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Lagrange wrote Euler that he believed that he had developed the true metaphysics of the calculus; at that time he seems to have been convinced that the use of infinitesimals was rigorous. Lagrange attempted to prove Taylor's Theorem, the power of which he was the first to observe, and then to develop the entire calculus from it.

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Joseph-Louis Lagrange
Leonhard Euler
Taylor's Theorem
Saturday, November 24, 1759