The present inquiry concerns ‘affective governmentality’ and is guided by the following question: How did affects become intelligible objects of knowledge and what enabled a scientific conception of affect to turn into a distinctive government of affect? In answering this question, the article first outlines how a lineage of thinkers used the speculative tools of geometry to conceptualize and deduce human affects. Through an analysis of Spinoza’s Political Treatise, the article then investigates how this geometric conception of affect enabled a productive and indirect government of affect. The article’s contribution to the study of affective governmentality is twofold: First, it advances the methodological claim that the proper register to study affects in governmentality is anatomo-political rather than biopolitical. Second, the analysis of the Political Treatise exemplifies, nuances and substantiates our understanding of how desires and affects were reconceptualized geometrically to conceive a government of bodies and souls.