Morgan Crofton

Morgan Crofton

Morgan Crofton (born June 27, 1826 in Dublin , † May 13, 1915 in Brighton ) was an Irish mathematician .

Origin and life

Crofton was born the son of a Protestant pastor (he succeeded the father of George Stokes , who was also a pastor in Skreen, County Sligo ). Crofton studied at Trinity College in Dublin . Since he was considering changing to the Catholic faith around this time (under the influence of the later Cardinal John Henry Newman in Birmingham), he did not want to accept the Fellowship of Trinity College offered him in 1848, which was connected with an ordination in the Anglican Church. In 1849 he became Professor of Theoretical Physics (Natural Philosophy) at Queen's College , Galway , but resigned in 1853 when he became Catholic. He then taught at a number of Jesuit schools in France. Later he was hired on the recommendation of James Joseph Sylvester at the Royal Military Academy in Woolwich , where he was also Sylvester's successor as Professor of Mathematics in 1870. In 1884 he retired and in 1884 became a professor at University College Dublin (the successor to the Catholic University in Dublin which Newman had founded), but left the lectures to John Casey, only took exams and continued to live in London. In 1895 he also retired here.

Crofton worked e.g. B. on geometry, James Clerk Maxwell's theory of trusses, but is best known today for his work in geometric probability theory (integral geometry), where Crofton's formula is named after him. It arose from consideration of Buffon's needle problem (Crofton On the theory of local probability . Transactions of the Royal Society, Vol. 158, 1868, p 181) and pushes the arc length s of a plane curve by an integral over the number of intersections with a randomly arranged “needle”, i.e. the (oriented) unit line, with the distance from the origin p and the orientation (angle ) as parameters. ${\displaystyle \gamma }$ ${\displaystyle n_{\gamma }}$ ${\displaystyle \varphi }$

${\displaystyle s(\gamma )={\frac {1}{4}}\int \int n_{\gamma }(\varphi ,p)\;d\varphi \;dp}$

The standardized density stands for evenly distributed lines in terms of position and orientation. For undirected lines you have to insert a factor of 2 accordingly. Crofton also derived formulas for the "hit count" of lines on express lines and two express lines simultaneously. ${\displaystyle d\varphi \;dp}$

He also wrote the influential article on probability theory in the 9th edition of the Encyclopædia Britannica of 1885 (Vol. 19, pp. 768-788).

In 1898 he was awarded an honorary doctorate from Trinity College, Dublin.

Fonts

• Lectures on the elements of applied mechanics , 1877
• with E. Kensington Tracts on Mechanics

literature

• Wilhelm Blaschke lectures on integral geometry , Chelsea 1949
• Obituary for Crofton, Proceedings London Mathematical Society Vol. 14, 1915, pp. XXIX

Web links

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