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History of Analytic Geometry (Dover Books on Mathematics), by Carl B. Boyer
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Specifically designed as an integrated survey of the development of analytic geometry, this classic study takes a unique approach to the history of ideas. The author, a distinguished historian of mathematics, presents a detailed view of not only the concepts themselves, but also the ways in which they extended the work of each generation, from before the Alexandrian Age through the eras of the great French mathematicians Fermat and Descartes, and on through Newton and Euler to the "Golden Age," from 1789 to 1850. Appropriate as an undergraduate text, this history is accessible to any mathematically inclined reader. 1956 edition. Analytical bibliography. Index.
- Sales Rank: #974540 in Books
- Published on: 2004-11-29
- Released on: 2004-11-29
- Format: Unabridged
- Original language: English
- Number of items: 1
- Dimensions: 8.46" h x .64" w x 5.62" l, .77 pounds
- Binding: Paperback
- 304 pages
Most helpful customer reviews
6 of 6 people found the following review helpful.
appreciate achievements of great minds
By W Boudville
Analytic geometry is where the maths student first encounters the combining of traditional Euclidean geometry with algebra. A profound mix, though perhaps most students won't appreciate it as such. Boyer shows how, slowly, the necessary ideas in analytic geometry came together. He traces the first stirrings back to the classical era of ancient Greece and Rome. But the greatest step may well have been due to Rene Decartes and his laying down of the x and y grid in two dimensions. Plus, of course, analytic geometry was necessary for the development of calculus, with the concept of a slope.
You probably are already familiar with all of the maths that the book covers. What Boyer offers is an appreciation of the great minds that preceded up and made these achievements.
3 of 3 people found the following review helpful.
A good book. A great book.
By Jonathan H.
Covers the subject neatly and rigorously. Good writer, has the rare skill of getting mathematics concepts across cleanly. Everyone I show this book to wants their own copy. So I give them mine and get another.
0 of 0 people found the following review helpful.
Well researched history of algebraicization of geometry
By Alan U. Kennington
This Dover book, "History of Analytic Geometry" by Carl B. Boyer, is a very competent history of the way in which geometry made many transitions from the Euclidean geometry of lines, circles and conics to the algebraic reformulations by Fermat and Descartes, finally to the arithmetization of geometry which we now take for granted.
Although the treatment is excellent, it seems to me that the subject of this book is a relatively dry, light-weight part of mathematics from the modern point of view, unlike the much more satisfying history of calculus, "The History of the Calculus and Its Conceptual Development", by the same author. Perhaps the reason for this apparent shallowness of coordinate geometry is that it is now so totally accepted in modern life, for example in longitude and latitude for the Earth, X and Y coordinates for computer monitor pixels or printer dots, and for every graph we ever see during school education or working life. Therefore to appreciate this history by Boyer, one must try to imagine the mind-set of pure mathematicians from Euclid to Euler, who believed that numbers and magnitudes are fundamentally different. Even in Euler's relatively modern 1748 work, "Introductio in analysin infinitorum", numbers are referred to as "lines" (in Latin), and this idea persisted well into the 19th century.
Boyer points out on pages 92 and 267 that it was only with the Cantor-Dedekind axiom that the identification of numbers with the points on a line is made explicit, late in the 19th century. But the "real number line" is taught at a very early age in the schools now as if it were obvious. What has happened is that numbers have developed so as to "fill in the gaps" on the "real number line" with algebraic and transcendental numbers. We take for granted now that numbers include decimal expansions to any number of significant digits, even infinite, but this was not part of mathematical understanding until the 19th century. This book shows how geometry was gradually, and sometimes painfully, developed until the algebraic and numerical viewpoints prevailed.
Some of the points in this book which I have added marginal notes for are as follows.
* Pages 13-14. It was Plato who required geometry to be concerned only with ruler and compass, not Euclid.
* Pages 17-18. According to Proclus and Eutocius, it was Menaechmus who discovered the 3 kinds of conics about 350 BC.
* Page 24. It was Apollonius who gave the names to the ellipse, parabola and hyperbola.
* Page 46. It was Oresme, about 1350 AD, who first published graphs with Y as a function of X.
* Page 57. Originally, cubic and quartic equations were solved using the geometry of conics, but from about 1550 onwards, there were algebraic methods to achieve the same objective. Stevin, about 1600 AD, said that anything which can be achieved with geometry can be done in arithmetic.
* Page 111. Wallis, about 1650, introduced negative abscissas (X coordinates), but "the significance of this step was not appreciated by his contemporaries".
* Page 133. Jean Bernoulli in 1692 was the first to use the term "Cartesian geometry" for geometry based on a coordinate system.
* Page 169. Clairaut in 1731 defined the distance between two points in 2 and 3 dimensional coordinate space as the square root of the sum of the squares of the coordinate differences.
Throughout the book, Boyer discusses the fundamental question of what distinguishes "analytic geometry" from the earlier geometry, whether it is the use of coordinates, the application of algebra or arithmetic to geometry, or the application of geometry to algebra or arithmetic, or maybe something else. Perhaps this is the weak point of the book. It is not really clearly stated what "analytic geometry" means. So it's difficult to know when it started, and which topics belong to calculus rather than analytic geometry.
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