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.

ISBN-13: 9781330337486
ISBN-10: 1330337484
Publication Date: 6/16/2015
Pages: 270
Rating:

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Publisher: Forgotten Books
Book Type: Paperback
Members Wishing: 0
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Genres:

Science & Math >> Mathematics >> Geometry & Topology

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Search - Principles of Geometry, Vol. 4 (Classic Reprint)

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