![Napier's Mirici Logarithmorum table for 19 deg](https://upload.wikimedia.org/wikipedia/commons/7/7c/Napier%27s_Mirici_Logarithmorum_table_for_19_deg.agr.jpg)
The term Napierian logarithm or Naperian logarithm, named after
John Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioann ...
, is often used to mean the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
. Napier did not introduce this ''natural'' logarithmic function, although it is named after him.
However, if it is taken to mean the "
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s" as originally produced by Napier, it is a function given by (in terms of the modern
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
):
:
The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as
:
or
:
In Napier's 1614 ''
Mirifici Logarithmorum Canonis Descriptio
''Mirifici Logarithmorum Canonis Descriptio'' (Description of the Wonderful Canon of Logarithms, 1614) and ''Mirifici Logarithmorum Canonis Constructio'' (Construction of the Wonderful Canon of Logarithms, 1619) are two books in Latin by John N ...
'', he provides tables of logarithms of sines for 0 to 90°, where the values given (columns 3 and 5) are
:
Properties
Napier's "logarithm" is related to the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
by the relation
:
and to the
common logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...
by
:
Note that
:
and
:
Napierian logarithms are essentially natural logarithms with decimal points shifted 7 places rightward and with sign reversed. For instance the logarithmic values
:
:
would have the corresponding Napierian logarithms:
:
:
For further detail, see
history of logarithms
The history of logarithms is the story of a correspondence (in modern terms, a group isomorphism) between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was w ...
.
References
*.
*.
*{{citation
, last = Phillips
, first = George McArtney
, isbn = 978-0-387-95022-8
, page
61 , publisher = Springer-Verlag
, series = CMS Books in Mathematics
, title = Two Millennia of Mathematics: from Archimedes to Gauss
, volume = 6
, year = 2000
, url = https://archive.org/details/twomillenniaofma0000phil/page/61
.
External links
* Denis Roegel (2012
Napier’s Ideal Construction of the Logarithms from the Loria Collection of Mathematical Tables.
Logarithms