Raymond attended Mount Allison Academy from 1885 to 1889. After graduating from the Academy, he entered Mount Allison College in Sackville where he studied both mathematics and music. He was awarded a B.A. in mathematics from the College in 1894 and he also received a Teacher's Diploma to teach violin. A few years after Archibald graduated, in 1903, Mount Allison College became Mount Allison University. He spent a year teaching mathematics at Mount Allison Ladies' College while he continued to study music, completing an Artist's Diploma (violin) in 1895.
After graduating from Mount Allison, Archibald entered Harvard University in 1895. He obtained a B.A. from Harvard in 1896, remaining there to study for his Master's Degree which was awarded in 1897. Following this Archibald studied in Europe, attending the University of Berlin in session 1898-99 and then the University of Strasbourg in 1899-1900. He was awarded his Ph.D. from Strasbourg in 1900 for a thesis The Cardioid and Some of Its Related Curves. This dissertation studied the cardioid, (x2 + y2 - ax)2 = a2 (x2 + y2) and also looked at its history. Archibald explained that the name cardioid was first used by de Castillon in a paper in the Philosophical Transactions of the Royal Society in 1741. It is the locus of a point on the circumference of circle rolling round the circumference of a circle of equal radius. In 1903 Archibald published the paper The Cardioid and Tricuspid: Quartics with Three Cusps on the work of his doctoral dissertation.
After completing his doctorate, Archibald returned to Canada in 1900 where he was appointed professor of Mathematics at Mount Allison Ladies' College in Sackville, New Brunswick. His mother Mary was vice-principal of the College until her death on 9 January 1901. Archibald served there for seven years, also acting as librarian, building the library from scratch to 12,000 volumes and writing by hand 30,000 catalogue cards. He was also head of the violin department. Already he showed his interest in collecting and cataloguing material, and he began to assemble calendars, student publications, photographs and memorabilia associated with Mount Allison. In 1905 he set up the Mary Mellish Archibald Memorial Library at Mount Allison and he continued to build the library throughout his life. In 1907 he was appointed as professor of mathematics at Acadia University, Wolfville, Nova Scotia, where he remained for a year before being appointed to Brown University in Providence, Rhode Island, United States. The university had begun as Rhode Island College in 1764 but had adopted the name Brown University in 1804 in honour of its benefactor Nicholas Brown. At this stage in his career Archibald made the decision that he could not keep his twin interests in music and mathematics both as part of his career and decided to concentrate on mathematics with music becoming a hobby. Archibald had been a professor at Acadia University but now he accepted the position of Instructor at Brown. He was to remain there for the rest of his life: as instructor from 1908 to 1911, assistant professor from 1911 to 1917, associate professor from 1917 to 1923, and professor from 1923 to 1943. After retiring in 1943 he was made professor emeritus. C R Adams was taught by Archibald at Brown during 1915-16. He writes :-
... he was in the prime of life and full of energy. He was striking in appearance, his hair wavy and beginning to grey, worn a little longer than was generally the custom; his eyes large and expressive. Always carefully groomed, he wore a high starched collar and stiff detachable cuffs. Often a white edging on his waistcoat gave an added touch of elegance to his dress. To me he was an impressive figure among the Brown faculty.We should mention a number of projects which Archibald undertook. We noted above the Mary Mellish Archibald Memorial Library which he founded at Mount Allison in 1905. Each summer he travelled from Providence to Sackville to spend time building up the collection which contains:-
... material on the Anglo-American folk song tradition; the development of folk music in Canada; ballad literature; art song; patriotic and college songs; and songs for children.However, Archibald collected other material at Mount Allison which he added to on his summer visits. This included:-
... records generated by Archibald in his activities as scholar, researcher, collector, historian of Mount Allison and by personal relationships with friends, family and fellow Allisonians. ... Scrapbooks contain correspondence, clippings, notices, posters, programs, memorabilia which document activities, students, faculty, buildings at Mount Allison, 1857 - 1935.His work as a librarian led to him making the mathematics library at Brown University one of the finest. He built up its holdings of journals and made a special feature of adding doctoral dissertations from leading American and European universities :-
Frequently he went to Europe for the summer, always provided with funds to spend for mathematical books for Brown. He carried on a flourishing correspondence with scientific book sellers throughout the world, and if a dealer offered an item for sale at less than it was worth he was quick to recognise a bargain and to cable an order for purchase.From 1921 to 1941 he was the librarian of the American Mathematical Society and is responsible for a remarkable growth in its library holdings.
His mathematical work was mainly in the history of mathematics. He published works such as Euclid's book On Division of Figures with a restoration (1915), Benjamin Peirce 1809-1880 (1925), Outline of the History of Mathematics (1932), The scientific achievements of Nathaniel Bowditch (1937), Semicentennial History of the American Mathematical Society 1888-1938 (1938, reprinted 1988), and Fifty mathematical table makers (1948). His interest and expertise in mathematical tables led to him becoming chairman of a National Research Council on tables in January 1939. He founded the journal Mathematical Tables and other Aids to Computation in 1943, which was published by the National Research Council of the National Academy of Sciences. Archibald became the first chairman of the editorial committee of the Journal. In 1944 he published, jointly with Bateman, A guide to tables of Bessel functions which occupied a whole issue of Mathematical Tables and Other Aids to Computation. Szegő writes:-
It contains a list of the tremendous literature on tables and graphs concerning the various Bessel functions. The great importance of such a collection is obvious.As a final comment on Archibald's publications, let us list the papers he published during the years 1945-50: Mathematical table makers - portraits, paintings, busts, monuments, bio-bibliographical notes I, II (1945, 1946); Material concerning James Joseph Sylvester (1947); Mathematical Table Makers. Portraits, Paintings, Busts, Monuments, Bio-Bibliographical Notes (1948); Bartholomaus Pitiscus 1561-1613 (1949); Rheticus, with special reference to his "Opus Palatinum" (1949); Obituary : R G D Richardson 1878-1949 (1950); The first translation of Euclid's "Elements" into English and its source (1950); and New information concerning Isaac Wolfram's life and calculations (1950).
Archibald was a delegate to the congress of universities of the British Empire in London in 1912. He was a delegate at the celebrations to mark the 700th anniversary of the founding of the University of Padua in 1922, and represented both the United States and Canada on the Euler Committee of the Swiss Society of Naturalists from 1922 to 1939. He was elected as an honorary member of the Polish Mathematical Society, the London Mathematical Society, the German Mathematical Society, and several other societies. He served on the council of the American Mathematical Society from 1918 to 1941 and he was president of the Mathematical Association of America in 1922.
In  the authors sum up their article as follows:-
Archibald was both a scholar of the old school and a gentleman of the old school ... He was brought up in the classical tradition with much emphasis on Latin and Greek. He had a very remarkable memory, and he carried with him at all times an enormous store of factual information in the fields of his interest.
Article by: J J O'Connor and E F Robertson