La Géométrie

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La Géométrie was published in 1637 as an appendix to Discours de la méthode ( Discourse on the Method ), written by René Descartes. In the Discourse, he 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 [1] (as well as, perhaps, considering the contemporary European social climate of intellectual competitiveness, to show off a bit to a wider audience).

Publishing process of production and dissemination of literature, music, or information

Publishing is the dissemination of literature, music, or information. It is the activity of making information available to the general public. In some cases, authors may be their own publishers, meaning originators and developers of content also provide media to deliver and display the content for the same. Also, the word "publisher" can refer to the individual who leads a publishing company or an imprint or to a person who owns/heads a magazine.

<i>Discourse on the Method</i> book by Descartes

Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences is a philosophical and autobiographical treatise published by René Descartes in 1637. It is best known as the source of the famous quotation "Je pense, donc je suis", which occurs in Part IV of the work. A similar argument, without this precise wording, is found in Meditations on First Philosophy (1641), and a Latin version of the same statement Cogito, ergo sum is found in Principles of Philosophy (1644).

René Descartes 17th-century French philosopher, mathematician, and scientist

René Descartes was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. He is generally considered one of the most notable intellectual figures of the Dutch Golden Age.

Contents

La Geometrie GeometryDescartes.JPG
La Géométrie

The work was the first to propose the idea of uniting algebra and geometry into a single subject [2] and invented an algebraic geometry called analytic geometry, which involves reducing geometry to a form of arithmetic and algebra 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.

Algebraic geometry branch of mathematics

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

Analytic geometry Study of geometry using a coordinate system

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

Geometry branch of mathematics that measures the shape, size and position of objects

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

The text

This appendix is divided into three "books". [3]

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., x, y, z, etc. are to denote unknown variables, while those at the start of the alphabet, a, b, c, 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, aa). He also breaks with the Greek tradition of associating powers with geometric referents, a2 with an area, a3 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." [4] 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 x and y. Known line segments are designated a, b, c, etc. The germinal idea of a Cartesian coordinate system can be traced back to this work.

Pappus of Alexandria Greek mathematician of Antiquity

Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection, and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, and was a teacher in Alexandria.

Cartesian coordinate system coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

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. [5] 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.

In mathematics, a quadratrix is a curve having ordinates which are a measure of the area of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circle.

Spiral curve which emanates from a point, moving farther away as it revolves around the point

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.

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 n has n roots. He systematically discussed negative and imaginary roots [6] of equations and explicitly used what is now known as Descartes' rule of signs.

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.

In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients, and that the difference between these two numbers is always even. This implies in particular that, if this difference is zero or one, then there is exactly zero or one positive root, respectively.

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. [7] 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 [it] for themselves."

Descartes is often credited with inventing the coordinate plane because he had the relevant concepts in his book, [8] 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, 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 this 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. [9]

See also

Notes

  1. Descartes 2006 , p. 1x
  2. Descartes 2006 , p.1xiii "This short work marks the moment at which algebra and geometry ceased being separate."
  3. this section follows Burton 2011 , pp. 367-375
  4. Pappus discussed the problems in his commentary on the Conics of Apollonius.
  5. Boyer 2004 , pp. 88-89
  6. he was one of the first to use this term
  7. Boyer 2004 , pp. 103-104
  8. A. D. Aleksandrov; Andréi Nikoláevich Kolmogórov; M. A. Lavrent'ev (1999). "§2: Descartes' two fundamental concepts". Mathematics, its content, methods, and meaning (Reprint of MIT Press 1963 ed.). Courier Dover Publications. pp. 184 ff. ISBN   0-486-40916-3.
  9. Boyer 2004 , pp. 108-109

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References

Further reading