History of Logarithms
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History
Predecessors
The Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of squares. However it could not be used for division without an additional table of reciprocals. Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers.
Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table.
In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity
or similar to convert the multiplications to additions and table lookups. However logarithms are more straightforward and require less work. It can be shown using complex numbers that this is basically the same technique.
From Napier to Euler
John Napier (1550–1617), the inventor of logarithms.
The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). Joost Bürgi independently invented logarithms but published six years after Napier. Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to Napier, remarked:
...the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years before Napier's system appeared; but ...instead of rearing up his child for the public benefit he deserted it in the birth.
—Johannes Kepler, Rudolphine Tables (1627)
By repeated subtractions Napier calculated (1 − 10−7)L for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10−5. Napier then
History
Predecessors
The Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of squares. However it could not be used for division without an additional table of reciprocals. Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers.
Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table.
In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity
or similar to convert the multiplications to additions and table lookups. However logarithms are more straightforward and require less work. It can be shown using complex numbers that this is basically the same technique.
From Napier to Euler
John Napier (1550–1617), the inventor of logarithms.
The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). Joost Bürgi independently invented logarithms but published six years after Napier. Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to Napier, remarked:
...the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years before Napier's system appeared; but ...instead of rearing up his child for the public benefit he deserted it in the birth.
—Johannes Kepler, Rudolphine Tables (1627)
By repeated subtractions Napier calculated (1 − 10−7)L for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10−5. Napier then