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Pure mathematics may sometimes be thought of as the abstract art form of the scientific evidence revealing the nature of our universe before we can even observe its truth.

For example, did you know that Neptune is the only planet in our solar system whose existence was predicted by pure mathematics and not telescopic sight?

In 1846, French astronomer, mathematically predicted that a large undiscovered planet was lurking beyond the orbit of Uranus. He was able to determine not only where it was located but also how much mass it had.

However, it was the German astronomer, , in the same year, at the Berlin Observatory, who found the elusive planet through a telescope by following Le Verrier’s observations and instructions.

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  (published on April 26, 2020 by Columbus Astronomical Society)

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  (published on May 22, 2011 by Marshall Ellenstein)

Paul Hewitt explains, in a humorous way, about the discovery of the planet Neptune.

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  (published on August 4, 2018 by Miki P)

This video illustrates the mathematical theory, calculations and mathematical modelling behind the prediction.

“The Discovery of Neptune in 1846 is regarded as one of the most legendary moments in the consolidation of classical mechanics. Discovering a distant world in the darkness of space by the use of mathematical reasoning became one of the late triumphs of the Enlightenment and Newton's vision. I used Rebound (a Python library for orbital mechanics) to show how Neptune perturbs Uranus's motion through 550 years. These perturbations were studied by Urbain Le Verrier and others to infer the existence of Neptune with high accuracy by the use of Newton's laws of motion and universal gravitation.”

The predictive powers of abstract mathematics …continued

In another example John Clerk Maxwell, a late fellow of Trinity College, Cambridge University, in 1859, published predicting the physical composition of Saturn’s rings using mathematical equations.

In 1855, Cambridge University had published an open competition to describe the rings of Saturn as being either completely solid rock or ice; entirely fluid; or made up of lots of individual, independently moving, particles that circled Saturn. The stipulation of the competition was that the description had be mathematically justified. As the rings were about 1.45 billion kilometers away, proving which hypothesis was the closest to reality was considered to be very difficult, if not impossible, to do at the time.

For a glimpse into the world of mathematical abstraction, see “” (published September 15, 2016 by SciShow).

SOME TANTALIZING EVIDENCE as to the scientific validity (using empirical data) of Maxwell’s mathematical proof came some 120 years later. On September 1, 1979, Pioneer 11 (launched in April 1973) became the first spacecraft to visit Saturn. The spacecraft passed within 21,000 km of the planet’s atmosphere.

After Pioneer 11 sent back photos and dates from its visit to Saturn, the rings were described as jet streams swirling around the equator of the planet at 300 miles per hour. During the Voyager 1 flyby in November 1980 (launched in September 1977), the rings appeared to break up into hundreds of small rings and each of the narrow rings seemed to be made of narrower structures. A deeper probe into the nature of Saturn’s rings was undertaken during the Voyager 2 flyby in August 1981 (launched in August 1977). It was discovered that what appeared to be hundreds of rings could better be described as waves in a sheet of icy particles.

(published September 17, 2013 by Muon Ray)

The DRAMATIC SCIENTIFIC PROOF of Maxwell’s mathematical conclusions back in 1859 came almost a century and a half later during the spacecraft Cassini-Huygens thirteen-year mission orbiting within the Saturn system from July 1, 2004 through September 15, 2017 UTC.

WATER VAPOUR GEYSERS found on ENCELADUS, one of Saturn’s moons

Enceladus is the sixth-largest moon of Saturn. It orbits within the second-outermost ring of Saturn, called the E ring, and was discovered in 1789 by William Herschel, a British-German astronomer who also discovered the planet Uranus within the sun’s solar system.

It is now known that the outermost E ring of Saturn consists of icy particles whose primary source of material is captured from the water vapour geysers emitting icy particles off the surface of Enceladus near its south pole. Data sent back from the Cassini space mission has revealed that Enceladus has a ocean between its rocky core and its icy crust.

For more on the discoveries of the Cassini mission, watch the NASA-held talks and news conferences from the agency’s Jet Propulsion Laboratory, in Pasadena, California.

“” – a von Kármán Lecture Series talk, original air date September 22, 2016 (published by the NASA Jet Propulsion Laboratory)

“” (published on September 14, 2017 by NASA)

On September 15, 2017, Cassini began its final entry into Saturn's atmosphere after its almost 20-year mission in space.

“” (published on September 15, 2017 by the NASA Jet Propulsion Laboratory). Please allow enough time for the recorded, originally live, broadcast to load.