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* La Géométrie* was published in 1637 as an appendix to

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* 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

**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.

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** 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.

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** 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.

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, *a*^{2} with an area, *a*^{3} 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** 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.

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.

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.

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] }

- ↑ Descartes 2006 , p. 1x
- ↑ Descartes 2006 , p.1xiii "This short work marks the moment at which algebra and geometry ceased being separate."
- ↑ this section follows Burton 2011 , pp. 367-375
- ↑ Pappus discussed the problems in his commentary on the
*Conics*of Apollonius. - ↑ Boyer 2004 , pp. 88-89
- ↑ he was one of the first to use this term
- ↑ Boyer 2004 , pp. 103-104
- ↑ 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. - ↑ Boyer 2004 , pp. 108-109

In geometry and algebra, a real number r is **constructible** if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge in a finite number of steps. Not all real numbers are constructible and to describe those that are, algebraic techniques are usually employed. However, in order to employ those techniques, it is useful to first associate points with constructible numbers.

In geometry, the **tangent line** to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve *y* = *f* (*x*) at a point *x* = *c* on the curve if the line passes through the point (*c*, *f* ) on the curve and has slope *f*'(*c*) where *f*' is the derivative of *f*. A similar definition applies to space curves and curves in *n*-dimensional Euclidean space.

**Franciscus van Schooten** was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes.

**Angle trisection** is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.

In mathematics, an **algebraic curve** is the set of solutions of a polynomial in two variables. An **algebraic plane curve** is an algebraic curve that is also a plane curve and therefore contained in an affine plane, such as the Euclidean plane, or the projective plane. In algebraic geometry, an algebraic curve is characterized as an algebraic variety of dimension one.

**Apollonius of Perga** was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.

**François Viète**, Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV of France.

The year **1649 in science** and technology involved some significant events.

**Johannes****Hudde** was a burgomaster (mayor) of Amsterdam between 1672 – 1703, a mathematician and governor of the Dutch East India Company.

In Euclidean plane geometry, **Apollonius's problem** is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga posed and solved this famous problem in his work Ἐπαφαί ; this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets.

**René-François Walter de Sluse** was a Walloon mathematician and churchman, who served as the canon of Liège and abbot of Amay.

In geometry, the **folium of Descartes** is an algebraic curve defined by the equation

**Derek Thomas** "**Tom**" **Whiteside** FBA was a British historian of mathematics.

In mathematics, **Hudde's rules** are two properties of polynomial roots described by Johann Hudde.

- Boyer, Carl B. (2004) [1956],
*History of Analytic Geometry*, Dover, ISBN 978-0-486-43832-0 - Burton, David M. (2011),
*The History of Mathematics / An Introduction*(7th ed.), McGraw Hill, ISBN 978-0-07-338315-6 - Descartes, René (2006) [1637].
*A discourse on the method of correctly conducting one's reason and seeking truth in the sciences*. Translated by Ian Maclean. Oxford University Press. ISBN 0-19-282514-3.

- Grosholz, Emily (1998). "Chapter 4: Cartesian method and the
*Geometry*". In Georges J. D. Moyal.*René Descartes: critical assessments*. Routledge. ISBN 0-415-02358-0. - Hawking, Stephen W. (2005). "René Descartes".
*God created the integers: the mathematical breakthroughs that changed history*. Running Press. pp. 285*ff*. ISBN 0-7624-1922-9. - Serfati, M. (2005). "Chapter 1: René Descartes, Géométrie, Latin edition (1649), French edition (1637)". In I. Grattan-Guinness; Roger Cooke.
*Landmark writings in Western mathematics 1640-1940*. Elsevier. ISBN 0-444-50871-6. - Smith, David E.; Lantham, M. L. (1954) [1925].
*The Geometry of René Descartes*. Dover Publications. ISBN 0-486-60068-8.

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

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