HISTORICAL BACKGROUND OF ANALYTIC GEOMETRY - MATHS - 2024

The historical antecedents of analytic geometry go back to the seventeenth century, when Pierre de Fermat and René Descartes defined its fundamental idea. His invention followed the modernization of François Viète's algebra and algebraic notation.

This field has its bases in Ancient Greece, especially in the works of Apollonius and Euclid, who had a great influence in this area of ​​mathematics.

The essential idea behind analytic geometry is that a relationship between two variables, such that one is a function of the other, defines a curve.

This idea was first developed by Pierre de Fermat. Thanks to this essential framework, Isaac Newton and Gottfried Leibniz were able to develop the calculus.

The French philosopher Descartes also discovered an algebraic approach to geometry, apparently on his own. Descartes's work on geometry appears in his famous book Discourse on Method.

This book points out that the compass and straight edge geometric constructions involve addition, subtraction, multiplication, and square roots.

Analytic geometry represents the union of two important traditions in mathematics: geometry as the study of form, and arithmetic and algebra, which have to do with quantity or numbers. Therefore, analytical geometry is the study of the field of geometry using coordinate systems.

History

Background of analytical geometry

The relationship between geometry and algebra has evolved throughout the history of mathematics, although geometry reached an earlier stage of maturity.

For example, the Greek mathematician Euclid was able to organize many results in his classic book The Elements.

But it was the ancient Greek Apollonius of Perga who predicted the development of analytic geometry in his book Conics. He defined a conic as the intersection between a cone and a plane.

Using Euclid's results on similar triangles and secants of circles, he found a relationship given by the distances from any point "P" of a conic to two perpendicular lines, the major axis of a conic, and the tangent at an end point of the axis. Apollonius used this relationship to deduce fundamental properties of the conics.

The subsequent development of coordinate systems in mathematics emerged only after algebra had matured thanks to Islamic and Indian mathematicians.

Until the Renaissance, geometry was used to justify solutions to algebraic problems, but there was not much that algebra could contribute to geometry.

This situation would change with the adoption of a convenient notation for algebraic relations and the development of the concept of a mathematical function, which was now possible.

Century XVI

At the end of the 16th century, the French mathematician François Viète introduced the first systematic algebraic notation, using letters to represent numerical quantities, both known and unknown.

He also developed powerful general methods for working algebraic expressions and solving algebraic equations.

Thanks to this, mathematicians were not completely dependent on geometric figures and geometric intuition to solve problems.

Even some mathematicians began to abandon the standard geometric way of thinking, according to which linear variables of lengths and squares correspond to areas, while cubic variables correspond to volumes.

The first to take this step were the philosopher and mathematician René Descartes, and the lawyer and mathematician Pierre de Fermat.

Foundation of analytical geometry

Descartes and Fermat independently founded analytic geometry during the 1630s, adopting Viète's algebra for the study of locus.

These mathematicians realized that algebra was a powerful tool in geometry and invented what is known today as analytic geometry.

One breakthrough they made was to surpass Viète by using letters to represent distances that are variable rather than fixed.

Descartes used equations to study geometrically defined curves, and stressed the need to consider general algebraic-graphical curves of polynomial equations in degrees "x" and "y".

For his part, Fermat emphasized that any relationship between the coordinates "x" and "y" determines a curve.

Using these ideas, he restructured Apollonius's statements on algebraic terms and restored some of his lost work.

Fermat indicated that any quadratic equation in "x" and "y" can be placed in the standard form of one of the conic sections. Despite this, Fermat never published his work on the subject.

Thanks to their advances, what Archimedes could only solve with great difficulty and for isolated cases, Fermat and Descartes could solve quickly and for a large number of curves (now known as algebraic curves).

But his ideas only gained general acceptance through the efforts of other mathematicians in the latter half of the 17th century.

Mathematicians Frans van Schooten, Florimond de Beaune, and Johan de Witt helped expand Decartes's work and added important additional material.

Influence

In England John Wallis popularized analytic geometry. He used equations to define the conics and derive their properties. Although he used negative coordinates freely, it was Isaac Newton who used two oblique axes to divide the plane into four quadrants.

Newton and the German Gottfried Leibniz revolutionized mathematics at the end of the 17th century by independently demonstrating the power of calculus.

Newton demonstrated the importance of analytical methods in geometry and their role in calculus when he asserted that any cube (or any third-degree algebraic curve) has three or four standard equations for suitable coordinate axes. With the help of Newton himself, the Scottish mathematician John Stirling proved it in 1717.

Analytical geometry of three and more dimensions

Although both Descartes and Fermat suggested using three coordinates to study curves and surfaces in space, three-dimensional analytical geometry developed slowly until 1730.

The mathematicians Euler, Hermann, and Clairaut produced general equations for cylinders, cones, and surfaces of revolution.

For example, Euler used equations for translations in space to transform the general quadratic surface so that its principal axes coincide with its coordinate axes.

Euler, Joseph-Louis Lagrange, and Gaspard Monge made analytic geometry independent of synthetic (non-analytic) geometry.

References

1. The development of analytic geometry (2001). Recovered from encyclopedia.com
2. History of analytic geometry (2015). Recovered from maa.org
3. Analysis (Mathematics). Recovered from britannica.com
4. Analytic geometry. Recovered from britannica.com
5. Descartes and the birth of analytic geometry. Recovered from sciencedirect.com