# The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler - Logarithms of Negative Numbers

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Euler's paper "On the controversy between Messrs. Leibniz and Bernoulli concerning logarithms of negative and imaginary numbers" [9] is the first publication in which the riddle of logarithms of negative numbers is solved. An English translation of this article by Stacy Langton is available by clicking here. Although d'Alembert, like Johann (I) Bernoulli before him, believed that $\ln{\left(-x\right)} = \ln{x}$ for positive real numbers $x,$ Euler showed that $\ln{\left(-x\right)} = \ln{x}+i(2n+1)\pi$ for any integer $n.$ Euler and d'Alembert had debated this at great length in their early correspondence, and although Euler was eventually able to persuade his correspondent that the logarithm is a complex-valued multi-function, d'Alembert never abandoned his assertion that $\ln{x}$ was a value of $\ln{\left(-x\right)}.$