Geometria

by René Descartes

This is an unabridged republication of the definitive English translation of one of the very greatest classics of science. Originally published in 1637, it has been characterized as "the greatest single step ever made in the progress of the exact sciences" (John Stuart Mill): as a book which "remade geometry and made modern geometry possible (Eric Temple Bell). It "revolutionized the entire conception of the object of mathematical science" (J. Hadamard).

With this volume Descartes founded modern analytical geometry. Reducing geometry to algebra and analysis and, conversely, showing that analysis may be translated into geometry, it opened the way for modern mathematics. Descartes was the first to classify curves systematically and to demonstrate algebraic solution of geometric curves. His geometric interpretation of negative quantities led to later concepts of continuity and the theory of function. The third book contains important contributions to the theory of equations.

Genres: Mathematics, Philosophy, Science, Nonfiction, Classics, France, Geometry

142 pages, Paperback

First published January 1, 1637

About the author

Meditations on First Philosophy (1641) and Principles of Philosophy (1644), main works of French mathematician and scientist René Descartes, considered the father of analytic geometry and the founder of modern rationalism, include the famous dictum "I think, therefore I am."

A set of two perpendicular lines in a plane or three in space intersect at an origin in Cartesian coordinate system. Cartesian coordinate, a member of the set of numbers, distances, locates a point in this system. Cartesian coordinates describe all points of a Cartesian plane.

From given sets, {X} and {Y}, one can construct Cartesian product, a set of all pairs of elements (x, y), such that x belongs to {X} and y belongs to {Y}.

Cartesian philosophers include Antoine Arnauld.



René Descartes, a writer, highly influenced society. People continue to study closely his writings and subsequently responded in the west. He of the key figures in the revolution also apparently influenced the named coordinate system, used in planes and algebra.

Descartes frequently sets his views apart from those of his predecessors. In the opening section of the Passions of the Soul , a treatise on the early version of now commonly called emotions, he goes so far to assert that he writes on his topic "as if no one had written on these matters before." Many elements in late Aristotelianism, the revived Stoicism of the 16th century, or earlier like Saint Augustine of Hippo provide precedents. Naturally, he differs from the schools on two major points: He rejects corporeal substance into matter and form and any appeal to divine or natural ends in explaining natural phenomena. In his theology, he insists on the absolute freedom of act of creation of God.

Baruch Spinoza and Baron Gottfried Wilhelm von Leibniz later advocated Descartes, a major figure in 17th century Continent, and the empiricist school of thought, consisting of Thomas Hobbes, John Locke, George Berkeley, and David Hume, opposed him. Leibniz and Descartes, all well versed like Spinoza, contributed greatly. Descartes, the crucial bridge with algebra, invented the coordinate system and calculus. Reflections of Descartes on mind and mechanism began the strain of western thought; much later, the invention of the electronic computer and the possibility of machine intelligence impelled this thought, which blossomed into the Turing test and related thought. His stated most in §7 of part I and in part IV of Discourse on the Method .

Reviews

[Orhan Pelinkovic · Rating 4 out of 5]

In La Géométrie (1637) I found Descartes' writing distilled and his mathematical demonstration lucid. I read the translation in English which alongside each page has a facsimile of the original French edition with a mid-17th century typeface design.

Descartes employs his avant-garde method of reasoning in the field of mathematics to display the method's versatility. In this unabridged edition, he presents an alternative system on how to approach, evaluate and think about mathematics. Descartes takes the ancient geometry that was constricted to a ruler and compass and transforms it into modern geometry. A geometry for which it was not necessary to draw lines, but enough to designate a letter for them. He appoints the letters a, b, and c for know values and x, y, and z for the unknown values which we use even now.

With this work, Descartes founded analytic geometry and laid the foundation for the development of calculus. His coordinate system we still use today to translate geometrical shapes into algebraic equations and vice versa, at last linking arithmetic and geometry into one.

[Roy Lotz · Rating 4 out of 5]

It is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery—Columbus when he first saw the Western shore, Pizarro when he stared at the Pacific Ocean, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method of coordinate geometry.

—Alfred North Whitehead, Introduction to Mathematics

Here I go again, writing a review for a book that’s way over my head. But someone has to do these things.

Descartes’ Geometry is a short book, originally appended to his famous Discourse on Method (the book in which he first proclaimed his momentous “cogito, ergo sum”), along with his Dioptrics and Meteorology. These three works were meant to showcase the success that Descartes had with his new way of thinking. I know nothing about the influence of the two works of science; but this book, which remained his only published work on mathematics, was groundbreaking.

Descartes effectively figured out how to marry geometry and algebra. He did this by assigning values to lines, and then creating equations out of them. It’s hard for us to really grasp how revolutionary this step was, considering that we are taught analytic geometry in middle school. But if you open a page of Euclid’s Elements —the book used to teach geometry for over 2,000 years—you will find that not a single proof has variables, equations, or numbers. So when Descartes managed, so to speak, to reduce the wild world of shapes and figures to the neat and tidy domain of arithmetic, it was nothing short of a revolution. The calculus of Newton and Leibniz would have been impossible without it.

Not surprisingly, it took a long while before this short book was properly understood. In part, this must have been due to its pathbreaking nature. But this was also partly Descartes’ fault. For one, he wrote in French, not Latin (which was, of course, the lingua franca of science). (Parenthetically, I would also like to say that this particular edition, published by Dover, has a facsimile of the original publication, which is pretty cool, if not particularly useful.) But another reason that the work wasn’t properly understood was because Descartes didn’t seem to want to be understood.

The Geometry presents little more than a sketch of his ideas, leaving numerous gaps in his arguments, and much to be figured out by the reader. In his own words, “I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave to others the pleasures of discovery.” It’s no coincidence that this book never replaced Euclid’s as the standard textbook. What’s more, for some reason Descartes thought it fitting to chose very complex equations to introduce his ideas. Some of them were so hefty that they filled up half a page. According to the Wikipedia article, Descartes was showing off, and I believe it.

Strange to say, but Descartes never uses the coordinate system named in his honor. But his entire way of doing things suggests such a system, so I can see why we call it after him. In any case, he at least includes his famous “rule of signs,” the shortcut for determining the number of positive, negative, and imaginary roots of an equation.

To sum up, I’m not sure I would recommend this work to anyone unless they have a decent background in math (which I lack), and are interested in the history of the subject. On the other hand, I found it a fascinating read, despite my frequent inability to follow his reasoning, just because of its historical importance. And if you can’t understand it either, don’t worry, you’re in good company. The best mathematicians of the day couldn’t understand it. It took a translation into Latin by van Schooten twelve years later, along with numerous explanatory notes, for it to finally receive its due recognition.

Posterity has judged you kindly, Descartes, despite your omitting much. But it sure doesn’t help your readability.

[Ray · Rating 3 out of 5]

It's confusing, to say the least. Descartes does not lay out his math very much so it's challenging to follow sometimes. However, his ideas are still quite interesting. My favorite was his comparison between the "ancient" maths and his math. He brings up many comparisons such as mechanics against geometry, and pen to paper style geometry against theoretical geometry. I only read till the second book so I still lack a big picture understanding, but I will make sure to keep these ideas in mind when reading the Discourse on Method.

[Tyler · Rating 5 out of 5]

If anyone is wondering about the mathematical genius of Descartes, I would recommend this book with his book on Polyhedra as shining examples. This one is particularly impressive, because within it is the rudiments of the Cartesian coordinate system, factoring polynomial equations, reducing and expanding polynomial equations, and graphing fourth and fifth dimensional figures by means of curved lines with exponential equations. There is a limit to the human understanding of these problems, and Descartes asymptotically approaches it in this absolute masterpiece of geometrical descriptions. The burning point and foci of the different mirrors in Optics is also treated here, of much more expansive detail. My only complaint, is the same one I will have for Galileo, which is that when they state the relations between different properties, they aren't really referencing any of the so-called 'ancient's that Descartes talks about in the first place. Kepler and the better mathematicians always referenced the propositions they were referring to. However, within this book, is a glimpse into the minds of not just Descartes, but the people who taught Descartes and everyone who read Geometry and was taught BY Descartes, for hundreds of years. Also, here and there Descartes will misidentify variables. He skips over things sometimes and this actually leads to the only really material error in the entire work, towards the middle of book 2 when he introduces e, and already has defined e as part of the framework of what he set up. This renders the rest of what he starts talking about vague, especially since he utilizes z, which is from the same faciendum of the problem, in the same equation as e.

But overall, incredible work, searching, and incredibly creative with equations. This man was an artist, not simply of thought, but of mathematics as well.

[Rod Van Meter · Rating 4 out of 5]

I do love the inclusion of the original facsimile. It's fun trying to puzzle out some of the 17th century French. Today, this is a book for math lovers only; most of the proofs and explanations (e.g., on how to find roots of polynomials, which remains an important problem for every high schooler and every STEM college student) are hard to follow, and while ground-breaking at the time probably not worth close study by most people today.

Random notes I took while reading:
Father of analytic geometry
Definer of Cartesian coordinates
Cavorted with Mersenne
Studied Galileo, 32 years his senior, as a student
Collaborated w Fermat, but Cartesian geometry not published until 1679, almost 30 years after his death
Independently discovered Law of Refraction, known since Ptolemy (2nd century), Alhazen (11th?), Ibn Sahl (10th), but now known as Snell's Law as most complete & accurate form. Interestingly, Descartes had speed of light backwards, he thought it went faster in dense media.
(Other roughly contemporaneous names in optics are Newton, Huygens, Fermat, Fresnel, Galilieo)
Descartes gave us our exponential notation, as well as the convention of x,y,z for variables and a,b,c for constants
Kepler introduced the term "focus" in geometry in 1604, but Descartes didn't adopt it, using "burning points" instead.
He also apparently considered it improper or insulting to make the book too easy to read.
We've all had profs like that...

[Jim · Rating 4 out of 5]

Descartes certainly proves his ability as a Mathematician in this book. While I struggled a bit with the graphic demonstrations, I was able to follow his Algebra, much more easily. Among other things, I learned about the usefulness of the Conic sections, and the method of using Algebraic Identities to solve 4th, 5th, and 6th degree equations. The author shares credit for some of the Identities with Cardan. I was delighted to run into something I learned a long time ago, and even helped my son understand when he was in school; Descartes Rule of Signs. This book is not for everyone. But I enjoyed it, and consider it a privilege to be taught by Decartes.

[Medhat ullah · Rating 4 out of 5]

It is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery—Columbus when he first saw the Western shore, Pizarro when he stared at the Pacific Ocean, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method of coordinate geometry.

[Jackson Snyder · Rating 4 out of 5]

Really cool stuff that he’s doing, basically creating algebra using geometry as a starting point. Really neat when I understood it, frustrating when I didn’t but I found a commentary that helped. Foundational math continues to amaze me.

[Federico Damian · Rating 4 out of 5]

It was a dense book, but an enlightening one. The beginning of analytic geometry!

[Florin · Rating 5 out of 5]

'[...] nădăjduiesc că nepoții mei îmi vor fi recunoscători nu numai pentru lucrurile pe care le-am explicat aici, ci și pentru cele pe care le-am omis voit, pentru a le lăsa lor plăcerea să le descopere.'

[JP · Rating 4 out of 5]

Book I: discusses construction of certain geometric figures using instruments, compares basic elements of geometry to basic operations in arithmetic; Book II: proves that the set of curved figures described by the range of 3rd order equations is limited to the known set of surfaces (circle, ellipse, hyperbola, parabola); also makes references to his own Dioptrique; Book III: "But it is not my purpose to write a large book. I am trying rather to include much in a few words.... I hope that posterity will judge me kindly, not only as to the things which I have explained, but also as to those which I have intentionally omitted so as to leave others the pleasure of discovery."

[Douglas · Rating 2 out of 5]

This is a classic in the history of mathematics. In this work the field of analytic geometry was introduced to the world. The book progresses through a series of example solutions to different problems in geometry so you can grab a pen and paper and work along with Descartes. This is a dual English/French edition with alternating pages.

[Paul Mamani · Rating 4 out of 5]

I never was goof with Math.
Mr descartes is very known genius in Peruvians schools.

That's why Ive chosen to read this book.
It hard to understand but interesting

[Trey Kennedy · Rating 4 out of 5]

More algebra than geometry.