Quadratic Equations and Astronomy

Astronomy is the study of the universe, which includes the stars and all of the other celestial bodies.  Copernicus proposed that the Earth orbited around the Sun rather than the Sun going around the Earth.  Copernicus said that the orbit of the Earth was a circle, because it is very close to a circle, and that is what others before him said.  Copernicus used a quadratic equation when he proposed that the Earth orbited in a circle around the Sun, and although his model was incorrect, his revolutionary ideas set in motion a chain of events that would eventually produce the greatest revolution in thinking that Western civilization has seen.  Copernicus provided the basis that Johannes Kepler used to calculate that the Earth’s orbit which actually ended up being an ellipse not a circle.  Kepler found some discrepancies between Copernicus’ theories and the predictions that he made from experimental data that he analyzed from the carefully collected observations of his predecessor Tycho Brahe.  Kepler came up with rules for planetary motion, which fit the observations perfectly.  These remarkable discoveries by Kepler helped to usher in the modern world.

Quadratic equations not only described the orbits along which the planets moved round the Sun, but also gave a way to observe them more closely.  A parabola is an ellipse with one focal point and the formula that describes the shape of a parabola is called a quadratic equation.  The fit between the ellipse, described by a quadratic equation, and nature seemed quite remarkable at this time.  It was as though nature said, “Here is a curve that people know about, let’s make some use of it.”  Understanding why this was the right curve had to wait till Galileo and then Newton.  The key to further advances in astronomy only came after the invention of the telescope.  Galileo’s telescope used lenses, the shape of which was formed by two intersecting hyperbolae.  Using his telescope, Galileo was able to observe the moons of Jupiter and the phases of Venus, both of which gave support to the Copernican theories.  Realizing how important quadratic equations were became the link to understanding acceleration.  Displacement is a quadratic function of time when acceleration is constant and Galileo was the first spot this link.

Newton was born in the year that Galileo died and he went on to totally transform the way that we understand science and the role that mathematics plays in scientific predictability.  The reflecting telescope was invented by Newton and it had a mirror for which each cross section takes the shape of a parabola!  Newton was inspired by the work of both Galileo and Kepler.  These scientific giants had accurately described phenomena of dynamics and celestial mechanics, but neither had formulated scientific explanations.  It was left to Newton to provide the mathematical explanation of the phenomena that they observed.  First, he formulated the three laws of motion, which explained Galileo’s observations.  Second, he described the fundamental law of gravitation, which was that two masses were attracted to each other by a force inversely proportional to the square of the distance between them.  By using geometrical arguments he was able to prove that such a law of force implied that the planets had to move around the Sun in a conic section.  Explaining gravity was a significant milestone in the development of astronomy.  Gravity is the force of attraction between all objects that tends to pull them toward one another. It keeps the moon in orbit around the Earth and keeps the Earth and the other planets of the solar system in orbit around the Sun.