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.