It is a truism that quantitative reasoning is a powerful tool of modern scientific research. Nevertheless there is apparently no systematic "philosophy of quantitative science" whereby its methodology is formalized or critiqued, and it seems to me that the field of statistics, centered as it is on numerical inference sensu lato, might be a suitable home for developing one. One strategy that I have found to be pedagogically useful studies historical examples of very effective persuasion by numerical arguments -- the rhetoric of numbers as coercive reasons for scientific judgments. For this talk I have selected four famous good examples: Jean Perrin's demonstration (1909) that atoms exist, Watson and Crick's discovery (1953) of the double helix, John Snow's proof (1854) that cholera is waterborne, and the recent (1985-1990) awareness that ordinary stomach ulcers are an infectious disease. Only the first two of these are couched in any quantitative theory, yet all four share a common framework of inference analogous to the Bayes/Laplace sort of "inverse probability," numerical reasoning from effect to cause. From comparisons among these four successful scientific research programmes we can draw interesting and subtle lessons about how numbers work as reasons for scientific judgments when they apparently work best.