During the early modern period, “mathematics” was generally understood to mean the study of number and magnitude, or of quantity in general. There were two varieties: “pure” mathematics and, using a term that became common around 1600, “mixed mathematics.” The former studied number and magnitude in abstraction, whereas the latter studied them in composite occurrence; that is, linked to (mostly material) objects. By 1700, mixed mathematics was extensive indeed: In the German philosopher Christian Wolff’s (1679–1754) paradigmatic Elementa matheseos universae (Elements of All Mathematics, 3rd ed., 1733–42), it comprised mechanics, statics, hydro-statics, pressure in air and fluids, optics, perspective, spherical geometry, astronomy, geography, hydrography, chronology, sundials, explosives, and architecture, both military and civil. By 1500, most of these fields were small if they existed at all; the rapid expansion of “mixed mathematics” is a characteristic feature of the early modern period. Compared with the mixed variety, pure mathematics had fewer domains. Wolff summarized it under the headings arithmetic, geometry, plane trigonometry, analysis of finite quantities (i.e., letter algebra and analytic geometry), and analysis of infinite quantities (i.e., differential and integral calculus); the last two were created in the seventeenth century.

In this chapter, we follow this early modern demarcation of pure mathematics; when using the term “mathematics,” unless explicitly indicated otherwise, we refer to pure mathematics so defined. The demarcation was in terms of the subject matter; it did not correspond to professional dividing lines. Few if any scholars identified themselves exclusively as pure mathematicians. Yet the principal stimuli for development in early modern pure mathematics were internal to its own traditions, stemming from classical and medieval pure mathematics.