Coble entered the Pennsylvania College of Gettysburg, the oldest existing Lutheran college in the United States founded in 1832, in 1893. The College was renamed Gettysburg College in 1921. Coble graduated with an A.B. in 1897 and, after spending a year as a public school teacher, he began his doctoral studies at Johns Hopkins University in 1898. He undertook research into algebraic geometry and was awarded his Ph.D. for his thesis The Relation of the Quartic Curve to Conics in 1902. His thesis supervisor was Frank Morley.
Coble was appointed an instructor in mathematics at the University of Missouri in 1902 and then was appointed to Johns Hopkins University in 1903 where he became Morley's research assistant. He published his doctoral dissertation as The quartic curve as related to conics in the Transactions of the American Mathematical Society in the year he took up the research assistant position in Baltimore. The Carnegie Institution of Washington had been founded in 1902 by Andrew Carnegie to contribute to various areas of scientific research and one of the first pieces of research the Institution supported was that of Coble and Morley. The funding was generous enough that Coble was able to use the grant to travel abroad and, as was the custom for American mathematicians at that time, he travelled to Germany where he studied at Greifswald University and Bonn University. His aim was to work with Eduard Study who was well known to the mathematicians at Johns Hopkins University since he had taught there in 1893. Study was the professor at Greifswald but he moved in the middle of 1904 to Bonn. Returning to the United States for the start of the 1904-05 session, Coble was appointed as an instructor in mathematics at Johns Hopkins University.
In 1905 Coble married Abby Walker Adams Whitney; they had four children. He was promoted to associate professor in 1909 but he left Johns Hopkins when offered a full professorship at the University of Illinois at Urbana-Champaign in 1918. Except for being a visiting professor at Chicago in 1919 and the year 1927-28 which he spent back at Johns Hopkins University, Coble remained at the University of Illinois for the rest of his career.
His interests in research relate to finite geometries and the group theory related to them, and to Cremona transformations related to the Galois theory of equations. Later in his career he studied the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface. His early papers, written while he was at Johns Hopkins University, include: On the relation between the three-parameter groups of a cubic space curve and a quadric surface (1906); An application of the form-problems associated with certain Cremona groups to the solution of equations of higher degree (1908); An application of Moore's cross-ratio group to the solution of the sextic equation (1911); An application of finite geometry to the characteristic theory of the odd and even theta functions (1913); and Point sets and allied Cremona groups (1915).
In 1929 Coble published the monograph Algebraic geometry and theta functions in the American Mathematical Society Colloquium Publications, being the tenth such volume. It became a classic and the American Mathematical Society republished it in 1961, then again in 1982. Some of his later publications include Configurations defined by theta functions published in the Duke Mathematical Journal in 1939. This paper reviews the invariant theory of Cremona transformations as developed by Coble in his earlier papers. A linear homogeneous transformation, with integral coefficients, is associated with a Cremona transformations and these transformations form a group which Coble studied. In particular, taking the integral coefficients modulo m, for some m, leads to a finite factor group. In 1940, also in the Duke Mathematical Journal, he published Trilinear forms. In the year before he retired he published Ternary and quaternary elimination (1946) which extends work by Macaulay and van der Waerden, and also extends work done by Morley and Coble some 20 years earlier.
In 1934 Coble became Head of the Department of Mathematics at the University of Illinois and continued in this role until he retired in 1947. He contributed to the war effort during World War II by running a special accelerated degree programme for navy personnel. By the time he retired in 1947 his health was already deteriorating due to Parkinson's disease. However he accepted a one year post at Haverford College but after teaching for one semester he found his increasing health problems too much to continue and he resigned. He was involved in a car crash in 1956 which left him unable to walk without assistance. He then moved to Lykens, Pennsylvania, where he lived out the final ten years of his life. He died in a hospital in Harrisburg, Pennsylvania.
Coble worked hard and effectively for the American Mathematical Society over many years. He was vice-president of the Society in 1917, then he edited the Transactions of the American Mathematical Society for five years from 1920. He was involved with editing the American Journal of Mathematics over many years between 1918 and 1933. However his most important work for the American Mathematical Society was done over the years 1933-34 when he was president. The Society at that time was in some financial difficulties and it was to Coble's great credit that he dealt so effectively with the problem. He wrote a letter as part of the campaign to put the Society on a proper financial footing which shows his position clearly:-
It would hardly seem just for a university, year after year, definitely to plan on a research program without also correspondingly arranging at least to pay in part for the publication of the results of such a program. It is only through the publication of the results of research that its support can be adequately justified. The universities which are constantly encouraging a larger and larger output of research and are thereby gaining more and more prestige might well be expected to contribute.Archibald in  gives some indication of Coble's hobbies, writing that he enjoyed:-
... golf, tennis, swimming, tramping, bridge, billiards, and light reading.
Article by: J J O'Connor and E F Robertson