MATHEMATICAL TABLES. The primary purpose of mathematical tables is to render the work of the professional computer in mathematics, engineering, astronomy, statistics, etc., less laborious than it would otherwise be. The arrangement and typography must be such that the minimum strain is imposed on the computer's eyes, for he may be called upon to use the table for hours at a stretch. The results tabulated are the "tabular re sults"; and the corresponding numbers, by which the table is en tered, are the "arguments." A table is one of single or double entry, according as it has one or two arguments. A table of logarithms of numbers is a table of single entry, the numbers being the "arguments" and the logarithms the "tabular results"; a simple multiplication table is one of double entry, giving the product xy as the "tabular result" corresponding to the "arguments" x and y.
The invention of logarithms in 1614 came as a great boon to computers (astronomers particularly), for it made calculations involving multiplications comparatively easy work. Since that time the majority of tables of special functions were, until quite recently, published giving the logarithmic instead of the natural values, but owing to the increasing utility of calculating machines, there is now a tendency to publish the natural values.
Common or Briggsian Logarithms of Numbers.—This system of logarithms is used for most practical purposes. The fundamental work which contains the results of the original cal culations is that of Briggs's Arithmetica Logarithmica (London, 1624) ; it gives the logarithms of the integers 1 to 20,000 and 90,00o to ioo,000 to 14 decimal places with interscript differences.
Briggs intended to publish the logarithms of the numbers 20,000 90,00o to 14 places, but before he completed this part of the table he was forestalled by De Decker in his Tweede Deel der Nieuwe Telkonst (Gouda, 1627) and Vlacq in his so-called Editio Secunda of Briggs's Arithmetica (Gouda, 1628), who gave the io—decimal logarithms of integers I.--Ioo,000 with differences. The tables of De Decker and Vlacq are identical, for the men were really partners in the speculation. For the majority of succeeding tables of logarithms of numbers, either the tables of Briggs or De Decker-Vlacq have been the sources, directly or indirectly. Very few recalculations have been made and for nearly 30o years the De Decker-Vlacq table, with its errors corrected, was the best ro place table of the logarithms of numbers. In 1794 Vega published a reprint of Vlacq's table; this so-place table, of which the ar rangement is not so good as Vlacq's, is very useful and is still in general use. Although Vega bestowed great care on the detec tion of errors, there are a number of last figure errors. The title is Thesaurus Logarithmorum Completus (Leipzig). Three photo graphic reprints have been published, two at Florence by the Istituto Geografico Militare in 1889, 1896, and the third by Stechert of New York in 1923. In the last one the reproduc tion is very poor and all the errors of the original appear. Duf field's up-place table (Washington, 1897) cannot be trusted, for, although he claims to have made a recalculation, practically all Vega's last figure errors appear. Peters in Zehnstellige Loga rithmentafel: Erster Band. (Berlin, 1922) gives the io-place logarithms of all numbers to ioo,000 with first differences and an auxiliary table, which shows corrections for second differences. The table is the result of a new calculation.