Editing Mathematical table

{{Short description|List of values of a mathematical function}}
{{multiple image
 | width = 150
 | footer = Facing pages from a 1619 book of mathematical tables by [[Matthias Bernegger]], showing values for the sine, tangent and secant [[trigonometric function]]s. Angles less than 45° are found on the left page, angles greater than 45° on the right. Cosine, cotangent and cosecant are found by using the entry on the opposite page.
 | image1 = Bernegger Manuale 136.jpg
 | alt1 = An old book opened to columns of numbers labeled sinus, tangens and secans
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 | image2 = Bernegger Manuale 137.jpg
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 }}
'''Mathematical tables''' are lists of numbers showing the results of a calculation with varying arguments. [[Trigonometric tables]] were used in ancient Greece and India for applications to [[astronomy]] and [[celestial navigation]], and continued to be widely used until [[electronic calculator]]s became cheap and plentiful in the 1970s, in order to simplify and drastically speed up [[computation]]. Tables of [[logarithm]]s and [[trigonometric function]]s were common in math and science textbooks, and specialized tables were published for numerous applications.

==History and use==
The first tables of [[trigonometric functions]] known to be made were by [[Hipparchus]] (c.190  &ndash; c.120 BCE) and [[Menelaus of Alexandria|Menelaus]] (c.70–140 CE), but both have been lost. Along with the [[Ptolemy's table of chords|surviving table of Ptolemy]] (c. 90 – c.168 CE), they were all tables of chords and not of half-chords, that is, the [[sine]] function.<ref name="mcs"/> The [[Āryabhaṭa's sine table|table produced by the Indian mathematician Āryabhaṭa]] (476–550 CE) is considered the first sine table ever constructed.<ref name="mcs">{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html |title=The trigonometric functions|last=J J O'Connor and E F Robertson|date=June 1996 |access-date=4 March 2010}}</ref> 
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 [[Madhava series|power series expansions]] of the sine and cosine functions by [[Madhava of Sangamagrama]] (c.1350 – c.1425), and the tabulation of a [[Madhava's sine table|sine table by Madhava]] with values accurate to seven or eight decimal places.

[[Image:Four-Place Mathematical Tables cover.jpg|thumb|right|upright=0.60|These mathematical tables from 1925 were distributed by the [[College Entrance Examination Board]] to students taking the mathematics portions of the tests]]

Tables of [[common logarithm]]s were used until the invention of computers and electronic calculators to do rapid multiplications, divisions, and exponentiations, including the extraction of ''n''th roots.

Mechanical special-purpose computers known as [[difference engine]]s were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions &ndash; that is, to compute large logarithmic tables. This was motivated mainly by errors in logarithmic tables made by the [[human computer]]s 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 [[HP-35|scientific calculator]]s, 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 the [[United States]] in 1938 as a project of the Works Progress Administration (WPA),  employing 450 out-of-work clerks to tabulate higher mathematical functions.  It lasted through World War II.<ref>{{cite journal|last=Grier|first=David Alan|title=The Math Tables Project of the Work Projects Administration: The Reluctant Start of the Computing Era|journal=IEEE Ann. Hist. Comput.|year=1998|volume=20|issue=3|pages=33–50|doi=10.1109/85.707573|issn=1058-6180}}</ref>

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 table]]s – remains commonplace today, especially in schools, although the use of [[scientific calculator|scientific]] and [[graphing calculator]]s as well as [[spreadsheet]] and dedicated statistical software on personal computers is making such tables redundant.

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 [[Lookup table|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 [[Space–time tradeoff|trades computing speed for the computer memory space]] required to store the tables.

==Trigonometric tables==
{{main|Trigonometric tables}}

Trigonometric calculations played an important role in the early study of astronomy.  Early tables were constructed by repeatedly applying [[trigonometric identities]] (like the half-angle and angle-sum identities) to compute new values from old ones.  

===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 above, 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.<ref>[[Abramowitz and Stegun]] Handbook of Mathematical Functions, Introduction §4</ref> 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 (geography)|meridian]] (indeed, any [[great circle]]) equals one [[nautical mile]] (approximately {{convert|1.852|km|mi|disp=or|abbr=on}}).

==Tables of logarithms==
[[File:Logarithmorum Chilias Prima page 0-67.jpg|thumb|A page from [[Henry Briggs (mathematician)|Henry Briggs]]' 1617 ''Logarithmorum Chilias Prima'' showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places.]]
[[File:Abramowitz&Stegun.page97.agr.jpg|thumb|Part  of a 20th-century table of [[common logarithm]]s in the reference book [[Abramowitz and Stegun]].]]
[[File:APN2002-table3-30deg.tiff|thumb|A page from a table of logarithms of [[trigonometric function]]s from the 2002 [[American Practical Navigator]]. Columns of differences are included to aid [[interpolation]].]]
Tables containing [[common logarithm]]s (base-10) were extensively used in computations prior to the advent of electronic calculators and computers 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 that differ only by a factor of a power of ten all have the same fractional part, known as the ''mantissa''. Tables of common logarithms typically included only the [[significand|mantissas]]; the integer part of the logarithm, known as the ''characteristic'', could easily be determined by counting digits in the original number.  A similar principle allows for the quick calculation of logarithms of positive numbers less than 1.  Thus a single table of common logarithms can be used for the entire range of positive decimal numbers.<ref>E. R. Hedrick, [https://archive.org/details/logarithmictrigo00hedriala Logarithmic and Trigonometric Tables] (Macmillan, New York, 1913).</ref> See [[common logarithm]] for details on the use of characteristics and mantissas.

===History===
{{main|History of logarithms}}
In 1544, [[Michael Stifel]] published ''Arithmetica integra'', which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table.<ref>{{Citation|first=Michaele|last=Stifelio|publisher=Iohan Petreium|location=London|year=1544|title=Arithmetica Integra|url = https://books.google.com/books?id=fndPsRv08R0C&pg=RA1-PT419}}</ref><ref>
{{springer  | title=Arithmetic  | id= A/a013260 | last=Bukhshtab  | first=A.A.   | last2=Pechaev | first2=V.I.}}</ref><ref>
{{Citation|title = Precalculus mathematics|author = Vivian Shaw Groza and Susanne M. Shelley|publisher = Holt, Rinehart and Winston|location=New York|year=1972|isbn=978-0-03-077670-0|page = 182|url = https://books.google.com/books?id=yM_lSq1eJv8C&q=stifel&pg=PA182}}</ref>

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'').<ref>{{Citation|author=Ernest William Hobson|title=John Napier and the invention of logarithms, 1614|year=1914|publisher=The University Press|location=Cambridge|url=https://archive.org/details/johnnapierinvent00hobsiala}}</ref> The book contained fifty-seven pages of explanatory matter and ninety pages of tables related to [[natural logarithms]]. The  English mathematician [[Henry Briggs (mathematician)|Henry Briggs]] visited Napier in 1615, and proposed a re-scaling of [[Napier's logarithm]]s to form what is now known as the [[common logarithm|common]] or base-10 logarithms. Napier delegated to Briggs the computation of a revised table.  In 1617, they published ''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. Prior to Napier's invention, there had been other techniques of similar scopes, such as the use of tables of progressions, extensively developed by [[Jost Bürgi]] around 1600.<ref name="folkerts">{{citation | last1 = Folkerts | first1 = Menso | last2 = Launert | first2 = Dieter | last3 = Thom | first3 = Andreas | arxiv = 1510.03180 | doi = 10.1016/j.hm.2016.03.001 | issue = 2 | journal = [[Historia Mathematica]] | mr = 3489006 | pages = 133–147 | title = Jost Bürgi's method for calculating sines | volume = 43 | year = 2016| s2cid = 119326088 }}</ref><ref>{{mactutor|id=Burgi|title=Jost Bürgi (1552 – 1632)}}</ref>

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.

==See also==
* [[Abramowitz and Stegun]] ''Handbook of Mathematical Functions''
*[[BINAS]], a Dutch science handbook
* [[Difference engine]]
* [[Ephemeris]]
* [[Group table]]
* [[Handbook]]
* [[History of logarithms]]
* [[Nautical almanac]]
* [[matrix (mathematics)|Matrix]]
* [[MAOL table book|MAOL]], a Finnish handbook for science
* [[Multiplication table]]
* [[Numerical analysis]]
* [[Random number table]]
*:[[A Million Random Digits with 100,000 Normal Deviates]]
* [[Ready reckoner]]
*[[Reference work|Reference book]]
*[[Rubber Book|Rubber book]] ''Handbook of Chemistry & Physics''
* [[Standard normal table]]
* [[Table (information)]]
* [[Truth table]]
* [[Jurij Vega]]

==References==
<references/>

* {{Citation | last1=Campbell-Kelly | first1=Martin | title=The history of mathematical tables: from Sumer to spreadsheets|title-link= The History of Mathematical Tables | publisher=[[Oxford University Press]] | series=Oxford scholarship online | isbn=978-0-19-850841-0 | year=2003}}

== External links ==
* {{Cite EB1911|wstitle=Table, Mathematical}}
* [http://locomat.loria.fr LOCOMAT] : A census of mathematical and astronomical tables.
{{Commons category|Mathematical tables}}

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