Bürgi's Kunstweg is a set of
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s invented by
Jost Bürgi
Jost Bürgi (also ''Joost, Jobst''; Latinisation of names, Latinized surname ''Burgius'' or ''Byrgius''; 28 February 1552 – 31 January 1632), active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, a maker of astronomica ...
at the end of the 16th century.
[Staudacher, S., 2014. Jost Bürgi, Kepler und der Kaiser. Verlag NZZ, Zürich.] They can be used for the calculation of
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
s to an arbitrary precision. Bürgi used these algorithms to calculate a
Canon Sinuum, a table of sines in steps of 2
arc seconds
A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of Angular unit, angular measurement equal to of one Degree (angle), degree. Since one degree is of a turn (geometry), turn (or complete rotat ...
. It is thought that this table had 8 sexagesimal places. Some authors have speculated that this table only covered the range from 0 to 45 degrees, but nothing seems to support this claim. Such tables were extremely important for
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
at sea.
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
called the Canon Sinuum the most precise known table of sines.
[Max Caspar, ''Johannes Kepler Gesammelte Werke'', Band XVIII, page 149-150, Letter from Kepler to Landgraf Philipp von Hessen, december 1623]
Bayerische Akademie der Wissenschaften
/ref> Bürgi explained his algorithms in his work Fundamentum Astronomiae
''Fundamentum Astronomiae'' is a historic manuscript presented by Jost Bürgi to Emperor Rudolf II in 1592. It describes Bürgi's trigonometry based algorithms called Kunstweg which can be used to calculate sines at arbitrary precision.Staudacher, ...
which he presented to Emperor Rudolf II. in 1592.
The principles of iterative sine table calculation through the Kunstweg are as follows: cells in a column sum up the values of the two previous cells in the same column
A column or pillar in architecture and structural engineering is a structural element that transmits, through compression, the weight of the structure above to other structural elements below. In other words, a column is a compression member. ...
. The final cell's value is divided by two, and the next iteration starts. Finally, the values of the last column get normalized. Rather accurate approximations
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
of sines are obtained after few iterations.
As recently as 2015, Folkerts et al. showed that this simple process converges indeed towards the true sines.difference calculus
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
.
References
{{DEFAULTSORT:Kunstweg (Jost Burgi)
Algorithms
Trigonometry