Mathematical Table

A mathematical table usually presents the values of some function y = f (x₁,..., x_n) for certain values of the independent variables. Examples of such tables include the multiplication tables memorized in grade school (y = x₁x₂, where x₁, x₂ = 1, 2,..., 9), tables of trigonometric functions, and tables of logarithms. Mathematical tables are important aids to computation and are used wherever calculations must be performed—for example, in mathematics, physics, chemistry, astronomy, engineering, and economics.

If the variables x₁,..., x_n are continuous quantities, a table for y = f(x₁,..., xn) gives the value of y only at certain values of the arguments—that is, the table gives the values y₁,..., y_N corresponding to the argument values (x₁,..., x_n)₁,..., (x₁,..., x_n)_N. The values of the function for unrecorded values of the arguments must be found by interpolation. Every mathematical table can be characterized by the following: its accuracy, that is, the number of significant digits or decimal places to which the values of the function are computed; the range over which the values of the arguments are taken; and the interval between adjacent argument values in the table.

The construction of a table of values of the function y = f(x₁,..., x_n) involves two basic tasks: the designing of the table and the computation of the values of f(x₁,..., x_n). In designing the table, the range of the arguments x₁,..., x_n must be chosen, and the argument values must be selected for which the table will give values of the function. Other matters that must be considered in designing the table include the way in which the material is to be arrayed and the question of whether already existing tables are to be made use of. The computations involved in making a table are not of a special nature. What is specific to the construction of a table is the need for careful checking of a large amount of numerical data, both during the computation process and in proofreading.

In designing a table, it must be decided how the desired number of values y₁,..., y_N are to be arrayed within the available space. This problem must be resolved in such a way as to make it as easy as possible to determine the value of the function f(x₁,..., x_n) for values of (x₁,..., x_n), including values not entered in the table. The range of the arguments is determined both by practical considerations and by the ease with which the function can be evaluated outside this range to the accuracy attained in the table. The interval between argument entries is selected so that interpolation of an acceptable order yields the desired number of significant digits or decimal places. Frequently used tables usually permit only linear interpolation. Special-purpose tables may permit interpolation of a higher order; interpolation of order greater than two, however, is undesirable and is rarely encountered. Auxiliary quantities that may be required for interpolation, such as differences, are usually included in the table. An important means of obtaining a smoother function, and thereby simplifying the construction of the table, is the substitution of arguments and the replacement of the original function by another function that is related to the original function by a simple equation. Through this technique, for example, interpolation may be simplified, or the number of table entries for values of the function may be decreased.

The first mathematical tables appeared in the earliest period of the development of mathematics. In Babylonia, for example, as long ago as 2000 B.C., tables of products of natural numbers and tables of such quantities as 1/n, n², n³, and n³ + n² were widely used. The tables were employed for a variety of calculations and permitted Babylonian mathematicians to solve rather complicated computational problems.

The first tables of transcendental functions appeared in ancient Greece as a result of the development of astronomy and the accumulation of a vast amount of observational data requiring mathematical processing. The Almagest of the second-century Greek mathematician Ptolemy contains the oldest extant trigonometric table. Ptolemy’s table gives the lengths of chords of a circle corresponding to arcs from 0° to 180°, at intervals of one-half degree. The chord lengths are expressed in fractions of the radius in the sexagesimal system. Interpolation is provided for by the inclusion of differences in the table.

Tables of trigonometric functions were prepared by Hindu mathematicians and mathematicians of the Near East and Middle Asia between the fifth and 11th centuries. Abu al-Wafa in the tenth century constructed tables of sines for every 10 minutes of arc with an accuracy of 1:60⁴; he also computed a table of tangents.

In Europe, the first large tables were constructed in the 15th century. The development of the natural sciences in the Renaissance stimulated European mathematicians and astronomers to make increasingly complete and accurate tables of trigonometric functions. The 15th-century astronomer and mathematician Re-giomontanus was the first to employ the decimal number system in tables. He computed a table of sines to seven places for every minute of arc. N. Copernicus also undertook the construction of trigonometric tables. In the first book of his De revolutionibus or-bium coelestium (On the Revolutions of Heavenly Spheres), which was published in 1543, he gave a five-place table of sines. Copernicus’ student Rhäticus began the computation of 15-place trigonometric tables for every 10 seconds of arc and for every second in the first and last degrees of the quadrant. The German mathematician B. Pitiscus expanded and completed the tables and published them in 1613. These tables became the basis for modern trigonometric tables.

The first table of logarithms of numbers was published by J. Napier in 1614. A table of antilogarithms was published in 1620 by the Swiss mathematician J. Bürgi. In 1617 the English mathematician H. Briggs published the first table of common logarithms; it gave the logarithms to eight places for numbers from 1 to 1,000. In 1624 he published a 14-place table of logarithms for numbers from 1 to 20,000 and from 90,000 to 100,000. The first tables of logarithms of numbers were quickly followed by tables of logarithms of trigonometric functions.

In 1633 the Dutch mathematician A. Vlacq published ten-place tables of log sin x and log tan x for every 10 seconds of the quadrant with differences. Briggs, also in 1633, published tables giving the natural sines to 15 places, tangents and secants to ten places, log sin x to 14 places, and log tan x to ten places at intervals of 0.01° from 0° to 45°.

The number of published tables sharply increased with the development of science, commerce, and navigation. Many more mathematical tables were published in the 18th century than in the 17th century. The 19th century saw not only an increase in the number of tables produced but also a considerable increase in the classes of functions covered. Special functions came to assume an important role in applications of mathematics. Tables of, for example, elliptic, hyperbolic, gamma, and cylindrical functions appeared. Such important mathematicians as L. Euler, A. Legendre, and K. Gauss assisted in the computation of mathematical tables.

Several times as many mathematical tables have been computed and published in the 20th century as in the entire preceding period. A particularly great number of tables have been prepared for various special functions. Some tables have been computed to an accuracy as great as 15–30 significant digits or decimal places. The preparation of tables has been closely associated with the development of computer technology. The photoduplication of computer-produced tables has practically eliminated the possibility of error.

Much work is being carried out in the USSR on the preparation of mathematical tables. In addition to individual publications, series of tables are being issued by the Institute of Mathematics of the Academy of Sciences of the USSR, by the Institute of Precision Mechanics and Computer Technology of the Academy of Sciences of the USSR, and by the Computer Center of the Academy of Sciences of the USSR. As the number of published tables increases, the efficient use of the tables and the planning of further work in this area require a systematization of tabulated data and a detailed description of existing tables.