Mathematical Tables - A Simple Example

A Simple Example

To compute the sine function of 75 degrees, 9 minutes, 50 seconds using a table of trigonometric functions such as the Bernegger table from 1619 illustrated here, one might simply round up to 75 degrees, 10 minutes and then find the 10 minute entry on the 75 degree page, shown above-right, which is 0.9666746.

However, this answer is only accurate to four decimal places. If one wanted greater accuracy, one could interpolate linearly as follows:

From the Bernegger table:

sin (75° 10′) = 0.9666746

sin (75° 9′) = 0.9666001

The difference between these values is 0.0000745.

Since there are 60 seconds in a minute of arc, we multiply the difference by 50/60 to get a correction of (50/60)*0.0000745 ≈ 0.0000621; and then add that correction to sin (75° 9′) to get :

sin (75° 9′ 50″) ≈ sin (75° 9′) + 0.0000621 = 0.9666001 + 0.0000621 = 0.9666622

A modern calculator gives sin (75° 9′ 50″) = 0.96666219991, so our interpolated answer is accurate to the 7-digit precision of the Bernegger table.

For tables with greater precision (more digits per value), higher order interpolation may be needed to get full accuracy. In the era before electronic computers, interpolating table data in this manner was the only practical way to get high accuracy values of mathematical functions needed for applications such as navigation, astronomy and surveying.