Born in Troy, Ohio, on November 2, 1871, the son of Samuel A. and Mary Brook Cairns, William graduated with an A.B. degree in 1892 from Ohio Wesleyan, Delaware, Ohio; and received his second A.B. (1897) and an A.M. (1898) degree from Harvard University. His Ph.D. degree in mathematics was earned from the University of Gottingen, Germany in 1907, having studied under Hilbert, Klein, and other well-known mathematicians.

Cairns's teaching career spanned four decades. From 1894 to 1896 Cairns was an instructor at the Troy High School, Troy, Ohio, and from 1898 to 1899 he taught at a high school in Calumet, Michigan. In 1899 he came to Oberlin as an instructor in Mathematics and Surveying. Although he was twice recruited early in his career by the University of Michigan, he remained at Oberlin, eventually becoming a full professor. He was named head of the Department of Mathematics in 1920 and served in this capacity until his retirement in 1939.

A commitment to the teaching of collegiate mathematics led Cairns to be active in professorial circles. In 1916, when the Mathematical Association of America was organized to deal with some of the problems in this discipline, Cairns became the first Secretary-Treasurer of the Association. He was also a representative of the Mathematical Association on the Council of the American Association for the Advancement of Science for nearly twenty years. He was elected Vice President in 1938. He was named president of the Mathematical Association in 1943. Subsequently, he was made an honorary President for his many contributions to the teaching of math and service to the organization.

Cairns, who taught mechanical drawing, descriptive geometry, and surveying statistics at Oberlin College, had as his research specialty the investigation of regular convex and star-shaped polygons as well as the history of mathematics itself. As a scholar, he contributed to the Bulletin of the American Mathematical Society (as writer and member of the editorial board), the American Mathematical Monthly, Science, School and Society, and Mathematical Teacher. He held a number of summer teaching appointments, including UCLA and the University of New Mexico.

He married Iva M. Crofoot (b. 1875) on August 25,1898. The first Mrs. Cairns taught Mathematics at the Oberlin Academy from 1902-1903, and later became Director of Home Economics at Oberlin High School. This union produced two children; daughter Mary Catherine and son Robert William. His second marriage to (Mrs.) Bertha Pope (d. 1964), Oberlin Class of 1930, took place in Oberlin on June 17, 1930. She had been head of Cranford Hall. Professor Cairns was fond of the outdoors, and he played golf at the Oberlin Golf Club. He died on July 15, 1955, in Pasadena, California, at the age of 83.