 Mathematical table

Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks.
A simple but common example is the multiplication table, which most people know from their early mathematics classes:
× 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5 6 7 8 9 10 11 12 2 2 4 6 8 10 12 14 16 18 20 22 24 3 3 6 9 12 15 18 21 24 27 30 33 36 4 4 8 12 16 20 24 28 32 36 40 44 48 5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 7 7 14 21 28 35 42 49 56 63 70 77 84 8 8 16 24 32 40 48 56 64 72 80 88 96 9 9 18 27 36 45 54 63 72 81 90 99 108 10 10 20 30 40 50 60 70 80 90 100 110 120 11 11 22 33 44 55 66 77 88 99 110 121 132 12 12 24 36 48 60 72 84 96 108 120 132 144
To find the result of 7×8, one looks in the left column to seven, then across the "sevenline" to eight. The easily found answer is 56. To find 9×3, one would swap the factors and find the equal product 3×9 (27) by the same technique.Contents
History and use
Tables of trigonometric functions were first known to be made by Hipparchus. Tables of common logarithms and antilogarithms were used to do rapid multiplications, divisions, and exponentiations, including the extraction of nth roots. Tables of special functions are still used. For example, the use of tables of values of the cumulative distribution function of the normal distribution – socalled standard normal tables – remains commonplace today, especially in schools.
Mechanical specialpurpose 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.
Creating tables is a common code optimization technique, and works as well for computers as humans. In computers, use of such tables is done in order to speed 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.
Tables of logarithms
A major type of mathematical tables are tables containing logarithms. Prior to the advent of computers and calculators, using logarithms meant using such tables, which were mostly created manually. Base10 logarithms are useful in computations when electronic means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base10) logarithms.
In 1617, Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adriaan Vlacq, a Dutch mathematician; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals.
Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error."^{[1]} An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega.
François Callet's sevenplace table (Paris, 1795), instead of stopping at 100,000, gave the eightplace logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in sevenplace tables. The only important published extension of Vlacq's table was made by Mr. Sang in 1871, whose table contained the sevenplace logarithms of all numbers below 200,000.
Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.
Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the French republican government of the 1790s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twentyfour places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792; and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." ^{[2]} Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.
For different needs, logarithm tables ranging from small handbooks to multivolume editions have been compiled:^{[citation needed]}
Year Author Range Decimal places Note 1617 Henry Briggs 1–1000 8 1624 Henry Briggs Arithmetica Logarithmica 1–20,000, 90,000–100,000 14 1628 Adriaan Vlacq 20,000–90,000 10 contained only 603 errors^{[3]} 1792–94 Gaspard de Prony Tables du Cadastre 1–100,000 and 100,000–200,000 19 and 24, respectively "seventeen enormous folios",^{[4]} never published 1794 Jurij Vega Thesaurus Logarithmorum Completus (Leipzig) corrected edition of Vlacq's work 1795 François Callet (Paris) 100,000–108,000 7 1871 Sang 1–200,000 7 See also
 Table (information)
 Abramowitz and Stegun Handbook of Mathematical Functions
 Jurij Vega
 Matrix
 Random number table
 Truth table
References
 ^ Athenaeum, 15 June 1872. See also the Monthly Notices of the Royal Astronomical Society for May 1872.
 ^ English Cyclopaedia, Biography, Vol. IV., article "Prony."
 ^ "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error.", Athenaeum, 15 June 1872. See also the Monthly Notices of the Royal Astronomical Society for May 1872.
 ^ English Cyclopaedia, Biography, Vol. IV., article "Prony."
 CampbellKelly, Martin (2003), The history of mathematical tables: from Sumer to spreadsheets, Oxford scholarship online, Oxford University Press, ISBN 9780198508410
External links
 http://locomat.loria.fr : A census of mathematical and astronomical tables.
Categories: Mathematical tools
 Mathematical tables
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