The typical precalculus book contains the obscure trigonometric identities known as the product-to-sum formulas. They usually get short treatment (or none) in a precalculus course because they are so rarely used. This is unfortunate since they have an interesting history. Before the invention of logarithms they were used to perform multiplications and divisions by a process known as prosthaphaeresis. Since the slide rule is a computational device based on logarithms, the authors wondered if a similar device based on prosthaphaeresis could be constructed. It can. They call it the "prosthaphaeretic slide rule".