Abstract. Leibniz and Newton made caviler use of infinitesimals in their writings on calculus in the 1600s. At this time many mathematicians viewed infinitesimals as problematic. One reason for this was that an infinitesimal quantity dx is often treated as being equal to zero, while at the same time one often divides by it in calculating derivatives dy/dx. This led mathematicians and philosophers to reject infinitesimals as "ghosts of departed quantities." About 150 years after Leibniz and Newton, Cauchy and Weierstrass developed a precise foundation for calculus which completely avoided infinitesimals. In a remarkable triumph of formal mathematics, in the 1960s Abraham Robinson showed that the modern techniques of mathematical logic could be use to construct a rigorous framework in which infinitesimal and infinite quantities can be treated in a precise way, justifying Leibniz's and Newton's intuition 260 years later. I will describe Robinson's construction of the hyperreals using the machinery of ultrapowers from mathematical logic.