This chapter shows that since his early days with Beeckman, Descartes had pursued a set of projects related to physico-mathematics, but far outstripping even it in potential scope and invested hopes. From 1618, Descartes had entertained an analytical, problem-solving oriented agenda in mathematics, which in these respects resembled his physico-mathematics, or so he thought. The parallels he perceived between his mathematical and physico-mathematical work triggered, in 1619–1620, his dream of a unified analytical approach to all mathematically based disciplines—practical, pure and physico-mathematical. To this project he appropriated the already circulating name "universal mathematics". However, that overheated conception quickly gave way to the even more encompassing mirage of a universal method, which remained with him from 1619 to the late 1620s, when, after his optical breakthrough, he picked up universal mathematics and method again in detail. These compounding enlargements of his original mathematical and physico-mathematical agendas are traced in this Chapter.