vol 4: Glossary
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Cartesian geometry
[Hazewinkel, Cartesian geometry] [Borowski, Cartesian geometry]
Cartesian geometry (also known as analytic geometry) was invented by Renee Descartes. It connects arithmetic and geometry by establishing a set of coordinate axes against which geometric figures may be measured. In one dimensional space, the only possible geometric figures are points and lines. By choosing a point as origin, and a unit of length, we may assign a number to every point on a line by measuring it against the x axis. In two dimensional space, as illustrated, two numbers are needed to address each point. In general, points in n-dimensional space are represented by ordered sets of n numbers.
We may also represent functions in Cartesian space. It is normal, for functions of one variable, to use the x axis to represent the domain of the function and the y axis its range.
Descartes' linkage between the ancient subjects of arithmetic and geometry enabled us to visualise arithmetic and algebraic ideas in geometric images and so laid the foundations for a very fertile period in mathematics. It is second nature for us now to a function as curves or other shapes in an appropriate Cartesian space.
Many of the deep questions in set theory, such as the cardinal number of the continuum, seem to have been motivated by the need to be able to assign a number to every point of a continuous line.
Further reading
Books
| Borowski, Ephraim J, and Jonathan M Borwein, HarperCollins Dictionary of Mathematics, Harper Collins 1991 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership.' Amazon back |
| Davis, Phillip J, and Reuben Hersh, Descartes Dream: The World According to Mathematics, Penguin 1988 Preface: 'We are concerned with the impact mathematics makes when it is applied to the world that lies outside mathematics itself; when it is used in relation to the world of nature or of human activities. This is sometimes called applied mathematics. This activity has now become so extensive that we speak of the "mathematisation of the world." We want to know the conditions of civilisation that bring it about. We want to know when these applications are effective, when they are ineffective, when beneficial, dangerous or irrelevant. We want to know how they constrain our ives, how they transform our perceptionof reality.' Amazon back |
| Descartes, Rene, and (Translated by David Eugene Smith and Marcia L Latham) , Geometrie , Dover 1956 Jacket: ' ... With this volume, Descartes founded modern analytical geometry. Reducing geometry to algebra and analysis and, conversely, showing that analysis can be translated into geometry, it opened the way for modern mathematics. ... This edition contains the entire definitive Smith-Latham translation of Descartes three books: Problems the Construction of which requires Only Straight Lines and Circles; On the Nature of Curved Lines; On the Construction of Solid and Supersolid Problems. Interleaved page by page with the translation is a complete facsimile of the 1637 French text, together with Descartes' original illustrations. ...' Amazon back |
| Gaukroger, Stephen, Descartes: An Intellectual Biography, Clarendon Press 1995 Jacket: 'Rene Descartes (1596-1650) is the father of modern philsoophy and one of the greatest of all thinkers. This is the first intellectual biography of Descartes in English; it offers a fundamental reassessment of all aspects of his life and work. ... Descares' early work in mathematics and science produced ground-breaking theories, methods and tools still in use today. This book gives the first full acount of how this work infomred and influenced the later phisosophical studies for which, above all, Descartes is renowned.... [It] offers for the first time a full understanding of how Descartes developed his revolutionary ideas. It will be a landmark publication, welcomed by all readers interested in the origins of modern thought.' Amazon back |
| Hazewinkel, Michiel, and (managing editor), Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
| Hilbert, David, and Leon Unger (translator, from the tenth German edition). Revised and Enlarged by Paul Bernays, Foundations of Geometry (Grundlagen der Geometrie), Open Court 1999 Jacket: 'Along with the writings of Hilbert's friend and correspondent Frege, Hilbert's Grundlagen der Geometrie is the major prop that set the stage for Russell and Whitehead's Principa Mathematica. Hilbert presents a new axiomatization of geometry, the reduction of geometry to algebra, and introduces the distinction between mathematics and metamathematics, with a new theory of proof. This edition is translated from the tenth German edition, including all the improvements which Hilbert derived from his own reflections and the contributions of other writers. Amazon back |
| Nakahara, Mikio, Geometry, Topology and Physics, Adam Hilger 1990 Jacket: 'Differential geometry and topology have become essential tools for many theoretical physicists. [this book] introduces the ideas of differential geometry and topology to postgraduate students and researchers of theoretical physics. ... Throughout the book there are explicit calculations and diagrams to clarify the abstract ideas involved. A large number of problems and exercises are included to help develop the reader's understanding of the subject.' Amazon back |
| Smith, David Eugene, and Marcia L Latham (translators), The Geometry of Rene Descartes, Dover 1954 Amazon back |
| Thomas Jr, George B, and Ross L. Finney, Maurice D. Weir, Frank R. Giordano, Thomas's Calculus, Addison-Wesley 2000 I used the second edition of Thomas' book in school and loved it. Since then it has gone through eight more editions, gained a small posse of additional authors and nearly doubled in size. It should still be pretty good! Amazon back |
Links
| Silvio Levy Cartesian coordinates in the plane 'In cartesian coordinates (or rectangular coordinates), the ``address'' of a point P is given by two real numbers indicating the positions of the perpendicular projections from the point to two fixed, perpendicular, graduated lines, called the axes.' back |
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