# The Prosthaphaeretic Slide Rule: A Mechanical Multiplication Device Based on Trigonometric Identities

Sher, David B., Nataro, Dean C., Mathematics and Computer Education

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 more powerful (prosthaphaeresis can't handle exponentiations), prosthaphaeresis quickly became a footnote to mathematical history.

A GEOMETRIC PROOF OF FORMULA 3

Of the many ways to prove (3), the following is the most useful in these circumstances because it suggests clearly how to make the prosthaphaeretic slide rule. It is sufficient for our purposes to consider only the case where A and B are acute angles, A > B, and A + B < 90 degrees. Figure 1, which follows, is of the unit circle in the first quadrant.

This figure shows how to construct the device:

1) Draw the first quadrant of the unit circle on a square surface. On the horizontal axis place a unit scale where O is at the origin and 1 is the radius of the unit circle. Subdivide the scale as finely as possible (tenths, hundredths, etc.)

2) Attach a rotor at the origin which has the same scale on it as the horizontal axis. The zero on this scale must be at the origin.

3) Attach to the square surface a slider that maintains a line perpendicular to the horizontal axis and can be moved left and right (much like the slider on a slide rule) always maintaining its perpendicularity.

The result is as follows:

We can now see that multiplying two numbers between 0 and 1 is easy. Let's do 0.6 × 0.5 as an example.

1) Position the slider so that it passes through 0. …

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