Logarithms turn multiplications into sums: to multiply two numbers, one adds their logarithms and comes back. The Laplace transform does something similar with differential equations: it swaps derivatives for multiplications. One solves an algebraic equation and then comes back. The origin Laplace came to this idea through probability. In the Analytic Theory of Probabilities, of 1812, he organized his calculations with generating functions: instead of working with a sequence of probabilities, he worked with a single function that stored them all. The continuous version of this idea is an integral that Euler had used earlier, but which Laplace explored systematically: