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Budan took up the study of medicine in Paris and received the title of doctor for a thesis entitled
Essai sur cette question d'économie médicale : Convient-il qu'un malade soit instruit de sa situation?
Budan is considered an amateur mathematician and he is best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have n real roots between two given numbers. The rule was in a memoir sent to the Institute in 1803 but it was not made public until 1807. A paper giving a proof was presented to the Academy in 1811 and published in 1822.
Fourier had independently discovered a rule which he taught in a course from 1797 but Fourier's version was not published until 1831, after his death.
References (4 books/articles)