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''La Géométrie'' () was
published Publishing is the activities of making information, literature, music, software, and other content, physical or digital, available to the public for sale or free of charge. Traditionally, the term publishing refers to the creation and distribu ...
in 1637 as an appendix to ''Discours de la méthode'' ('' Discourse on the Method''), written by
René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
. In the ''Discourse'', Descartes presents his method for obtaining clarity on any subject. ''La Géométrie'' and two other appendices, also by Descartes, ''La Dioptrique'' (''Optics'') and ''Les Météores'' (''Meteorology''), were published with the ''Discourse'' to give examples of the kinds of successes he had achieved following his method (as well as, perhaps, considering the contemporary European social climate of intellectual competitiveness, to show off a bit to a wider audience). The work was the first to propose the idea of uniting algebra and geometry into a single subject and invented an
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
called analytic geometry, which involves reducing
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
to a form of
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
and
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
and translating geometric shapes into algebraic equations. For its time this was ground-breaking. It also contributed to the mathematical ideas of Leibniz and Newton and was thus important in the development of calculus.


The text

This appendix is divided into three "books". Book I is titled ''Problems Which Can Be Constructed by Means of Circles and Straight Lines Only.'' In this book he introduces algebraic notation that is still in use today. The letters at the end of the alphabet, viz., , , , etc. are to denote unknown variables, while those at the start of the alphabet, , , , etc. denote constants. He introduces modern exponential notation for powers (except for squares, where he kept the older tradition of writing repeated letters, such as, ). He also breaks with the Greek tradition of associating powers with geometric referents, with an area, with a volume and so on, and treats them all as possible lengths of line segments. These notational devices permit him to describe an association of numbers to lengths of line segments that could be constructed with straightedge and compass. The bulk of the remainder of this book is occupied by Descartes's solution to "the locus problems of Pappus." According to Pappus, given three or four lines in a plane, the problem is to find the locus of a point that moves so that the product of the distances from two of the fixed lines (along specified directions) is proportional to the square of the distance to the third line (in the three line case) or proportional to the product of the distances to the other two lines (in the four line case). In solving these problems and their generalizations, Descartes takes two line segments as unknown and designates them and . Known line segments are designated , , , etc. The germinal idea of a
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
can be traced back to this work. In the second book, called ''On the Nature of Curved Lines'', Descartes described two kinds of curves, called by him ''geometrical'' and ''mechanical''. Geometrical curves are those which are now described by algebraic equations in two variables, however, Descartes described them kinematically and an essential feature was that ''all'' of their points could be obtained by construction from lower order curves. This represented an expansion beyond what was permitted by straightedge and compass constructions. Other curves like the quadratrix and spiral, where only some of whose points could be constructed, were termed mechanical and were not considered suitable for mathematical study. Descartes also devised an algebraic method for finding the normal at any point of a curve whose equation is known. The construction of the tangents to the curve then easily follows and Descartes applied this algebraic procedure for finding tangents to several curves. The third book, ''On the Construction of Solid and Supersolid Problems'', is more properly algebraic than geometric and concerns the nature of equations and how they may be solved. He recommends that all terms of an equation be placed on one side and set equal to 0 to facilitate solution. He points out the factor theorem for polynomials and gives an intuitive proof that a polynomial of degree has roots. He systematically discussed negative and imaginary roots of equations and explicitly used what is now known as Descartes' rule of signs.


Aftermath

Descartes wrote ''La Géométrie'' in French rather than the language used for most scholarly publication at the time, Latin. His exposition style was far from clear, the material was not arranged in a systematic manner and he generally only gave indications of proofs, leaving many of the details to the reader. His attitude toward writing is indicated by statements such as "I did not undertake to say everything," or "It already wearies me to write so much about it," that occur frequently. Descartes justifies his omissions and obscurities with the remark that much was deliberately omitted "in order to give others the pleasure of discovering tfor themselves." Descartes is often credited with inventing the coordinate plane because he had the relevant concepts in his book, however, nowhere in ''La Géométrie'' does the modern rectangular coordinate system appear. This and other improvements were added by mathematicians who took it upon themselves to clarify and explain Descartes' work. This enhancement of Descartes' work was primarily carried out by
Frans van Schooten Frans van Schooten Jr. also rendered as Franciscus van Schooten (15 May 1615 – 29 May 1660) was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes. He translated La Géométrie in Latin and wrote c ...
, a professor of mathematics at Leiden and his students. Van Schooten published a Latin version of ''La Géométrie'' in 1649 and this was followed by three other editions in 1659−1661, 1683 and 1693. The 1659−1661 edition was a two volume work more than twice the length of the original filled with explanations and examples provided by van Schooten and his students. One of these students, Johannes Hudde provided a convenient method for determining double roots of a polynomial, known as Hudde's rule, that had been a difficult procedure in Descartes's method of tangents. These editions established analytic geometry in the seventeenth century.


See also

* Claude Rabuel


Notes


References

* * *


Further reading

* * * *


External links

*
Project Gutenberg copy of ''La Géométrie''
* Bad OCR
Cornell University Library copy of ''La Géométrie''
* Archive.org
''The Geometry of Rene Descartes''
* Facsimile (fr)

{{DEFAULTSORT:Geometrie 1637 books Mathematics books Mathematics literature 1637 in science Works by René Descartes History of geometry