How does one get this idea [for the proof of Sylvester’s Theorem]? The answer is: I don’t know! It is like asking: How did Michelangelo do this?
Mathematics is not about answers, it’s about the questions you ask.
11 pm // After procrastinating all day, now I truly enjoyed the enlightening moment of realizing that a catenary has the form of cosh(x). Loving the beauty of theoretical mechanics.
Mathematics is the taming of the infinite
“The other planets dutifully followed the laws of planetary motion, but Uranus appeared to violate them. Breaking Kepler’s laws, Uranus moved too quickly for decades, then at the right speed, then too slowly. The observations weren’t easily dismissable, but their physical cause was unknown. An additional planet beyond Uranus, gravitationally tugging on it, offered a potential solution. Determining the mass, orbital parameters, and location of an unseen world presented incredible calculational challenges.”
On March 11, 1811, Urbain Le Verrier was born. As a mathematician of tremendous skill in France, he had only a passing initial interest in astronomy, until the 1840s, when the influential François Arago suggested that he take up the puzzle of Uranus’ orbit, which appeared to violate the laws of planetary motion. Le Verrier theorized that if there were an outer planet beyond Uranus with the right mass and orbital parameters, it could cause these observed orbital anomalies. On August 31, 1846, Le Verrier composed a letter detailing his predictions and sent it to the Berlin Observatory. On September 23, the letter arrived. That very night, the portion of the sky where Le Verrier claimed a new planet should be was clear, and less than one degree away from his location, there it was: the planet Neptune.