In high school, if not sooner, one learns theorems established more than two millenia ago by Euclid and Archimedes. I am thinking of the theorems expressed today by the equations
𝐶 = 2π𝑟,
𝐴 = π𝑟²
for the circumference and area of a circle whose radius is 𝑟, and
𝐴 = 4π𝑟²,
𝑉 = (4/3)π𝑟³
for the surface area and volume of a sphere whose radius is 𝑟. One may also learn about the curves that Apollonius called parabola, ellipse, and hyperbola and that are given today by instances of the general quadratic equation
𝐴𝑥² + 𝐵𝑥𝑦 + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0.
Such notation as this was introduced in the seventeenth century by Descartes, who apparently used it also to understand the curve given by the cubic equation
(𝑦² − 𝑎²)(𝑦 − 𝑏) = 𝑎𝑥𝑦.
Decartes showed that the curve in question could be generated as in the animation below, where a parabola slides along its axis, and a straight line has one point fixed and one point moving with the parabola, and the points of intersection of this straight line with the parabola give us the desired curve.
Polytropy 