Taneesha Iqbal Analytic Geometry

Taneesha Iqbal

Analytic Geometry

Analytic geometry is the study of geometry through the use of a coordinate system. Before this form of geometry was invented, geometry was primarily synthetic, which focused on the properties of figures based on a set of given axioms and construction with only a compass and straightedge. With analytic geometry, the coordinate system allows for a connection between the numeric distinctions of algebraic expressions and the more abstract concepts of synthetic geometry. The most fascinating characteristic of analytic geometry comes from how centuries of mathematicians were very close in discovering this newfound geometry.

One of the earliest indications of analytic geometry is dated back around 350 B.C.E with Greek mathematician Menaechmus. Menaechmus is most well-known for his discover of conic sections, or intersecting a cone with a plane at different angles to produce parabolas, ellipses, and hyperbolas. His discovery would lead him to perform some feats, most notably intersecting a plane multiple times to create circles such that it would look similar to z²=xy and intersect them with a parabola similar to y²=z. Menaechmus would make two mean proportional from the circles and parabola to solve for square roots. However, since no distinct value of y or z was given, Menaechmus solved for all the possible square roots versus a single one. Later that century, Apollonius of Perga formalized Menaechmus’s discovery of conic sections and coined terms for these sections in his publish work Conics, which many future mathematicians will use as reference in discovering analytic geometry and later Calculus. In Conics, Apollonius found a relation of a parabola such that the distance of any point on that conic corresponds to a pair of perpendicular lines using the major axis its conic and its tangent at a point. The distances resemble coordinates and the relation represents the quadratic equation for a parabola. However, Apollonius’s model failed to consider the notion of negative numbers. If this rudimentary model of the Cartesian plane considered the existence of negative coordinates and had distinct number values, then there would have been a greater chance of analytic geometry being discovered about two decades prior.

While Greek mathematicians focused primarily on the geometrical aspect of analytic geometry, Persian mathematician Omar Khayyám focused on the connection between geometry and algebra around the 11^th century. In order to close the gap between these two major mathematical topics, Khayyám would first create a cubic equation using the edge of a cube as a variable. From this cubic equation, Khayyám would create a geometrical construction of curves and solve for solutions to the equation. Khayyám was efficient in discovering a relationship in real numbers and spurred the notion that algebra and geometry were not separate concepts which would, in turn, help spur the discover of analytic geometry in the 16^th century.

A large part of why analytic geometry was not discovered earlier was because algebra was yet to be formalized. Around the 16^th century, French mathematician François Viète introduced the first systematic algebraic notation that did not solely describe an unknown quantity. Instead, Viète used letters as a substitute for constant coefficients and parameters in an algebraic equation. This notion would allow for a more systematic approach in the formal manipulation of algebraic expressions. In other words, future mathematicians would no longer depend on geometric figures or geometric intuition to solve problems. The missing gap in connecting Viète’s algebraic approach with geometry would involve finding a relation between algebra and geometry such that linear equations correspond to lengths, quadratic equations corresponding to squares, cubic equations corresponding to volumes, and equations with higher powers not currently not having a physical form. This gap was closed by the cofounders of analytic geometry, René Descartes and Pierre de Fermat.

In 17^th century France, René Descartes and Pierre de Fermat independently discovered geometry through distinct yet very similar methods. Descartes was most well-known for creating the Cartesian plane. His discovery was made when he was in bed and described the position of a fly on his ceiling in correspondence to his walls. The distance of the fly from the walls would help Descartes create a coordinate system that is most commonly used today. By using his coordinate system, Descartes would create a method in analytic geometry that would involve using a geometric curve to derive an algebraic equation from the curve. This method would lead him to discover that with all the points on the curve, the product of the distances from the points to a certain line is equal to the product of the distances to other lines. Meanwhile, Fermat had the opposite approach in discovering analytic geometry. While Descartes started with a geometric curve, Fermat would start with an algebraic equation. From this equation, Fermat would emphasize the relation of x and y coordinates and used the findings from Apollonius’s Conics to find that any quadratic equation in x and y can be put into the standard form of one of the conic sections. Since Descartes’s method proved to be more complicated with equations that had a degree greater than 3, Fermat’s approach is more widely accepted in today’s use of analytic geometry with modern day primary or secondary school students using an algebraic equation to draw a geometrical figure on the Cartesian plane.

Analytic geometry, along with many other forms of math, is a bit peculiar in that it took centuries of mathematicians to discover a concept that is taught to children. Analytic geometry would eventually be used to form other variations of math, such as calculus, showing that math is continually building off each other. Maybe in the distant future, children will be learning calculus as children now learn about coordinates.

Sources

Bogomolny, Alexander. “What Is Analytic Geometry?” Interactive Mathematics Miscellany and Puzzles, 2017, www.cut-the-knot.org/WhatIs/WhatIsAnalyticGeometry.shtml.

Mardisa, K.V. “Omar Khayyam, Rene Descartes and Solutions to Algebraic Equations.”

Maths.leeds.ac.uk, 19 Nov. 1999, www1.maths.leeds.ac.uk/~sta6kvm/omar.pdf.

O'Connor, J.J, and E.F Robertson. “Menaechmus.” history.mcs.st-Andrews.ac.u, School of Mathematics and Statistics University of St Andrews, Scotland, Apr. 1999, www. history.mcs.st-and.ac.uk/Biographies/Menaechmus.html.

O'Connor, J.J, and E.F Robertson. “René Descartes.” history.mcs.st-Andrews.ac.u, School of Mathematics and Statistics University of St Andrews, Scotland, Nov. 2014, www. history.mcs.st-and.ac.uk/Biographies/descartes.html.

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