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Ne craignez point, Monsieur, la tortue ('Sir, don't fear the tortoise')

Fishburn and Hughes: From a letter written by Leibniz in January 1692 to Simon Foucher (1644-1696), a philosopher who applied the Cartesian method of doubt in the quest for truth. The letter stresses the need to illustrate the working of all accepted 'axioms' to further the progress of science. In particular Leibniz asserts the axiom that 'nature does not make jumps', from which it follows that all matter is infinitely divisible. With regard to motion, Leibniz agrees with Foucher that all space is infinitely divisible and adds that infinitely divisible space exists in a time which is also infinitely divisible. In the contest with the tortoise, Achilles need not Tear the tortoise': the total time (and total distance) necessary for Achilles to catch up with the tortoise can be expressed as the sum of an infinite geometric progression in which each term is smaller than the previous one. While the number of terms is infinite, because the terms become infinitely small, their sum is a finite quantity. At that point Achilles reaches, and begins to overtake, the tortoise. This provides a mathematical resolution of Zeno's famous paradox. The ideas expanded in this letter reflect Leibniz's earlier work on the infinitesimal calculus. Pierre Menard, Author of the Quixote