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. Specialized tables were published for applications such as astronomy, celestial navigation and statistics.
A simple example
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:
- sin (75° 10′) = 0.9666746
- sin (75° 9′) = 0.9666001
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 :
- sin (75° 9′ 50″) ≈ sin (75° 9′) + 0.0000621 = 0.9666001 + 0.0000621 = 0.9666622
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).
History and use
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.
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 of 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.
Tables of logarithms
Tables containing common logarithms (base-10) were extensively used in computations prior to the advent of computers and calculators because logarithms convert problems of multiplication and division into much easier addition and subtraction problems. Base-10 logarithms have an additional property that is unique and useful: The common logarithm of numbers greater than one all have the same fractional part, known as the mantissa. Tables of common logarithms typically included only the mantissas; the integer part of the logarithm, know as the characteristic, could easily be determined by counting digits in the original number.
The fractional part of the common logarithm of numbers greater than zero but less than one is just 1 minus the mantissa of the same number with the decimal point shifted to the right of the first non-zero digit. But same mantissa could be (and was) used for numbers less than one by offsettng the characteristic. Thus a single table of common logarithms can be used for the entire range of positive decimal numbers. See common logarithm for details on the use of characteristics and mantissas.
The method of logarithms was publicly propounded by John Napier in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). The book contained fifty-seven pages of explanatory matter and ninety pages of tables related to natural logarithms. The English mathematician Henry Briggs visited Napier in 1615, and proposed a re-scaling of Napier's logarithms to form what is now known as the common or base-10 logarithms. Napier delegated to Briggs the computation of a revised table, and they later published, in 1617, Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a table for the first 1000 integers calculated to the 14th decimal place.
In 1624 his Arithmetica Logarithmica, appeared in folio, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places (1-20,000 and 90,001 to 100,000). This table was later extended by Adriaan Vlacq, but to 10 places, and by Alexander John Thompson to 20 places in 1952.
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." 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 seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place 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 seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.
Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions. Briggs completed a table of logarithmic sines and logarithmic tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places, and the tangents and secants for the same to ten places; all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica. Tables logarithms of trigonometric functions simplify hand calculations where a function of an angle must be multiplied by another number, as is often the case.
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 twenty-four 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."  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 multi-volume editions have been compiled:
|1617||Henry Briggs, Logarithmorum Chilias Prima||1–1000||14||see image|
|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|
|1792–94||Gaspard de Prony Tables du Cadastre||1–100,000 and 100,000–200,000||19 and 24, respectively||"seventeen enormous folios", never published|
|1794||Jurij Vega Thesaurus Logarithmorum Completus (Leipzig)||corrected edition of Vlacq's work|
|1795||François Callet (Paris)||100,000–108,000||7|
The computational advance available via common logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker.
- Abramowitz and Stegun Handbook of Mathematical Functions
- Difference engine
- Group table
- History of logarithms
- Nautical almanac
- Multiplication table
- Random number table
- Table (information)
- Truth table
- Jurij Vega
- The longitude of Philadelphia City Hall
- Abramowitz and Stegun Handbook of Mathematical Functions, Introduction §4
- J J O'Connor and E F Robertson (June 1996). "The trigonometric functions". Retrieved 4 March 2010.
- E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913).
- Stifelio, Michaele (1544), Arithmetica Integra, London: Iohan Petreium
- Bukhshtab, A.A.; Pechaev, V.I. (2001), "Arithmetic", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Vivian Shaw Groza and Susanne M. Shelley (1972), Precalculus mathematics, New York: Holt, Rinehart and Winston, p. 182, ISBN 978-0-03-077670-0
- Ernest William Hobson (1914), John Napier and the invention of logarithms, 1614, Cambridge: The University Press
- Bruce, I. (2002). "The Agony and the Ecstasy: The Development of Logarithms by Henry Briggs". The Mathematical Gazette 86 (506): 216–227. doi:10.2307/3621843. JSTOR 3621843.
- "The Difference Method of Henry Briggs". Retrieved 2012-04-24.
- Athenaeum, 15 June 1872. See also the Monthly Notices of the Royal Astronomical Society for May 1872.
- English Cyclopaedia, Biography, Vol. IV., article "Prony."
- Roy, A. E. (2004), Orbital Motion (4th ed.), CRC Press, p. 236, ISBN 9781420056884,
In G. Darwin's day, logarithm tables came in different sizes
- "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 Glaisher, in Monthly Notices of the Royal Astronomical Society for May 1872, pp255-262.
- Campbell-Kelly, Martin (2003), The history of mathematical tables: from Sumer to spreadsheets, Oxford scholarship online, Oxford University Press, ISBN 978-0-19-850841-0
- http://locomat.loria.fr : A census of mathematical and astronomical tables.
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