 In this section we will begin by showing how Greek mathematics studiously avoided the use of infinity in the presentation of its results by making use of the method of exhaustion (3.1). Then we will look at the widespread use of infinitary objects and procedures in 17th-century mathematics (theory of indivisibles and points at infinity in geometry (3.2), infinitesimals in the calculus (3.3)) and Galileo’s problem of extending counting to infinite collections (3.4). By the early 18th-century mathematics had undergone its first “infinitistic revolution” (the second is associated with the name of Cantor, see section   3). Infinity had become a pressing foundational problem, and this will lead us to section   3.