Toccata
Augustin-Louis Cauchy was born in Paris as the French Revolution began to stir. Politics would flavour the rest of his life as his strongly held views swung him in and out of favour with the succession of authorities who took over France in the wake of the revolution. One colleague went so far as to declare that Cauchy was "mad and there is nothing that can be done about him." Nonetheless, nobody denied his mathematical genius. Cauchy published more papers than any mathematician but Leonhard Euler, covering nearly every topic in mathematics and applied mathematics.
Two of his most famous results are vector inequalities. Vectors can be most simply imagined1 as straight arrows which have particular lengths and directions. I could, for instance, describe an arrow-vector as "2 centimetres up". The sum of vectors can be found by laying the vectors to be added up "head-to-tail" and drawing a new vector from the starting point to the end point. This new vector is the sum of the
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