Xu Guangqi (1562–1633), Li Zhizao (1565–1630), and Yang Tingyun (1557–1627) — together known as the “Three Pillars” of Catholicism in late Ming China — promoted the Western mathematics brought to China by the Jesuits as superior to contemporary Chinese mathematics. Xu, the most prominent among them, in a preface for their purported “translation” of a European mathematical treatise, denounced Chinese mathematics as “tattered sandals” to be “discarded,” because Western mathematics was, he claimed, in every way superior. I argue that they did not themselves believe these claims, for they purloined what is arguably the most difficult mathematics in their “translation”—fangcheng, or what we would now call linear algebra—from the very Chinese mathematical treatises Xu execrated. I trace the mathematical practices behind these fangcheng problems, reconstructing these practices as solved on two-dimensional counting boards. I show how adepts were able to compute solutions to difficult problems involving n conditions in n unknowns with only counting rods and an understanding of simple two-dimensional patterns. I present evidence that specialized fangcheng problems with solutions so arcane that they can serve as “fingerprints” circulated across the Eurasian continent, and can be found in Italian texts from the thirteenth century. The advent of the Jesuits in China was thus hardly the “first encounter” of China and the West. Instead, world history of science must trace the global circulation of scientific practices.