The Development of Analytic Geometry

Overview

The fundamental idea of analytic geometry, the representation of curved lines by algebraic equations relating two variables, was developed in the seventeenth century by two French scholars, Pierre de Fermat and René Descartes. Their invention followed the modernization of algebra and algebraic notation by François Viète and provided the essential framework for the calculus of Isaac Newton and Gottfried Leibniz. The calculus, in turn, would become an indispensable mathematical tool in the development of physics, astronomy, and engineering over the next two centuries.

Background

The relationship between geometry and algebra has evolved over the history of mathematics. Geometry reached the greater degree of maturity sooner. The Greek mathematician Euclid (335-270 b.c.) was able to organize a great many results in his classic book, the Elements. Algebra was a far less organized body of ideas, drawing on Babylonian, Egyptian, Greek, and Hindu sources, and dealing with problems ranging from commerce to geometry. Until the Renaissance, geometry might be used to justify the solutions to algebraic problems, but there was little thought that algebra would shed light on geometry. This situation would change with the adoption of a convenient notation for algebraic relationships and the development of the concept of mathematical function that it permitted.

To illustrate both the importance of notation and the function concept, we could consider one of the classic problems in algebra, solution of the quadratic equation. In modern notation such an equation would be written Ax² + Bx + C = 0. Here it is understood that A, B, and C represent numbers, x represents the unknown quantity to be found, and the small 2 appearing in the first term means that the unknown x is to be squared or multiplied by itself. While the solutions to some forms of this equation were known to the ancient Babylonians, the notation was not fully developed until the work of the French mathematician François Viète (1540-1602), who standardized the use of letters to represent both constants and variable quantities. Given this notation, it is then an easy thing to think about the equation as having the form f(x) = 0. Where the function f(x) = Ax² + Bx + C. One can then think of a second variable, say y, being defined by the function, y = f(x) = Ax² + Bx + C, so that we have a relation between the two variables x and y that can be studied in itself.

The essential idea behind analytic geometry is that a relation between two variables, such that one is a function of the other, defines a curve. This idea appears to have been first developed by the French lawyer and amateur mathematician Pierre de Fermat (1601-1665). In his book Introduction to Plane and Solid Loci, written in 1629 and circulated among his friends but not published until 1679, he introduced the idea that any equation relating two unknowns defines a locus or curve. Fermat allowed one of the variables to represent a distance along a straight line from a reference point. The second variable then denoted the distance from the line. Fermat went on to derive equations for a number of simple curves including the straight line, the ellipse, the hyperbola, and the circle. Since Fermat did not consider negative distances, he could not display the full curves, but other mathematicians would soon overcome this problem.

The French philosopher René Descartes (1596-1667) also discovered an algebraic approach to geometry, apparently independently. Descartes was one of the dominant intellectual figures of the seventeenth century, best known as a philosopher, the author of several important physical theories, and a major contributor to mathematics. Descartes's work on geometry appears as one of the three appendices to his famous book Discourse on the Method of Rightly Conducting Reason and Reaching the Truth in the Sciences. The other two appendices are on optics and meteorology. As the title suggests, Descartes saw mathematics primarily as a route to sure knowledge in the sciences.

In his appendix on geometry, Descartes began by pointing out that the compass and straight-edge constructions of geometry involve adding, subtracting, multiplying, dividing, and taking square roots. He proposed assigning a letter to represent the length of each of the lines appearing in a construction, and then writing equations relating the lengths of the lines, obtaining as many equations as there are unknown lines. Finding the unknown lengths then becomes a matter of solving the set of equations thus obtained.

After thus showing that algebra can be applied in solving classic geometric problems, Descartes then discussed solving problems that have curves as their solution. In this type of problem there are not enough equations to determine all the unknown quantities, and one ends up with a relation between two unknowns. It is at this point that Descartes suggested using the length away from a fixed point on a given line to represent x, and the distance from x on a line drawn in a fixed direction to represent y. If the fixed direction is chosen at a right angle to the first line, we obtain the modern system of rectangular or Cartesian coordinates, named after Descartes.

Descartes then proposed that any equation involving powers of x and y describes an acceptable geometrical curve and showed that those special curves known as conic sections—the circle, ellipse, hyperbola, and parabola—are all described by algebraic equations in which the highest power of x or y is two. The study of such curves was gaining in importance as a result of discoveries in physics and astronomy, particularly the discovery by the German scientist Johannes Kepler (1571-1630) that the planets move not in perfect circles or combinations of perfect circles but in ellipses. Further, Kepler had shown the planets do not move at constant speed but at a speed that varies with their distance from the Sun. Analytic geometry provided a useful description of the shape of such orbits. An explanation of the actual motion would soon follow as Isaac Newton (1642-1727) proposed his laws of motion and universal gravitation and developed the techniques of the calculus to apply them.

The two basic problems of the calculus are easily expressed in terms of analytic geometry. The first is finding the tangent line to the curve described by y = f(x) at any point, and the second is finding the area between a segment of the curve and the line y = 0. Solving these problems leads directly to the solution of two others: finding the values of x for which y = f(x) is a minimum or maximum, and finding the length of a segment of a curve. Fermat had solved the problem of finding tangents and the associated problem of finding maxima and minima by 1629. When Descartes' Geometry appeared in 1637, Fermat criticized it for not including a discussion of tangents or maxima and minima. Descartes replied that these results could be readily obtained by anyone who understood his work, and that Fermat's work showed far less understanding of geometry than his own. The dispute over the importance and priority of Fermat's and Descartes' contributions would eventually subside, with each man acknowledging the other's contributions.

The full development of the calculus would be achieved by Newton and German scientist Gottfried Wilhelm Leibniz (1646-1716), working independently of each other. As in the case of Fermat and Descartes, a dispute as to priority broke out, but in this case a more bitter and long lasting one.

In 1696 the Swiss mathematician Johann Bernoulli (1667-1748) published a problem in applied calculus as a challenge to other mathematicians. The problem, known as the brachistochrone, is to find the curve along which a sliding bead will move from one point to another in the least time with gravity as the only external force. The answer is an upside-down version of the cycloid, the curve generated by a point on the circumference of a wheel as it rolls along a level surface. In 1697 Bernoulli was able to publish his own solution along with those obtained from four other mathematicians, Newton and Leibniz among them. Newton's solution had been submitted anonymously, but this did not fool Bernoulli, who said, "I recognize the lion by his claw."

Impact

Analytic geometry represents the joining of two important traditions in mathematics, that of geometry as the study of shape or form and that of arithmetic and algebra, which deal with quantity or number. This combination was needed if the physical sciences were to progress beyond the notions of Aristotelian philosophy about perfect and imperfect motions to a natural philosophy based on observation and experiment. It is not surprising then that both Fermat and Descartes were concerned with the scientific issues of their day, both with optics in particular, and Descartes more generally with all areas of physics and astronomy.

The techniques of calculus, built on the insights of analytic geometry, have become the fundamental mathematics of the physical sciences and engineering. With the addition of differential equations, which represent a further development of the basic ideas of calculus and analytic geometry, the mathematical framework has proven robust enough to incorporate the new areas of thermal physics and electromagnetism in the nineteenth century and quantum theory in the twentieth. Thus the modern university curriculum for future scientists and engineers always includes several semesters devoted to analytic geometry and calculus.

DONALD R. FRANCESCHETTI