John Machin

John Machin

John Machin (* 1680 in England , † June 9, 1751 in London ) was an astronomer and mathematician with a professorship at Gresham College in London. He is known because of his arctan formula for the circle number, discovered in 1706 ${\displaystyle \pi }$

${\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}$,

which with the aid of the Taylor expansion of the arctangent results in a rapidly converging series for the numerical calculation of . He had used this to calculate 100 decimal places from . The Leibniz series ${\displaystyle \pi }$ ${\displaystyle \pi }$

${\displaystyle \arctan 1={\frac {\pi }{4}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots }$

is not suitable for a numerical calculation because of its slow convergence. An appreciation of this discovery is that it represents the first essentially new method for the numerical calculation of according to Archimedes . This Archimedean method was still used by Ludolph van Ceulen around 1600 to calculate 35 decimal places. Today many more arctan formulas are known for of Machinian design. ${\displaystyle \pi }$ ${\displaystyle \pi }$

literature

• Siegfried Gottwald , Hans J. Ilgauds, Karl-Heinz Schlote: Lexicon of important mathematicians. Harri Deutsch, Thun 1990, ISBN 3-8171-1164-9 .

Web links

• John J. O'Connor, Edmund F. Robertson :  John Machin. In: MacTutor History of Mathematics archive .