Kepler's Impossible Equation

Johannes Kepler developed this equation in 1605 after discovering that planets sweep out equal areas in equal times on elliptical orbits. Astronomers use the equation to predict planetary positions at future times.

Johannes Kepler discovered that planets sweep out equal areas in equal times, establishing his second law of planetary motion. This law applies to all bodies in elliptical orbits around a central gravitational focus.

Johannes Kepler published the Rudolphine Tables in 1627 after years of computation using his iterative method to solve his unsolvable equation. These tables became the standard reference for astronomical predictions.

Johannes Kepler developed a simple iterative algorithm to estimate planetary positions when he could not solve his equation algebraically. This method enabled him to create the Rudolphine Tables published in 1627.

Johannes Kepler discovered that planets follow elliptical orbits around the Sun, replacing the centuries-old assumption of perfect circular motion. Mercury has the highest eccentricity among the known planets at 0.21.

Isaac Newton developed this method around 1670, making a significant improvement to Kepler’s iterative approach. Newton used geometric constructions to estimate curve slopes, creating a method now applied to numerous problems beyond astronomy.

Joseph Louis Lagrange developed this theorem in the late 1700s while attempting to solve Kepler’s equation and similar problems. His work created a method to generate Taylor series for inverse functions.

Pierre-Simon Laplace computed this limit in his massive five-volume treatise Mécanique Céleste published beginning in 1798. Laplace was aware that Lagrange’s inversion did not always work but did not rigorously justify his claim until Cauchy developed convergence theory.

Augustin-Louis Cauchy developed this elegant theory in the early 1800s, leading the formalization and unification of calculus. In 1831, he presented his paper on series convergence at the French Academy of Sciences, where Laplace reportedly ran home to check his own work for divergent series.

Frederick Wilhelm Bessel developed these special functions in the decades after Lagrange applied his inversion theorem to Kepler’s equation. Bessel took a fresh mathematical approach to the orbital mechanics problem.