This dissertation examines the pedagogical project of Christopher Clavius (1538-1612) as a key step in the development of modern mathematics. In it, I show that Clavius united two contemporary approaches to mathematics: one that saw the field as an abstract way of discovering universal truths, and one that saw the field as an art, that is a tool for practical purposes. To do so, he combined pure and applied mathematics throughout his textbooks. The union of mathematics as a science and mathematics as an art was motivated by the needs of the nascent Jesuit school system in which Clavius was the professor of mathematics at the flagship school, the Collegio Romano. This unification permeated Clavius’s work, leading him to write textbooks on practical mathematics in addition to his commentaries on pure mathematics and theoretical astronomy. Moreover, Clavius combined the different aspects of mathematics within his individual textbooks. This is apparent in his 1574 commentary on Euclid’s Elements, a text that formed the foundation for Jesuit mathematics education. In this textbook, Clavius’s changes to and commentary on the Euclidean text along with his diagrams show the pure abstract forms of mathematics to have potential applications in both sciences, like theoretical astronomy, and arts, like cartography. Through a comparison of Clavius’s commentary on Euclid to two other closely contemporary commentaries on the same text, one by Federico Commandino (1509-1575) and the other by Sir Henry Billingsley (d. 1606), I show that Clavius’s combination of the abstract and physical facets of mathematics created an image of mathematics on par with philosophy as well as a versatile tool for philosophers and artisans alike. This vision of mathematics combines those found in Commandino’s and Billingsley’s commentaries, which respectively emphasize mathematics as a science and mathematics as an art. In so doing, Clavius provided his readers with a realist approach to mathematics, paving the way for increasingly more mathematical descriptions of the world that emerged during the Scientific Revolution and that relied on progressive advances in abstract mathematics.