Peter Anstey has criticized the traditional distinction between empiricism and rationalism. He proposes replacing it with a late seventeenth-century distinction: that between experimental and speculative natural philosophy. But the natural philosophy of Galileo Galilei cuts across this experimental-speculative divide. It resembles speculative natural philosophy insofar as its probative force depends upon a priori, mathematical reasoning. But it also resembles experimental natural philosophy insofar as the principles of such reasoning are tested against experience. To understand Galileo’s scientific reasoning we need to supplement the experimental-speculative distinction with a third category: that of a mathematically-oriented style of early modern natural philosophy.