At the outset of an influenza pandemic, early estimates of the number of secondary cases generated by a primary influenza case (reproduction number, R) and its associated uncertainty can help determine the intensity of interventions necessary for control. Using a compartmental model and hospital notification data of the first two waves of the Spanish flu pandemic in Geneva, Switzerland in 1918, we estimate the reproduction number from the early phase of the pandemic waves. For the spring and fall pandemic waves, we estimate reproduction numbers of 1:57 (95% CI: 1:45, 1:70) and 3:10 (2:81, 3:39), respectively, from the initial epidemic phase comprising the first 10 epidemic days of the corresponding wave. Estimates of the variance of our point estimates of R were computed via a parametric bootstrap. We compare these estimates with others obtained using different observation windows to provide insight into how early into an epidemic the reproduction number can be estimated.