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Isaac Barrow is largely responsible for that preservation and promulgation of the Euclidean tradition which invigorated the physical science and mathematics of Newton and others, and allowed for an ongoing engagement with classical Greek mathematics. Barrow's philosophy of mathematics remains relevant to many key issues still at the forefront of modern philosophies of mathematics. The tradition of mathematics as a liberal discipline, which is the main topic of this book, reaches back to the educational ideals of Plato and Aristotle, and was brought to an early state of near perfection in the Euclidean school of Greek geometry. Within this framework mathematics is practiced as the means of training and cultivating our formal powers: the intellect and imagination function together to achieve a high degree of familiarity with the geometrical structures, patterns and relations that we find embedded in the world around us. The ultimate goal is liberation of the spirit from exclusive preoccupation with mundane concerns.However, we remain at the same time anchored to a realistic philosophical conception of the world, in which our mathematical notions arise by abstraction from real, objective properties of things outside ourselves. This practice of mathematics as a liberal discipline is exemplified nowhere more completely than in the work of Isaac Barrow. The interplay between these modern philosophies and liberal educational ideals forms a recurrent theme of this book. Separate chapters are devoted to Barrow's development as a translator of Greek mathematics and as a mathematician in his own right; to his contributions to the birth of the infinitesimal calculus; to his conception of geometry as the most fundamental and realistic form of mathematics; to his treatment of physical space and time; and to the philosophical and theological implications of the constructive ideal which lies at the heart of classical geometry. ... Read more

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Customer Reviews (1)

neat computer program for proofs
Most of us first encountered Euclidean geometry in primary school. At those ages, it is a branch of maths that can be taught reasonably rigourously. And from which, there are all these memories of how diagrams are used in proofs. What Miller does here is refresh and expand on this diagrammatic approach. Providing a good review of the fundamentals of geometry, if nothing else.

But the present intrudes, in the form of an elegant computer program with a simple text and graphics output. This program can, with some assistance, find, that is, prove some conjectures. It's really awesome when it gives results! ... Read more

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Develops a simple non-Euclidean geometry and explores some of its practical applications through graphs, research problems, and exercises. Includes selected answers. ... Read more

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Publisher: London Pub. by Harrison for the University of St. AndrewsPublication date: 1911Subjects: Geometry, Non-EuclideanNotes: This is an OCR reprint. There may be numerous typos or missing text. There are no illustrations or indexes.When you buy the General Books edition of this book you get free trial access to Million-Books.com where you can select from more than a million books for free. You can also preview the book there. ... Read more

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This scarce antiquarian book is a selection from Kessinger Publishings Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the worlds literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

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Originally published in 1901.This volume from the Cornell University Library's print collections was scanned on an APT BookScan and converted to JPG 2000 format by Kirtas Technologies.All titles scanned cover to cover and pages may include marks notations and other marginalia present in the original volume. ... Read more

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This work offers insights on various facets of geometry through development of four geometric themes: ellipse, the shape or shadow cast by a cirlce; all three types of conic sections - the ellipse, the parabola and the hyperbola; certain properties of geometric figures related to the problem of finding the largest areas that can be enclosed by a curve of given length - the isoperimetric problem; and a development of the non-Euclidean geometry in the Poincare disc model of the hyperbolic plane. ... Read more

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A geometry math book. ... Read more

Customer Reviews (2)

A Simple Non-Euclidean Geometry and Its Physical Basis
This book does an excellent job of describing Galilean Geometry.I know of no other books on the topic, and this one was hard to find.It was used as a reference in an advanced mathematics course on geometries in 2010.There are a handful of points that were confusing in the text, but for the most part it is a fine book for self-study.There is an introduction, two main chapters on Galilean geometry, and a shorter final chapter on Minkowskian geometry.Each chapter ends with about twenty problems and several open ended challenge problems.About 25% of those problems have answers or hints in the back of the book, but they tend to be fairly cryptic.Pages 214-288 are three appendices describing a set of nine geometries to which the ones presented belong.

The level of the text calls for a solid mastery of trigonometry, but requires no calculus, although a few limits appear in the appendices.Also required is a fair bit of mathematical insight and creativity.The subject is not for the faint of heart;of the several geometries studied in the course, this was by far the most involved.The book has particular appeal to physicists, as Galilean geometry is rarely even mentioned anywhere else, but is useful as a way of comparing classical and Einsteinian relativity.

Thorough Galilean geometry
Galilean geometry is the study of the x-t plane under the motions that correspond to changes in time origin and change of uniform motion coordinate system. Although we introduce this geometry on physical grounds, we quickly move on to study this geometry in its own right without any physical motivation. Distance in Galilean geometry should mean distance in space and not in time so the distance between two points is the distance between their projections on the x-axis. Therefore circles in Galilean geometry are pairs of lines t=constant. The angle between two lines can be defined in terms of the section they cut out of a circle centred at their intersection, so in this way we have angles in Galilean geometry. But circles can also be defined in terms of angles: a moving point on a circle makes a constant angle with any two fixed points of the circle (half the central angle). Applying this definition in Galilean geometry gives a new type of curve which we call cycles. Circles and cycles together are in many ways the natural analog of Euclidean circles. We see this in particular when we study circle inversion. Euclidean circle inversion can be understood in terms of stereographic projection: put the plane tangent to the south pole and the eye at the north pole and project up on the sphere, then interchange the plane and the eye and project back. In Galilean geometry the sphere must be replaced by a cylinder (since, e.g., the center of the circle t=-c, t=+c is the line t=0 and thus maps to a line at infinity). In the chapter called "Conclusion" we look at how our ideas carry over to the geometry of space-time in special relativity, again without much concern for physics. As before, we use physics only to derive the motions of our geometry, in this case the Lorentz transformations, and then quickly go on to consider things of purely geometric interest. For instance, we see that the difference in signs between space and time components of a Lorentz transformation implies that a circle will be a hyperbola x^2-t^2=0 and the analog of stereographic projection requires a hyperboloid. ... Read more

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This book is a collection of surveys and exploratory articlesabout recent developments in the field of computational Euclideangeometry. Topics covered include the history of Euclidean geometry,Voronoi diagrams, randomized geometric algorithms, computationalalgebra, triangulations, machine proofs, topological designs,finite-element mesh, computer-aided geometric designs and Steinertrees. This second edition contains three new surveys coveringgeometric constraint solving, computational geometry and the exactcomputation paradigm. ... Read more

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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more