Calculus is often called Infinitesimal Calculus, even when it is developed using limits and there are no infinitesimals. The historical confusion between the two formulations is finally untangled. These seven chapters can be covered in seven lectures in a formal course or in seven sessions for self-instruction. The opening chapter "All the Preliminaries" lays the groundwork for the rest of the book. The next chapters. "What is the Problem?" and "Developing a New Intuition", make plain the mathematical obstacles that caused a two- century delay between the invention of calculus and the first rigorous formulation of the subject. The succeeding chapters, "Limits: A First Attempt at Rigor" and "Limits: Rigor That Works" show why the more obvious definition does not work and why the modern definition solves that problem. The final chapters, "Continuity: The Key to Everything" and "Derivatives: Putting It All Together", launch the student into calculus with the most solid foundation possible.