Mathematics goes ballistic: Benjamin Robins, Leonhard Euler, and the mathematical education of military engineers.

Efforts to understand the trajectory of cannonballs are an interesting example of the tensions between practical and theoretical knowledge. Although Galileo's 1638 parabolic trajectory was an important theoretical step forward, field gunnery practice was guided by Tartaglia's 1537 'mixed motion' model until the eighteenth century. In 1742, Benjamin Robins published New principles of gunnery, and revolutionized the study of ballistics by suggesting the projectile's initial velocity—not its range—was the appropriate parameter to consider in accounting for air resistance. In 1745, Leonard Euler produced a German translation of New principles, adding his own extensive commentary. Euler's annotated translation quickly became a standard text—Napoleon Bonaparte studied ballistics from the French version—thereby influencing the education of artillery officers and, eventually, of all engineers. This paper surveys the contributions of Robins and Euler to mathematical ballistics theory, examines the influence of these developments on the education of eighteenth-century military engineers, and considers the extent to which the history of ballistics theory supports the thesis that the drive to reconcile practical knowledge with theoretical knowledge can be a critical element in shaping mathematical theory. We close with comments concerning the use of this history in today's classroom. [ABSTRACT FROM AUTHOR]