However ill health disrupted Morley's undergraduate course and he was forced to take an extra year because of these health problems. Morley only achieved the eighth place in the First Class Honours. To say 'only' here may seem strange since this was an extremely good result in an examination which saw Mathews first and Whitehead fourth. Richmond writes in , however:-
Ill health beyond all doubt had prevented him from doing himself justice, but the disappointment was keen. In middle life he was loath to speak of his student days...Morley graduated from Cambridge with a B.A. in 1884 but his relatively poor performance meant that he had no hope of a fellowship. He took a job as a school master, teaching mathematics at Bath College until 1887. This was an important period for Morley since he was able to overcome his health problems and with the improvement in health came a renewed confidence in his own mathematical abilities. He settled in the United States and was appointed an instructor at the Quaker College in Haverford, Pennsylvania, in 1887. The following year he was promoted to professor. At Haverford, Morley worked, not with others at the College, but with the mathematicians Scott and James Harkness, both also graduates of Cambridge, England, who were at Bryn Mawr which was close to Haverford.
In 1889 he married Lilian Janet Bird, who was a musician and poet. We should note that the marriage produced three sons, Christopher (born in Haverford on 5 May 1890), Felix, and Frank, who all went on to become Rhodes scholars. Christopher Darlington Morley (1890-1957) became a novelist, and his works include The Trojan Horse, Kitty Foyle and The Old Mandarin . Felix M Morley (1894-1982) became editor of the Washington Post and was also president of Haverford College from 1940 to 1945. Frank Vigor Morley (1899-1985) became a director of the publishing firm Faber and Faber but was also a mathematician who collaborated with his father for over twenty years.
Morley was appointed Professor of Mathematics at Johns Hopkins University in 1900. The University had been a leading one in the United States over the period while Sylvester worked there, but he had left in 1883. The strong graduate programme in mathematics which had been set up there continued to flourish but by 1900 it had begun to decline and Morley's appointment was a very definite attempt to reinvigorate the programme. Certainly he proved an excellent choice leading by example and supervising 48 doctoral students over his years at Johns Hopkins. Coble writes that Morley made it (see  or ):-
... a cardinal point to have on hand a sufficient variety of thesis problems to accommodate particular tastes and capacities.We should look now at Morley's mathematical achievements. He wrote papers mainly on geometry but also on algebra. We mentioned that while at Haverford he had collaborated with Harkness. They jointly authored a text A treatise on the theory of functions which was published in 1893 and revised as Introduction to the theory of analytic functions in 1898. These were excellent advanced level texts published at a time when very few such advanced mathematics books were being produced in the United States. Many years later, in 1933, he published Inversive geometry written jointly with his son Frank V Morley.
Morley's own favourite among his geometry papers was On the Lüroth quartic curve which he published in 1919. He is perhaps best known, however, for a theorem which is now known as Morley's Theorem:-
If the angles of any triangle be trisected, the triangle, formed by the meets of pairs of trisectors, each pair being adjacent to the same side, is equilateral.Morley loved posing mathematical problems and over a period of 50 years, starting in his undergraduate days, he published over 60 problems in the Educational Times. Most are of a geometric nature. Here is an example, see :-
Show that on a chess-board the number of squares visible is 204, and the number of rectangles (including squares) visible is 1296; and that, on a similar board with n squares in each side, the number of squares is the sum of the first n square numbers, and the number of rectangles (including squares) is the sum of the first n cube numbers.We mentioned that Morley was a chess enthusiast while at school and, indeed, he was an exceptionally good chess player, so the problem above reflects one of his hobbies. He played at the highest level and beat Lasker on one occasion while Lasker was World Chess Champion.
Zund writes about the significance of Morley's mathematical work:-
Today much of Morley's research seems of less than compelling significance, and one is tempted to regard his interests as those of a talented amateur - an artist who took delight in small and beautiful things - rather than those of a professional mathematician. Yet, whatever the significance one chooses to attach to them, Morley must be given credit for both finding and solving such questions.Morley made a major contribution to mathematics in the United States. He undertook editorial work for the Bulletin of the American Mathematical Society and the American Journal of Mathematics while at Haverford. Later, when at Johns Hopkins University, became the editor of the American Journal of Mathematics and held this position for 30 years. In 1919-20 he served as president of the American Mathematical Society.
He is described by Cohen in  as:-
... a striking figure in any group. Deliberate in manner and speech, there was a suggestion of shyness about him. He was generally very well informed and interested in a strikingly wide range of subjects. He was of an artistic temperament. While many of his papers and lectures seemed involved to the uninitiated, they all possessed a characteristic artistic charm.His son, Frank V Morley, gives this description of his father:-
... 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.)
Article by: J J O'Connor and E F Robertson