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What is the earliest formal equation of magnetism?

The history of discovery of magnetism can trace back to Thales of Miletus. Although many people had discovered, they did not formalize an equation.

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As with the previous question on electricity it is ambiguous as to what constitutes "formal equation" or "magnetism". Light is an electromagnetic wave, and geometric optics became mathematical in antiquity, Descartes even introduced wave optics in luminiferous ether in 1638. Peregrinus discovered that lodestones align needles along lines of longitude between poles on the stone back in 1269, which one can turn into an "equation" if one so wishes. Gilbert's theory of magnetism appeared in 1600 and was qualitative, but many of his observations can also be written as "equations", as can be Descartes's later on.

Perhaps the first explicit mathematical law is mentioned in Newton's Principia:"power of magnetism... in receding from the magnet, decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations". The law was confirmed by the publishers of Principia's 1742 edition Le Seur and Jacquier. In 1750 Michell clarified that in a magnet "each pole attracts or repels exactly equally" and "attraction and repulsion of magnets increases, as the squares of the distances from the respective poles increase". See Whittaker's History of the Theories of Aether and Electricity

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