(B) HISTORY OF ALGEBRA

(xiii) Theory of Equations. Augustin-Louis Cauchy (1789-1857).

Theory of Equations. -- That every numerical equation has a root -- that is, some quantity in a numerical form, real or imaginary, which, when substituted for the unknown quantity in the equation, shall render the equation a numerical identity -- appears to have been taken for granted by all writers down to the time of Lagrange. It is by no means self-evident, nor is it easy to afford evidence for it which shall be at the same time convincing and free from limitations. The demonstrations of Lagrange, Gauss, and Ivory, have for simplicity and completeness given way to that of Cauchy, published first in the Journal de l'Ecole Polytechnique, and subsequently in his Cours d'Analyse Algébrique.

Augustin Cauchy (1789-1857) and some of his mathematical/algebraic work in the background
(French postage stamp issued in 1989)

The demonstration of Cauchy (which had previously been given by Argand, though in an imperfect form, in Gergonne's Annales des Mathématiques, vol. v.) consists in showing that the quantity which it is wished to prove capable of being reduced to zero, can be exhibited as the product of two factors, one of which is incapable of assuming a minimum value, or, in other words, that a less value than one assigned can always be found, and therefore that it is capable of acquiring the value zero. This argument, if not absolutely free from objection, is less objectionable than any of the others. The reader may consult papers by Airy and De Morgan, in the tenth volume of the Transactions of the Cambridge Philosophical Society.





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