Mathematical table
Encyclopedia
Before calculator
s 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:
To find the result of 7×8, one looks in the left column to seven, then across the "seven-line" 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.
s were first known to be made by Hipparchus
. Tables of common logarithm
s 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 – so-called standard normal table
s – remains commonplace today, especially in schools.
Mechanical special-purpose computers known as difference engine
s 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 calculator
s, 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.
A major type of mathematical tables are tables containing logarithm
s. Prior to the advent of computer
s and calculator
s, using logarithms meant using such tables, which were mostly created manually. Base-10 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., base-10) 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." 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 function
s.
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:
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...
s 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
Computation
Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...
. Tables of logarithms and trigonometric functions were common in math and science textbooks.
A simple but common example is the multiplication table
Multiplication table
In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....
, which most people know from their early mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
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 "seven-line" 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.
History and use
Tables of trigonometric functionTrigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s were first known to be made by Hipparchus
Hipparchus
Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
. Tables of common logarithm
Common logarithm
The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...
s 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
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
of the normal distribution – so-called standard normal table
Standard normal table
A standard normal table also called the "Unit Normal Table" is a mathematical table for the values of Φ, the cumulative distribution function of the normal distribution....
s – remains commonplace today, especially in schools.
Mechanical special-purpose computers known as difference engine
Difference engine
A difference engine is an automatic, mechanical calculator designed to tabulate polynomial functions. Both logarithmic and trigonometric functions can be approximated by polynomials, so a difference engine can compute many useful sets of numbers.-History:...
s 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
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
in part to produce specialized mathematical tables for aiming artillery
Artillery
Originally applied to any group of infantry primarily armed with projectile weapons, artillery has over time become limited in meaning to refer only to those engines of war that operate by projection of munitions far beyond the range of effect of personal weapons...
. From 1972 onwards, with the launch and growing use of scientific calculator
HP-35
The HP-35 was Hewlett-Packard's first pocket calculator and the world's first scientific pocket calculator . Like some of HP's desktop calculators it used reverse Polish notation. Introduced at US$395, the HP-35 was available from 1972 to 1975.Market studies at the time had shown no market for...
s, most mathematical tables went out of use.
Creating tables is a common code optimization
Optimization (computer science)
In computer science, program optimization or software optimization is the process of modifying a software system to make some aspect of it work more efficiently or use fewer resources...
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
Lookup table
In computer science, a lookup table is a data structure, usually an array or associative array, often used to replace a runtime computation with a simpler array indexing operation. The savings in terms of processing time can be significant, since retrieving a value from memory is often faster than...
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
Space-time tradeoff
In computer science, a space–time or time–memory tradeoff is a situation where the memory use can be reduced at the cost of slower program execution...
required to store the tables.
Tables of logarithms
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s. Prior to the advent of computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
s and calculator
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...
s, using logarithms meant using such tables, which were mostly created manually. Base-10 logarithms are useful in computations when electronic means are not available. See common logarithm
Common logarithm
The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...
for details, including the use of characteristics and mantissas
Significand
The significand is part of a floating-point number, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.-Examples:...
of common (i.e., base-10) logarithms.
In 1617, Henry Briggs
Henry Briggs (mathematician)
Henry Briggs was an English mathematician notable for changing the original logarithms invented by John Napier into common logarithms, which are sometimes known as Briggsian logarithms in his honour....
published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
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
Adriaan Vlacq
Adriaan Vlacq was a Dutch book publisher and author of mathematical tables. Born in Gouda, Vlacq published a table of logarithms from 1 to 100,000 to 10 decimal places in 1628 in his Arithmetica logarithmica. This table extended Henry Briggs' original tables which only covered the values...
, a Dutch
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
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." An edition of Vlacq's work, containing many corrections, was issued at Leipzig
Leipzig
Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...
in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega
Jurij Vega
Baron Jurij Bartolomej Vega was a Slovene mathematician, physicist and artillery officer.-Early life:...
.
François Callet's seven-place table (Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, 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
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
, 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 function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s.
Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony
Gaspard de Prony
Gaspard Clair François Marie Riche de Prony was a French mathematician and engineer, who worked on hydraulics. He was born at Chamelet, Beaujolais, France and died in Asnières-sur-Seine, France.-Education and early works:...
, by an original computation, under the auspices of the French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
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
Cubic function
In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
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:
| Year | Author | Range | Decimal places | Note |
|---|---|---|---|---|
| 1617 | Henry Briggs Henry Briggs (mathematician) Henry Briggs was an English mathematician notable for changing the original logarithms invented by John Napier into common logarithms, which are sometimes known as Briggsian logarithms in his honour.... |
1–1000 | 8 | |
| 1624 | Henry Briggs Arithmetica Logarithmica | 1–20,000, 90,000–100,000 | 14 | |
| 1628 | Adriaan Vlacq Adriaan Vlacq Adriaan Vlacq was a Dutch book publisher and author of mathematical tables. Born in Gouda, Vlacq published a table of logarithms from 1 to 100,000 to 10 decimal places in 1628 in his Arithmetica logarithmica. This table extended Henry Briggs' original tables which only covered the values... |
20,000–90,000 | 10 | contained only 603 errors |
| 1792–94 | Gaspard de Prony Gaspard de Prony Gaspard Clair François Marie Riche de Prony was a French mathematician and engineer, who worked on hydraulics. He was born at Chamelet, Beaujolais, France and died in Asnières-sur-Seine, France.-Education and early works:... Tables du Cadastre |
1–100,000 and 100,000–200,000 | 19 and 24, respectively | "seventeen enormous folios", never published |
| 1794 | Jurij Vega Jurij Vega Baron Jurij Bartolomej Vega was a Slovene mathematician, physicist and artillery officer.-Early life:... Thesaurus Logarithmorum Completus (Leipzig Leipzig Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing... ) |
corrected edition of Vlacq's work | ||
| 1795 | François Callet (Paris Paris Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region... ) |
100,000–108,000 | 7 | |
| 1871 | Sang | 1–200,000 | 7 |
See also
- Table (information)Table (information)A table is a means of arranging data in rows and columns.Production % of goalNorth 4087102%South 4093110% The use of tables is pervasive throughout all communication, research and data analysis. Tables appear in print media, handwritten notes, computer software, architectural...
- Abramowitz and StegunAbramowitz and StegunAbramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards...
Handbook of Mathematical Functions - Jurij VegaJurij VegaBaron Jurij Bartolomej Vega was a Slovene mathematician, physicist and artillery officer.-Early life:...
- MatrixMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
- Random number tableRandom number tableRandom number tables have been used in statistics for tasks such as selected random samples. This was much more effective than manually selecting the random samples...
- A Million Random Digits with 100,000 Normal DeviatesA Million Random Digits with 100,000 Normal DeviatesA Million Random Digits with 100,000 Normal Deviates is a 1955 book by the RAND Corporation. The book, consisting primarily of a random number table, was an important 20th century work in the field of statistics and random numbers...
- A Million Random Digits with 100,000 Normal Deviates
- Truth tableTruth tableA truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
External links
- http://locomat.loria.fr : A census of mathematical and astronomical tables.