Frank Morley (1860-1937)

One of the most sparkling gems in all of elementary geometry is Morley's Theorem:

The three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle.

If the reference triangle is labeled ABC, then the equilateral triangle is called the Morley triangle of ABC. Closely related are other Morley triangles and Morley centers.

Born in Woodbridge, Suffolk, England, Morley received the Sc.D. degree from Cambridge University. He moved to Pennsylvania in 1887 and taught at Haverford College until 1900, when he became chairman of the mathematics department at Johns Hopkins University.

The pencil portrait of Morley seen here is used by permission of the estate of Felix M. Morley. (Felix Morley, son of Frank and Lilian Morley, was editor of the Washington Post (1933-1940), won a Pulitzer Prize (1936), and was President of Haverford College (1940-1945).)

In 1900, Morley's Theorem first appeared in the literature as a special case of a theorem in the first issue of Transactions of the American Mathematical Society. For an impressive summary of related results and a list of 116 references, see

Cletus O. Oakley and Justine C. Baker, American Mathematical Monthly 85 (1978) 737-745.

In his youth, Morley won a number of medals at chess tournaments. Of his three sons, one of them, Frank Vigor Morley, became a mathematician and the author of a book entitled My One Contribution to Chess (B. W. Huebsch, New York, 1945). There, the great geometer is vividly portrayed —

... then he would begin to fiddle in his waistcoat pocket for a stub of pencil perhaps two inches long, and there would be a certain amount of scrabbling in a side pocket for an old envelope, and then there would be silence for a long time; until he would get up a little stealthily and make his way toward his study—but the boards in the hall always creaked, and my mother would call out, "Frank, you're not going to work!"—and the answer would always be, "A little, not much!"—and the study door would close.

(It wasn't hard to gather that my father was working at geometry, and I knew pretty well what geometry was, because for a long time I had been drawing triangles and things; but when you examined the envelope he left behind, what was really mysterious was that there was hardly ever a drawing on it, but just a lot of calculations in Greek letters. Geometry without pictures I found it hard to approve; indeed, I prefer it with pictures to this present day.)

These paragraphs are quoted by permission of the estate of Frank V. Morley, mathematician and writer. F. V. Morley wrote a review for the second-most widely read geometry book of all time: Flatland, a Romance of Many Dimensions, by Edwin Abbott Abbott, first published in 1884, reprinted many times; e.g., in 1998 by Penguin Books. In The Saturday Review of Literature, Oct. 30, 1926, F. V. Morley writes:
Dr. Abbott was out for fun when he wrote his friendly little geometrical romance, and it is good to see that the old wine is no worse for its new bottle. It is still a pleasant tonic, and an excellent stimulant ...
Another son of Frank and Lilian Morley was the famous author, Christopher Morley. All three sons were Rhodes scholars.
Here is a photograph of Frank Morley lecturing at Haverford College, sometime before 1901. Courtesy of the Haverford College Archives.

Morley was President of the American Mathematical Society during 1919-1920 and editor of the American Journal of Mathematics during 1900-1921. He was an immensely effective teacher, and forty-five students earned their doctoral degrees under his immediate supervision. An accomplished chess player, Morley once beat the algebraist Emmanuel Lasker during the time that Lasker was world chess champion.

Morley's life and works are the subjects of two articles:

Arthur B. Coble, "Frank Morley - In Memoriam," Bulletin of the American Mathematical Society 44 (1938) 167-170.

H. W. Richmond, "Frank Morley," Journal of the London Mathematical Society 14 (1939) 73-78.

Morley's publications, beginning with his co-authored books, are listed here:


A Treatise on the Theory of Functions (with James Harkness), Macmillan, London, 1893.

Introduction to the Theory of Analytical Functions (with James Harkness), Macmillan, London, 1898.

Inversive Geometry (with his son, F. V. Morley), G. Bell, London, 1933. Reprinted, Chelsea, New York, 1954.


A nine-line conic, Messenger of Math. 15 (1886), 190-192.

Some properties of confocal conics and a derived cubic, Messenger of Math. 16 (1887) 181-185.

On critic centres, American J. of Math. 10 (1888) 141-148.

On plane cubics which inflect on crossing their asymptotes, Messenger of Math. 17 (1888), 51-57.

Note on geometric inferences from algebraic symmetry, American J. of Math. 10 (1888) 173-174.

On the geometry of a nodal circular cubic, American J. of Math. 11 (1889) 307-316.

On the epicycloid, American J. of Math. 13 (1890) 179-184.

On the kinematics of a triangle of constant shape but varying size, Quart. J. of Math. 24 (1890) 359-369, 386.

On the covariant geometry of the triangle, Quart. J. of Math. 25 (1891) 186-197.

On adjustable cycloidal and trochoidal curves, American J. of Math. 16 (1894) 188-204.

Note on the congruence 2^(4n) = ((-1)^n)(2n!)/(n!)^2, where 2n+1 is prime, Annals of Math. 9 (1894) 168-170.

Apolar triangles on a conic, Bull. American Math. Soc. 1 (1895) 116-124.

Note on the theory of three similar figures, Bull. American Math. Soc. 1 (1895) 235-237.

On the generalization of Weierstrass' equation with three terms, Bull. American Math. Soc. 2 (1896) 21-22.

Note on the common tangents of two similar cycloidal curves, Bull. American Math. Soc. 2 (1896) 111-116.

A generating function for the number of permutations with an assigned number of sequences, Bull. American Math. Soc. 4 (1897) 23-28.

A construction by ruler of a point covariant with five given points, Math. Annalen 49 (1897) 596-600.

On a regular rectangular configuration of ten lines, Proc. London Math. Soc. 29 (1898) 670-673.

Some polar constructions, Math. Annalen 51 (1899) 410-416.

On the Poncelet polygons of a limaçon, Proc. London Math. Soc. 29 (1898) 83-97.

On a regular configuration of ten line pairs conjugate as to a quadric, Bull. American Math. Soc. 5 (1899) 252-253.

Note on the sphero-conic, Math. Gazette 1 (1900) 249-250.

The no-rolling curves of Amsler's planimeter, Annals of Math. 1 (1900), 21-35.

On the metric geometry of the plane n-line, Trans. American Math. Soc. 1 (1900) 97-115.

The value of [an integral], Bull. American Math. Soc. 7 (1901) 390-392.

On a point in Sylvester's theory of canonical forms, Bull. American Math. Soc. 7 (1901) 206.

On the series 1 + (p/1)^3 + {p(p+1)/2}^3 + ... , Proc. London Math. Soc. 34 (1902) 397-402.

Orthocentric properties of the plane n-line, Trans. American Math. Soc. 4 (1903) 1-12.

Projective coordinates, Trans. American Math. Soc. 4 (1903) 288-296.

On the geometry whose element is the 3-point of a plane, Trans. American Math. Soc. 5 (1904) 467-476.

On a plane quintic curve, Proc. London Math. Soc. 2 (1905) 114-121.

On two cubic curves in triangular relation, Proc. London Math. Soc. 4 (1907) 384-392.

On reflexive geometry, Trans. American Math. Soc. 8 (1910) 14-24.

Plane sections of a Weddle surface [with J. R. Conner], American J. of Math. 31 (1909) 263-270.

Contact curves of the plane quintic curve, Johns Hopkins Univ. Circular 2 (1912) 69-73.

On the extension of a theorem by W. Stahl, Prov. 5th Internat. Congress of Math., Cambridge (1912) 2, 11-17.

An extension of Feuerbach's theorem, Proc. Nat. Acad 2 (1916) 171-173.

The Lüroth quartic curve, American J. of Math. 41 (1919) 279-282.

The curve of ambience, American J. of Math. 46 (1924) 193-200.

The eliminant of a net of curves, American J. of Math. 47 (1925) 91-97.

On an equation of planar motion, American J. of Math. 47 (1925) 98-100.

Note on Neuberg's cubic curve, American Math. Monthly 32 (1925) 407-411.

On a differential inversive geometry, American J. of Math. 48 (1926) 144-146.

New results in elimination [with A. B. Coble], American J. of Math. 49 (1927) 463-488.

Extensions of Clifford's chain theorems, American J. of Math. 51 (1929) 465-472.

On algebraic inversive invariants [with B. C. Paterson], American J. of Math. 52 (1930) 465-472.

On the intersections of trisectors of the angles of a triangle, J. Math. Assoc. Japan 6 (1924) 260-262.

It has been a pleasure to correspond with several grandchildren of Frank Morley in the United States, England, and Scotland, who provided helpful materials for this site. I thank Mrs. Louise Morley Cochrane, especially, and friend-of-the-family Thomas Banchoff of Brown University. Mrs. Cochrane's recent book and Professor Banchoff's website (mostly geometry) are listed below.

Louise Cochrane, Adelard of Bath: the first English scientist, British Museum Press, London, 1994.

1st and 2nd Morley Centers
Thomas Banchoff's Home Page
Morley Family Papers (1890-1986), Special Collections, Haverford College
Clark Kimberling Home Page