History of Calculus

What new concepts were brought by Lagrange, and then Cauchy, to the calculus and what did they reject about previous approaches. In what ways did this lead to a more rigorous approach to the calculus? Were there any remaining gaps?

The eighteenth century saw many developments of the calculus introduced by both Newton and Leibniz. Nonetheless, there were mutters of discontent from a few regarding the very foundations of both Newton’s and Leibniz’ interpretations of calculus. Berkeley is thought to have made the most significant criticisms regarding their methods with great validity [K]. Many mathematicians sought to consolidate the theories of calculus but it wasn’t until Lagrange in the late eighteenth century and subsequently Cauchy in the nineteenth century that the most serious of criticisms were adequately addressed.

The chief critic of the apparent “absurdity” in the methods of calculus was Berkeley. He objected firstly to Newton’s Method of Fluxions based on the use of a non-zero increment in calculations only for it to be then treated as being of zero value; a practice he considered contradictory. He went on to attack the Method of Differentials favoured by the continental mathematicians. Berkeley struggled, in particular, to conceive the notion of infinitesimals that was generally accepted. Maclaurin in Britain and Euler on the continent attempted to quell the doubts put forward by Berkeley, but, in retaining the key concepts of fluxions and infinitesimals were unsuccessful in ridding the calculus of the ambiguity it suffered.

Lagrange removed all reference to all aspects of calculus he believed to lack proper definitions: this included infinitesimals, fluxions, zeros and even limits [K].

Bibliography:
J. F. Scott, A History of Mathematics, Taylor & Francis LTD (1960)
V. J. Katz, A History of Mathematics: Brief Edition, Pearson (2004)
D. E. Smith, History of Mathematics: Volume 1, Dover Publications Inc (1958)
D. E. Smith, History of... [continues]

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