Cavalieri or Simpson? A History of the Familiar Quadrature Rule

Summary: Simpson’s rule, a well-known quadrature method, is widely used due to its simplicity of application. Perhaps for this very reason, unlike other quadrature formulas, it is referred to by various names: Cavalieri-Simpson by Italian mathematicians and Kepler rule by German-speaking mathematicians. In this work, we aim to investigate the contributions of all mathematicians who led to the development of this formula. This will be a journey through the history of mathematics, taking us across various historical periods from Kepler’s era to the mid-1700s, and then concluding at the beginning of the 1900s when the names of Cavalieri and Simpson were officially linked by Peano.