 When Descartes’ mathematical researches commenced in the early seventeenth century, mathematicians were wrestling with questions concerning the appropriate methods for geometrical proof and, in particular, the criteria for identifying curves that met the exact and rigorous standards of geometry and that could thus be used in geometrical problem-solving. These issues were given an added sense of urgency for practicing mathematicians when, in 1588, Commandino’s Latin translation of Pappus’s Collection (early fourth century CE) was published. In the Collection Pappus appeals to the ancient practice of geometry as he offers normative claims about how geometrical problems ought to be solved. Early modern readers gave special attention to Pappus’s proposals concerning (1) how a mathematician should construct the curves used in geometrical proof, and (2) how a geometer should apply the methods of analysis and synthesis in geometrical problem-solving. The construction of curves will be treated in 1.1 and analysis and synthesis in section 1.2 below.