Mathematical Tables - History and Use

History and Use

The first tables of trigonometric functions known to be made were by Hipparchus (c.190 BC – c.120 BC) and Menelaus (c.70–140 CE), but both have been lost. Along with the surviving table of Ptolemy (c.AD 90 – c.168), 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 as 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.

Tables of common logarithms were used until the invention of computers and electronic calculators to do rapid multiplications, divisions, and exponentiations, including the extraction of nth roots.

Mechanical special-purpose computers known as difference engines were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions – i.e. to compute large logarithmic tables. This was motivated mainly by errors in logarithmic tables made by the human 'computers' of the time. Early digital computers were developed during World War II in part to produce specialized mathematical tables for aiming artillery. From 1972 onwards, with the launch and growing use of scientific calculators, most mathematical tables went out of use.

One on the last major efforts to construct such tables was the Mathematical Tables Project that was started in 1938 as a project of the Works Progress Administration (WPA), employing 450 out-of-work clerks to tabulate higher mathematical functions, and lasted through World War II.

Tables of special functions are still used; for example, the use of tables of values of the cumulative distribution function of the normal distribution – so-called standard normal tables – remains commonplace today, especially in schools.

Creating tables stored in random access memory is a common code optimization technique in computer programming, where the use of such tables speeds up calculations in those cases where a table lookup is faster than the corresponding calculations (particularly if the computer in question doesn't have a hardware implementation of the calculations). In essence, one trades computing speed for the computer memory space required to store the tables.