Prosthaphaeresis

Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximating products using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally-applicable way of approximating products quickly. Its name comes from the Greek prosthesi and afairo, meaning addition and subtraction, two steps in the process.

Decreasing the error

If all the operations are performed with high precision, the product can be as accurate as desired. Although sums, differences, and averages are easy to compute with high precision, even by hand, trigonometric functions and especially inverse trigonometric functions are not. For this reason, the accuracy of the method depends to a large extent on the accuracy and detail of the trigonometric tables used.

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For example, a sine table with an entry for each degree can be off by as much as 0.0087 if we just choose the closest number; each time we double the size of the table we halve this error. Tables were painstakingly constructed for prosthaphaeresis with values for every second, or 3600th of a degree.

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Inverse sine and cosine functions are particularly troublesome, because they become steep near -1 and 1. One solution is to include more table values in this area. Another is to scale the inputs to numbers between -0.9 and 0.9. For example, 950 would become 0.095 instead of 0.950.

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Another effective approach to enhancing the accuracy is linear interpolation, which chooses a value between two adjacent table values. For example, if we know the sine of 45° is about 0.707 and the sine of 46° is about 0.719, we can estimate the sine of 45.7° as:

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: 0.707 × (1 - 0.7) + 0.719 × 0.7 = 0.7154.

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The actual sine is 0.7157. A table of cosines with only 180 entries combined with linear interpolation is as accurate as a table with about 45000 entries without it. Even a quick estimate of the interpolated value is often much closer than the nearest table value. See lookup table for more details.

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