strongly resisted by doctors at the time, who were unwilling to wait for tests to determine if current treatments were effective, or discard treatments if they were found ineffective. Gradually Louis' methods gained acceptance as doctors began recognizing the "numerical method" added objectivity to treatment approaches and improved outcomes. The "numerical method" involved the use of averages of groups of patients with the same illness to determine what should be done with individual cases of that illness. As part of the study, Louis emphasized the importance of the similarity of patients beyond mere disease, and attempted to account for factors such as age of patients in different treatment groups, diet, severity of illness and other treatments used beyond bloodletting. Louis also emphasized the importance of population (rather than individual) comparisons in the belief that differences between individual patients would "average out" in the group, though he failed to grasp the importance of randomization to ensure this. Opponents argued that individual cases were too diverse to be combined into statistically average groups; Louis responded by pointing out even individual cases have commonalities, and by claiming each case is unique, no progress could ever be made within medicine. Louis acknowledged his own research included too few cases for absolute certainty in the treatment methods, and his student later stated that once a total of 500 cases had been accumulated, then certainty could be reached. Louis' first study of the topic was of the use of bloodletting in pneumonia, selecting 77 patients with a very similar form of pneumonia. After determining the timing of onset, duration and frequency of death rates of the disease, Louis then analyzed the timing of the bloodletting as either early (1–4 days since the start of the illness) or late (5–9 days). Based on these patients, Louis found that those who were bled early recovered earlier than those bled late, but also died at greater rates. Based on his findings, Louis concluded bloodletting was only a valid use in the late stages of disease. The ultimate impact of Louis' studies on the practice of medicine is hard to evaluate, as the practice of bloodletting was already beginning to decline when he published his results. Little is known of Louis' training in math or medicine, or how he developed his "numerical method". At the time of his practice, the mathematician Pierre-Simon Laplace was extremely influential and had introduced the