Throughout the four centuries preceding the rise of Tīmūr, science had been actively cultivated in many parts of the Iranian plateau by practitioners who were in the forefronts of their respective disciplines. The dissipation of political power which attended the breakup of the ‘Abbasid empire entailed also a dispersal of the scientists concentrated at Baghdad. But lines of inquiry opened by them were pursued by scholars supported at the various later dynastic centres. During this time trigonometry emerged as an independent branch of mathematics. The foundations of geometry were repeatedly re-examined, and steps were taken towards generalising the concept of number; solutions were given for extensive classes of algebraic equations. Advances were made in mathematical geography, and in astronomy, both in theory and in observation. In optics, a start was made in explaining the phenomenon of the rainbow. It is the object of this chapter to give an account of the extent to which these activities were continued during the century and a half between, say, 1350 and 1500. In point of fact, most of what follows describes the accomplishments of one man, Jamshīd Ghiyās al-Dīn al-Kāshī (d. 832/1429), working at one place, Ulugh Beg's Samarqand observatory, and in one subject, numerical analysis. The names of a number of other scientists at Samarqand are known, and many more can be assembled from other localities, but only the writings of Jamshīd transcend competence and are contributions to knowledge.