## Tartaglia's Wager and Halley's Comment

Charles Groetsch
University of Cincinnati

Our story begins in a taverna in Verone; the year is 1531.  Self-taught mathematician Nicolo Tartaglia is engaged in a friendly, but spirited argument with a gunner in the service of the local duke.  The theoretician Tartaglia holds that a cannon fired at 45 degrees achieves a maximum range.  But the experienced gunner is convinced that the maximum range occurs at some angle less than 45 degrees.  Using elementary calculus and a fair amount of algebra and trigonometry, we settle this old argument, at least for the simplest mathematical model of resistance.  If time permits, we will pick up the historical thread again a century and a half later, and use our result to prove the truth of a cryptic remark of Edmond Halley concerning the range of a slow projectile.