The Calculus Diaries - Jennifer Ouellette [10]
Sometime in the 1620s, Fermat first encountered a work of Apollonius of Perga called Plane Loci, exploring two-dimensional curves. Fermat set about proving (in the rigorous, mathematical sense) some of his ancient colleague’s results. He discovered that geometric “statements” of the ancient Greeks could also be rendered algebraically—essentially translating them into x’s, y’s, and the other accoutrements of symbolic equations.
Any geometric object—a square, a triangle, a curved line—can be represented by an equation. These are the formulae we all had to memorize in geometry class: a circle is y 2 + x2 = a 2, for example, while Archimedes’ killer parabola is y = ax2. Points on a graph are noted as sets of numbers inside parentheses (x, y) representing a point in space. The x indicates how far along the horizontal axis a point is located from a point of origin (0). The y does the same on the vertical axis. If you generate enough points from the equation and connect the dots, you end up with a curve. The more points you plot on your Cartesian grid, the smoother the resulting curve will be.
We know these as Cartesian coordinates because the introverted Fermat procrastinated on polishing his work into a publishable format; his ideas didn’t appear until 1637, with the publication of Introduction to Plane and Solid Loci. That same year, Descartes covered much of the same ground in a separate treatise entitled simply, Geometry. Born in 1596, Descartes lost his mother to tuberculosis when he was barely one year old. His father, a member of his provincial parliament, trusted his son’s education in philosophy and mathematics to the Jesuit priests at a college in La Flèche. But after earning a degree in law in 1616, Descartes “abandoned the study of letters,” opting instead to travel the world to gain as varied experience as possible.
Yet he retained an interest in philosophy and mathematics, and actively pursued knowledge in both. One day, the story goes, he lay on his bed watching a fly buzzing through the air. Descartes realized that its position at any moment could be described by three numbers representing its distance along each of three intersecting, mutually perpendicular axes (corresponding to the lines formed by the intersection of the room’s walls in a corner). This insight formed the basis of the Cartesian coordinate system. Descartes—along with Fermat—used this coordinate system to turn figures and shapes into equations and numbers.
While both Fermat and Descartes independently conceived of the underlying notion of translating between curves and algebraic expressions, people liked Descartes’ treatise a bit better, mostly because his notation was easier to use. But Fermat is the one who realized that it worked both ways: He could also turn an equation into a graph, and work with the resulting curve to glean insights that might not be readily apparent from simply studying the abstract algebra.
Most notably, Fermat realized that converting the expression into geometry made it easier to find the largest and smallest value within a given range—the maximum and minimum, as we call them today. At any point on a curve, it is possible to draw a straight line that just touches it at exactly that point, called the tangent. You simply study the line that is tangent to the curve at the point of interest and determine its slope.
If the slope of that tangent line is positive (slanting upward from left to right), the expression is increasing; if negative (slanting downward), the expression is decreasing. The steeper the slope, the faster the expression is increasing or decreasing. Where are the maxima and minima? Wherever the slope of the tangent line flattens out to zero (becomes horizontal) along that curve. Like Archimedes before him, who stopped just short of inventing integral calculus, Fermat came within a hair’s breadth of inventing differential