PROSTHAPHAERETIC SLIDE RULE: A MECHANICAL MULTIPLICATION DEVICE BASED ON TRIGONOMETRIC IDENTITIES, THE

Mathematics and Computer Education, Winter 2004 by Sher, David B, Nataro, Dean C

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

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 0.6 on the horizontal axis.

2) Position the rotor so that its 1 coincides with the slider.

3) Without disturbing the rotor's position, move the slider until it passes through 0.5 on the rotor's scale.

4) The product (0.3) is at the intersection of the slider and the horizontal axis.

As Example 1 shows, any multiplication can be reduced to multiplying a pair of numbers between 0 and 1 and then adjusting the decimal point.

The Examples show how any division problem can be reduced to one of the two procedures above, followed by an adjustment of the decimal point.

THE AUTHORS' EXPERIENCE AND A HISTORICAL QUESTION

The authors and all those acquainted with them know that they should never be allowed near any tools that could cause injury, i.e., any tools at all. Nonetheless, we felt an obligation to build the device anyway (see Figure 4). This we did, using a two foot square piece of plywood for the base and long thin wooden laths for the slider and the rotor. We used a computer to print out the scales (good move!). The pivot for the rotor remained wobbly despite all of our efforts and we had to add a second slider because the rotor would always move when the position of the first slider was changed. The jerrybuilt nature of the sliders caused general amusement in the two talks we have given on this subject. In spite of all this, we consistently achieved two significant figures of accuracy in our calculations.

It is clear that competent woodworkers or metalworkers could do much better by starting with a larger square base and by being . . . well . . . competent. At least four significant figures could be obtained on a consistent basis - an accuracy that is sufficient for many applications. Tycho and others in the prelogarithm age would have saved a great deal of labor by mechanizing their calculations with such a device. Yet, we are unaware of their ever having done so. Neither of the authors is a mathematical historian, but based on our readings of mathematical history we have yet to discover a hint of a computing device such as ours. If anybody knows of the existence of such a device we would be grateful for the information.

REFERENCES

1. Michael Sullivan and Michael Sullivan III, Precalculus: Graphing and Data Analysis, 2nd Edition, by, Prentice-Hall, Upper Saddle River, NJ (2001).