Log table

Mathematical tables are lists of numbers showing the results of calculation with varying arguments, before calculators were cheap and plentiful, people would use such tables to simplify and drastically speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics.

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:

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 :

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.

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 (indeed, any great circle) equals approximately one nautical mile (1.852 km or 1.151 mi).

The first tables of trigonometric functions known to be made were by Hipparchus (c.190 – c.120 BCE) and Menelaus (c.70–140 CE), but both have been lost. Along with the surviving table of Ptolemy (c. 90 – c.168 CE), they were all tables of chords and not of half-chords, i.e. the sine function. The table produced by the Indian mathematician Āryabhaṭa is considered the first sine table ever constructed. Āryabhaṭa's table remained the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table, culminating in the discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama (c.1350 – c.1425), and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.

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