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Vulcan was proposed to explain a small perturbation in Mercury's orbit from the path predicted by classical mechanics, technically called perihelion precession.
During Mercury's orbit, its perihelion advances by a small amount each orbit. The phenomenon is predicted by classical mechanics, but the observed value differed from the predicted value by the small amount of 43 arcseconds per century.
This idea and the name "Vulcan" was postulated by the French mathematician Urbain Le Verrier in 1859, closely following his spectacular success in "discovering" the planet Neptune in the same way — using only calculus. Various persons and astronomers around the world attempted to prove the existence of the said planet.