page_title: William Chapple (surveyor)
section_title: Contributions to mathematics
hierarchical_section_title: William Chapple (surveyor) / Contributions to mathematics
context_page_description: William Chapple was an English surveyor and mathematician. His mathematical discoveries include the first proof of the existence of the orthocentre of a triangle, a formula for the distance between the incentre and circumcentre of a triangle, and the discovery of Poncelet's porism on triangles with a common incircle and circumcircle. He was also one of the earliest mathematicians to calculate the values of annuities.
context_section_description: Andrea del Centina writes that "To illustrate the work of Chapple, whose arguments are often confused and whose logic is very poor, even for the standard of his time, is not easy especially when trying to keep as faithful as possible to his thought." Nevertheless, Chapple made several significant discoveries in mathematics.
  Euler's theorem in geometry gives a formula for the distance  between the incentre and circumcentre of a circle, as a function of the inradius  and circumradius :
  An immediate consequence is the related inequality . Although these results are named for Leonhard Euler, who published them in 1765, they were found earlier by Chapple, in a 1746 essay in The Gentleman's Magazine. In the same work he stated that, when two circles are the incircle and circumcircle of a triangle, then there is an infinite family of triangles for which they are the incircle and circumcircle. This is the triangular case of Poncelet's closure theorem, which applies more generally to polygons of any number of sides and to conics other than circles. It is the first known mathematical publication on pairs of inscribed and circumscribed circles of polygons, and significantly predates Poncelet's own 1822 work in this area.
  In 1749, Chapple published the first known proof of the existence of the orthocentre of a triangle, the point where the three perpendiculars from the vertices to the sides meet. The orthocentre itself was known previously, but Chapple writes that its existence was "often taken for granted, but no where demonstrated".
  Chapple learned of the problem of valuation of annuities through his correspondence with John Rowe and Thomas Simpson, and carried out this valuation for Courtenay. In this, he became one of the first mathematicians to work on this problem, along with Simpson, Abraham de Moivre, James Dodson, and William Jones.
caption: Three altitudes of a triangle meet at the orthocentre