John Casey: a Founder of Modern Geometry


John Casey (1820-1891).

Next Tuesday – 12th May – is the 200th anniversary of the birth of John Casey, a notable Irish geometer. Casey was born in 1820 in Kilbeheny, Co Limerick. He was educated in nearby Mitchelstown, where he showed great aptitude for mathematics and also had a gift for languages. He became a mathematics teacher, first in Tipperary Town and later in Kilkenny [TM186; or search for “thatsmaths” at ].

Although having no advanced education, Casey pursued his interests in mathematics in his leisure time. His work came to the attention of George Salmon and Richard Townsend of Trinity College Dublin. They persuaded him to enter Trinity College, which he did in 1858. He was elected a Scholar in 1861 and graduated the following year.

Mathematical Papers

John-Casey-SequelCasey took a position as Science Master at Kingstown School (Dun Laoghaire) and continued with his mathematical studies. In 1873 he was appointed professor at the Catholic University in St Stephen’s Green, later reconstituted as University College Dublin. He declined an offer of a chair at Trinity College: being a man of deep religious convictions, he felt obliged to support the development of Catholic education in Ireland.

Casey produced several research papers on geometry while studying at Trinity College. In his career, he published more than twenty mathematical papers, many appearing in the “Philosophical Transactions of the Royal Society” and in the “Proceedings of the Royal Irish Academy”. He also wrote six textbooks on geometry that were well received and widely used. In one of these, “A Sequel to Euclid”, published by Hodges, Figgis & Co in 1881, Casey included extensions to the theorems of Euclid that he had discovered in his own research. In consequence, he is regarded as one of the founders of the modern geometry of the triangle and circle.

Casey’s Theorem

Some two thousand years ago the Greek astronomer Claudius Ptolemy established a result in Euclidean geometry: for a quadrilateral inscribed in a circle, the product of the diagonals is equal to the sum of the products of the opposite sides. In symbols, if the sides, in order, are A, B, C and D and the diagonals are M and N, then AC + BD = MN. This theorem was extended by Casey: he replaced the vertices of the quadrilateral by circles within and tangent to the outer circle, and proved a result for the common tangents that greatly generalized Ptolemy’s theorem.


Casey’s Theorem relates the lengths of tangents between four circles within a larger circle [figure adapted from (thanks to Maurice O’Reilly)].

Another theorem of Casey is important for graphic designers. For four points A, B, C and D in order on a line, with the circles of diameter AC and BD intersecting at point O, the angle BOC is called Casey’s angle. Its importance is that it is a perspective invariant, and this helps artists to produce realistic images and video game designers to achieve verisimilitude.


Casey’s Angle [Applet written by Maurice O’Reilly, DCU].

Casey married in 1847 and had two sons and two daughters. He received many distinctions in recognition of his mathematical work. In 1866 he was elected as a member of the Royal Irish Academy, later serving as its Vice-president. In 1875 he was made a Fellow of the Royal Society. He was awarded honorary degrees by Dublin University and by the Royal University of Ireland. He died, aged 70 years, in 1891 and is buried in Glasnevin Cemetery.


Gow, Rod, 2000: John Casey, 1820-1891. In Creators of Mathematics: the Irish Connection. Editor Ken Houston, UCD Press, Dublin, ISBN: 1-900621-49-5.

Applets written by Maurice O’Reilly, DCU: 

(1)  Applet to illustrate Casey’s Theorem

(2)  Applet to illustrate Casey’s Angle (parallel projection)
(3)  Applet to illustrate Casey’s Angle (projection from a point)

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