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INTRODUCTION
The typical precalculus book contains the obscure trigonometric identities known as the product-to-sum formulas [1, p. 470]. 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. We call it the "prosthaphaeretic slide rule".
SOME HISTORY
Formula 3 is attributed to the Arab mathematician ibn-Yunus (d. 1008). [2, p. 264]. All three formulas were known in Europe by the Sixteenth Century, [see 3, pp. 456-462 for fragments on this subject by Clavius and Pitiscus] By the end of that century they were used extensively at Tycho Brahe's observatory for astronomical calculations. The level of accuracy was great because excellent trigonometric tables of up to fifteen decimal places existed at that time. [2, p. 340]
The similarity between prosthaphaeresis and logarithms is striking. This is no coincidence. It is believed that Napier learned of prosthaphaeresis from a friend who had visited Tycho's observatory in 1590. [2, p. 342] Napier saw that exponents also have interesting product to sum properties and, thus inspired, began his great work on logarithms. Since logarithms are easier to use and...