''Fundamentum Astronomiae'' is a historic manuscript presented 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 ...
to
Emperor Rudolf II
Rudolf II (18 July 1552 – 20 January 1612) was Holy Roman Emperor (1576–1612), King of Hungary and Croatia (as Rudolf I, 1572–1608), King of Bohemia (1575–1608/1611) and Archduke of Austria (1576–1608). He was a member of the Hous ...
in 1592. It describes Bürgi's
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
based
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 called
Kunstweg which can be used to calculate
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 at arbitrary precision.
[Staudacher, S., 2014. Jost Bürgi, ''Kepler und der Kaiser''. Verlag NZZ, Zürich.]
General
Bürgi took special care to avoid his method becoming public in his time. However,
Henry Briggs (mathematician)
Henry Briggs (1 February 1561 – 26 January 1630) was an English mathematician notable for changing the original logarithms invented by John Napier into common (base 10) logarithms, which are sometimes known as Briggsian logarithms in his ho ...
(1561-1630) was acquainted with the method, likely via a link to
John Dee
John Dee (13 July 1527 – 1608 or 1609) was an English mathematician, astronomer, astrologer, teacher, occultist, and alchemist. He was the court astronomer for, and advisor to, Elizabeth I, and spent much of his time on alchemy, divinatio ...
who knew
Christoph Rothmann, a colleague of Bürgi at the court.
Method
Bürgi used these algorithms, including multiplication table in
sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
system, to compute a ''
Canon Sinuum'', a table of sines to 8 sexagesimal places in steps of 2
arc seconds. 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. Bürgi's method only uses additions and halving, his procedure is elementary and it converges from the standard method.
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. The iterative algorithms obtains good
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 after few iterations, but cannot be used on large subdivisions, because it produces very large values. This was an early step towards
difference calculus.
Ursus, his friend wrote in his 1588 ''Fundamentum astronomicum'', “I do not have to explain to which level of comprehensibility this extremely deep and nebulous theory has been corrected and improved by the tireless study of my dear teacher, Justus Bürgi from Switzerland, by assiduous considerations and daily thought.
..Therefore neither I nor my dear teacher, the inventor and innovator of this hidden science, will ever regret the trouble and the labor which we have spent.”
Bürgi writes, “For many hundreds of years, up to now, our ancestors have been using this method because they were not able to invent a better one. However, this method is uncertain and dilapidated as well as cumbersome and laborious. Therefore we want to perform this in a different, better, more correct, easier and more cheerful way. And we want to point out now how all sines can be found without the troublesome inscription
f polygons namely by dividing a right angle into as many parts as one desires.”
See also
*
Almagest
The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it canoni ...
*
Plimpton 322
Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a table ...
References
External links
Bürgi's tables in his Fundamentum Astronomiæ (ca. 1586-1587)
History of mathematics
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