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Synopsis
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 representation 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 σ be the conic, through the points I, J, which contains the three intersections of the lines p, q, r; then this conic σ 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 t...
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- PublisherForgotten Books
- Publication date2018
- ISBN 10 1330337484
- ISBN 13 9781330337486
- BindingPaperback
- LanguageEnglish
- Number of pages266
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Principles of Geometry, Vol. 4 (Classic Reprint)
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Paperback. Condition: New. Print on Demand. This book delves into the fascinating connections between geometry in different dimensions, showcasing the elegance and interconnectedness of mathematical principles. By exploring relationships between theorems in two, three, four, and even five dimensions, the author reveals how seemingly disparate concepts are intricately linked. Through a series of detailed examples and insightful commentary, the book guides readers through a journey of discovery, starting with the familiar geometry of circles and lines in a plane and culminating in the complexities of higher-dimensional spaces. Readers will encounter intriguing geometric figures, such as the "double-six" theorem and Kummer's surface, and witness how their properties can be understood through the lens of different dimensions. The author's approach not only illuminates the intrinsic beauty of these geometric objects but also demonstrates the power of unifying mathematical concepts across various dimensions. Ultimately, this book offers a profound appreciation for the depth and interconnectedness of geometry, revealing how the exploration of higher dimensions can shed new light on even the most fundamental geometric principles. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Seller Inventory # 9781330337486_0
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Principles of Geometry, Vol 4 Classic Reprint
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PAP. Condition: New. New Book. Shipped from UK. Established seller since 2000. Seller Inventory # LW-9781330337486
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Principles of Geometry, Vol. 4 (Classic Reprint)
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Paperback. Condition: Brand New. 268 pages. 8.90x5.60x0.40 inches. This item is printed on demand. Seller Inventory # zk1330337484
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