Ludolph van Ceulen

Ludolph van Ceulen

Ludolph van Ceulen (born January 28, 1540 in Hildesheim , † December 31, 1610 in Leiden ) was a fencing master and mathematician .

Life

Ludolph van Ceulen moved to Delft when he was a child . In 1594 he founded a fencing school in Leiden . Pieter Bailly, a fencing master at the school, wrote a manuscript about fencing with just a rapier in 1602 for Prince Moritz of Orange , who sponsored the school. In the same year there was a conflict between van Ceulen and Bailly, because, contrary to agreements, he also gave fencing lessons outside of his school, which the city finally forbade him.

In 1600 van Ceulen was appointed the first professor of arithmetic , surveying and fortification at the engineering school attached to the University of Leiden . The teachers at the engineering school had a low reputation at the university, as they often came from practical experience and, like van Ceulen, for example, had no university education. They also taught in the local language rather than Latin. Van Ceulen's works were only translated into Latin by his pupil Willebrord Snell after his death .

Ludolph number

The replica of the tombstone

Ludolph van Ceulen is still famous today for the calculation of the circle number to 35 decimal places , the first progress after the calculation to 16 places by the Persian mathematician Jamschid Masʿud al-Kaschi in 1424. Up until the 19th century it was also known as Ludolph's number . He spent much of his life doing these calculations and having the 35 places engraved on his headstone . The original tombstone was lost in the 19th century, but a replica was placed in the Pieterskirche in Leiden on July 5th, 2000. Van Ceulen's pupil Snellius , the translator into Latin and editor of his works, noticed in 1621 that this accuracy could have been achieved with half the calculation. Christiaan Huygens then provided the mathematical proof . ${\displaystyle \pi }$ ${\displaystyle \pi }$

Van Ceulen's calculations

In his research on the number of circles, Archimedes assumed regular polygons inscribed or circumscribed in a circle with the radius (unit circle), the so-called exhaustion method . The higher the number of corners of these polygons, the closer they approach the circle from inside and outside . Archimedes started with the regular hexagon, continued with the dodecagon , then with the 24, 48, 96 and so on. Each time the side lengths of the inscribed and the circumscribed corner have to be recalculated. With the help of the ray theorem and the Pythagorean theorem, Archimedes found the following relationship between two successive side lengths and : ${\displaystyle r=1}$ ${\displaystyle s_{n}}$ ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle s_{2n}}$ ${\displaystyle s_{n}}$
${\displaystyle s_{2n}={\sqrt {2-{\sqrt {4-s_{n}^{2}}}}}={\frac {s_{n}}{\sqrt {2+{\sqrt {4-s_{n}^{2}}}}}},\qquad S_{n}={\frac {2s_{n}}{\sqrt {4-s_{n}^{2}}}}}$

Archimedes probably used the left recursion formula. A (simple) transformation results in the mean recursion formula, which is more favorable for numerical calculations (cancellation) and which comes from more recent times.

Archimedes obtained the inequality through the inscribed and circumscribed 96-sided (i.e. n = 6 · 2 · 2 · 2 · 2) and from it . ${\displaystyle 3{\tfrac {1137}{8069}}<\pi <3{\tfrac {1335}{9347}}}$ ${\displaystyle 3{\tfrac {10}{71}}<\pi <3{\tfrac {1}{7}}}$

The corresponding polygonal circumference differs less and less from the circumference as it increases . So the numerical value of is an increasingly better approximation for Van Ceulen calculated according to this principle up to the inscribed 2 ⁶² corner (a polygon with about 4 trillion sides) and thus gained the approximate value over the course of 30 years: 3.141 592 653 589 793 238 462 643 383 279 502 88. ${\displaystyle u_{n}=n\cdot s_{n}}$ ${\displaystyle n}$ ${\displaystyle U=2\pi \cdot r=2\pi }$ ${\displaystyle u_{n}/2}$ ${\displaystyle \ \pi .}$

Fonts

De circulo & adscriptis liber (1619)

• Ludolph van Colen: Proefsteen Ende Claerder nieleggingh dat het claarder bewijs: (so dat ghenaempt is) op de gheroemde ervindingh vande quadrature of the Cirkels een onrecht te know gheuen / end gheen waerachtich bewijs is; Kort claar bewijs cvon denomin iegelsiewe ghe zyn diameter te groot is the end of our Zuler de quadratura circuli des zeluen vinders onrecht is ; Amsterdam, 1586
• Ludolf van Ceulen: Van den circkel. Because in gheleert you will find the naeste proportions of the circkels-diameter tegen synen omloop, because door all circkels (with all figures, often land with cromme lines) can be quite ghemetem. ... Nor de tables sinum, tangentium, end of secantium ... Ten laetsten van Interest, met alderhande tables daer toe serving, met het ghebruyck, door veel constighe example gheleerdt, ... ; Delf dead: ghedruckt by Ian Andriesz Boeckvercooper woonende aen't Maret-Veldt in't Gulden ABC, 1596 ( Retro digitized )
• Ludolf van Ceulen: De arithmetic en geometrical fondamenten. Met het ghebruyck van dien in veele different constighe quests, soo geometrice door lines, as arithmetice door irrational ghetallen, oock door the rule coss, end de tables finuum ghesolveert ; Leyden, Ioost van Colster, Iacob Marcus, 1615
• Fundamenta arithmetica et geometrica cum eorumdem usu in variis problematis geometricis, partim solo linearum ductu, partim per numeros irrationales et tabulas sinuum et algebram solutis , authore Ludolpho a Ceulen, ... e vernaculo in latinum translata a Wil. Sn. [Willebrordo Snellio], 1617
• Ludolphi a Ceulen de Circulo et adscriptis liber, in quo plurimorum polygonorum latera per irrationalium numerorum griphos, quorumlibet autem per numeros absolutos secundum algebricarum aequationum leges explicantur ... , Omnia e vernaculo latina fecit et annotationbrordus illustrus Wille19

literature

• Moritz Cantor :  Ceulen: Ludolph van C. In: General German Biography (ADB). Volume 4, Duncker & Humblot, Leipzig 1876, p. 93.
• Kurt Vogel :  van Ceulen, Ludolph. In: New German Biography (NDB). Volume 3, Duncker & Humblot, Berlin 1957, ISBN 3-428-00184-2 , p. 186 ( digitized version ).
• Friedrich Katscher: Some discoveries about the history of the number Pi as well as the life and work of Christoffer Dybvad and Ludolph van Ceulen. Austrian Academy of Sciences, Memoranda 116. Vienna 1979.

Web links

• A Chronology of Pi - Pre computer calculations of π. April 25, 2011, accessed March 25, 2013 . 
• Today's calculation of Pi. Accessed March 25, 2013 . 
• John J. O'Connor, Edmund F. Robertson :  Ludolph van Ceulen. In: MacTutor History of Mathematics archive .
• Henk JM Bos: Ludolph van Ceulen en de uitdaging van de wiskunde. (PDF; 307 kB) In: Nieuw Archief voor Wiskunde, vol. 1, pp. 240-253. September 2000, accessed on March 25, 2013 (Dutch, "Ludolph van Ceulen and the challenge of mathematics"). 

Individual evidence

1. ↑ www.math.uu.nl: Biography - Ludolph van Ceulen, Leiden University (Dutch)
2. ↑ RMTh.E. Oomes, JJTM Tersteeg, J. Top: Het grafschrift van Ludolph van Ceulen. (PDF; 660 kB) In: Nieuw Archief voor Wiskunde, vol. 1, pp. 156-161. June 2000, accessed on March 25, 2013 (Dutch, "On the grave and epitaph of van Ceulen"). 
3. ↑ http://www.math.uu.nl/wiskonst/ruziesceulen/biovc.html biography