places. The first two are the easiest because they each only require two tables. Using the second formula, however, has the unique advantage that if only a cosine table is available, it can be used to estimate inverse cosines by searching for the angle with the nearest cosine value. Notice how similar the above algorithm is to the process for multiplying using logarithms, which follows these steps: scale down, take logarithms, add, take inverse logarithm, scale up. It is no surprise that the originators of logarithms had used prosthaphaeresis. Indeed the two are closely related mathematically. In modern terms, prosthaphaeresis can be viewed as relying on the logarithm of complex numbers, in particular on Euler's formula 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. For example, a sine table with an entry for each degree can be off by as much as 0.0087 if we just round an angle off to the nearest degree; each time we double the size of the table (for example, by giving entries for every half-degree instead of every degree) we halve this error. Tables were painstakingly constructed for prosthaphaeresis with values for every second, or 3600th of a degree. 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. Another effective approach to enhancing the accuracy is linear interpolation, which chooses a value between two adjacent table values. For example, if we know that 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 0.707 × (1 − 0.7) + 0.719 × 0.7 = 0.7154. 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 entries without it. Even a quick estimate of the interpolated value is often much