When looking up at the night sky with the naked eye, it is possible to spot five planets easily: Mercury, Venus, Mars, Jupiter, and Saturn (six if you consider the one you are standing on). In 1781, an astronomer named William Hershel was observing with his telescope and noticed a greenish object which had been previously catalogued as a star by numerous other people.
He started mapping its movement and, soon, it was clear that it was, in fact, a planet orbiting far out in our solar system, beyond Jupiter and Saturn. This newly discovered planet was named Georgium Sidus. This was later changed to Uranus to keep the terminology of Roman gods.
It is actually possible to see Uranus slightly without a telescope, but the sky has to be exceptionally dark, and you need to have very sharp eyesight. The other Ice Giant, Neptune, is impossible to see without a telescope; therefore it was a lot harder to discover.
After observing Uranus for a few decades’, astronomers noticed that the orbit was different compared to the predicted calculations. A French mathematician named Urbain Le Verrier concluded that this was due to the gravitational influence of another very massive planet which had yet to be discovered.
He later did the math to find where this mysterious planet was. This had never been done before because, for the first time, the location of the object was unknown; therefore the calculations were utterly different to usual. In the past, scientists had to determine the gravitational pull of an object based on the location of the other planets.
This time, it was the other way around therefore the unknown variable was different making the orbital mechanics’ calculations a lot more complicated. In 1846 Le Verrier sent his calculations with the exact position where to find the new mysterious planet to the Berlin Observatory.
Here, astronomer Johann Galle read the letter and immediately went to use the telescope and found the planet on the very same night. The calculations were extremely accurate. Neptune had been observed for the first time.
Le Verrier had beaten the competition by only two days. Another mathematician, called John Couch Adams, had made very similar predictions. Unfortunately for him, he had just been too slow.
Neptune was then also observed over time, and it was noticed that, just like Uranus, it was not in the predicted position in its orbit. Based on what had happened in the past, a ninth massive planet was predicted.
This led to a treasure hunt which resulted in the discovery of Pluto which was nowhere near the expected mass; therefore, the discovery of Pluto was an accident, making Neptune the only planet to have been discovered using mathematics.
In 1989, the Voyager 2 spacecraft flew past Neptune and Uranus. It turned out that the mass of the planets had been miscalculated from earth. When the new masses were used in the orbital mechanics’ calculations, the two planets were precisely where they were supposed to be therefore solving a mystery that had kept scientists awake for more than a century.