Editing Mathematical table (section)

===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 above, 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.<ref>[[Abramowitz and Stegun]] Handbook of Mathematical Functions, Introduction §4</ref> 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.

To understand the importance of accuracy in applications like navigation note that at [[sea level]] one minute of arc along the Earth's [[equator]] or a [[Meridian (geography)|meridian]] (indeed, any [[great circle]]) equals one [[nautical mile]] (approximately {{convert|1.852|km|mi|disp=or|abbr=on}}).