ABSTRACT
This chapter extends the idea of covariation to rates of change—a central idea in calculus. When Newton thought about instantaneous rates of change, his methods were much more intuitive than methods students are taught in school today. His concepts of “fluents” and “fluxions” leverage the idea of covariation, without the need for a formal definition of derivative. However, other mathematicians mocked those concepts as “ghosts of departed quantities” because they lacked a rigorous theoretical foundation until the invention of nonstandard analysis three hundred years later. In the meantime, mathematicians developed formal definitions of limit and continuity, which constitute the standard analysis used in most calculus classes today. Although Newton’s calculus functioned fine without either foundation, his methods foreshadow those foundations while suggesting ways they might connect to students’ intuitive ways of operating.
