The family moved from Leith to Aberdeen when Alexander was young and he received his early schooling and training as a pupil-teacher in that city attending the North Parish School, Aberdeen, and the Grammar School, Old Aberdeen. He returned to Edinburgh where he studied at the Church of Scotland Training College and the University of Edinburgh. He graduated from the University with an M.A. with First Class Honours in Mathematics and Natural Philosophy in 1886. His undergraduate career had been outstandingly successful and he was awarded the Sir David Baxter Scholarship in Mathematics, and the Neil-Arnott Bursary as the best student in his year in the Physical Laboratory. He also was awarded a B.Sc. degree from the University of Edinburgh.
In 1888 Alexander Morgan married Isabella Duthie (born in Aberdeen in about 1861). They had sons William (born about 1892) and Alexander (born about 1896).
On graduating M.A., Morgan was appointed a Lecturer in Mathematics at the Church of Scotland Training College, Edinburgh. In 1897 he was awarded a D.Sc. by the University of Edinburgh for his thesis The geometrical representation of elliptic integrals of the first kind. John Clark writes in  about Morgan's work at the Training College:-
His own personal influence with the students was very great and wholly good. There must be today  hundreds of workers in the field of Education, in Scotland and elsewhere, who look back with feelings of lively gratitude to their association with Dr Morgan in the Training College, and with sincere affection too for the man himself.In 1903 Morgan became Principal of the Church of Scotland Training College. He held this position until 1920 when the Provincial Training College was created from the union of the two Church Training Colleges. Morgan then became Director of Studies at the Provincial Training College until he retired in 1925.
John Clark writes about Morgan's influence on Scottish Education :-
For many years he took a prominent part in the counsels of the Education Institute, becoming President of the Institute in 1911 - 12, and Convener of the 1916 - 17 Reform Committee whose report had an important bearing on the Education Act of 1918.After he retired in 1925, Morgan continued to work hard for education. He served as Convener of the Business Committee of the General Council of the University of Edinburgh for several years. He also represented the General Council on the University Court from 1932 to 1939. He also took on the role of Honorary Secretary of the Extra-Mural Adult Education Committee. For his great contributions to the University he was awarded an honorary LL.D. in 1937.
In 1888 Morgan married Isobel Duthie; she died in 1941. Their son, Lieutenant-General Sir William Duthie Morgan, K.C.B., was Commander-in-Chief Central Mediterranean Forces.
Morgan joined the Edinburgh Mathematical Society in November 1887. He was President during session 1898-99. He submitted papers to the Society such as 0n the Geometrical Representation of Elliptic Integrals of the First Kind (Friday 12 February 1897). He was elected a fellow of the Royal Society of Edinburgh on 4 May 1896. He was proposed for the fellowship by George Chrystal, Simon S Laurie, John Sturgeon Mackay, and Peter Guthrie Tait.
Morgan's death was reported in The Scotsman on Tuesday 19 March 1946:-
Former Principal of Moray House
Dr Alexander Morgan, O.B.K., M.A., F.R.S.E.. a former Principal and Director of Studies of Edinburgh Provincial Training College, Moray House, and for many years an outstanding figure in Scottish educational circles, died at his home in Edinburgh on Sunday night.
Under Dr Morgan's chairmanship, the Scottish Education Reform Committee (1916-17) issued a report which exerted a powerful influence on educational advances, including the Act of 1918. His zest for education was great, but it was not confined in a narrow pedagogic channel, and the many teachers who passed into the schools of the country from Moray House had reason to be grateful for the wide culture and sympathies of an inspiring Principal. After his retirement from the Provincial Training College, his skill as an organiser and his experience as an administrator were freely put at the disposal of the General Council and the Court of the University of Edinburgh.
A son of the late William Morgan, Leith, he was born at Leith in 1860. His early education was received at North Parish Church School, Aberdeen, and Old Aberdeen Grammar School, and after a period as a pupil-teacher in Aberdeen he went through two years of professional training in the Church of Scotland Training College, Edinburgh.
Proceeding to Edinburgh University, he graduated M.A. with first class honours in mathematics and natural philosophy, and gained the Sir David Baxter Scholarship in mathematics and the Neil Arnott bursary as the most distinguished student of the year in the physical laboratory. Later he took the B.Sc. degree and in 1897 he received the degree of D.Sc. for original investigation in mathematics, conferred for the first time for research in that department.
Immediately after graduation in 1886 he was appointed a lecturer in the Edinburgh Training College, became Principal in 1903, and when the office of Director of Studies was created he combined the duties of that office with those of Principal. He was largely responsible for the closer connection established between that joint office and the University Chair of Education, and also for the institution of the Chair of Education. He retired in 192S.
From 1932 till 1939 Dr Morgan was one of the assessors elected by the Genera Council to the Edinburgh University Court and before joining the Court he rendered valuable service for some years as Convener of the Business Committee of the General Council. In addition lo those services, the University was indebted to him for his laborious and fruitful researches into the University records, part of the results of which were published in the volume of Charters, Statutes, and Acts in 1931, and for his work on the Extra-mural Committee. Dr Morgan was a strenuous worker in the cause of adult education and a voluminous writer on educational subjects. In 1937 the University of Edinburgh conferred upon him the honorary degree of LLD.
For many years Dr Morgan was a leading member of the Educational Institute of Scotland, and was president in 1912. He was a member of the Scottish Savings Committee and received the O.B.E. in 1931. He was at one time chairman of the Morningside Ward Liberal Committee and was a member of the Scottish Liberal Club.
Dr Morgan's wife died in 1941. Their son, Lieut. General Sir William Duthie Morgan, has had a distinguished war record. He succeeded Field-Marshal Sir Harold Alexander as Supreme Allied Commander, Mediterranean Theatre, last October, and is now C. in C. Central Mediterranean Forces.
Article by: J J O'Connor and E F Robertson